Saturday, July 31, 2021

Are we made of math? Is math real?

[This is a transcript of the video embedded below.]

There’s a lot of mathematics in physics, as you have undoubtedly noticed. But what’s the difference between the math that we use to describe nature and nature itself? Is there any difference? Or could it be that they’re just the same thing, that everything *is math? That’s what we’ll talk about today.

I noticed in the comments to my earlier video about complex numbers that many people said oh, numbers are not real. But of course numbers are real.

Here’s why. You probably think I am “real”. Why? Because the hypothesis that I am a human being standing in front of a green screen trying to remember that the “h” in “human” isn’t silent explains your observations. And it explains your observations better than any other hypothesis, for example, that I’m computer generated, in which case I’d probably be better looking, or that I’m a hallucination, in which case your sub consciousness speaks German und das macht igendwie keinen Sinn oder?

We use the same notion of “reality” in physics, that something is real because it’s a good explanation for our observations. I am not trying to tell you that this is The Right Way to define reality, it’s just for all I can tell how we use the word. We can’t actually see elementary particles, like the Higgs-boson, with our own eyes. We say they are real because certain mathematical structures that we have come up with describe our observations. Same thing with gravitational waves, or black holes, or the particle spin.

And numbers are just like that. Of course we don’t see numbers as objects walking around, but as attributes of objects, like the spin that is a property of certain particles, not a thing in and by itself. If you see three apples, three describes what you see, therefore it’s real. Again, if that is not a notion of reality you want to use, that’s totally okay, but then I challenge you to come up with a different notion that is consistent and agrees with how most people actually use the word.

Interestingly enough, not all numbers are real. The example I just gave was for integers. But if you look at all numbers with infinitely many digits after the decimal point we don’t actually need all those digits to describe observations, because we cannot measure anything with infinite accuracy. In reality we only ever need a finite number of digits. Now, all these numbers with infinitely many digits are called the real numbers. Which means, odd as it may sound, we don’t know whether the real numbers are, erm, real.

But of course physics is more difficult than just number. For all we currently know, everything in the universe is made of 25 particles, held together by four fundamental forces: gravity, the electromagnetic force, and the strong and weak nuclear force. Those particles and their forces can be mathematically described by Einstein’s Theory of General Relativity and Quantum Field Theory, theories which have been remarkably successful in explaining what we observe.

For what the science is concerned, I’d say that’s it. But people often ask me things like “what is space-time?” “what is a particle?” And I don’t know what to do with questions like this.

Space-time is a mathematical structure that we use in our theories. This mathematical structure is defined by its properties. Space-time is a differentiable manifold with Lorentzian signature, it has a distance measure, it has curvature, and so on. It’s a math thing. We call it “real” because it correctly describes our observations.

It’s a similar story for the particles. A particle is a vector in a Hilbert space that transforms under certain irreducible representations of the Poincare group. That’s the best answer we have to the question what a particle is. Again we call those particles “real” because they correctly describe what we observe.

So when physicists say that space-time is real or the Higgs-boson is real, they mean that a certain mathematical structure correctly describes observations. But many people seem to find this unsatisfactory. Now that may partly be because they’re looking for a simple answer and there just isn’t one. But I think there’s another reason, it’s that they intuitively think there must be something more to space-time and matter, something that distinguishes the math from the physics. Something that makes the math real or, as Stephen Hawking put it “Breathes fire into the equations”.

But those mathematical structures in our theories already describe all our observations. This means just going by the evidence, you don’t need anything more. It’s therefore possible that reality actually is math, that there is no distinction between them. This idea is not in conflict with any observation. The origin of this idea goes all the way back to Plato, which is why it’s often called Platonism, though Plato thought that the ideal mathematical forms are somehow beyond human recognition. The idea has more recently been given a modern formulation by Max Tegmark who called it the Mathematical Universe Hypothesis.

Tegmark’s hypothesis is actually more, shall we say, grandiose. He doesn’t just claim that actually reality is math but that all math is real. Not just the math that we use in the theories that describe our observations, but all of it. The exponential function, Mandelbrot sets, the number 18, they’re all real as you and I. If you believe Tegmark.

But should you believe Tegmark? Well, as we have seen earlier, the justification we have for calling some mathematical structures real is that they describe what we observe. This means we have no rationale for talking about the reality of mathematics that does not describe what we observe, therefore the mathematical universe hypothesis isn’t scientific. This is generally the case for all types of the multiverse. The physicists who believe in this argue that unobservable universes are real because they are in their math. But just because you have math for something doesn’t mean it’s real. You can just assume it’s real, but this is unnecessary to describe what we observe and therefore unscientific.

Let me be clear that this doesn’t mean it’s wrong. It isn’t wrong to say the exponential function exists, or there are infinitely many other universes that we can’t see. It’s just that this is a belief-based statement, not supported by evidence. What’s wrong is to claim that science says so.

Then what about the question whether we are made of math? Well, you can’t falsify this hypothesis. Suppose you had an observation that you can’t describe by math, it could always be that you just haven’t found the right math. So the idea that we’re made of math is also not wrong but unscientific. You can believe it if you want. There’s no evidence for or against it.

I want to finish by saying I am not doing these videos to convince you to share my opinion. I just want to introduce you to some topics that I think are thought-stimulating, and give you a starting point, in the hope it will give you something interesting to think about.


  1. mm... if we observe three of "something" than that 3-something most or many people consider to be real.

    Ad maybe also this "three" connected to and taken apart from the "something" many or some might consider to be real.

    Two questions pop up: what about the number 3 not connected to "something", just as an abstract (plationian) entity? Do most persons consider this abstract number 3 to be real or not? Some might, some might not, it seems to me a philosophical issue.

    Secondly, the mind notoriously almost always puts everything in three dimensions of space and one dimension of time. That is fine with Newtons Laws, but we run into difficulties (and in my case headache) if we try to model General Relativity and Quantum Theory in our limited minds or brains.

    That creates a kind of havoc with the common sense definitjon that something is considered real because of our observations. These observations become indirect and reality becomes like a veiled bride. And this veil is heavily shrouded in math....

    Maybe we need another definition of "reality" not so strongly connected to our observations...

    These are just some thoughts, just freewheeling.

    1. 3 is a nice number for juggling. Siteswap 3 is like a regular braid, or a standing wave traveling in time. And in space, the situation is ideal. 3 is just large enough to be interesting and small enough to comprehend. For juggling in general, the time dimension crashes through combinatorics. The spatial dimension crashes through graph theory. Though knots prefer 3 dimensions, juggling is not just braid theory, which might be considered a 3rd perspective after time and space. Thesis, antithesis, and synthesis; 3 elements of philosophy like a dreamspace of string theory.

    2. This comment has been removed by the author.

    3. I think one of the messages of this video is to explain, like Feynman also did, that to avoid your headache you must not try to map the “real” of science onto the “real” in your mind.
      Yes, I understand that this is very unsatisfactory, but this is the best that science can do at the moment.
      I do not think that math is shrouding things. I think it is the opposite. I think the math is indicating that the underlying mechanism is a logical mechanism. I avoid my headache with that thought.

  2. An excellent video! You had me laughing out loud (too loud!) with your German-speaking-subconscious hallucination comments.

    1. Hi Terry, I Google-translated her remark in German: 'and that kind of makes no sense, does it?'
      Given my dreams, that is something I would do, if I haven't already done so then forgotten, given how absolutely nonsensical my dream life can be. The language would not have previously existed, though.

    2. Hi C Thompson,
      I want to tell you, and you math geeks, about a repetitive dream that I used to have.
      I’d be walking in a park and come across a bench with an open book lying face down on it. I’d pick the book up and turn it over to see what was written inside. But I’d always wake up as I was turning it over before I could read it. The dream persisted for a long time, and I learned to recognize it as a dream and would try to force myself to remain asleep so that I could read the book.
      One day it worked!!! I turned the book over and it was full of maths equations that I couldn’t understand. I never had the dream again.

      P.S. How are you doing?

    3. Hi Jonathan, I'm good, just relaxing at Mum's place with my cat. I'm enjoying being out in the countryside and eating and sleeping as I please, talking with Mum and having my usual strange sleep salad dreams.
      I hope you're keeping cool. :)

    4. C Thompson, wow, thanks! I did not realize that Google provided voice translations. In retrospect that makes perfect sense given how good their voice recognition is. I do use the very cool text-image-translation feature they provide, since for for multi-lingual Zoom meetings it lets me use my phone to read a comment without noodling around with the computer image. That one is arguably a vastly more difficult converstion task than voice translation!

      Mr. Jonathan Camp, your story reminded me of two dreams. The first one was back in college when I was taking several math classes. I was in my room sleeping badly, and in my dream (and in real life) I was cold because I had kicked the covers off. In the dream my cover had become a matrix, and if I solved the matrix-blanket problem I would get warm again. I solved it multiple times, and got very frustrated when I nonetheless stayed cold.

      The other was when I was in a hospital and, shall we say, not in very good shape (I have myasthenia gravis, nicely controlled now). The combination of drugs I was getting induced incredibly detailed, brightly colored hallucinations, and in one of them I was researching my own condition, looking for insights in a large book that might help my treatment. The book was filled with vivid, excrutiatingly detailed script, and as I looked at it I realized “All this text is just nonsense being generated by my brain. I will never find any answers here because my brain doesn’t contain the information I need for my own treatment, so even if I could read this it has no value.”

      One result of that period of time was bafflement on my part: Why would anyone want to be in a state of mind where everything they did was unreal and disconnected with reality? My illness-force experience of this world of psychedelic hallucinations was uniformly unpleasant and distressing. I like reality, it’s a gift!

    5. I just used the text translator and copy-pasted. I have it permanently set to German for Dr. Hossenfelder's and some of her followers' German writing.
      My German is mostly from a semester's worth in 8th grade, I didn't learn much more. I know how to pronounce the words but I can't ask where the toilets are, so Dr. Hossenfelder etc. are light-years ahead of me there. I do want to memorise Dr. H's phrase and casually toss it out in conversation.

      I've a friend who has had vivid hallucinations involving complex geometry and mystic imagery, which he's drawn amazingly detailed artwork inspired by these visions.

      Also at Jonathan Camp: If anyone is interested in dreams in general, one of my favourite sites is the World Dream Bank, which focuses on dreams and dream-based art but is also an almost life-long record of the bloke who runs it. Be aware though that there's much potentially-offensive and not-safe-for-work but there are categories like Space, Mathematics and Science that play with ideas in bizarre and entertaining ways.

  3. Hi Sabine, this is a great video. As you rightly note, a large portion of the mystery is circumscribed by definition: how do we define what it means to be real?

    You define 'real' to mean 'that which is a good explanation for our observations' because this seems to you how we use the word in everyday language. I'm going to take your challenge to come up with another definition that is consistent and agrees with how most people use the word. I think it is a better definition than the one you are using.

    That which is 'real' is 'non-deceptive in its appearance and requires no further explanation to match it with our observations no matter how much it is analysed.'

    Here are some examples of how this better matches common usage:

    * Most people say that dreams are unreal while waking life is reality.

    * Most people say that illusions are unreal while something that is not an illusion and non-deceptive is real.

    * Most people say that the psychotic hallucinations are unreal while those perceptions of neuro-typical people are to be regarded as real.

    * Most people say that the hairs that one sees because of cataracts of the eye are unreal because they require further explanation... ie, the hairs are unreal.

    * Most people say that a rope that is mistaken for a snake on a moonlit night is unreal even though someone walking along might jump in fright after initially seeing it.

    I hope you can see that my preferred definition better meets these examples?

    With this updated def. I would say that none of the Higgs boson, gravitational waves, Black holes, or particle spin are real. They emphatically do meet our observations, and perhaps they are our best explanations, but they are still deceptive and require further elaboration. In this same way, math is not real. Even the math that best explains our observations or that which we use to predict the future behavior of physical systems is unreal.

  4. Math is not real. It is the reverse, math is derived from reality by humans. There is only our-math. Reality is more universal. Very dangerous going the other way so are those doomed s-theorists

  5. Very interesting and convincing. I like to say that math is just thinking, thinking is math, but I may have to expand that a bit.

    Platonism always struck me as going too far. Take the old fox-goose-rice example, where you have to cross a river with them in a boat which will only carry you plus one of them. I consider that a math problem, to be solved by considering all the cases and finding the one that works. Under Platonism, it seems to me all those possible cases, right and wrong, would be part of the Platonic universe, which seems somewhat silly to me. That is, there is bad math and good math, and what doesn't work (i.e., bad) in one situation might work in another, so who decides what is and isn't in the Platonic universe?

    Whereas in your concept, math is part of our universe, and good and bad are determined by what works and doesn't work right here. I like that better.

    Neal Stephenson's novel "Anathem" is based on the Platonic Universe concept, though, and he makes it seem vaguely plausible. Enough to carry a good story, anyway. I consider it his best novel, better than "Snow Crash", better than "Cryptonomicon".

    1. There is a confusion in Sabine’s video which I tried to explain on her Twitter post for this video. There is a difference between the platonic concept of maths, and how it’s used in physics or in your example. From a platonic perspective, there is a sense in which maths has a real nature beyond our use of it. Yes, there are areas of maths that don’t seem to be fundamental to nature, but the core areas including everything from pi to imaginary numbers are seen as discoveries. Yes they can be partial, such as Euclid, but mostly it’s an uncovering of something that is changeless and absolute.

      Sometimes physicists feel they “discover” a theory, but in reality they create a better way to describe and predict what nature does. It will always be an abstraction, not the thing itself. However platonic entities are the thing itself, and it’s the ways we describe and use them which is the abstraction.

      I’ve probably not explained this very well, but I do think this is an important distinction which is not clear in Sabine’s video.

  6. Maybe math is not about numbers and structures but more about relations between such entities. This is what makes reality mathematical.

  7. Alice in Wonderland and rabbit holes come to mind. Is there a reality beyond my perception ... I assume so? When I say an apple is red, I really mean I perceive it as red.

    I agree that mathematical descriptions can provide some incredibly accurate descriptions of reality and provide some interesting ways of thinking abut this thing called realty.

    Are mathematics and reality and one and the same? I will suspend belief for the moment.


  8. “I see three apples!”

    What could be a more definitive example of the Tegmark idea that math is reality? Of course integers are real!

    A fruit farmer enters the room and informs you that the fruit on the left is a quince, not an apple. Are there still three apples?

    You get annoyed by such nitpicking (apple picking?) and decide to eliminate any confusion by building a mathematically formal apple recognition system,

    Instead, the situation gets worse.

    Aspects of recognizing apples that are easy for a brain whose ancestor’s lives depended on finding edible fruits prove complicated for a formal system. You find you need programs to recognize the concept of an “object” and sophisticated optical spectrum scanners to analyze its surface. Even then, the results are ambiguous.

    You give up and try counting simpler objects. Atoms! What could be simpler than atoms? And indeed, you find that the criteria for defining entities with smaller total numbers of quantum-level properties are less complex than gigantic entities such as apples. Reality becomes more countable at the fermion level. Relief!

    But hold on: Your sensor equipment became truly massive to get that level of resolution of reality. Hmm. It’s almost as if there is a trade-off, one in which “simple” integer counting requires either a great deal of forgiveness and sloppiness in defining the object (e.g., apples) or a great deal of sensory perception augmentation to get close to more precise, quantum-defined numbers (atoms).

    That’s odd. Aren’t integers supposed to be the most straightforward and precise of all mathematical constructs? Why do they require so much equipment to perceive and define? Is more going on here?

    John Wheeler once famously claimed that reality is made from bits [1], the computer science basis of integers. Here’s what Julian Barbour says to that [2]:

    “Wheeler’s thesis mistakes abstraction for reality. Try eating a 1 that stands for an apple. A ‘bit’ is merely part of the huge interconnected phenomenological world that we call the universe and interpret by science; it has no meaning separated from that complex.”

    So back to the question of Sabine’s excellent video: Is math real?

    The universe certainly has rules that enable incredibly complex constructs in the natural world. You are one such construct. If these rules are what you call math, math is as real as anything we can discern with our senses. But I would argue that a much better name for this particular set of rules is physics.

    Why are smoothness and limits in calculus so difficult to prove formally? It’s because both derive from the inability of quantum mechanics to support infinite information density. This limit seeps into our math almost by osmosis. You cannot prove such properties in any meaningful way without first tipping your hat to the brutally unforgiving limits on detail imposed by quantum mechanics.

    Even more unsettling is this: In sharp contrast to the already-given rules of physics, the rules of mathematics depend inextricably on the levels of cognitive and information processing complexity seen in entities such as cells (amazing things, cells), humans, and computers. Thus, as demonstrated by the unexpected complexities of defining integers, math and cognition form an inextricably bound duality.

    Is math real? Maybe. But only if you first accept that math is a subset of both physics and cognition, one that is necessarily guided and limited by cognitive processes. We have a more humble and more resource-conscious name for this precise form of cognition in computer science. We call it programming.


    [1] Wheeler, A. Toward “It From Bit”. Quantum Coherence and Reality Conference, University of South Carolina, Columbia, December 10-2, 1992.

    [2] Barbour, J. Bit from It. FQXi Essay Contest 2010-2011 (2011).

    1. Terry, I feel like I'm halfway between a physically unique entity and a bunch of traits that can be recognised and explained as a conglomerate of habits, Attention Deficit Hyperactive Disorder, bad life experiences glued together with Douglas Adams references, song lyric quotes, etc, and now I'm also attached to everything by maths rules. This is a comment to cogitate on.

      @Jonathan: There is no spoon!

    2. "It's easy to eat apple pie but hard to eat pi apples!" (not by Groucho Marx)

  9. The problem is largely that mathematics is not an empirical subject. Contrary to Sesame Street and the old film “Donald Duck in Math Land,” mathematical objects do not lie around us for direct observation, detection or measurement.

    In the case of three apples, we use the number 3 to describe a quantity of objects that are categorically the same or similar. We use this independent of whether the apple is a Fuji apple or a golden delicious. So there is something at work here with our ability to make such categorical assignments. Grothendieke and Etale developed a form of cohomology of categories to understand at least within mathematics how this happens. How this occurs with our assignment of mathematics to physical systems is not as clear.

    I think spacetime is built from entanglements. With quantum mechanics we have the issues of action per ħ and in statistical mechanics we have entropy per k. The action gives us a time derivative ∂S/∂t = iH, a form of the Hamilton-Jacobi equation and similarly with entropy, ∂S/∂t = (∂S/∂β)(∂β/∂t) = C, which is complexity and ∂β/∂t for black hole gravitation is a form of the geodesic deviation equation. Indeed, the Schrödinger equation is a form of geodesic deviation. Strominger worked a hydrodynamic approach to general relativity, and what is curious is the ratio of viscosity to entropy is η/S = 4π. The viscosity of spacetime η = s√(ρc^4/8πG) is a direct measure of the quantum entropy that form space or spacetime by von Neumann S = -k Tr[ρlog(ρ)], Here ρ in the viscosity equation is the vacuum energy density and in the second ρ means the density matrix ρ = |ψ〉〈ψ|.

    Of course, in doing this I shift the geometric meaning of spacetime from Riemannian geometry to the geometry of entanglements that involves Riemann in addition to Teichmuller and Mirzakhani. So, this still leaves us a missing ontological junction between the physical world and mathematics.

    1. I find myself in rare but total disagreement with Dr. Crowell's statement that math is not empirical. The case I like to point to is Andrew Wiles' proof of Fermat's Last Theorem. It was based on an empirical conjecture of a correspondence between the characteristic numbers of two different fields of mathematics. Proving that conjecture was true was the last link in a long chain leading to the proof, and it arose from empirical observation of the calculated characteristic numbers. I have found some others agreeing with my position, as follows:

      Greg Chaitin, co-founder of Kolmogorov -Chaitin Information Theory:

      “For years I’ve been arguing that information-theoretic incompleteness results inevitably push us in the direction of a quasi-empirical view of math, one in which math and physics are different, but maybe not as different as most people think. As Vladimir Arnold provocatively puts it, math and physics are the same, except that in math the experiments are a lot cheaper!”

      Tim Johnson, at the Magic, Maths and Money Blog:

      A more sophisticated misunderstanding relates to the way mathematics is conducted. The error originates in how mathematicians present their work, as starting with definitions and assumptions from which ever more complex theorems are deduced. This is the convention that Euclid established in his Elements of Geometry and led Kant to believe that synthetic a priori knowledge was possible. Euclid actually started with Pythagoras’ Theorem, and all the other geometric ‘rules’ that had emerged out of practice, and broke them into their constituent parts until he identified the elements of geometry. It was only having completed this analysis did he then reconstruct geometry in a systematic way in The Elements. Today the consensus within mathematics is that the discipline is analytic, from observations, not synthetic. Outside of mathematics there persists a belief in the power of pure deductive, synthetic a priori reasoning.

      John Von Neumann:

      “Mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.”

      Scott Aaronson on a mathematical breakthrough in complexity theory:

      "It’s yet another example of something I’ve seen again and again in this business, how there’s no substitute for just playing around with a bunch of examples."

    2. Jim V: I thought about mentioning this idea of mathematical empiricism that Chaitin has written about. Time and so forth caused me to not mention this. This is a type of empiricism, and with theorem proving assistant algorithms such as Coq this is becoming more prevalent. I think this is still qualitatively different from standard scientific empiricism. The measurement of the orbit of a planet to frame classical mechanics or the properties of an atom involves systems that are completely independent of anything we fabricate. What mathematics we employ in theoretical developments are what we impose, not what nature imposes. Mathematical empiricism involves the use of symbolic structures that are then cast as algorithms. Computers, computer languages etc are not something nature generates.

      There is of course the issue that Seth Lloyd brought to the fore. He makes the argument that the universe is a sort of computer. I tend to look at this a bit more guardedly. The Lie algebra of a gauge interaction has properties parallel to logical operations. This means that complex interactions can have some complex processing-like structure. However, for the most part this is just a random set of operations. The set of all possible coin tosses of length N has 2^N possible binary strings. Some of these could be binary codes you run on a computer. The set of elementary particles in the universe with a gauge group containing R roots, R = 2 for SU(2) and R = 6 for SU(3) (these have 1 and 2 weights filling out the dimension of these), There are then R^N possible symbol strings. In the observable universe N = 10^{80} or for those entering a black hole N ≈ 10^{60}, and so the set of possible strings means there are computing algorithms. Of you enter a black hole you could sample all of these in principle.

      The SU(3) root system connects with the hexacode Golay code.

      Given though that these occur by randomness, it is not quite the same as a person framing a mathematical problem on a computer. To think these are equivalent might be compared to a sort of Platonism, and I am completely agnostic on that.

      As Garrison Keillor put it about “Guy Noir,” One man on the 13th floor of the Atlas Building seeks answers to life’s perplexing questions. Guy Noir Private Eye.” The quote is something like that. I see the relationship between physics and mathematics as something we may never know, and I suspect we can never know. It is something we can discuss in metaphysical conversations over scotch and cigars.

    3. Papa57: It is not clear what bearing this has on the relationship between physics and mathematics.

    4. @JimV
      "Euclid actually started with Pythagoras’ Theorem, and all the other geometric ‘rules’ that had emerged out of practice, and broke them into their constituent parts until he identified the elements of geometry. It was only having completed this analysis did he then reconstruct geometry in a systematic way in The Elements."

      Nice example. And exactly that happens during the development of thinking as described by (some good) physiologists (which also agrees with personal observations and contemplations). The first phase of reflexes, stimuli, is habitually skipped and later forgotten.

      When a child (some at least) first learns the placeholder (the name), e.g. Vadim, and from observation after others - "Who did this? Peter did this." - learns to answer when asked, "Who did this?" with, "Vadim did this". While in effect it means such and such stimuli led to processing and to such and such response. As when a parent says, "Do that." it is the parent who stimulates the reflex, a child processes and responds in some way.

      After awhile it starts to associate and identify with the placeholder and learns to mark it cumulatively as 'I', i.e. starts to skip the initiating stimuli (as if he was initiator of the action from the beginning, identification with agency, or 'I') and takes the processual phase of the reflex (stimulus-process-response) to be his [= appropriating agency] will. This 'will' being the feeling of the processual phase of the reflex (while the brain computes output based on all known to it input). When brain computes the expected output - it feels as pleasant. When not - not pleasant, so tries to reach some closure.

      So 'I' and 'free will' are born exactly out of forgetfulness about the first phase of the reflex, namely stimuli, and appropriating initiation of action to itself (so pride and shame are born). As if it suddenly magically happened all-ready out of some miracle well (pun intended for Zen connoisseurs). Then writing some tractates ceaselessly about 'greatness of The Will' and how it 'makes us human so special'. While in effect is just a habituated forgetfulness.

      "I see the relationship between physics and mathematics as something we may never know, and I suspect we can never know."
      Never is a long time. As I understand Jonathan Gorard does some research on this subject and finds some interesting patterns from mapping math & physics from the perspective of that "proof space" (myself been thinking that 'insights' in that space may be represented by lightlike paths, and voila! That also riffs with your entanglement spaces & knowledge sharing). And calls it... metamathematics :-) But I wouldn't count it alongside philosophical musings on metaphysics, it's just another level of math abstraction.

    5. Geometry is classified as mathematics. There are several areas of mathematics, and subjects such as number theory and algebra have little in the way of mental pictures, while geometry has mental imagery. Of course, we know there are connections between these two, where algebra often describes geometric constructions. The Langland’s conjecture means that number theory may become as central to the geometry of physics as algebra is now.

      Space and spacetime is an oddity. If it is the emergence from quantum entanglement, this is suggestive of space as a purely mental construction. As I outlined above the perspective of spacetime as a hydrodynamic fluid and as defined by entropy of entanglement gives a ratio of this fluid viscosity and entropy η/S = 4π. I also think that entanglement of states is conserved in spacetime for these states on paths, think of a sum over paths in a path integral, as geodesic on the spacetime. Proper time is then a parameterization of paths that conserve entanglement. Time and space are then emergent structures from entanglement. Does this mean that we should just dismiss space and spacetime as just an illusion?

      Whether space and spacetime are considered ontologically real may depend on your perspective on this. The Fermi and Integral spacecraft found that spacetime is smooth down to a scale lower than the Planck scale. This measurement was due to the simultaneous detection of a range of electromagnetic radiation from very distant, billions of light years, burstars. As such this a choice of measurement with an extreme IR probe. An extreme UV probe may likely find spacetime very broken or foamy by contrast. This is not unlike a choice of measurement of a quantum state. In a duality between reality and locality, most measurements are set up to select realism of measurements without locality. Of course, of late the opposite form of the Bell inequalities has been on interest, but for most work we choose locality. I would then say much the same of space and time. I then would say an operational perspective on this would say that space, time and spacetime are real or “real enough” for all practical purposes.

      If these IR probe measurements of EM radiation from distant source continue to hold up, we gain some confidence that our standard ideas of space in point-set topology and calculus are reasonable models. The mathematics then has operational “truth-value,” and mathematicians may not be errant in saying this geometry is real. Of course, we cannot prove this. The dual UV perspective is more discrete or knot topological in foam and the like. This dual perspective is operationally real as well. It is interesting how much of string theory and related physics has such close analogues in condensed matter physics with lattice and finite element structures.

    6. I don't think of spacetime as mere illusion but a precise abstraction suitable for its purposes (for an emergence of a specific type of observer, which is currently implicit, and of specific theory of observations and appropriate geometry). So "real enough" is fine with me.

      I enjoy your considerations about hydrodynamic but some areas seem to dissonate. If spacetime is emergent (which I think it is), then there is something funky going on with matter (don't know the Higgs mechanism and incompetent to think of spinors, didn't assimilate math, but thinking of some kind of frequency difference which zaps in the dynamic field, I enjoyed Penrose's stress that in QM in Hamiltonian instead of momentum we use the differential *operator*; but then he goes on to construct this very peculiar twistor object, which contains intrinsic and extrinsic attributes from the get go while representing 'pure holomorphicity', then as a result gets rid of massless field equations in that formalism... that's some witchcraft, unfortunately cannot check the whole structure, but it's also surprising in similar ways).

      So eventually the observed spacetime behavior is smooth, but isn't it because of choice of states, our computation of normalized states and something in-between *may* be lost. So regarding entanglement states thinking proceeds as follows, it may be alright that entanglement is conserved for geodesics in spacetime. But is this necessarily the only way? I.e. may it also be conserved for some non-local processes (from the point of view of a specific measurement, i.e. knowledge intrinsic to the observer and in the measurement, so otherwise-entangled)?

      In that scenario, what the observer sees as non-local in spacetime happens in some projection space by definition (alert for hand-waving crap, but intuition is like that - some fiber bundle, with image on fibers representing knowledge of the observer and the projection of that image representing emergent spacetime). Yet, non-locality is only conditional for the type of the observer as spacetime is apparent, provisional (meaning, the whole 'observer vector space' from the bundle, not just a single image on it, not single human knowledge, but all knowledge of that kind). So entanglement is conserved, yet the state is not necessarily always observed as local, so not always on spacetime geodesics, but lives according to the entanglement space laws.

      That of course might mean 'will never know' and may be hidden by computational irreducibility as effective expression of this phenomenon will be not 'how and what' to observe but 'where to look' in the first place, i.e. something termed by scientific or mathematical intuition. And all working methods on paper will be expressed in constructive math (including model of *local* observer, i.e. local consciousness), yet there will be that 'hunch' for non-ergodic patterns which may stem from that spacetime non-local part.

      Of course, that is not new. In fact, it's exactly the principle of Brahman (universal knowing, or 'differential operator', or computation; btw, not unlike Jewish basic principle). Where it emanates as a type of personal deity with the world (spacetime - an instance of that type, it's irrelevant here whether worlds are one or many, the base space of fiber bundle), a type of personal soul with the body (local human consciousness - an instance of this type, or the image on the vector space of the fiber bundle).

      In fact, definition of 'real' in Upanishads is 'that, which is unchanging among change'. Some tried to ontologize into a thing to simplify matters (Terry's 'bits' analogy). But what is meant more akin to the universal function of 'knowing'.

      Oh, boy... :-)

  10. I read James Gleick's book on Chaos Theory as a young adult and it blew my mind in a way nothing else had, it was a major paradigm shift for me. (Now I have smaller ones once every few Saturdays.)
    That so many different disparate structures and phenomena in the Universe can be described and modeled that includes how the world functions on a mathematical level in a way I was previously unaware of was wondrous.

    All of the 'math things' that many of you describe here feel like they're on another layer of reality that I can sometimes see through the deep fog of my ignorance (that I am slowly dissipating) and I glimpse those same wonders.

    I think mathematics is real/exists insofar as it is something thought of, used and recorded. We're well aware though that a description of something may not correlate with reality as it manifests. *looks at String Theory and certain political ideologies, while thinking at least String Theory hasn't actually harmed anyone yet*.

    But then, I've photographed my bowl of miso soup in a restaurant because the settled solids and seaweed pieces reminded me of a Calabi-Yau manifold, so... :)

  11. To me the question "Is math real?" is one of those pseudoscientific quandaries with no meaningful way to resolve.

    Some say that math would exist, I suppose in some form of Platonism - the theory that numbers or other abstract objects are objective, timeless entities, independent of the physical world and the symbols used to represent them, even if there were no intelligent life in the universe. Even if that were the case, what purpose would math serve? Could math alone create stars, planets, rocks, much less living thinking beings?

  12. Three things:
    A. Maths are human-made and do not actually exist in Nature.
    B. Aren't mathematicians employed by physicist to create the maths necessary to falsify (if I'm using the term concretely) their hypothesis?
    C. Are there any other animals in the world that demonstrate the use of maths in their daily lives?

    1. Hi jonathan.
      All good things are...
      - Regarding Point C (the other two points are too high for me) I think:
      In a certain way all of them.
      Flagellates alone, which move towards a light source, apply "the knowledge of nature" about gradients in order to improve their chances of reproducing...
      Another example: cells in general apply the concept from inside and outside...
      And the "knowledge" about the validity of the efficiency of structures that follow the Fibonacci is realized innumerable times in nature, too.
      - [ three... ]° (x)

    2. At least some species of mammals can count to a simple degree, but I don't know if that's the extent of their abilities.

  13. Thank you for giving me something interesting to think about.
    That is why I'm here.

  14. If one chooses to think of both math and science as human inventions then it could be asked "Is science or math more important?" It's hard to imagine we could have science, at least modern day science, without the support of mathematics, and yet math has grown and evolved with ever new discovery in science, making it even more useful in the study of science. The title of a book published in 1951 by Scottish-born mathematician and science fiction writer Eric Temple Bell (1883-1960) 'MATHEMATICS Queen and Servant of Science" seems to capture the essence of the dual importance of each.

  15. It seems that the concepts of 'real' and 'existence' (implicitly linked with identity with a concept of an independent agent, i.e. 'I', and its inevitable derivative of 'free will') are what causes most issues in considerations of that sort.

    To be more precise, people make assumptions (or take some things for granted, so just don't know that they took some initial assumptions) and make further propositions while staying in zeroth epistemology. "C'mon, everyone knows what we are talking about." While in fact 'reality' and 'existence' are words which may cover very broad maps of meaning and hide some unclear areas. So most of the time they cover unexplored areas taken for granted as obvious.

    Same relates to questions of 'why' category (i.e. extending to 'what', i.e. search for essence-of-things, beginning-end, grand-purpose, etc., like the mentioned 'yes, but what space-time *really is*'?). Attempt to conceptually grasp what something *really is*. It may be alright as thought stimulating exercise or conversation (being aware of the process of abstracting, in order to examine some process in detail). The trouble is - we only make sense of things in relations, technically through a functional structure of relations of what relates to what and how (one may add extensionally but it would complicate matters unnecessarily). And concept (and accordingly any definition, however good-bad it may be) does not cover the structure of relations.

    In that sense, one may separate empty concepts (illusions) from abstractions. The first category may be quite elaborate but eventually dissonates with the observed phenomena in some major ways (or superseded by more precise structure). So, fairy tales, myths, religions as instruments to 'explain' life (and socio-engineer behavior of a group of people) are the earliest examples of that sort of thinking. They also played the function of sharing of knowledge in the group (so not only primitive superstitions or instruments of control, not so simple). The second category is scrupulously and painfully built from observations and elaborate developments in attempt to capture the relational structure of the processes around. So represents the real, actual, relevant knowledge.

    Yes, dichotomy is not so black and white so some phenomena that started as 'explainers' (superstitions, etc.), i.e. as empty concepts, will be examined, studied and find their way into knowledge, but through means of developing a proper relational structure (language), i.e. condensing of abstractions, most others will be revoked (majority do not hold to the principle of energy conservation).

    In that sense, empty concepts are illusions, while abstractions are worked out and written down relations, which represent the best knowledge we got so far. The difficulty of a beginning thinker is to distinguish one from another (and current systems of education do not help in that) that abstractions are really developed, like vehicles or airplanes, which required many bright men, thousands of work-hours, etc., to condense knowledge. I.e. they are literally built, represent something tangible. In that sense a precise design of an airplane is as 'real' (if not more so, as it potentially contains more degrees of freedom) as the implemented product itself. But due to the fact that it was literally developed out of the graveyard of unsuccessful prototype models (and not on 'spirits of ancestors' as people in Cargo Cult believe). So that developed knowledge of relations is 'real'. I like to think of math as the science of relations (yes, science).

    But it's still puzzling that many (even) bright people still formulate those questions in such terms (of 'real' and 'existence') and keep confusion going.

  16. I’m in complete agreement Sabine, though it seems to me that you haven’t provided the curious with an answer to their question. If mathematics isn’t in itself “real” (contra Plato and Tegmark of course), then what shall we call it? I have a suggestion.

    Mathematics may be termed “real”… in the capacity of a human language. This is to say in the sense that French and English are also “real”. Math is a strange language since while natural languages evolved into our species, math was recently invented under the advent of civilization. (If math would have evolved into us then I think we’d automatically know the sorts of things that a pocket calculator tells us when we punch in the right keys.)

    It’s strange to me how rarely people in academia speak of mathematics as a language. This seems to open the door to all sorts of funky notions.

    1. There is a range of interpretations of mathematics. Most mathematicians think there is some sort of objective aspect to mathematics that is outside human constructions. This is in some way Platonism, and there is no way to prove this. Brouwer advanced intuitionism, which says mathematics is the creation of the human mind. So mathematics is in that setting just a set of rules similar to chess.

      There are animals that understand numbers. Corvid birds and parrots can count and even add and subtract. This means mathematics is trans-species. Is it something other intelligent life works? If we find signals from some ETI out there that has numerical or mathematical coding, then I think there is more universality.

      I have sympathies with the idea that mathematics has some universality to it that is beyond any mental processing. The problem is that I suspect we can never understand how this comes about.

    2. Thanks Lawrence,
      It may be that if we fully endorsed mathematics as a language construct then this would also suggest how any universality might exist to mathematics. Try this:

      First observe that every statement in the language of mathematics may also be phrased in the language of English. Why? Because English is infinitely expandable and we tend to talk about the math we do. So any new mathematical symbol will also be given an English name so we might reference it orally.

      Second, does it matter that non-humans are sometime able to develop symbolic representations from which to think and even communicate? I’m not sure this changes anything. A dog might know its name, or a crow might even count. Symbols may be tools for other animals just as they are for us. I’d expect advanced alien species to both name themselves and count, and not because names or numbers are beyond invented constructs, but rather because conscious forms of function should tend to have certain similar desires.

      So that might be a reasonable answer. In our language of mathematics there should be various tools that tend to be useful for other reasonably advanced form of conscious function, and so if advanced enough they should also tend to develop the language that we call mathematics.

    3. What's wrong with just calling it unreal and leaving it at that? Math isn't real. It requires human explanation and elaboration to have any awareness at all of it. See my definition above for 'real' and ask yourself whether it meets the expectations of worldly language.

    4. Take a look above. I just wrote a post on the reality of geometry in light of how it may be emergent from quantum entanglements. There is a lot there I will no repeat here.

      I agree in one sense that an operational perspective this is possible. I would say that probably if we received a message from some extraterrestrial intelligence (ETI) that we may have some confidence that math in universal. First off, they are using the electromagnetic field in communications technology. This means these beings are using Maxwell theory, which is represented mathematically. If they also encode things in some form of numerics this also suggests a universality to mathematics.

      I would say then from an operational perspective that mathematics appears, not proven mind you, to have some possible universality. In this sense this may be good enough FAPP.

      BTW, when it comes to dogs, I am amazed how a creature that has such complex social behavior and considerable memory abilities are so hopelessly unmathematical. Dogs have almost no spatial reasoning. For evidence of that, just tie two or more dogs up outside and watch. Even one dog gets wound up and seems unable to just walk the other winding direction. Dogs also display no numerical ability.

    5. Manyoso,
      You can always defines something to not be real and argue that this is done from a generally useful definition, though someone else might define it to be real and argue that this is done from a generally useful definition. From my position it seems productive to accept either definition to try to understanding if anything useful is being said in a given case.

      I’d say that most people here would call English “real” in the sense that it’s the language that we’re speaking right now. Furthermore when I tell you “2 + 2 = 4”, it might be said that math is real in the same sense. Right?

    6. Thanks Lawrence, that’s interesting. I’m not entirely sure if you’re agreeing or disagreeing with me however. I’m saying that mathematics exists as a language that the human created. Furthermore I’d expect any ETI that is advanced enough to also develop what we’d call the same language, as well as various other terms that may be translated into English. I propose that any universality that we might observe here should stem from our similar need/desire for such lingual tools.

    7. Philosopher Eric: I would say most physicists consider mathematics as a sort of language, or a set of rules similar to those of chess. Though there are departures from math and games. I tend honestly to think mathematics is more than this, but I have no proof.

      Noam Chomsky developed transformational grammars that are math-models of languages. These turn out to be applicable in computer science more than in understanding human language. I might conjecture that human language is some informal set of structures with some mathematical roots.

      When it come to ETI, unfortunately I suspect the SETI effort will probably never find any hint of such. I suspect it is too rare and distant. Too bad in many way if so.

    8. I think I understand Lawrence. Your advanced grasp of mathematics tells you that it’s far less arbitrary that something like the game of Chess. So you’d rather not say that math is only a language, but have no proof of this. Well note that an ETI shouldn’t need Chess to advance, though it should need mathematics. To me that seems like a plausible way to support your inclination. It seems to me that math should have been discovered rather than invented like one of our games. Still I’m hesitant to call math anything more than a language even still. All sorts of platonic foolishness seem to result from that, such as the ideas of Tegmark.

      It sounds like we’re in agreement on the Fermi paradox. Consider my own answer to this riddle. Yes there should be countless places out there that harbor robust ecosystems that produce advanced intelligences. There are two things which make me doubt that we’ll ever have any direct evidence of this however.

      One is that advanced intelligences should tend to kill themselves off pretty fast in a geological sense once they become powerful enough. Apparently we’re the first such generation on Earth, and I’d expect more to come after us. The time of the next one should depend upon how much life we kill off beyond ourselves. The net effect should be tiny blips of intelligent EM radiation that quickly degrades to nothing intelligible in space, separated by long periods of silence even here.

      Secondly, despite all the sci-fi fun we have regarding space exploration, I think we’re confined to this ecosystem just as the intelligent products of other ecosystems should be confined to their’s. Some realize how fragile biology is, and so imagine that our robots could become self sustaining elsewhere. I doubt this however. Even conscious robots should require the resources of this planet to become self sustaining.

    9. Strange: To me what Sabine said did answer the curious and what you say is equivalent to what Sabine says.

    10. I would say the difference between mathematics and games is that games are finite. Some years back the number of possible checkers games was computed. The number of possible chess games also should be finite. Most of mathematics is not closed in this manner, where induction and other methods involve a sort of infinitude.

      When it comes to humanity I would almost say we are collectively obsessed with committing mass suicide. What is disappointing is that it is clear we are pretty badly screwing this up. In fact we almost could not do much worse. If one pauses for a moment and think about the big problems, pollution, energy, resource loss, and societal ones such as nuclear weapon proliferation, drug abuse and the like, through my lifetime I would say we have not solved anything. We have done some ameliorative actions and solved parts of these, but largely through my lifetime I would witness to the fact we humans have not solved a single damned thing.

      Maybe other ETI are not so collectively insane. If we communicate with then it is clear they would control nature and use resources. Whether they become addicted to this as we have is unknown. I think one thing that will prevent ETI communications is they are rare, and potentially the closest one might be 10s or 100s of millions of light years away.

    11. Lawrence,
      That sounds right to me. Consider my own assessment of why we seem so collectively fatalistic. It’s that personally feeling good each instant is what constitutes the value of existing for anything. I consider this the purpose which evolution uses to drive the conscious form of function.

      Before consciousness, life should have essentially functioned robotically and thus without purpose. While non-conscious life could deal well with “closed” environments reasonably well (such as a checkers board), it needed a purpose based element to advance further regarding the open ended circumstances that were more standard. This purpose I speak of is sometimes referred to as “qualia”. The essential difference between us and other conscious forms of life I think, is that language and hard science have made us extremely powerful before we could grasp sufficiently effective ways to use it.

      Though I do suspect that we’ll kill ourselves of in the next 100,000 years or so, I also have some hope that we’ll get a reasonable way down that road. I suspect that our soft sciences will finally begin progressing and so help balance our amazing power with better understandings of our nature itself. If our nature gets reasonably figured out academically, I suspect that a true world government, along with continued technological advancement, could spare us for quite a while. I presume that other intelligent life out there exist under similar constraints.

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    2. Hi Steve,

      I was trying to come up with a cogent question yesterday and failed.
      What I was trying to ask was, what would maths be evidence of, and in what form?

  18. I saw an exchange between Tegmark and Massimo Pigliucci who made this point, that his idea was metaphysics rather than science and Tegmark agreed.

    Tegmark does sometimes talk as though the mathematical universe was actually science.

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    1. I should learn to wait until I've fully thought out what I want to say before I push the send button.

  20. In continuation on the topic of metaphysics, I looked up a well-directed excerpt from a reflection of an ingenious physiologist Ivan Sechenov, "Who is to elaborate on the problems of psychology, and how?" where he examines metaphysical developments in thinking (in the end of 19th century) and which may be of interest:

    "But why does the metaphysical method of studying psychical phenomena lead to absurd deductions? Does the fallacy lie in the logical form of metaphysical reasoning, or only in the objects of reasoning?

    We are already acquainted with the logical side of reasoning: it consists in comparing two objects (i.e., either two concrete forms or the whole and one of its parts, or two parts of one and the same form, or of two separate forms) and in their commensuration from the point of view of similarities, dissimilarities, causal relationships, etc. Besides, we can detect by intuition any, at least serious, fallacy in logical reasoning; in such cases we say: "the inference is illogical", "the reasoning is inconsistent", etc. Metaphysics, however, cannot be accused of inconsistency; otherwise its doctrines would not have held sway for such a long time. On the contrary, it is the consistency of metaphysical reasoning, along with the universality of the problems it undertakes to solve, that attracts most. Hence the error must lie in the objects of metaphysical investigation. This circumstance is of extreme importance to us, because it convincingly shows that the real substrata of all psychical processes are invariable, no matter whether our reasoning is based on reality or on pure metaphysical abstractions.

    But what kind of error is contained in the objects of metaphysical investigation?

    When the metaphysician in his desire to obtain more profound knowledge ignores the world of real impressions (which for him are a kind of profanation of the essence of things by our sense organs), and turns of necessity to the world of ideas and concepts (since there is not other place to which he can retire), and does so with the conviction that that which is truly ideal, that is, the least real, is what really matters, he inevitably deals with abstractions; he forgets that these abstractions are fractions, i.e., conventional values, and, without a moment's hesitation, objectivises or transforms them into essences. I say, and I say it with deep conviction and without any exaggeration, that the metaphysician tries to prove that 1/2=1, 1/10=1, 1/20=1, etc. He does the very thing a mathematician would do if he were to take it into his head to isolate a mathematical point or an imaginary value without acknowledging their conventional character. What is more, conventional mathematical values even in their isolated form are still abstractions, while the ultimate objects of metaphysics, or its essences, are products of decomposition not of real impressions but of their verbal expressions. This is the second deadly sin of metaphysics, a striking example of which is confounding the name of an object, i.e. mere sounds, with the object itself, for instance, the name Peter with the man Peter; this lapse is rooted in the peculiarities of language and in the attitude of the human mind towards its elements."

    So, in essence, when nerds don't have unis/labs/places to hang out and be at peace, they are getting pissed off by people around and start churning heavy metaphysics... Moral? Better to keep nerds happy.

  21. I wrote to Max Tegmark, and asked him:

    “Bearing in mind that I am a poet whose work often reflects a fascination with science and not the other way 'round, does the following make any sense at all?

    Having just finished [your book] Our Mathematical Universe, I would say that the MUH [your Mathematical Universe Hypothesis] holds that our universe, while far more complex, is no more real than a triangle — and I do not mean a triangular physical object, nor three objects forming the vertices of a triangle nor three objects forming the sides of a triangle — but a triangle. ”

    He replied:


  22. If the maths is real, do stars have consciousness?

    1. I 'fess up - ArtPulseDynamics asked me that question as a joke after watching the video; I dared him to ask here. It was funny at the time.

  23. Dear Sabina. This made me giggle. I think, reading the last paragraph, that even you started to doubt yourself.

  24. It's quite natural to assume that a multiverse of all possible algorithms exists. It's doubtful whether a multiverse of all mathematical objects as we conventionally define them exists, simply because most of them require an infinite amount of information to be specified unambiguously. See e.g. this article:

    The reason why this is plausible is because we are algorithms ourselves. Your existence is due to the universe running via your brain whatever algorithm defines your identity. But the implementation doesn't matter. If someone were to run an exact simulation of your brain, then that would generate the exact same conscious experience that the real brain generates, provided the simulation is precise enough to capture the algorithm precisely.

    We can then consider a thought experiment where not only the brain is simulated but also the rest of the body and the local environment, causing you to be conscious of the virtual environment generated by the computer instead of the real world. The simulation then doesn't have to run in real time. The only thing that matters is that the simulation has to actually run. But because the way the simulation is implemented doesn't matter, this means that the computer can render it in some scrambled form without that affecting the conscious experience.

    Since our own universe counts as a computation rendering our consciousness, it then follows that the scrambling due to applying a time evolution operator should be irrelevant. This then implies externalism because if we assume that the present moment exists, then applying a time evolution operator to the present moment maps it to future or past states that then exist in a scrambled form inside the present moment. Conscious experiences of past or future observers therefore exist.

    One can then speculate that quantum mechanics is an effective theory of a multiverse of algorithms. In the conventional formulation a system on which we intent to perform a measurement should be scribed by a complete set of commuting observables. But one can argue that strictly speaking we only ever observe our own brain states, therefore there should exist a commuting set of observables for brain states. and this is then going to represent the algorithm implemented by the brain.

    Such a set of observables then defines a sector of the multiverse where a particular observer is present. One can then speculate that the quantum multiverse is not the real multiverse, but that the real multiverse is the set of all algorithms. Quantum mechanics then yields a local linear approximation of this multiverse.

    1. You see, there is a dig difference between, "we are algorithms ourselves" (ontology, for both 'us', 'algorithms' and identity between them) and, "we can be represented by algorithms" (epistemology, if 'us' is taken weakly).

      In that perspective, we can talk about good enough emulation of local conscious experience (AGI). Yet, it may not be just about the skin-bag (not meaning anything extra-goo-goo like soul, but information shared or linked through the environment is enough), so we may discover (a conjecture) that the computational effort that is needed to cross that ultimate barrier is either more expensive than is available to us (or not worth it, as it's cheaper to work directly with bio-material already plugged-in into the environment) or computationally irreducible in the paradigm of computation (which may in itself turn out to be the best paradigm of all, of which I'm not so sure).

      So, if "it quacks like a duck", it only means that "FAPP of current human knowledge, it quacks like a duck". As we may never know that we may trigger some extinction event just by eradicating ducks, because they were transferring some wasp parasite, helping some bees to survive, then those bees pollinate some plant, and yada-yada-yada. So Nature may be running top speed already, we may only humbly ask for a ride. Some observations support this:

      I have't checked the paper, but if you map 'the present moment' (whatever it is) to states, for future you will need to select a finite basis and normalize beforehand what you want to know according to the energy available, so that's already losses. And past may not be recoverable at all (WF collapse for entangled states). And in order to talk about past or future conscious observers you need a theory of the conscious observer (even local is enough as any other may not be feasible). In that sense, it's not clear what algorithm in theoretically unknown space (multiverse, entangled space, spinor network, twistor space, whatever) is and it's doubtful that such construct (if ontologically defined) may generalize well. The best what is currently done is groping for an elephant (attempts to recover QFT & GR) by its waste products. Or pray to the almighty Differential Operator.

      Infinities and singularities may be regarded by local observers (=limited computation) e.g. *as* asymptotically running functions (as well as real/transcendental numbers, etc.), which may be used with care in order to evaluate some limits, induce, negate unfruitful areas, etc. We don't necessarily need to "forget all about Cantor crap" and all switch to constructive math. Whatever works. All our math is hacking after all. Just some remember it, and some seem to forget it.

      In other words, to get an egg one needs a universe. I am suspecting that all such attempts to ontologize anything on the macro scale and replicate it will hit the wall of the second law.

      PS Yet, there is some new voodoo (which does not seem to be what it claims, namely to "evade the second law", maybe locally under specific conditions, etc., interesting if Sabine might find it interesting and cover it):

  25. Galileo Galilei said that the language of the universe is writing in mathematics.
    I would add that like any other language, mathematics can be use to write fiction.

    1. ... Which might be more interesting to mathematicians and some physicists than to anyone else, likely.

    2. Although I shouldn't presume that because I don't actually know.

  26. For me, any definition of reality must minimally include some reference to observations *and* reproducibility. Something along the lines of "that which can be observed and reproducibly verified by others".

    Numbers seem to fit that definition but numbers can't be observed independently of the thing you are observing which makes them tricky. Does something need to be observable independently to be real? Is that even possible?

    An analogous example is the colour red. Real or not? People can reproducibly observe three apples to be red, but where or what is the redness of a thing? You can describe those apples down to the fundamental particles and you won't find red. Just some photons with a particular (number!) range of wavelengths to which we attach the word red. Red is never independent of those photons so is it real?

    So what about the three-ness of those apples? Like others have said above (and like the redness) three is just a property of the apples, like red. Three is no less real than red and also can't be observed independently. The difference is that three is far more widely applicable as a property. Does that make it more of less real than red?

    1. Dear Plato,

      So if there is "nothing" (no dimensions, no space, no time, no "somethings" (which might create spacetime)), there is no abstract Plationian number threes?

      Can an abstract number 3 exists other than as a representation (for instance in a brain?)?

      So no universe, no plationan "space" either?

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  28. Hi Sabine,

    as I see it, physics needs math and a process of calculus for prediction, and concept for humans to communicate and (most of the time) to believe they understand something to the presupposed "real thing". And existence is the only point that cannot be denied - hence a "reality".

    But physics from its beginning is about separating objects, variables, fields, observables, whatever is needed. So, after hundreds of years of separation physics is stuck in it, and the only unity that still appears in physics is the mathematical structures of the calculus (and theory).

    So with respect to your question, I first understand that the unity of nature - which can be seen anywhere any time - cannot get rid of with math. That is first level, but the question also relates to the most scary philosophical question, which is also the foundation of religions: why is there something rather than nothing? And it is answered with the preexistence of math. So it looks to me like math becomes another name for God - (maybe with logical power instead of magical in the eye of a scientist). I guess Tegmark not a religious person. But should write Math ?


  29. I tend to believe math was invented not discovered. Yet the famous Fibonacci sequence has captivated mathematicians, artists, designers, and scientist for centuries. Also known as the Golden Ratio, its ubiquity and astounding functionality in nature suggest its importance as a fundamental characteristic of the universe.

    Leonardo Fibonacci came up with the sequence when calculating the ideal expansion of rabbits over the course of one year. Today its emergent patterns and ratios (phi = 1.61803...) can be seen from the microscale to the macroscale, and right through to biological systems and inanimate objects. While the Golden Ratio doesn't account for 'every' structure or pattern in the universe, it's certainly a major player.

    Some examples are the in the structures of many types of seeds, flowers, fruits and vegetables, tree branches, shells, hurricanes, faces, spiral galaxies, even the microscopic realm is not immune to Fibonacci. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates phi 1.6180339.

    Ref: 15 Uncanny Examples of the Golden Ratio in Nature

  30. Hi Sabine,

    I find your examples somewhat confusing. First, you claim that not all real numbers are "real". Then, you say that space-time, as a differentiable manifold, is "real". Doesn't the existence of a differential manifold presuppose the existence of real numbers?

    1. Well, physicists arguably construct space-time over the real numbers. But ask yourself if you'd notice any difference if you replaced the real with the rational numbers. Does completeness play any role for anything observable? I don't think it does.

    2. Hi, Sabine.
      I don't want to poop currants again, but wasn't the opposite kind of your main argument against the simulation hypothesis?*
      As usual: no need to answer me included. ^.^ ,
      The fact that Heisenberg made it clear to us that complete observability is not possible underpins your claim here, but also reminds me to the definition: "Truth is the sum of the influences that bring you to your next decision."
      I still like the idea, even if it doesn't represent to the usual opinion.

    3. "I don't want to poop currants again, but wasn't the opposite kind of your main argument against the simulation hypothesis?"

      In case you mean the discreteness argument, the rational numbers are dense in the real numbers. That wasn't my main argument though.

    4. Sabine, you said:

      Does completeness play any role for anything observable? I don’t think it does.

      I agree emphatically, but the full implications of that simple statement on physics and math are surprisingly broad and profound.

      For physics, if completeness never impacts observable physics, then models that begin conceptually at the continuum level and “back off” in complicated ways to accommodate the imprecision in the real universe cannot be the best way to represent reality. But these models work! Yes, emphatically. The trick is to realize that the stepwise and thus inherently imprecise algorithms used to implement such models are also capturing subtle details of the physics involved and must be explicitly included in the models.

      Quantum mechanics even provides a delightfully generic heuristic for doing just that, which is this: The level of detail is proportional to available mass-energy.

      An example of this would be to merge General Relativity with the algorithmic calculation processes used to make actual GR predictions. This merger provides GR with new variables that previously assumed space always has infinite precision, regardless of available mass-energy or other factors.

      For example, if you link the algorithmic side of GR to the assumption that matter creates spacetime — which I would suggest is not that radical since you cannot “see” any aspect of spacetime without first having matter — then the ability to bend spacetime also changes. In cosmic voids devoid of matter, space loses resolution and becomes a bit blocky and excruciatingly flat. That’s another way of saying that cosmic voids are explosive. They lose the ability to complete with regions of space where matter content enables curvier and thus more attractive space. The universe becomes less stable at large scales, not more. At galactic levels, such variable levels of “space resolution” could easily play a role in why galaxies have such a lovely diversity of beautiful forms. MOND? Probably, since spacetime stiffness would also extend the reach of large gravity wells. But unlike standard MOND, this approach would not violate GR, only give it new parameters.

      My brother Gary, a lawyer, came up with an excellent phrase for capturing the idea that matter creates spacetime and thus affects its level of resolution:

      “In spacetime, matter matters.”

      For math, if completeness never impacts reality, then a good deal of physics-related math that is assumed to be okay goes out the window. My favorite example is the entire topological concept of manifolds. (Somewhere out there: “What? Terry, dude, manifolds are rock solid and the basis of all sorts of physics, including General Relativity!” Add spicy words to taste. :)

      The problem is that continuum thinking allows shapes such as balloons in 3-space to be “thinned down” until the inner and outer faces merge at some infinitesimally small scale, thus creating a 2-space. Topologists then discard the embedding space and focus only on the internal connectivity properties of lower-dimensional space. It’s even called a 2-sphere, which can be confusing since the surface of a 3-ball is a 2-sphere!

      One cannot do infinitesimal manifold surface mergers in a universe that lacks completeness. The best alternative is polarized manifolds with embedding spaces, such as a 2-sphere as the polarized (ball on one side, air on the other) surface of a 3-ball. Would such a shift impact physics theory? Sure. If nothing else, it would make all embedding space very much real. A hypersphere universe would require more than four dimensions, for example. Other impacts, such as on particle physics, are much more subtle.

      So again: While the assertion that mathematical completeness has no impact on anything observable seems innocuous, the devil is in the details. This particular feisty little devil ends up flipping many rather significant issues in physics and math on their heads.

    5. It is a similar question with the simulation argument. Would you notice the difference if all the things you observed were approximated using computable numbers? What calculation would you make to show that it wasn't, if all you have to work with is also computable numbers?

    6. Hi Sabine,
      An even more basic question: is the number π "real" according to your definition?

    7. Tamás,

      No, Pi isn't real because you'll never need all those digits. I actually used this as an example in the video.

    8. Robin,

      "It is a similar question with the simulation argument. Would you notice the difference if all the things you observed were approximated using computable numbers? What calculation would you make to show that it wasn't, if all you have to work with is also computable numbers?"

      Well to say the obvious that depends on how good the approximation is. But I don't know what you think this has to do with the simulation argument. The laws of nature aren't numbers.

    9. Sabine,

      I find this interesting because for me, the existence of π is a textbook example of a "good explanation for our observations" (of course it is true that we do cannot observe all the digits, but we can prove that they must be there.)

    10. @Tamas:

      I don't think manifolds presuppose the existence of real numbers; it simply turns out that in our description of them, historically speaking, they were required.

      To be more precise, it ought to be possible to characterise the category of manifolds by a list of properties. This is is what would normally be called a 'universal' property in category theory. I put the term 'universal' in quotes as I find this term confusing; personally, I find the term 'characterising' more descriptive. It's not the first time that the naming of a concept in mathematics is obscure. Maths is hard enough without making it harder by not naming concepts well.

    11. "it is true that we do cannot observe all the digits, but we can prove that they must be there"

      no you can't

    12. @Mozibur: I was referring to differentiable manifolds. You can try to define what "differentiable" means without using the field of real numbers, but isn't that unnecessary nuisance? It is so much simpler with reals.

      @Sabine: maybe there is a misunderstanding, but you define a mathematical object to be real if it offers "a good explanation for our observations".

      Observation: if I try to draw circles, no matter how small or large, and try to measure the ratio of the circumference and the diameter, then I get slightly different values, always close to 3.14.

      Explanation: in Euclidean space and for perfect circles, the ratio does not depend on the size of the circle. We call this ratio π. Since our world locally resembles Euclidean space, my circles somewhat resemble perfect circles, and my measurements are more-or-less precise, I always get something close to π.

      Don't you agree that this is a good explanation for my observations? Do you have a better explanation?

    13. All the Maths only exists in our minds as far as we know.It is just an artefact of the evolved brain in the physical world, which then unsurprisingly works well in describing the physical world.

      1, π, a differentiable manifold with Lorentzian signature, a vector in a Hilbert space that transforms under certain irreducible representations of the Poincare group, are all apparently precisely defined although not known to be consistent.

      The maths structures fit the physics observations only up to the precision of measurement. So π is as real as a differentiable manifold with Lorentzian signature.

    14. Terry Bollinger1:35 PM, August 01, 2021

      I still haven't completed my homework from your last comments, but..

      "For math, if completeness never impacts reality, then a good deal of physics-related math that is assumed to be okay goes out the window. "

      Isn't the proof in the pudding? If a better model for the physical data is achieveable by moving to discrete Maths, where is this better model with its better fitting results?

    15. Steven Evans 5:36 AM, August 03, 2021

      That is a meaningful point of view. What I find confusing is Sabine's claim that a differentiable manifold with Lorentzian signature is real, while π is not real.

    16. Tamas,

      I do not know a theory that explains our observations which does not use a differentiable manifold. I know that if you cut off Pi after 10^300 digits, that'll still describe our observations.

    17. Sabine,

      My take on this is that Pi as a mathematical object is conceptually simpler than "the first 10^300 digits of Pi" as a mathematical object. So while we can use the latter to describe our observations, it will just become a more complicated description.

    18. Sabine Hossenfelder7:23 AM, August 03, 2021

      "I know that if you cut off Pi after 10^300 digits, that'll still describe our observations. "

      But isn't the 2 in E=mc^2 conceptual and experimentally E=mc^1.99999999, say, would do?

      Equally for some circular orbit C=πD is conceptual, but C = 3.141592654 x D would do experimentally.

      Isn't the point that we have reasons to think that the 2 and the π are the safest numbers to put in the theories in terms of maintaining their validity however much precision in measurement increases? But as concepts, 2 and π both only exist in our minds as far as we know?

    19. Steven Evans said:

      Isn't the proof in the pudding? If a better model for the physical data is achieveable by moving to discrete Maths, where is this better model with its better fitting results?

      Yes: If such ideas have merit, they must produce experimentally verifiable predictions by which they can be tested. They should also provide simpler and more computationally efficient models of known phenomena.

      Probably the best test candidate for mass-indexed, precision-aware algorithms is predicting the large-scale structure of the universe. No other topic in physics offers a more extreme disparity in the spatial distribution of ordinary matter.

      The hand-wavy prediction is that adding an algorithm-level scaling factor to code that implements general relativity equations should enable accurate prediction of large-scale cosmic structure, and do so without the need for dark matter or dark energy.

      The net impact of such a scaling factor in GR code would be to make the emptiest regions of spacetime "blockier" and less capable of supporting gravity. The resulting rule of thumb is this: The larger a cosmic void grows, the more aggressively it expands. Instead of dark matter pulling things in, emptier regions in general and cosmic voids in particular should push ordinary matter away from them, while ordinary matter will try to stick to itself.

      Thus regions containing matter and regions empty of matter should behave somewhat like incompatible fluids, with one of the fluids -- the one that contains ordinary matter -- "stickier" than the other fluid. That difference alone should provide testable differences in predictions of how the universe looks at very large scales.

      Closer to home, matter-indexing of quantum field theory could provide an intriguing new twist on renormalization, specifically in how and why the importance of some virtual loops fades off. By "intriguing" I mean it sort of flips the entire interpretation upside down? The total mass-energy of available in a situation becomes a fundamental given, not a derived value. How "real" virtual loops become then depends on how distant they are in derivation from the mass-energy that enables their existence. If mass indexing is valid in the context of quantum field theory, then the math should end up simpler and have fewer arbitrary assumptions, yet still produce the same predictions of particle and field interactions.

      One final note: While precision-aware approaches have a concept of scale or granularity attached to them, they are definitely not the same as discrete (e.g., cellular automata) approaches. Bits, precise numbers, and well-defined lattices are all emergent, limited-resolution artifacts in a low-resolution holographic universe. Anything "discrete" thus cannot be fundamental in such a framework.

    20. Addendum: I just realized that my own assessment of a matter-indexed universe as "two immiscible fluids, one self-sticky, the other one expanding," has a nicely mundane interpretation:

      The large-scale universe should look like a fluffy loaf of bread.

    21. Steven,

      "But isn't the 2 in E=mc^2 conceptual and experimentally E=mc^1.99999999, say, would do?"

      Yes, but those are both rational numbers so I don't get the point. And in any case as you certainly known E=mc^2 isn't the right equation, it's actually a scalar product of two vectors, that is, a contraction with a two tensor, and the reason there's a two in that exponent is the same reason graviational waves have spin 2. Without that two, all of General Relativity wouldn't work. Thus, the 1.9999999 might be compatible with some observations, but not with the vast majority of them because that'd make the theory inconsistent.

    22. Sabine,

      In my view, Pi as a mathematical entity is simpler than "the first 10^300 digits of Pi" as a mathematical entity. The most economic way to define "the first 10^300 digits of Pi" is to define Pi first. Thus, if you replace Pi by the first 10^300 digits in your theories, then you get a more complicated theory, not a simpler one. Besides, just as in case of replacing 2 by 1.99999, this would make the math inconsistent.

    23. Let me return to this question of Sabine for a moment:

      "Sabine Hossenfelder 8:42 AM, August 01, 2021

      Well, physicists arguably construct space-time over the real numbers. But ask yourself if you'd notice any difference if you replaced the real with the rational numbers. Does completeness play any role for anything observable? I don't think it does."

      Actually there is a huge difference. Consider the function that is 0 on rationals smaller than Pi, and 1 on rationals larger than Pi. If we consider only rational numbers, then this function is continuously differentiable, and its derivative is the all-0 function. So you can have arbitrary fluctuations in differentiable functions with all-0 derivative. I think this is a huge difference.


    24. Terry Bolinger said:
      “Thus regions containing matter and regions empty of matter should behave somewhat like incompatible fluids, with one of the fluids -- the one that contains ordinary matter -- "stickier" than the other fluid. That difference alone should provide testable differences in predictions of how the universe looks at very large scales.”

      In fact, that’s the conclusion reached in this paper
      and a series of follow up papers published by H. Sato and K. Maeda in the early eighties. And that leads to my own assertion that all avenues to an overarching theory that combines Newton/Einstein and MOND haven’t been fully explored.
      If there were only one Void in the Universe and one gravitationally bound matter structure the expanding Void would just push on it and that’d be the end of it. But, there are thousands of Voids and thousands of matter structures. So when an expanding Void ‘pushes’ against a matter structure there are others ‘pushing’ back from other directions. And while the net effect is overall expansion, what’s under appreciated are the junctions between the expanding Voids and the bound matter structures. I contend that junction is curved! It lenses, looks and acts like a weak gravitational field. And though it’s too weak to alter the dynamics of the densest regions, it can affect the less dense regions. And like you said, no Dark Matter is needed.

    25. @Terry
      "Somewhere out there: “What? Terry, dude, manifolds are rock solid and the basis of all sorts of physics, including General Relativity!”"

      BI (Before Internet) folks think stones are real and quantum states are creepy. AI folks think quantum states are real and stones are creepy. Transitional folks think they are in an asylum, so anything goes :-)

    26. Tamas,

      "Actually there is a huge difference. Consider the function that is 0 on rationals smaller than Pi, and 1 on rationals larger than Pi. If we consider only rational numbers, then this function is continuously differentiable, and its derivative is the all-0 function. So you can have arbitrary fluctuations in differentiable functions with all-0 derivative. I think this is a huge difference."

      Of course it's MATHEMATICALLY a huge difference whether you define a function over the real or rational numbers. I was, needless to say, referring to the difference for our observations.

      "In my view, Pi as a mathematical entity is simpler than "the first 10^300 digits of Pi" as a mathematical entity. The most economic way to define "the first 10^300 digits of Pi" is to define Pi first."

      Possibly correct, but physicists don't actually use the first 10^300 digits of Pi. I didn't quite anticipate I'd have to spell this out, sorry. I don't know what inconsistency you might be referring to.

    27. Brad, thanks, what a fascinating reference! And it's from way back in 1983! Here's a quick quote:

      "For example, the perturbed region with a density less than the critical density expands forever but the closed universe itself shrinks to zero volume within a finite time."

      I wonder if Roger Penrose is familiar with these papers? Their idea the universe shrinks to zero volume as a void expands is at least reminiscent of Penrose's latest CCC concepts. Penrose might find these papers quite interesting.

      I will definitely look at these papers in more detail. It's interesting that their premise seems surprisingly modest: the geometry of a closed universe unavoidably leads to void instability.

      Have you published anything on your colliding-voids variant of the idea?

      Again, thanks!

    28. Sabine,

      "Of course it's MATHEMATICALLY a huge difference whether you define a function over the real or rational numbers. I was, needless to say, referring to the difference for our observations."

      The mathematical difference that I mentioned implies, among other things, that we can no longer use differential equations to describe our observations (or at least we have to make some annoying extra effort to get rid of all the pathological solutions).

      "Possibly correct, but physicists don't actually use the first 10^300 digits of Pi. I didn't quite anticipate I'd have to spell this out, sorry. I don't know what inconsistency you might be referring to."

      I don't see how physicists' current computational methods have anything to do with what is real and what is not. The mathematical inconsistency is simply the contradiction we get with the definition of the sine function if we assume that Pi is rational.

    29. Tamas,

      "The mathematical difference that I mentioned implies, among other things, that we can no longer use differential equations to describe our observations (or at least we have to make some annoying extra effort to get rid of all the pathological solutions)."

      That's just wrong. For solving a differential equation it's completely irrelevant whether you postulate the functions are defined over the real or rational numbers. Your example above isn't even in the solution space. Of course if you have a function over the rational numbers it doesn't have values for the real numbers, so what "pathological" solutions are you referring to? The solutions are physically entirely indistinguishable, which is my entire points.

      "The mathematical inconsistency is simply the contradiction we get with the definition of the sine function if we assume that Pi is rational."

      Of course you do not assume that Pi is rational. What are you even talking about?

    30. sorry, I meant irrational when I wrote real

    31. I think my argument can be summarized as follows:
      The question is not whether we _need_ irrational numbers to describe our observations - it is whether we can use them to give _simpler_ (and thus better) mathematical models that describe our observations. I'm trying to convince you that this is the case: replacing the reals by rationals sometimes makes the mathematical models more complicated.

    32. "Your example above isn't even in the solution space. Of course if you have a function over the rational numbers it doesn't have values for the real numbers, so what "pathological" solutions are you referring to?"

      My example didn't have values on irrational numbers. Let me repeat: Consider the function that is 0 on rationals smaller than Pi, and 1 on rationals larger than Pi. This is defined only on rationals.

    33. Tamas,

      I understand what you say. I am saying it's wrong. It makes no difference whether you use the rational or real numbers for anything in physics. Using the real numbers makes nothing simpler - it makes no difference. The easiest way to see this is that physicists never ever use any properties of the real numbers specifically (that rational numbers wouldn't also have) for anything.

      As to your example, I don't know what you think this shows. Can you define discontinuous functions on the rational numbers? Yes, you can. So what?

    34. Sabine, it still seems to me that you misunderstand my example. It is
      - defined only on rational numbers
      - continuous (every rational number has a neighborhood where the function is constant)
      - its derivative is 0 everywhere (again, every rational number has a neighborhood where the function is constant).

    35. Sabine,
      Some of my answers are not showing up, so it is a bit hard to argue, but I'll try. You are right that "physicists never ever use any properties of the real numbers specifically". So why do their rational computational methods for solving differential equations work, if there are pathological solutions over the rationals? Here is why:

      1) We know from math that, under certain assumptions, the differential equations have a unique smooth solution over the reals. Crucially, this is not true over the rationals, where the smooth solutions can be rather pathological, as my example shows.

      2) We can also prove using math that, again under certain assumptions, the physicists' rational computational methods converge to this unique smooth real solution. We can even give quantitative bounds on the rate of convergence.

      So, you are right, the physicists only use rational numbers, but if we want to understand why this works, then we have to use math with real numbers.

    36. Addendum 2:
      The large-scale universe should resemble Ciabatta bread [1], only stringier.

      Voids in Ciabatta bread expand fastest while enclosed by bubble walls that contain expanding gases. However, in a Ciabatta universe, there is no gas pressure. Thus, the rate of void expansion is driven solely by the absence of matter, not the presence of walls. This difference means that in a Ciabatta universe, walls are unstable with respect to collapse into filaments, and filaments are unstable with respect to collapse into “compact,” roughly spherical galaxy superclusters.


    37. Tamás to Sabine:
      … I’m trying to convince you that … replacing the reals with rationals sometimes makes the mathematical models more complicated.

      Tamás, I have a question for you: Can you give me a single example of a physics experiment that used real numbers to predict the results?

      If your instant first thought was “yes,” please consider what you are saying. Every numeric prediction ever made in physics, whether back in the days of hand calculation or more recently by computer, was made using finite numbers of digits. Likewise, every fraction calculated in any physics problem used ratios of finite strings of digits. Even the decimal fractions used to describe such values are nothing but large integer numbers over powers of ten.

      Another name for these omnipresent ratios of finite numbers of digits is rational numbers.

      Hiding the rational number foundations of all physics calculations by calling the more abstract parts “real numbers in equations” and the iterative, rational-number parts that do the actual work “algorithms that find approximate the equations” does not make the rational numbers disappear, nor does it simplify anything.

      In fact, I would point out that a head-in-the-sand strategy regarding real numbers has done astonishing damage to physics by encouraging sloppy models that are chock full of calculation noise posing as “theory.” Calculaton noise is what you get when you extend your predictive precision far beyond the ruthlessly hard information limits established a century ago by quantum mechanics. Examples of such damage include: string theory; many-worlds nonsense (that one’s not even quantum, it’s just “I flunked coding theory!” infinite wave noise); the idea that we might be in a simulated universe because, hey, you know, you can stack simulations without regard to resource limits; the idea that an electron is a “real” point hiding somewhere in its Schrödinger wave function, versus the softer, fuzzier, and energy-paradox-free Dirac field (think orbitals); and even erudite Kruskal-Szekeres coordinates (ouch!) with their “free” infinite density of points at the center, resulting in a Jedi mind trick that falsely convinces readers that K-S has “solved” the infinite time dilation paradox of black hole event horizons.

      All of the above assume, at levels so implicit that physicists usually are not aware of it, that Platonic perfection is part of reality and thus “free” when modeling that reality. Quantum mechanics when expressed in terms of available information does not support this premise. Ironically, even wave models of quantum mechanics fall into this trap by showing pristine, exquisitely precise waves of probability when from an information perspective, there are at most just a few bits of “real” structure available.

    38. Time for a confession: I’m responsible for addicting physicists to real numbers!

      Well, maybe not just me, but certainly my ilk. We are an evil lot, we electronics and computer and software types. Decades ago, we looked out on a world chock full of engineers and scientists and mathematicians and game players (especially game players) who shared a deep yearning for the perfection of Platonic structure, for the reality of real numbers, for the soft differentiable smoothness and seductive curves of manifold smoothness, and thought, “Wow! What an opportunity to fleece some rubes!”

      And so it began. At first, we offered our clientele just a taste, just a bit here, a few more bits there, enough to get them hooked. Then we poured it on! They all wanted perfection, a world that doesn’t exist, a Reality of Reals, smooth and seductive, offering infinite precision at no cost, the stuff of dreams. So did we give it to them? Of course we didn’t! Come on, no such world exists! But oh, how almost-real and almost-free we could make that world seem, just by offering a little more silicon for just a few more bucks, Visa and Mastercard accepted! Dreams Come True, Reals Made Reals, Differentials Made Smooth. After Jurassic Park and Terminator 2, we even got the movie industry hooked!

      And every bit (heh, get it, “bit”?) of our supposed supply of “real” real numbers was a Unix pipe (heh, get it, “Unix pipes?”) dream! All we really gave them for their oh-so-many bucks was lots of bits, rational numbers, and time-bound iterative algorithms. But how they fell for it! Even physicists who should have known became so entranced with our costly-contrived cotton-candy visions of Platonic reality that they started spouting off about simulated universes inside of simulated universe because, you know, real numbers must be free after all, wow! (It helped that the bills for new computers always went to accounting, not them.)

      And so the sad conclusion: We, the computer industry that knows in gruesome detail what is going on under the hood, chose to use our knowledge to addict the entire world to the fantasy of real numbers while conning them with nothing but rational ones. We put a computer in every pocket, typically at the cost of many chickens indeed. If we did it over, might we have had mercy and left physicists out of our numbers-con, thus saving the world a few billion bucks in wasted research money and lives? Yeah, maybe. Plus university physicists can be so darned slow in paying our invoices! So forget physicists, it’s still gamers that are the true core of our real numbers con. Long live MMOG!

    39. Tamas,

      Sorry, I don't know what you mean. You have constructed the function so that clearly the left limit to Pi isn't the same as the right limit. Why do you think the function is continuous? There is no \delta for which, etc etc

    40. Hi again Tamas,

      Sorry, I found a whole bunch of comments (from you and other people) in the junk folder. Not sure why. In any case, they should all have appeared now.

      It occurs to me there's an easier way to answer your question. If you are worried about us supposedly getting flooded by discontinuous solutions to differential equations, why do you think we never have this problem when we solve the equations numerically which we arguably don't do on the real numbers. (Not even on the rationals.)

    41. Sabine Hossenfelder 1:14 AM, August 05, 2021

      The function is continuous because Pi does not exist, so it cannot be discontinuous there. I think the point is that in math, you cannot both have your cake and eat it, i.e., you cannot say that only rational numbers exists, and at the same time pretend that their topological space is a line. No, the topological space of rational numbers is a much more complicated structure, that's why we have these strange continuous functions.

    42. Sabine Hossenfelder 1:34 AM, August 05, 2021

      I think I have answered this as well as I can in Tamás 5:37 PM, August 04, 2021. In a nutshell, we don't have this problem because we can mathematically prove that the numerical solution methods converge to unique smooth solution that exists over the real numbers.

    43. Tamas,

      I hadn't seen your earlier answer, sorry. I still don't know what you want though. You seem to agree that the difference between defining a function on the reals and rationals doesn't matter for the physics, so what's your point?

      As to the discontinuity. In the end it's a matter of definition. If you don't want to call the function discontinuous because the point isn't in the space you defined the function over, then call it singular. I'd argue that since there is a sequence (defined on the rational) that converges to a single value whereas the value of the function doesn't, it's discontinuous (and hence non-differentiable).

      Either way you put it we seem to agree that such cases of course won't show up in solutions to differential equations just by replacing R with Q, hence you don't need R.

  31. I've often thought about the apparent paradox of infinite precision and true infinities in pure math, and the limited opportunity for the real universe to make anything of these things due to the limited amount of energy available to it.

    I reached a sort of compromise based on this energy saying the potential is there, but to explore it must take energy, therefore math is as real as we decide to make it. The unexplored depths of real numbers will wait forever and will always be there if we decide to go searching.

  32. Suppose that math is the investigation of axiomatic systems. It doesn't really make sense to ask whether math is real; we should ask whether a particular axiomatic system is a complete and accurate model of our universe. It seems that no empirical evidence will ever be able to tell us whether we need all of the real numbers or whether we need the axiom of choice. There may be infinitely many axiomatic systems that are consistent with empirical evidence, or we might find that we are always refining our axiomatic models, getting closer and closer to the perfect model, but never reaching the final model, because it is impossible to fully describe the universe with axioms. It seems there are questions we will never be able to answer.

  33. I’ve always believed that the set of axioms chosen for our “math” matter (as shown by Kurt Godel).

    A particular “choice” “limits” what you can do with any given system based on those axioms.

    So how do we know that we even have the “right” set of axioms?

    And I think we know there are truths inexpressible by the axioms we use. Do you think that affects physics in some way?

  34. I think the imagination of reality we have is based on our sensual perception of our surroundings: The retina producing images, the cochlea forming the sounds and so on.
    Mathematics is just a production of a very odd structure we call consciousness and until yet I have found no satisfactory definition for it.
    But apart from undefinable, it produces mathematics, doesn’t it?
    So in the end I believe mathematics as a product of our consciousness - that some parts of it describe the reality we percept (even the reality only our technical detectors can percept) is a mystery that I have been wondering for all my life (65 long years).

    Klaus Gasthaus

  35. @Sabine
    Thanks for an interesting and balanced post.
    As I understand it (in shorthand) reality in math is some function of utility. I find that easy to accept from practical viewpoint and life experience. So say some math is useful to explain a hypothesis but not make a prediction (yet) or to actually lead to actualizing the math physically. A possible conclusion might be that the utility is subjective. Not very useful or insightful thought but still..Maybe tendency to jump to motivation isn't completely wrong?

  36. Godel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of probability in formal axiomatic theories. These results published by Kurt Godel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
    Ref: Wikipedia 'Godel's Incompleteness Theorems'

    For the sake of argument let's suppose Godel's theorems do prove that it's impossible to find a complete and consistent set of axioms for all mathematics. What does that say about theories in science? In modern day theoretical science all theories are based on postulates, which in turn are based on mathematical axioms. That would seem to put a damper on finding a complete and consistent Universal Theory of Everything in science.

  37. It's at least possible to offer a working definition of 'real' that facilitates a discussion - but only within a specific context.

    The word 'physical' is more problematic. I'd say it means nothing. Its a placeholder for a concept many people seem to wish they had; but it's really just sound in the air.

    1. Agreed. The video is great, but the entire conversation is rooted in the word 'real' which is wide open to subjective interpretation and misinterpretation.

    2. As I said, if you want to use a different definition of "real" fine, but then please make sure it's consistent and meaningful.

    3. Would you be satisfied with: "What is real is what can be measured"?*

    4. Who measures? What is a measurement? Do you actually need to measure something or does it just have to be possible to measure it. How do you know it is possible to measure it?

      I don't know what this definition even means.

    5. I think I gonna take that as a NO! ^.^ ,

    6. On second thought...
      1) any field
      2) energy transfer
      3) in this context it has to happen
      4) I was hoping that you could tell
      Do you like currant cake?*

    7. Good morning, muck, der and Dr. Hossenfelder.

      I was typing out an answer to Sabine full of explanations then I remembered I was replying to a physicist with a degree in mathematics.

      I'll just add, I think the measurement is taken by someone or something that collects and collates the data - the observer who hears the tree fall in the forest and registers the resulting vibrations as the sound the tree made.

      Muck, should we be suspicious of currant cake? :)

    8. Good morning, C Thompson (8)
      A little mistrust rarely hurts, as long as it doesn't lead to prejudice in the next step, I think... and I think that collecting the data by a sensor in your example is actually another "measurement"...
      at least according to my definition (x I'm really not the type who loves to define things
      but I actually really like currant cake x)

    9. Got it.
      I'm wondering why one would not want to poop currants, figuratively speaking. Google is too literal with its results.

    10. (8) °[
      I was wondering where to write this. So many brilliant thoughts that one would like to build on. I think I'll stick with the currants...
      "You probably think I am "real". Why? Because the hypothesis that I am a human being... explains your observations.
      And it explains your observations better than any other hypothesis...", [from the on top linked YouTube clip ~ 0:37 - 0:56]
      How can one be sure that this statement is true?*
      I mean: Sabine claims that this is "the best we got", regarding our observations.
      I dare to think that this is just one side of the medal, or in other words a cherrypicked hypothesis, which misses an - at least for me - important aspect: The "self interest" {but I realy do not like this term, so let me say:} "the hope of self-realization due to interaction" or more simple "the wishfull-thinking" of the recipient.
      I think it is not only the "fits the observation best"-assumption but also (if not primarily) the assumption which gives the observer the maximum of possible interest...
      With the intended meaning of: interest <> that what is between.
      Following this tain of thought it becomes - at least for me - even more clear that the "Multiverse-Hypothesis" is not only unscientific but also without any interest because it terminates the process of self-realization...
      at least outside the community of its believers and wanna make believers.

  38. I guess for me the problem with the Platonist route is a horse & cart thing something like this: Does an electron consult the standard model to decide how to move? As I see it, the electron just get bumped along by what around it and "cares not one whit" for the Standard Model. We know by observation that the way electrons get bumped has regularities so we can can construct a mathematical artefact to describe what they do. Actually, a series of artefacts, currently the Standard Model, later on probably something else. The next version might use a quite different formulations, like particles>waves>fields>the next thing, with a different gestalt and a different set of consequences.

    From my point of view, the equating of reality with equations is a kind of conceit leading to anything, like the idea that other universes must exist because you have written down an equation. The most egregious example of this in physics (absolute doozies abound elsewhere) would have to be Everettian Many Worlds that generates a complete universe, or at least a light cone, whenever anything at all happens, and actually, when things don't happen but might have. All in service of some equation. Models and out-there reality are completely different things. We can't talk or think about reality - or do physics - without models but they are just different things.

  39. Sabine,

    "But what’s the difference between the math that we use to describe nature and nature itself? Is there any difference?"

    There is an obvious difference. If you want to go from A to B you need a car. A description of a car is not enough.

    A car you can drive is real. It can take you from A to B.

    A picture of that car is also real. You can look at it, place inside an album, etc. But the picture is different from the car, you cannot drive it.

    A mathematical description of that car is also real, but it's real as a mathematical description, not as a car. You can store it on a computer, copy it, make a simulation using it, etc. but you cannot drive it.

    1. Andrei,

      I think you're confused about what "description" means. Of course we do not have a description of a car that is as complex as a real car and that does the same thing as a car etc. A picture of a car is but a rough description of some features of a car.

      You have to ask yourself the following: What properties does a car have that you cannot describe by math? The fact that no one *does* describe a car by math is rather irrelevant.

    2. Even if the description would include all details, down to subatomic particles you still cannot drive it. It's just some data on a computer or paper. A perfectly accurate description of a car is still not a car.

      There is no property of a car that cannot be described by math. But that description, even if perfect, is still not a car. You can use the description to build a car, true, but the building process is an absolute requirement. The description only contains the information needed to build the car, you need actual matter (electrons, protons and other particles) to make a car.

    3. "There is no property of a car that cannot be described by math. But that description, even if perfect, is still not a car."

      If you can't find a difference between the description and the real thing thing then talking about a difference isn't scientific.

      As I already said, you are confused about what a "description" is. It's not just an equation or a list of properties. It's the complete mathematical structure. I hope you understand that an equation isn't the same as a solution to the equation which isn't the same as the embedding of that solution into space-time?

    4. Btw, Tegmark explains this very nicely in his book.

    5. Sabine,

      “If you can't find a difference between the description and the real thing thing then talking about a difference isn't scientific.”

      There is a difference. A car moves, powered by its engine, it evolves/changes in time. A description of the car, no matter how complex is static. A mathematical structure corresponding to a car does not move powered by the mathematical structure corresponding to its engine. There is no time evolution there. An electron accelerates in a magnetic field. The mathematical structure corresponding to an electron does not.

      Tegmark understands this problem, and tries to solve it by an appeal to the block universe concept in chapter 11 of his book, “Is time an illusion?” You would expect to find here an explanation for this illusion of time, but there is nothing there. He says:

      “If the history of our Universe were a movie, the mathematical structure would correspond not to a single frame but to the entire DVD”

      The problem is that a DVD is not a movie. You need a DVD player, which is a real device, evolving in time, to transform the static information burned on the DVD into a movie. Tegmark does not explain anything, he simply asserts that an observer inside the mathematical structure we call the universe would have this “illusion of time”. He does not deduce our observations from its postulate. He postulates our observations as well.

      It’s the same trick employed by the many-worlds proponents. They can’t explain Born’s rule from their postulates, so they simply assert that our observations are in agreement with Born’s rule.

      A scientific theory should explain what we observe starting from its postulates, not postulate the observations themselves.

      In the chapter “Description Versus Equivalence” Tegmark makes the argument you are referring to:

      “Remember that two mathematical structures are equivalent if you can pair up their entities in a way that preserves all relations. If you can thus pair up every entity in our external physical reality with a corresponding one in a mathematical structure (“This electric-field strength here in physical space corresponds to this number in the mathematical structure,” for example), then our external physical reality meets the definition of being a mathematical structure—indeed, that same mathematical structure.”

      OK, let’s analyze his “argument”

      P1. two mathematical structures are equivalent if you can pair up their entities in a way that preserves all relations. – OK, I agree with that

      P2. “you can thus pair up every entity in our external physical reality with a corresponding one in a mathematical structure” – OK, I agree with that too.

      C. From P1 and P2: “our external physical reality meets the definition of being a mathematical structure”

      Can you spot the problem here? P1 is about two mathematical structures, not about physical reality, so C does not follow. Nice try, Dr. Tegmark!

    6. Andrei,

      A function of time is a mathematical structure.

      And, no I don't see a problem with that argument. It's logically correct.

    7. @Andrei
      "A car moves, powered by its engine, it evolves/changes in time. A description of the car, no matter how complex is static."

      Think how you've developed the knowledge of math in the first place, what it takes to share it and teach, i.e. to propagate further, and you will see a dynamic picture, though of a different character.

      The design represented by abstractions itself is constantly evolving (in its own 'knowledge space'). People are just habitually thinking that "abstractions are just words, not real", so some fancy of imaginative quick brain that got lucky while others didn't. Or "Don't know the meaning of the word? Look in the dictionary!" Yet, it does not work that way. If I look "Dirac's sea" in the dictionary w/o appropriate developed structure of knowledge, I will start questioning sanity of people who do that stuff (or my own, it doesn't matter FAPP).

      I think the best analogy for abstractions that we haven't got in the discourse (and words in general) is that it's not some static textual concept, but we can approach it as a program that one has to install (yet carefully, as there are also viruses and malware) in order to learn something. It does create a possibility for operational usage, but that may as well not be expressed. Yet, if one does not learn the design of a car (or does not repeat the whole development in knowledge which that design compresses in itself), there is no car. That type of knowledge does require certain organized processes of high level of cooperation and concentration of bright brains.

      If you still say 'that's just mental!' - then it just does not contain any information concerning how would you propagate knowledge (e.g. "how to teach Cargo Cult people to build real airplanes? how to organize their society (with an assumption that most adults will not already change their views, they are calcified, so won't uninstall the malware)? where to begin, even?"). Yet even that argument is not precise, as when we learn new information, new synaptic connections are physically forming in the brain, and the most funny part in that is that we rarely understand how will they actually manifest (yes, AI and neuroscience will help, but up to a point, and there will be "coming back to the drawing board" moment).

      So at the very least, you can always say that by learning the design of a car (math), you do change you synaptic connections, so there are dynamic processes that are involved (and which you can check already), it's only that they are not what you expect, so to speak, not under you nose.

    8. I guess the argument above may lead astray. The import is that P2 is possible at all, i.e. compression can be done from observations to a structure of functional relations. Nature enables that feature! And transfer of knowledge is therefore possible, which is the source of wonders (at least to me, as it's not at all obvious feature).

      So if it wasn't a priori in-built feature in Nature, the compression itself would be under question (in fact, I don't think there would be anything that would start pondering about it). So, although I don't know anything about Max Tegmark, it's understandable that some people wonder about that miracle and postulate mathematical foundations to Nature itself. So what we call math is just a small subset that we've pried so far from Nature.

    9. Andrei, to me it seems you are mixing up two different “Real”s or realities. What I think Sabine is talking about is the reality of a (mathematical) model and not about an physical reality that you seem to be talking about.

    10. Vadim,

      "The design represented by abstractions itself is constantly evolving (in its own 'knowledge space')."

      The design is not evolving by itself. You need a physical brain to evolve it. The car is moving by itself (assuming it's autonomous). So the design cannot be the same thing as the car. There is a correspondence, but not identity.

      Likewise, a DVD does not play itself. You need a physical DVD player to do that. There is a correspondence between the movie and the information stored on the DVD, but the DVD is not a movie.

      Sabine said that "a function of time is a mathematical structure." True, but you still need a physical device, a computer, to make it into an evolving structure. Again, there is a correspondence between the function and the changing graph on the computer screen, but the function is not the changing graph.

      Tegmark simply postulate that it is in fact the case that the DVD IS the movie and no DVD player is necessary. The DVD is just undergoing the illusion of being played on a DVD player.

      Tegmark does not explain how a static structure can have illusions, and why we have the illusion of time and not some different one. As such, his theory is devoid of any value.

      "The import is that P2 is possible at all, i.e. compression can be done from observations to a structure of functional relations. Nature enables that feature!"

      If you are able observe something your brain has to be able to store that information, so, at the vary least, a description in terms of brain patterns must be possible. Hence, the fact that we can provide descriptions for our observations does not look unexpected to me. But anyway, my point is that Tegmark fails to provide any evidence that physical reality IS math. As my above examples are showing, a correspondence does not imply identity.

    11. @Andrei
      "The design is not evolving by itself. You need a physical brain to evolve it. The car is moving by itself (assuming it's autonomous). So the design cannot be the same thing as the car. There is a correspondence, but not identity."

      The point is that it's the same with the car (it does need brain, i.e. the fact that it is a kind of a "flywheel" (memory) that just reproduces environmental stimuli is enough). You see, the argument is not about going into symbolic space (or conceptual) vs perceptual space, and not about their successive products (reduced to abstractions, that are not static, so represented by relations), but that both spaces manifest patterns, which are captured by relations, sometimes curious and surprisingly similar according to mathematical structures. And that is what I was implying.

      The design of a car is not static, because the car is not what it seems when perceived by monkey. And if you postulate that 'autonomic detached from the network AI controlled car independent of its environment, evolution of neg-entropy pumping molecules organized in curious patterns, etc. is more real than operation of all the structures that are necessary to produce them' you are simply returning to square one, i.e. you are equating such an appearance to Nature, which... it is! :-) Some say, "it's just a product of human thought" but it's half-baked head in the sand argument. Do you know where did this thought come from? And to whom? Do you really think you have agency over it?

      Concerning correspondence and identity. Identity in itself is a murky operation (it often may relate *a section* of one process to another process, so making something static and then forgetting about it, if person who is using it is not careful). Especially when we take some postulates out of context (that's why I prefer considerations, postulates or axioms, i.e. definitions, are rarely useful for thinking). I don't know what Max Tegmark considers in his text, but P2 is semi-correctly formulated the way I see it, i.e. "...reality *meets the definition* of being a mathematical structure", which to me reads as "for all we know, phenomena *can be seen as* mathematical representation, so they must rely on some orderly structural foundation". I.e. my light mistrust goes to "definition", but it can simply mean, "Nature exhibits orderly, hence, mathematical behavior". Yet, if he strongly identifies and strongly ontologizes such concepts, it's indeed a confusion. Sometimes it's used because it sells well or easier to get a grant, compare two statements, "You are an automaton which can be uploaded through the wires! And I work to make it a reality!" with, "FAPP, conscious behavior can be emulated and represented by model, which can be uploaded through wires".

      Which in colloquial tones may be expressed as designs, cars, triangles, manifolds, trees, apples, you, I, universe are all "real enough" in that sense. They are abstractions. Yet, of different orders. It's just often difficult enough to express something in words, so I tend not to pick and choose by excerpts (while the import seems coherent). And I thought it was, so attempted to add few cents. But I personally do not like the concept of "real" (and "existence") itself, as it doesn't tell you anything and for synthetic generalization one can always use Nature (or Universe, Cosmos, life, etc.). But I'm not "against it" as it can be used as thinking process generator and helps to bring up other matters and clear air a bit. So like a philosophising operator (oh goodness, not a philosopher).

  40. This comment has been removed by the author.

    1. Jonathan, I tried to organise an MRI scan at an imaging centre with staff trained to work with patients with pacemakers in Canberra (closest to Mum's place outside of Sydney, which has a rampaging COVID-19 outbreak) and then to organise the right doctor from the hospital I was in to expedite my appointment and I couldn't manage to think of what to say properly so Mum did it for me.
      Concurrently, I am perfectly capable of cogitating on such abstractions as 'Is maths real? What is 'real', even?' and following along somewhat, so go figure.
      I'm here all month. :}

      I think 'it's aliens!' is now a catchphrase of the S.H. fandom.

    2. I'm leaving my now non-sequitur reply to the removed comment to baffle future readers.

  41. Anyone at all interested in the Multiverse, or who wants to discuss it, should read an excellent book about the Multiverse written by someone who is a) extremely knowledgeable on it and many other topics and b) has no personal stake in the debate. We need to be careful that opinions are not based on the loudest sound-bites.

    1. Phillip Helbig8:58 AM, August 02, 2021

      "To begin with, it seems hard to deny the possibility in principle that a multiverse theory might hold"

      So what is the scientifically testable formulation of a multiverse theory?
      Oh, he doesn't have one - it's not even a scientific question.

      And how do we calculate the "possibility" of a theory, even of "just" being non-zero and so possible "in principle"?

      Oh, he doesn't know, because he's just spewing bullsh*t.

      More complete drivel from another hopeless moron and the CUP are only too happy to stick a "cool" graphic on the cover and hawk yet more of this stinking rubbish.

      After reading the 210 pages of this comic what extra facts will we learn about the physical universe?
      Answer: zero

      This guy and the CUP are outright frauds. Just churning out vaguely plausible sounding, pseudo-intellectual rubbish at the taxpayers' expense. Disgraceful, incompetent parasites the lot of them.

    2. The multiverse is a consequence of inflation. Inflationary cosmology has some empirical support. It is consistent with the structure of the CMB and the Λ-CDM predictions appear to be in line, at least with respect to the accelerated expanding universe as some physical vacuum with expansion that transitioned from an unstable vacuum with a much larger acceleration. That inflationary acceleration was 60-efold or about 10^{26} expansion in 10^{-30}sec.

      The multiverse comes about when one considers that the transition to the physical vacuum occurred in a causal bubble, and was not something that occurred throughout the de Sitter spacetime. So this inflationary bubble as a transition from an unstable vacuum to a physical stable vacuum should then just be one of a vast number of these.

      Inflation is consistent with the data, though as yet we do not have data on B-modes or other clear evidence, and the multiverse is then a consistent derivation from inflation. As a result the multiverse is probably at least a 50% proposition.

    3. Phillip Helbig8:58 AM, August 02, 2021

      Galileo taught us 450 years ago that all that matters is how well the model fits the data. Have Simon Friederich and the CUP still not received the memo?

      So what Simon Friederich feels it "seems hard to deny" is irrelevant.
      "God", the character from an Iron Age fairy tale which bizarrely he mentions, is meaningless and irrelevant.
      And the noddy little Philosophy 101 ideas like the "inverse gambler's fallacy" are of no interest to Physicists with the LHC, LIGO, Hubble, etc.

      So this book is a menage a trois of pre-Galilean confusion, Iron Age myth and undergraduate "philosophy", not "excellent".

      ** There is no model in existence that includes fine-tuning/a multiverse which fits the data better than a model without them **

      That's the point, as made by Galileo half a millenium ago.

    4. The inverse gambling fallacy may bolster your case. If you witness an unlikely outcome, say throwing 6 coins and they all come up heads, the fallacy is to assume there has been some prior set of trials. The error can be seen in an application of Bayes' rule

      P(X|Y) = P(X) P(Y|X)/P(Y).

      If you observe a single trial that is exceptional then P(Y|X) = P(Y), the trial is independent of previous trials, and this leads to P(X|Y) = P(X). Our learning of X does not influence Y. This in ways takes the fine tuning and argument by design down a notch or two.

      Your arguments here amount to a lot of yelling and thrashing around, but really do little to add any real content.

      The multiverse has some physics basis to it. This in no way is a proof for other cosmologies. I suspect the great majority of these are off-shell terms in quantum cosmology. This means they can be dismissed as physically real. I do not know whether this argument can remove all other cosmologies and leave only the observable one.

    5. Lawrence Crowell6:18 PM, August 03, 2021

      We've been though this before, Lawrence.

      Inflation and the multiverse are not falsifiable theories so are unscientific. There are 100 versions of inflation predicting B-modes and another 100 that don't.

      Consistent with the data? So is God.

      Λ-CDM? Contains a lot of energy in early unobservable times. Doesn't make it true even if it's the current best model. Revisit Galileo - no jumping to Platonic conclusions.

      Transitioning from unstable vacua to physical vacua in innumerable bubbles?? All unfalsifiable pseudoscience.

      "As a result the multiverse is probably at least a 50% proposition."

      You would be having a laugh if you'd written 1%.
      The multiverse is not even a scientific proposition. It's currently meaningless.

    6. Who says these are not falsifiable. Inflation most certainly is falsifiable. If there is something found in the universe that falls outside of inflation then it is wrong. The multiverse, while it is a consequence of inflation, is more nuanced. The main prospect is for finding evidence of the multiverse is to find signatures of interactions between different different pocket worlds.

      You are making category errors. One does not need to directly observe something. All one needs is to observe and measure consequences of something. In the case of vacua transition, that is the mechanism of inflation. So far that is consistent with data, and with further work maybe detection of B-modes can reach the 5-sigma level. That is all one needs. We do not observe quantum waves, but rather measure physics to obey properties predicted by quantum waves.

      If multiverse is not testable, then neither is a single universe hypothesis.

    7. Lawrence,

      Inflation is not falsifiable because the word "inflation" doesn't define the theory. You can chose any potential you want and make that fit to whatever observation comes.

      And in any case, finding evidence for inflation in *our* universe tells you nothing about the reality of universes that we can't observe.

      Lots of physicists are seriously confused about this. Just because you have math for something in your theory doesn't mean it's real. We assign reality to something when we observe it. If you can't observe it, you have no rationale for calling it real. Seems to be difficult to grasp.

    8. Lawrence Crowell7:08 AM, August 04, 2021

      But the inverse gambler's fallacy is irrelevant to physics anyway as we only have 1 observed example of physics and cannot say whether it is exceptional or not. No probabilities can be calculated.

      Fine-tuning and the argument by design are unfalsifiable and therefore not scientific, so they are on the bottom notch from the get-go. Again, inverse gambler's is irrelevant.

      Remove all other cosmologies? What other cosmologies? There is only 1 observed cosmology.

      Inflation, the multiverse, fine-tuning, string theory, "God" are all unfalsifiable and therefore unscientific. They all deal with "spaces" beyond physical data and so you can tweak the "theories" at will to try to make them join with the actual physical data. From a scientific POV they are all literally meaningless.

      God created the universe 4,500 years ago. But the universe is at least 13.7 bn years old. Oh, OK, God created the universe 13.7 bya.

      This is the kind of blatant moving of goalposts that is being perpetrated in all these "theories". It is not Physics.

      The CUP book ignores what Galileo told us half a millenium ago, bizarrely mentions a mythical character from an Iron Age fairy tale, and introduces irrelevant ideas from undergraduate philosophy.

      It should not be understated how utterly, utterly ludicrous it is that this book has been published. Given its content, it should have been written with a quill on parchment, or maybe published on stone tablets.

      What will CUP be publishing next:
      "Voodoo A concert pianist's take"?
      "Witchcraft and sorcery Your local newsagent's outlook"?

      It's anybody's guess.

  42. “This idea is not in conflict with any observation. The origin of this idea goes all the way back to Plato, which is why it’s often called Platonism … “

    Plato’s basic understanding was that the world is determined by structures. The objects which we see are more or less unimportant stuff. And the rules which we observe in the world are not physical laws as we understand it since Newton but the consequence of the dominance of these structures. Like the motion of the planets which follows the basic structure “circuit”.

    It was of some influence on physics that the German educational system one century (or so) back was based on this position of Plato. When it was detected with some helplessness that particles behave differently than expected, Werner Heisenberg stated that the only solution for physics could be to go back to the concept of Plato. So he developed and enforced a structure-based understanding of it in his version of quantum mechanics, in contrast to Schrödinger and de Broglie who wanted a more physical solution.

    Similarly Einstein was influenced by this spirit (even though he didn’t like it and left the school early). Also Einstein developed a structure-based relativity in contrast to Lorentz who as well aimed at a more physical solution.

    So, Plato is more a part of our physical world view than most of us are aware of.

    1. “So, Plato is more a part of our physical world view than most of us are aware of.”

      antooneo, that’s an excellent observation. Any mathematical use of words such as “point,” “line,” or “surface” also amounts to an invocation of the perfect structures of Platonism, since none of these concepts have exact physical representations under the known rules of our universe.

      When Plato postulated such perfect structures, his hypothesis was so effective at explaining observations that it remained largely unchallenged for nearly two millennia. It was not until the 1920s that the overwhelming evidence for atomicity and quantum blurriness destroyed any serious hope for uncovering real-world examples of Platonic perfection. Yet as you noted, even quantum theory founder Heisenberg reacted to emerging quantum theory not by embracing uncertainty but by recommitting himself to the path of Platonic perfection and structure.

      But why did Heisenberg take such a position in the face of quantum uncertainty?

      My suspicion — nothing more — is that even though Heisenberg, by his statement, “formed his mind” by studying Plato, it was the subtler influence of Newton and Leibniz’s calculus that most inclined him towards recommitting to Platonic perfection. For both Heisenberg and most scientifically inclined folks, the deeper question is this: How can the centuries-old masterpiece of calculus with its intrinsic reliance on infinitely detailed lines and surfaces work so well if there is not some deeper Platonic world of pure structure residing at the end of its infinitesimal limits? Aren’t the mundane algorithms of numeric calculation nothing more than imperfect lenses for glimpsing that ultimate, timeless perfection?

      Here’s a different interpretation: Far from accessing some timeless world of perfect structure, the calculus is just a type of compiler. Its rules transform one formal expression into a new formal expression that gives the same result — has the same “meaning” — when it is “executed.” After all, taking a limit merely creates a new formalism that, with the application of sufficient calculation resources, gets you closer a bit closer to the still-unreachable goal of Platonic perfection. In this interpretation, the issue of limits becomes a heuristic — and not always a good one, as demonstrated by the false dualities of manifolds — for ensuring that meanings of the two forms remain compatible even when iterated to “infinity.”

      For anyone dedicated to Platonic perfection, the most unsettling feature of this interpretation is that the Taylor series you use to calculate a result may be closer to the actual physics than any timeless expression of classical Platonic perfection can ever hope to be.

    2. Terry Bollinger, you say:

      “Any mathematical use of words such as “point,” “line,” or “surface” also amounts to an invocation of the perfect structures of Platonism, since none of these concepts have exact physical representations under the known rules of our universe.”

      If Plato’s concepts do not have exact representations in our universe, are they then necessary for us for understanding the universe? Maybe these exact representations are beyond that what our universe contains.

      And has his hypothesis really been so effective? Take the example which I have mentioned: In the history, the followers of Plato did not see a reason to leave the Ptolemaic system. Because they did not see a reason to ask, WHY the planetary motion is as it is. It was Newton to find this, and Newton’s goal was not to find more abstract structures but to understand the rules of the mechanical motion.

      And yes, I find the view of Plato in our present physics, but not as an advantage. If we look at quantum mechanics in the way of Heisenberg and at relativity in the way of Einstein, it reflects Plato in the way that both acted to find structures but failed to find the causes of the physical observations. And I see here the master reason why present physics is in a type of a deadlock.

      You suspect that the world has somewhere a perfect structure which we have to find. Maybe it is that way, even though I doubt this. But I see the blockage of the development of a better understanding if this is our only or our exclusive view. For example it is possible to understand particle properties by direct understanding and make calculations yielding precise results, even though Heisenberg has stated that this is impossible and should not even be attempted. And on the other hand it is possible find the physical causes of relativity which helps to solve open problems like dark matter and dark energy. But unfortunately well-educated physicists do not even attempt to do this.

      So, Plato is still with us, but in my view this is more a load than an advantage.

    3. "How can the centuries-old masterpiece of calculus with its intrinsic reliance on infinitely detailed lines and surfaces work so well if there is not some deeper Platonic world of pure structure residing at the end of its infinitesimal limits?"

      To those who feel this way (which I understand is not Dr. Bollinger's position), I would reply that calculus is just the limit of finite-difference systems as the minimum increment goes to zero. Therefore in a universe governed by finite-difference equations, provided the minimum increments were small enough not to force themselves on people's attention, the notion of calculus would still be natural to derive. (And in fact, calculus is used as an approximation for many discrete systems, such as fluid flow and electricity.) I for one do not see the logical necessity assumed by the above quote. It seems to me the inverse of an argument that since large numbers exist, infinity must exist, in some higher plane.

      I much prefer to think that our universe has certain properties which allow certain things and relationships to exist and work, and others not to work, so our Platonic universe and our physical universe are one and the same. For example, conservation of energy on the macroscopic scale says that some things tend to exist long enough to count, so integers work. In another universe, this might not be possible, or one plus one might equal zero, if every particle was its own anti-particle. So in order for the Platonic plane to apply to all conceptual universes it might have to contain contradictory relationships such as 1=-1. If we limit it to things which work well enough to be useful (if only conceptually) in this universe, then, voila.

    4. antooneo:
      You suspect that the world has somewhere a perfect structure which we have to find.

      Please read my entire comment. Respectful restatement of an idea with which one disagrees more than anyone else on this planet is not quite the same as being an advocate of that idea... :)

    5. JimV:
      One little point. You say:

      “I much prefer to think that our universe has certain properties which allow certain things and relationships to exist and work, and others not to work, … . For example, conservation of energy on the macroscopic scale says that some things tend to exist long enough … .”

      The conservation of energy seems to me a good example for this discussion about Plato. We know this physical law, but has anyone anytime asked for the cause of it? To my knowledge not. And also this is part of the follow-up of Plato. On the other hand, I know a particle model from which the conservation of energy follows. That has an important consequence. Quantum mechanics use the model of exchange particles to describe forces. Now the permanent emission of these exchange particles (for instance for the electric force) means a permanent violation of the energy law because any exchange particle can transfer energy onto a charged object, maybe after very long time and distance. In the view of this, it makes sense to deduce the energy law from the internal set up of a particle because in this case the law is only effective for structures from an elementary particle upward, and so there is no logical conflict with this exchange process.

  43. (1+2¹+3²+5³+1/2¹*3²/5³)⁻¹ = 137,036⁻¹

  44. Nope, math is just a language like any other.
    We all speak it. Some to a greater extent than others, and there are different dialects. I cite Reverse Polish Notation as one of them. I speak 4-function math with little understanding beyond that. Others have a much greater vocabulary than I do.
    Just like spoken languages, maths change with time. Wasn’t the order of operations different before the twentieth century? New *words* are added occasionally to aid in the connivance of ideas. Isn’t 𝚿 just shorthand for a much longer wave function equation? It didn’t exist until that Schrodinger guy came along and added it to our vocabulary. Just like spoken words, 𝚿 probable has several different meanings depending on the context.
    Maths can be used to write stories, and that includes fiction. It can be used to write history in financial ledgers. It can be used to write forecasts. It’s used to describe Nature around us. Someone once said that figures don’t lie, but liars can figure. So yes it is used to deceive others as well. Pyramid schemes come to mind.
    Someone else once said that mathematics is the universal language. I say, universal only if we’re talking about planet earth. Other species may find our calculus as childish.
    If we find something that can’t be described by maths, it’s probably aliens.

    1. I like this one. :)
      (Shroedinger = 'That cat guy again'

    2. Random cogitation:

      I was watching Olympic show-jumping earlier and contemplating how different mathematic modalities (is that the word?) are like how different subjects have different specialised vocabulary to distinguish and describe different things.

      Perhaps. it's like how someone familiar with horses (as a child I was obsessed with equines) can look at the coats of different animals and think: chestnut, sorrel, dun, bay, blood bay, liver - but someone else might look at the same variety of colourings and think, 'brown, or brown and black.'
      Many of the people here know of a variety of breeds and colourings of these maths, but I'm looking at the field and thinking, 'it looks like ... math things'

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  46. I always think of physics as being maths with units, where it's important to know the quantity you're talking about, and not just the mathematical equation. So Pythagoras' theorem has a nice mathematical form, which is useful when dealing with lengths, but not quite so useful when dealing with densities or magnetic fields. I know that many equations in physics work with dimensionless quantities where the units are spirited away but, under the bonnet, there is always some actual physical quantity at play. If you see three apples, then three describes part of what you see; but you also need the units, apples in this case. I remember at school when I would get zero marks for the answer to a physics question whenever I left off the units. Overall I think I am more impressed with the world being explicable in terms of four base units (mass, length, time, electric charge - and the other few) than being explicable in terms of mathematics. And, of course, in any mathematical equation describing the world, the units have to balance. So, along with the question "Is maths real?", perhaps we should also ask the question "Are units real?".

  47. What about solving a quadratic equation involving motion and one of the two solutions is negative?

  48. I think the imagination of reality we have is based on our sensual perception of our surroundings: The retina producing images, the cochlea forming the sounds and so on.
    Mathematics is just a production of a very odd structure we call consciousness and until yet I have found no satisfactory definition for it.
    But apart from undefinable, it produces mathematics, doesn’t it?
    So in the end I believe mathematics as a product of our consciousness - that some parts of it describe the reality we percept (even the reality only our technical detectors can percept) is a mystery that I have been wondering for all my 65 years.

  49. A precursor to Plato was Pythagoras, however, we know so little about his philosophy that it's hard to say anything concrete. However, Plato was known to belong to certain Pythagorean circles and I think it is fair to say that his philosophy is heavily influenced by Pythagoreanism. In fact, I'd say, if you're interested in what Pythagorean philosophy is like, read Plato.

    I'd also distinguish mathematical Platonism from Platonism per se as the former is a more recent creation and a much truncated version of Platonism. Plato thought of mathematics as real and a stepping stone to his theory of forms/ideas, whose steps are the dialectic. Hegel described this dialectic in his *Phenomenology*, but starting in the reverse order, from Being/Non-Being or what the Pythagoreans and Plato would have called the Monad or the One. Two of the lower forms of the One is the form of the Good and Justice. In Christian or Islamic Platonism this is identified with the attributes of God/Allah. It's why when Martin Luthor King said, "the universe bends towards justice", he meant something like this and not simply as nice sounding language. As an Islamic Platonist, I wholeheartedly concur.

    For Plato, mathematics is an aspect of neccessity. In ancient Greek religion, this would be Ananke, commonly described as holding a spindle in order to weave the fabric of reality. Physics would then be the art of physical neccessity and physicists are driven to find the irreducible minima of physical reality and the best way to describe it. Since Einstein, this has been held to geometric. But now I'm not so sure. Einstein himself regarded the geodesic equation in General Relativity as a unification of gravity and inertia, rather than a geometric phenomena.

    I'd also like to point out that the philosophical position in direct opposition to mathematical Platonism is mathematical nominalism. This states that mathematical entities ate not real and only name things. For example, the number two is an abbreviation that describes two things, for instance, two bowls, two chairs or two trees.

    I think it pays to name philosophical concepts in the same way it pays to name things in physics, especially when they're common enough concepts, because they are common to all. Feynman thought so when he retired his idiosyncratic notation for tan - a large T with the overhanging bar stretched over the argument.

    Finally, I think it's worth adding that the mathematical forms in Platos philosophy is impressed upon the substance of the world. In a sense, it can be described as part of natural law. And although it's real, its reality should be distinguished from the matter of the world. How these two differing ontological categories actually interact is a puzzle, rather like the mind body distinction in Cartesian philosophy. Except of course, in the latter both mind acts upon body and body acts upon mind; whilst in the former, the action is only one way - mathematical form is the acting agent and matter the substance that allows itself to be acted upon by mathematical form. Presumably this is why Aristotle conceptualised his notion of force in the manner he did, that is without concieving that there could be a reaction (back-reaction). So in a sense, we can say neccesity *forces* matter to act in the way that it does ...

  50. ... Finally, I think it's worth adding that the mathematical forms in Platos philosophy is impressed upon the substance of the world. In a sense, it can be described as part of natural law. And although it's real, its reality should be distinguished from the matter of the world. How these two differing ontological categories actually interact is a puzzle, rather like the mind body distinction in Cartesian philosophy. Except of course, in the latter both mind acts upon body and body acts upon mind; whilst in the former, the action is only one way - mathematical form is the acting agent and matter the substance that allows itself to be acted upon by mathematical form. Presumably this is why Aristotle conceptualised his notion of force in the manner he did, that is without concieving that there could be a reaction (back-reaction). So in a sense, we can say neccesity *forces* matter to act in the way that it does.

    And as a final punchline, I'd say that freedom cannot obviously be an aspect of neccessity - it is antithetical to it, that is, it is its opposite. It is an aspect of the Good, as freedom is a good (as is neccessity!) And it manifests itself in the freewill of human beings (and animals - and less obviously, plants). This is one way of resolving the paradox of human free will arising from a deterministic world. In fact, I'd say what we call freedom is a dialectic between pure freedom and pure neccessity. Pure freedom doesn't manifest itself in this world. It's what Heraclitus would have called the unity of opposites. Aristotle refers to it too, and this is why he says that all philosophers (preceding him) would say that the roots of Being are contraries, his name for these unities. It's also why *sublation* figures so prominently in his philosophy, it's his name for the unity of opposites. And then of course Marx borrowed the concept for his material dialectic - he cut out the spiritual. It's why he's said to have turned Hegel upside down. But what else is new? Every modern philosopher had been busy doing the same: Kant, Schopenhauer, Freud, Jung & Nietzsche and more recently, Dawkins and modern physicists (or as I like to call them, neo-Epicureans. After all, one of the first athiest philosophies was that of Democritus, adumbrated by Epicurus, valorised by Lucretious and revived in the renaissance and evangelised in the modern era. Although none of the earlier philosophers are as thorough going materialists as those of the modern era. They still believed in some kind of spiritual/divine reality).

    And Ok, really finally - the One of Pythagoreanism can be identified with the Dao of Daoism and the Two of Pythagoreanism, aka the unity of opposites, can be identified with the taijitu, the Ying-Yang symbol in Daoism. The same one that Bohr put on
    his coat of arms when he was knighted with the motto Contraria Sunt Complementa - contraries are complementary. Personally, I find it eye-opening that two different philosophical traditions have come up with essentially the same philosophy of reality. This is real metaphysics and not the sad little strawman that modern athiests are busy knocking down, again and again.

    (Sorry for the long post).

    1. Thank you for sharing your ideas, Mozibur. :)

    2. It's difficult for me to walk through ontological forests of the ancient Greeks with clear cut edges and identities inhabiting the worlds, but at times amusing in its own way as they indeed set the cultural discourse on rails, which is not much evolved since (surprisingly, considering relativity), so in a sense they present all the thoughts one may encounter in leisurely conversation just expressed more fully.

      But even in the model you presented, that which is expressed by "free will", "freedom" (as another aspect of the Good) is then not per se necessarily what is meant by it. I.e. from a local unit perspective, it's not, "I'm free to do what I want!" but the Good's freedom (God's will), so to speak. As it then means the freedom is a dynamic characteristic in accord with the Good. (even by definition, can something closed and static in whatever space be free?) In that case, a unit's will (whether in accord with the Good or not) is only an appearance. So one necessarily ends up splitting wills and dichotomies of 'unreadiness' and 'readiness' to understand what freedom is really about. And that would be quite alright if it be understood that way (and not dogmatically pursued and propagated as the only way). The difficulty is - it doesn't. And if a scheme doesn't work as it is already (meaning not more simple to digest and accessible to any discourse), why not check what our developments in physiology and otherwise tell us instead. Which also of necessity will bring humility to anyone, yet the person will be more aligned with the modern language and structure of knowledge (as it's not by itself, at least in principle, more complex than metaphysics those Greeks are weaving).

      Another comment is on the opposites. I haven't read enough to comment what they thought. But Aristotle in Ethics seems to stress and develop the idea of the mean. That is, neither plus nor minus. One has to find equilibrium. But it's not just moderation, but that something tangible develops out of that discrimination of the mean and that results in ethos of a person. In other words, ethos is not given, not guaranteed just because one is born human, i.e. can only be considered as potentiality. But must be worked out. That is completely in accord with Buddha's the Middle Way. I.e. it's not stupid moderation, normalization same for all, but something that must be worked out individually given one's conditions. And that is profound.

      Considering relations, I find that part most revealing and deep, even though they dance around postulating ontologies. But one principle that Aristotle mentions is implicitly elucidating of his approach that ontologies by themselves are *only instrumental* (but that may be my reading). The principle is: "Arguing towards the first principles, not from them." Basically, indicating that it is a theory, a model, adequate for an occasion (e.g. implanting understanding of ethics to students, etc., whatever the situation might have been). But acknowledging that it's always approaching the unknown and must be done with diligence.

      Considering further developments. As if nothing new happened in the discourse. I think it's not so, just doesn't seem to get such a wide acclaim. E.g. Alfred Korzybski considered and integrated many confusing subjects, I assume Whitehead, Wittgenstein and other math philosophers contributed to everyday language (albeit subtly). Overall, thinking is just very-very conservative (whatever seems to be the case). It does not change w/o work.

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  52. I agree with Mr. Jonathan Camp12:13 PM, August 02, 2021

    Math is a human language. And like any human language it does three things at once: it describes what is observed, it limits/defines the scope of observation, and it justifies observation&description.
    Ho and behold, that’s physics does: 1) experiments and observation, 2) computer simulations, and 3) math. Math is used both to analyse the data and to specify the design of a simulation. Withe math (a language!) it thus shortcuts the circuit of inquiry and justifies itself. Like any human language it is a tautological enterprise. In the transcription of Sabine’s video it is summed up perfectly:

    “…something is real because it’s a good explanation for our observations.”

    Physics can nor will ever close the gap between the language-independent physical world we live by and the language-dependent inquiry we make. Sabine’s double question in the video “are we made of math?/is math real?” leaves out the gap.

  53. Math is real? No, and it is rather important to see the difference!
    It is a question of metabolism, throughput in calories. You can’t “breath fire” into an equation, except metaphorically. Music, in this sense, is more real than mathematics.

    Now, I am in awe of mathematics, a feeling perhaps akin to walking through a medieval cathedral, such integrated complexity and reminder of something beyond my understanding. But for all its high flight, there are strings attached. To serve in physical theory, mathematics must be grounded, be exactly referenced and replicable in ‘real’ object or process. Thus, kilogram is determined by an exactingly machined chunk of platinum alloy Pt-10Ir and the interval between tick and tock is defined as being equal to the time duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.

    The International System of Units is an exacting and expensive endeavor that serves to secure the abstraction of mathematics to those things which breath fire.

  54. I’m reminded of an M&M Candies commercial.
    On Christmas Eve our two M&M Candies, Yellow & Red, walk into their living room and discover Santa Clause standing there. The Yellow one exclaims He is real! Faints and falls over.
    Santa turns around and upon seeing the M&M candies, exclaims, They are real! Faints and falls over.

    Am I to live in fear that someday I’ll walk into my kitchen for a midnight snack and find a quadratic equation standing there?

    1. Suppose a series of equations were identified that correctly explained the creation, appearance, motion, smell, taste, sound and digestion of the M&M. In such a circumstance what would the distinction be between the M&M and those equations? I suspect that you are caught up in an error by assuming that those equations wouldn’t be taking you into account and therefore couldn’t be real. I say this because your final sentence gives you the observer a prominent but false role. The M&M would exist as it does whether you found it in the middle of the night or not. Take yourself out of the picture completely and Dr. Hossenfelder’s observation may seem clearer to you.

    2. @Jonathan:

      You'll wander into the kitchen and unthinkingly open the fridge and gaze within, realising that you proved the assertion of 'no free will' correct because you didn't even know you were fridge-ward until its palid interior light was upon you.

      As you prepare a snack, you'll muse that your brain made a pretty pantomime of choosing the ingredients. As you munch, your jaw will seem a device mechanical, of angles and forces. You feel slightly disembodied as your skeleton and musculature, blood and gristle, array themselves in an elaborate series of mere pivots, beams, swivels and pipes.

      Several hours later, you'll rip the most satisfying wind and laugh, blowing the ghastly spectre of disembodied mathematics clean away.

  55. A good explanation for our observations doesn’t mean that its real.
    Some of us have recently discussed the future possibility of AI cranial implants. It would be neat to have a chip installed in my brain that would turn me into a maths wizard. Or would it? Other than to impress my friends, what would I do with it? I already know all the maths that I need for my daily life. I can use a calculator, spreadsheet, and program computers for almost anything else beyond that.
    But I want to get to my point about maths not being real here. They are real in the sense that maths is a human-made language and nothing more. But that’s not my point.
    I like intelligent women! So do you, and don’t tell me that you don’t. But there is a difference between real intelligence and implants. I don’t know about you, but I prefer real.

    1. Think about something you believe is real and try to explain why you think it's real. Maybe then you'll understand why I say what I say.

    2. Hi Jonathan (8)
      Yesterday I watched "Ghost in the Shell" in the 2017 film version.
      In the story, our conventional understanding of the term implantation is turned upside down. The question of the meaning of 'knowing one's own story' is also asked...
      The story draws the conclusion that the character of the heroine, without knowing the way in which she was formed, asserts itself over an illusion that has been given to her in order to make her controllable.
      I think the story is great and worth to be mentioned here. ^.^ ,

    3. Sabine,
      I have variety of hammers which I consider to be real, could send pictures. The reason I believe them to be real is that when I accidentally hit my thumb (not so often these days) I feel pain due to the transfer of energy to soft tissue containing an abundance of nerve endings.
      While have suffered considerable discomfort due to your postings on the subject of determinism, I have yet to be hit with an actual equation that causes pain.

      My point is, the criteria for distinguishing between what is real and any of its abstract representations,is energy throughput in appropriate metric. That is an observable condition.
      So, is math real? You posed the question and did not seem to come down firmly on the affirmative.
      I do not understand why you say what you say.

    4. Dear Dr. Hossenfelder,
      Wow! Sabine Hossenfelder has asked me a question!!
      “Think about something you believe is real and try to explain why you think it's real. Maybe then you'll understand why I say what I say.”
      The question may be rhetorical, but I feel an obligation to reply. No off the cuff comment here, only to be deleted later. This is my opportunity and I will not back down from it.
      Yes, Sabine, I have been paying attention. I accept your test.
      I would ask for your patience as I seldom have more than an hour when I post to your blog. I will compose my answer off-line and upload it later.
      You will not be disappointed.


    5. "The reason I believe them to be real is that when I accidentally hit my thumb (not so often these days) I feel pain due to the transfer of energy to soft tissue containing an abundance of nerve endings.

      The pain is an observation (so is the visual input etc) which is well explained by an object you call "hammer".

      Whether you have an equation for that is entirely irrelevant.

  56. This comment has been removed by the author.

    1. I deleted this comment because it was in the wrong place.
      I reposted it where it belongs.

  57. One can have 1/3, which is rational and infinite decimal number. One needs infinite precision to measure something as 1/3. Square root of two... the same. Can one say something that is square root of two? Yes,  take a square of length one. Can one define something of length one? No, because immersed in the reals, one needs infinite precision. Nobody has a tool to measure with infinite precision.

    So the use of real numbers, or rationals or complex numbers systems... to be used as models is based upon the idealization  that one has in principle infinite precision. But this is just an idealization or an approximation. There is also a metaphor on this- But all this works well, until now.

    If we say that on the table are two apples... we do not see the problem of the idealization. Two apples or three apples are totally different... or not? After all, an apple is also an idealization, because an apple is composed of molecules that can fly partially on air, making the apple fuzzy, and mixing with the other apple... There is also a metaphor and idealization here.  And still, we are able to perceive two apples rather than three apples with the maximum precisio possible, which is +- 1. And if there is just half an apple? This is not an apple, one can say. If not, one can imagine marbles... which are much harder :).

    All this is to say that at least integer arithmetics is real, it is present coherently and undoubtedly in Nature. Humans (among other beings) can recognize it and the arithmetics. It can be the case of beams that can undoubtedly recognize further mathematical structures? Can we perceive them as we can perceive arithmetic? It could be. The capacity of perception of mathematical structures in Nature should then be taken as a sufficient criterion for real mathematical structure.

    Thus between the spectrum that arithmetic is real and all mathematics is real, must be the truth.

  58. "So when physicists say that stuff is real, they mean that a certain mathematical structure correctly describes observations."

    This clear description of what “real” means in science is what I find missing is virtually all outreach science communications while I think this is critical if the purpose of this outreach is promoting understanding of science.

    Take for example the book “A Brief History Of Time” from Steven Hawking.
    In chapter 1 it says basically the same: “a theory is just a model that exists only in our minds and does not have any other reality”.
    But then in chapter 2 this seems to be contradicted by things like: “We must accept that time is not completely separate from and independent of space” and “the fact that space is curved”.
    It is then left upon the reader, mostly laypeople mind you, to remember that this is just a model that does not have any other reality.
    No wonder many laypeople get confused about science.

    But I get the impression it is worse that this. I get the impression that many physicists are also confused about this and I wonder if Steven Hawking was one of them?

  59. Yes, Sabine’s video/transcipt is perfectly clear. Verbatim:
    - “We have no rationale for talking about the reality of mathematics that does not describe what we observe, therefore the mathematical universe hypothesis isn’t scientific”
    - “So the idea that we’re made of math is also not wrong but unscientific. You can believe it if you want. There’s no evidence for or against it.”
    - “Just because you have math for something doesn’t mean it’s real.”

    BTW, I guess no hands-on researcher will take her/his model for anything else than scientific [I.e. language-dependent] reality. But I agree, Leon, that scientist (physicists included) tend to be sloppy in making/explaining the difference. If I’m not mistaken it was the (not so) late physicist Steven Weinberg who quipped that the universe speaks in numbers. I bet he meant that the objects he observed provided answers in terms of the numbers that his inquiry was based upon.

    Oh well. I guess our host SH reads this too. Signora Hossenfelder: sing us a song about it. 😎

  60. Sabine,

    I get your point about trying to explain why anything we think is real, is actually real. Rene Descartes (1596,1650) famously said "I think, therefore I am". That is most likely the only thing anyone can ever be certain of.

    1. Howard,

      Yes, that's right. But we arguably don't only use the word "real" for our own thoughts. Even if you're a solipsist, we organize our thoughts in other categories, talking about things and their properties and so on.

  61. Dear all,

    If we take the perspective from our brains, there are (at least) two "realities".

    One is the "outside world" which we perceive through our senses (eyes, ears, nose, touch, skin and such). Where different senses confirm one another, we safely assume "something is out there". The belief is strengthened when we communicate with others who confirm that they have similar perceptions and convictions. I think that most people don't belief that this "world outside" is just cooked up by our brain, of by our act of observing something. That the moon is there, even when no-one is looking.

    The interface of our senses might be a bit tricky at times. When a bomb explodes it creates waves in the air, but when there is no creature to perceive these waves as sound, than there was no sound, there were only waves. But these are the details.

    There is a second "reality" in our brain, which does only indirectly result from our senses. These are our emotional feelings and thoughts and so on. Mathematics are a result of our thoughts and in my view firmly belong in this "second reality in our brains".

    Sometimes the mind plays tricks with us, and some people see or hear things which are not really there. But again, these are details.

    In our day-to-day living experiences where our attention lies, all the realities in our brain are combined in one.

    It is sometimes a bit difficult to dis-entangle what is cooked up by ourselves, and what is safe to assume to be the direct result of something "out there".

    So we get all kind of discussions about the nature of reality. Is God real, is math real. I think a "brain-based analysis" could bring these discussions on a somehow firmer footing, or at least offer a scientific perspective.

  62. If Euclid (...probably lived in the 3rd century B.C.) was still looking for plausible intuition for mathematical foundations and thus made an interdisciplinary connection that could be evaluated as right or wrong, in modern mathematics the question of right or wrong does not arise. Euclid's definitions are explicit, referring to extra-mathematical objects of "pure contemplation" such as points, lines, and surfaces. "A point is what has no width. A line is length without width. A surface is what has only length and width." When David Hilbert (1862 - 1943) axiomatized geometry again in the 20th century, he used only implicit definitions. The objects of geometry were still called "points" and "straight lines" but they were merely elements of not further explicated sets. Hilbert said that instead of points and straight lines one could always speak of tables and chairs without disturbing the purely logical relationship between these objects.
    But to what extent axiomatically based abstractions do couple to real physical objects is another matter altogether. Mathematics does not create (new) phenomenology, even if theoretical physicists like to believe this within the framework of the standard models of cosmology and particle physics.

  63. This comment has been removed by the author.

  64. Sabine Hossenfelder12:25 AM, August 04, 2021

    I see. Physicists don't currently need the irrationals but do need the rationals.


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