Saturday, December 05, 2020

Is Infinity Real?

[This is a transcript of the video embedded below]

Is infinity real? Or is it just mathematical nonsense that you get when you divide by zero? If infinity is not real, does this mean zero also is not real? And what does it mean that infinity appears in physics? That’s what we will talk about today.

Most of us encounter infinity the first time when we learn to count, and realize that you can go on counting forever. I know it’s not a terribly original observation, but this lack of an end to counting because you can always add one and get an even larger numbers is the key property of infinity. Infinity is the unbounded. It’s larger than any number you can think of. You could say it’s unthinkably large.

Okay, it isn’t quite as simple because, odd as this may sound, there are different types of infinity. The amount of natural numbers, 1,2,3 and so on is just the simplest type of infinity, called “countable infinity”. And the natural numbers are in a very specific way equally infinite as other sets of numbers, because you can count these other sets using the natural numbers.

Formally this means a set of numbers is similarly infinite as the natural numbers, if you have a one-to-one map from the natural numbers to that other set. If there is such a map, then the two sets are of the same type of infinity.

For example, if you add the number zero to the natural numbers – so you get the set zero, one, two, three, and so on – then you can map the natural numbers to this by just subtracting one from each natural number. So the set of natural numbers and the set of the natural numbers plus the number zero are of the same type of infinity.

It’s the same for the set of all integers Z, which is zero, plus minus one, plus minus two, and so on. You can uniquely assign a natural number to each integer, so the integers are also countably infinite.

The rational numbers, that is the set of all fractions of integers, is also countably infinite. The real numbers, that contain all numbers with infinitely many digits after the point, is however not countably infinite. You could say it’s even more infinite than the natural numbers. There are actually infinitely many types of infinity, but these two, those which correspond to the natural and real numbers, are the two most commonly used ones.

Now, that there are many different types of infinity is interesting, but more relevant for using infinity in practice is that most infinities are actually the same. As a consequence of this, if you add one to infinity, the result is still the same infinity. And if you multiply infinity with two, you just get the same infinity again. If you divide one by infinity, you get a number with an absolute value smaller than anything, so that’s zero. But you get the same thing if you divide two or fifteen or square root of eight by infinity. The result is always zero.

I hope there are no mathematicians watching this, because technically one should not write down these relations as equations. Really they are statements about the type of infinity. The first, for example, just means if you add one to infinity, then the result is the same type of infinity.

The problem with writing these relations as equations is that it can easily go wrong. See, you could for example try to subtract infinity on both sides of this equation, giving you nonsense like one equals zero. And why is that? It’s because you forgot that the infinity here really only tells you the type of infinity. It’s not a number. And if the only thing you know about two infinities is that they are of the same type, then the difference between them can be anything.

It’s even worse if you do things like dividing infinity by infinity or multiplying infinity with zero. In this case, not only can the result be any number, it could also be any kind of infinity.

This whole infinity business certainly looks like a mess, but mathematicians actually know very well how to deal with infinity. You just have to be careful to keep track of where your infinity comes from.

For example, suppose you have a function like x square that goes to infinity when x goes to infinity. You divide it by an exponential function, that also goes to infinity with x. So you are dividing infinity by infinity. This sounds bad.

But in this case you know how you get to infinity and therefore you can unambiguously calculate the result. In this case, the result is zero. The easiest way to see this is to plot this fraction as a function of x, as I have done here.

If you know where your infinities come from, you can also subtract one from anther. Indeed, physicists do this all the time in quantum field theory. You may for example have terms like 1/epsilon, 1/epsilon square and the logarithm of epsilon. Each of these terms will give you infinity for epsilon to zero. But if you know that two terms are of the same infinity, so they are the same function of epsilon, then you can add or subtract them like numbers. In physics, usually the goal of doing this is to show that at the end of a calculation they all cancel each other and everything makes sense.

So, mathematically, infinity it interesting, but not problematic. For what the math is concerned, we know how to deal with infinity just fine.

But is infinity real? Does it exist? Well, it arguably exists in the mathematical sense, in the sense that you can analyze its properties and talk about it as we just did. But in the scientific sense, infinity does not exist.

That’s because, as we discussed previously, scientifically we can only say that an element of a theory of nature “exists” if it is necessary to describe observations. And since we cannot measure infinity, we do not actually need it to describe what we observe. In science, we can always replace infinity with a very large but finite number. We don’t do this. But we could.

Here is an example that demonstrates how mathematical infinities are not measurable in reality. Suppose you have a laser pointer and you swing it from left to right, and that makes a red dot move on a wall in a far distance. What’s the speed by which the dot moves on the wall?

That depends on how fast you move the laser pointer and how far away the wall is. The farther away the wall, the faster the dot moves with the swing. Indeed, it will eventually move faster than light. This may sound perplexing, but note that the dot is not actually a thing that moves. It’s just an image which creates the illusion of a moving object. What is actually moving is the light from the pointer to the wall and that moves just with the speed of light.

Nevertheless, you can certainly observe the motion of the dot. So, we can ask then, can the dot move infinitely fast, and can we therefore observe something infinite?

It seems that for the dot to move infinitely fast you’d have to place the wall infinitely far away, which you cannot do. But wait. You could instead tilt the wall at an angle to you. The more you tilt it, the faster the dot moves across the surface of the wall as you swing the laser pointer. Indeed, if the wall is parallel to the direction of the laser beam, it seems the dot would be moving infinitely fast across the wall. Mathematically this happens because the value of the tangent function at pi over two is infinity. But does this happen in reality?

In reality, the wall will never be perfectly flat, so there are always some points that will stick out and that will smear out the dot. Also, you could not actually measure that the dot is at exactly the same time on both ends of the wall because you cannot measure times arbitrarily precisely. In practice, the best you can do is to show that the dot moved faster than some finite value.

This conclusion is not specific to the example with the laser pointer, this is generally the case. Whenever you try to measure something infinite, the best you can do in practice is to say it’s larger than something finite that you have measured. But to show that it was really infinite you would have to show the result was larger than anything you could possibly have measured. And there’s no experiment that can show that. So, infinity is not real in the scientific sense.

Nevertheless, physicists use infinity all the time. Take for example the size of the universe. In most contemporary models, the universe is infinitely large. But this is a statement about a mathematical property of these models. The part of the universe that we can actually observe only has a finite size.

And the issue that infinity is not measurable is closely related to the problem with zero. Take for example the mathematical abstraction of a point. Physicists use this all the time when they deal with point particles. A point has zero size. But you would have to measure infinitely precisely to show that you really have something of zero size. So you can only ever show it’s smaller than whatever your measurement precision allows.

Infinity and zero are everywhere in physics. Even in seemingly innocent things like space, or space-time. The moment you write down the mathematics for space, you assume there are no gaps in it. You assume it’s a perfectly smooth continuum, made of infinitely many infinitely small points.

Mathematically, that’s a convenient assumption because it’s easy to work with. And it seems to be working just fine. That’s why most physicists do not worry all that much about it. They just use infinity as a useful mathematical tool.

But maybe using infinity and zero in physics brings in mistakes because these assumptions are not only not scientifically justified, they are not scientifically justifiable. And this may play a role in our understanding of the cosmos or quantum mechanics. This is why some physicists, like George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should be formulating physics without using infinities or infinitely precise numbers.

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1. I remember when I was a kid, many decades ago, some of us nerdy types excitedly read "One, Two, Three, Infinity" by George Gamow. Gamow covered some of the same territory you just did although I'm guessing progress has been made in the last half century or so.. If I recall correctly, Gamow said that only three "transfinite" numbers had as yet been identified. The lowest order of transfinite is the number of all integers. The second is the number of all real numbers. The third is the number of curves one can draw through space. Again, I'm relying on memory, but I believe he said nobody had figured out a fourth transfinite. Has that changed? Are there now more transfinite numbers than their used to be?

1. There are the aleph's of George Cantor, where aleph's is countable infinity. Aleph1 is the first uncountable transfinite number. It is also the continuum, or Bernays and Cohen showed it to be unprovably consistent with set theory. It is a form of Godel's theorem. It also turns out a y alephn also works.

For physics and computation only aleph0 and aleph1 are at all possibly relevant. There many alephs. The beyond those are least inaccessible cardinals, due to Uman. Beyond this, which is beyond classical set theory, are nonstandard set theories. I have little idea of those things.

2. Lawrence Crowell1:39 PM, December 05, 2020

"Aleph1 is the first uncountable transfinite number. It is also the continuum, ... It also turns out a y alephn also works."
Not quite true. While Cohen showed pretty much any n > 0 is possible depending on the model, there are other restrictions e.g. by Konig's theorem n cannot be ω.

"For physics and computation only aleph0 and aleph1 are at all possibly relevant. "
Mmmm, Turing birthed Computer Science out of considerations of Hilbert's Entscheidungsproblem and Godel's incompleteness theorems. Computability, set theory and logic are intimately connected. E.g. the question of termination of certain computer programs can be reduced to the question of existence of large cardinals.

3. So, to pursue the point, there is nothing special about the infinity of number of curves in space? I believe there was supposed to be a proof, analogous to the "diagonal proof" but different, that this infinity is of a larger sort than the infinity of real numbers.

4. Christopher8:48 PM, December 13, 2020

"there is nothing special about the infinity of number of curves in space? "
Apparently the cardinality of the set of curves in R^n is the same as that of R. See the answer here:

https://math.stackexchange.com/questions/2550313/is-aleph-3-the-cardinality-of-all-the-surfaces-that-exists

2. While I am not familiar with the works of any of the physicists listed in the video, I am profoundly of the perspective that infinities and infinitesimals are never anything more than unreachable limits of processes — algorithms, really — that obey certain convergence smoothness rules. To me, the concept of infinitely smooth pre-existing spacetime makes no more sense than postulating the pre-existence of an infinitely elaborated Mandelbrot set. Ironically, quantum mechanics practically screams the message of there being a bottom to details by forcing physics always to pay more energy to get more detail. It's just that our primate architecture neural systems are so committed to classical thinking that we force such an interpretation even when it really makes no sense.

The solution to the problem of infinities is both oddly simple and deeply disturbing: All that is needed is to abandon two or three millennia of falsely believing that mathematical and physical spaces are both absolute and infinitely detailed, and face the simple truth that everything we have been doing has actually been an algebra not of infinitesimals, but of finite, resource-limited algorithms.

Thus while it's easy to postulate Planck foam as an extremum of pushing space-fabric calculations to unimaginable extremes, the experimental reality has always been that such extreme and elaborations require impossible concentrations of mass-energy. It is mathematically, physically, and philosophically more efficient to assume instead that the universe uses its own versions of convergent algorithms to implement physics; that is, that we live in a universe not of particles, but of convergent algorithms, one in which infinities of all sorts need not apply.

I've mentioned my personal acronym for this view a couple of times here: PAVIS, for physics as virtual instantiation software. Quantum mechanics in PAVIS is literally just the uncertainty that arises because the fabric of classical reality is mass-energy limited and incomplete. Delightfully, quantum computing (QC) in PAVIS turns out to be not much more than making natural computations more efficient by getting rid of as much as possible of the noise — the irrelevant details — created by the rich information content of classical matter.

Solutions arise in well-constructed QC for much the same reason that rubber rafts with amorphous, ill-defined forms can safely navigate the rapids that would shred classically rigid boats. Forget the classical rigidity of von-Neumann inspired qubits: If you really want to see QC hum, take the approach of biomolecules. These combine quasi-classical problem setups with the quantum blurriness around the edges to schmooze themselves smoothly and efficiently into impossibly unlikely reaction paths. That, my friends, is where the real power of quantum computing lies, rather than the horribly strangled and insanely classical-like concept of qubits. Like pilot waves and point particles, qubits are far more a reflection of our deep fear of the profoundly different than they are of embracing how the universe really operates in the quantum realm.

Finally, algorithm-first PAVIS physics transforms certain interpretations of mathematics. That's because to develop an algorithm-first interpretation it is also necessarily to assume absolute conservation of mass-energy as a first principle, one deeply linked to virtual pair production. Magnificent constructions such as Noether's theorem remain magnificent, but they also get flipped upside down into what I've sometimes here called Rehteon's theorem: the idea that the smoothness of differential mathematics emerges from conservation, rather than the other way around. Conservation encourages smoothness by chipping away algorithmic extrema in much the same way that extreme, thinned-out points and edges on a conserved clump of clay eventually tend to collapse back in on themselves.

1. "PAVIS - physics as virtual instantiation software ... algorithm-first PAVIS physics transforms certain interpretations of mathematics"

Pretty cool!

Here are recent articles by Nicolas Gisin on physics with intuitionistic vs. classical mathematics:
https://arxiv.org/abs/2011.02348 [4 Nov 2020]
https://arxiv.org/abs/2002.01653 [4 Feb 2020]

2. In the end, the universe must work. That single fact should eliminate things like singularities.

3. > This is why some physicists, like George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should be formulating physics without using infinities or infinitely precise numbers.

Do we have the mathematics to do this?

1. No, because we don't know how to get GR right without a continuum.

2. Wait, so what about the entirety of calculus? Differential forms are infinitely small.

3. Another question is - is the type mathematics we bring to bear on physics a metaphysical assumption?

Realistically, what choices do we have? Do we have a lot of choices of independent axioms? E.g., the independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF), leaves us with a choice, but I doubt this choice matters for physics.

4. COOL !! I Just now noticed my (pending) post below is likely an Answer to ^this query :

S: > This is why some physicists, like George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should be formulating physics without using infinities or infinitely precise numbers.

Q: Do we have the mathematics to do this?

Reply: 1. Sabine Hossenfelder12:25 PM, December 05, 2020
No, because we don't know how to get GR right without a continuum.

See answer in my Post below (now pending)

5. I think the mathematics is a work in progress. They make a good argument as to why it is needed. They don’t mention GR as I recall.

“Physics without Determinism: Alternative Interpretations of Classical Physics”
https://arxiv.org/pdf/1909.03697.pdf

“Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?”
https://arxiv.org/abs/1803.06824

6. Getting to GR without a continuum: Mathematically, we'd have to look at the continuous GR as an approximation of the deeper discrete reality. Like the Navier-Stokes equations are an approximation of the behavior of a continuous fluid that is really composed of discrete objects.

The problem is that without knowing what the discrete components of space and time are, we don't know how to "round off" the equations to discreteness.

We are doing math with Reals when reality is likely dealing with only Integers. e.g. if space is composed of discrete space-particles and gravity comes in units of gravitons, then space cannot be infinitesimally curved, and gravitational effect cannot be infinitely small. There is a 0 gravitational effect, and a single unit of gravitational effect, and nothing in-between.

Or perhaps what seems like a half unit of gravitational effect is a 50% duty cycle of unitary gravitational effects; like a light flickering with an average of 50% lit.

7. Dr. A.M. Castaldo12:39 PM, December 10, 2020

If you could re-write GR as a discrete theory you'd get the same answers. Continuous Maths is essentially covering any discrete theory down to as small as you like, and you'd never spot the difference at the precision of physical measurements.

"Integers" also contains lots of assumptions about order and commutativity of combination, etc. There is no evidence at all that reality is dealing with integers.

GR predicts the subtle phenomenon of g-waves which have been apparently detected by LIGO, yet there is a discrepancy between the predictions of GR as currently applied and the simplest case for gravity of orbiting stars in galaxies. This is an incredible state of affairs. This is what needs sorting out, not tweaking the Maths.

8. Steven Evans:

"Continuous Maths is essentially covering any discrete theory down to as small as you like, and you'd never spot the difference at the precision of physical measurements."

This isn't a priori clear. Please see my earlier post about singular limits.

9. I see. So taking limits in the Maths could be hiding a lurking significant discontinuity in the Physics.

10. Yes. In particular the continuum is by many believed to be necessary for quantum mechanics. (I am not sure I buy the argument, but it's a long discussion.)

11. Dr. Hossenfelder: Why would a continuum be necessary? QM already has discrete quantum jumps, the orbital shells around a nucleus. If a single orbital shell also consisted of a billion discrete points, could we detect the difference?

12. If you are interested in founding physics of mathematical finitism I suspect some coomlementarity between continuum and infinite with finite sets is the best bet.

13. Dr Castaldo,

You are referring to measurement spectra of certain systems. I am referring to transformations in the Hilbert space. The relevant paper is this.

14. > "The relevant paper is [Hardy 2008]."

Possibly an interesting read for historians of science. It feels a bit like an eighteenth century scientist expounding the axioms of phlogiston theory. Impressive and exquisitely useless. (2008!)

15. The paper is from 2001. Seems your reading of its content wasn't any more attentive.

16. The PDF that I downloaded (twice) from the link that you provided says "February 1, 2008". Not that it makes any difference.

4. Sabine, you stated the following:
-------
“But is infinity real? Does it exist? Well, it arguably exists in the mathematical sense, in the sense that you can analyze its properties and talk about it as we just did. But in the scientific sense, infinity does not exist.”
-------
I disagree. For there exists the presence of the infinite nothingness (or infinite void) that is forever giving-way to the expanding reality of the universe (or the expanding reality of a multiverse if such a thing is real)

In other words, as a thought experiment, if it were somehow possible to gather all of the matter of not only our universe, but of every possible context of reality imaginable (be it objective or subjective, local or non-local, eminent or transcendent) into one centralized location (or “lump”), and then somehow double it,...

...then clearly there would need to literally exist some sort of “arena-like” containment medium that, again, is forever giving-way to the doubling (or tripling, or quadrupling) of the matter.

And because we cannot imagine this infinitely receptive “arena” of nothingness as containing any sort of wall that would halt the exponential expansion of matter,...

...then it (the “arena”) seems to be a tangible and “observable” aspect of reality (if only by inference) that is indeed infinite.
_______

1. Science is not validated by imagination. How is your "ininite nothingness" required to explain any specific observation?

2. Jim Birch asked the following question:
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“Science is not validated by imagination. How is your "ininite nothingness" required to explain any specific observation?”
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It all seems self-evident to me, nevertheless, humans have made the “specific observation” that the universe appears to have begun at a stage when it was no larger than a single atom. And that over the course of 13.8 billion years, that infinitesimal “speck” has expanded to the point where it is now estimated to be approximately 93 billion light years in diameter.

In which case, reason dictates that “something” on the other side of the all-encompassing “film” of this expanding bubble of reality, is giving-way...

(or making-room, or relenting, or whatever you want to call it)

...to the presence and expansion of matter.

Furthermore, reason also dictates that you could add any number of 93 billion light year in diameter universes into the above scenario (say 10^500) and you still could not exhaust the holding capacity of whatever it is that is making-room for their presence.

In fact, one can deduce that the holding capacity of whatever it is that is making-room for those 10^500 universes is so vast and boundless that it would make those 10^500 universes seem to be no larger than that initial atom mentioned above.

Hence (and to reiterate what I stated in my initial post), whatever it is that is making-room for the presence of matter, it seems to be a “tangible and observable” aspect of reality...

(“observable” by inference – loosely similar to how “dark matter” is observable by inference)

...that is indeed infinite.

(At the very least, it is an alternate vision of the concept of infinity that has a little more “meat and utility” to it, as opposed to mathematical infinities.)
_______

3. Keith D. Gill11:30 AM, December 06, 2020

"In which case, reason dictates that “something” on the other side of the all-encompassing “film” of this expanding bubble of reality, is giving-way..."

If 4D space-time is a closed surface like the surface of a 5D sphere, then it won't have another "side" or an "all-encompassing film".
I don't think it's even empirically known that space is expanding, just that observable matter and energy now take up a lot more space than they did 13.7 bya.

"one can deduce"
10^500 is finite not infinite. Physically, no infinity has been observed. Mathematically, infinities have to be assumed, there is no known way of deducing them from simpler axioms. So no-"one can deduce" infinity.

4. -------
Steven Evens:
“I don't think it's even empirically known that space is expanding,...”
-------
So then, the generally accepted conclusion that according to the Big Bang theory, the universe was once the size of an atom but is now 93 billion light years in diameter, is not an indication that some sort of expansion has taken place?

You seem to be crossing over into Cartesian territory where the only thing that one can be empirically certain of is the existence of one’s own “I am-ness.”

-------
Steven Evans:
“...just that observable matter and energy now take up a lot more space than they did 13.7 bya.
-------
I’m not sure of what you are getting at, for you seem to be confirming my point.
-------
Steven Evans:
“10^500 is finite not infinite.”
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Right. However, it should have been obvious that I was simply using the example of the vast, yet finite number of 10^500 universes as a point of comparison to help visualize the boundless (infinite) holding capacity of, again, whatever it is that is making-room for those 10^500 universes (or 10^500,000,000 universes, pick any number).

-------
Steven Evans:
“Physically, no infinity has been observed.”
-------
Right. And that’s why I tried to make it clear that what I am suggesting can only be “inferred.”

-------
Steven Evans:
“Mathematically, infinities have to be assumed, there is no known way of deducing them from simpler axioms. So no-"one can deduce" infinity.”
-------
Definition of deduce:
---
Deduce
/dəˈd(y)o͞os/
verb,
1. arrive at (a fact or a conclusion) by reasoning;

Synonyms: conclude, infer, draw the inference, extrapolate, glean, divine, intuit, etc., etc..
---
By that definition, I have deduced ("intuited") that the absolute nothingness that resides above and outside of what we call "material reality," is a clear and obvious representation of something that is infinite.
_______

5. Steven Evans wrote:
-------
“Physically, no infinity has been observed.”
-------
In my initial response to that assertion, I basically agreed with it.

However, upon deeper reflection, I suggest that as we stand on the earth and look out into space, then no matter which direction we cast our gaze, we are looking into a real and existent infinity.

And that is why I disagreed with Sabine’s assertion...
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Sabine wrote:
“...in the scientific sense, infinity does not exist...”
-------
...for clearly, it does indeed exist.

6. Keith D. Gill11:00 AM, December 09, 2020

" then no matter which direction we cast our gaze, we are looking into a real and existent infinity."

You've got very good eyesight.
https://en.wikipedia.org/wiki/Universe
Diameter Unknown.
Diameter of the observable universe: 8.8×10^26 m

No infinite quantities have ever been observed in nature. A simple fact.

In Maths, infinities have to be assumed in the axioms.

"...for clearly, it does indeed exist."

Nobody knows that. You can't just decide the answer without any evidence. On the plus side, you could be in the running for the next UK Astronomer Royal.

7. Steven Evans wrote:
“No infinite quantities have ever been observed in nature. A simple fact.”
-------
I disagree.

As I stated earlier, as one looks into any direction of the open space that stretches out beyond this planet, we are indeed looking into a real and existent (observable) context of infinity.

I mean, just because there may be a “few” galaxies of suns and planets standing in the way of our gaze, along with an assumed boundary to this universe (a boundary that exists by reason of it all beginning with a “Bang” 13.8 billion years ago),...

...then other than your valid issue of none of us possessing good enough eyesight to see the true depth of the infinity we are gazing into, it is nevertheless there.

Not meaning to sound like the proverbial broken record here, but infinity exists in the form of, again, the boundless arena of absolute nothingness that is forever making-room for the expansion of matter.

The problem is that you are hamstringing your ability to perceive the infinity I am describing by limiting your concept of “nature” to only that which the calipers of science can grasp and measure.

Now with all of that being said, I fully admit that if you combined the worth and importance of what I am suggesting with that of a \$1.50, it will get you a ride on the city bus. Ha!
_______

8. Keith D. Gill3:59 PM, December 10, 2020

"we are indeed looking into a real and existent (observable) context of infinity."
This is not known to be the case. I just pointed to the information about the size of the observable universe. It is finite.

"along with an assumed boundary to this universe"

There is no known boundary to the universe. It's topology is unknown. It could be a closed surface in which case it has no boundary. Nobody knows.

" (a boundary that exists by reason of it all beginning with a “Bang” 13.8 billion years ago),..."

It is not empirically known that the observable universe had a beginning. It is empirically known that the observable universe expanded from a miniscule extent to what is observed today over 13.8 billion years. It is just speculation that the observable universe had a beginning (not unreasonable speculation, granted).

"none of us possessing good enough eyesight to see the true depth of the infinity we are gazing into, it is nevertheless there."

You can't see it but it's there? You are not on strong ground here.

"but infinity exists in the form of, again, the boundless arena of absolute nothingness that is forever making-room for the expansion of matter."

Nothing is the absence of anything for a start. And there is no "boundless arena" of anything known to be "making room" for the expansion of matter. You need to read some topology. There are many possibilities for a closed universe.

"The problem is that you are hamstringing your ability to perceive the infinity I am describing"

What you are describing is not perceivable - that's why I can't perceive it. You think the universe is a growing sphere, expanding into something, but that is absolutely not known to be true and there are many other possibilities topologically.

"by limiting your concept of “nature” to only that which the calipers of science can grasp and measure.""

Everything we know about nature comes from Natural Science. It's the only known method for providing information about Nature. What you can imagine is irrelevant.

"if you combined the worth and importance of what I am suggesting with that of a \$1.50, it will get you a ride on the city bus. Ha!"

I suspect I'd be forced to take Shanks's pony....;)

9. Steven Evens:
Me: "we are indeed looking into a real and existent (observable) context of infinity."
You: “This is not known to be the case. I just pointed to the information about the size of the observable universe. It is finite.”

Finite compared to what?

Steven Evans Wrote:
“There is no known boundary to the universe.”

Yet you just insisted that the universe is “finite,” which, in my book, implies a boundary.

Steven Evans wrote:
“Nothing is the absence of anything for a start. And there is no "boundless arena" of anything known to be "making room" for the expansion of matter. You need to read some topology. There are many possibilities for a closed universe.”

You just asserted that there is no known boundary to the universe, yet now you are speaking of the many possibilities for a “closed universe,” which, again, implies the existence of a limiting boundary.

You can’t have it both ways.

Besides, we are talking about two different things here – the “somethingness” of universes and matter, and the “nothingness” that seems to be constantly making-room for the somethingness.

Steven Evans wrote:
“Everything we know about nature comes from Natural Science. It's the only known method for providing information about Nature. What you can imagine is irrelevant.

Tell that to one of the greatest natural scientists of all time who allegedly stated the following:

“Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution.” – Albert Einstein
_______

10. Keith D. Gill10:59 AM, December 11, 2020

https://en.wikipedia.org/wiki/Surface_(topology)#Closed_surfaces

"A closed surface is a surface that is compact and without boundary. Examples are spaces like the sphere, the torus and the Klein bottle."

You personally can't seem to imagine the surface without it being embedded in 3D space, because that's all you've experienced, but this is the surface of a sphere and it has no boundary:

x^2 + y^2 + z^2 = 1

It's perfectly possible that 4D space-time is the closed surface of a 5D sphere, say, and therefore has no boundary. It's a simple fact of topology.

"Finite compared to what?"
Finite as in it doesn't go on forever.

"and the “nothingness” that seems to be constantly making-room for the somethingness."

Nothing is the absence of anything. Nothing can't make room or do anything else. That's what we mean by "nothing". As I said, if the universe is a closed space, it can be finite and not have a boundary. Empirically speaking, whether the universe is infinite or not is unknown, as the wikipedia article states and all physicists accept.

"“Imagination is more important than knowledge."

Ahh, I see, you think because you can imagine something it makes it real. Like fairies or trolls?
Einstein used his imagination to do Natural Science. Einstein's theories are accepted as relevant because they were confirmed by **observation** in the physical world, not because they appeared in his head.

Here are links to a resource called a "dictionary" where you can look up the meaning of words you don't understand and thereby communicate with other humans:

https://www.merriam-webster.com/dictionary/finite
https://www.merriam-webster.com/dictionary/nothing
https://www.merriam-webster.com/dictionary/imagination
https://www.merriam-webster.com/dictionary/natural%20science

11. P.S.
"Yet you just insisted that the universe is “finite,” which, in my book, implies a boundary."

The observable universe is finite (possibly a tautology), but whether the universe is finite or infinite is "unknown".

FYI:
https://www.merriam-webster.com/dictionary/unknown

12. Steven Evans wrote:
“Ahh, I see, you think because you can imagine something it makes it real. Like fairies or trolls?....Here are links to a resource called a "dictionary" where you can look up the meaning of words you don't understand and thereby communicate with other humans:”

Steven, it’s unfortunate that you have now found it necessary to resort to cheap insults. Nevertheless, thanks for the conversation.
_______

13. Keith,
You did write that you didn't know what "finite" meant, so I pointed you in the direction of the relevant resource. I also told you several times that topologically the universe could be a closed surface and therefore be finite and not have a boundary. I even gave you a link to the relevant Wikipedia articles.

Have a nice day.

5. I love the infinity topic. (Continuing fractions are crazy weird.) Great video.

Suggested topic: The history of the ether. All the attempts to detect it from stellar aberration observations, Airy’s water filled telescope, Fizeau’s experiment, to Michelson-Morley make a fascinating tale. The fact that Einstein’s relativity was more influenced by the alternative explanations of induction in Maxwell’s equations shows phenomenal insight. New ether-like concepts like quintessence and the Higgs field seem revive the possibility that ‘the vacuum ain’t nothing’.

6. Sabine, while I generally hold the same view, that infinities are a mathematical convenience, I'm wondering how one would reconcile it with the fact that for the canonical commutation relation [x,p]=iℏ to hold, the operators x and p have to be described by infinite dimensional matrices. No finite approximation does the trick, as far as I can tell.

1. ? Of course there are finite dimensional matrices that fulfill this relation, the simplest are the Pauli matrices. Non-commutativity isn't remotely as weird as people think it is. What is weird is that we only teach commutative math in school.

2. Let me clarify my question. The commutator of Pauli matrices is not an identity, but the remaining Pauli matrix, which is traceless. [x,p] is a multiple of the identity, which is not traceless, and that is a property of infinite-dimensional matrices only. My question is whether this is an indication of infinities being "real" in some sense, or if there is a different way to understand it.

3. Sergei: For the canonical commutation relation [x,p]=iℏ to hold, the operators x and p have to be described by infinite dimensional matrices.

Sabine: There are finite dimensional matrices that fulfill this relation, the simplest are the Pauli matrices.

I think Sergei was referring to the fact that the canonical commutation relation pq - qp = ihI is impossible in finite matrices, noting that the trace of finite matrices is commutative, so the trace of the left side is zero for any finite matrices, whereas the trace of the identity matrix on the right side is obviously not zero. It holds for the canonical variables like linear position and momentum because they are represented by infinite dimensional matrices... indeed the terms become infinitely large. The Pauli spin matrices aren't a counter-example to this, because they are of the form [sx,sy]=sz, etc.

4. I see what you mean, sorry for the misunderstanding. To address your question, it seems to me one would have to think about whether the difference (between large but finite and infinite) is observable. I suspect the answer is "no", but I wouldn't know off the top of my head how to prove it. It seems to me one almost certainly has to draw on some particular properties of the operators under question. It's an interesting point, thanks for bringing it up.

5. Hi Amos, Yes, sorry for the misunderstanding.

7. This comment has been removed by the author.

8. Thanks very for this video, I really enjoyed watching it :) I wonder if you could say something about the "infinity" problem in so-called quantum electrodynamics? Roughly speaking, how did they "remove" it? What exactly does "renormalization" mean? My knowledge in this has been limited to dummied down "pop-culture" books on the subject. Thanks and regards. ~ Ronian

1. It is briefly mentioned in the video. You identify the type of infinity and then subtract it suitably to leave a finite result. It's a historical accident that we do it in this weird way.

9. Is infinity real? I suppose it depends on the definition of real.

1. Which I talked about extensively last year.

10. “But is infinity real? Does it exist? Well, it arguably exists in the mathematical sense, in the sense that you can analyze its properties and talk about it as we just did. But in the scientific sense, infinity does not exist.

That’s because, as we discussed previously, scientifically we can only say that an element of a theory of nature “exists” if it is necessary to describe observations. And since we cannot measure infinity, we do not actually need it to describe what we observe. In science, we can always replace infinity with a very large but finite number. We don’t do this. But we could.”

Concerning existence in a mathematical sense, I agree, and I would also add that things like ordinals and cardinals "exist" in the sense that they "∃", when described within a first order logical system like ZF.

But in a physical sense, I would like to just submit this: if you agree that at least in principle our observations could result in arbitrarily large numbers, then shouldn't we need the infinity right there, precisely to "describe our observations"? If we don't need infinity, shouldn't we start out by saying that we can only ever have finitely many observations, and our entire scientific approach should take that into account? And if we believe something like "we can always take another measurement", have we not made a commitment to at least the countable infinity?

11. Addendum: I looked up the current research work of George Ellis, Tim Palmer, and Nicolas Gisin. In the recent works of the first two — brilliant fellows no doubt — I saw at best only a vague connection to the question of why infinities cannot exist. I’m sure it’s in there somewhere, I just didn’t catch any particularly clear, insightful statement of it.

It’s not the first time I’ve had that frustration. Physics for decades and real mathematics for centuries have unintentionally encouraged profligate use of infinities (think MWI for example), and conversely have tended to drive out any presumption that nature might actually be… well… efficient in some horribly, embarrassingly pragmatic fashion. Instead, huge fiscal and intellectual resources have ended up applied to such sad-sack silliness as MWI, which if you think about it a bit doesn’t even pass the muster of the basic information density constraints taught in undergraduate radio-frequency engineering. It is certainly not a quantum theory since it doesn’t quantize anything, like Einstein had to do to solve the ultraviolet catastrophe. Does it really matter that MWI uses matter waves instead of only electromagnetic waves? No, it does not. When indifference to finite limits becomes endemic across large swaths of untestable theorizing, it generates a lot of noise, and make the real signal in what people are saying hard to discern.

In contrast, Nicolas Gisin clearly “gets” the issue of finite resources versus infinities, even if he uses the unfortunate label of “intuitionist” theory. The label makes his detailed and very hard-nosed form of pragmatism sound like some sort of wishy-washing discussion of who has the better intuition for maths.

Here’s a snippet of a talk that Gisin gave at Université de Genève on Oct 7, 2020:

https://youtu.be/IWlixRKdq1M?t=1189 to 1344 (19:49 to 12:44)

The whole talk is worth a careful viewing. What Gisin is saying is not what you are likely to expect, and that makes it hard to hear what he is saying rightly.

One questioner asked him three times, in variant ways, whether Gisin’s assertion that finite volume can only have finite information means that space is “discrete”.

You can hear (or at least I think I can) the carefully guarded frustration in Gisin’s negative answers. Gisin knows that trying to make space into a finite array of bits means you are completely missing his real point. Discrete space is another way of saying that the underlying structure must still at some level still be classical, like an array of bits in a computer memory. That is not what Gisin is trying to say. He keeps trying to bring it back to the real issue of how there isn’t classical determinism of any kind going on, neither the smooth continuum version or the seemingly safer discrete-array form.

I would tend to express this same issue more in terms of information theory and lack of resolution. If you don’t have enough bits, then absolutely no matter what you do, something about the nature of both time and history has to get blurry, just as a video coming in over a low-bandwidth connection will be blurrier than one arriving over an optical fiber. Whether it sounds heretical or not doesn’t matter, because in the end that is all quantum uncertainty can be: the darkness and blurriness that inevitably comes upon you when you finally reach a scale of detail where the bits run out. Our universe is always finite in mass-energy, at least locally, so it must necessarily get blurry — get quantum — at its atomic and particle edges. Nothing else is possible. This is the same blurriness that Gisin is addressing, just using a different set of terminology. He understands that a finite universe is necessarily a fuzzy universe, and not in some casual or sloppy way, but in a way that reaches so far down that we have a special name for it: quantum mechanics.

1. A very interesting read, and a video :)

Some things come to mind, at least tangentially related, concerning a finite volume of space (or space-time) only having a finite amount of information. Exploring this question seems like jumping the gun to me, because, I would suggest, the total amount of "information" we have about a physical state of the universe if finite. There are only finitely many measurements done to date. Each measurement is different, but at the end of the day the state of knowledge is discrete. Many measurements are just detection versus non-detection, that's one bit, others are rational numbers with finite precision. If that's the only way human physicists can make "measurements", then the amount of accessible information is finite in every space chunk simply because all of it is finite, period.

And if that feels a bit like cheating, consider also this. Like you say, the amount of energy within a finite (compact?) region of space is finite, and so perhaps there's also a "resolution" limit on how much of that energy can cascade into the human brain as bits over a period of space-time that is a typical human "measurement".

Perhaps the quantum nature of physics on small scale is due to how we humans think about physics, and not only because of how the world actually is. Human brain seems to contain only a finite amount "scientific" information at any time, but it does not have to mean that physics which enable it have the same property. It would be straightforward to design a mathematical universe with real numbers and whatnot that would within itself allow for finite approximations to classical computers to evolve and walk around and try to figure things out, and perhaps we are in a similar situation.

Thanks again for a fantastic, very stimulating read :)

2. I appreciate your comments and underlying pragmatism. And for me, the ideas of Gisin et al regarding determinism and an open future are a welcomed breath of fresh air, like a knee to the neck being lifted. That may seem hyperbolical, but for me it is not.

“It is mathematically, physically, and philosophically more efficient to assume instead that the universe uses its own versions of convergent algorithms to implement physics; that is, that we live in a universe not of particles, but of convergent algorithms, one in which infinities of all sorts need not apply.”

I don’t understand this. Is it possible to view particles themselves as algorithmic elements, simple steps in a much broader calculation?

I do question your equating quantum uncertainty with a lack of bandwidth at the fringe. I wonder if it is instead revelatory of something deeper. The fact that as your certainty of measurement of one quantity increases only while that of the other proportionately declines suggests an underlying stratum which we can selectively approach by choice of device.

And, I wonder at (if I understood him) Gisin’s proposition that a random number generator creates new bits of information. For one thing, how much information does a random string actually contain unless it is in external relationship, a code echoed? Isn’t the information of a bit string actually in the relationships between the elements, one to another that can extend to the entirety of a finite string. The question perhaps is whether that relational property can extend to an infinite string.

The other question is that, in moving from the abstract realm of Shannon bits to that of a physical dynamic, a bit has to ‘do’ something, in Bateson’s words, “make a difference.” What does that look like? Well, in particular there are particles with varying degrees of ‘hang time.’ They are physical dynamics that endure (proton half-life =1.67 x 10^34 years). One can wonder if they are located on some vast preexisting timeline or whether, in sum, what they are ‘doing’ is actually creating time with all its multitude of tempos.
Is it useful to make an analogy between the internal relationships of a bit string and that of a physical composite of particles? In order for an aggregate of particles to ‘hang’ in time there must be a dynamic internal relationship between the parts, a physical logic that is consistent within itself and, in contrast to Gödel's second incompleteness theorem, can demonstrate its own consistency by its constancy. There are gene sequences in the human genome (for that matter all plant and animals) that are thought of have endured more that 3.5 billion years.

So, a scattershot of comments here, hope something’s in the black.

3. Terry Bollinger11:03 PM, December 05, 2020

"I saw at best only a vague connection to the question of why infinities cannot exist."
It's an empirical question. Nobody knows if space-time is infinite. Is time going to suddenly stop one day?

"Physics for decades and real mathematics for centuries have unintentionally encouraged"
Mathematics is very clear what it means by infinity. Potential infinity is assumed in the Peano Axioms and actual infinity is assumed in ZFC. It's all very much intentional and rigorous.
Physicists are also well aware that infinite Maths is just a tool.

"think MWI for example"
The problem with MWI as a physical theory is that there is no empirical evidence for it. That's all.

"doesn’t even pass the muster of the basic information density constraints taught in undergraduate radio-frequency engineering."
Constraints which are not known to apply to quantum physics. There is no refutation of MWI, only a lack of evidence.

"In contrast, Nicolas Gisin clearly “gets” the issue of finite resources versus infinities,"
Empirically it is not known whether the physical universe is finite or not. Let's stick to saying what we actually know.

"whether Gisin’s assertion that finite volume can only have finite information means that space is “discrete”."
Information theory may be a useful tool, but there is no evidence that it is an ontology. Finite volumes are not empirically known to have infinite anything.

"He keeps trying to bring it back to the real issue of how there isn’t classical determinism of any kind going on, "

There is, it's just emergent that's all. And superdeterminism hasn't been ruled out of quantum theory. Again, best to just state what we actually know to be true.

"Our universe is always finite in mass-energy, at least locally, so it must necessarily get blurry — get quantum — at its atomic and particle edges. "
We can all claim that it was obvious that the underlying reality had to be quantum after the fact. If you want to make a case for information theory applied to Physics, tell the Physicists something they don't already know.

"Nothing else is possible. "
Superdeterminism is possible.

"He understands that a finite universe is necessarily a fuzzy universe,"
Empirically, we don't know if the universe is finite or fuzzy.

4. Don Foster, Steven Evens, and Philip Thrift earlier: Excellent comments and questions, and nicely from both directions! I can't answer today, but will put in a reply in a day or so. Don, from your reference I note that Nicolas Gisin works with Flavio Del Santo, whom I know a bit and very much respect. It's important to see this community grow.

5. Your 'algorithm first' ideas are entirely in detail. Algorithms process data. It is the data itself that is meaningful.

Let's take the lambda calculus as a paradigm for describing computation/algorithms. The lambda calculus is famously Turing complete and has no data primitives. Rather, the programmer who would use the lambda calculus abstractly assigns certain patterns found in the lambda calculus as his/her preferred data primitives. Without this *choice* of the programmer, the algorithms themselves are meaningless.

So in your 'algorithms first' PAVIS, what are the data primitives?

6. Steven Evans,

“Superdeterminism is possible.”

Consider this. The reason math works is that the universe, in some yet to be fully understood fashion, calculates what happens next across a vast manifold wavefront of present moments like a huge parallel computer. The mathematics of physics arises as piecemeal replication of that innate substrate of calculation. But, with all due deference to experimentalists who conceive, design and fabricate the devices that have wonderfully refined our equations, the universe has managed to both sustain and profoundly articulate itself independent of our mathematics.

The proposition of Gisin et al, is that the precision of our present mathematics projects upon the time evolution of our universe a fidelity that it does not actually reflect or require. The most current atomic clock is thought to lose only one second in thirteen billion years, a constancy made possible only by the considerable effort made to insulate it from the vagaries of its environs. Empirically contrast that with a biology that is both consistent in its process and yet adaptive to every possible environmental extreme.

It is quite possible I misread the nature of superdeterminism and its ability to explain the universe we observe, but I see its suppositions as conjectural and an actual hindrance to clearer understanding.

So, is it possible to prove superdeterminism?

7. [TERRY RESPONSES, 1 OF 4]

A few responses, in text-occurrence order, likely with more typos than usual :)

Philip Thrift: Thanks for the excellent Gisin references! Gisin has unexpectedly become my favorite mathematician. I’m not used to saying “Yes!” frequently in my head when reading most mathematical works, but in this case I’ve found I do.

Peter John: You make a good point that a working universe severely limits the idea that any kind of true singularity can exist.

Ivan Zaigralin: Thanks, and you are spot-on about observational limitations imposed by necessarily finite instrument and observer measurements. As you aptly note, this limitation unavoidably means that the underlying universe thus could then be infinitely detailed and thus real number based, yet still “register” to all measurements as indeterminant. In fact, both the pilot wave model and superdeterminism rely on this interpretation.

Along those same lines, in a universe with Bell violations pilot wave models are necessarily superdeterministic due to their use of true-point particles that are guided by the pilot wave. The pilot wave becomes a bit of local interpretive fluff in such a model, since it’s always the infinitely long particle path (a quantum, hidden-variable version of a worldline) that must pre-encode all experimentally observed Bell violations.

So why do I suggest this is not the case? It’s pretty simple, really: Point-like limit cases still require infinite energy to instantiate. Thus to uphold the well-verified principle of classical mass-energy conservation, any process that models the quantum world by first assuming the reality of pure states must also simultaneously “subtract out” the increasing energies invoked as you approach such states. That in a nutshell is why to this day the most inexplicable anthropic number in all of physics is how our universe ended up perched on the mindbogglingly narrow edge of a positive/negative net vacuum energy knife. A conservation (unitarity) first approach simply trims off all such detours to infinity by what amounts to self-observation and self-collapse via mass-energy conservation. That in turn implies that wave collapse — which I would emphatically argue is the true heart of quantum mechanics, with waves being nothing more than the statistical outcome of wave collapse when it is applied at every scale — is the deeper constraining principle underlying why techniques such a renormalization even work.

Don Foster:

>… “Is it possible to view particles themselves as algorithmic elements, simple steps in a much broader calculation?”

Yes, and here’s why: What we call “particles” are never anything more than path decisions relative to classical physics. You do not experimentally get a clearly visible electron particle going through one of two slits, you get a new wave that “decides” (is recorded) to emanate from only one slit, with all other histories erased. The electron coming out of that slit is actually more wave like, not less, due to the more limited aperture (same principle as in telescopes). Decisions are translatable to bits, and thus to outcomes (or more properly ongoing states) of algorithms.

>… “I wonder if … a lack of bandwidth at the fringe … is instead revelatory of something deeper.”

Good question! The reason for me making the assertion that lack of resolution is all that is going on an argument of conciseness: If you do nothing more than ask what happens if finite mass encodes finite bits, you quickly start getting something that looks just like quantum mechanics. Is that a sufficient argument? Of course not! But it’s a pretty powerful starting point for exploring quantum mechanics in new ways, ones that don’t simply assume infinite details and perfect states because such idea are more amenable to how our brains like to work (and that really is an issue here).

(continued in next comment)

8. [TERRY RESPONSES, 2 OF 4]

Don Foster response continued:

>… “I wonder at (if I understood him) Gisin’s proposition that a random number generator creates new bits of information.”

Try this: Entropy increases. Since entropy is information, my own physics-flavored interpretation of Gisin on that point is that he’s talking about entropy in an interestingly abstract way. I would say it this way: The universe keeps getting more classical, that is, the range of possible futures into which it can evolve keeps getting narrower over time.

>… “a bit has to … “make a difference.” … One can wonder if … physical dynamics [such as the way protons] endure … are ‘doing’ [something by] creating time with all its multitude of tempos.

That is an excellent observation!

Yes indeed, information — persistence of state — begets information. It is the existence of durable fermions, in particular of electrons, protons, and neutrons, that is the flatly essential first step for creating a “state machine” with sufficient persistent complexity and internal relationships to enable what we perceive as space and time. The fact that such fermions can have well-defined locations in xyz space is pretty much where the concept of information and classicality begins. Also, notice that the only way fermions can achieve such localization — and enable classical space, time, and information — is via an almost continual process of multi-scale wave collapse. It’s this self-collapse process that prevents the matter-wave equivalent of the ultraviolet catastrophe from occurring. The other more common name for the matter-wave catastrophe — that is, for the degeneration of the universe into noise that is no longer correlated to its actual mass, energy, and information content limits — is MWI. That’s why collapse-free MWI is really a statement of the wave-proliferation problem, rather than an attempt to solve it. So once again, it’s wave collapse that ends up being the fulcrum point of reality, by enabling “location in spacetime” to have information-rich meaning for particles such protons.

>… “In order for an aggregate of particles to ‘hang’ in time there must be a dynamic internal relationship between the parts, a physical logic that is consistent within itself and, in contrast to Gödel's second incompleteness theorem, can demonstrate its own consistency by its constancy.”

I mentioned briefly the importance of “virtual pairs” to any conservation-first approach to the mathematics of physics. That’s because if you are serious about absolute conservation, you obviously cannot just start creating somethings out of nothings. Instead, you must use pairs of somethings and anti-somethings, always with a net return to null existence and energy, as with virtual particle pairs.

Large-scale existence requires that at least one of these pairs, more specifically the positive and negative mass-energy pair, must be aggressively separated by the overall dynamics of the creation-annihilation system. Carroll-Chen in 2004 [1] (first by far!) and Boyle-Finn-Turok in 2018 [2] models, and for that matter my own personal notes in 2007, implement these ‘aggressive separation dynamics’ by using time/anti-time pairs — opposite vectors for pushing positive and negative universes away from each other in time. The Carroll-Chen model, intriguingly, is spherical, while I and Boyle-Finn-Turok used binary models, although I considered the spherical mode for one day and then went back to binary.

That’s the macro level. The micro level answer to your question is next.

[1] https://arxiv.org/abs/hep-th/0410270
[2] https://arxiv.org/abs/1803.08930

(continued in next comment)

9. [TERRY RESPONSES, 3 OF 4]

Don Foster response continued:

After you get mass-energy persistence is when your question about “dynamic internal relationship between the parts” kicks in. At the level of atoms, mutual relationships play a vital role is in multi-scale self-observation by thermal, information-rich matter. That sounds exotic, but all it really means is that atoms stay in your body and such in no small part because they are immersed in astronomically complex bath of phonons (quantized sound, heat, information) that almost continually collapses the locations of atoms and keeps them from turning into Schrödinger waves.

Here also is there the very concept of “point-like particles” becomes delightfully nefarious and destructive: Because folks habitually assume there is a “real” particle lurking somewhere a wave packet, folks unavoidably are also assuming 99.999…% of this continual, multi-scale wave packet collapse process as if it were a given. And then we wonder why the “boundary” between classical and quantum is so hard to define? Treating wave collapse as real and as something that is going on continually and at every level of “mundane” classical physics, specifically in any situation in which information (sound, heat, information for starts) is part of the dynamics, is a bit of prerequisite for everyone moving on to some deeper understanding quantum mechanics.

>… “There are gene sequences in the human genome (for that matter all plant and animals) that are thought of have endured more than 3.5 billion years.”

Yes, since they occur unchanged in forms that diverged that long ago. If information and information persistence is fundamental to physics, it puts a different spotlight on how life uses DNA to preserve information over time, doesn’t it? If nothing else, it may point to deeper use of fundamental physics by life than we’ve realized. I would point to biomolecular leveraging of quantum computing as another example.

Steven Evans:

>… “Physicists are also well aware that infinite Maths is just a tool.”

I agree completely! If there issues, its more when our decidedly classically-oriented minds try to apply these excellent tools in ways that inadvertently mask over critical issues such as continual wave self-collapse in thermal matter. On that point for example, just thinking “particle” is a infinity-class cheat, both mathematically and experimentally. We need to get beyond that by looking at collapse both a lot more seriously and a lot more mathematically. (And yes, sure, creating new maths for a rigorous understanding of what wave collapse is not likely to be easy. But that this century-long mystery might well prove to be at the very heart of why we still don’t understand QM fully is a surprise… how?…)

>… TB: ‘Nothing else is possible.’ SE: ‘Superdeterminism is possible.’

Surprisingly, I rather like superdeterminism, mostly because I think it is the most accurate, honest, and in-your-face interpretation available for what experimental quantum mechanics in a universe of special and general relativity actually means, e.g. it forces one to realize that Minkowski was necessarily a pre-quantum superdeterminist. I am delighted that Sabine et al may be getting near (fingers crossed, this is a very tricky area) some experimental ideas for testing this interpretation. I always recall Bell’s statement that one of the reasons why he found his inequality is the sharpness of reasoning that the pilot wave model gave him. Nice!

And with that said, I must also confess that my PAVIS / dark wave packet interpretation of the universe as a naturally occurring state machine, one with immensely complex relationships that we only interpret as space and time, is probably as far away from superdeterminism as one can get. The real point in both cases is to develop such ideas until actual experiments can tell us what is correct. What a radical concept! :)

(continued one more time for my response to manyoso)

10. [TERRY RESPONSES, 4 OF 4]

manyoso:

>… “Algorithms process data. It is the data itself that is meaningful. Let's take the lambda calculus [in which] the programmer … abstractly assigns certain patterns … as … data primitives. Without this choice … the algorithms … are meaningless. So in your ‘algorithms first’ PAVIS, what are the data primitives?”

I am fascinated that you brought up the lambda calculus, since it is the closest algebra of which I am aware to what I was talking about.

The creation-annihilation operators of particle physics are a subset of the algebra needed, but for PAVIS these operators need to be both generalized and cascaded to create the complex structures that serve as the persistent data (e.g. the protons) within of the overall state space. As with the lambda calculus, it is these patterns that serve as data primitives for this “pi” (pair) calculus. A primitive is any pattern that cannot immediately self-annihilate (like a virtual particle pair) due to some level of mismatch between otherwise canceling patterns. Hydrogen atoms are a lovely example of such a mismatch, since their electric charges cancel ultimately have a common origin and can cancel, but the electrons and quarks cannot. Thus you end up with almost-cancelled hydrogen atoms.

The more complicated part is the chaotic branches of the pi tree, the ones that get sliced and diced by so many operators that unwinding them become statistically unlikely. That’s actually the self-observation, self-collapsing part, and it also drives the critical dynamic of causal time-pair emergence. Without that, you cannot get pattern groups far enough apart to persist long enough to do thing like get cross-link the charge branches into protons and electrons. It’s very difficult to do any of this without creating at least two universes, and the 2004 (2004, wow) Carroll-Chen model suggests how to create an infinite number of such time vectors with a spherical symmetry.

Getting this to work for space seems to require some form of rather remarkably duality between fermions and space itself, that is, the idea that at a pretty deep level of the lambda (or pi) tree the half-spin spatial obstinacy of fermions is deeply linked to both the number and energy (“far is free”) behavior of 3-space. I used to think that the fact that 3 shows up both in the number of confined color charges and unlimited spatial dimensions was just a coincidence, but in a tree in which both emerge from a common split that’s actually unlikely: they are far more probably connected in a very deep way indeed.

Both wave collapse and smoothness emerge from the dynamics of new pair trees tending to branch of from conserved numbers such as mass, but also becoming more fragile and collapse-prone (unitarity) as they disperse those same numbers over xyz space, which again is just another shared component of the tree. Unitarity and multivariate intersection number combinations [1], which prioritize the more “real” virtual particles, seem relevant.

So is lambda calculus and its focus on structure relevant to PAVIS and dark function, infinity-adverse approaches? Oh yeah. The pi version may need to be even more restrictive, but lambda is not a bad start at all. For numeric calculations, though, approaches that first work on leveraging the absence of much classical-level information detail in these structures would seem worthwhile.

[1] https://arxiv.org/pdf/2008.04823.pdf

11. Don Foster11:10 AM, December 09, 2020

"The reason math works is that the universe, in some yet to be fully understood fashion, calculates what happens next across a vast manifold wavefront of present moments like a huge parallel computer."

There is no evidence of this. And I suspect assuming this, if you could define it exactly, would not lead to any knowledge about Physics.

"the universe has managed to both sustain and profoundly articulate itself independent of our mathematics."

AFAIK, the physical is real and came first and Maths is abstract and came later. So I suppose I agree.

"The proposition of Gisin et al, is that the precision of our present mathematics projects upon the time evolution of our universe a fidelity that it does not actually reflect or require."

Well, really it's just a continuation of a search for the theories that best describe and efficiently categorise physical phenomena. Of course, the theories are not isomorphic to reality. Also, with regards to a necessity for discrete models, Dr. H. links below to a paper which posits that the continuum might be necessary to describe the quantum. So it's all still an open question, it seems.

"The most current atomic clock is thought ... contrast that with a biology ... extreme."

Yes, there's Physics then there's Chemistry then there's biology. It is amazing, you are right. You can make some really cool models with Lego bricks, too.

"but I see its suppositions as conjectural and an actual hindrance to clearer understanding."
"So, is it possible to prove superdeterminism?"

http://backreaction.blogspot.com/2020/02/guest-post-undecidability.html
I believe the situation is that it may be partially testable, so it is conjectural. I would have thought superdeterminism provides an explanation more commensurate with human experience of determinism and therefore easier to "understand" compared to say, MWI, which transcends human experience.

12. Terry Bollinger4:05 PM, December 09, 2020

" I must also confess that my PAVIS / dark wave packet interpretation of the universe as a naturally occurring state machine, "

13. "Superdeterminism" per se is not testable. Superdeterminism is a property of certain types of models. You can only test the models themselves. It's the same with eg, supersymmetry, or determinism for that matter. These are properties of models, but these properties alone are not testable.

I have argued that superdeterministic models make a fairly general prediction that is largely model-independent in, say, the same way Bell's theorem is a fairly model-independent prediction about hidden variables theories. I have explained this in several of my papers in detail, but the brief summary is that sequences of rapidly repeated measurements in small systems should be correlated.

Problem is, no one wants to do the experiment. (Or, well, I thought I'd found someone, and then COVID happend. I'll get back to this eventually.)

14. Terry Bollinger4:05 PM, December 09, 2020

Your FCXi essay is interesting. The search for exact but concise descriptions of the universe as the search for Kolmogorov complexity minimum. String Theory as "trampoline effect". The limitations of using group theory.

Has this kind of information theory approach had any success though in providing knowledge about the physical universe?
Relatedly, Dr. H.'s recent post about the Black Hole Information Loss Paradox stated that this paradox could just be framed in terms of reversibility (connected to exp(iπ)+1=0 ?? ;), and the information theory framing isn't necessary.

Anyway, your message has changed the state of the recipient.

15. I see. So rather than turn your supercomputer "up to 300" and announce physical results to the press based on a video game, you actually intend to try to confirm this consequence of your conjecture by experiment?

This is a novel approach.

Maybe you should mention this novel scientific method to Crazy Luke and he can pass it on to the other inhabitants of La-la Land, Geraint Lewis and Brian Schmidt....

16. Terry Bollinger,
Thanks very much for your comments and taking time to put mine in perspective. I admit to having only a Braille-like appreciation of some of the physics and certainly the mathematics. This topic has diverged into many different discussions with general focus on the intersection of math and physics. I am forming a reply, but, given the breadth of comments and my limited understanding, I seem to be having trouble to find the point of best leverage.
That said, as a somewhat detached observer and given the relatively arcane subject matter at hand, I often wonder, “What drives so much curiosity?”

17. Terry Bollinger,
Thanks again for your comments. We are discussing issues tangential to the notion of physical infinities and how they affect what actually happens next (my primary interest here). At some point I would like to consider two propositions that relate:

• Any time metric is most usefully seen as a formal axiomatic system and hence Gödel limited.

• The universe contains time independent systems.

“What we call “particles” are never anything more than path decisions relative to classical physics.”
“… folks habitually assume there is a “real” particle lurking somewhere [in] a wave packet…”

Here you seem to favor the wave depiction over particle as being more fundamental, but as you note, the particular expression depends on relationship with the environs, be it measurement device or nuclear neighborhood? I was very interested to find out how that works. As you put it, particle expression occurs:

“… in no small part because they are immersed in astronomically complex bath of phonons (quantized sound, heat, information) that almost continually collapses the locations of atoms and keeps them from turning into Schrödinger waves.”

That is something I can envision as a reoccurring, dynamical refresh of existing form with its own integral tempo. I also see as implicit the notion of a crucible, a dynamic of parts sustained within a systemic boundary. It seems relevant here that a free neutron has a half-life of only about 10.5 minutes. But, it that the way to look at it?

Out of habit, I checked the etymology of ‘system:’ https://www.etymonline.com/search?q=system

"the whole creation, the universe," from Late Latin systema "an arrangement, system," from Greek systema "organized whole, a whole compounded of parts," from stem of synistanai "to place together, organize, form in order," from syn- "together" (see syn-) + root of histanai "cause to stand," from PIE root *sta- "to stand, make or be firm."

So, the question I have is whether the universe is one system with one clock on the wall (something I see as implicit within any deterministic, unitary time evolution) or whether subsystems with space-like separation have some measure of ontological integrity as a sustained dynamic each with its own little clock on the wall.

You mention “path decisions.” May we consider path as the salient, most revelatory observable? Does the universe have one given path with certain outcome or is it an interaction of many paths with outcome yet to be determined?

Rooting that question in the biology of complex, self-organizing systems and their interactions, is it always a given in any instance whether fang meets fur?

12. "Take for example the size of the universe. In most contemporary models, the universe is infinitely large."

This statement is in agreement with Wikipedia, which, in discussing the Big Bang, describes the original singularity as "apparently infinitely dense". However, this would seem to be mistaken. Given that the Universe is a physical construct, and knowing that all physical things are inherently finite, the proper definition of the singuarity should be "immeasurably dense".

1. J. Benjamin9:49 AM, December 06, 2020

"describes the original singularity as "apparently infinitely dense"."
There is no empirically observed "original singularity" and there is no "apparent" infinity. The singularity and the infinities appear in **models** and are almost certainly showing the limits of the models. It is not even known if the observable universe had a beginning.

"knowing that all physical things are inherently finite,"

This is not empirically proven. Nobody knows whether space-time is finite or not.

" the proper definition of the singuarity should be "immeasurably dense". "
The "singularity" is only known to exist in the model not reality. The very early universe is simply unaccessible to observation currently.

This is the misunderstanding made in Lewis and Barnes` stupid book, "The Csomic Revolutionary's Handbook", the follow-on from their first stupid book. Consistent, if nothing else.

13. As others have rightly suggested, infinity is a concept, not a number, which is why there are so many examples of numerical absurdities showing up on YouTube, the most infamous being the "-1/12" quantity in string theory. My own suggestion would be to never include infinity in any expression involving an equal sign.

14. Bill10:25 AM, December 06, 2020

"As others have rightly suggested, infinity is a concept, not a number, "
Every number is an abstract concept. Even 1 does not exist in reality.

"the most infamous being the "-1/12" quantity"

Mathologer does a good explanation on YouTube of what is meant by 1+2+3+...=-1/12

" in string theory."
The trouble with string theory is not the maths used, but that there is no empirical evidence to support it, so it's almost certainly not empirically true.

"My own suggestion would be to never include infinity in any expression involving an equal sign. "
But cardinal arithmetic is possible with the infinite cardinals.

Infinity is probably the least dangerous mathematical concept for Physicists - if infinity appears in their models, they know the Maths is probably misrepresenting reality.

1. Of course the concept of infinity is dangerous for the quest to understand our physical world. For it does create a whole new world, mischievously called Cantor's Paradise. What you prove is valid in that mathematical universe, not in ours (any contrary observation is welcome). And most dangerously the infinity is a viral concept. Once you assume it somewhere in the theory, it pops out elsewhere with new paradoxes and weirdness. Zeno paradoxes `fixed` by calculus and limits, Banach-Tarski `fixed` by no-choice, -1/12 and its friends. are so-called `renormalization`. Needlessly to say, none of these paradoxes couldn't be created without the concept of infinity. Don't you feel that this is just a whack-a-mole game and no different than adding epicycles over epicycles to the theory.

2. "Zeno paradoxes `fixed` by calculus and limits"

Not to my satisfaction. As I understood Zeno, he showed that movement through continuous space requires getting to the end of a series of points which has no end. He was definitely not saying that an infinite sum can't have a finite limit. His simplest example, the Arrow, shows that it can (distance L = L/2+L/4+L/8+...). The issue is how to reach that limit by any physical process which sums the terms--such a process would never end. As I understand it, Democritus understood what Zeno was saying and extended it to matter (continuous matter must be an illusion), i.e., atomic theory. Calculus did not eliminate atoms nor prove that continuous motion exists. Calculus is the limit of discrete math as the discrete increment goes to zero, and such is a good approximation to any discrete system (such as fluid flow) when the increment is very small.

Maybe there is an answer to Zeno's point which I am not seeing, but the epsilon-delta limit process of calculus is not it. The fact that a limit exists does not prove that the limit is reachable by a physical process. To my knowledge, no mathematician or computer ever has or even will physically sum an infinite series all the way to its end. They use other tricks, such as the algebraic solution of the Geometric Series. Or they simply stop after some number of terms and decide the result is close enough. Is that what nature does (under the assumption of a continuous system)?

The answer that continuous motion is an illusion produced by a large but finite number of small discrete changes (as in a motion picture) seems simpler and more likely to me.

3. Dogan Ulus4:56 AM, December 07, 2020

"Of course the concept of infinity is dangerous"
I was being a little tongue-in-cheek, but I think Physicists see infinite quantities appearing in their models as a red flag - no-one believes in infinite densities or temperatures in the very early universe.

Plenty of problems in Maths and computability rely on assuming large cardinals for their resolution. Much of Grothendieck's work, for example, depended initially at least on the existence of an inaccessible cardinal.

"What you prove is valid in that mathematical universe, not in ours "
Yes, that's the key point always.

"Once you assume it somewhere in the theory, it pops out elsewhere with new paradoxes and weirdness."

I don't think infinity is assumed in physical theories so much as it appears due to convenient assumptions like point particles or smooth motion.

" Zeno paradoxes `fixed` by calculus and limits,"
I don't think anyone is claiming that Zeno's paradoxes are "fixed" by calculus, just calculus is a useful tool for ignoring them.

" Banach-Tarski `fixed` by no-choice,"
Again, B-T is true for R^n with Choice; physicists are aware that the moon is not R^n though.

" -1/12 and its friends. are so-called `renormalization`. "

Can you do QED and QFT without renormalization? Dunno. Again this was originally necessary fiddling of the model due to initial assumption of point particles, I think.

"Needlessly to say, none of these paradoxes couldn't be created without the concept of infinity. "

I think the villain has often been zero in Physics models - point particles of zero extent, space of zero extent leading to singularities in black hole and Big Bang models.

"Don't you feel that this is just a whack-a-mole game and no different than adding epicycles over epicycles to the theory. "
The difference is that someone came up with a better model than epicycles, while no-one has come up with better models than QED or QFT. If a physicist replaces infinity with -1/12 in a model and the model correctly describes observation, what's the problem? Feynman had no idea why his diagrams worked, but they did.

4. JimV4:58 PM, December 07, 2020

" but finite number of small discrete changes (as in a motion picture) seems simpler and more likely to me."

Why more likely? Empirically it's simply not known. Don't Planck limits come into play and it all becomes physically meaningless?

5. @Steven
Actual infinity, the continuum (infinite collection of zero-size points), and real numbers (infinite precision numbers) are in the same package. Facades of the some concept. One cannot separate from one from the other. This Platonic world is nicely described as Cantor's Paradise.

> "Much of Grothendieck's work, for example, depended initially at least on the existence of an inaccessible cardinal."
Then I'm sorry that Grothendieck's results (no matter how much he worked) is not valid if an inaccessible cardinal doesn't exist (as we argue here). This is the Achilles' heel for many mathematicians and physicists. This the same for the theory of relativity and quantum mechanics. They belong to Cantor's Paradise.

"If a physicist replaces infinity with -1/12 in a model and the model correctly describes observation, what's the problem?"

An over-used example but it's still the best. Ptolemaic model of planetary motion **was** describing the observation better than early Heliocentric model. And it's much later shown that you can describe any planetary motion by using a certain number of epicycles. So the Ptolemaic model was powerful, beautiful, popular, fitting to observations, but real? This is where a physicist starts working. Unfortunately, "shut up and calculate" mentality is detrimental to the physics.

6. Dogan Ulus4:39 AM, December 08, 2020

"Actual infinity, the continuum (infinite collection of zero-size points), and real numbers.. is nicely described as Cantor's Paradise."
"Cantor's Paradise" was the phrase used by Hilbert to refer to Cantor's discovery of different orders of infinity, not the continuum or the real numbers. By the time of Cantor's work, the real numbers and calculus had been put on a rigorous footing by Weierstrass, et al. Weierstrass' Oasis, maybe?

"Then I'm sorry that Grothendieck's results (no matter how much he worked) is not valid if an inaccessible cardinal doesn't exist (as we argue here)."

Sure. But not all G's work depends on the existence of an inaccessible cardinal, and you can't show that the concept of an inaccessible cardinal is any more contradictory than the concept of 1.

"This is the Achilles' heel for many mathematicians and physicists. This the same for the theory of relativity and quantum mechanics. They belong to Cantor's Paradise."

Relativistic and quantum phenomena are empirical facts. Your computer proves the latter. I think you are extending the concept of Cantor's Paradise well beyond it's usual meaning.

"Unfortunately, "shut up and calculate" mentality is detrimental to the physics."
I think you'd struggle to formally refute the instrumentalist position. Clearly, Einstein's theory of gravity has more explanatory power than Newton's. But is space-time "real"ly bent by matter? I don't think anyone knows what that means beyond comparison with a football warping a duvet. If you write the equations as if "matter bends space-time" you get the right answers, that's all.

7. @Steven Hilbert was surely a great mathematician that knows about Cantor's work on the actual infinity, its historical background, and its implications. Cantor's work was a peak point in the formalization of the continuum. Mathematics was finally freed from its physical grounding and could be complete as envisaged. This would be one single Truth (but it could't, as shown by Godel). One should read the history and initial assumptions of relativity theory and quantum mechanics from this formalist perspective of the era.

8. Dogan Ulus9:06 AM, December 08, 2020

"One should read the history and initial assumptions of relativity theory and quantum mechanics from this formalist perspective of the era."

Just to be very clear..

The functioning of computer chips depends on the quantum behaviour of the electron - so the quantum behaviour of the electron is confirmed quintillions of times a second.

Clocks on satnav satellites in Earth orbit literally run more quickly than clocks on the ground - again, this is confirmed millions of times a second.

These are observed, empirical facts more certain than my own existence to me. They have nothing to do with Cantor's infinite ordinals.

9. @Steven

Yes, those validations are required for any theory in the beginning. But paradoxes and weirdness are more important once the initial phase of the theory passed. Then counterexamples are much more valuable than confirmations... Otherwise Mercury retrograde confirms the theory of epicycles in every couple month. But we know that theory is not significant anymore.

Also relativity and quantum mechanics are much worse than Newton's regarding false positive predictive power (things that theory says possible but no slight evidence so far). I think the concept of infinity is #1 reason for that (The more rigorous adherence, the more weird results).

10. Grothendiecks work relied on category theory where we can say *all* sets, or *all* groups etc etc. Since Russell, we know that the use of *all* requires care. This is at the root of the required inacessible cardinal or, equivalently, Grothendieck universes. Arguably, we can alternatively use higher order logic which in terms cardinality strength lies between ZFC & ZFC + one inaccessible cardinal.

11. @JimV: I fully agree with your assessment about Zeno & the calculus. I'm glad to see that I'm not the only one!

12. Mozibur12:01 PM, December 09, 2020

" relied on category theory where we can say *all* sets, or *all* groups etc etc."

"In his [Ernst, 2015],Ernst shows that any theory that allows the formation of the category of all graphs and that includes the required mathematical staples is in fact inconsistent."

Category Theory appears to have ended up with a similar "all" problem to Set Theory.

13. Mozibur8:18 AM, December 10, 2020

" I fully agree with your assessment about Zeno & the calculus"

You "fully agree"?

So where is the evidence for JimV's "more likely" below, if you "fully agree":

" but finite number of small discrete changes (as in a motion picture) seems simpler and more likely to me."

For example, below Dr. H. links to a paper which suggests the continuum could be necessary for quantum theory.

I'm not having a dig at JimV, just asking the question. But I suspect you "fully agree" because I didn't accept all of JimV's comment and you are still sore about Gentzen.

The standard of truth in Maths is proof, and in Physics observation. Your personal feelings about people shouldn't influence your assessment of evidence.

Get over Gentzen. The proof assumes ε_0 . Everyone knows that.

14. Belated reply to Steven Evans' "Why more likely? Empirically it's simply not known. Don't Planck limits come into play and it all becomes physically meaningless?"

"Physically meaningless" seems like an exact description of what happens in the interval between two points in a discrete system. Anyway, I say "simpler and and (hence) more likely" because it resolves Zeno's Paradox in an understandable way (to me, and "simpler" was as stated in my personal opinion), and it makes the following prediction: if there is a minimum, discrete distance increment ds which can be traveled in a minimum discrete time increment dt, then the universe has a speed limit ds/dt--nothing can go faster than that; and the universe does seem to have a speed limit (c). (There could also be such a limit in a continuous system but some other assumptions would be required to impose it.) Occam's Razor, and all that. (I'm not sure if Mario's Sharp Rock applies though.)

I understand there is an issue with the fineness of experimental data on the Lorentz Transform relationship, but Dr. Aaronson at "Shetl-Optimized" has assured me that this is still explainable in a discrete, quantum system. Unfortunately he did not give a detailed explanation for this, as he seemed to consider it obvious (under some quantum mechanical model).

Finally, the universe is no doubt much stranger than I can imagine so no one need care about my personal opinions about it. People are free to have their own opinions on the matter, but not to disrespect Zeno casually where I can see it without hearing mine. (Unless and until someone does convince me that Zeno has been refuted.)

(Thank you to Mozibur for also making me not alone in that opinion--as to Zeno, that is. I did not take it as an endorsement of my preference for discrete systems.)

15. JimV8:53 PM, December 12, 2020

Thanks for the reply. I see - so you're saying such a discrete theory is at least consistent with observation.

15. Hi Sabine !!!
On this day, it took a lot just to get through to you to say hello..
More than that, I hope you're doing well, and everyone you know.

I'm still standing.

16. What would physicists do without differential equations? Without Dirac's delta-function? Without the concept of point masses? (The three-body-problem is hard even with this simplification.)

Considering limits is indispensable. It is beneficial because it removes irrelevant or unknown parameters. Fluid mechanics was created before the size of molecules was known. The size and type of molecules is irrelevant in fluid dynamics -- a few macroscopic parameters like density and viscosity suffice. (It took the genius of Albert Einstein to think of a situation where the finite size of molecules does have observable consequences.)
Electrons must surely have some length scale well below the classical electron radius, but it is irrelevant in QED. Renormalization ensures that all results are independent of some possible fine structure. (When some day an effect will be discovered that is not described by QED, physicists will be excited to finally have a handle on that microscopic scale that permits electrons to have spin.)

Differential equations must not be thought of as exact representations of reality. This is obvious for fluid mechanics. Schrödinger's equation is an extreme case. Too many people seem to view it as an embodiment of some ultimate truth, which it is not.

17. In terms of Gisin's physics with intuitionistic math, I found this. (A bachelor's thesis, so maybe the future of physics is "finitely-precise"!)

Indeterministic finite-precision physics and intuitionistic mathematics
(Bachelor's thesis, Radboud University, Nijmegen, 31 July 2020)

pdf: https://www.math.ru.nl/~landsman/Tein.pdf

1. Physics without infinite precision means no calculus, no spacetime, no quantum mechanics as we know today. It's not just about making some objects finite but every single of them. And it must start from the mathematical model of the continuum.

Brouwer and its intuitionism have several fair points but it does attack from a quite weak position in my opinion. Yes, the Law of Excluded Middle should not be valid over infinite collections whereas the actual problem is the existence of such collections at the first place. Hence this could be seen as yet another (but unpopular) attempt to patch the concept of infinity.

Hermann Weyl's attempt to formulate a new continuum based on arithmetics (his book Das Kontinuum is pretty critical about the infinity) was quite promising but I think it does still seek a concilation with 2500 years old tradition.

However remember all these happened before the rise of computational sciences, which are based on discrete structures and sequences, without carrying the baggage of milleniums. Perhaps this would be the foundation of new physics to be built upon. So I kinda agree with the general direction of Stephan Wolfram (though not being a fan of cellular automata).

2. We will have to see. There has been a number of these ideas, but they have died on the vine.

Even finite mathematics causes fits. Between 10^{10^{10}} and 10^{10^{10^{10}}} are integers that have no possible description. There are not enough atoms or qubits on the observable universe.

There have been recent announcements on accurate measurement of the fine structure constant. We might ponder whether this pure number has infinite precision, at least in principle. yet, there is not enough energy in the universe to measure this. It is also questionable whether there is any meaning to radiative correction term below the Planck scale in length.

I tend to think the odd relationship between physics and mathematics is here to stay. It is odd because we are unclear on why mathematics works, but why there are these gaps with large numbers and infinity.

3. @Dogan Ulas:

I was initially quite taken by intuitionism because it showed we could get by with dropping an assumption that we generally take for granted: the law of the excluded middle. But eventually it dawned on me that it could be very useful. For example, it allows the introduction of actual infinitesimals that we can reason with. This means a tangent is exactly what we intuitively think it is: an infinitesimal arrow, or in the language of synthetic differential geometry where this notion is pressed into service, microlinearity. I think the whole theory is quite renarkable and it's a pity it's not as well known as it should be.

4. @Mozibur I usually do not argue over the LEM since there is no question on its validity over finite collections. Extending the LEM over actually infinite collections is the part questioned by intuitionists, but I see this is just an unfruitful attempt to tame the Infinity, which I think no such a thing possible. Instead one should drop the Infinity itself.

A tangent, represented as a non-zero extent of the space where you can refine unboundedly (but never to be completed) may be harder in math. But it's no less intuitive than imagining a zero-size point. Rigorous studies on the latter pointy representation showed us that such imagination is very much divergent...

18. Infinity is not a number, at least not in the standard sense. It is more a property of a set. Most ideas of infinity are that they are a cardinality. If one in ZF set theory has the axiom of choice there are ordinal infinities.

The axiom of choice permits ordering of elements of a set. For w a stand in for omega, the ordinality of a set, we have 1 + w = w. The "Hilbert hotel" trick of shuffling elements shows this. However w + 1 requires adding at the end of the ordering. This is not the same.

1. Lawrence Crowell6:33 AM, December 07, 2020

"Infinity" may not be a number, but ℵ_n and ω are cardinals and ordinals, at least allowing such infinite cardinals and ordinals hasn't thus far led to a contradiction. And in general cardinals and ordinals are what most people mean by natural numbers.
Even for basic arithmetic you have to assume 0 exists, and successors, and induction, and it can't be proved consistent. How is that situation any better than for infinite cardinals and ordinals? 1 and ω are equally abstract ideas.

2. Most of number theory does not apply to transfinite numbers. These are categorically distinct.

3. Fair point.

4. @Lawrence Crowell: Frege originally defined numbers as the class of all such sets. Here, the number two refers to the set of all two things. Arguably, Freagean numbers are properties of sets. Of course Russell showed we had to be careful about using the quantifier *all* but nevertheless ...

19. Thank you for this original topic.

The word infinity makes me think of black holes, Schwarzschild metric and infalling observers.

An infalling observer is reaching the event horizon at a moment which – from the point of view of all outside observers – is infinity, that is the end of the world.

All other infalling observers and all other infalling objects will – from the point of view of all outside observers – reach the event horizon at a moment which is infinity, that is the end of the world. So it seems that there is a sort of simultaneity, all infalling observers and all infalling objects are reaching the event horizon at the same moment which is infinity.

But not enough: The same is true for all other black holes of the universe, that means that all infalling objects of the universe are crossing (sort of) simultaneously their respective event horizon – from the point of view of all outside observers.

Will the whole universe end at event horizons? This is not very probable, with view to the theories of acceleration of the expansion of the universe. But it would be a strange idea which would give rise to the assumption that behind the event horizon there is a world which begins after the infinity of our world.

As I wrote, it is unprobable that something like this would happen. But it is an interesting mathematical model of two worlds, the second world is beginning after the infinity of the first world, and infinity is a very precise moment.

20. What happens if the mathematics you use to "describe observations" requires infinity and can't establish its validity with finitary methods? There are allegedly such results in finite combinatorics that can't be established without use of an infinite set. I suppose those get ruled out from being used in physics?

1. You can probably convert such a finite combinatorial problem into a physical problem, just not one Physicists are interested in. I suspect most of Physicists' problems with the infinite arise from dividing by zero.

21. Hi Sabine. !!!

a bit of art
on a cloudy day.

Up and down the stairs she Strode.
Round and round a widow walk.
She couldn't eat , she couldn't talk.
Her gaze fixed on the vastness of a sea.

Up and down the stairs she Strode.
- round and round that widow's walk.

Her heart cried out about to break.

waiting for a ship
that was late.

1. Where's the art - it looks kinda artless...

2. You may, or may not understand this.

but,. at this moment.

-- Kids in the Hall rule.

22. Interesting video, as always.

Just before the 5-minute mark in the video, there is a limit as x approaches zero, which should be corrected to a limit as x approaches infinity.

1. Hi Santo,

Yes, I am sorry about this. I put a note in the info below the video. Unfortunately, I can't fix it in the video.

23. - Mathematical concepts are usually not real in a 'physics environment' , neither is the number 3, which when seen from a mathematical viewpoint has my forks to be able to translate into the physical world. The idea that nature could contain infinities is false. Let alone that people usually forget that for instance saying that the universe is infinite-large is not exactly defined because there are many classes of mathematical infinities.

1. Marc E11:07 AM, December 08, 2020

"The idea that nature could contain infinities is false."

Empirically it is simply not known whether space-time is infinite or not. Is time suddenly going to stop at half 3 this afternoon because space-time is full?

2. Complete nonsense (with humble apologies)

3. Marc E5:47 AM, December 13, 2020

I'm happy to admit so if you can provide evidence.

So can you provide evidence that the universe (not the observable universe) is finite?

Methinks not.

4. Marc E5:47 AM, December 13, 2020

OK, you are not completely wrong. If space-time doesn't have a limit, it doesn't necessarily make it infinite.
On the question of whether nature could contain infinities, I'm inclined to accept Dr. H.'s statement that this is unscientific.

Is the universe infinite? This question is unscientific because we can only access the observable universe and we can't measure infinities.

And equally: Did the observable universe have a beginning? is an unscientific question due to finite precision of measuring equipment.

So the idea that nature could contain infinities isn't demonstrably "false" but unscientific at this juncture, and therefore not worth considering.

24. The relationship between infinity and both ordinary mathematics and physics is subtle. We might be tempted to think the ll difference between infinity and finite numbers is some signature of how infinity is forbidden as a direct observable.

In physics the proverbial infinity is the 1/r divergence. The classical radius of the electron, found by equating the potential energy with mass-energy, was a way of trying to get rid of this divergence. More advanced methods of renormalization are used. As scale is adjusted what is phyically relevant can be removed from the infinity, which in turn is ignored.

T-duality is a correspondence between the momentum k = 1/r of a field confined to some region and the momentum-energy of a string wound around a circle or torus of radius r. T-duality is a feature of mirror symmetry. This is a correspondence between symplectic geometry and indices in algebraic geometry. Clearly the limit r ---> 0 corresponds to divergent field that corresponds to zero field.

The braid group is a central extension of SL(2,C). This is the set of linear fractional transformations z ---> (az + b)/(cz + d) which is T-duality for a = d = 0 and b = c = 1. Kaufman showed there are braid and knot structures with quantum entanglement. The Morse or Floer topogical indices are obstructions to unitary transforms between entanglement types, such as the notorious monogamous theorem.

There is then some correspondence between the counting of invariant curves in an algebraic geometric setting as moduli for entanglements and symplectic geometry for integrable many body theory.

Of course there is infinity waiting in the wings. The limit points are divergences. There are also singular points on the corresponding manifolds, which in string theory are Calabi-Yau spaces. In this structure are fibration af spaces with spaces, and these divergences involve the reduction of the space of fibration. This appears to be a natural case to exclude, or at least some ancillary condition imposed to restrict things away from that.

25. The various types of infinity are perfectly valid mathematical entities. We just have to be careful how to use them. And while I don't agree with Tegmark's Mathematical Universe, the possibility of something along those lines is very plausible, and in that sense a physical infinity could be as important as any other physical or mathematical constant. I think there is some metaphysical connection between mathematical entities and the physical universe, and that connection underlies why mathematics and the physical world are consistent with each other to such a high degree of precision. Since infinity plays a central role in calculus, the same connection would extend to the entity of infinity.

1. I'm glad to see that I'm not the only one who finds Tegmarks universe a little indigestible. In my view, he's a neo-pythagorean in suggesting that the underlying reality of the world is mathematical. But Plato was also a Pythagorean and whilst he also pointed us towards an underlying mathematical reality - and interestingly based upon triangles just like causal set theory but of course they are very different too - he also asserted that there were higher notions of Beauty, Justice and the Good which he indentified with the One.

My view on why mathematics is so ubiquitous in the physical sciences is that it characterises neccessity and neccesity has an obvious role to play in the physical world whereas in the human world, what is most significant is those attributes of Beauty, Justice and the Good come into fruition along with the sense of human freedom. Of course, we are still bound by neccessity, as we have bodies and so we are physical creatures but we think of what is new & novel here that we don't discern so easily in the physical world.

2. Noether's theorem tells us best why physics is so tied to mathematics. With physics we have quantities that are conserved. Noether showed how observation principles are equivalent to symmetries. At some kernel this is probably one reason mathematics is so central in physics.

26. There is no unambiguous concept of infinity in mathematics either. This follows from the fact that whatever mathematics one is dong, it can always be reduced to a game played with symbols. We can only ever manipulate a finite number of symbols using a finite number of rules.

This means that even if we interpret certain mathematical objects as being infinitely large in a certain sense, it can always be reinterpreted in finitistic terms, making the standard interpretation in terms of infinite concepts ambiguous.

Another way to see this is, is to consider a thought experiment where a very large computer simulates an entire civilization of people including mathematicians. These people then live in a digital simulation and they are then able to develop mathematics. Nothing would stop them from defining the real numbers, and constricting the proof that the set of real numbers is not countable.

Nevertheless, everything the digital mathematicians do is always reducible in terms of strictly finitistic terms, because ultimately it's all rendered by a finite state machine.

1. As Max Tegmark wrote:

"So if we can do without infinity to figure out what happens next, surely nature can too—in a way that's more deep and elegant than the hacks we use for our computer simulations. Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I'm betting that we also need to let go of it."

Max Tegmark ("What scientific idea is ready for retirement?")
https://www.edge.org/response-detail/25344

2. I want to start by saying that I agree with your mathematical statements, and in particular with what you are saying about simulating mathematicians by a finite state machine. I also want to add that it is rather peculiar that the standard mathematical model for a computer, the Turing machine, has infinite memory and never makes mistakes, while all the computers we use are finite state, and probabilistic.

As for how all this relates to physics and life in general, I'd like to offer a different perspective :)

I am a logician, not a professional physicist, but I can safely say that we don't have a shred of empirical evidence that we live inside a finite state machine. I think it's merely possible, but nothing points this way in the empirical sense. If anything, as others have said in this discussion, we don't seem to have a single major physical theory that doesn't use real numbers, so the evidence is at best a mixed bag.

"This follows from the fact that whatever mathematics one is dong, it can always be reduced to a game played with symbols. We can only ever manipulate a finite number of symbols using a finite number of rules."

I consider myself a formalist at heart (in the first approximation), and I believe and agree that's how we prove things, but that's not nearly all mathematicians do. One of the things we do is we come up with novel axiomatic systems. There are no formal rules for that, and in the language of mathematics, at least, there are infinitely many finite axiomatic systems. Another thing we do is we come up with definitions. And notation. And taxonomy. Aspects like that can best be characterized as poetic and artistic, and they rub shoulders with physics in a sense that applied mathematics excites our brains for obvious reasons. If counting the beans allows you to get more beans without your tribe realizing what's going on, you know how it goes...

27. Wait so what about the measure theory infinity, or the Alexandroff compactification infinity? I don't think they represent any sort of cardinality do they? I'm no mathematician so I'm confused now.

1. You're right in saying that measure theoretic infinity and Alexandrov compactification infinity are not examples of cardinalities. They are other notions of how infinite is used in maths. The former is about the density of the points, and the latter is adding a point at infinity. We compactify by collecting all the unbounded sets together and calling it the point at infinity and adding it to the set in question. Density speaks for itself.

2. Yes, of course. I'm just a bit critical of the video in the sense that it seems to transition from talking about cardinalities, to talking about infinity algebra (measure theory) to talking about limits of sequences or functions. I think the distinction between these different infinities is important, but they all seem to have been merged.

28. This means that even if we interpret certain mathematical objects as being infinitely large in a certain sense, it can always be reinterpreted in finitistic terms, making the standard interpretation in terms of infinite concepts ambiguous.

It does no such thing. Rendering one thing in another form doesn't make the original "ambiguous."

29. Coming in late to is infinity 'real' but ... aren't probability amplitudes continuous? And, given that they are, since there are an infinite number of points in any interval [a, b] where a is strictly less than b, isn't this an example of an infinite number of somethings in the real world? So ... how do you enumerate all possible amplitudes for a given system ;-)

I guess what I'm getting at here is that any procedure that constructs some infinite subset of a nondegenerate interval will, at best, take an infinite number of discrete steps to do so; given the physical constants of this universe, that means the computation will take an infinite amount of time. IOW, verifying that there is some physical instantiation of some order of infinity in our universe will take an infinite amount of time to do so. So a) is our universe in fact timelike infinite, and if so, b) are we able to take advantage of this to perform an infinite number of computations?

30. Hi Sabine !!!
Hope you enjoyed my art, I enjoyed your video.

Now,
All Right.

you brought up a number of times in your video , the term
" in practice ".

That's interesting to me, as I am an experimental scientist.

Now, what happens if we equate Infinity to zero.
for practical purposes.

I've seen many amazing equations and calculations that ended in Infinity or came full circle to zero.

-and were thrown out like garbage.

that's a mistake.

The key here it's to take a step or two back and segment some of these equations and calculations
- regardless how they end up.
(For practical purposes).

- it's being done now

as we speak.

Ultimately, you have baby
- and bathwater.

1. Baffling. Infinity ain't zero. Aren't you confusing this with the infinitesimal?

2. um,

no.

31. @Sabine:

I fully agree with your assessment of the 'scientific' infinity. But I'd like to add my tuppence on the mathematical infinity.

Personally, I think Aristotle (I know this is going back a bit) had the best characterisation. He said it was characterised by its inexhaustability. And he distinguished this from the All which is just everything. Although he wasn't aware of the set theoretic paradox of the universal set uncovered by Russell, I don't think he'd be particularly surprised - otherwise why have two ideas for roughly the same thing? The All is something we can only gesture towards but cannot define because we run into the inexhaustability of the infinite. If you can characterise infinity then, by Aristotles notion, there's bound to a bit - rather a lot - which we've missed.

Take for instance the infinities of set theory so beloved by a certain stripe of mathematician. All cardinalities are the cardinalities of sets. But the collection of *all* sets is not a set, it is a class. This is our All, surely?

But then hold on, I can hear someone say, isn't the cardinality of the class of all sets larger than the cardinality of any set? Well, you'd think so; but not if we decide to not define the cardinality of classes, or even classes.

This sounds artificial, and to my mind it is. Lets say we add classes and also its cardinality. Well, I can hear someone from the audience say, haven't we found our All and since it has a cardinality, then we have a size and this is the greatest infinity by definition - and so they conclude triumphantly, Aristotle is wrong!

But lets not be too hasty. Definitions aren't everything. After all, if we have classes surely we can iterate what we've just done? For sure ... that no one's done it is not a sign of its impossibility but that no-ones interested. Personally I put this down to the hegemony of ZFC. And its this that proves Aristotle is correct. Because if we follow this thought we can then do as we've just done above and complete it, say nit to classes, sincevthat names been used up, say to collections. But then again we can do the same...

In other words, cardinalities are barely the beginning of infinity. There is much, much more. I think Cantor knew this when he talked about the Absolute Infinite.

To my mind, this is roughly like an energy svale, we say a theory is effective upto a certain ebergy scale; likewise, a mathematical theory of sets is effective upto a certain scale along the infinite.

Tgos leads me upto a second notion of the infinite which is much more qualitative but I thjnk just as essential, if not more. It's by William Blake. He said, if the doors of perception were cleansed, we would see everything as they really are, infinite. Aldous Huxley was so impressed by this that he named a book after it, likewise Freeman Dyson. To my mind, Blake had an intuitive grasp of what the infinite really meant ...

1. Take a look at this paper, its worth studying. I am a Fuzzy Logic guy and can live with uncertainty until two valued T or F becomes clear. "Indeterminacy and ‘The’ Universe of Sets: Multiversism, Potentialism, and Pluralism" Neil Barton∗ 8 April 2020† https://philarchive.org/archive/BARIAT-15

32. I just want to add that I think that Newton felt exactly the same with his famous quote on collecting pebbles by the sea when the great sea of truth lay vast and immeasurable in front of him. Some people think he was being modest. I don't : he was stating exactly what he saw.

33. Dr. Hossenfelder;

Your topics are always of the greatest interest to many, and they elicit outstanding responses from so many smart and articulate people. I always read them with great interest and an open mind as I try to learn more. I was intrigued with the thought of taking a measurement. Any distance scale has a start point and an endpoint, and we know that there are an infinite number of positions between these two points. Thus there cannot be a most accurate, smallest distance or exact measurement for anything. In this case we simply accept a measurement as the best estimate for the purpose of our use. And this is OK. But what about the Planck length and all of the things that are occurring down at this length scale? If we accept infinitely small then one-dimensional strings and additional dimensions are not the smallest things occurring because by definition we can go even smaller. And, if we can go smaller, then there has to be a possibility that strings are comprised of something even smaller, and maybe there are even more dimensions at an even smaller scale. Maybe down at the distance of the Planck length squared there is another whole string theory for the first one? Yes I am being silly, but if string theory and the Planck Length are going to be where we stop going smaller there needs to be a physical and/or mathematical reason for this other than "renormalization." ( I joke again)

And, I guess I have to say it, we can make a corresponding argument for the probability amplitude of a particle wave as well as so many other aspect of quantum physics.

I guess the next question would be, yes, mathematics does have infinities but why does every concept of math have to have an existence in reality? Why can't math be infinite but our reality and our universe finite? Maybe nature has placed real limits on things for us.

As always, thanks again Dr. Hossenfelder to your time and topic presentations in this blog.

1. >why can't math be infinite but our reality and our universe finite?

I agree. But I'd add that the mathematical world is real and so part of the universe in an extended sense. This is called, mathematical realism or mathematical platonism. I think of our world as having higher existences than just what we see. Of course, Platonism is a much richer and more profound idea than mathematical platonism. To Plato, it's simply the first step of his dialectic. It's a pity that Platonism has been reduced to simply the first step. I'd also add, given what Plato said about truth and the cave has been vindicated in todays social media world where truth has been abandoned with the kind of pathologies we are seeing today. I also hold Nietzsche responsible for this as he was intent on upending Zarathrustha. Well, I went back to Zarathrustha to find out what he did say, and he spoke of 'right thought, right mind' and the 'truth'. The kind of things I hold dear.

2. "But I'd add that the mathematical world is real"

If "1" is physically real, show me it. Maths is not known to be physically real.

3. "But I'd add that the mathematical world is real"

I see what you mean and agree, with the caveat that reality varies in proportion to metabolism, its energetic throughput.

As to truth, its meaning and present-day pathologies, it is disconcerting to find among its definitions:

“ a fact or belief that is accepted as true: the emergence of scientific truths.”

That seems to make truth itself a matter of metabolism and hence mutable through propaganda.

34. I am not sure if some has mentioned in this discussion the following, but in mathematics, there are number systems containing numbers of infinite size. Surreal numbers are an example (famous) where being w an infinite number side,
1+w \neq w etc... and where the rules of arithmetics transfer to numbers of infinite size.

Another example is in hyperreal numbers, which is the foundation for Robinson's non-linear analysis.

Another example is Levi-Civita field of numbers.

As you see, infinite can be a number; there are many different infinite numbers.

Personally, I think that an infinite can be as real as an spherical apple can be. Both are concepts through which we deal and understand our experiences. I also think that reducing the concepts on hand, a tendency driving by a kind of purity, in the construction of explanations, is not very good strategy. In particular, it seems to me that the criteria that such or such theory is wrong because it contains an infinite quantity is arbitrary and I think also it is missleading.

35. I enjoy your talks! These articles have helped my understanding of infinities.
https://arxiv.org/pdf/2007.04812.pdf
https://philarchive.org/archive/BARIAT-15
http://homepages.math.uic.edu/~kauffman/Laws.pdf
Also a study of surreal numbers

36. "Infinite nothingness" is an oxymoron. If nothingness is infinite, then it is something, not nothing. Surely only something can be infinite.

37. Thanks for the video. A bit off-topic but I have a suggestion for a video - could you do a video about Mach's principle? Is there such a thing as absolute rotation? Thanks!

1. Rotation is acceleration; it's always absolute.

38. Terry Bollinger,

I would welcome your review of comments on the nature of information. You note that:

“Entropy increases. Since entropy is information, my own physics-flavored interpretation of Gisin on that point is that he’s talking about entropy in an interestingly abstract way. I would say it this way: The universe keeps getting more classical, that is, the range of possible futures into which it can evolve keeps getting narrower over time.”

Shannon’s metric for the entropy in bits per symbol of a discrete signal has proven its utility in improving the fidelity and capacity of communication channels. It is a measure of lack of internal relationship between one symbol of a string and its neighbors, sometimes distant. With a symbol set as given, the entropy in bits/symbol of a text string is maximal when each symbol is entirely independent of the other and is diminished by any degree of certainty one symbol conveys as to what symbol is next. Once again, it is a measure of internal relationship and is, by premise, noncommittal on external relationship, silent on the nature of any meaning being conveyed between sender and receiver.

If we accept the notion of bit from it, then we can backtrack a bit, if not to Genesis, then at least to the Gospels. If our bit string is from a phonetic language then each symbol conveys a phoneme, the “distinct units of sound in a specified language that [in combination] distinguishes one word from another.” When we speak words, we are creating distinct sound waves. Whether they convey meaning or not is of course a matter of external relationship with a listener. In any case, it is useful to note that largest portion of the human motor cortex is devoted to running, not the hands, but the complex assemblage of muscles and tissues involved in modulating a base tone into the particulars of a certain set of distinct phonemes. The vocal tract has more than a score of mutable parts and each phoneme is produced by a particular arrangement of this constraining 'ductwork' (here including amplification through resonance)

Thus, while Shannon’s bits per string is maximal when there is no constraining relationship in which one symbol is in part limiting the next, we find that each phonetic symbol has directly related antecedents of constraining processes. There is the physical process of creating distinct sound patterns, the constraints of syntax in ordering the words, the words to convey meaning and beyond that, the long, winding channels of a language shared among people traversing sometimes difficult terrain.

Ultimately, is information in the content or in the container?