[Still from the 1956 movie The Ten Commandments] |

No one has any idea why mathematics works so well to describe nature, but it is arguably an empirical fact that it works. A corollary of this is that you can formulate theories in terms of mathematical axioms and derive consequences from this. This is not how theories in physics have historically been developed, but it’s a good way to think about the relation between our theories and mathematics.

All modern theories of physics are formulated in mathematical terms. To have a physically meaningful theory, however, mathematics alone is not sufficient. One also needs to have an identification of mathematical structures with observable properties of the universe.

The maybe most important lesson physicists have learned over the past centuries is that if a theory has internal inconsistencies, it is wrong. By internal inconsistencies, I mean that the theory’s axioms lead to statements that contradict each other. A typical example is that a quantity defined as a probability turns out to take on values larger than 1. That’s mathematical rubbish; something is wrong.

Of course a theory can also be wrong if it makes predictions that simply disagree with observations, but that is not what I am talking about today. Today, I am writing about the nonsense idea that the laws of nature are somehow “inevitable” just because you can derive consequences from postulated axioms.

It is easy to see that this idea is wrong even if you have never heard the word epistemology. Consequences which you can derive from axioms are exactly as “inevitable” as postulating the axioms, which means the consequences are not inevitable. But that this idea is wrong isn’t the interesting part. The interesting part is that it remains popular among physicists and science writers who seem to believe that physics is somehow magically able to explain itself.

But where do we get the axioms for our theories from? We use the ones that, according to present knowledge, do the best job to describe our observations. Sure, once you have written down some axioms, then anything you can derive from these axioms can be said to be an inevitable consequence. This is just the requirement of internal consistency.

But the axioms themselves can never be proved to be the right ones and hence will never be inevitable themselves. You can say they are “right” only to the extent that they give rise to predictions that agree with observations.

This means not only that we may find tomorrow that a different set of axioms describes our observations better. It means more importantly that any statement about the inevitability of the laws of nature is really a statement about our inability to find a better explanation for our observations.

This confusion between the inevitability of conclusions given certain axioms, and the inevitability of the laws of nature themselves, is not an innocuous one. It is the mistake behind string theorists’ conviction that they must be on the right track just because they have managed to create a mostly consistent mathematical structure. That this structure is consistent is of course necessary for it to be a correct description of nature. But it is not sufficient. Consistency tells you nothing whatsoever about whether the axioms you postulated will do a good job to describe observations.

Similar remarks apply to the Followers of Loop Quantum Gravity who hold background independence to be a self-evident truth, or to everybody who believes that statistical independence is sacred scripture, rather than being what it really is: A mathematical axiom, that may or may not continue to be useful.

Another unfortunate consequence of physicists’ misunderstanding of the role of mathematics in science are multiverse theories.

This comes about as follows. If your theory gives rise to internal contradictions, it means that at least one of your axioms is wrong. But one way to remove internal inconsistencies is to simply discard axioms until the contradiction vanishes.

Dropping axioms is not a scientifically fruitful strategy because you then end up with a theory that is ambiguous and hence unpredictive. But it is a convenient, low-effort solution to get rid of mathematical problems and has therefore become fashionable in physics. And this is in a nutshell where multiverse theories come from: These are theories which lack sufficiently many axioms to describe our universe.

Somehow an increasing number of physicists has managed to convince themselves that multiverse ideas are good scientific theories instead of what they de facto are: Useless.

There are infinitely many sets of axioms that are mathematically consistent but do not describe our universe. The only rationale scientists have to choose one over the other is that the axioms give rise to correct predictions. But there is no way to ever prove that a particular set of axioms is inevitably the correct one. Science has its limits. This is one of them.

Mathematics is only a pencil. You have to be aware who is using it. Today physics is lost in the mind. Only consciousness can develop physics. Entire cyclotron physics is false.

ReplyDeleteAmrit, mathematics already IS "who is using the pencil".

Delete(Sorry, Dr. Hossenfelder, if the post becomes a video I will comment further on YouTube, I no longer comment blogs as close to 100 thousands of my posts have disappeared without a trace for in the last 20 years. Try googling me: you will find nothing except what makes me look bad. I am done with that fight. No witnesses who spoke for me.)

WELL SAID Sabine!

ReplyDeleteA CERN Quanta article "Why the Laws of Physics Are Inevitable" was put online yesterday at https://www.quantamagazine.org/how-simple-rules-bootstrap-the-laws-of-physics-20191209/

ReplyDelete... which seems to be relevant to your new post.

Can anyone clarify why the article says that a spin 3/2 particle is required by SUSY? (I am an amateur.)

It seems to me that a spin 1/2 field could convert bosons to fermions and vice versa. So why the need for spin 3/2?

Yes, thank you, needless to say, I read the Quanta piece.

DeleteIf supersymmetry is right and gravity is mediated by a spin-2 particle (the graviton) then this spin-2 particle must have a partner particle (the gravitino) which has spin 3/2.

A spin 1/2 field does not "convert bosons to fermions", unless you mean that spin 1/2 fields interact with bosons.

Scientific models have to be falsifiable by observations, but science can never prove the correctness of the models due to limited observations. At most, scientists can only say the models are not falsified by existing observations up to now.

ReplyDeleteThere is more to it I think, because there are also models that can never be falsified because they can always be adjusted to gainsay any evidence that comes along. They are in effect, designed to be immune to falsification. I think one of the best examples is the quantum multiverse, in that it is founded on the claim that realities arise in 'just the right way' to always reproduce the probabilities predicted by QM.

DeleteYou can look at a theory, and if it appears that it cannot be falsified (not by the impracticality of a test) but because it can always evade the outcomes of any test, then there are good grounds for asserting that the theory is unscientific.

If a theory is falsifiable then there needs to be some claims that depend on that theory. Even though we cannot go back and see dinosaurs, we know that evolution predicts that we should see the kinds of fossils and genetic evidence we observe. If we found modern cat and rabbit fossils among dinosaur fossils then we now that evolution cannot explain that.

Yes, there is more to it. There are degrees of maturity for a scientific model. At one extreme, there is no evidence or observation yet to support or falsify a model. At other extreme, we cannot count ALL observations in universe to support a model. The more observations to support a model, the more mature of the model.

DeleteWhen a scientific model without any evidence to support, the model is basically at the stage of scientific fiction, not yet to the stages of nonfiction. For instance, multiverse and string theory are all scientific fictions.

The bottom line is that science can only prove a model to be false, but cannot prove a model to be true 100%. Mathematical logic considers axioms or assumptions as non-provable facts. But science considers axioms or assumptions as defeasible facts and need to be retracted based on further evidence. While mathematical logic is monotonic logic, scientific logic is based on non-monotonic logic.

"The maybe most important lesson physicists have learned over the past centuries is that if a theory has internal inconsistencies, it is wrong. By internal inconsistencies, I mean that the theory’s axioms lead to statements that contradict each other. A typical example is that a quantity defined as a probability turns out to take on values larger than 1. That’s mathematical rubbish; something is wrong."

ReplyDeleteYes, but such things have lead in the past to the right answer. E.g. scattering off a barrier using the 1D Klein-Gordon results in things that look like probabilties doing just that. A bit more thought explained the necessity of antiparticles.

That's why I have been repeating for years that physicists should try to solve problems of mathematical inconsistencies. I have literally written a book about it.

Delete

ReplyDelete...mathematics works so well to describe nature...I came across this ironic quote from Russell recently:

"Physics is mathematical, not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover. For the rest our knowledge is negative."It seems that there is some progress to be made regarding non-linearity. Or what I call very naively becoming. I was wondering whether some problems or some limits might be related to the mathematical language of change, versus the appearance of new feelings. But emergence is certainly not the enemy of reductionism, it is just difficult to know what is felt and how in various situations.

Felker,

DeleteWe know quite a bit about non-linearity: we cam, for example, solve multiple non-linear polynomial equations in multiple variable in a very straightforward way (google "GrÃ¶bner bases"). We have various means to solve non-linear differential equations, etc.

The problem seems to be rather that it is just a very big field. Linear equations are a small area.

Non-linear equations areeverythingelse.Sort of like vertebrates vs. invertebrates: lots more species of invertebrates, partially because they are

everythingthat is not a vertebrate.Felker also wrote:

>Or what I call very naively becoming. I was wondering whether some problems or some limits might be related to the mathematical language of change, versus the appearance of new feelings.

Not quite sure what you mean there. Again, non-linearity is sort of a grab-bag term, just as invertebrate is. We wouldn't say that "invertebrate" really explains anything (except, presumably, noting that the creature lacks a backbone), and similarly all that "non-linearity" really explains is that the thing isn't linear.

By the way, you mentioned on another thread that you did not see the last couple of messages between you and me: when comments reach a count somewhere around 200 on Sabine's blog, a little message pops up near the bottom that says "Load more..." You need to click on that or you can't see the latest comments.

I failed to notice that for a while, and so I thought that Sabine was blocking some of my comments that were perfectly innocuous. (Sorry, Sabine!)

Dave

Dave,

Delete« …similarly all that "non-linearity" really explains is that the thing isn't linear. »That seems logical;-)

I was awkwardly trying to make a distinction between change and becoming, and vaguely came to the idea that linearity vs non-linearity would help me to phrase it. But I am still confused, and it may be just rubbish.

Yes, I knew this trick about the 200 messages but I had completely forgot it! Thanks for having remind me that.

Referring to Max Tegmark like idea: if all was fundamentally mathematics, redused to primes and symmetries, it would be obvious that many premises changed:

ReplyDelete- no need for axioms (universe is mathematics)

- no need for inflation (math can affect homogenous development)

- no need for guessings or unpredictability (math can... wait a minute!)

I said math can affect homogenous development - it requires a restrictive regulations...

In fact because the measurable continuum can obviously have many various occurences, we have to pause to think that althought there couldn't be indeterminism there can be something deeper. In the mechanism of mathematical pattern which define what happens there could be some regulations to make choises so that it seems to be indeterminism but it would be something else: deliberate evolutionary choice, consciousness.

Eusa wrote:

Delete>In fact because the measurable continuum can obviously have many various occurences, we have to pause to think that althought there couldn't be indeterminism there can be something deeper. In the mechanism of mathematical pattern which define what happens there could be some regulations to make choises so that it seems to be indeterminism but it would be something else: deliberate evolutionary choice, consciousness.

You mean the real number line is

conscious?Oh my God -- and I've never even sent it a Christmas card!

Seriously,, I think you've stretched meaning beyond the breaking point.

Dear Dave,

DeleteThat's really the serious question:

- which is more reasonable to study; manyworlds or evolutionary choice?

Since axioms of physics are not set in stone, what about the type of logic we use to carry over truth values within mathematical models of physics. Can it be that we have right axioms of physics, but wrong type of logic in use (classical instead of many-valued, constructive or quantum, i.e. the one nature is based upon), and so theorems in mathematical physics are not actually theorems of nature itself.

ReplyDeleteJulius,

DeleteThere is nothing wrong with classical logic. The conclusion that it has to be replaced is based on bad arguments. QM can be very well understood within the realm of classical logic.

Logic can be various. I understand the logic named "classic" is on its own. QM logic is different. It's of course consistent as a working logic must always be. Physics is the way to find new logic again and again.

DeleteMammal brains is prepared to accept only one logic at time - that's the issue you need to see through.

Eusa,

DeleteCan you point out a situation in QM where classical logic fails?

There is quantum logic. It does not obey the distributive property. Let p be a proposition on momentum of a particle and x and y be propositions on position of a particle. Suppose we say p is the momentum is in the interval [1, ½], x is the proposition the momentum is in the interval [-1, 1] and y is the particle is in [1,2].

Deletep ↔ [1, ½]

x ↔ [-1, 1]

y ↔ [1,2]

Now now we form the proposition p and (x or y) that is the particle is in the momentum interval [0,1/2] and the position [-1, 2]. Using the Heisenberg uncertainty the spread is then 3/2 that is greater than 1 which we take to be Ä§. Now consider the distributive of this the particle proposition (p and x) or (p and y) that the particle has a spread ½ that is less than the Heisenberg spread, so this is false.

I think I have this right. So that is the main deviation from Boolean logic. The curious situation though is that we have to measure quantum states and observables within the context of classical physics and correspondingly Boolean logic.

In classic mechanics bodies are under neutral motions, in QM there are antipodal charges and entanglement correlation. A region of validity is a key aspect. Then there is QED logic, QCD logic, etc. Every theory have its logic and range of validity. No issue here.

DeleteLawrence Crowell,

DeleteLet me know if I've got this right!

p and (x or y)=true means that it is possible to prepare a set of particles so that all of them have the momentum in the [1, ½]interval and all of them will pass through a hole that is 3 units wide.

p and x = false means that it is not possible to prepare a set of particles so that all of them have the momentum in the [1, ½]interval and all of them will pass through a hole that is 2 units wide. Only some of them will pass.

p and y = false means that it is not possible to prepare a set of particles so that all of them have the momentum in the [1, ½]interval and all of them will pass through a hole that is 1 unit wide. Only some of them will pass.

Imagine that we split a 3 units hole, using a very thin wire, in two holes, one 2 units wide, the other, obviously 1 unit wide. Our particles, with momentum in

[1, ½] interval will get through this device, but only some of them will pass through each opening. So, it seems to me that p and (x or y) is equivalent with (p and x) or (p and y).

Eusa,

DeleteClassical electromagnetism is part of classical physics, right? So charged particles are not a concept unique to QM.

All theories we have obey the laws of classical logic. Where is your counterexample?

The proposition p and x and the p and y can be true individually, but in the OR condition they both can't be true.

DeleteLawrence Crowell,

Delete"The proposition p and x and the p and y can be true individually, but in the OR condition they both can't be true."

Obviously they cannot be both true because you cannot find the particle in the [-1,1] interval and in the (1,2] interval at the same time, right? But we don't need them to be true at the same time, one is enough, because A V B = true if at least one of the statements, A or B is true.

Eusa wrote:

Delete>Logic can be various. I understand the logic named "classic" is on its own. QM logic is different.

Eusa, I'll take "quantum logic" seriously when its proponents actually use it prove theorems.

Andrei: This is the odd thing. You can have either proposition true, but not on an or condition. This is one way of seeing a departure from classical logic.

DeleteLawrence Crowell,

DeleteI don't get your point at all. You claimed that there is something wrong with classical logic because p and (x or y) is true, yet (p and x) or (p and y) is false. Nevertheless, I have shown you that this is not the case. When the experiment is analyzed properly we see that in fact (p and x) or (p and y) is also true. So your counterexample failed. As a result, no departure from classical logic is revealed.

I think your error originates in the fact that the uncertainty principle does not imply that no particle with momentum in the [1, ½] interval will be found in the [-1,1] or (1, 2] position intervals, only that not all of them will be. Some will be in [-1,1], others in (1,2] so there is nothing wrong with a classical understanding of the experiment. In fact, a similar argument may be presented in classical physics. If you have a gun with an accuracy of 3 units at some distance all bullets will hit a 3 units target but some will be confined to a 1 unit region while others in a 2 unit region. How is this in contradiction with classical logic?

Yeah, I see your issue. I was a bit hasty. I should have done this with

Deletep ↔ [1, 1/3 ]

x ↔ [-1, 1]

y ↔ [1,2]

then p and (x or y) says p has a spread of 1/3 and x or y a spread of 3 and the Heisenberg uncertainty with the condition this is 1 which works. On the other hand p and x has a spread 2/3 and p and y has a spread of 1, The latter works but the form does not because it is less than the uncertainty spread. OK, I think I have it right now.

The main point is that the distributive property does not work. Popper in his book on quantum mechanics and realism argues that a measurement has the effect of making Î”pÎ”x ≥Ä§ into a localization or collapse so this is violated for the system plus apparatus. This is somewhat in line with Bohr's ideology.

Sorry about the confusion. I often write these first thing in the morning while coffee is taking the cobwebs away from my mind.

Sorry, early morning again. I meant that p and y have a spread of 1/3. So it is false too. With the or condition both must be false. It is like two switches in parallel.

DeleteLawrence Crowell,

DeleteMy objection had nothing to do with the specific numbers you've chosen. I'll try to explain my point again.

It is possible to prepare an ensemble of particles so that a momentum measurement on any of them will be in [1,1/3] and a position measurement will be in [-1,2]. So, we say that the proposition:

P1: p and (x or y) is true.

OK, then we notice that it is not possible (because of uncertainty principle) to prepare an ensemble of particles so that a momentum measurement on any of them will be in [1,1/3] and a position measurement will be in [-1,1]. So, we say that the proposition:

P2: p and x is false.

Again, we notice that it is not possible to prepare an ensemble of particles so that a momentum measurement on any of them will be in [1,1/3] and a position measurement will be in [1,2]. So, we say that the proposition:

P3: p and y is false.

Your argument is the following:

P4: According to classical logic P1 = P2 V P3

P5: P1 is true

P6: P2 and P3 are false so P2 V P3 is also false

From P4, P5 and P6 it follows that true=false, so classical logic has been refuted. Do you agree that this is your claim?

The error in this line of argument (assuming I did not misrepresented it) is the following:

The uncertainty principle only refers to the whole ensemble, not to each individual particle. So, it is entirely possible that a particle with momentum in [1,1/3] will be found in [-1,1], just like it is possible that another particle with momentum in [1,1/3] will be found in [1,2]. In fact, it is obvious that if the particle is found in [-1,2] it HAS to be either in [-1,1] or in (1,2]. Do you agree with this? So, for each individual particle either P2 or P3 must be true, so in fact P2 V P3 has to be true. The paradox has been removed, classical logic is just fine.

I made a couple of small errors, but I fixed that and the argument I think still holds. The Heisenberg uncertainty principle holds for a single quantum state or particle. So while p or (x and Y) is true, (p and x) or (p and y) is false because (p and x) violates the HUP and (p and y) also violates the HUP. You only have TRUE = FALSE if you insist the distributive property holds.

DeleteLawrence Crowell,

DeletePlease specify what in my previous comment you disagree with, because I have explained in a detailed way why your argument does not hold (in the last two paragraphs).

It is not different particles found in the two intervals. It is the same particle. The HUP is an obstruction to classical Boolean logic.

DeleteLawrence Crowell,

Delete"It is not different particles found in the two intervals. It is the same particle."

A particle is never found in more than one location. When the experiment is done for each particle there will be a single dot on the screen. If this dot is in the [-1,2] interval it must be either in [-1,1] or in (1,2], right? So there is no contradiction between the experimental results and classical logic.

It is possible to violate the uncertainty principle in the past. There is nothing stopping you to measure very accurately the position of a particle in a momentum eigenstate. What you cannot do is to prepare a particle in a state where both momentum and position are accurately known. But your lack of ability to prepare such a state does not mean that such a state cannot exist.

So, while you cannot prepare a particle with a momentum spread of 1/3 and position in [-1,1], after the experiment is done, you will find some particles that had the momentum spread of 1/3 were actually detected in [-1,1]. So, even if P2 violates the uncertainty principle it is not always false. Sometimes is true, sometimes is false, but when P2 is true P3 is false and when P2 is false P3 is true, so P2 V P3 is always true, in agreement with classical logic. The uncertainty principle has been violated, but this is OK, because it is in the past.

Your comments indicate you need to rethink QM a bit. With two slits what passes through is the quantum wave, which may or may not be ontological, and statistics on the detector end or screen illustrates wave interference. If one tries to localize which slit the particle passed through the wave interference is lost.

DeleteYou comment about the past violation appears to contradict the Wheeler Delayed Choice Experiment. QM is weird and attempts to apply classical reasoning to it run into contradictions.

Lawrence Crowell,

DeleteI propose you to put aside for now the two slit and delayed choice experiments and focus on your example. Simple question:

Do you agree or disagree that a particle prepared with a momentum spread of 1/3 can be detected in the [-1,1] position interval?

Please justify your answer.

Sabine Hossenfelder wrote"There are infinitely many sets of axioms that are mathematically consistent but do not describe our universe. The only rationale scientists have to choose one over the other is that the axioms give rise to correct predictions. But there is no way to ever prove that a particular set of axioms is inevitably the correct one. Science has its limits. This is one of them."

ReplyDeleteThis is a profound truth of all human knowledge.

Mathematics is all about conceived relationships between objects. That some of the conceived objects happily reflect physically describable phenomenon is fortunate for human understanding.

One may philosophically assert that relationships between objects exist whether or not physics or mathematics describes the relationship. The irony is that no mater how clever and reproduceable our calculations the objects themselves remain a mystery beyond our understanding, After all we never see the moon we only perceive the light.

Do I understand you correctly that the "inevitability" in the Quanta piece only means that given the assumptions about spin, the laws of physics can be mathematically derived, which, however impressive, is not "inevitability" until it can be shown spin, itself, is inevitable. And there's no justification for that assertion. Presumably, then (?) one must show that spin arises from some more fundamental property.

ReplyDeleteSpin plus a whole bunch of other assumptions, notably about quantization and what it means to be "physical." Many of these supposed inevitabilities bring in a host of assumptions that people never write down, eg about the structure of the vacuum, or the validity of Lorentz invariance or the "natural" size of certain contributions and so on.

DeleteThis is by no means to say that I think this is the wrong thing to do -- the very opposite is the case, I think it's the right thing to do -- it's just that phrasing axiomatic approaches in terms of "inevitability" is nonsense.

Much of this reminds me of the arguments of that great philosopher of science, Charles Peirce. That the laws of nature are not inevitable was expressed by Peirce in the word "tychism."

ReplyDeleteRe: "No one has any idea why mathematics works so well to describe nature .."

ReplyDeleteIt seems to me that mathematics is human thought in a formal, codified form which is (usually) applications or extensions of previous thought, and (usually) checked and agreed-with by expert thinkers. The fundamentals of math include the ability to count, to do logic, and to remember past events.

It also seems to me that "doing the math" as described above is an obviously valuable survival trait and as such a natural focus for evolution of complex organisms. Trivial example: a pack of 13 lions walk into some concealing brush around a water hole. You are thirsty but wait. Later, 12 lions leave the water hole.

Therefore I do not see any ineffable mystery in the usefulness of math (thought) in modeling how this universe works. It would be much more surprising to me if evolution had spent billions of years selecting for some mental ability which was useless for understanding the universe, and even more surprising if then that ability was the tool of choice for understanding the universe's mysteries.

So in a way, and without meaning any offense, it reminds me a little of Douglas Adams' parable of the puddle of water which found it miraculous that the hole it occupied fit its shape so precisely.

(No doubt you are thinking of cases where some branch of mathematics was developed long before it was found to have useful applications, but if any comprehensive statistical analysis of this has been done and found to exceed random likelihood I am not aware of it.)

(Or perhaps it is a semantics issue of some kind.)

(Disagreement with the post: 10 euro fine.)

The principles of physics are simply ways we have of representing what we observe in some mathematical order. There is no reason to think that nature “obeys” these as laws, a term that is otiose for the most part. It is though not surprising how we understand what we observe is cast in mathematical form. After all we measure things, these are represented by numbers or geometric measures, and these then fit into mathematical rules. Nobody in the 19th century saw quantum mechanics and relativity coming, and the system of rational or classical mechanics seemed to be almost divine in its basis. So what scheme is used at any time may be superseded by a more general system.

ReplyDeleteWhy are the principles of physics what they are? There really can never be an answer to that question. Why is momentum conserved? Why is it conserved as a four-vector and not a three-vector as had been previously thought? Why is energy and matter arranged in quantum waves? Now that is a really tough one, and we have no idea.

When it comes though to the multiverse, the local breakdown of the vacuum in a deSitter spacetime with a large Î› is a pretty basic consequence of inflation. Inflation also has some measure of observational support, even if the big fish in B-modes has yet to be confirmed. So I would not say this is useless. Of course verifying the existence of these other pocket worlds will not be easy. These pocket worlds may interact with each other, and it is then possible there are signatures of such. The CMB may bear scars of such.

"what we observe is cast in mathematical form"

DeleteAnd mathematics is a (human-invented) language that evolves over the centuries, updated today by programming languages (like invading tribes). It is not a language with vocabularies and rules set in stone (like what Moses was given on Mount Sinai).

Mathematics is more than human invented. Some birds such as in the raven, Crow and Jay family and parrots are rather numerant. Some mammals have the same and good spatial reasoning. With dogs they are F math students, but good in language and social skills.

DeleteIt should not be surprising math is the lingua Franca of nature. Since the days of counting head of cattle to measuring Higgs bosons and gravitational waves we have numerical evaluations.

Now we might ask why numbers measures are Archemedian, well for 99.9% of cases and there is a bit of theory saying otherwise, instead of nonstandard arithmetic logic. That we can't answer, at least not easily.

"It should not be surprising math is the lingua Franca of nature."

DeleteFirst: "There aren’t any mathematical objects."

Syllabus: Philosophy of Applied Mathematics

Cian Dorr and Hartry Field

http://www.nyu.edu/projects/dorr/teaching/AppliedMathSyllabus.pdf

Second: We don't have (yet) in our (human) possession all the math language(s) that we need to have in the first place to understand nature.

Third, nature does not seem to have studied Mathematics and theoretical physics to build itself; if we continue along the path that mathematics is the fundamental structure; then the birds would have to study aerodynamics to be able to fly

DeleteHaving theories in mathematical form, internal consistency, and the unreasonable effectiveness of mathematics in physics all amount to the same thing. Mathematics here means any system of axioms and rules that's internally consistent. As we said in the previous video, the anthropic principle says we can only exist in a universe that's consistent and constant enough to support our evolution. Given that we exist, it's inevitable that the universe is mathematical.

ReplyDeleteI don't get why anyone would think this particular set of physical laws or the commonly used set of numbers and operators are inevitable. If quantum mechanics was our everyday experience we'd invent math where 2+2 = interference. The only question, as discussed, is if the evolution of structure is inevitable or in some sense consistency confers reality to the universe as per Max Tegmark. I'm sympathetic to both ideas, because if the support for structure is a fluke you have to invent multiverses or accept that we're unlikely.

As I recall, one of the salient moves in the "unreasonableness" discussion is that mathishness is simply closely linked to what ways of thinking are effective for human/primate / mammal / vertebrate / animal life use. Our tools work for us by the same token that we are today's flavor along the billions of years of life on earth. In that light, the observation that our math seems to work well for us is a bit like the weak anthropic principle.

ReplyDeleteMikeP wrote:

Delete> In that light, the observation that our math seems to work well for us is a bit like the weak anthropic principle.

We certainly have non-mathematical ways of talking about the world: e.g., describe your favorite flavor of ice cream.

On the other hand, is there any way of talking

preciselyabout anything complicated that does not require math? Try to describe actuallymakingyour favorite flavor of ice cream and I am pretty sure the recipe has to be mathematical at least in the sense of being quantitative.So, if the universe is capable of being described in precise terms, I suspect that is equivalent to saying it can be described mathematically.

Einstein asked if God had any choice when he made the universe. Perhaps the answer is that God had to use math, unless God is just a very imprecise, sloppy fellow.

Dave,

DeleteThe ice cream metaphor presents an interesting slant on why math by itself is an inadequate way of approaching science. If all you want to do is reproduce your favorite version of strawberry ice cream then a quantitative recipe is all that is needed. If however, you want to find your favorite flavor of strawberry ice cream by experimentation, then a single recipe provides no avenue for exploration. For exploration a qualitative list of both necessary and possible ingredients is needed, opening a landscape of strawberry ice cream.

I don't want to beat the metaphor up just point out that a qualitative list of possible ingredients carries more information and potentiality than a precise recipe. With precision comes a loss of information.

bud rap,

DeleteIf your goal is to

knowwhat is in a particular batch of ice cream, then you want the recipe.The goal of physics is to know as well as we can how the physical world works. That takes math.

Admittedly, in the "softer" sciences -- biology and, especially, the social and behavioral sciences -- our understanding is not precise enough to codify everything in math, and maybe it never will be. Feynman was quite vocal about what he called "cargo-cult science" (I was in the audience when he gave this talk) in the social and behavioral sciences in which they superficially imitate physics, without any real content.

But in physics the mathematical approach has worked well enough that anyone who is unwilling to approach the math really cannot fully understand what is going on. A lot of us -- ranging from Steve Hawking to me -- have tried hard to explain ideas in physics using as little math as necessary.

But if you think you can really grasp current theories in physics (or, for that matter, the best physics theories of three hundred years ago) with no math at all, you are just fooling yourself.

Of course, if your goal is

notto understand as well as you can how the physical world works but rather to come up with amusing ideas that intrigue and engage a lot of non-scientists, then you can just ignore everything I just wrote!Dave,

DeleteIf your goal is to know what is in a particular batch of ice cream, then you want the recipe.But if your recipe contains ingredients that do not exist, of what value is your recipe? If the ice cream exists but the alleged ingredients do not, shouldn't the conclusion be that the recipe is wrong.

So no part of the physicists' description of nature ("laws", etc.) is fundamentally inevitable, but the nature of nature may or may not be inevitable, not that we're ever likely to know?

ReplyDelete"There are infinitely many sets of axioms that are mathematically consistent but do not describe our universe."

ReplyDeletealso

There are infinitely many sets of axioms that are mathematically consistent THAT DO describe our universe.

What makes you think so?

DeleteSabine,

ReplyDelete"...statistical independence is sacred scripture, rather than being what it really is: A mathematical axiom, that may or may not continue to be useful."

I see that you oppose the idea of granting a "sacred scripture" status to statistical independence, which is OK. However, I would go even further and deny even the status of a mathematical axiom. Unlike the postulates of QM or GR which have not been shown to be false in any experiment, the statistical independence is clearly false in a plethora of situations, like all systems described by fields (QFT and GR included). It's only true when rigid body mechanics with contact forces only is a good approximation, but even in this case it needs not to be assumed as it can be deduced.

I am afraid that, by whatever set of axioms, it is inevitable that basically any equation proves that nothing is nothing.

ReplyDelete

ReplyDelete"But the axioms themselves can never be proved to be the right ones"That's pretty much the definition of "axiom".

Axioms are assumed so don't require a proof, but the point of the blog post is about finding axioms for already existing structures e.g. finding the axioms for arithmetic or physics. How do you know when you've got the right axioms? In physics you can never know because of the finite precision of measurements and the possibility of unknown unknowns. In Maths it can be done because they can prove things exhaustively and concepts can be perfectly precise.

DeleteI was thinking "I agree with everything", until you lost me here:

ReplyDelete"Another unfortunate consequence of physicists’ misunderstanding of the role of mathematics in science are multiverse theories.

This comes about as follows. If your theory gives rise to internal contradictions, it means that at least one of your axioms is wrong. But one way to remove internal inconsistencies is to simply discard axioms until the contradiction vanishes.

Dropping axioms is not a scientifically fruitful strategy because you then end up with a theory that is ambiguous and hence unpredictive. But it is a convenient, low-effort solution to get rid of mathematical problems and has therefore become fashionable in physics. And this is in a nutshell where multiverse theories come from: These are theories which lack sufficiently many axioms to describe our universe.

Somehow an increasing number of physicists has managed to convince themselves that multiverse ideas are good scientific theories instead of what they de facto are: Useless."

This sounds like a

non sequitur. What does the stuff before the second occurrence of "multiverse" have to do with the Multiverse?"There are infinitely many sets of axioms that are mathematically consistent but do not describe our universe. The only rationale scientists have to choose one over the other is that the axioms give rise to correct predictions. But there is no way to ever prove that a particular set of axioms is inevitably the correct one. Science has its limits. This is one of them."I think that this is some sort of misunderstanding, as it describes one type of Multiverse (and not a popular one at that).

Whether or not you agree with him, I think that Max Tegmark does a good job of explaining the different types of Multiverses in

. He also mentions fine-tuning in the context of the Multiverse, whileOur Mathematical UniverseLewis and Barnesdiscuss the Multiverse in the context of fine-tuning. I know that you've read the former. I don't think that you've read the latter. If you need a copy, I have a spare one. (Whatever your opinion, I'm sure that readers here would enjoy it.)Phillip,

DeleteI have no idea why I would be interested in reading Lewis and Barnes as I have explained to you sufficiently often that fine-tuning arguments are mathematically ill-defined and therefore scientifically useless.

I also do not know what your difficulty is with understanding the paragraphs you quote. If you have a theory that does not describe our universe but a large number of universes, you are lacking an axiom that specifies the theory so that it makes predictions for what we observe. That is the case for all multiverse theories. That people even talk about them as if they were scientific demonstrates they don't understand what the purpose of a scientific theory is.

The only way you can coax predictions out of such an ambiguous theory is to put the missing axiom(s) back in somehow. Multiverse people do this by postulating many more axioms as would be necessary, which still does not make the theory scientific.

In case you still have trouble comprehending this, assuming that Lambda is a constant and has value a \pm b is simpler than inventing a long story about an infinite number of universes with a measure and an observational bias from which you derive in the end a lambda, which comes out wrong by a factor ten, so then you have to make your measure more complicated etc. And all of this just to get the same result you could have gotten by just writing down Lambda = a \pm b. Occam would weep over this nonsense.

I regard Tegmark and Lewis & Barnes as bat-sh*t crazy. Well at least Tegmark did some top rank physics with cosmology and CMB, but I think his math-universe idea is pure nonsense. Lewis & Barnes are doing a sort of Tipler confusion between science and theology.

DeleteThe multiverse is something for the toolbox. For all we know the dS inflationary spacetime is a quantum or semi-classical system and all these other cosmologies are quantum corrections on this observable universe.

Note that few people besides Max talk about his mathematical universe (though, interestingly, Olaf Stapledon alluded to something very similar). The "normal" multiverse is Max's Level II.

DeletePeople seem to get worked up over the fact that Barnes is a theist. The arguments can be correct and he can interpret them wrongly. Creationists use the adaptations of living things to show how smart God is; that doesn't mean that those adaptations don't exist (and, of course, are due to evolution). Similarly, will you claim that it is wrong to believe that the Hubble constant is between 60 and 80 just because Barnes believes it?

DeleteOne of the interesting things about the book is that Lewis can interpret the same arguments in a different (and, in my view, better) light.

Lawrence Crowell12:57 PM, December 11, 2019

Delete"I regard Tegmark and Lewis & Barnes as bat-sh*t crazy. "

Amen.

Phillip Helbig 12:00 PM, December 12, 2019

Delete"The arguments can be correct"

What arguments????????????

There are no arguments!

"If the universe could be different then blah blah blah" is not an argument, it's a speculation. There is **zero** evidence that the universe can be any different than it has been observed to be.

What is the matter with you? Nobody can be this thick.

The idea the universe could be very different is not really testable. In the multiverse setting it is possible I think that these other cosmologies, except for maybe some tiny number or measure over the sample space of vacuum expectations possible, are virtual or off shell quantum corrections. The universe is "real" because it has a classical aspect to it. The smaller the vacuum energy Î› the more classical spacetime might be. So maybe only for very small Î› can we say a cosmology has what we might call classical ontology.

DeleteThis has been a bane for M-theorists trying to calculate the vacuum energy of spacetime by working gauge fluxes through wrapped D-branes. It is a higher dimensional form of Gauss' law in the end. It is possible to calculate smaller values, but they are still enormously large. The 123 order of magnitude problem is difficult to understand this way. With the possible 10^{1000} Calabi-Yau configurations it is not easy to navigate to the one appropriate for the observable universe. Easy compactifications, such as toroidal compactification, clear does not pertain to the observable world.

With theological arguments the difficulty I see is that one replaces an unknown about the world with an infinite unknowable. This is begging the question and appealing to a special pleading of sorts as an answer. If people want to believe in some ultimate form of mystical entity, whether the absolute nothingness or Tao or an ultimate consciousness, they are at liberty to do so. However, one can't appeal to these things as an answer to a scientific unknown.

I fully agree with your viewpoints ; some practical points I use to confirm. Our math has way to many freedoms w.r.t what's possible in nature which lead to 'constructs' such as string-theory. For which an abundant mistake was introduced that dimensions got physi-sized (!) in order to make it work (for example). A mathematician from the 18h century would have called this blasphemy. The same deep collision lays in the foundations of Quantum Theory where we used an absolute tool (Math from Newton's age) to describe the small. But that-math was only derived from our examples in the Big World....

ReplyDelete

ReplyDelete"I have no idea why I would be interested in reading Lewis and Barnes as I have explained to you sufficiently often that fine-tuning arguments are mathematically ill-defined and therefore scientifically useless."Maybe because one can properly judge a book only after reading it?

Sure, I don't read crackpot books. One has to have some sort of filter. But you are going out on a limb by essentially calling not only Lewis and Barnes but such respected cosmologists as Martin Rees, Bernard Carr, and Steven Weinberg crackpots. While famous people can be wrong (I have tilted at such windmills myself; progress is slow, I know), in order to be convincing one has to show, exactly, why they are wrong, for which knowledge of the original claims is essential.

Some, but not all, famous people read this blog. But proper scientific debate is still more in refereed journals than on YouTube. (As my late history teacher used to say, just an observation, not a judgement.) Why not write a "fine-tuning is bullshit, the multiverse is bullshit" paper and publish it in, say,

Foundations of Physics(the editor-in-chief does read this blog, at least occasionally)? Of course, that would not prove that you are right, but it would bring the debate up to a higher level than that of blog comments. :-|Phillip,

Delete"But you are going out on a limb by essentially calling not only Lewis and Barnes but such respected cosmologists as Martin Rees, Bernard Carr, and Steven Weinberg crackpots.I didn't say any such thing. Stop putting words into my mouth.

While famous people can be wrong (I have tilted at such windmills myself; progress is slow, I know), in order to be convincing one has to show, exactly, why they are wrong, for which knowledge of the original claims is essential."I wrote a book about this and I wrote a paper about this and I have done my part of the job. If you still don't get it, too bad for you.

Phil,

DeleteIf I can jump in, you wrote:

>Why not write a "fine-tuning is bullshit, the multiverse is bullshit" paper and publish it...

Personally, I think such a paper would get a *yawn* from most physicists because

it is what they already think.I don't think the average physicist who is studying energy levels in Si or whatever thinks of the landscape/multiverse as anything except sci-fi.Remember the physics conference a while back where a show of hands was taken on whether the audience believed in all of this? Polchinski was shocked that most did not (I think his exclamation was "Holy s***!").

And one does not have to disdain an author to not read his book: I would much prefer to work my way through Weinberg's books on QFT and QM, which I still have not found time to do, than read a book on the landscape/multiverse.

There has to be a

strongpositive reason to read a particular book, simply because any bright person has averylong list of books he or she would like to read.All the best,

Dave

Lewis and Barnes are crackpots. They claim that there is evidence for universal fine-tuning when there is none. It is not known that the universe can be any different to how it has been observed to be.

DeleteBernard Carr admits there is no evidence for fine-tuning, that it is speculation.

Martin Rees was paid 1 million pounds by the Templeton Foundation, so wittering on about fine-tuning and the multiverse for the past 4 decades without providing a single piece of evidence hasn't been a complete waste of time for him.

Steven Weinberg: "It is still too early to tell whether there is some fundamental principle that can explain why the cosmological constant must be this small.""

That's the point - for any purported fine-tuning, there may be a physical explanation - we just don't know. Which leaves the fine-tuners to make their speculations and see if it leads to any empirically verifiable knowledge. It hasn't so far.

Conclusion: The universe is not known to be fine-tuned. There is not known to be a multiverse. No knowledge has been gained from these speculations.

Very, very obvious and simple. But somehow beyond your ability to understand.

Phillip Helbig 6:07 AM, December 11, 2019

Delete"Maybe because one can properly judge a book only after reading it?"

But interestingly in this case, people who haven't read the book (myself, Dr H. and others) have judged it correctly, while people who have read the book (you, Brian Schmidt, Marcelo Gleiser) have written utter drivel about it and completely failed to see the blindingly obvious fact that there is no evidence for its claims.

Why is that?

Well the book is a thick filter. If you can't tell from the title and the page of contents that it's utter drivel then you are thick and might well read it, making yourself thicker in the process.

Phillip Helbig6:07 AM, December 11, 2019

Delete"Sure, I don't read crackpot books."

Yes, you do. You read "A Fortunate Universe: How Crazy Luke Wishes Baby Jesus Loved Him".

Physicist Dave wrote: "I would much prefer to work my way through Weinberg's books on QFT and QM, which I still have not found time to do, than read a book on the landscape/multiverse." I certainly concur ! Until studying Weinberg's volumes, QFT really did not make sense to me. But, after studying string theory books (Green, Zweibach, Becker, West--in that order) I can say string theory still makes little sense to me. Regards axiomatization in the physical sciences, one can see how difficult such an enterprise is by reading the beautiful monograph by Giles: Mathematical Foundations of Thermodynamics (1964).

Delete

Delete"Martin Rees was paid 1 million pounds by the Templeton Foundation, so wittering on about fine-tuning and the multiverse for the past 4 decades without providing a single piece of evidence hasn't been a complete waste of time for him."While I think that Rees should have rejected the prize, or donated it to some sceptic foundation or whatever, he was as surprised as anyone that he got it, so the claim that he believes in the Multiverse because he won the Templeton Prize is logically wrong. Or do you believe in some sort of strange teleology?

Phillip Helbig 12:04 PM, December 12, 2019

DeleteYou don't seem to be able to understand English. Rees believes in the Multiverse for the same reason as you - he doesn't understand basic logic or the fundamental principle of empirical science. His 4 decades of waffle aligned with the agenda of Templeton to subvert the need for empirical evidence in physics, so they rewarded him and he helped their fraudulent cause by taking the money.

You still fail to provide any evidence of fine-tuning across several comments, focusing instead on complete irrelevances. Are you afraid to write down your views because they will be torn to pieces?

Gary Alan wrote to me:

Delete>But, after studying string theory books (Green, Zweibach, Becker, West--in that order) I can say string theory still makes little sense to me.

I'm currently working through Zwiebach and have Becker as the next task. My ultimate goal is to be able to explain string theory as we explain normal physics (it may or may not have anything to do with the real world, but it would be nice to have it explained in a way that can be understood by normal physicists!).

Of course, one does not always achieve one's ultimate goal...

Delete"Are you afraid to write down your views because they will be torn to pieces?"It appears to me that you know my views rather well. I would argue that you misunderstand them, but spelling them out probably wouldn't help.

Actually, I plan to write them down and submit them to a respected journal. I post a link after the corresponding paper has appeared.

In the meantime, I'm glad to be in the company of Weinberg, Schmidt, Carr, and Rees (though he should have rejected the prize---maybe he'll give me a slice to make up for that).

There are many crackpots---just take a look at viXra. Why do you spend so much time trying to convince people here in the comments threads?

Phillip Helbig 5:53 AM, December 13, 2019

Delete"I would argue that you misunderstand them,"

Go on then. Argue it. I've been asking you to do that for 6 months, but you keep running away like you're scared.

"but spelling them out probably wouldn't help."

LOL! Very convincing excuse. It would help you learn some basic logic and the fundamental principle of empirical science.

"In the meantime, I'm glad to be in the company of Weinberg, Schmidt, Carr, and Rees"

Weinberg is the only first rate physicist in that list and he doesn't think there is evidence for fine-tuning for the very obvious reason which you have never addressed:

Steven Weinberg: "It is still too early to tell whether there is some fundamental principle that can explain why the cosmological constant must be this small.""

"Actually, I plan to write them down and submit them to a respected journal."

If you can't even address this simple point of basic logic, you are clearly going to be wasting your time writing a paper.

ReplyDelete"In case you still have trouble comprehending this, assuming that Lambda is a constant and has value a \pm b is simpler than inventing a long story about an infinite number of universes with a measure and an observational bias from which you derive in the end a lambda, which comes out wrong by a factor ten, so then you have to make your measure more complicated etc. And all of this just to get the same result you could have gotten by just writing down Lambda = a \pm b."You are completely missing the point. That might be OK for some quantities. At some level, maybe things just are as they are. In the case of fine-tuned quantities it is more problematic. Of course, you can just claim that there is no fine-tuning but, again, this is a strong claim. The Multiverse can explain why some (but, perhaps, not all) quantities are fine-tuned.

This is really no different than the fact that no explanation is required for why there are six continents, but one is required to explain why the Earth is at the right distance from the Sun so that there is liquid water. The answer is a type of multiverse: there are many planets around many stars, so of course life based on liquid water is on an Earth-like planet. Trivial, really.

Of course, we now know that there are exoplanets, and have expected so for a long time. But suppose that someone back in the Middle Ages would have proposed this explanation. Would he have been a non-scientific crackpot?

Phillip,

Delete"The Multiverse can explain why some (but, perhaps, not all) quantities are fine-tuned."No, it can't, as I have evidently unsuccessfully tried to tell you many time. Making assumptions about a probability distribution over a space will always be more complicated than just assuming the values of certain constants. There is no simpler explanation than a constant of nature.

As I have also said many times, if you have trouble comprehending this, you only need to look at what scientists do in practice. No one who works with LCDM instead works with measures on the multiverse.

If there wasn't liquid water on earth, no one would care!

DeleteDr. Sabine,

Delete"Making assumptions about a probability distribution over a space will always be more complicated than just assuming the values of certain constants. There is no simpler explanation than a constant of nature"

I'm not sure this is as straightforward as you present. Assuming the value of certain constants is certainly more specific and less general than considering a probability distribution, but is that the only goal?

Imagine that I cataloged all the (essentially infinite) possible measurements one could make about the universe as well as the corresponding measured values, using an individual measuring process to define the constant and the actual value as that constant's value. According to your position, would this not be the 'simplest' and therefore best description of reality? Surely the goal of science is to come up with models that are both as verifiably accurate and as general as possible.

Phillip Helbig 6:17 AM, December 11, 2019

DeleteYou have no evidence of "fine-tuned quantities". There may be physical explanations in the future. You have no idea whether the universe can be any different to the observed one. Of course, within the universe there are known fine-tunings (pianos) and multiverse-like effects (lottery winners, water on Earth maybe). However, that is not to say that the universe itself is fine-tuned or that there is anything so unlikely about the universe that it suggests a multiverse. You have provided *zero* evidence of this, as ever.

"Of course, you can just claim that there is no fine-tuning but, again, this is a strong claim. The Multiverse can explain why some (but, perhaps, not all) quantities are fine-tuned."

You are making a false dichotomy. You are claiming that there is evidence of fine-tuning. I and others are pointing out there is no evidence, so there is no reason to think fine-tuning is true. We are not claiming that fine-tuning has been refuted. This is basic, basic logic. You are extremely confused on this matter.

"but one is required to explain why the Earth is at the right distance from the Sun so that there is liquid water. The answer is a type of multiverse:"

But we know that planets can be at distances from their star where water doesn't exist, either too close or too remote. We do NOT know, however, that the universe could be any different to the one observed or that there is anything unlikely about the nature of this universe. Why do you keep making these blatant, trivial logical blunders? You need empirical evidence to support a physical theory, not a useless analogy.

"Of course, we now know that there are exoplanets, and have expected so for a long time. But suppose that someone back in the Middle Ages would have proposed this explanation. Would he have been a non-scientific crackpot?"

He probably would have been tortured to death by the Catholic Church.

If he had had no reason for his claim then it would just have been a lucky guess. Maybe you and crazy Luke will turn out to be right by luck, because you certainly don't have any evidence. You are claiming that there is evidence that the universe is fine-tuned, but you have provided none. You just assume the universe can be different to how it has been observed, assume some random probability distributions for certain constants without any apparent reason, and then conclude from these baseless assumptions that the observed constant has almost zero probability and is therefore fine-tuned. No evidence to support the assumptions; no new empirically verified knowledge resulting from the assumptions.

It is complete and utter drivel. You are cranks.

This comment has been removed by the author.

DeleteA hierarchy of mathematical structures that are discovered by humans exists that starts from a founding structure and evolves into more complicated structures that together constitute a model, which shows the structure and behavior of the physical reality that we can observe

ReplyDeleteMathematics restricts the extension of the hierarchy to more complicated levels. The founding structure acts like a seed from which only one type of plant can grow.

In this way the founding structure restricts the extension of the hierarchy such that it results in a model that describes the kind of structure and behavior of reality that we can observe.

This indicates that physical reality applies similar mathematical structures that implement its structure and behavior

It all began with counting and the invention (some might say 'discovery') of the integers.

DeleteThere are some mathematicians (like Doron Zeilberger at Rutgers) who find the invention of "the integers" (as an actual entity consisting of an infinite number of things, as commonly taught in many school rooms) as being a bit ridiculous. :)

DeletePhilip Thrift 5:53 AM, December 12, 2019

DeleteThere are millions of mathematical proofs which depend on the axiom of infinity (i.e. the existence of an actual infinity), and many of these theorems are open to being disproved by a counter-example e.g. Fermat's Last Theorem. I suspect Doron Zeilberger has never come up with such a counter-example. So either all the counter-examples are hanging out beyond the reach of supercomputers, or by pure coincidence all the proofs depending on infinity just happen to give the right answer despite being flawed, or Prof Zeilberger doesn't have a point. I'm thinking the last reason is correct.

The proper way to do physics is to construct a mechanical model and then describe it mathematically.

ReplyDelete(Or to impose a mechanical model on a successful mathematical formalism. )

I'm reminded of an odd take that Gestalt psychologists had-- they were trying to discover what they took to be "laws of perception". The premise: there ARE laws of perception and if we knew them we could describe every possible perception. That universe of possible perceptions would have to include results of every possible experiment that a scientist could perform. An interesting project. Mathematics, in their scheme, suggested a fundamental ordering principle of the perceptual apparatus. But there would have to be other restrictions, since mathematics alone gives too many schemes, not all of which conform to what we might observe. This would be an inside-out way to approach understanding of the universe for sure. I know they studied optical illusions a lot. They didn't allow for the possibility that some independently existing "universe" might contain important structures that might not be part of any possible perception. Since our perceptual apparatus evolved to meet primate survival needs, it would be surprising to find that it could represent everything that might exist.

ReplyDeleteI fail to understand the phrase "scientifically useless." As with other terminology (such as the word "beauty") it appears as if it is "in the eye of the beholder." Steven Weinberg wrote: "at the time--1967-- I proposed my theory, there was no experimental evidence for or against it, and no immediate prospect of getting any."(1974,Sci Am, 231).Based on his words,would we not-- at that time-- have considered his 'model' as "scientifically useless" ? Andrzej Trautman writes: "much of the language of theoretical physics is sufficiently imprecise to allow vivid disputes between authors who attribute different meanings to the words they use." (1981, Yang-Mills and Gravitation, A Comparison). Utiyama's 1956 paper is interesting: "thus, we have obtained the expression for the covariant derivative without using the concept of parallel displacement."(Phys Rev 101). Mathematician S.S. Chern: "...this 'confluence of divergent paths' of mathematics and physics is indeed a mysterious phenomena" and "...beyond simple description."(page 336,Lectures on Differential Geometry). If I have a point to emphasize buried in all these quotes, it is that the "scientific usefulness" of an idea, a model, a mathematical construct, is not always (or, necessarily) grasped all at once during the time in which it is proposed.

ReplyDelete"I proposed my theory, there was no experimental evidence for or against it, and no immediate prospect of getting any."

DeleteThe point is that he admits and presumably at the time admitted it was speculation with no empirical evidence. There are fine-tuning, multiverse and String Theory advocates who don't admit their ideas are pure speculation.

Sabine wrote:

ReplyDelete“It is the mistake behind string theorists’ conviction that they must be on the right track just because they have managed to create a mostly consistent mathematical structure.”

No respectful string theorist thinks that “they must be on the right track”. Having a toy model of a consistent quantum gravity is just one motivation to research the theory farther.

Udi,

DeleteMaybe you should read some interviews with Witten because he very definitely leaves me with the impression he thinks he is on the right track.

Sabine wrote:

Delete“Maybe you should read some interviews with Witten because he very definitely leaves me with the impression he thinks he is on the right track.“

From your transcript of Graham Farmelo interviewing Witten, Witten says:

“string slash M theory is the only really interesting direction we have”

So he thinks it is an interesting direction, but he does not claim that he “MUST be on the right track”. Nor does he make any claims that string theory is “inevitable”.

Udi,

DeleteI didn't say he says it's inevitable. I said he clearly thinks he is on the right track and so do all string theorists I know. They have zero awareness for how strongly their conviction depends on having made a particular set of assumptions that have no deeper justification. Also, I find this discussion deeply uninsightful and have no interest in continuing it. If you want to say you have a better opinion of string theorists than I do, fine. Point taken. But I don't know how that's interesting.

Sabine wrote:

Delete“I didn't say he says it's inevitable.”

You wrote about “string theorists’ conviction that they MUST be on the right track” in a blog post that criticizes the “inevitability of the laws of nature”. I challenge you to find any respectable string theories that makes such claims.

“I said he clearly thinks he is on the right track and so do all string theorists I know.”

Naturally string theorists think their research is a promising approach – that is why they choose to study it. This is very different from thinking that it is inevitable or must be true.

“They have zero awareness for how strongly their conviction depends on having made a particular set of assumptions that have no deeper justification.”

Again you are making baseless allegations. Clearly string theorists know what are the basic assumptions of their theory is, otherwise they wouldn’t be able to study it. They are convinced that their approach is promising, they are not convinced that it MUST be true.

“If you want to say you have a better opinion of string theorists than I do, fine. Point taken. But I don't know how that's interesting.”

It is not about my opinions. You are making baseless claims against string theorists. I am trying to show you that your opinions are without merit.

Udi,

DeleteDo you follow Lubos' blog? He seems pretty sure string theory

mustbe true!I myself like Lubos and often agree with him: e.g., I think he has a more balanced view of Russia than many Americans do.

He is though a bit more confident in his judgments about the future of physics than I am.

Lawrence, Udi, and some others:

DeleteSeveral of you posted rather unfriendly comments about Lubos in response to Dave's above remark. I don't think this is a fruitful discussion and I will not approve further comments on the topic. Could you please stick with the physics, thanks.

Shouldn't the more realistic question be rather:

ReplyDeleteAre the laws of PHYSICS inevitable?

If someone believes, as I do, that our phycics is unifiable,

there is at most only one single (I would say - universal) LAW of NATURE (if any at all). And as far as I was able to investigate (and understand) it, it is inevitable (it concerns the energy transfer).

On the other side, the laws of physics, as I understand them, are never inevitable. Our physics, better or worse, unified or not, is always only a myth, an illusion, our human description of how we understand the Nature.

In 1964, John R. Platt wrote in the journal Science:

ReplyDelete“Many –perhaps most– of the great issues of science are qualitative, not quantitative, even in physics and chemistry. Equations and measurements are useful when and only when they are related to proof; but proof or disproof comes first and is in fact strongest when it is absolutely convincing without any quantitative measurement.

Or to say it another way, you can catch phenomena in a logical box or in a mathematical box. The logical box is coarse but strong. The mathematical box is fine-grained but flimsy. The mathematical box is a beautiful way of wrapping up a problem, but it will not hold the phenomena unless they have been caught in a logical box to begin with.”

No free will AND no inevitable laws of physics?

ReplyDeleteWhat do we get?

Cheers,

mj horn

Try to imagine the model. Then go from there.

ReplyDeleteYou are awesome Sabine, you cut through all of the hocus pocus.

ReplyDeleteThank you

ReplyDeleteExcellent article, yes, and it also happens outside the scientific world; the utilitarian value of gold has become an absolute intrinsic value; When things have been useful for a long time, they magically acquire a certain value of their own.

I’m admittedly writing this reply after only reading the first sentence,

ReplyDelete“No one has any idea why mathematics works so well to describe nature”. I’ve read your blog for years and I am not sure why you would start that way? Clearly it is the most precise form of communication humans have by many orders of magnitude over any other. Everyone will agree on what the quantity 2 is, therefore if an equation makes a prediction and the predicted quantity is consistently accurate we know at the very least it works at approximating nature. You’ve said many times here about interpreting equations where the process cannot be observed is useless scientifically, which I agree; however we all can agree if the result at least approximates nature or not. There is no other language where everyone will always apply the exact same meaning to a precise conclusion.Louis,

DeleteYou are misunderstanding the mystery. The mystery is not why, if you know that there are regularities in the universe and that we are able to infer these regularities mathematics is a good tool to do that. The mystery is why the universe is this way to begin with.

Why is there something rather than nothing?

DeleteMathematics is simply logic, developed in a way that is practical for our limited brains to work with. What would the universe be like if it were not logical? You're already taking a logical universe as a given when searching out inconsistencies in our theories.

DeleteBut is there any good reason to expect that we could describe so much behavior already with so few axioms of relative simplicity? Maybe that's the mystery.

Sabine,

DeleteI think Louis has a strong point here: if there is

anyprecise way of describing nature at all, what could that precise language be except the language of mathematics?This has been brought home to me in the various discussions with the non-scientists here in your comment threads where they spin long statements in English that we scientists know mean nothing in terms of the math. But they won't learn the math, and, so, without any math or science background, they truly do not realize that their long-winded, wordy diatribes do not actually mean anything.

All the best,

Dave

'is' implies that something exists in a certain place in a certain time. If time and place are emerging entities than this question only is meaningful when time and place are there. What raises the question can there be causation without time?

Delete"Why is there something rather than nothing?"

DeleteThere are many, many nothings. You mean "why is there a something as well as lots of nothings?"

Dave,

Delete(a) as I said above, the mystery is why there is any precise way of describing nature and one that we can understand and discover to begin with, not that, if there is such a way, why mathematics is useful for this description

(b) Just because you and I cannot think of anything better than math doesn't mean there is nothing better.

(c) As I wrote here (under a pseudonym) one doesn't need math to do science. Science is all about using one system to predict another system. Math is a way to do that, but it's not necessary.

It seems to me that math and physical theories co-evolved as humans developed the ability to think abstractly. So it's a sort of anthropic principal of math; it works because that's how it evolved...the universe is mathematical simply because it's describable. Beyond that, of course, is a mystery...

DeleteSabine,

DeleteYou wrote:

>"(a) as I said above, the mystery is why there is any precise way of describing nature and one that we can understand and discover to begin with, not that, if there is such a way, why mathematics is useful for this description"

Agreed.

But is a Nature that cannot be described precisely even possible? I don't know of course. But if you or I tried to describe such a Nature, we'd probably do it mathematically, which would sort of be self-refuting!

You also wrote:

>"(b) Just because you and I cannot think of anything better than math doesn't mean there is nothing better."

Hard to prove a negative But it is at least a plausible hypothesis that the reason humans cannot think of a precise way to describe things without math is because that is the only way to do it.

You also wrote:

>"(c) As I wrote here (under a pseudonym) one doesn't need math to do science. Science is all about using one system to predict another system. Math is a way to do that, but it's not necessary."

Sure: before the DNA revolution, biology was not that mathy. But, the more precise biology has become the more math has intruded. And, of course, biology is based on chem which is based on physics -- each step down the scale, the more mathy it becomes.

By the way, I am not defending our colleagues who are "lost in math." I trust you agree with me that special relativity really cannot be done without math (at least algebra) and yet the modern understanding of relativity (Minkowski diagrams and all that) is not

justmath: it is very physical.And, in fact, one of the tasks that has always interested me is finding out what the lowest level of math is that can be used to clearly explain some area of physics.

Re your essay, I fear I find Tegmark's views sort of mildly interesting sci-fi: Greg Egan does this sort of thing better as real sci-fi.

The basic point I am making you in fact stated in your essay:

>"Words are malleable, they can be played, they can be abused. Languages evolve and adapt. Writers make a living from recreating language over and over again and we admire the novelty. But the very reason that makes language socially useful makes it so unsuitable for science. It’s imprecise, open to interpretation, dependent on a large number of unrecorded factors. As they say, if a thousand people read a book, they read a thousand different stories.

>"Mathematics on the other hand does not suffer from these shortcomings. Mathematical structures are defined to have certain properties. They’re not open to interpretation and don’t depend on the context. Mathematics is therefore highly reproducible and precise. If a thousand people read Einstein’s field equations, they read the same equations."

Indeed.

As someone who almost became an economist rather than a physicist, I also agree with your point that what we might call "math envy/ physics envy" in the social and behavioral sciences has often been counter-productive. If we do not have a precise understanding of some field, we should not pretend otherwise.

By the way, I am not equating math with equations or mathematical symbols. I am currently reading Andrew Wallace's classic

Differential Topology: First Steps: more words than equations or math symbols, but still indisputably math. We use math symbols because we often find that enhances communication and thought, but it is the precision, not the symbols, that is distinctive of math.Math should always be our servant, not our master. But, at least in physics, it certainly is a necessary servant.

Mathematical symbols stand for concepts which can only be formulated in words.

DeleteThe only indisputable math is analytic logic which is really just an empty tautology.

A = A

Sabine,

DeleteRe

“the mystery is why there is any precise way of describing nature and one that we can understand and discover to begin with”:Human beings have developed the ability to represent things with symbols, so it is natural that physicists would represent the law of nature relationships they found with maths symbols. But, well before physics, I would think that the only reason that maths relationships make sense to human beings is because we are made out of similar types of relationships.

Then there is the issue of how human beings know anything anyway. But apart from the special high-level analysis provided by the human brain, I would think that the issue of how human beings know anything is the same issue as how the micro-world “knows” law of nature relationships and numbers: for the whole system to work, I would think that there has to be a continuity of knowledge from the micro-world to the macro-world.

DeleteIt amazes me how one speaks of the "great" precision of mathematics; much of this is based on statistics, probabilities, uncertainty principle; most of the time we have no choice but to settle for the most precise imprecision possible; and other times to be very precise we have to charge the pointer with an energy that we are not able to produce.

PhysicistDave wrote:

Delete“Sabine,

I think Louis has a strong point here: if there is any precise way of describing nature at all, what could that precise language be except the language of mathematics?

This has been brought home to me in the various discussions with the non-scientists here in your comment threads where they spin long statements in English that we scientists know mean nothing in terms of the math.

...they truly do not realize that their long-winded, wordy diatribes do not actually mean anything.”

----------------

Dave, it is one thing to rightfully employ the language of mathematics to “describe” nature.

However,...

(and to what I believe is Sabine’s point)

...it is something else altogether to assume that the descriptive tool being used (i.e., mathematics) also provides an answer as to “why” a describable reality even exists.

You seem to harbor a simmering hostility to any sort of philosophical speculations regarding the implications of what science is uncovering.

That makes no sense to me.

I mean, why not leave every (reasonable) approach on the table until the irrefutable truth of reality is finally (if ever) discovered?

Sabine Hossenfelder wrote:

Delete“Dave,

(a) as I said above, the mystery is why there is any precise way of describing nature and one that we can understand and discover to begin with,...

(b) Just because you and I cannot think of anything better than math doesn't mean there is nothing better.”

----------------

Those are wise statements, Sabine, something of which I am very glad to hear you declare.

What Dave doesn’t seem to appreciate is that the scientists who so intensely focus on the bark, and the limbs, and the leaves of a particular tree, sometimes need a non-scientist around to remind them of the forest.

Uh-oh, should that have been expressed in mathematical language for it to mean anything?

Man, this is a tough crowd!

Keith D. Gill wrote to me:

Delete>It is something else altogether to assume that the descriptive tool being used (i.e., mathematics) also provides an answer as to “why” a describable reality even exists.

Keith, I was not critiquing you: you can philosophize as you wish. I

wascriticizing some posters here who are quite certain of certain aspects of current theories in physics that they are in fact completely wrong about, simply because these posters will not bother to learn any of the math that gives those theories content.It is arrogance combined with ignorance that I have criticized.

I will add that, within the framework of physics, "why" tends to mean "Please show from basic principles (preferably with a fairly transparent derivation) that this must be true."

E.g., "why" is Kepler's constant-area law true for planetary orbits? Well, the constant-area law is just another way of stating conservation of angular momentum, and conservation of angular momentum can be proven to be true from Newton's laws and the fact that the gravitational force is a "central force" -- i.e., it is directed towards the Sun at the center of the solar system.

You don't like that kind of an answer to "why" questions? Then you probably should not be asking physicists or talking about physics!

Keith also wrote:

>You seem to harbor a simmering hostility to any sort of philosophical speculations regarding the implications of what science is uncovering.

You think so? Maybe if you give a particular example, I can answer more clearly.

I will say that lots of "philosophical" commentary on science, sometimes even by famous philosophers (e.g., Popper's commentary on quantum mechanics in his

The Open Universe) simply betrays an utter ignorance of the relevant aspects of science. And much "philosophical" commentary is worse than that: it is "not even wrong."I am certainly

notclaiming philosophers are always wrong. Indeed, recently the philosopher Tim Maudlin had a public tiff with the brilliant Nobel laureate Gerard 't Hooft over the issue of locality and Bell's theorem.Having studied the matter for decades, I myself am quite certain Maudlin is right and 't Hooft is wrong. There are nowadays a significant number of philosophers, like Maudlin, who have gone to a great deal of effort to really

learn the physics. But they are still a minority among philosophers.Keith also asked me:

>I mean, why not leave every (reasonable) approach on the table until the irrefutable truth of reality is finally (if ever) discovered?

Because most of what non-physicists say about physics is nonsense, and it just muddies the waters: modern physics is tough enough without fouling the waters.

As I have emphasized, science progresses by ruthlessly, relentlessly annihilating ideas that can be shown to be false.

But most people find this to be inhumane: most people think that other people who try to destroy their cherished beliefs are really not very nice people, not the kind of people you would like to have as friends, co-workers, neighbors, etc.

And they have a point.Or, to use your own words:

>"Man, this is a tough crowd!"

Yes, indeed: any group of honest scientists is.

For better or worse, that is how science works, how it has created more non-obvious knowledge about reality in five centuries than the human race had accumulated in all the millennia leading up to that time.

I like this ruthless annihilation of human beliefs that do not stand up to empirical tests. I doubt that most human beings will ever come to feel the same way.

All the best,

Dave

DeleteDave, I don't know if you realize that science is the current source of myths; The Big Bang, multiverses, 10 or 20 dimensions, quantum entanglement, energy and dark matter, asymmetry of matter and antimatter, the genome and the entire complex system of functioning of a cell; These are all sources of science; but also of myths; It has always been that way, and in order to have science, there must first be questions, fantasy, philosophy, hypothesis, theories, a basic science, a scientific result is not reached without a whole productive process of ideas; when science finishes describing all the "How" theology will still have all "Why"

This reminds me of something Weinberg wrote about. He had collaborated on a study of linearity in quantum mechanics, that amplitudes simply add. IIRC, he tried variations where an additional exponent was applied after amplitudes were summed, and found that any exponents other than 1 created normalization problems. So he opined that in some sense this one aspect of QM could be seen as inevitable, and he wondered if in the future we might begin to see aspects of a final theory as similarly inevitable.

ReplyDeleteIn light of your article though, I have to admit that searching for or evaluating proposed theories on some basis of inevitability is unlikely to be fruitful. That quality of a theory is better to probe after general acceptance, perhaps as much as an exercise to find extra wiggle-room in an existing theory. Like in modified gravity theories: Does the equivalence principle uniquely determine Einstein's equations? Looks like the answer there is "no." And not that even the equivalence principle is sacrosanct, but it feels nice to pursue a kind of conceptual reductionism in this way.

I’m certain that there are huge fundamental problems regarding the foundation of particle physics which are unexpressed respectively suppressed. You don’t have to be a particle physicist to get the “picture”.“See” for example the following documented facts…

ReplyDeleteThe nonexistent spin of quarks and gluons

The first assumption was, due to the theoretical specifications of the mid-1960s that in the image of the SM the postulated proton spin is composed to 100% of the spin components of the quarks. This assumption was not confirmed in 1988 in the EMC experiments. On the contrary, much smaller, even zero-compatible components were measured (Î”Î£ = 0.12 ± 0.17 European Muon Collaboration). Also the next assumption that (postulated) gluons contribute to the proton spin did not yield the desired result. In the third, current version of the theory, quarks, gluons and ... their dynamical-relativistic orbital angular momentum generate the proton spin.

On closer inspection, already the second readjustment has the putative “advantage” that the result in the context of lattice gauge theory and constructs, such as "pion clouds", algorithmically "calculated", can’t be falsified. But this purely theoretical based construction obviously does not justify classification of the quarks as fermions. No matter how the asymmetrical ensemble of unobservable postulated theoretical objects and interactions is advertised and will be advertised in the future, the quarks themselves were never "measured" as spin-½ particles.

Summary in simple words: It is possible to create a theory-laden ensemble of Quarks and “other” theory objects and their postulated interactions, but the Quark itself - as an entity - has still no intrinsic spin -½ in this composition. That means that Quarks aren’t fermions, no matter what the actual theoretical approach would be! This is a basic, pure analytical and logical statement.

Generally speaking: If one postulates that a theoretical entity has an intrinsic value but one discovers that one needs to add theoretical objects and postulated interactions to get the desired intrinsic value, one has to admit that ones entity has no physical characteristic as such.

Interesting blog post about axioms. I am surprised that none of the previous comment-writers mentioned Godel's Incompleteness Theorem, in this context. Godel's theorem is described in https://en.wikipedia.org/wiki/Kurt_G%C3%B6del#Incompleteness_theorem

ReplyDeleteand in other references.

But according to Arthur C.Clarke (can't remember the book's title), one consequence of Godel's theorem is: in a system based on axioms (among other things), there will be effects in that system that cannot be predicted from a knowledge of such axioms or a knowledge of other aspects of that system.

Thus, there are fundamental limits to knowledge that have nothing to do with the limitations of the human brain - which kind of agrees with Dr. Hossenfelder when she writes: "But the axioms themselves can never be proved to be the right ones and hence will never be inevitable themselves. You can say they are “right” only to the extent that they give rise to predictions that agree with observations."

The Ptolemy system is made based on observations converted into Mathematics, it was able to give exact predictions; because no matter what the challenge of the observations is, mathematics always finds a resource to describe it; the only bad thing It had was those rare behaviors of the planets; otherwise it worked, even had a kind of anthropic principle associated with it, "We were the center of the universe, therefore it was made for us." Today we have a quantum physics, based primarily on observations, with a mathematical model tailored to the observations and it is able to give exact predictions; The only bad thing it has is that rare Ptolemic behavior of the particles. Uhmmm!,I'm not going to go out of line

ReplyDeleteBootstrap theories start with spin, and that is a bit of a problem.

ReplyDeleteWhy? Because spin has units of angular momentum, that is, of mass times distance squared per unit of time. The problem is that once you have assumed the existence of mass, distance, and time, working together in quantum Planck units, you have already

assumedmost of known physics. It then simply becomes an elaborate game of banging and crashing and jiggling various combinations of these three deeply fundamental concepts against each other until, finally, symmetries and equations emerge that meet all known experimental classical and quantum constraints for those very same concepts.And it is then a surprise that known physics "inevitably" pops out as a result?

This isn't really so, but I can see how someone could conclude that. People like Penrose with his spin networks and later twistors, and Finkelstein with his quantum logic, would not have spent so much time and effort on the matter had this been true. To give a simple counterexample from dusty math, the idea of a point (e.g. in spacetime) need not be the basic element even of a classical geometry. In Pluecker's line geometry, the point is a derived concept - it's the lines that are the basic elements of the geometry, and the space is the (projective) manifold of lines. You do not think of the lines themselves as aggregations of points, although you can get a good visualization of the structure that way. So the hope is to create the spacetime labels as being derived from more primitive objects.

Delete-drl

Spin/angular momentum is the only motion/energy that cannot be "transformed away". Hence, it is actually more fundamental than lineaer motion, mass, etc.

DeleteGreg Feild: I agree that spin is deeply fundamental. An example that stands out in my mind is the strangely high efficiency of algorithms that embed particle states in Wigner phase space (e.g. Sellier signed particles) versus ones that embed in location space (e.g. Feynman QED). Wigner space is more spin-compatible due to the equal footing it gives location and momentum coordinates, and I suspect (but do not know) that this indirectly helps algorithms in Wigner space converge more quickly. Wigner space algorithms can collapse QED-equivalent calculations from months on a supercomputer down to minutes on a high-end laptop. That still amazes me, even though I had anticipated and begun looking for examples of rapid quantum convergence algorithms when Google Scholar first pointed out the Wigner examples to me.

Delete-----

drl: Thank you for bringing up PlÃ¼cker coordinates. I had never heard of this nicely straightforward predecessor to Grassmann and Clifford algebras, and I find its projective approach to representing location space intriguing.

There is a different and more subtle point here, though.

When in your post I read "… the idea of a point (e.g. in spacetime) need not be the basic element even of a classical geometry", my reaction was "

Yes!" But when I next read "… it's the lines that are the basic elements of the geometry, and the space is the (projective) manifold of lines," my reaction was "No!"Why? Because no matter how you slice or dice them, lines wind up being defined algorithmic by two nominally infinitely precise points. Thus you don't really get rid of the problem of precise points, but instead shift them to a different space.

I did not use to think this way, but nowadays I'm solidly in the camp of believing that unexamined point-first thinking is intellectually and computationally inefficient, counterproductive, and paradoxical. It causes our quick-to-abstract primate brains to overshoot reality way too easily, so that we end up talking about and computing things that simply do not exist. A universe that is quantum at its lower end simply cannot be as point-like as many of our algorithms assume, including even some very good algorithms such as Feynman's QED. So is it all that unreasonable to postulate that the best and most efficient algorithms should be no more point-like that quantum-fuzzed reality? Notably, point-first thinking also leads to most of the paradoxes and many of the odd dualities of mathematics, since if you begin with unreachable infinite limits as your axiomatic "pure states", you've introduced non-sequiturs and potentially grotesque inefficiencies before you even started. Bad logic, that.

For a more powerful and efficient axiomatic starting point that avoids points, try instead to view physics as virtual instantiation software (

"Pavis"). By this I just mean that most of the Platonic and classical realities we accept so blithely as precise, mathematically perfect concepts are in fact never anything more than unreachable asymptotic limits that our primate brains latch onto to simplify processing and increase our chances of survival in a hostile universe. In the Pavis axiom set, the universe is a naturally occurring hierarchy of virtual-pair quantum fluctuations that at various points stepped on each other's toes and ended up creating various forms of persistence — information — which in turn made them forgot how to return to zero. If nothing else, it's fun and surprisingly effective to think of Newton's Third Law as nothing more than a virtual momentum pair whose demise has been thermodynamically postponed until the end of the universe… :)Consciousness must be at the heart of any uncased first cause - it is immaterial, and therefore can have no beginning and no end. Its all about information, and self-aware consciousness.. without it we have only unrealized potentiality...

ReplyDeleteGriffiths and Hilton (1970): "the point we stress is that the selection of axiom systems is not arbitrary but is done for good reason...in any case, moreover, the system must be logically consistent, and experience shows that it is quite difficult to be both arbitrary and logically consistent in the selection of axioms." (page 597, Classical Mathematics).

ReplyDeleteThere are no "laws of nature." What we see and discover are the patterns by which objects, events, phenomena, become apparent and take on real persistence. At the fundamental level, that of pure energy, to my mind the existence of an endless electromagnetic field, there are only a few ways in which fundamental elements can assemble to generate larger, more complex structures. The forms and nature that ultimately appear are multiples of those fundamental patterns, the assembly rules of real objects, events, phenomena. What is the antonym of "recursive"?

ReplyDeleteIn your imagination, put yourself in the ultimate, finest grain element of a fractal, then build it up through its stages to the level of "ponderable matter." This is the process by which reality is built. The steps are: reverberation, reinforcement, resonance, up to a phase transition into a stable state. Repeat, then repeat again. It is a repetitive process out of which the universe we know has arisen. It's structure, it's rules of organization, are what we perceive as the underlying order.

I think we need to get rid of the concept of "laws." Laws imply imposition, intentionality. Our true discoveries are the nature and range of these patterns.

DeleteHello C Scurlock, I think that there must be some kind of dynamic field in a state of equilibrium that produces a regularity, and this regularity produces a monotony described in physical constants; reason why the space, that seems inert and immaterial, hides a dynamic that establishes conditions to the rest of the particles; These are not very different from space, they only present some kind of asymmetry with this.

Just change "mathematics" here to "mathematical physics":

ReplyDelete"It is high time that we move over [from the 'Greek' (Euclidean, Axiomatic, Deductive. ...) approach] to the much more democratic and egalitarian (and far less boring!) 'Babylonian' (Algorithmic, Inductive, Experimental,...) approach to mathematics" and "the whole notion of proof will lose its centrality, and the keyword 'algorithm' will inherent the earth."

Doron Zeilberger

http://sites.math.rutgers.edu/~zeilberg/Opinion174.html

I had to go back and reread an article by T.D. Lee "Time as a Dynamical Variable" (1985, pages 38-64, Shelter Island 2). Here, T.D. Lee concludes: "For more than three centuries we have been influenced by the precept that the fundamental laws of physics

ReplyDeleteshould be expressed in terms of differential equations. Here, we try to explore the opposite--difference equations; difference equations are more fundamental and differential equations are regarded as approximations. With today's rapid progress in computing technology, this may well be an opportune period for such a radical departure from the traditional view." This is an interesting attempt to 're-derive' physics from an alternative starting point. Rovelli also attempts to re-derive physics 'from scratch' in his book Quantum Gravity (2007). The quote from Griffiths and Hilton (mathematicians) is apt: "experience shows it is quite difficult to be both arbitrary and logically consistent in the selection of axioms." Physicists should read Griffiths and Hilton.

Gary Alan: I was intrigued by your quote of T.D. Lee that difference equations may provide a more appropriate approach than differential equations for modeling physics. In fact, here's my offhand attempt to use that idea to create an inverse of Noether's theorem:

ReplyDeleteRehteon's theorem: For every quantum or classical quantity whose conservation over time has been assured by statistically irreversible thermodynamic entanglement (observation), there exists a sequence of difference equations of mass times area over time (action) for which the asymptotic limit of infinitely entangled, infinitely smooth physical space defines a differentiable symmetry.