Mathematics: Intuition\'s Consistency Check by Ken Wharton

Dear Ken Wharton,

Your point has been well argued in the essay: For self-consistency check, physicists have to depend on mathematics (there is no alternative). By winnowing out models that are not self-consistent, we have arrived at a set of theoretical models (in physics), which are essentially mathematical. So it is no mystery that there is some convergence or overlap between physics and mathematics.

I agree with your above arguments. But why is it that mathematics can act as a tool for checking self-consistency in physics? Or what exactly is the role of mathematics in the domain of physics? My essay deals with this question. I invite you to read my essay: A physicalist interpretation of the relation between Physics and Mathematics.

I disagree with your view that intuitions can be counter productive (though you have not ruled out the role of intuition completely). I would say that there should be two checks: (i). checking for mathematical consistency (ii). checking for physical consistency. The former, for intuition based models and the latter, for counter-intuition based models.

You say, " The .......... intuitions have guided us astray". I disagree. I argue that the equations of QM and GR are correct, but the counter intuitive 'physical structures' interpreted from these are incorrect. These 'structures' will not qualify the 'check for physical consistency'. There can be alternate physical interpretations that agrees with these equations. Finiteness theory proposed by me is one such example; whether it the 'right theory' or not is another question.

Dear Ken,

I really enjoyed your thought provoking and solidly argued essay.

I agree with your conclusion "Physics and math therefore share the same essential need for internal consistency, and this makes mathematics the first step on the road to a viable physical theory." and the hope for the future "It remains to be seen where physics goes next, but it seems likely that our models will find uses for even more unusual mathematical structures". I would like to add "bring on the models"!

Best of luck in the contest, your essay was well worth spending some time on.

Regards,

Ed Unverricht

Dear Ken,

The clarity of your writing makes all your points all the more convincing, and (not that this necessarily should be a criterion for rating essays) it would be difficult to object to most of what you say.

My comments here are solely for further consideration. I agree with Jose Koshy's comment (darn, now I have to look at Jose's essay) that perhaps physicists' intuition ought to be given greater weight, despite what you say about them (frequently misguiding us). The reason for that is that in the absence of patterns, paradigms, structures, etc. that make sense to us and which we can manipulate in our minds, physics may become a (proverbial) victim of its own success.

Here is how it would work. Say that things get so esoteric that only computers can generate theories and proofs (by now you know where this is heading), and humans can no longer see why any of it makes sense, nor can they follow the proofs. Computers may propose new theories, but those will still have to be tested against the real world. Given that there can be a myriad of computer-generated theories, and that we will not know which ones to follow up on first, it will present us with various quandaries. Experiments take time, and may only get more expensive in the future. We will not be able to afford to test all of the new theories, nor be able to pick only viable ones for experimental confirmation.

Unless we come up with a ToE to program the computer with before switching to autopilot, we will become dependent on the computer to divine the correct theory solely from mathematics. I am extremely skeptical about any claims that "mathematics alone" can figure out the universe without checking with it (the universe), and I am not getting on an airplane without a human pilot sitting somewhere near the front.

I would also like to draw your attention to a part of Sophia Magnusdottir's essay. You may have already seen it. She proposes that an "analog" method might obviate the need for mathematics in at least parts of physics, perhaps even those not well matched to known mathematics. An example might be adiabatic "computing" already in use. This could certainly work in principle, but since you will still have to measure at least the relative "scale" between the two systems (one being the subject, and the second being the solver) to get the result or prediction you are after, it may not provide all the tools to do physics.

Your central point appears to be unassailable. For at least any foreseeable time horizon, mathematics will be indispensable for physics.

En

Dear Ken,

In your really excellent, clear and pleasant to read essay I cannot find statements to disagree. So forgive me, please, the lack of a major controversy.

In your argument "In both fields, the cave entrances still need to be invented by people: mathematical axioms and physical hypotheses have to be dreamed up in the first place..." I would only correct "invented" into "uncovered".

There were many attempts to formulate axioms in physics (D. Hilbert, J. von Neumann, L. Nordheim, H. Weyl, E. Schrﾃｶdinger, P. Dirac, E. P. Wigner and others). All these efforts failed. That is a pity however a deductive system can consist not only of axioms but also of already established theorems. As far, obviously, theorems were reserved exclusively for mathematics. That means that we could use theorems only in the case we accept the reality is isomorphic to the specific mathematical structures that are covered by that theorem. Following that conclusion I propose to use the geometrization conjecture, proved by Perelman (so it is the theorem in geometry). We have here the set of eight Thurston geometries. We can treat them as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime. As you probably know, the geometrization conjecture states that every closed 3-manifold can be decomposed into pieces that each have one of the eight types of geometric structure, resulting in an emergence of some attributes that we can observe. The Thurston geometries include: S3, E3, H3 (constant curvature) that all three are homogeneous and isotropic, and five more exotic Riemannian manifolds, which are homogeneous but not isotropic (S2 ﾃ-- R, H2 ﾃ-- R, SL(2, R), Nil and Solv geometry). The constant curvature geometries arise as steady states of the Ricci flow, the other five arise naturally where the dynamics of the Ricci flow is more complicated and where topological changes (neck pinching or surgery) happen. I attribute the geometries to fundamental interactions and fermions, except of five exotic ones. I call this concept Geometrical Universe Hypothesis (paraphrasing Tegmark's MUH). It makes the testable prediction referring to 5 exotic geometries. In this short essay I present only very general sketch that delivers the initial conditions. Many mathematical details you can find e.g. in: Torsten Asselmeyer-Maluga, Helge Rose, On the geometrization of matter by exotic smoothness, http://arxiv.org/abs/1006.2230v6.

If you are interested, you can find more details in my essay.

I would appreciate your comments. Thank you.

Your essay deserves the highest rating as it is undervalued.

Jacek

Dear Ken,

Great read and very well argued piece and very instructive discussion on time. I really enjoyed your craft. If you allow me to indulge myself to integrate your concept with KQID below. You wrote on option B that I choose to discuss: "(B) Only t is ordinary time; T is not." KQID: time t (iτLx,y,z) is relativistic times according to KQID relativistic and holographic Multiverse Ψ(iτLx,y,z, Lm), whereas T (Lm) is absolute digital time T ≤ 10^-1000seconds in which everything is in the state of block NOW where time-past-present-future merged. And T is the clock rate of our Existence computed, simulated and projected by Qbit (00, +, -) as the omni-mathematician, the Grand Wizard Merlin, Planck's matrix of all matter and KQID-Maxwell's infinite being with infinite memory. This hologram Existence is the 3-D relativistic (t) as the fetus of time, and T is permanently pregnant with 3-D time as Existence. In other words, t is inside T and we only experience relative times in 3D time simply because T is ticking regularly in asymmetrical Minkowski's Multiverse timeline as history. it is really 3-D in the 0-D or Lm(Tn...) where space is 3-D time moving perpendicularly in 0-D T. Yes, as you inferred Existence is infinite, that requires infinite math and physics that is deeply paradoxical in its nature (see Richard Shoup's essay) and it must follow its own self-consistent "mathematical rules that seem strange and esoteric to us humans."

Extremely fine jewelry craftsmanship, I would vote highly and please comment and rate my piece.

Thank you,

Leo KoGuan

Hi Ken,

Nice essay, and I am pleased to see that you have so elegantly managed to weave in another argument for the block universe. One thing that you have not addressed in your essay is the demarcation criteria for mathematics. Sure, mathematics proceeds from systems of axioms, but not all systems of axioms count as mathematics. There are many systems of axioms that would lead to boring or overly complicated theories, and mathematicians have a knack for avoiding exploring those ones.

If mathematics just consisted of exploring all and any axiom systems then I agree with you that it would be no surprise to find that mathematicians had already explored the structures needed for physics. However, in actual fact, mathematicians have a lot of guiding principles for what constitutes a mathematical theory, including elegance, unifying power etc. One has to explain why the mathematics developed according to those principles is likely to show up in physics, rather than just any axiom systems. My answer to this is that mathematicians' criteria for what counts as mathematics ultimately stem from the natural world, so it should be no surprise that those theories later show up in our descriptions of the natural world. I would like to know what you think about this.

Dear Ken,

As someone knowledgeable and respected in the community and haven treated the subject of axioms/hypotheses so well in your essay, would you volunteer an opinion on three things.

1. Can geometric objects (points, lines, surfaces and bodies) perish or are they eternally existing things?

2. I see from a comment that you are an advocate of block universe. Can block universe perish or it eternally existing?

3. I have a hypothesis that you may want to view and it is this:

"the non-zero dimensional point does not have an eternal existence, but can appear and disappear spontaneously, or when induced to do so".

This is related to question 1 above and attempts to exorcise the millennia old Parmenidean spell cast on our mathematics and physics, when he said, "How can what is perish?", and allow that whatever exists can perish.

4. You said and I agree that, "There's no point in pursuing hypotheses that are internally inconsistent; if a hypothesis leads to self-contradictory implications, it's ruled out before the first experimental test".

In this regard, someone in this community (Armin Shirazi) wrote a paper on the 'photon existence paradox', which seems to ask that in a block universe since time does not flow or elapse for a photon, then the time of emission of a photon is the same as the time of its absorption, how then can photon exist?

Best regards,

Akinbo

Dear Ken,

Thank you for a very interesting essay.

While I agree that consistency has to be a central concern for mathematics and physics, there is one point that I think is over-stressed. The role of axioms in mathematics is really something that only people in foundations take an interest in. For the general mathematician, axioms do not play a role. Most analysts or applied mathematician could probably not list off the axioms of their field. Even Georg Cantor who founded and worked in set theory did not use axioms. In some sense Godel's incompleteness theorem says that axioms are all inadequate.

My paper offers a mechanism for finding both the consistency of mathematics and physics and shows how these mechanism are the same. The essay is called "Why Mathematics Works So Well".

I really enjoyed your analysis of the flow of time intuition.

All the best,

Noson Yanofsky

Dear Ken,

pardon me for barging into your discussion with Akinbo, but because my name was mentioned in reference to this issue with null-frames, I feel the need to clarify some issues, but first I'd like to comment on your response to that point.

"But since the stopwatch is an imaginary construct which measures nothing, that hardly seems to be relevant; that stopwatch has nothing to do with time."

I think this statement is misleading because it seems to switch the meaning of the word "nothing" mid-way. It seems to suggest that there is nothing that would be measured by such an imaginary construct, but that is not true: It would measure something, namely the spacetime interval traversed by an object associated with c. The fact that this happens to be zero does not mean that there is nothing that is being measured. To give an analogy: If I try to measure the number of coins in my pocket, and my pocket happens to be empty, and I say : "I measured nothing", that does not mean that there was nothing that was measured by me because I did make a measurement, namely the number of coins in my pocket.

I think that we should be extra careful in using words with multiple meanings like "nothing", to avoid confusion.

"To say that "time doesn't elapse for a photon" is therefore misleading and wrong; there are certainly a non-zero number of cycles between emission and absorption."

I find this statement curious. It is certainly true that in any time-like frame the number of cycles between emission and absorption can be counted and amounts to a number greater than zero. But I don't think you can use this to say anything about what is being observed in a null-frame because doing so implies a transformation from a time-like interval to a null-interval, and you know better than most that this is not permitted. So this leaves only a statement about a mechanism from within the null-frame to discern the number of cycles. What do you propose it to be? This is not a rhetorical question, I genuinely would like to know because whatever you give as an answer could then be used to countenance the statement that in such frames the moment of "coming into existence" (say, emission of a photon) is the same as the moment of "going out of existence" (e.g. absorption of a photon), because you could insert these as distinct events in between.

Now to the clarifications: I introduced the issue that I labeled as the "existence paradox" within the context of a paper in which I used this as a starting point for arriving at a key argument for deriving the invariance of c in different frames, such that the invariance would seem intuitive (provided one accepts the starting assumptions of the derivation).

The paper was originally entered into the first FQXi contest:

http://fqxi.org/community/forum/topic/329

But an updated version which corrects some minor errors can be found here:

http://deepblue.lib.umich.edu/handle/2027.42/83152

From my discussions with Akinbo it is clear that he thinks that this problem is grounds for rejecting special relativity. I disagree. I think it is not only not grounds for rejecting SR, but also points to an implication of the theory that has still, 110 years after it was introduced, escaped the physics community. In my experience the reason for that is that most refuse to deal with the problem by denying that there is a problem (A time-honored way of solving problems for which one has no answer). For example, I see people asserting that it is "meaningless" or that it "makes no sense" to talk about lightlike frames because no spacetime observer can transform to such frames. But I think this is simply false: You can meaningfully and sensibly talk about such frames by referring to them as frames to which no spacetime observer can transform. Even your response to Akinbo evinces such denialism.

Now, lest I be charged with only criticizing and failing to make any constructive suggestions, I feel obligated to mention what I think is the approach to understanding this. But I am aware that this is your blog post, so I will try to keep this mercifully brief.

I tried to confront this issue head-on in this paper:

http://vixra.org/pdf/1306.0076v2.pdf

I believe the fact that objects associated with null geodesics have only 3 independent components is the key issue in connecting the foundations of special relativity directly to the foundations of quantum theory. In fact, in my current entry, I introduced a novel object, an "incomplete spacetime vector" which, like such objects, has only 3 independent components, but unlike them, it is not constrained to the boundary of a light-cone, but rather "fills" the interior in the manner of a potentiality. I then proved that in the non-relativistic limit, this object reduces to the Feynman path integral provided one aspect of it is identified with a transition probability amplitude. The paper is here:

http://fqxi.org/community/forum/topic/2474

Again, please pardon me for inserting myself into your discussion with Akinbo, I will try to make up for it by leaving a comment that deals exclusively with your essay.

Best wishes,

Armin

Dear Ken,

I have just read your essay and would like to offer the following feedback:

Section 2: Your argument presented here is very similar to part of the argument I gave for why it should not be surprising to see that mathematics is very effective in physics. We seem to both consider the fact that consistency in the mathematics and in our models of reality is essential to be a key ingredient of the explanation.

Section 3: I find your "Cartoon" graph rather nifty. My own bias, though, is that whatever aspect of nature seems counterintuitive merely reflects our lack of "understanding" of that aspect, where I am using the word in Feynman's sense, but generalized to any area of physics (not just QM).

Let me give an example: Consider an experiment in which a helium balloon is attached inside a normal car, say to the floor, but not quite reaching the ceiling, and imagine a group of people were asked which way the balloon would sway once the car starts and accelerates forward. I believe it is reasonable to suppose that most people's intuitions would lead them to predict that the balloon would accelerate towards the back of the vehicle once it accelerates forward. When the car accelerates forward and the balloon is seen to accelerate forward as well, most people would probably experience a violent clash between their intuitions and their observation. However, once they learn that this behavior results because the density of the balloon's interior is less than that of the surrounding air, and that in fact something does accelerate towards the back, namely the mass of air which displaces the balloon, then their intuitions would completely flip. It would no longer seem intuitive to them that the balloon should accelerate toward the back because they would have gained "understanding" about this small aspect of nature.

I believe that every single counter-intuitive aspect of nature is like that, and that being satisfied with a counter-intuitive mathematical description is falling short of seeking that sort of "understanding", which, I had imagined, was the primary reason scientists chose their career path to begin with.

If my belief is correct, then your cartoon is actually just a "slice" of a dynamic cartoon in which what counts as intuitive and what counts as non-intuitive is not static but changes over time (perhaps sometimes drastically so, when there are scientific revolutions), and that your description is more biased toward the earlier phase in which there are still many, many aspects of nature which still seem counter-intuitive to us.

Section 4A:

First, I must admit that I find the question about the "flow of time" somehow not nearly as interesting as some other foundational questions, and so I have not invested all that much thought into it. But even so, I am still skeptical of your claim "But if you try to look for this perfectly-obvious flow of time in our standard descriptions of physical phenomena, you won't find it."

Let me offer the following as a possible counterexample, neglecting gravity (I am not sure that this is really is a counter-example, but I am offering it in the hope that if I am wrong, I will have learned something):

[math]\frac{c}{\gamma}=c \frac{d\tau}{dt}\equiv\frac{ds}{dt}=\lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}[/math]

This seems to map to our sensory perception that an instant just "flowed by" the traversal of an infinitesimal "distance" along the spacetime interval, and as such seems to be a physical description of it. From a block universe point of view of course what matters for most practical purposes is just the worldline, but it is not clear to me that this would somehow falsify or negate the above as a physical description of our perception of time as a flow. If it doesn't, then this still stands as a counterexample. In fact, the absence of a discussion of the viability of a possible parameter tau(t) or s(t) in section 4b. seems rather glaring to me.

Overall, I think your essay was very well-written and raised some food for thought. I also liked the cave analogy. I hope you found my comments useful.

Best wishes,

Armin

Dear Ken,

I am certain that you did not expect to find yourself in a discussion with me because you answered one of Akinbo's questions, and since this is your blog, if at any time you wish to terminate the discussion I will honor your wish. As you have responded to my comments and ended your response with a question, I would like to offer a response, however.

"This is getting perilously similar to a discussion about how many angels could dance on the head of a pin. There's no watch, there's no precise (mathematical!) definition of such a watch, so these questions can't be addressed"

I think there is a fundamental difference between the imaginary construct of a clock moving at c in space, and the imaginary construct of angels dancing on a pin. Proper time is a well-defined quantity in SR, and we know that those proper times which correspond to time-like intervals can always be (in principle) measured by means of a clock in the rest frame. Given that, the construct of a clock moving at c involves a reasonable extrapolation of the theory. In contrast, angels dancing on a pin do not involve a reasonable extrapolation of any accepted physical theory because none of them use angels as a conceptual building block.

I think that all but equating these two constructs, which are dissimilar in an essential way, is the sort of response which deflects from the issue I am raising. However, I am heartened by your question:

"Do you have a good reason why anyone should consider such a transformation in the first place, let alone try to draw any conclusions from it, infinities and all?"

The fact that you are asking me this indicates to me that you are at least willing to consider the issue, and for that I am grateful.

The short answer is: yes, because not doing so is not scientific.

However, in order for the short answer to make sense I will have to give a somewhat longer answer. First, let me clarify that by "considering such a transformation" I do not mean that such a transformation is possible, or that its current description in terms of infinities is false, or anything like that. In fact, I think we agree on all matters of fact pertaining to SR, and even on the immediate inferences that can be drawn from them. For example, I completely agree with your statement:

"If that doesn't include the invariant number-of-cycles, then it's not a frame in which one can discuss our universe." provided you are willing to substitute "spacetime rest frame" for "frame" (I believe you would be ok with that, if not, please correct me).

Where we part ways (or, more generally, where I seem to part ways with most other physicists) is on the question of whether these immediate inferences constitute the end of the line of inquiry. I believe they do not. Let me try to articulate the issue as clearly as I can:

According to SR, objects associated with c in space cannot be associated with rest frames in spacetime, a direct consequence of the speed of light postulate. If one tried nevertheless to imagine what it would be like to associate a hypothetical rest frame with an object described by v=c, then a reasonable extrapolation of the mathematics of SR indicates that all events separated in time would in such a frame be compressed to a point. This implies that an observer in such a hypothetical frame would "observe" the moment of his "coming into existence" to be the exact same as the moment of his "going out of existence" (say emission and absorption of a photon), which implies that an observer in such a frame would observe his own duration of existence in spacetime to be exactly zero. Now, this is NOT a problem yet.

The difficulty appears when one considers how we would use this extrapolation if we did not already know that there are in fact, in the real world, entities associated with v=c. Surely, in that case we would interpret this extrapolation to mean that objects associated with v=c do not exist, and consider any of the relevant reasonable extrapolations as theoretical evidence for that. The problem is that such objects do in fact exist, and their existence seems to run counter to what we would have taken as a prediction of the theory had we not already known of their existence (this is what I labeled as the "existence paradox" where "paradox" is meant in the sense of a puzzle which is counterintuitive but still consistent with the theory).

My charge of denialism is that there is a genuine problem deserving of explanation in SR, in the sense of a discrepancy between an apparently reasonable interpretation of what the theory predicts and what we observe in the real world, which is at present almost universally treated by physicists as if it were a non-problem.

The history of SR itself gives an example of how such denialism can delay progress in science: Surely there were physicists before Einstein who realized that observers in motion relative to each other observe different electromagnetic fields, but it does not seem to have bothered them. As a result, they lost the chance to discover SR.

If it is acknowledged that there is a real problem here (in the sense of, say, the twin paradox, prior to someone finding a solution for it), then one can undertake the usual course of scientific inquiry:

1. Attempt to formulate a question that leads to a hypothesis which can be investigated

2. Formulate a scientific hypothesis

3. Investigate whether the hypothesis is refuted or supported by the available evidence.

Denying that there is a problem blocks this scientific process. I hope my short answer makes sense now.

There remains one issue I need to address, namely whether this particular problem is really the sort of problem that lends itself to the process I just outlined. The only way I can answer this is by giving the results of my own inquiry as a sort of "existence proof" that this can be done. My aim is not to convince you that the explanations I have arrived at are the right ones, it is merely to convince you that there is a real problem in SR that is universally ignored.

1) The question I formulated was: Is it possible to associate a rest frame (which obviously cannot be a spacetime rest frame) with objects characterized by v=c in such a way that it is consistent with all relevant reasonable extrapolations of SR? (finding such a frame would then allow us to "understand" why the existence paradox is not a real paradox)

2) I formulated as a hypothesis that such frames are associated with objects which exist in a 2+1 dimensional analog of spacetime such that there exists no function which maps their position in this lower-dimensional analog to a position in space (the existence of a such a function would contradict both SR and QM: SR because it implies that one can associate a position vector in space, and hence a spacetime rest frame with such objects, and QM because one could use such a map to construct a sequence of position vectors i.e. a sequence of "unmeasured" positions).

3. The evidence I have gathered so far could optimistically be considered as corroborative and pessimistically as merely consistent with the hypothesis. I will give two arguments based on SR(there are more, but this post is already unreasonably lengthy, please pardon).

a. The fact that in such frames, by another reasonable extrapolation of the mathematics of SR, the spacelike basis vector in the direction of motion and the timelike basis vector both converge to a lightlike vector and therefore become parallel indicates that in such frames spacetime is a linearly dependent vector space, which in turn implies that the dimensionality associated with such a frame is lower than the dimensionality of spacetime. A potential problem with this argument is that since lightlike vectors have zero magnitude, they can be considered both parallel and orthogonal, but I believe that as long the orthogonality does not negate their also being parallel, and the implication that follows (If am mistaken on this, I would appreciate a correction).

b. If one takes the "missing dimension" to correspond to the direction of motion in space, and takes

[math]\beta^2 + \frac{1}{\gamma^2}=1[/math]

as an axiom, then this straightforwardly implies the invariance of the speed of light: From the fact that the object is intrinsically lower-dimensional such that it lacks extent in the direction of motion follows that

[math]L_0=\frac{L}{\gamma}=0[/math]

as observed in every spacetime frame, which implies

[math]\frac{1}{\gamma}=0[/math]

in every spacetime frame, which implies

[math]\beta=1[/math]

in every spacetime frame.

Again, please pardon the excessive length of this post, I think I have pretty much said everything I wanted to say on this matter, so I'd expect further posts, if there will be any, to be substantially shorter.

I would appreciate finding out whether this extended argument had any effect on your views on this issue or not.

Best wishes,

Armin

A quick addendum regarding my purported counterexample:

Of course the coordinate time refers to the time passing for a moving observer, but this does not mean that this equation is not applicable, it only means that the situation you described in your essay pertaining to the flow of time describes the special case ds/dtau=c. Unfortunately, c has a strong connotation as speed in space, perhaps the convention ds=ic dtau is a better choice for this purpose.

Best,

Armin