Peter Jackson: on 4/24/15 at 15:44pm UTC, wrote Ken, Thanks, I now see the subtlety of your point which was on...
Thomas Ray: on 4/23/15 at 1:40am UTC, wrote Hi Ken, I think the titles of our essays say the same thing in different...
Ken Wharton: on 4/22/15 at 22:42pm UTC, wrote Dear Peter, Thanks for your careful reading, and interesting comments... ...
Peter Jackson: on 4/22/15 at 13:38pm UTC, wrote Ken, Interesting essay, the basics well argued but it left me unconvinced...
Ken Wharton: on 4/21/15 at 21:53pm UTC, wrote Dear Marc, Thanks for your very nice words! On your question, I've been...
Marc Séguin: on 4/21/15 at 2:15am UTC, wrote Dear Ken, Thank you for an entertaining and thought-provoking essay. In my...
James Hoover: on 4/17/15 at 16:54pm UTC, wrote Ken, I am revisiting essays I’ve read to assure I’ve rated them. I...
TOPIC: Mathematics: Intuition's Consistency Check by Ken Wharton [refresh]
Author Ken Wharton wrote on Mar. 13, 2015 @ 21:07 GMT
There is a well-noted overlap between mathematics and physics, and in many cases the relevant mathematics was developed without any thought of the eventual physical application. This essay argues that this is not a coincidental mystery, but naturally follows from 1) a self-consistency requirement for physical models, and 2) physical intuitions that guide us in the wrong directions, slowing the development of physical models more so than the related mathematics. A detailed example (concerning the flow of time in physical theories) demonstrates key parts of this argument.
Ken Wharton is a professor in the Department of Physics and Astronomy at San Jose State University. His research is in Quantum Foundations.
Jose P. Koshy wrote on Mar. 14, 2015 @ 10:37 GMT
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.
Ed Unverricht wrote on Mar. 15, 2015 @ 06:52 GMT
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.
Ed Unverricht
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...
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.
Thanks for your interest. I didn't mean to completely dismiss intuition -- it's crucial for both physicists and mathematicians, even if often misleading. I was mainly targeting innate intuitions that have been selected for by evolution, which have not proved to be a particular benefit to fundamental physics. Certainly you might concede that *some* of our innate intuitions might have occasionally led us astray...?
And yes, experimental checks are a crucial part of science, assuming that's what you mean by "physical consistency". That goes for whether an idea is "intuitive" or not! ;-)
Jose P. Koshy replied on Mar. 18, 2015 @ 05:03 GMT
I agree with you that some of our innate intuitions have led us astray in the past. In my opinion, non-intuitive ideas like QM and GR also have led us astray.
You say, "And yes, experimental checks are a crucial part of science, assuming that's what you mean by "physical consistency". By physical consistency, I meant checking whether it conforms to 'intuition'. Experiments can confirm proposed mathematical relations and also our intuition based ideas. However, an 'experimentally verified mathematical relation' cannot give a clear physical picture, because we can have different physical interpretations based on 'that mathematical relation'. My argument is that out of the different interpretations, it is possible to select one that conforms to intuition. But from the time of Newton, the 'simplest mathematical structure' based on the given mathematical relation, is being assumed as the real physical structure.
Dear En Passant,
Good points; you might see my agreement about the importance of intuition in my post above. I suppose in your future scenario it would become even more important! (Is there a science fiction story along these lines? Sounds like a promising read.)
Concerning your interesting example of "physics without mathematics", using a physical system to basically do the math for you. To me that still seems like a mathematical tool, in that you could only get out useful info to the extent that you could map some of the physical processes onto mathematical concepts. I suppose it could be a partial-black-box technology, in that the calculations might not be do-able any other way, but I would still classify it as "physics using mathematics", even so. After all, calculus was essentially a black-box technology for hundreds of years before they really figured out exactly why it worked. Still, thanks for an interesting point that I'll need to think and read about more!
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...
There were many attempts to formulate axioms in physics (D. Hilbert, J. von Neumann, L. Nordheim, H. Weyl, E. Schrödinger, 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 × R, H2 × 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, .
I'm far more willing to accept your point on the "overly complicated" axioms than the "boring" ones. I would imagine that the reason certain axioms lead to 'boring' results is that they're isomorphic to some other well-understood bit of mathematics. (Or are you saying they're so boring they don't even count as mathematics at all? Regardless, I think you'll agree that the more interesting cases lie on the complicated side.)
I don't have much experience with axioms that might lead to something so "complicated" that mathematicians generally ignore them, but I'm eager to read more about some of the examples you mention in your own essay. I guess the question I have right now is this: are these complicated systems considered uninteresting math, that's generally avoided except by perhaps a very few people, or are they not considered math at all?
My one tiny bit of experience on this front has come about after finding quaternions so useful for spin-1/2... I've now read a bit about octonions and a tiny, tiny bit about what happens if you try to extend such algebras to 16-element systems. From what I can gather, the 16-case gets so convoluted that it's arguably not even an algebra at all, but a few people have tried to develop such systems all the same. If you're basically arguing that the ugly features that have steered mathematicians away from 16-element "not-quite-algebras" make it unlikely that they will ever find physical applications... Well, I think I agree with you!
However, (again, from the tiny bit I've read), one reason these 16-element systems are uninteresting is that they essentially contradict themselves for certain multiplications, manifesting as discontinuities. In this case, "overly complicated" might simply boil down to "inconsistent at certain points", which would then be covered by my essay; the key feature I stress about math is its internal consistency. Still, I presume that your examples probably are more than just mere inconsistency, so stay tuned for a better response!
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?
Hi Akimbo,
Thanks for the interesting questions.
I'm not exactly sure what to make of your first few questions... But if the universe came to an end, so that the block wasn't infinite in time, the finite block would still be said to "exist", regardless. (I'm using "exist" in a timeless sense, which is the only sensible sense to refer to the block universe.) The year 1984 "exists" in the block, even though we now consider it the past. (And so does the year 2084, even though we now consider it the future!)
On #3, I think you might find my essay's dissection of the (terrible) "time is a river" analogy of some use. If you're imagining things "appearing" and "disappearing", then you're imagining ordinary time running in the background, when it sounds to me like you perhaps don't want to imagine any dimensions at all.
On #4, one shouldn't confuse the fact that time doesn't "flow" with the fact that time *does* elapse on clocks. The block-universe representation of a clock shows a cyclic structure extending in the time-direction. If you think about it, that's exactly what the block-universe representation of an electromagnetic wave looks like; a cyclic structure extending in the time direction (and also a spatial direction). 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 suppose if one insists on thinking of a photon as a little person holding a stopwatch, and that person can somehow travel at lightspeed, then sure: that stopwatch won't budge. 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.)
"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).
http://fqxi.org/community/forum/topic/329
But an updated version which corrects some minor errors can be found here:
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:
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:
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.
> 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.
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...
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... at least not in any widely-agreed-upon framework.
I'm not certain what you mean by a "time-like frame", although I imagine you mean the perspective of an agent on a time-like worldline. But since all agents are on time-like worldlines, why the caveat? (I'm certainly *not* sure what you mean by a "null frame"; more on this below.) I suppose I'd be willing to consider "space-like" frames, as viewed from the perspective a tachyonic agent, but only in terms of tachyonic clocks, not to be confused with ordinary clocks. And it seems to me that tachyonic clocks would measure lengths, not time, although I haven't thought about such things too carefully.
But that's entirely beside your point; you would like to consider these mysterious "null frames". Now, in the case of EM waves, the number of cycles between emission and absorption is not only greater than zero, it's exactly the *same* number of cycles in every frame that describes the same events. If you tell me there is some other "frame" where this number is different, I'd respond that it's therefore not a frame at all. The whole point of different reference frames is to discuss the same events from different perspectives. If that doesn't include the invariant number-of-cycles, then it's not a frame in which one can discuss our universe.
Note, I'm not saying we should ignore such frames simply because it's not physically possible to transform into them. Even mathematically, boosting into a frame moving at exactly c encounters so many singularities and infinities that it can no longer be said to describe the events in our universe. (One loses the cycle count for EM waves, for example.) It's certainly not a reversible transformation. 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?
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...
"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
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
as observed in every spacetime frame, which implies
in every spacetime frame, which implies
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.
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".
Dear Noson,
Very interesting point... I'm somewhat surprised to hear that about general mathematicians! Still, I had hoped it was clear I was using "axiom" in a broader sense: any new entry point into new mathematics, even if built upon other mathematics, is effectively a new set of axioms (at least for the purposes of my essay).
The analogy I used in the essay was another set of caves, "deeper in" from other caves. The "axioms" in this case wouldn't be fundamental, but would still be rules used for deductive cave-exploration. Does that help?
Your own essay sounds quite interesting; I'll try to get to it soon!
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...
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):
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.
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.
I'm in full agreement with you about an evolving intuition; my main target was innate intuitions, but these of course can be corrected and changed (hopefully in the right direction!). Some intuitions, though, are harder to change than others.
As far as your 'flow of time' example goes, you're describing motion through space: a flow of a particle, not a flow of time. In the terminology of my essay, setting T=t(time), t=\tau (proper time) would fall into category A); two time-parameters that have a well-defined relationship can't be used to describe one objectively changing with respect to the other. In fact, that was the very foil I had in mind when I wrote that bit of the essay, even though I didn't mention \tau(t) explicitly.
If you think about it, your equation doesn't describe anything objective; every single term on both sides are agent-dependent. Furthermore, as you noted in your follow-up, the time t is measured by an agent at rest with respect to the moving worldline in question. As soon as you fixed this problem, both of your t's became identical, and the statement became a meaningless tautology.
One last thought: If time flows, it flows in a particular direction. To get a flow you therefore need to break time-symmetry, and you won't find that in SR.
Very beautiful essay. I think your cartoon of idea-space is very suggestive, and should help us when we want to trade mathematical consistency for intuitiveness. Here seems to be a complementarity between consistency and intuitiveness, pretty much like Bohr's complementarity between truth and clarity. I like your example of an idea that seems to be supported by our intuition and couldn't become mathematically consistent, as well as the closing statement "The future of physics may lie in a counter-intuitive direction, but at least we know it will be framed in the language of mathematics".
Author Ken Wharton replied on Apr. 13, 2015 @ 16:10 GMT
Thanks for the kind words, Cristi... Although I wouldn't say there's a general tradeoff between consistency and (innate) intuitiveness; in *general* I would say they're unrelated, or if anything, perhaps even tend to go together. True, since the intuitive ideas tend to get explored first, the "promising frontier" for physics has been in the non-intuitive direction for some decades, now. But that's just because the intuitive and consistent ideas have already been explored, not because there aren't any intuitive ideas that are also consistent.
Dear Professor Wharton:
Perhaps it would be useful to distinguish between (a) a phenomenon which is not representable by a consistent mathematical structure and (b) a supposed phenomenon which is inconsistent or incoherent and therefore impossible. In particular, this distinction might be helpful for discussing what seems to be the difference between time and space. Granted, what seems to...
Perhaps it would be useful to distinguish between (a) a phenomenon which is not representable by a consistent mathematical structure and (b) a supposed phenomenon which is inconsistent or incoherent and therefore impossible. In particular, this distinction might be helpful for discussing what seems to be the difference between time and space. Granted, what seems to be special or unspacelike about time cannot be represented mathematically. Granted, furthermore, the standard metaphors for the distinctive feature or features of time are in some ways more misleading than illuminating. I can agree with your critique of the images of flow and passage. Nonetheless, I would contend that the lack of a mathematical representation should not be taken as leading to the conclusion that the supposed distinctive characteristics of time are non-existent. The warranted conclusion is, I think, somewhat more complicated: either those characteristics do not exist, or they exist in the physical world without being mathematically representable, or they exist only subjectively, that is, only in experience. On the last alternative, it remains true that the distinctive features of time are real, even though they are not part of physical reality. In that case, any attempt at a complete account of the nature of things would still be under the obligation of trying to explain those features.
At this point I think another distinction might be helpful. This is the distinction between intuition and experience. An intuition, or intuitive belief, is something that we are inclined to believe. A familiar example is the belief that, if we drop a heavy cannon ball and a much lighter pebble from the top of the leaning tower of Pisa, the cannon ball will reach the ground first. But an experience is something different. Here is an example: “The Moving Finger writes; and, having writ, / Moves on: nor all thy Piety nor Wit / Shall lure it back to cancel half a Line, / Nor all thy Tears wash out a Word of it.” This verse does not state an intuitive belief. It tries to describe exceptionless features of experience. It is hard to know what people might believe, intuitively or otherwise, about time travel, the irrevocability of the past, and similar topics. But time is an experienced part of reality, and it is experienced as something very different from a dimension of space. I do not know how the experience arises. Maybe it derives from other entities and forces which in themselves lack distinctive temporality. Obviously, intuitive beliefs can be overruled. They often have been and often should be overruled. But experience is something other than beliefs, and therefore experience has to be treated differently.
You give 3 options; the first and last I'm okay with. But not "they exist in the physical world without being mathematically representable". If there's no consistent mathematical framework in which they can be discussed, then the very notion is inherently inconsistent and won't have a physical counterpart. (IMHO...)
As for the last option, that our perception of time is real as far as *experience* is concerned, that's fine, but that makes this issue a consciousness-problem, not a physics problem. I wish physicists would recognize this and leave it alone. (Unless, I suppose, they're also going to be building useable models of consiciousness... but that's still not physics.) Looking to "other entities and forces" to explain one aspect of our conscious experience seems to be like a huge mistake, mixing lower-level and higher-level concepts in a way that seems wholly and utterly implausible -- especially because those "other entities" don't seem to show up at any of the intermediate levels between physics and consciousness (mesoscopic physics, chemistry, biology, neural networks, etc.).
I intended to comment this essay earlier, then I forgot. As it would mainly repeat other arguments already here, I only added things to my review page. Instead I will just react here to the above discussion.
You wrote "If there's no consistent mathematical framework in which they can be discussed, then the very notion is inherently inconsistent and won't have a physical counterpart". What about applying this logic to the wavefunction collapse, that is the thing I see to show up at an intermediate level between physics and consciousness (thus contradicting your claim that no such a thing shows up): we have no properly coherent mathematical framework to specify how it may happen. To deny it as physically real, would mean to adopt the many-worlds interpretation, wouldn't it ? But if it is real, and if being real would require having a consistent mathematical framework, does it means such a framework needs to be someday discovered ?
Apart from this, as I explained in my essay, I do not consider the characters of consciousness as more "higher level" than the physics concepts, as I see consciousness as part of the foundation of physical reality.
Alexey/Lev Burov wrote on Apr. 12, 2015 @ 21:56 GMT
Pondering on your conclusion
"So while physics and math do have a striking degree of overlap, this is hardly some cosmic coincidence. The necessity for consistency in physical models, along with some mistaken human intuitions, can mostly explain the largest questions"
I asked myself: writing this, did you keep in mind that "the laws of nature are described by beautiful equations", as Wigner's brother-in-law put it? If yes, how might this explanation look like? If not, wouldn't the key part in the physics-mathematics relation be lost?
Dear Alexey,
I agree with your sentiment, but unfortunately "beauty" is a bit too subjective of a premise to start with when looking for an objective explanation. Even "simplicity" and "elegance" have subjective aspects, but maybe "efficient" is the right starting point. We humans (or at least some of us) find it beautiful when a very wide range of phenomena can be explained with a few efficient concepts. Asking why this is in fact the case is an excellent question, but I wouldn't say it's the key part of the mystery.
I say this because even if I provided a good explanation for why there are a few rules that explain everything, that would really only apply to the most fundamental physics from which everything else emerges. Such an explanation wouldn't cover higher-level, effective- or emergent- physics, for which mathematics is certainly still important, and this mysterious overlap between math and physics continues. I would say that the most use of higher-level math actually takes place at this higher level, where any "ultimate efficiency" arguments don't really apply.
Furthermore, I'm convinced we haven't gotten down to the truly fundamental level yet, in any of our theories except maybe perhaps GR. So at this point, I see pretty much all of physics as a higher-level approximation, and speculating about the efficiencies of the fundamental level that may be waiting for a discovery is just... well... speculation! :-) Although I am convinced, as are most physicists, that any ultimate explanation will indeed be efficient.
Ken,
Math is developed w/o any thought of the physical application due to a 1) self-consistency requirement for physical models and 2) misguided physical intuitions.
How is the non-logical axiom with a role in theory-specific assumptions handled for 1 and 2?
A lot of thought-provoking concepts in your essay.
My essay () only sets out to show connections of mind, math and physics with the stellar achievement leading to quantum biology, the LHC, and DNA.
Thanks for sharing your ideas.
Dear Ken Wharton
You have very creative approach to this topic.
I wrote in my essay that I do not believe in absolutely wrong intuition of people. I thought mostly specifically that quanum randomness has some explanation background as opposition to positivists (Roger Schlafly in this contest). I claim that quantum randomness and free will (in panpsychism) are the same things. This is physical intuition and some type of mathematical intuition. (I also claim that interpretation of QM must exist.) Are you positivist and you disagree to these two my claims?
At the other side I admitted that our intuition is connected only Newtonian physics, but also with math and logic which we learn from childhood.
According to your intuition about time flow: I claim that time is one the most fundamental notions, connected also with panpyscism. (This is claimed also by Sylvain Poirer on this contest). Thus space in space time is only another form of time. I claim that panpsychism is only in absolutely primitive form, thus feeling of flow of time and physical time, as we know it from equation do not mean essential difference.
I also claim that speed of time is dependent of dimensionless masses of particles (G and hbar are mostly things of agreement of measurement units. Mostly, not absolutely ...) Thus, time flow is coupled with mass. Thus, spacetime without rest matter does not exist.
In this contest, you can find also essays which claims that time does not exist ...
My essay
Alma Ionescu wrote on Apr. 16, 2015 @ 13:56 GMT
Thank you for a very nice read! I enjoyed the classical view and style and the witty sense of humor that is displayed throughout the essay.
You are making perhaps the most balanced analysis of how the physical universe looks from an intuitive point of view and the counterintuitive realities that it hides. The two-time trap is an exquisite dissection of the insurmountable problems that one will stumble upon when trying to explain the flow of time, which is the most intuitive concept in the world and seemingly indisputably ingrained at the very core of the perception of existence. In the light of this exposition, your conclusion is indeed indisputable, that whereas we are particularly attached to our intuition, the laws are ultimately decided, rigged as you so charmingly say, without any regard for our opinions.
Wish you best of luck in the contest! Should you have time to read my essay, your comments are more than welcome.
I am revisiting essays I’ve read to assure I’ve rated them. I find that I rated yours on 4/14, rating it as one I could immediately relate to. I hope you get a chance to look at mine: .
Member Marc Séguin wrote on Apr. 21, 2015 @ 02:15 GMT
Thank you for an entertaining and thought-provoking essay. In my opinion, among those submitted by FQXi members, yours is the one that presented the most interesting arguments relating mathematics to physics.
I really liked your cave analogy, especially the physicists being "chased out of caves by the monsters of experimental falsification". Your analysis of the role of...
I really liked your cave analogy, especially the physicists being "chased out of caves by the monsters of experimental falsification". Your analysis of the role of intuition in physics is very interesting, as is your "cartoon" of idea-space. I am intrigued by your statement that "mathematicians are simply better explorers of new and strange ideas" than physicists, because they are less unencumbered by physical intuitions. I wonder what you think of philosophers. Since they do not (usually) even bother to translate their theories into mathematical language, does this make them even "less unencumbered" and more likely to be the first to explore some ideas that could ultimately become essential aspects of "proper" physics? Or are they so unencumbered that they are likely to be stuck in wild geese chases? (I am thinking of philosophers like Robert Nozick or David Lewis (or "amateur philosopher"-scientists like Rudy Rucker and Hans Moravec) that have entertained on more or less purely logical grounds the idea that the whole of physical reality could be a subset (or even equivalent to) the whole of abstract mathematical reality.)
I found your discussion of the flow of time very interesting, and I agree with you that to describe the flow of time on purely mathematical grounds one would need a two step system. I think that it should be possible to use a mathematical structure as a "clock" to define another mathematical structure that has the correct properties to describe a physical universe where time flows. If we ever succeed in describing the flow of time within a self-contained explanatory scheme, I think it will have to be something like that.
In any case, I agree with you that the future of physics should lead us deeper into mathematics.
All the best, and good luck in the contest!
Thanks for your very nice words!
On your question, I've been quite impressed by the philosophers of science who I interact with professionally. (In fact, these days I seem to be interacting with more philosophers than physicists!). The top of that field seems to be filled with people who know their mathematics far better than I do, and even when some of their arguments are not couched in mathematics, they are often couched in formal logic.
Now, of course this isn't always true. Some of the unnamed people to whom I directed my flow-of-time rant are philosophers who hold that there is some objective flow of time (the A-theory of time, the philosophers call it, as opposed to the Block B-theory). These arguments are not couched in mathematics or careful logic, I would claim. Your examples may apply in this category as well. But just because someone is doing philosophy, it does not *necessarily* mean that they are not using mathematics; there's a lot of really good philosophy of science for which mathematics is an essential tool.
Best, Ken
"The only caves that physicists find worth entering are the structurally-sound ones that have been mapped out by the mathematicians."
This infers that that no worthwhile hypothesis can be conjectured unless the maths pre exist it. More worrying it seems to imply there's...
This infers that that no worthwhile hypothesis can be conjectured unless the maths pre exist it. More worrying it seems to imply there's no room for a physical mechanistic hypothesis to be developed ('intuitive' or not) which can then be used to derive the mathematical description. I couldn't believe you really meant that, but then you wrote;
"Physics without mathematics would be a surefire route to failure." Of course maths is one requisite, but is it a 'PRE'requisite as your hypothesis suggests?
You then again seem to leave the solid ground and suggest it's more likely; "math is just less biased than physics, more willing to explore new ideas." Which I must admit is counter to most of my experience. Do you suggest that's my intuition being wrong?
Nowhere do you identify that apparently self consistent maths can contain flaws and mislead us or not fully correspond to the physical processes for which it's invoked. I identify some important cases of this in my own essay, one relating to QM apparently not previously recognized. I hope you may review and comment.
However I agree the QM case (rarely I find) IS consistent with your 'maths first' hypothesis, where indeed the very possibility of any 'classical' mechanism has been rejected by most.
I also agree time doesn't flow and has no 'direction'. It seems a horrendous 'category error' to endow it with material attributes. But does that example prove that intellectual rationalisation, deduction and induction (intuition?) are near valueless any more than fallacious mathematical proofs show that maths is useless?
Of course I agree your basis premise that maths is an (one) essential tool and 'consistency check' but is it the only one? must it pre-exist, and is it guaranteed always correct? As Alma points out in her excellent essay, in mathematics any inconsistency would be shot dead on sight. In physics this often isn't the case and inconsistencies become acceptable by familiarity! I think that may be the greatest danger of the assumptions you hypothesize, perhaps responsible for ever deeper theoretical entrenchment.
> This infers that that no worthwhile hypothesis can be conjectured unless the maths pre exist it.
I certainly didn't mean to imply this; I do think it's generally true, but not necessarily true. Hopefully I made it clear eventually that one might develop new mathematics to handle new hypotheses. So...
I certainly didn't mean to imply this; I do think it's generally true, but not necessarily true. Hopefully I made it clear eventually that one might develop new mathematics to handle new hypotheses. So no, it's not an *absolute* pre-requisite.
> "math is just less biased than physics, more willing to explore new ideas." Which I must admit is counter to most of my experience.
It's not counter to mine, but then again I might be overly biased, as I've been finding it hard to push unusual physics ideas. I would say that when it comes to dramatic changes, leaping into some quite different framework, math is more fearless than physics, for the reasons I outline in the essay.
>Nowhere do you identify that apparently self consistent maths can contain flaws and mislead us or not fully correspond to the physical processes for which it's invoked.
I guess I didn't dwell on the fact that most mathematically-consistent physics ideas don't pan out, but that's absolutely true. (It's not the math that misleads us; it's the incorrect physics!)
> I also agree time doesn't flow and has no 'direction'. It seems a horrendous 'category error' to endow it with material attributes. But does that example prove that intellectual rationalisation, deduction and induction (intuition?) are near valueless any more than fallacious mathematical proofs show that maths is useless?
Just because our innate intuitions are near-valueless when it comes to deep theoretical physics doesn't mean that we can't build up new intuitions that *are* useful. What the example shows is that math can steer us away from making serious mistakes, even if our innate intuitions insist on making them. (In the same way, math can tell us which are the fallacious proofs, even if several people make the same mistake.)
> Of course I agree your basis premise that maths is an (one) essential tool and 'consistency check' but is it the only one?
Certainly not! More important by far is experimental verification.
> As Alma points out in her excellent essay, in mathematics any inconsistency would be shot dead on sight. In physics this often isn't the case and inconsistencies become acceptable by familiarity! I think that may be the greatest danger of the assumptions you hypothesize, perhaps responsible for ever deeper theoretical entrenchment.
Very interesting point... And I worry greatly about people accepting inconsistencies due to familiarity, esp. in quantum theory. Maybe physics should strive to be a lot more like math in this regard, along with being more willing to explore a wider parameter space of ideas.
Thanks, I now see the subtlety of your point which was on 'acceptability' of new concepts rather than the plethora there actually is. In that case I agree entirely. The truth is certainly out there, but, the acceptablity of ANYTHING outside the old inconsistent doctrines seems inversely proportional to the number, an innundation! of ideas, and laziness in assessment. There's simply no system in place to prevent ever deeper theoretical entrenchment.
I'm entirely convinced, as are a growing number, that the 'discrete field' model I describe is an immensely powerful advancement of understanding. It's now been 'out there' for some years, is unfalsified and the only criticism it's received is that it's 'different' to the flawed current model. Bless 'em!
But if you, a respected professor, can't get your theory noticed what chance me!? I've been 2nd in the community scoring before and had 4 top 10 hits, and no mention in the judges, so I'm sure you also have more chance than me here. I have had 2 papers in minor journals and hhad expected a few professors to notice (the 'discrete field' model) and join in to help, but everyone seemingly has their own pet theory so it's every man for himself!
Ces't la vie. Did you see the redshift video (90 mins blue shifted into 9mins)