Mathematics: Intuition\'s Consistency Check by Ken Wharton

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

Dear Armin,

I too would like to see a response from people familiar with this matter. I wonder whether you would be willing to pose your question in an FQXi blog accessible to a larger audience (here, not too many people will find it - no disrespect intended).

Now that I saw your writing, I plan to make time to read your essay.

You are right, any inconsistencies in accepted theories must be investigated (if uncovered by qualified people). Even if answers aren't now available (let's suppose), such inconsistencies need to be looked at again regularly and not just "papered over." Obviously, no one is willing to scrap SR over this particular thing (if unresolved), but it may well lead to new ideas (and solutions).

Just for my own curiosity, what is the evidence that such objects exist (v=c), and what is the duration of their existence, and how is that measured?

Thanks.

En

Dear Ken,

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".

Best wishes,

Cristi Stoica

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 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.

Best wishes,

Laurence Hitterdale

Dear Ken,

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?

Best regards,

Alexey Burov.

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 (http://fqxi.org/community/forum/topic/2345) 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.

Jim

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

Best regards

Janko Kokosar

Dear Ken,

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.

Warm regards,

Alma