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.

      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.

      Hi Laurence,

      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.).

      Best,

      Ken

      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 (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

      Ken,

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

      Jim

      Dear Ken,

      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 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!

      Marc

        Dear Marc,

        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

        Ken,

        Interesting essay, the basics well argued but it left me unconvinced over a few questions. You write;

        "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 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.

        Thanks for the valuable opportunity to study and better consider those seemingly increasingly common views by expertly expressing them.

        Sincerely

        Peter

          Dear Peter,

          Thanks for your careful reading, and interesting comments...

          > 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 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 again,

          Ken

          Hi Ken,

          I think the titles of our essays say the same thing in different words.

          And in your final comments, you say " ... mathematicians can be fearless explorers without being viewed as heterodox; physicists tend to wait for experimental reasons to venture in non-intuitive directions."

          This is another thing I said in different words, to David Hestenes, whom so many of us hold in high esteem -- that while I much appreciate the mathematical contributions of Hestenes the physicist, I think it is too difficult for most research mathematicians to surrender the freedom of rational idealism that pure mathematics affords us. We want to prove theorems without any thought of physical applications, or even if we're doing anything useful.

          That's why I think both you and Hestenes have done such a good job of demarcating mathematics and physics, with the purpose of showing how they independently correspond. Going too far either way, as you say and imply, can betray both our intuitions and even perhaps our sanity.

          Highest mark, and I hope you get a chance to drop by my forum.

          Best,

          Tom

          Ken,

          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)

          9 min video glimpse of the holy grail. I'd greatly value your comments.

          Best wishes

          Peter

          a month later

          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.

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