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RECENT POSTS IN THIS TOPIC

**Sylvain Poirier**: *on* 6/9/15 at 14:44pm UTC, wrote I intended to comment this essay earlier, then I forgot. As it would mainly...

**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... ...

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FQXi FORUM

May 26, 2022

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Mathematics: Intuition's Consistency Check by Ken Wharton [refresh]

TOPIC: Mathematics: Intuition's Consistency Check by Ken Wharton [refresh]

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.

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.

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Your point has been well argued in the essay:

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.

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Dear Ken,

I really enjoyed your thought provoking and solidly argued essay.

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

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

Regards,

Ed Unverricht

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I really enjoyed your thought provoking and solidly argued essay.

I agree with your conclusion "

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

Regards,

Ed Unverricht

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Dear Ken,

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

My comments here are solely for further consideration. I agree with Jose Koshy’s comment (darn, now I have to look at Jose’s essay) that perhaps physicists’ intuition...

view entire post

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...

view entire post

report post as inappropriate

Dear Jose,

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! ;-)

Best,

Ken

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! ;-)

Best,

Ken

Dear Ken Wharton,

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.

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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.

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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!

Cheers,

Ken

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!

Cheers,

Ken

Dear Ken,

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

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

view entire post

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...

view entire post

report post as inappropriate

Dear Ken,

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

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

Thank you,

Leo KoGuan

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

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

Thank you,

Leo KoGuan

report post as inappropriate

Hi Ken,

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

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

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

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

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Hi Matt,

Interesting point! I guess I should have read your essay before submitting my own... I'll post in your topic later in the week, after I'd had a chance to read yours more carefully, but here's my first take:

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'...

view entire post

Interesting point! I guess I should have read your essay before submitting my own... I'll post in your topic later in the week, after I'd had a chance to read yours more carefully, but here's my first take:

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'...

view entire post

Dear Ken,

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

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

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

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

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

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

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

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

Best regards,

Akinbo

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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:

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,

4. You said and I agree that,

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

Best regards,

Akinbo

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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.)

Best,

Ken

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.)

Best,

Ken

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...

view entire post

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...

view entire post

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Dear Armin,

> 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...

view entire post

> 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...

view entire post

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

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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

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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!

Best,

Ken

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!

Best,

Ken

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...

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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...

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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

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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

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Hi Armin,

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.

Best,

Ken

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.

Best,

Ken

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

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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

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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...

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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...

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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

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

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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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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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.

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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.

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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.

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

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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

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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

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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

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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

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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,

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

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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

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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

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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...

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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...

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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

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...

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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...

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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...

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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...

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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

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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

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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

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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

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