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RECENT POSTS IN THIS TOPIC

Peter Jackson: on 4/17/15 at 16:17pm UTC, wrote Rob, Now I'm down to speed-reading a short essay is often merciful. Yours...

James Putnam: on 4/1/15 at 22:51pm UTC, wrote I am informed that I already rated your essay. I did remember that I read...

James Putnam: on 4/1/15 at 22:43pm UTC, wrote Dear Robert McEachern, Great Paragraph! "So where should one start, in...

Joe Fisher: on 4/1/15 at 18:26pm UTC, wrote Dear Dr. McEachern, I thought that your engrossing essay was exceptionally...

Anonymous: on 3/16/15 at 0:44am UTC, wrote Robert H. McEachern, It has been rewarding reading your essay entries and...

Robert McEachern: on 3/5/15 at 16:16pm UTC, wrote Eckard, I think the title of your essay hits the nail on the head. It is...

Eckard Blumschein: on 3/5/15 at 6:31am UTC, wrote Dear Rob, You wrote "Natural Philosophy (Science) had stagnated for 2000...

Robert McEachern: on 2/26/15 at 16:03pm UTC, wrote Roger, Don't confuse the truth of the "starting points" or axioms, with...


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FQXi FORUM
December 13, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: Demystifying the Connection Between Physics and Mathematics by Robert H McEachern [refresh]
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Author Robert H McEachern wrote on Jan. 21, 2015 @ 21:13 GMT
Essay Abstract

The fundamental distinction between Mathematics and Physics, arises from the distinctly different nature of their “starting points”. The usefulness of mathematical methods, for describing observable behaviors, depends upon the complexity of those behaviors, which in turn depends upon their information content.

Author Bio

Robert H. McEachern was educated as an AstroPhysicist. He then worked for several years as a Geophysicist, during which time he became interested in signal processing theory. He then spent the rest of his career developing signal processing algorithms for application to communications systems, and sensor systems. He is now retired.

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Demond Adams wrote on Jan. 22, 2015 @ 20:58 GMT
Robert,

You state an excellent argument. Einstein did not appreciate the abstract nature of Mathematics until he needed it to advance GR. Mathematics does not conform to the strict confines of reality like physics, mathematicians are allowed the freedom to test the unbounded limit of their imagination within the logic of numbers. This is similar to an artist (Jackson Pollock) painting abstract images, or a Jazz musician violating fundamental music theory. Nevertheless, it is the job of physicists to translate these abstractions into the fundamental laws of realism.

Best Regards,

D.C.Adams

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Author Robert H McEachern replied on Jan. 24, 2015 @ 15:15 GMT
I agree that physicists translate mathematical abstractions into physical laws. But as in any language, some translations are better than others. One measure of "better" is how well the supposed laws fit the data. But another, more subtle one is related to the fact that language symbols (either math or non-math) do not convey their own meanings. Words are treated like indices, and their meaning is look-up from the receiver's memory. Hence, if a translator employs symbols that are not already known, a priori, to the receiver, the translation is not going to be very effective.

In physics, particularly quantum physics, this has resulted in the "interpretation" problem. The laws seem to fit the data very well. But what do the laws mean?

Physicists have long suffered from the delusion that the meaning should reside entirely within the laws; hence, studying the laws ought to reveal their meaning. But it does not reside there, anymore than the meaning of a Jackson Pollock painting resides entirely within the painting. Ultimately, that is why complex observers seem to behave different from the entities they observe; they assign meanings to things, like, paintings and physical laws, which have no intrinsic meanings.

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Demond Adams replied on Jan. 24, 2015 @ 20:11 GMT
"What does it all mean?" Einstein asked..."It means nothing," Chaplin replied.

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Edwin Eugene Klingman wrote on Jan. 22, 2015 @ 21:53 GMT
Dear Rob,

Your essay, as I've come to expect, is excellent. You do so much, so well, in three pages that I only wish you had gone on for nine!

As a specific example, I found most interesting your observation that "Measurements have most and least significant digits. Indices do not." You clearly illustrate this via the example of an analog TV treating the received signals as...

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Author Robert H McEachern replied on Jan. 24, 2015 @ 15:39 GMT
Edwin,

As you know, I have more than a little interest in Bell's Theorem. So I will read your essay soon, and give my major comments there. Here I will only say that your statement that "the relations are independent of actual physical measurements, they depend only on indices." seems to oversimplify the situation. The traditional view is that the QM measurements themselves (after compensating for noise/errors) only take-on a few discrete values. Hence they are similar to very short (perhaps even just a single bit) indices. The interesting point is that when indices become so short, the distinction between measurements and indices vanishes. That is ultimately why such measurements seem to exhibit characteristics unlike more classical cases, in which the measurements need not be treated as indices.

Rob McEachern

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Edwin Eugene Klingman replied on Jan. 24, 2015 @ 21:14 GMT
Dear Rob,

I believe some noise has entered our communications. In particular, a crucial word has been dropped. I say "In quantum commutation relations… the relations are independent of actual physical measurements, they depend only on indices". "Commutation" is a key and crucial qualifier, and I believe it makes the statement true, not oversimplified, but I've not yet had time to fully investigate this. As commutation relations are key axioms for some quantum mechanics formulations, I think this is significant. As stated, I intend to think further on this.

Also, I'm not sure that the traditional view is that QM measurements themselves only take-on a few discrete values. Neither the position nor the momentum of the free particle is quantized. It is 'bound' or 'constrained' systems that are forced to have discrete solutions. Quantum action h-bar does not, in my opinion, require quantized energy, momentum, or position, in and of itself. Boundaries bring these about.

So I'm still of the opinion that commutation relations (which apply to the momentum and position of a free particle) are based on indices, not measurements, as per my above comment.

You never fail to bring new insights to the party,

Best,

Edwin Eugene Klingman

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Author Robert H McEachern replied on Jan. 24, 2015 @ 21:47 GMT
"Neither the position nor the momentum of the free particle is quantized." That is true, but in the QM cases of interest, the amount of information contained in their product is quantized, and to the minimum possible value - one bit. That is what the uncertainty principle is all about:

If you can measure the momentum to 100 bit accuracy, then you will only have 1/100 bit accuracy in position. The interesting point is that "information" content does not depend on one measurement's accuracy, or the other's, it depends on the product of both; the time * bandwidth product, or the position * momentum product. The bits of either single measurement, are too highly correlated to be "information" at all, they are merely data bits, not information bits.

Rob McEachern

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Colin Walker wrote on Jan. 23, 2015 @ 00:40 GMT
Hi Robert,

I was delighted to find your essay starting off, as does mine, with 'reductio ad absurdum' as an illustration of the difference between mathematics and physics. It had not occurred to me that the terminology of mathematics might not fit physics. I would say that physics is mathematics on shaky ground.

It was a good read, and I thought the theme of information relevance was a good one.

Best regards,

Colin

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Author Robert H McEachern replied on Jan. 24, 2015 @ 16:01 GMT
Colin,

I am glad to hear you say that "It had not occurred to me that the terminology of mathematics might not fit physics." I don't think it has occurred to many people. Pointing it out was my primary reason for writing the essay. Math is an end, unto itself. Physics is not. When physicists lose sight of that fact, they have drifted so far across the border between physics and metaphysics, that they have lost sight of that border, and are no longer in the realm of physics, at all. As fun and interesting as mathematically-based metaphysical speculations may be, they remain entirely outside the realm of physics, until an observation of something, across the border in the physical realm, guides the wanderers back to the land of physical reality.

Rob McEachern

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Sujatha Jagannathan wrote on Feb. 16, 2015 @ 09:43 GMT
Somewhere the paper searches for perfection, but it fails to depict the 'Big Picture'!

Sincerely,

Miss. Sujatha Jagannathan

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Richard Lewis wrote on Feb. 18, 2015 @ 12:07 GMT
Hello Robert,

I did like your objective to demystify the apparently mysterious connection between maths and physics by considering the information aspects.

Also a very apt survey of the limitations of the application of maths to physics and other fields.

My own take on the subject is to see the apparent mystery as our lack of understanding of fundamental reality and by finding the correct interpretation of reality we will solve the mystery of why maths is so effective.

Regards

Richard

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Author Robert H McEachern replied on Feb. 18, 2015 @ 21:26 GMT
Hi Richard,

I think there are two different, but equally interesting issues:

(1) A lack of understanding of fundamental reality, caused by failing to correctly interpret the meaning of the mathematics, that is being used to describe physical reality. I addressed this issue in my 2012 FQXI essay.

(2) Mathematics is only effective at describing a limited subset, of the set of all observable phenomenon - those in which the observations contain only small amounts of information. That is the subject of my current essay.

I think there are greater mysteries lurking in (1) than in (2). This is because, even when we discover equations that perfectly describe the observations, that discovery, in and of itself, provides few clues as to why those equations, rather than some others, should be the ones that work. The equations merely describe how things behave, but not why they behave, as they do; the cause of the effect, is not contained within any mere description of the effect, regardless of how accurate that description might be.

Rob McEachern

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basudeba mishra wrote on Feb. 19, 2015 @ 12:56 GMT
Dear Sir,

How do you say that the value of the starting axioms and postulates of mathematics is not measured by their truth? In fact these are set up based on universal observation, which cannot be wrong, though it may be sometimes wrong in other fields. The conclusions of mathematical operations are always logically consistent – though it may or may not be interesting. The truth content...

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Author Robert H McEachern replied on Feb. 19, 2015 @ 16:14 GMT
Basudeba,

In mathematics, axioms and postulates, are not based on observations. That would be self-contradictory, since they are not physically observable phenomenon. One might observe physical approximations to such abstract entities, but the abstractions themselves cannot be observed. "The conclusions of mathematical operations (correctly performed) are always logically consistent", as...

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Roger Schlafly wrote on Feb. 26, 2015 @ 00:20 GMT
You lost me when you said that Math does not care about truth. Can you give an example or the sort of math statement where no one cares about its truth? When mathematicians publish papers, aren't they intending to demonstrate truths?

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Author Robert H McEachern replied on Feb. 26, 2015 @ 16:03 GMT
Roger,

Don't confuse the truth of the "starting points" or axioms, with the truth of some theorem that has been demonstrated, based upon those axioms. I am only talking about the former.

Here is a quote from the first sentences of Wiki-axiom:

"An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning."

Note the difference between the classical and modern definitions; the latter no longer requires an axiom to be true. That does not mean it is false. It means that interesting starting points may not have the property of being either true or false.

For example, one might choose as a "Starting Point", the statement "Let N and M be prime numbers." That statement, is neither true nor false. But the statement "N*M is a prime number", is false.

Rob McEachern

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Eckard Blumschein wrote on Mar. 5, 2015 @ 06:31 GMT
Dear Rob,

You wrote "Natural Philosophy (Science) had stagnated for 2000 years, partially as a result of the ancient greeks modeling science on mathematics (Deductive logic, rather than Inductive logic), and consequently not being sufficiently motivated to verify the truth of their starting points, via observations" and a "stagnation in contemporary theoretical Physics ... for the same reason"

My essay adopts a more detailed and partly different interpretation of the 2000 years by historians of mathematics. Wallis' numbers between -oo and +oo instead of ancient non-zero numbers were certainly motivated by curiosity rather than by observation.

When you are often writing "starting points" or "initial conditions" you seem to agree with me that the use of boundary conditions does usually not fit to the description of processes. Shouldn't you support (or refute) my belonging reasoning, too?

BTW, I quoted your 2012 essay.

Regards,

Eckard

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Author Robert H McEachern replied on Mar. 5, 2015 @ 16:16 GMT
Eckard,

I think the title of your essay hits the nail on the head. It is indeed the unwarranted interpretations, slapped onto the equations of mathematical physics, that cause all the problems in understanding the nature of reality.

Where we differ, seems to be that you believe that avoiding the usage of particular mathematical techniques, will solve the problem, whereas I believe...

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Anonymous wrote on Mar. 16, 2015 @ 00:44 GMT
Robert H. McEachern,

It has been rewarding reading your essay entries and forum contributions. It has been challenging to respond to your questions. Your educated clear thinking is powerful.

James Putnam

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James A Putnam replied on Apr. 1, 2015 @ 22:43 GMT
Dear Robert McEachern,

Great Paragraph!

"So where should one start, in the search for anything that might be either “interesting” or “true”, regarding the connections between Mathematics and Physics? One obvious starting point, is with the observation that mathematics seems “unreasonably” effective in describing the observations of concern to physicists, but much less so, in most other fields. What are the characteristics of these fields, that correlate with this fact? In a word, complexity. Or, more precisely, the effectiveness strongly correlates with the information content of the observational data, that the mathematics is being used to quantitatively describe; the lower the information content, the more effective mathematics is, at precisely describing it. Physics deals only with very low information content phenomenon (the “laws” of physics do not require very many bits, or other symbols, to specify them) in comparison with phenomenon at the opposite end of the complexity spectrum, like life. Consequently, it is not surprising that a low information content symbology, like sets of mathematical equations, succeed at capturing the essence (the information content) of one, but not the other."

I have now rated your essay. Good luck!

James Putnam

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James A Putnam replied on Apr. 1, 2015 @ 22:51 GMT
I am informed that I already rated your essay. I did remember that I read it earlier. I thought I waited to rate it. Sorry I can't help a second time. :) Your overall rating is unjustifiably low.

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Joe Fisher wrote on Apr. 1, 2015 @ 18:26 GMT
Dear Dr. McEachern,

I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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Peter Jackson wrote on Apr. 17, 2015 @ 16:17 GMT
Rob,

Now I'm down to speed-reading a short essay is often merciful. Yours was the opposite (whatever that is?) and I was pleased to also savour it at walking pace.

I don't think we need to fill 9 pages to earn a high score! We know from the past that we agree on fundamentals. I too identify flawed 'logic' and starting points, going on to specifics (I think you'll love my exposure of the great red and green sock switch con trick!)

Ref our interest in Bell's theorem, after the contest I'd be honoured you might also read this analysis of the false assumption he knew must exist. Quasi-classical Entanglement, Superposition and Bell Inequalities.

I like your thinking, your writing style and your content. Very well done.

Peter

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