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The Unreasonable Effectiveness of Mathematics in the Natural Sciences


Genady
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10 hours ago, Genady said:

Here is one of many things Penrose has to say about Platonic world of mathematics:

Roger Penrose is the son in another of those dynasties of scientists (including mathematicians).
He is also foremost an analyst.

I see litle to disagree with in your extract, though elsewhere in that book he lapses into his own unproved speculations, particularly about QM.
His writing is, however, very dense, so one should always take careful note of the caveats he adds.

I do however offer a counterexample to this unlimited statement. I have emboldened some key phrases.

Quote

Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the world
.............    
The important point about these models, for our present discussion, is that they are basically purely abstract mathematical models. The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers. 

 

The only proof of the four colour theorem we have is basically the method of exhaustion.
This method follows a different pathway from the one defined in the extracted passage.
If you do not understand this please ask.

11 hours ago, Genady said:

Here we agree. And Eugene Wigner thought so as well. His article was just for posing a question to be answered. I disagree with your answer. I do agree with Penrose's attitude about mathematics and extend it to answer the Wigner's question. 

 

I do not understand how you both agree and disagree with what I said   ?

 

11 hours ago, Genady said:

You have said what you wanted to say, and I have said what I wanted to. I don't think we need to keep going circles. Maybe somebody else will contribute something different.

You are new here so I will forgive you for this comment.

I am entitled to make what civil comment I choose about your posted material and I will continue to point out where you are in error or misunderstand something.
I will also explain in as much detail as you like why I think this to be so.
That is what a discussion forum is for.

 

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25 minutes ago, studiot said:

You are new here so I will forgive you for this comment.

I am entitled to make what civil comment I choose about your posted material and I will continue to point out where you are in error or misunderstand something.
I will also explain in as much detail as you like why I think this to be so.
That is what a discussion forum is for.

 

May I reply point by point? Starting with the last point?

Thank you. I didn't mean it as an offence. I didn't mean to shut you up. Please, continue commenting and explaining. Am I obligated to reply? That is what I think is not always necessary, is it? May I reply only on comments that are of interest to me? Somebody else might be interested and reply on other comments. In my opinion, discussions, unlike arguments, don't have to end in agreement or disagreement. They can just give a food for further thought.

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22 minutes ago, Genady said:

May I reply point by point? Starting with the last point?

Thank you. I didn't mean it as an offence. I didn't mean to shut you up. Please, continue commenting and explaining. Am I obligated to reply? That is what I think is not always necessary, is it? May I reply only on comments that are of interest to me? Somebody else might be interested and reply on other comments. In my opinion, discussions, unlike arguments, don't have to end in agreement or disagreement. They can just give a food for further thought.

A discussion with rancour.

+1

If you are asking a question then clearly it is in your own interests to provide extra detail to those prepared to answer but needing to know more.
Often their gift in knowing the subject better is knowing what questions to ask.

If you are presenting a report on something eg in the scientific news section then the onus is on you to add sufficient summary to allow others to evaluate the subject presented.

If you are presenting a hypothesis or conjecture, it is up to you to introduce such supporting material as may be needed, including answering questions or objections from the membership on the presented material.

Material to introduce general discussion can be presented as a question or statement, either way,  supporting background and explanation aids the discussion.

Taking (or agreeing to take) one point at a time can be very productive.

 

Edited by studiot
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17 hours ago, Genady said:

62 years ago, on February 1960, Eugene Wigner concluded his article of the above title:

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

I think it might be interesting to discuss, if there is a better understanding of this miracle now. (Leave the issue of deserving it alone, please :) )

On the one hand, it is amazing that math describes these basic laws of nature, but on the other hand, it seems required, by the very definition of what we mean by a law of nature.

What would physics look like if certain phenomena were not following some mathematical description? Are there examples of such? Would we even recognize the behavior if it didn't follow some pattern that could be described with math?

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4 hours ago, exchemist said:

I suppose what you are saying is that there is something about pure mathematics that corresponds to the physical world

Yes, this is one direction of thought. For example, see the reference here:

14 hours ago, TheVat said:

Max Tegmark, a modern neoPlatonist, seems to find a mathematical universe reasonable.   His book is very provocative and carefully argued, though not persuasive to me.  

https://en.wikipedia.org/wiki/Our_Mathematical_Universe

 

I don't like this direction of thinking. I rather think in the opposite direction, similar to Penrose and to this:

1 hour ago, swansont said:

On the one hand, it is amazing that math describes these basic laws of nature, but on the other hand, it seems required, by the very definition of what we mean by a law of nature.

In other words, it is not math that corresponds to the physical world, but rather physical theories that correspond to math. Physical theories, to be good theories, need precision, rigor, clear concepts, they need to be free of human prejudices, psychology, vague language, etc. Math has exactly these attributes. Physical laws are described mathematically by necessity.

To this argument (which is Penrose's one, not mine) I add a bit of biology. For example, humans, like all other mammals, have an area of the brain responsible for computing a "mental map" of the surroundings. It evolved because it gives obvious advantages to the organism. And of course it computes specifically 3D maps. Why would it do any other kind? We never needed to deal with any other kind. Thus, we are good in visualizing in 3D. Now, studying phenomena beyond our immediate experience, we stumbled upon other kinds. We cannot use our visual imagination and intuition, and our everyday language, to describe them. Mathematics is the only available tool.

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2 hours ago, swansont said:

On the one hand, it is amazing that math describes these basic laws of nature, but on the other hand, it seems required, by the very definition of what we mean by a law of nature.

What would physics look like if certain phenomena were not following some mathematical description? Are there examples of such? Would we even recognize the behavior if it didn't follow some pattern that could be described with math?

It is often forgotten that Physics abounds with data for which there is no theoretical basis or Law which states such and such 'must' have this or that value.

A large amount of Professor Millikan's highly successful and readable book discusses the decades of impediment cause by a lack of knowledge of the value of e/m.
Although we have now measured it and moved on, to this day we still can't demonstrate why it has this value and no other.
Of course there is plenty of theory as to the consequences of this value.

In fact pretty well every scientific equation and formula in existence contains such values ( mostly constants) which just are what they are and we have to measure them empirically.

Physics, of course, is not the only Science where rational thinking holds sway.

The description of crystal forms in granite given in Professor Swinnerton's delightful book is a masterpiece of rational thinking, without any Mathematics whatsoever in evidence.

 

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10 minutes ago, studiot said:

It is often forgotten that Physics abounds with data for which there is no theoretical basis or Law which states such and such 'must' have this or that value.

The highly successful Standard Model has about two dozens of such numbers that need to be just put in by hand. One of the criteria for a good theory Beyond Standard Model is to have fewer of those.

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2 minutes ago, studiot said:

It is often forgotten that Physics abounds with data for which there is no theoretical basis or Law which states such and such 'must' have this or that value.

A large amount of Professor Millikan's highly successful and readable book discusses the decades of impediment cause by a lack of knowledge of the value of e/m.
Although we have now measured it and moved on, to this day we still can't demonstrate why it has this value and no other.
Of course there is plenty of theory as to the consequences of this value.

Does that mean the behavior is not following mathematical relationships? 

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1 minute ago, swansont said:

Does that mean the behavior is not following mathematical relationships? 

What behaviour ?

I said there are mathematically definable consequences.

Would these consequences not be different if the value was ten times different ?

5 minutes ago, Genady said:

The highly successful Standard Model has about two dozens of such numbers that need to be just put in by hand. One of the criteria for a good theory Beyond Standard Model is to have fewer of those.

Put into what ?

Does the 'standard model ' predict the values of such numbers ?

If so why bother to have them ?

Why not just use the numbers themselves ?

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3 minutes ago, studiot said:

What behaviour ?

I said there are mathematically definable consequences.

Would these consequences not be different if the value was ten times different ?

The behavior of the laws of physics, as per the OP

Behavior being different is not the same as the behavior not following laws/being described by math 

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17 minutes ago, studiot said:

It is often forgotten that Physics abounds with data for which there is no theoretical basis or Law which states such and such 'must' have this or that value.

On the other hand, sometimes these numbers turn out to be not requiring any theoretical basis. For example, Kepler worked hard to find a geometrical principle explaining why Solar System has five planets. Of course, it turned out that there are more than five, but moreover, it turned out that this number as well as the planets' given orbits, are just an incidental consequences of the Solar system evolution.

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1 minute ago, swansont said:

The behavior of the laws of physics, as per the OP

Behavior being different is not the same as the behavior not following laws/being described by math 

So What ?

 

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1 minute ago, Genady said:

On the other hand, sometimes these numbers turn out to be not requiring any theoretical basis. For example, Kepler worked hard to find a geometrical principle explaining why Solar System has five planets. Of course, it turned out that there are more than five, but moreover, it turned out that this number as well as the planets' given orbits, are just an incidental consequences of the Solar system evolution.

Yes that is true.

So

1) There are numbers of no special consequence in Physics. That is there are parts of Mathematics that have no special meaning in Physics.

2) There are facts (numbers) in Physics which have special meaning that have no special meaning in Mathematics.

Both leading to the conclusion that there is incomplete overlap between Mathematics and Physics.

Is that not rational thinking ?

3 minutes ago, swansont said:

One is the topic of discussion, the other is not

OK I consider myself driven off the forum.

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1 minute ago, studiot said:

Yes that is true.

So

1) There are numbers of no special consequence in Physics. That is there are parts of Mathematics that have no special meaning in Physics.

2) There are facts (numbers) in Physics which have special meaning that have no special meaning in Mathematics.

Both leading to the conclusion that there is incomplete overlap between Mathematics and Physics.

Is that not rational thinking ?

This is all correct. The overlap between Mathematics and Physics is far from being complete. Especially, from the Mathematics perspective (I might be just a bit biased :) ). Mathematics is much-much larger than the part used in Physics. The OP, however, was why there is such an overlap at all and quite a big and essential one, from the Physics perspective.

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36 minutes ago, Genady said:

In other words, it is not math that corresponds to the physical world, but rather physical theories that correspond to math. Physical theories, to be good theories, need precision, rigor, clear concepts, they need to be free of human prejudices, psychology, vague language, etc. Math has exactly these attributes. Physical laws are described mathematically by necessity.

To this argument (which is Penrose's one, not mine) I add a bit of biology. For example, humans, like all other mammals, have an area of the brain responsible for computing a "mental map" of the surroundings. It evolved because it gives obvious advantages to the organism. And of course it computes specifically 3D maps. Why would it do any other kind? We never needed to deal with any other kind. Thus, we are good in visualizing in 3D. Now, studying phenomena beyond our immediate experience, we stumbled upon other kinds. We cannot use our visual imagination and intuition, and our everyday language, to describe them. Mathematics is the only available tool.

Plus one. Are you quite sure English is not your native language?  You could have fooled me.

  In discussion, this argument for the effectiveness of models is sometimes called the evolutionary argument.   To adapt to a challenging environment, our modeling of reality has to maintain a high degree of correspondence with what is "out there." At least, insofar as it helps us navigate the macro world of tigers, waterholes, snakes, poison plants, wire coral, etc.

The small pitfall of the evolutionary argument is that it only requires models to be effective on a very pragmatic level.  We could imagine a creature that needs lots of water.  It perceives a pool of water as a jiggling purple tetrahedron that sings "I Feel Pretty" in a lovely soprano voice.  It responds by going towards the singing, and drinks, and lives another day.  Now, though adaptive, we might assert that this model of a pool of water is not very realistic, and is more like a useful hallucination than a good map of what is out there in the world.   That illustrates why we have to approach our mathematical representations of reality with healthy caution and not be carried away by Tegmark-ian notions like "the multiverses are composed of math." 

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4 minutes ago, TheVat said:

Are you quite sure English is not your native language?  You could have fooled me.

Yes, the third one. There is one more after that. Thanks a lot.

5 minutes ago, TheVat said:

That illustrates why we have to approach our mathematical representations of reality with healthy caution and not be carried away by Tegmark-ian notions like "the multiverses are composed of math." 

Right. Or by endlessly repeated (esp. in a pop-science) reference to "mathematical beauty".

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Sorry for neglecting the forum for such a long time. It is a stressful time (corona (homeoffice) and stress at work), and I seldom have the peace to do more 'thinking' for this forum.

Here an observation about this topic:

I wonder if it is really so astonishing that math is so effective describing the world around us. In my opinion we need only two aspects of nature to more or less guarantee that we can use math to describe it:

  • regularities in natural phenomena, to begin with simple phenomena like the yearly rising of the Nile, sun sets etc. I cannot imagine a regularity that cannot described mathematically. If somebody can, please give an example.
  • the existence of 'natural kinds', like water molecules. Simply said, if you acquired knowledge about one water molecule, you know it is valid for all water molecules.

Life would would be impossible without these. So just add the smallest bit of anthropic reasoning (if above aspects of nature would not be the case, no observers could exist) and your are done. No?

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1 hour ago, Eise said:

Sorry for neglecting the forum for such a long time. It is a stressful time (corona (homeoffice) and stress at work), and I seldom have the peace to do more 'thinking' for this forum.

Here an observation about this topic:

I wonder if it is really so astonishing that math is so effective describing the world around us. In my opinion we need only two aspects of nature to more or less guarantee that we can use math to describe it:

  • regularities in natural phenomena, to begin with simple phenomena like the yearly rising of the Nile, sun sets etc. I cannot imagine a regularity that cannot described mathematically. If somebody can, please give an example.
  • the existence of 'natural kinds', like water molecules. Simply said, if you acquired knowledge about one water molecule, you know it is valid for all water molecules.

Life would would be impossible without these. So just add the smallest bit of anthropic reasoning (if above aspects of nature would not be the case, no observers could exist) and your are done. No?

What an excellent post. +1

And isn't 'regularity'  a superb word since not only does it apply to so many different branches of Mathematics in some wya, but it is a general word that has not been hijacked by any discipline.

Regularity appears algebra in Group theory, in geometry in polygons and other figures, in the solution of differential equations many of which are important in Physics such as Schroedinger, the Wave eqaution.

May I suggest that electrons would be a better example than water molecules since H2O and D2O have some different physical properties ?

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I always had problems with Wigner's motto. I would totally agree with the great man, had he picked a word other* than "unreasonable". To me, it's not unreasonable. A big part of understanding Nature is about quantifying it, and then measuring it. So, were the great Wigner alive, and would he bother to listen to me, I'd probably ask him, 'What do you mean "unreasonable?". I think this is very much in the vein of what Swansont said.

* Alternative list:

amazing

fortunate

etc.

 

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9 minutes ago, joigus said:

I always had problems with Wigner's motto. I would totally agree with the great man, had he picked a word other* than "unreasonable". To me, it's not unreasonable. A big part of understanding Nature is about quantifying it, and then measuring it. So, were the great Wigner alive, and would he bother to listen to me, I'd probably ask him, 'What do you mean "unreasonable?". I think this is very much in the vein of what Swansont said.

* Alternative list:

amazing

fortunate

etc.

 

Now I see where all (my) confusion in the earlier discussion came from -- I never seriously related to the word "unreasonable" and saw it just as metaphoric. In the article Wigner talks only about unexplained effectiveness.

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Well I think it is neither unreasonable, nor unexplained.

'Unsuprising' would be theonly un-word I would choose.

 

The whole issue reminds me of (I think it was Strange who first said it here) of the story of the puddle.

The puddle who woke up sentient one morning and said to himself. "Wow look at that, this hole fits me so well, it must have been made for me!"

 

:)

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