The Unreasonable Effectiveness of Mathematics in the Natural Sciences

[Genady]

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  )

[studiot]

26 minutes 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  )

What is 'the Language of Mathematics' ?

As far as I know it does not extend to miracles.

Edited January 3, 2022 by studiot

[Genady]

1 minute ago, studiot said:

What is 'the Language of Mathematics' ?

From the article I think he means, mathematical concepts.

2 minutes ago, studiot said:

As far as I know it does not extend to miracles.

The "miracle" is its appropriateness etc.

[studiot]

1 minute ago, Genady said:

From the article I think he means, mathematical concepts.

Go on, it's your thread.

[beecee]

4 minutes ago, studiot said:

What is 'the Language of Mathematics' ?

Isn't it often said that mathematics is the language of physics?

[Genady]

Just now, studiot said:

Go on, it's your thread.

Do you mean you want to know my take on it?

1 minute ago, beecee said:

Isn't it often said that mathematics is the language of physics?

Yes. Wigner asks, why. It is not generally developed as such by mathematicians.

39 minutes ago, studiot said:

Go on, it's your thread.

First, I think he had exaggerated in the title, "Natural Sciences". Most natural sciences rather use applied mathematics, for obvious reasons, we need to calculate things just like in engineering. So, the question is limited not even to physics, but to fundamental physics. This is where the fundamental laws are intrinsically mathematical, are based on purely mathematical concepts, such as Hilbert space in QM.

[beecee]

This article, may add more to this debate......

https://www.sciencealert.com/the-exquisite-beauty-of-nature-reveals-a-world-of-math

 

(If of course this is the case, it re-enforces even more, the actions of scientists like Carl Sagan, in attempting mathematical communications with plaques on the Pioneer and Voyager probes)

[exchemist]

58 minutes ago, Genady said:

Do you mean you want to know my take on it?

Yes. Wigner asks, why. It is not generally developed as such by mathematicians.

First, I think he had exaggerated in the title, "Natural Sciences". Most natural sciences rather use applied mathematics, for obvious reasons, we need to calculate things just like in engineering. So, the question is limited not even to physics, but to fundamental physics. This is where the fundamental laws are intrinsically mathematical, are based on purely mathematical concepts, such as Hilbert space in QM.

I should have thought that what he should really have marvelled at is the patterns, i.e. the order that we perceive in nature. Human mathematics is just a way of modelling that order, it seems to me. Such constructs as Hilbert space are just our models, after all.     

Edited January 3, 2022 by exchemist

[Genady]

5 minutes ago, beecee said:

This article, may add more to this debate......

https://www.sciencealert.com/the-exquisite-beauty-of-nature-reveals-a-world-of-math

 

(If of course this is the case, it re-enforces even more, the actions of scientists like Carl Sagan, in attempting mathematical communications with plaques on the Pioneer and Voyager probes)

It is strange that the article doesn't mention Plato. This is the idea of Platonic world of pure concepts. (I assume the Roger Penrose's take on it.) It exists (somehow?) by itself. In the sense that, for example, there is infinite number of prime numbers, regardless if we know that or not, and moreover, regardless of existence or non-existence of the Universe.

Mathematics is an investigation of that world.

7 minutes ago, exchemist said:

I should have thought that what he should really have marvelled at is the patterns, i.e. the order that we perceive in nature. Human mathematics is just a way of modelling that order, it seems to me. Such constructs as Hilbert space are just our models, after all.     

Yes, he marveled at that as well. But he argues, and I agree with this, that mathematics is not developed to model them, physics does.

Let's take a simpler example than Hilbert space, complex numbers. They were not developed in math to model any pattern in nature. But they are absolutely essential to describe quantum laws.

[exchemist]

55 minutes ago, Genady said:

It is strange that the article doesn't mention Plato. This is the idea of Platonic world of pure concepts. (I assume the Roger Penrose's take on it.) It exists (somehow?) by itself. In the sense that, for example, there is infinite number of prime numbers, regardless if we know that or not, and moreover, regardless of existence or non-existence of the Universe.

Mathematics is an investigation of that world.

Yes, he marveled at that as well. But he argues, and I agree with this, that mathematics is not developed to model them, physics does.

Let's take a simpler example than Hilbert space, complex numbers. They were not developed in math to model any pattern in nature. But they are absolutely essential to describe quantum laws.

Well yes, but that's just because complex numbers are a good way to represent periodic phenomena: being numbers in 2D, a fixed radius rotating is handy for waves etc. via projections along real and imaginary axes and so forth. 

I don't see that it is surprising that items from the mathematical toolkit are useful to model aspect of the order  in nature. It is the order that is the issue, really. 

[Genady]

Let's take another example, number π. It appeared in math while investigating circles and triangles. But it kept and keeps popping up almost everywhere in math and physics, in places that have no direct (immediate, obvious) connections to circles and triangles. I think, with complex numbers, the π, and many other mathematical concepts we just stumbled upon something big and important -- just like looking for a shorter way to India, finding America.

BTW, there are uncountably infinite number of transcendental numbers, but only two appear everywhere (almost) in math and physics, π and e. 

[exchemist]

4 minutes ago, Genady said:

Let's take another example, number π. It appeared in math while investigating circles and triangles. But it kept and keeps popping up almost everywhere in math and physics, in places that have no direct (immediate, obvious) connections to circles and triangles. I think, with complex numbers, the π, and many other mathematical concepts we just stumbled upon something big and important -- just like looking for a shorter way to India, finding America.

BTW, there are uncountably infinite number of transcendental numbers, but only two appear everywhere (almost) in math and physics, π and e. 

Where π is concerned, it is not surprising that circles, spheres and periodic phenomena occur in nature. Finding exponential processes in nature is not a surprise either. 

[Genady]

2 minutes ago, exchemist said:

Where π is concerned, it is not surprising that circles, spheres and periodic phenomena occur in nature. Finding exponential processes in nature is not a surprise either. 

The Gaussian integral is not necessarily related to circles, spheres, and periodic phenomena. However, the number π is there:

[exchemist]

12 minutes ago, Genady said:

The Gaussian integral is not necessarily related to circles, spheres, and periodic phenomena. However, the number π is there:

True. And Gaussian distributions do crop up in physics. (Euler's Identity shows a connection between e and π which I suspect may be somewhere at the bottom of this, but I don't pretend to be a mathematician) Is all this unreasonable?   

Edited January 3, 2022 by exchemist

[Genady]

1 minute ago, exchemist said:

True. And Gaussian distributions do crop up in physics. (There is a connection between e and π of course, in Euler's Relation.) But is all this unreasonable?   

Each example separately is not unreasonable. What is "unreasonable" (I'd rather say, asks for a root cause explanation) is the deep connection between these two originally not connected worlds, mathematical concepts and physical phenomena.

7 minutes ago, exchemist said:

(Euler's Identity shows a connection between e and π which I suspect may be somewhere at the bottom of this, ...

No, it is not related to Euler's identity.

[studiot]

I really can't see where all this is going.

Yes numbers and the theory of numbers form an important part of Mathematics.

But they are not fundamental concepts, although a numbering system, very different from our own, was probably the earliest maths to 'studied'.
Note that the Australian aborigines have only 3 numbers  one, two, many.

I agree that numbers are very important in applied maths since this often deals with quantities.

But what about the rest of Mathematics ?

And what about the (physical or engineering) subjects Mathematics cannot tackle ?

What about the difference between synthesis and analysis ?

 

Edited January 3, 2022 by studiot

[Genady]

8 minutes ago, studiot said:

But what about the rest of Mathematics ?

Examples of not numbers? Sure:

• Riemannian geometry and GR
• Linear algebra and quantum mechanics
• Group theory and elementary particles

[studiot]

7 minutes ago, Genady said:

Examples of not numbers? Sure:

• Riemannian geometry and GR
• Linear algebra and quantum mechanics
• Group theory and elementary particles

 

None of the above are any more fundamental than number theory is.

I note from your posts that English is not your first language, although your English is very good.

Is there some trouble understanding my posts since you are not directing your answers at the questions or comments I make ?

If so I am very happy to expand on or clarify my comments.

 

 

Edited January 3, 2022 by studiot

[Genady]

1 minute ago, studiot said:

 

I note from your posts that English is not your first language, although your English is very good.

Is there some trouble understanding my posts since you are not directing your answers at the questions or comments I make ?

Yes, you are right about my English, and thank you for calling it "very good."

I think I understood your other questions. I didn't reply, except for the "not numbers" one, because I did not understand how they are related to the topic. 

For example, "...they are not fundamental concepts...". I don't think so, but it just doesn't matter here. They are mathematical concepts, that is the point here.

Or, "what about the (physical or engineering) subjects Mathematics cannot tackle ?" The question is about the fundamental subjects where math is extremely effective and necessary, not about other subjects.

"What about the difference between synthesis and analysis ?" I don't know. What about it?

 

[studiot]

16 minutes ago, Genady said:

Yes, you are right about my English, and thank you for calling it "very good."

I think I understood your other questions. I didn't reply, except for the "not numbers" one, because I did not understand how they are related to the topic. 

For example, "...they are not fundamental concepts...". I don't think so, but it just doesn't matter here. They are mathematical concepts, that is the point here.

Or, "what about the (physical or engineering) subjects Mathematics cannot tackle ?" The question is about the fundamental subjects where math is extremely effective and necessary, not about other subjects.

"What about the difference between synthesis and analysis ?" I don't know. What about it?

 

Thank you , now we are getting somewhere.

This thread is discusses the language of Mathematics.

4 hours ago, Genady said:

"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.

Yet we are onto two pages and you have yet to mention set theory,  mathematical processes, mathematical proofs  .......

 

In answer to your question, analysis and synthesis are processes.

Much of Mathematics at a fundamantal level, (and therfore its language) is about processes.

When you analyse something you are working on something that is already there.

You can describe it, categorise it, in many mathematical ways.
You can put numbers to it, you can measure it.

When you synthesise something you are trying to make or establish something that does not yet exist.
In my experience that is generally a more difficult task.

Mathematical processes such as proofs often fall into one of these two categories.

Edited January 3, 2022 by studiot

[Genady]

17 minutes ago, studiot said:

Thank you , now we are getting somewhere.

This thread is discusses the language of Mathematics.

Yet we are onto two pages and you have yet to mention set theory,  mathematical processes, mathematical proofs  .......

 

In answer to your question, analysis and synthesis are processes.

Much of Mathematics at a fundamantal level, (and therfore its language) is about processes.

When you analyse something you are working on something that is already there.

You can describe it, categorise it, in many mathematical ways.
You can put numbers to it, you can measure it.

When you synthesise something you are trying to make or establish something that does not yet exist.
In my experience that is generally a more difficult task.

Mathematical processes such as proofs often fall into one of these two categories.

I read that you talk here mostly about applied mathematics. It is not a part of the question. 

The last sentence is about mathematical processes such as proof. This also is not a part of the question. The mathematical results are. As I've answered in the beginning of the thread, the topic is, mathematical concepts.

P.S. Could you please limit your posts to one question at a time? This would help to stay focused, and might eliminate a need for some other questions.

Edited January 4, 2022 by Genady

[TheVat]

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

 

[studiot]

21 minutes ago, Genady said:

I read that you talk here mostly about applied mathematics. It is not a part of the question. 

The last sentence is about mathematical processes such as proof. This also is not a part of the question. The mathematical results are. As I've answered in the beginning of the thread, the topic is, mathematical concepts.

 

You clearly understand different structures of Mathematics and of language from the ones I understand.

Without suitable structures and language little Mathematics and even less applications can be done.

For instance those using applications tend to use the fundamentals of set theory such as uniqueness and closure without realising they do so.

Group theory is based on such requirements They are not concepts they are requirements. The structure is organised this way for good reasons.

There is nothing miraculous, fortuitous or gifted about this. It is quite deliberate on the part of Mathematicians.
And it is from pure maths, not applied maths.

 

Edited January 4, 2022 by studiot

[Genady]

26 minutes 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

 

Thank you, I will check it.

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

This was an extraordinary idea for its time, and it has turned out to be a very powerful one. But does the Platonic mathematical world actually exist, in any meaningful sense? Many people, including philosophers, might regard such a ‘world’ as a complete fiction— a product merely of our unrestrained imaginations. Yet the Platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the world— or, rather, of certain aspects of the world— and these models may be tested against previous observation and against the results of carefully designed experiment. The models are deemed to be appropriate if they survive such rigorous examination and if, in addition, they are internally consistent structures. 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. 
If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato’s world to be in any sense absolute and ‘real’. Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics, we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselves, with a reality that lies beyond what each individual can achieve?

Penrose, Roger. The Road to Reality (p. 12). 

21 minutes ago, studiot said:

 

You clearly understand different structures of Mathematics and of language from the ones I understand.

Without suitable structures and language little Mathematics and even less applications can be done.

For instance those using applications tend to use the fundamentals of set theory such as uniqueness and closure without realising they do so.

Group theory is based on such requirements They are not concepts they are requirements. The structure is organised this way for good reason.

 

We do have quite different views on mathematics, I think. My view is much closer to the one of Penrose, which is partially described in the quote above.

21 minutes ago, studiot said:

There is nothing miraculous, fortuitous or gifted about this.

 

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. 

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.

Edited January 4, 2022 by Genady

[exchemist]

11 hours ago, Genady said:

Each example separately is not unreasonable. What is "unreasonable" (I'd rather say, asks for a root cause explanation) is the deep connection between these two originally not connected worlds, mathematical concepts and physical phenomena.

No, it is not related to Euler's identity.

OK, you will know better than I, I'm sure.

I suppose what you are saying is that there is something about pure mathematics that corresponds to the physical world and this is not necessarily what one might expect.  I shall have to think about that.