The Unreasonable Effectiveness of Mathematics in the Natural Sciences

[Eugenes]

Hi folks. I am a student and i have to do an essay about a topic that intrigues me.

I read the quote " The Unreasonable Effectiveness of Mathematics in the Natural Sciences" ( of E.Wigner) on the book "Is God a mathmatician" of Mario Livio, and that hit me.

Maths in fact can seem useless or a mere tool of applied sciences, but in reality, often it has anticipated some evolutions of physical theories and every branch of it has been applied in some way. One example could be the knot theory that, from the theory of mathematics it has been applied to the study of DNA (for whom interested: http://www.tiem.utk.edu/~gross/bioed/webmodules/DNAknot.html).

What do you think? Do you know a physical, chemical, biological area that have applied some theorem that math have discovered long time before?

[ajb]

You see this quite a lot in some areas of physics. For example, lots of the results found in quantum information theory are related to group theory, representation theory and combinatorics. I attended talks this weekend that many of the physical questions were answered by already known mathematical results. The tick is knowing enough about mathematics and the physics that one is interested in to spot these links. Many physicists don't know much mathematics.

[Eugenes]

Yes i see what you mean, physics is math applied, and thats a fact since from Galileo. but the thing becomes mindblowing when math is a step ahead of physics or other sciences,when old new branches of math that doesn't seemed so helpful became fondamental.another example is the non-euclidean geometry that turned out to be the basis of the theory of relativity. I'd love finding other examples

[studiot]

What intrigues me is the "unreasonableness" part.

 

What is unreasonable about it and why?

Edited April 27, 2015 by studiot

[Greg H.]

What intrigues me is the "unreasonableness" part.

 

What is unreasonable about it and why?

That's what I was wondering, tbh

[Acme]

What intrigues me is the "unreasonableness" part.

 

What is unreasonable about it and why?

That phrase is mostly a catchy hook used as a title to an article, though it is an argument made by some folk with disdain for math. (As we all too often encounter here at the forum. As if. )

The author explains:

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

...The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

Wigner in fact concludes the article contradictorily to the title saying,

...Let me end on a more cheerful note. 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. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

As to the OP and asking for math preceding utility or application I'm still thinking...

[Eugenes]

That phrase is mostly a catchy hook used as a title to an article, though it is an argument made by some folk with disdain for math. (As we all too often encounter here at the forum. As if. )..

Well maybe it is, but nobody can deny that mathematics is to nature as words to poetry

[Acme]

 

That phrase is mostly a catchy hook used as a title to an article, though it is an argument made by some folk with disdain for math. (As we all too often encounter here at the forum. As if. )..

Well maybe it is, but nobody can deny that mathematics is to nature as words to poetry

 

I have no idea what that means or how it relates to my post. In any regard, people can & do deny all manner of things concerning mathematics and I have no intention of going down any such rabbit hole. Good luck on your essay and inasmuch as it's a writing assignment, keep in mind what's important is how you write and not what you write.

[studiot]

 

Do you know a physical, chemical, biological area that have applied some theorem that math have discovered long time before?

 

 

What you haven't told us is the context of your essay, given the title.

 

The earliest mathematics, even such things as the requirement to measure the seasons, planetary conjunctions etc all derived from the practical or observational before the mathematics.

 

The earliest example I can think of of mathematics preceding the application was the introduction of complex numbers in the 16th century by Cardano, some 400 years before was applied to alternating electric current theory.

Edited April 27, 2015 by studiot

[imatfaal]

The concept of non-Euclidean geometry - Bolyai-Lobachevsky, Gauss, Reimann

[swansont]

You see this quite a lot in some areas of physics.

 

Indeed. There are many examples where one branch has "discovered" some mathematical tool is useful, only to find that the math was being used in some other branch, or was well developed within mathematics. I've seen it happen in timing analysis, when we hired some mathematicians to help develop some advanced modeling. The scientists/engineers described the problem they were working on, and the math guy said "Oh, that's XXX" — a well-developed bit of math, and they dredged up thirty-year-old papers to learn about it.

[ajb]

...but the thing becomes mindblowing when math is a step ahead of physics or other sciences,

There are plenty of things in mathematics that seem devoid of physics and many results that have little impact on physics. That said, it does seem that just about every branch of mathematics has some impact on physics, but not every result therein.

 

...when old new branches of math that doesn't seemed so helpful became fondamental.

Physics has always been and I think always will remain a big source of inspiration for mathematicians. Many interesting and important branches of mathematics have their roots in physics, or at least it was their early application to physics that drove further development. You example below is a good one.

 

...another example is the non-euclidean geometry that turned out to be the basis of the theory of relativity.

And I would add the basis of just about all physics in one way or another.

 

I'd love finding other examples

As I have said, a lot of mathematics does have its roots in physics, so finding examples of mathematics that originally had no motivation from physics may not be so clear-cut.

 

Anyway, another example of mathematics that is well-developed and has found applications in physics could include things of a topological nature; homotopy-coherent structures (A, C and L infinity algebras) and homological algebra for example. These kinds of structures are important in quantum field theory and deformation quantisation.

 

Other examples could include the very abstract ideas found in category and higher category theory. For sure categorical ideas are found throughout physics and again especially quantum field theory. Related to this are ideas from algebraic geometry, such as sheaves, ringed spaces and so on. All these have pushed our idea of 'geometry' and have been applied in various forms to physics.

 

In reverse, it is also very interesting when physical ideas enter mathematics. Good example of this is the work of Donaldson, which is closely related to Seiberg–Witten theory. This has to do with the study 4d manifolds using (super)Yang-Mills instantons. There are several other places where supersymmetric theories interface with geometry, maybe we could discuss this better another time. For now, I would just like to point out Witten's use of supersymmetric mechanics in his proof of the index theorem.

[overtone]

Some number theory was being done in China and in Greece in the third century AD, and found its first practical applications centuries later. One can argue that it was based on the need to, say, determine the length of the hypotenuse of a triangle for surveying or manufacturing or construction purposes, but most of it had no such uses and the rest probably wasn't actually practical for those purposes for a very long time.