Mathematics and physics

We would like to have your comments on the interplay between physics and mathematics. There is Galilei's famous dictum from the beginning of the scientific revolution, which says that the Laws of Nature are written in the language of mathematics. Why is it that the objects of mathematical creation, satisfying the criteria of beauty and simplicity, are precisely the ones that time and time again are found to be essential for a correct description of the external world? Examples abound, let me just mention group theory and, yes, your Index Theorem!

SINGER: There are several approaches in answer to your questions; I will discuss two. First, some parts of mathematics were created in order to describe the world around us. Calculus began by explaining the motion of planets and other moving objects. Calculus, differential equations, and integral equations are a natural part of physics because they were developed for physics. Other parts of mathematics are also natural for physics. I remember lecturing in Feynman's seminar, trying to explain anomalies. His postdocs kept wanting to pick coordinates in order to compute; he stopped them saying: "The Laws of Physics are independent of a coordinate system. Listen to what Singer has to say, because he is describing the situation without coordinates." Coordinate-free means geometry. It is natural that geometry appears in physics, whose laws are independent of a coordinate system.

Symmetries are useful in physics for much the same reason they're useful in mathematics. Beauty aside, symmetries simplify equations, in physics and in mathematics. So physics and math have in common geometry and group theory, creating a close connection between parts of both subjects.

Secondly, there is a deeper reason if your question is interpreted as in the title of Eugene Wigner's essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences^[3]". Mathematics studies coherent systems which I will not try to define. But it studies coherent systems, the connections between such systems and the structure of such systems. We should not be too surprised that mathematics has coherent systems applicable to physics. It remains to be seen whether there is an already developed coherent system in mathematics that will describe the structure of string theory. [At present, we do not even know what the symmetry group of string field theory is.] Witten has said that 21st century mathematics has to develop new mathematics, perhaps in conjunction with physics intuition, to describe the structure of string theory.

ATIYAH: I agree with Singer's description of mathematics having evolved out of the physical world; it therefore is not a big surprise that is has a feed back into it.

More fundamentally: to understand the outside world as a human being is an attempt to reduce complexity to simplicity. What is a theory? A lot of things are happening in the outside world, and the aim of scientific inquiry is to reduce this to as simple a number of principles as possible. That is the way the human mind works, the way the human mind wants to see the answer.

If we were computers, which could tabulate vast amounts of all sorts of information, we would never develop theory - we would say, just press the button to get the answer. We want to reduce this complexity to a form that the human mind can understand, to a few simple principles. That's the nature of scientific inquiry, and mathematics is a part of that. Mathematics is an evolution from the human brain, which is responding to outside influences, creating the machinery with which it then attacks the outside world. It is our way of trying to reduce complexity into simplicity, beauty and elegance. It is really very fundamental, simplicity is in the nature of scientific inquiry - we do not look for complicated things.

I tend to think that science and mathematics are ways the human mind looks and experiences - you cannot divorce the human mind from it. Mathematics is part of the human mind. The question whether there is a reality independent of the human mind, has no meaning - at least, we cannot answer it.

Is it too strong to say that the mathematical problems solved and the techniques that arose from physics have been the lifeblood of mathematics in the past; or at least for the last 25 years?

ATIYAH: I think you could turn that into an even stronger statement. Almost all mathematics originally arose from external reality, even numbers and counting. At some point, mathematics then turned to ask internal questions, e.g. the theory of prime numbers, which is not directly related to experience but evolved out of it.

There are parts of mathematics where the human mind asks internal questions just out of curiosity. Originally it may be physical, but eventually it becomes something independent. There are other parts that relate much closer to the outside world with much more interaction backwards and forward. In that part of it, physics has for a long time been the lifeblood of mathematics and inspiration for mathematical work. There are times when this goes out of fashion or when parts of mathematics evolve purely internally. Lots of abstract mathematics does not directly relate to the outside world.

It is one of the strengths of mathematics that it has these two and not a single lifeblood: one external and one internal, one arising as response to external events, the other to internal reflection on what we are doing.

SINGER: Your statement IS too strong. I agree with Michael that mathematics is blessed with both an external and internal source of inspiration. In the past several decades, high energy theoretical physics has had a marked influence on mathematics. Many mathematicians have been shocked at this unexpected development: new ideas from outside mathematics so effective in mathematics. We are delighted with these new inputs, but the "shock" exaggerates their overall effect on mathematics.

Next: Newer developments

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