Open Questions: Mathematics. Eugene Wigner was a physicist. Ironically, Bertrand Russell, who was a.

Mathematics may be defined as the subject in which we never. It is very likely, however, that Russell didn't mean to deprecate mathematics. Mathematics, rightly viewed, possesses. The point is that the ultimate nature of mathematics itself is something that. It may be that these two apparently. This circumstance. Philosophers of mathematics over the centuries - - and they have been at.

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Such questions won't concern us here. Instead, we'll consider the question.

What is it that. mathematicians study? Essentially, two things are obvious: numbers and. That has been true since the time of the ancient Greeks, and even. Egyptians and the Babylonians. You won't go. far wrong if you consider number and geometry still to make up the core of. To some extent we might add to this the study of logic itself.

Of course. mathematicians have, at least since Euclid, relied heavily on logic. But. it was only with the work of Gottfried Leibniz (1. This project matured slowly, with important.

George Boole (1. 81. Bertrand Russell (1. Into this merged the set theory developed by.

Georg Cantor (1. 84. Logic and set theory are now well. Perhaps logic will.

If you wish to be a little more abstract, you might say that mathematics. Geometry was originally about things like points. Much later, by no longer. In another direction, by considering the relationship. Similarly, abstractions of the concept of number led to modern algebra - -. These diverse mathematical constructs are all examples of. In another direction.

Leibniz and Isaac Newton (1. Already you can see that the line dividing the mathematics of geometrical. Geometry. except for its most abstract incarnation as topology, depends on. But in the other direction, mathematical analysis depends on topology for.

Analysis has also evolved from dealing with functions. Euclidean" space, to functions which "live". This process can. Euclidean geometry". We will. discuss noncommutative geometry. Another way in which the line dividing the study of number from the.

Mathematically, one speaks of symmetry using the language of "group theory". Lie groups". Perhaps another way, then, to describe what mathematics is "about" is to. These may be patterns which can be. In this case, we are talking about mathematical biology or.

This is the stuff of "applied" mathematics. However, and without meaning to detract from applied mathematics, this.

Kähler. manifolds", "Banach spaces", "Hopf algebras", and "cohomology. That's roughly as far as it seems worthwhile to discuss what mathematics. So lets turn now to more. It is probably fair to say that.

There is a general awareness of the sorts of things that researchers in. However, this is not so, by and large, for mathematics. Anyone with a good.

Yet. undergraduate courses in these topics hardly touch on the questions. At the same time, there. But this situation need not.

It is quite possible to sketch out the lay of the mathematical. It has to be admitted to begin with that boundaries between different. In some cases, specialized branches disappear. And of course. new branches appear from time to time. In mathematics, "chaos theory". Nevertheless, the main branches have been fairly stable over the last.

What we can recognize as the main branches often have. But the modern form of these branches. There's good reason to think that these branches will. Mathematical analysis. The branch whose modern shape goes back the furthest, perhaps, is. It can be traced back to the calculus of Newton. Leibniz. The primary object of study is functions - - correspondences.

For example, in physics, the motion in space of a physical. Newton's laws of motion were formulated in terms of such functions and. Integration of functions. Integration is used to define the area of irregularly shaped objects and. Calculus, as formulated by Newton and Leibniz was exceptionally useful. About 2. 00 years, or more, were required.

This process is still going on. In. quantum field theory, for instance, there is a notion of integration (Feynman.

Set theory and point set topology were both byproducts of putting analysis. They provided the framework which made it possible.

Point set topology was an abstraction from geometry which retained almost. In these terms, a continuous function is simply one which. Physical situations are still modeled, just as they were in Newton's. That is, the unknown in the. The equation is usually. What is required is to find functions that satisfy the. For a Newtonian. equation of motion, the function which is a solution describes the.

One would like to find all possible functions that satisfy the. In some cases, there may be a unique function which is. The simplest differential equations of Newtonian mechanics involve.

These are usually. But things can become complicated quickly. When several. interacting objects are considered, the equations involve multiple. The two main new physical theories of the 2.

Einstein's equation, Schrödinger's equation. Dirac's equation, etc.) Not surprisingly, perhaps, such equations are. But there are techniques. In many cases, it may not be possible to prove that solutions. Physically this would mean that the. These difficulties are not an inadequacy. Even for a classical.

Navier- Stokes equations. One way to look at the development of mathematical analysis in the 2.

The first. step is to take a different point of view on differential equations and. The second step is to add both.

Taking these two steps, one gets what is known as. David Hilbert (1.

Hilbert problems" in 1. His work gave us "Hilbert space".

Any finite dimensional vector. Hilbert space. The more interesting examples of Hilbert spaces, however. The axioms of linear algebra define the algebraic structure.

Hilbert space. These axioms allow any two elements to be added together. Any element can also be multiplied. Finally, the axioms specify that there is a "scalar product" between. The inner product is very important, because it's not only an algebraic. It does this. by making it possible to define a "norm" on the space, which is like.

This, in turn, makes it possible to define a "metric". The distance between two elements is. Given the machinery of Hilbert space, it is possible to translate a. Hilbert space. What this buys for you is the ability to talk about. This is because it is possible to prove powerful. Hilbert space, such as the "spectral theorem". This theorem states that for certain types of operators there exist.

In. other words, the operator may change the length but not the direction of. Such elements are called "eigenvectors" of the. What is. so useful about that?

Well, for instance, in quantum mechanics it. And there are ways. Mathematics normally progresses by generalizing useful results. Given. Hilbert space as a model, there are generalizations that don't have.

Hilbert space, such as the inner product. But one. can still consider vector spaces that are assumed to have a norm, without.

Or, taking things a step further, one may suppose merely the existence. There is a good. reason why such generalizations are worthwhile to make. The reason is. that important theorems may be proven in the more general case, without.

The consequence is that the proofs are. So far in talking about functions we haven't been specific about. Normally, in problems. Rn. with real coordinates).

But from a mathematical perspective. By complex numbers we. Most of the concepts of.

But some of the results are strikingly. For example, if a complex function has even a. Functions of this kind. Complex analysis was developed largely in the 1.

Augustin- Louis Cauchy (1. Bernhard Riemann.

Another issue that is more prominent with complex functions. This situation requires. Riemann worked out how.

Riemann surfaces" in the process. This work provided. Most major issues in complex analysis were resolved by the early.