Research in pure mathematics

Pure mathematics is the study of a priori truth --- facts that hold because they must hold, because logic mandates them. You could also say that these are things which are tautologically true, or true "by definition." But that does not mean they are necessarily simple or trivial. Is it obvious that there are infinitely many prime numbers? Not if you've never seen the proof. And yet this is true solely as a consequence of the definition of prime numbers.

The infinitude of primes has been known for over two thousand years. Mathematics extends as far back as written history.

Mathematical truths are tautological --- so they can never be falsified. Because of this, the discipline of mathematics is fundamentally cumulative. It keeps building on itself, creating more and more elaborate structures. All of the "easy" things in math were discovered very long ago and are now known to every educated person. More advanced topics are hard to explain if one has to jump over intermediate material, as if you tried to explain division to someone who did not already know multiplication. Actually, much advanced math logically cannot be explained to laymen: once you had covered all of the necessary intermediate material, the listener would be an expert himself!

Understanding advanced math requires patience, but it does not require a tremendous intellect. Although mathematical research is cumulative, it periodically undergoes unifying simplifications, so that the subject does not really get more complicated as one goes deeper into it. Anyone who can do long division is probably capable of learning as much math as they want, if they're willling to spend the time.

It is difficult to imagine having an industrial civilization without calculus, or any civilization without arithmetic. What about more advanced recent topics in mathematics? Here are some applications:


• vector calculus in electricity and magnetism
  
• Boolean algebra in computer science
  
• finite-state automata in linguistics
  
• differential geometry in physics (general relativity)
  
• game theory in economics
  
• Banach spaces in engineering (control theory)
  
• group theory in chemistry
  
• number theory in cryptography
  
  
Actually, in many cases the real mathematical theory --- the body of theorems which are regarded by mathematicians as central --- is not used very much in applications; but the basic concepts are crucial. Perhaps the greatest contribution of modern mathematics to science is to provide good definitions.

That is harder than it sounds! Who would have come up with Boolean algebras, if not us? But we had to come up with them. Modern mathematics without Boolean algebras (or topological spaces, or groups, or Banach spaces, or varieties) would have an obvious gaping hole.

Fermat's last theorem --- the statement that there are no whole number solutions to the equation xⁿ + yⁿ = zⁿ with n > 2 --- doesn't appear to have any applications, in itself, and perhaps never will. But the ideas that were developed (by Andrew Wiles and others) in order to prove it are stronger candidates for future usefulness. This is one reason why mathematicians are so interested in proving theorems: often, the ideas that underlie a proof are more significant than the conclusion.

There are also whole areas of mathematics, like set theory, which don't seem likely to ever have much practical significance (though you can never be sure). These subjects are needed only for the internal coherence of mathematics. But they are the exception, not the rule.

One clear lesson from the past is that subjects which initially seem unrelated sometimes end up being closely linked, and the most important application of any given branch of mathematics often has nothing to do with its historical origin. It just happens that a lot of what we do does end up being useful in one way or another, something Eugene Wigner referred to as the "unreasonable effectiveness" of mathematics in the natural sciences. Conversely, mathematicians have always derived, and continue to derive, great inspiration and insight from work in other fields, especially physics.

Definitely. The twentieth century has witnessed a vigorous development of abstract mathematics. Today's mathematics differs from math in the year 1900 at least as much as today's physics differs from the physics of that time.

Hilbert's twenty-three problems give an indication of the progress math has made in the past hundred years. These problems were posed by the great German mathematician David Hilbert at the International Congress of Mathematicians in 1900. Although some of Hilbert's problems are vague, the more definite ones have mostly been solved --- with one big exception: the Riemann hypothesis. This is considered the most outstanding unsolved problem in pure mathematics at the moment.

In my specialty, C*-algebras, the central problem is to find a rigorous formulation of quantum field theory. For instance, the "renormalized" series that are used in quantum electrodynamics (the quantum theory of the electromagnetic field, QED) do not converge. In plain language, physicists have a sophisticated theory of QED which was originally developed in the late 1940s by Schwinger, Feynmann, and Tomonaga. It typically makes predictions in the form of infinite series, and just the first few terms of the series (or even just the first term) yield highly accurate results. But adding more terms makes the answers less accurate, and the series eventually diverge. Thus, the accuracy of the initial calculation shows that the theory must be "right" in some sense, but taken literally it is nonsensical.

To learn what C*-algebras have to do with this, take a class from me or read my book.

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