The previous article about the relations of mathematics and physics was here.
Many mathematicians are rigor addicts. Bourbaki are their true ancestors. What distinguishes a mathematician from a physicist is that she really enjoys when her proofs are proved so that the statements become completely undisputable and independent of any arguments that are not based on pure logic.
This is not the kind of ideas I want to talk about in this text. The main reason is that rigor is about the form how the ideas are expressed and dealt with. It says very little about the bloody content. Moreover, rigorous proofs typically require the mathematicians to make many choices when they are building their proofs and these choices are not unique. A physicist could say that these choices are analogous to gauge choices in gauge theories. These choices are not yet "physically" or "invariantly" important. The truly important results - the essence of mathematics - is the quotient of the proof and the groupoid of all possible changes of the chosen details that constitute the proof. Clearly, many technologically gifted mathematicians will disagree because they consider every trick used during a proof to be a piece of art, but I think that they are not right.
Every proof is a skeleton that contains not only the ultimate beauty of the mathematical structures but also a lot of bones that will eventually have to be removed as soon as the beauty becomes a part of our life and culture, a part that is taken seriously. These small bones can play an important role sociologically because they are often necessary for a result to be accepted; however, their role is destined to diminish with time.
Moreover, it is possible to design a computerized system that can verify, with a complete certainty, that a proof written in a certain language is indeed correct. That might be interesting but I am surely not the only one who thinks that mathematical insights are more than a dull sequence of mechanical procedures.
What kinds of ideas are there in the abstract world of mathematics?
I am going to start with mathematical logic and set theory. This subfield is a sophisticated version of the science about the logical paradoxes such as the paradox of a liar. No doubt, this field is interesting. It has consequences for philosophy. From a physicist's viewpoint, however, it deals with ideas that are more likely to be important for picky bureaucrats rather than Gods who create the worlds. One of the reasons is that the validity of statements such as the axiom of choice cannot be tested experimentally in any world that qualitatively resembles ours. Gödel's theorems about various system of axioms can be viewed as the culmination of this subfield.
Another related subfield is computer science, classical combinatorics, game theory, abstract "nonlinear" algebra, and closely related problems. The virtue of the mathematical structures studied by this subfield is that their meaning is completely transparent although it may require a lot of effort to find many conclusions about them. Nevertheless, the focus is on discrete objects and the methods to study them are discrete, too.
The previous sentence distinguishes computer science from another subfield that deals with discrete objects, namely number theory. It is the part of mathematics that studies the properties of integers such as the properties of the set of primes. The difference from computer science is that while number theory focuses on discrete objects, it does not avoid continuous tools to find the answers. Various combinatorial coefficients are encoded in generating functions. The distribution of primes is encoded in the Riemann zeta function. It is a function of a complex variable whose detailed analytic properties can be studied by continuous methods but they still have important consequences for the discrete questions.
I view the relevance of continuous structures for discrete questions as a very deep discovery in mathematics. This link is behind many relations between objects - even pairs of discrete objects - that could not have been foreseen in the past.
Now we have gotten to calculus. It studies mathematical functions of real and complex variables, their sums, integrals, and differential equations. Surely, the statements about them can be written using a finite number of characters. But they talk about functions - objects from a set that is not countable. Addition, subtraction, multiplication, exponential, and logarithm are the basic operations and all of their finite combinations that can include complex numbers are the so-called elementary functions. Functions that solve easy enough differential equations with simple enough boundary conditions - although they can't be expressed using the operations from the last sentence - are the "simple" special functions. One can combine them into more convoluted structures and study their properties, too. Again, the level of complexity can increase indefinitely but surely many readers share my feeling that we are not missing much if we don't have a name for the most general solution of a 4-th order differential equation with polynomial coefficients of 7-th order.
There are various subfields of mathematics that I will call "interdisciplinary" in this classification that include the application of the rules of calculus to discrete structures such as the finite fields based on prime integers as well as exotic number systems such as non-Archimedean numbers that are neither discrete nor continuous in the usual sense. These interdisciplinary subfields use the methods of both adjacent subfields. I also chose to remove statistics and probability from my classification because of the overly applied character of these fields.
The last category I want to mention is linear algebra combined with geometry and modern geometry. The focus here is on linear structures or properties of other structures that are inherently linear. Lie groups and their linear representations, vector bundles and their exotic generalizations, index theorems, algebraic topology and topological invariants, homology, cohomology, and K-theory fit squarely into this category. No doubt, this category is heavily overlapping with theoretical physics and string theory has made the links even tighter. Many people assume that this category of mathematics is guaranteed to strengthen as the research of string theory continues.
Are we missing something? Quite likely, I have omitted an important subfield of mathematics. But imagine that I have not. Is it possible that future mathematicians will study structures that don't belong to either of the categories above? What kind of concepts could such a new branch of mathematics use?
Which branches of mathematics will gain importance in the future? It seems likely that the extended "computer science" category will always be growing because it studies an infinite set of ideas where new questions - more complex questions - can always be found and pursued. In this sense, the extended "computer science" is analogous to biophysics. In both cases, the closeness to applied sciences is obvious, too.
But let us forget about this category and think about "pure mathematics" only. Is it possible that in 50 years, the leading mathematicians will look at a completely new category of questions? Or is it possible that most mathematicians will study extensions of something that we currently consider marginal - such as the p-adic numbers? Of course it is possible. But do you have an idea how the new structures and the basic theorems that will make them interesting will look like?
Moreover, I believe in some sort of complete unification of pure mathematics - and pure mathematics with theoretical physics - in the idealized future. In other words, when string theory is completely understood, it could include all truly nice mathematical structures and it could maximally trivialize all important proofs of mathematics.
Already today, when you ask a physicist about a proof of a certain theorem - for example Jacobi's obscure identity - the physicist will probably think about supersymmetry in the RNS formalism where the identity simply follows from spacetime supersymmetry. Of course, we know many examples like that. When you embed an interesting mathematical structure into string theory (or a simpler model from theoretical physics), the entire toolkit of theoretical physics is available for your considerations and many proofs can become very efficient.
Could it be that all interesting mathematical structures that satisfy interesting nontrivial constraints can be isolated in string theory or one of its limits? I wouldn't be surprised. Another questions is: could you ever construct rigorous proofs of theorems based on the straightforward and powerful intuition and experience of string theory?
The answer to the last question clearly depends on the set of axioms that you are allowed to start with. When you invent a rich structure in theoretical physics that contains a mathematical object as a tiny portion of its spirit, you always face an uncertainty: does the rich structure actually exist? For example, is there an S-matrix for every value of the moduli in type II string theory that obeys all the known constraints and reproduces all the known limits?
Even though we don't have yet proofs of such assertions that would be rigorous according to the usual mathematical standards, theoretical physicists know very well why they're more or less sure about these statements, at least in the highly supersymmetric sectors of the landscape.
This fact makes it legitimate to postulate the existence of string theory, its vacua, and the known dualities as axioms. Such a system of axiom is almost certainly consistent. Of course, dualities are normally used by most string theorists as axioms because when they reduce a new physical phenomenon to the old dualities supported by overwhelming bodies of evidence, they are finished.
I certainly don't want to claim that extending the axiomatic system by new insights that have been supported by overwhelming evidence is the only way to make progress. The physical observables in string theory are in principle rather ordinary objects and the statements about them should be provable with the full standards of mathematics if we needed to do it. Nevertheless, extending the system of axioms is one of the ways to go.
Many facts about string theory that we would add to such a system of axioms may look provincial, local, and ad hoc. Such an axiom is only relevant in one or several corners of the "landscape". What we would be happy to find, of course, are other axioms that are more fundamental and that imply many of the individual, local axioms about particular vacua and their dualities.
The possibly primordial axioms we can imagine today are the consistency criteria of quantum gravity. I think that many more people should try to find a proof of various implications from the general rules of quantum gravity to more specific insights of string theory. For example, prove that in 9 flat dimensions, if you have supersymmetry, you can always find a scalar field whose vev may be used to decompactify a 10th dimension. Prove that the rank of a gauge group in a 9-dimensional supersymmetric theory of quantum gravity can't exceed 18. Prove that a supersymmetric theory in 5 or more dimensions inevitably contains solitonic strings, they can be quantized, and they can be weakly coupled at a point of the moduli space. Prove that in 4 dimensions, a comparably powerful result holds. Prove string theory. Neither of these nontrivial assertions is true in effective field theory but with additional conditions from quantum gravity added, they could be true and possibly provable in finite time.
Such a full or partial proof that quantum gravity implies string theory could also be used as a template for other proofs in mathematics whose goal would be to classify all interesting mathematical structures: the classification of compact simple Lie groups remains an inspiring textbook toy model. It is very likely that we will never find a complete classification of all interesting fundamental ideas in mathematics and physics - and probably not even a definition of such a grandiose entity. But it is conceivable that we can get much closer to this goal than what we can dream about today.