Why mathematics?

Mathematics is perhaps the purest of all the pure mental endeavors of humankind. During the times of Euclid and Pythagoras, mathematics was seen as a pure mental exercise that could deliver truth and nothing but truth. Today, this view is subject to a minor modification: the ``truth'' is related to a set of starting propositions (called axioms). An axiom can be, in some cases, completely lacking intuitive content and beyond intuitive or empirical verification. In most of the cases, axioms are, however, propositions which we take as self-evident. From this view, the relevance of mathematics to our topic can be seen the following two ways:

The (apparent?) sense of absoluteness and universality of mathematics on its own and its relationship to thoughts;

The efficacy of pure mathematical argument on physical reality (by way of sophisticated theoretic physics).

For one thing, mathematics is seen by many as an exact deductive science which has its own reality. But unlike other disciplines in natural sciences, they think, a theorem is absolutely and universally true. As long as a theorem is proved by a mathematician, all mathematicians should be able to prove (at least to verify) the theorem as well and the theorem is considered simply proved. The strong belief that mathematics forms a consistent unity may justify our calling it mathematical realism. In this sense, mathematical objects (such as numbers, theorems, proofs, etc.) exist on their own and have objective existence independent of the minds of mathematicians. We may call them ``mathematical reality.'' According to this position, the job of mathematicians, exactly as their colleagues in physics, is to discover the hidden reality, so that the truth can ``fall into place.''

It is indeed this fascinating belief that has raised an interesting question: what exactly are the rules of mathematical reasoning and why don't the outcomes contradict each other? This is a topic of mathematical logic. Many questions are answered positively in this domain – mathematically Interestingly, as by-products of this discipline, different ``logics'' have been discovered (or developed). For example, the first order intuitionist logic that turns down the law of double-negation can be still shown to be compact and complete. Nevertheless, there are also many puzzling and pessimistic results, for example Gödel's Incompleteness Theorem .

It turns out that the development of mathematical logic has in many ways also aided the growth of modern computer science -- formal language, automata theory, proof theory, and recursion theory, to name a few area strongly influenced by mathematical logic. Moreover, it was the ambition of a branch of computer science -- artificial intelligence (AI) that again brought to light profound problems about the definition of mind. This, no doubt, will have significant impact on natural language understanding and/or processing. In fact, it is because of our customary way of treating logic (indeed, classical first order logic) as a better way of reasoning (for some, it is the perfect way) and taking other everyday reasoning (non-monotonic, modal, context-sensitive) as frictional or impure forms thereof that has led to many difficulties in AI (see Chapter 1 for examples).

We have to see that mathematics plays a crucial role in our contemporary understanding of physical reality. In a sense, this role is active and somewhat tyrannical. For one thing, mathematics is not just a crucial tool for describing experiments or observation. Rather, the description and prediction power of mathematics is attributed to Nature's agreeing with mathematics. Einstein, for example, spent more than eight years of his lifetime devoted to the development of the General Theory of Relativity without the slightest clue from physical experiments and observations. The ultra-high agreement of the General Theory of Relativity to observed data in some areas  certainly suggests that it is not merely a matter of luck. There must have been something in Einstein's mind that held the key to the mystery of the universe.

Indeed, many important discoveries of today's physics are guided by mathematical theories rather than the other way around (Gedanken experiments with pencil and paper alone are in principle mathematical exercises). The role of experiments is to confirm or refute an existing mathematical theory. The job of experiment is therefore passive in this sense. An experimentalist physicist will not be surprised to see outcomes predicted by a mathematical theory. On the contrary, she is surprised when the phenomenon predicted by the theory is not there.

An observation of the power of logic/mathematics renders a naive sub-symbolic approach highly implausible. For one thing, the sub-symbolic school is an alternative view seeing frictionless reasoning as an idealized version of a more subtle classical physical activity and attacking the difficulties of AI from the bottom up. In light of the efficacy of mathematics and logic, it is hardly imaginable that a mental framework emerging from this classical substrate may give rise to a highly abstract understanding of multidimensional geometry, for example.

(Author unknown)