Case Study I: Zilber’s Field December 15, 2007Posted by Alexandre Borovik in Uncategorized.
Many mathematical objects are exceptionally rigid, we cannot change them at will, they offer resistance. The “robustness” of a mathematical object (one may also talk about a dual concept: “robustness” of a mathematical theory which describes the object) has been explicated as a formal mathematical notion, categoricity, in the mathematical discipline called model theory (more precisely, in model-theoretic stability theory). One of the recent results in that theory, Boris Zilber’s work on the Schanuel Conjecture, can be described (but perhaps not formulated in full detail) at a level of elementary algebra/calculus/number theory. It poses a significant philosophical challenge which, to the best of our knowledge, so far has been entirely ignored by the philosophers of mathematics.
The Schanuel Conjecture is about transcendental numbers. It says that if you have complex numbers which are linearly independent over the rationals, and consider the system of numbers
then the latter has transcendence degree at least . The conjecture contains in itself a huge number of known results. For example, if one takes , the conjecture says that the system
has transcendence degree at least one, which means that is a transcendental number — a classical result of transcendental number theory.
Zilber took a number of natural (and known) algebraic properties of complex numbers and the exponentiation function for axioms (so that “exponentiation” is understood as a map which satisfies
added to them, as a further axiom, the formulation of the Schanuel Conjecture (still unknown) and proved:
[1.] The axioms are consistent, that is, they have a model, an algebraically closed field of characteristic 0 with a formal exponentiation function , such that all these axioms are satisfied in (and recall that all axioms, with a possible exception of the Schanuel Conjecture, are satisfied in the field of complex numbers with he standard exponentiation).
[2.] There is exactly one, up to isomorphism, such field of cardinality continuum.
And now we can formulate some questions:
- Why would almost every mathematician (with the possible exception of intuitionists and ultrafinitists like Alexander Yessenin-Volpin — but they have an honourable excuse) immediately agree that of course it should be true that and that therefore the Schanuel Conjecture should be true? What is the basis of this belief in “it should be true”?
- Why does Zilber’s theorem have a suspiciously foundational, metamathematical feel about it?
- Zilber’s field had been built by a version of the Fraisse-Hrushovski amalgam method, non-constructive and seriously transcendental. Why are most mathematicians prepared to believe that , despite the two objects having completely different origins?
The appearance of the Fraisse amalgam method on the scene should remind us that even in the countable domain we have a wonderful and paradoxical example of seemingly incompatible constructions leading to the same “universal” object. The famous random graph can be constructed probabilistically by coin tossing: vertices of the graph are natural numbers, and for every pair of vertices we toss a fair coin and, if we get heads, we connect the vertices and by an edge. The same graph can be constructed by a totally deterministic procedure: we take for the set of vertices the set of all prime numbers congruent to , and draw an edge between prime numbers and if is a quadratic residue modulo . In all cases the resulting object is the same, THE random graph. And — last by not least — the random graph can be constructed as a Fraisse amalgam of finite graphs (which actually explains the first two constructions).
Is the acceptance of actual infinity just the price that mathematicians are prepared to pay for the convenience (and beauty) of the “canonical” objects of mathematics?