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The aims of the project, as seen by a mathematician December 15, 2007

Posted by Alexandre Borovik in Uncategorized.
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Most uses of infinity in mathematics are restricted to the recycling of a small number of “canonical” and ubiquitous structures. Can this phenomenon be explained by intrinsic mathematical reasons, or is it rooted in mathematicians’ “flight from uncertainty” and their desire for a (perhaps misleading) sense of security when working in the infinite domain?

Some recent results bring this issue into the focus of philosophical and metamathematical research. In the subsequentt posts, we outline four concrete case studies taken from mainstream research in mathematics and computer science.

The first one concerns Boris Zilber’s work on Schanuel’s Conjecture; it raises a range of puzzling questions about the nature of one of the most classical and mainstream objects in mathematics, the field \mathbb{C} of complex numbers. It is also one of the rare cases when philosophical considerations directly affect mathematicians’ belief in the validity of a concrete mathematical statement.

Another example is the metamathematical thesis “explicit = Borel” brilliantly illustrated by Simon Thomas on concrete group-theoretic problems. The thesis would not be so convincing without Kuratowski’s Theorem which gives a strikingly short list of standard Borel spaces. Here, the canonicity of particular infinite objects generates explanatory power in metamathematical discourse.

The third example is intentionally set on a deceptively elementary level: we discuss various versions of natural numbers as they appear in computer science and computer programming. One cannot get rid of the impression that if one of the paradigmatic principles of mathematics is its orientation towards unique canonical structures, computer science enjoys the unlimited freedom of creating a potential infinity of versions of even the most mundane everyday objects. Of course, this is due to the fact that the objects and structures of computer science are man-made. But what about the objects and structures of mathematics? Are they man-made too? This forces us to revisit the classical disputes — and not only of Platonists and formalists in philosophy of mathematics, but also the ancient dispute between realists and nominalists.

We believe that the issue of the identity and “sameness” of mathematical objects precedes that of infinity: infinity of what? We argue that the prominence of a small number of “canonical” structures is a key point in understanding the relations between the human mind and infinity, and they could not be understood without asking the same question in the finite domain — why, given the potential infinity of objects and concepts, does (finite) mathematics reuse a surprisingly limited range of structures?

The issue of mathematical identity is taken up in our fourth example. Here we look at a program, known as `higher-dimensional category theory’, one of whose central ideas is that to grasp identity properly we need to employ structures with infinitely many layers, \omega-categories. Although, at least a dozen definitions of these entities have been proposed, most researchers in this field are convinced they will be found to be the ‘same’ in a sense consistent with the program’s ideas.

The project, philosophical and metamathematical by nature, will be shaped as a dialogue between a mathematician and a philosopher. It should result in the formulation (perhaps even proof) of explicit mathematical conjectures which would move the issues of “canonicity” and “ubiquity” from purely philosophical into the mathematical realm. Even more importantly, our aim is to shift the discourse about infinity in mathematics from the rarefied heights of esoteric abstraction into the realm of the daily job of mainstream mathematicians.

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