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`Theoretical' Experimentation


While there is an ongoing crisis in mathematics, it is not as severe as the crisis in physics. The untestability of parts of theoretical physics (e.g., string theory) has led to a greater reliance on mathematics for `experimental verification'. This may be in part what led Arthur Jaffe and Frank Quinn to advocate what they have named `Theoretical Mathematics' (note that many mathematicians think they have been doing theoretical mathematics for years) but which we like to think of as `theoretical experimentation'. There are certainly some differences between our ideas and theirs but we believe they are more of emphasis than substance. Unlike our initial experiment where we are working with and manipulating floating point numbers, `theoretical experimentation' would deal directly with theorems, conjectures, the consequences of introducing new axioms.... Note that by placing it in the realm of experimentation, we shift the focus from the more general realm of mathematics, which concerns itself with the transmission of both truth and insight, to the realm of experimentation, which primarily deals with the establishment of and transmission of insight. Although it was originally conceived outside the experimental framework, the central problems Jaffe and Quinn need to deal with are the same. They must attempt to preserve the rigorous core of mathematics, while contributing to an increased understanding of mathematics both formally and intuitively.

As described in Arthur Jaffe and Frank Quinn's ``Theoretical Mathematics'': Toward a Cultural Synthesis of Mathematics and Theoretical Physics it appears to be mainly a call for a loosening of the bonds of rigor. They suggest the creation of a branch of theoretical (experimental) mathematics akin to theoretical physics, where one produces speculative and intuitive works that will later be made reliable through proof. They are concerned about the slow pace of mathematical developments when all the work must be rigorously developed prior to publication. They argue convincingly that a haphazard introduction of conjectorial mathematics will almost undoubtedly result in chaos.

Their solution to the problems involved in the creation of theoretical (experimental) mathematics comes in two parts. They suggest that

theoretical work should be explicitly acknowledged as theoretical and incomplete; in particular, a major share of credit for the final result must be reserved for the rigorous work that validates it. ([9] p.10)
This is meant to ensure that there are incentives for following up and proving the conjectured results.

To guarantee that work in this theoretical mode does not affect the reliability of mathematics in general, they propose a linguistic shift.

Within a paper, standard nomenclature should prevail: in theoretical material, a word like ``conjecture'' should replace ``theorem''; a word like ``predict'' should replace ``show'' or ``construct''; and expressions such as ``motivation'' or ``supporting argument'' should replace ``proof.'' Ideally the title and abstract should contain a word like ``theoretical'', ``speculative'', or ``conjectural''. ([9] p.10)
Still, none of the newly suggested nomenclature would be entirely out of place in a current research paper. Speculative comments have always had and will always have a place in mathematics.

This is clearly an exploratory form of mathematics. But is it truly experimental in any but the Baconian sense? The answer will of course lie in its application. If we accept the description at face value, all we have is a lessening of rigor, covered by the introduction of a new linguistic structure. More `mathematics' will be produced but it is not clear that this math will be worth more, or even as much as, the math that would have been done without it.

It is not enough to say that mathematical rigor is strangling mathematical productivity. One needs to argue that by relaxing the strictures temporarily one can achieve more. If we view theoretical (experimental) mathematics as a form of Galilean experimentation then in its idealized form `theoretical' (experimental) mathematics should choose between directions (hypotheses) in mathematics. Like any experimental result the answers will not be conclusive, but they will need to be strong enough to be worth acting on.

Writing in this mode, a good theoretical paper should do more than just sketch arguments and motivations. Such a paper should be an extension of the survey paper, defining not what has been done in the field but what the author feels can be done, should be done and might be done, as well as documenting what is known, where the bottlenecks are, etc. In general, we sympathize with the desire to create a `theoretical' mathematics but without a formal structure and methodology it seems unlikely to have the focus required to succeed as a separate field.

One final comment seems in order here. `Theoretical' mathematics, as practiced today, seems a vital and growing instititution. Mathematicians now routinely include conjectures and insights with their work (a trend that seems to be growing). This has expanded in haphazard fashion to include algorithms, suggested algorithms and even pseudo algorithms. We would distinguish our vision of `experimental' mathematics from `theoretical' mathematics by an emphasis on the constructive/algorithmic side of mathematics. There are well established ways of dealing with conjectures but the rules for algorithms are less well defined. Unlike most conjectures, algorithms if sufficiently efficacious soon find their way into general use.

While there has been much discussion of setting up standardized data bases to run algorithms on, this has proceeded even more haphazardly. Addressing these issues of reliability would be part of the purview of experimental mathematics. Not only would one get a critical evaluation of these algorithms but by reducing the problems to their algorithmic core, one may facilitate the sharing of insights both within and between disciplines. At its most extreme, a researcher from one discipline may not need to understand anything more than the outline of the algorithm to make important connections between fields.

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