Responses to ``Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics'', by A. Jaffe and F. Quinn''

AMS Bulletin 30 (1994), pp. 181-182

Abstract. It is argued that the information-theoretic incompleteness theorems of algorithmic information theory provide a certain amount of support for what Jaffe and Quinn call ``theoretical mathematics''.

[For the complete set of responses to Jaffe and Quinn in the AMS Bulletin, click here.]

One normally thinks that everything that is true is true for a reason. I've found mathematical truths that are true for no reason at all. These mathematical truths are beyond the power of mathematical reasoning because they are accidental and random.

Using software written in Mathematica that runs on an IBM RS/6000 workstation [5,7], I constructed a perverse 200-page exponential diophantine equation with a parameter N and 17,000 unknowns:

Left-Hand-Side(N) = Right-Hand-Side(N).

This work is part of a new field that I call algorithmic information theory [2,3,4].

What does this have to with Jaffe and Quinn [1]?

The result presented above is an example of how my information-theoretic approach to incompleteness makes incompleteness appear pervasive and natural. This is because algorithmic information theory sometimes enables one to measure the information content of a set of axioms and of a theorem and to deduce that the theorem cannot be obtained from the axioms because it contains too much information.

This suggests to me that sometimes to prove more one must assume more, in other words, that sometimes one must put more in to get more out. I therefore believe that elementary number theory should be pursued somewhat more in the spirit of experimental science. Euclid declared that an axiom is a self-evident truth, but physicists are willing to assume new principles like the Schrödinger equation that are not self-evident because they are extremely useful. Perhaps number theorists, even when they are doing elementary number theory, should behave a little more like physicists do and should sometimes adopt new axioms. I have argued this at greater length in a lecture [6,8] that I gave at Cambridge University and at the University of New Mexico.

In summary, I believe that the information-theoretic incompleteness theorems of algorithmic information theory [2,3,4,5,6,7,8] provide a certain amount of support for what Jaffe and Quinn [1] call ``theoretical mathematics''.

References

1. A. Jaffe and F. Quinn, Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. 29 (1993), 1-13.
2. G. J. Chaitin, Algorithmic information theory, revised third printing, Cambridge Univ. Press, Cambridge, 1990.
3. G. J. Chaitin, Information, randomness & incompleteness---Papers on algorithmic information theory, second edition, World Scientific, Singapore, 1990.
4. G. J. Chaitin, Information-theoretic incompleteness, World Scientific, Singapore, 1992.
5. G. J. Chaitin, Exhibiting randomness in arithmetic using Mathematica and C, IBM Research Report RC-18946, June 1993.
6. G. J. Chaitin, Randomness in arithmetic and the decline and fall of reductionism in pure mathematics, Bull. European Assoc. for Theoret. Comput. Sci., no. 50 (June 1993), 314-328.
7. G. J. Chaitin, The limits of mathematics---Course outline and software, IBM Research Report RC-19324, December 1993.
8. G. J. Chaitin, Randomness and complexity in pure mathematics, Internat. J. Bifur. Chaos 4 (1994), 3-15.