Putting a price on reliability

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In the last two sections we talked about the importance of communicating insights within the mathematical community. There we focused on the process of mathematical thought but now we want to talk about communicating insights that have not been made rigorous.

We have already briefly talked about Wilf and Zeilberger's algorithmic proof theory and its denial of insight. In this section we will discuss the implications of this theory and D. Zeilberger's philosophy of mathematics as contained in Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture ([16]). It is probably unfortunate but perhaps necessary that the two voices most strongly advocating truly experimental math are also at times the most hyperbolic in their language. We will concentrate mostly on the ideas of Doron Zeilberger but G. J. Chaitin should not and will not be ignored.

We will begin with D. Zeilberger's ``Abstract of the future''

We show in a certain precise sense that the Goldbach conjecture is true with probability larger than 0.99999 and that its complete truth could be determined with a budget of 10 billion. ([16] p. 980)

Once people get over the shock of seeing probabilities assigned to truth in mathematics the usual complaint is that the 10 billion is ridiculous. Computers have been getting better and cheaper for years. What can it mean that ``the complete truth could be determined with a budget of 10 billion?'' What is clear from the article is that this is an additive measure of the difficulty of completely solving this problem. If we know that the Reimann hypothesis will be proven if we prove lemmas costing 10 billion, 2 billion and 2 trillion dollars respectively, we can tell at a glance not merely what it would `cost' to prove the hypothesis but also where new ideas will be essential in any proof. ( This assumes that 2 trillion is a lot of `money'.)

The introduction of `cost' leads immediately to consideration of a trend that has over taken the business world and is now intruding rapidly on academia: a focus on productivity and efficiency.

It is a waste of money to get absolute certainty, unless the conjectured identity in question is known to imply the Riemann Hypothesis ([16] p. 980)

We have taken this quote out of its context (Wilf and Zeilberger's algorithmic proof theory of identities) [16] but even so we think it is indicative of a small but growing group of mathematicians who are asking us to to look at not just the benefits of reliability in mathematics but also the associated costs. See for example A. Jaffe and F. Quinn in [9] and G. Chaitin in [5]. Still, we have not dealt with the central question. Why does D. Zeilberger need to introduce probabilistic `truths'? and how might we from a `formalist' perspective not feel this to be a great sacrifice?

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