Abstract
We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like.
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Notes
All of these themes are considered more closely in Van Kerkhove (2005).
Thurston (1994).
Van Bendegem (2004) discerns and explains in more detail.
A third popular procedure is abduction, or the prima facie logically unsound but practically successful inference from effect to cause.
For further philosophical discussion of possible types of mathematical experiment, we refer to Van Bendegem (1998).
See Pòlya (1945) and further publications by the same author.
Wilson (2003) is a good introductory historical source.
Highly improbable as it was, one should add. A simple mathematical argument shows that any counterexample to x n + y n = z n must satisfy n < x, y, z. As FLT had already been empirically checked up to n = 125,000, years before Wiles, any possible counter-examples were by then considered unaccessible by calculation only, and would have instead required considerable mathematical ingenuity and creativity themselves. For a general introduction to FLT, see Singh (1997).
For an elaborate treatment of the virtues of proving, beyond the mere establishment of theorems, see Rav (1999).
A state-of-the-art can be found in Wang (2004).
A result of Takeuti (1978) shows that a large part of analysis can be coded into elementary number theory on the condition that only predicative definitions are used. Should a proof of Goldbach’s Conjecture satisfy this condition, then there must exist a proof using only the proof methods of elementary arithmetic. However, such a proof will be highly complex, convoluted and near impossible to understand, hence of no explanatory value at all. It is in this sense that a proof of the conjecture will almost certainly have to go beyond elementary arithmetic.
For an historical overview, see Echeverría (1996).
Oliveira e Silva (2007).
Following this principle, the universe is assumed to be smooth, i.e. isotropic and homogeneous, on a very large scale.
Elkies (1988).
As a matter of fact, mathematical realists, Platonically oriented or otherwise, often feel the need to complement the notion of formal proof with such concepts as mathematical intuition. The most famous of them was Kurt Gödel, who wrote: “We do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception,...” (Gödel 1983, p. 484).
Projects in which are invariably met with suspicion, ever since Appel and Haken. Consider the open matter of whether the computer aided proof by Thomas Hales of Kepler’s Conjecture, stating that no packing of congruent spheres can have a density greater than that of the face-centred cubic packing, can be accepted or not. See Hales (2000) and Aste and Weaire (2000).
The paradigmatic example is the Classification Theorem of Finite Simple Groups, also known as the Enormous Theorem, according to which all simple groups of finite order can be classified into one of four types. The purported proof, the result of a programme launched and coordinated by Daniel Gorenstein (1923–1992) from the early 1970s, now in its entirety consists of some fifteen thousand journal pages, spread over hundreds of articles (some of which still unpublished), and written by dozens of authors. See Solomon (2001).
Corfield (2004, pp. 41–42).
Corfield (2004, p. 40).
See http://www.ericr.nl/wondrous/pathrecs.html for an overview of pathrecords.
For more detail and further elaboration on various aspects of mathematical inquiry in connection with this conjecture, see Van Bendegem (2005).
With this peculiarity in mind that the probabilistic statements in question, i.e. of the type P(s) = x, s being a mathematical conjecture and 0 ≤ x ≤ 1, do have the status of absolute results.
More details about probabilistic methods for recognizing primes can be found in the third chapter of Crandall and Pomerance (2001), on which we have largely drawn for this brief overview.
This paragraph has been mainly based on Goldreich (1996).
“Though such space-times are problematic in various ways, they are, we contend, not beyond the pale of physical possibility. If the Creator had a taste for the bizarre we might find that we are inhabiting one of them” (Earman and Norton 1996, p. 250). See chapter four of Earman (1995) for a detailed treatment, as well as the more recent papers (Németi and Andréka 2006), addressing the physical possibilities of future computers exhibiting “beyond-Turing power” in the light of recent astronomical results, and (Welch, to appear), offering a deep philosophical analysis of computational limits in Malament–Hogarth universes.
This is a mathematical variant of a general supertask, which challenges one to perform an infinite number of operations in a finite time span, the most famous example being of course that of the Thomson lamp. Having available a limited time span, from t = 0 to t = 1, and the particular lamp being switched on at \(\hbox{t}=\frac{1}{2}\), off at \(\hbox{t}=\frac{3}{4}\), on 3 at \(\hbox{t}=\frac{7}{8}\), and so on, the set task is that of determining whether the lamp will be on or off at t = 1. See Laraudogoitia (2004).
Nevertheless, it is possible to point to more subtle differences. Take first order predicate logic. It is undecidable, for it has no decision procedure. But, unlike in our universe, in Malament–Hogarth space it is possible to go through all (i.e., countable infinitely many) potential proofs of a given proposition.
Wigner (1960).
Hersh (1991).
The paper arousing this controversy was Jaffe and Quinn (1993).
Zeilberger (1993, pp. 980–981).
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Our appreciation goes to two anonymous referees, for direct comments on previous versions and valuable suggestions. The former author also acknowledges support by the Alexander von Humboldt Stiftung, Germany.
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Van Kerkhove, B., Van Bendegem, J.P. Pi on Earth, or Mathematics in the Real World. Erkenn 68, 421–435 (2008). https://doi.org/10.1007/s10670-008-9102-5
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DOI: https://doi.org/10.1007/s10670-008-9102-5