Abstract
Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
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- 1.
See (Davis & Hersh, 1981)
- 2.
Almost verbatim quote from Bonsall (1982, p. 13)
- 3.
There are exceptions. Consider the pons asinorum proof found by Gelernter’s program, which showed that the angles of an isosceles triangle are equal by noting that triangle ABC is congruent to triangle ACB (i.e., its mirror image) (Hofstadter 1999). While it can certainly be called an ingenuous move, that appreciation is not shared by the program itself and did not play a role in its reasoning or discovery.
- 4.
Loosely adapted from (Hofstadter 1981/1985, p. 76–77)
- 5.
Loosely adapted from a quote in (Hofstadter 1981/1985, p. 75)
- 6.
Adapted from a quote by Wittgenstein in (Avigad 2008, p. 330–331)
- 7.
By defining understanding by its constitution (physical or mental) or by an undefined wonder property, one could sideline all entities one isn’t keen to attribute understanding to (e.g., computers, other ethnicities, genders, or species) by marking out an inevitable difference in constitution or by simply denying the property (e.g., “humans can grasp meaning, computers can only pretend to” or “humans are conscious, but an artificial replication would be a zombie”) without specifying what makes the difference relevant. Such implicit chauvinism is much harder to substantiate if one must mark a difference in mathematically valuable performance. While still possible to deny certain performances the “mathematically valuable” attribute for chauvinistic reasons, one will be faced with the more demanding task of convincing a mathematical community which performances to (not) value.
- 8.
Vervloesem (2010) even argues that conceptual shortcomings could be the main reason why computer proofs are still only on the fringe of mathematical practice. Enriching this aspect would lead to increasingly interesting (and more easily readable) proofs.
- 9.
Adapted from a quote in (Hofstadter 1982, p. 15)
- 10.
Dennett’s (1986/1998) metaphor
- 11.
This section is loosely based on Dennett’s (2013) thought-experiment “Who is the author of Spamlet?” The mathematics is purely fictional.
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Delarivière, S., Van Kerkhove, B. (2017). The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding. In: Sriraman, B. (eds) Humanizing Mathematics and its Philosophy. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-61231-7_16
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