Question

Descartes, IIRC, somewhere says something about the vagaries of memory influencing our justification for believing in our memory, and thence for believing in proofs involving many steps that we have to mnemonically track as we go along. Not that our justification is entirely negated, but just weakened modulo hyperbolic doubt, say. Now, with respect to the preface paradox directly, does this paradox indicate that a properly modest mathematician shouldnotcompletely accept a proof that goes beyond a certain (arguably vague) boundary of length? I've read that some proofs go over 200 pages, for example. Granted, typesetting will be a factor in this kind of situation to the extent that irregular typesetting can inflate a text's size beyond necessity, but I rarely see full academic writers performing that kind of inflation operation (I can remember one and only one dissertation I read where the font was Courier and the spacing was egregious, so what probably could've been a 300-page document ballooned to ~600 pages). But so I doubt that most long mathematical proofs appear to be so long mainly on account of how they were physically jotted down. But at any rate, then, perhaps more intriguingly, does the viability of long mathematical proofs undermine the preface paradox instead?

Answer 1

Long mathematical proofs can be subject to the vagaries of memory and raise questions about their reliability. The preface paradox highlights the need for caution when accepting lengthy proofs.

Step-by-step explanation:

Long mathematical proofs can be subject to the same challenges as other forms of memory-based beliefs due to the vagaries of memory. René Descartes discussed this concept, known as hyperbolic doubt, which suggests that our justification for believing in our memory weakens when it comes to proofs involving many steps that require mnemonic tracking. While this does not completely negate our justification, it raises questions about the reliability of long mathematical proofs.

The preface paradox refers to the idea that a person should not completely accept a proof that goes beyond a certain boundary of length. This boundary is subjective and can vary depending on the complexity and typesetting of the proof. While some long proofs may appear inflated due to irregular typesetting, most lengthy mathematical proofs are inherently complex. Therefore, the viability of long mathematical proofs does not undermine the preface paradox but rather highlights the need for caution and careful analysis when assessing their validity.