Abstract

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses.

Highlights

  • I mean an inference from particular instances to a generalisation

  • Frege articulates his scepticism about enumerative induction in mathematics in the following passage, in which he unfavourably compares arithmetical inductions to scientific ones: In ordinary inductions we often make good use of the proposition that every position in space and every moment in time is as good in itself as every other

  • Many mathematicians and philosophers are sceptical about the value of enumerative inductive evidence in arithmetic, especially in the absence of further supporting evidence

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Summary

Introduction

I mean an inference from particular instances to a generalisation. So from ‘ A1 is F’, ‘ A2 is F’, ..., to ‘All As are F’. Let’s suppose that, in this third case, your only evidence (or relevant knowledge) for truncated-RH consists of 4 × 1018 randomly chosen values of s drawn from the set of complex numbers of modulus less than or equal to 1. Once more, all these instances corroborate the hypothesis. Not so in scenarios 2 and 3, where the sample points—the evidence—don’t seem biased in any obvious way, as they are drawn randomly from the entire set To put this in a broader context, let’s compare this form of inductive scepticism to some others.

Sceptics
How much does it matter?
Varieties of size-scepticism
Frontloading of evidential value I
Frontloading of evidential value II
C-scepticism
Questioning the framework
S-scepticism and u-scepticism
Other orders
Conclusion
Full Text
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