Abstract
We further explore the pluralist’s conception of proof. In particular we contrast it to the so-called axiomatic conception of proof. The pluralist adopts an analytic conception of proof. Two claims are defended. One is that all proofs can be viewed as analytic. The second is that it is preferable to do so. The reason it is preferable is that proofs open our eyes to exploration not only towards further proofs in the same formal system, but analytic proofs also invite us to question the axioms and the contexts of proof. We exercise our sceptical caution, to lead us to much more fundamental types of exploration than we would have engaged in had we viewed proofs as axiomatic.
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- 1.
The notion of what counts as mathematics, can be geographically and historically extended if we ignore our modern, rigid, conception of proof. A suggestion, made by Bishop is to think of mathematics as an essential tool for coping with the environment. Coping includes: counting measuring, locating, designing, playing and explaining. If explaining is short of proving, then proving is not central to mathematics as practiced in many cultures, and most of the time by most of us. This point comes from François and van Kerkhove (2010, 129).
- 2.
It is understood that rules of inference and axioms are usually inter-definable, and therefore a natural deduction proof or sequent calculation are also axiomatic proofs.
- 3.
The page number for the quotation is not given, nor is it very important.
- 4.
As (François and van Bendegem 2010, 117) remark, “We all know… that “real” mathematical proofs hardly reach this high quality level [of an axiomatic proof]. (The usual claim (sic) [observation?] is that the first chapters of any introductory book in whatever area of mathematics satisfy this standard, but from the third chapter onward the standard is left behind).”
- 5.
An algebro-geometric proof uses algebraic tools which were ‘alien to the intuitions’ of the Italian geometers at the time. The algebraic tools were developed by Zariski, Weil and later by Serre and Grothendieck, after the death of Enriques (Mumford 2011, 250).
- 6.
Miller defended such a view in unofficial conversation at the Logic Colloquium meeting in Sophia in 2009.
- 7.
It is probably more customary to question the material conditional, since it receives a lot of attention in the literature. Instead, here I choose to question classical conjunction, in order to stretch the point, and show that nothing I sacred.
- 8.
The conventionalist version of this runs: if we insist that the symbol ‘&’ together with the rules implicitly defining it are a convention, then we are free to change said convention. Think of Hilbert: the symbols are essentially arbitrary, and are implicitly defined completely, and only, by the axioms and rules which mention them. Hilbert had an axiomatic view of proof. Regardless of what fuels one’s axiomatic view, under it we would be stuck with the proof and could make no further moves.
- 9.
For a long time, in books on the history of mathematics, there was the view that mathematics was to be identified with what was developed in Europe. Other remarkable developments made “outside” were recognised only if they fed in to European mathematics. Moreover, mathematics is cumulative, and once something is proved it is forever true – in fact, it always was true. This view is being questioned by present ethnomathematics (François and Van Kerkhove 2010) and by revisions in, and new views towards, mathematics (François and Van Bendegem 2010). The new views are pluralist.
- 10.
If we take the truth-table definition to be a stipulation, then that delimits the context when the rule may be deployed. This is fine temporarily, but sooner or later, it will be possible to go beyond the stipulation and ask what alternatives there might be and to what extent they make sense in particular contexts.
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Acknowledgement
I should like to thank Aberdein for useful comments on an early version of this chapter.
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Friend, M. (2014). A Pluralist Approach to Proof in Mathematics. In: Pluralism in Mathematics: A New Position in Philosophy of Mathematics. Logic, Epistemology, and the Unity of Science, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7058-4_12
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