Abstract
Rigour in proof is of utmost importance for the pluralist, since he has no solid ontology to ground his theory, and his conception of ‘truth’ is also relative (to a theory). In the first section we look at the pluralist’s motivation for rigour. In the second section, we develop a characterisation of rigorous proof. There are several characterisations varying over the account of meaning we attach to mathematical claims and axioms. In the third section, we evaluate the characterisation with reference to our motivation. Lastly, we draw some general conclusions for the pluralist. With the analysis we discover that rigour is a regulative ideal, sensitive to philosophical inclinations.
This chapter is co-written with Pedeferri. A less pluralist oriented version of the chapter is in the form of a paper also co-written with him (Friend and Pedeferri 2012).
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-94-007-7058-4_16
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Notes
- 1.
There is an impredicativity in the account of the pluralist. This is also a subject for future study.
- 2.
For a discussion of motivations for rigour which are not necessarily pluralist see Friend and Pedeferri (2012).
- 3.
‘Science’, as used here, is not confined to physics, chemistry and biology. Rather, it means any form of rigorous, earnest and open enquiry.
- 4.
For example, the proof uses Eilenberg-MacLane-Steenrod axiomatization of homology and cohomology in topology, yet it is a problem “in number theory”. Nor is the proof restricted to these areas of mathematics. In 2009, Mark Kisin simplified Wiles’s proof “so it does not really use algebraic geometry, but is still all about the ‘cohomology’ that Grothendieck invented and which descends through Cartan and Serre from Eilenberg-MacLane-Steenrod.” (McLarty, personal correspondence, 2010).
- 5.
There are exceptions. Under a formalist influence, a mathematician will try to hide the genesis of a proof or idea. Thus, he presents the proof as standing on its own.
- 6.
As a side note, it is interesting to think of reductio proofs as giving us the limitations of internal implications. Think about reductio proofs as telling us that if we introduce notion x (expressed as a wff), then the theory becomes absurd. Here, since we are sensitive to the paraconsistent logicians, ‘absurdity’ can mean non-triviality, rather than non-contradiction.
- 7.
Wright refers to this as the width of cognitive command, and the greater the width, the greater the objectivity of the notion (Wright 1992). Squaring the pluralist account with Wright’s sensitive treatment of objectivity is the subject of future research.
- 8.
This is because the ‘probability’ being evoked here is subjective probability, and depends on knowledge and experience.
- 9.
To complicate matters, we should also recall that the minima of one person is not the minima of another: so, for example, we might show that we only need three axioms and one rule of inference to generate a proof of a certain conclusion. Someone else might say that this is not minimal in an interesting sense, since some of the axioms are very powerful. It would therefore be better, according to that person, to show that the same conclusion can be deduced, maybe using more axioms or rules, each of which is less powerful. Reverse mathematics, as it is practiced, makes particular assumptions about what counts as ‘minimum’. The choice can be questioned.
- 10.
It can also be a disadvantage (Goethe and Friend 2010, 285).
- 11.
There are some delicate exceptions. For example, the rule of induction is not as strong as the second-order axiom of induction. There are some theorems we can prove using the axiom which we cannot prove using the corresponding rule. The difference has to do with the universal quantifiers in the axiom, since this is different from the implicit ‘quantifier’ of when we can apply a rule. Tait pointed this out to me in conversation at the AMS (American Mathematical Society) meeting March 2012. These subtleties need not concern us here.
- 12.
There is a limit. Because of the ‘naturalness to humans’ clause, a logic with the Sheffer stroke as the only connective is technically nice, but is not very natural because of the awkward match with natural language. Moreover, different linguistic communities will have very different notions of what counts as natural, for example concerning negation.
- 13.
This second part is not explicitly acknowledged by Dummett et. al. because they wanted meaning to be strictly additive. That is, it should be possible to understand one rule in isolation of the others, in order to respond to the manifestation requirement, dear to Dummettian anti-realists. We think that put starkly, this is naïve. Context does a lot of work in helping us to understand a term or concept. Put another way: while it is quite right that we can take a formal set of rules, and modify them one at a time, it does not follow that the meaning of, say, conjunction is necessarily unaffected if we ‘only’ explicitly modified our rule for conditional introduction.
- 14.
Constructivist mathematicians and logicians try to give an elimination and introduction rule to one connective symbol at a time. This is in order to separate the meaning of, say, conjunction from that of, say, disjunction. One motivation for this is that if we find that we have problems with the system – it generates unwanted conclusions, we can make minimal re-adjustments. All of this is correct, however, I do not think we can escape the idea that part of the meaning is implicit in the other rules in the formal system. See Appendix 2, where we discuss Prior’s rules for the Tonk connective.
- 15.
I am aware that some readers will start to feel distinctly nauseous at this suggestion. If you feel this way, then please, rest a while, have a wee dram, and when you feel a little stronger, consult the section on nausea in Chap. 11.
- 16.
The view is that the exercise of giving a formal representation of an idea, or group of ideas, is an exercise in deepening understanding, since it gives us something relatively precise and fixed to measure our original concepts against.
- 17.
In this respect the pluralist distinguishes himself from a Brouwerian intuitionist who locates all meaning and real mathematics in the mind and never on the written/typed, page. For the Brouwerian, formal representation fixes and therefore necessarily distorts real mathematics.
- 18.
For interesting conjectures of how metaphor and symbol influence mathematical development see Johansen (2010, 193–194).
- 19.
We could make the notion of proof more complex by allowing impredicative definitions or contextual definitions, where a contextual definition is one where the biconditional of definition is within the context set by quantifiers, and where one side of the biconditional of definition is an equivalence relation.
- 20.
Enriques’ most important proof was that the “characteristic series of a “good” complete continuous system of curves on a smooth algebraic surface F is complete. (Here “good” in the old Italian terminology means not superabundant, or, in modern terms, the first cohomology group of a divisor class should be zero).” (Babbitt and Goodstein 2011, 244.) The proof had a gap. This was not ‘filled’ until the appropriate algebraic tools were developed much later. This is one of the many interesting cases of mathematicians feeling that they are right, and being proved right much later.
- 21.
There was a rather heated dispute between Enriques’ and Severi who criticised his proof in print.
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Friend, M. (2014). Rigour in Proof. 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_8
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