MATERIA PRIMA

In (Larvor 2012), Larvor sets out the case for the existence of essentially informal arguments (from which we have essentially informal proofs too, as those proofs which involve essentially informal arguments). Hereby we can identify a substantial area which is not covered by the traditional approaches to the philosophy of mathematics, which also requires investigating math- ematical practice. The philosophy of mathematical practice can then gain traction and make the perspectival shift being advocated clear. Larvor’s

main proposals are that essentially informal arguments are distinguished from formal arguments by theircontent-dependence and, furthermore, that a proper account of content-dependence will include a broadening of the picture of inference beyond merely linguistic argument to a full class of

inferential actions. This is because the content of mathematics, and that which we act on inferentially, is not limited to language alone.

In what way are essentially informal proofs content-dependent? Larvor argues that the content of a proof connects to some domain the proof is located in, and that this domain has some class of acceptable inferences that can be employed in proofs in this domain. A proof is then valid if all of the inferences used in the proof are acceptable in the domain, and have been applied properly etc. Importantly, formal rules are acceptable across all domains (such as modus ponens)16, but Larvor’s picture allows us to also have domains with more contentful moves which are not generally applicable across all domains. While this might mean some inferences are specific to a very restricted domain, many are in fact acceptable across a broad range of domains without this being so broad as to include all domains. A content- dependent proof will then make use of these content-dependent inferences.

Further clarification is needed, though. For instance, the above has not yet told us about what content is or what counts as a domain. We would be in big trouble if the domains were so fine-grained to have it that each purported proof is located in its own domain, with the acceptable inferences being precisely those employed, as this could trivialise mathematics and the whole notion of proof. The right answer seems to be that domains are particular areas of mathematics, with particular frameworks for acceptable inferences established through mathematical practice, though it would be wise to follow L¨owe & M¨uller in including some context-dependence in this. The content is merely the subject-matter of the proof, which connects to the domain in the straightforward sense that, for example, a geometrical proof reveals that we are working in the domain of geometry and thus authorises the use of geometrical moves in the proof.17

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There is, here, the obvious question of which logic determines the rules that are acceptable across all domains, or whether there even is such a thing. I don’t think it would be a bad thing for the position being advocated here if there is indeed no universal background logic. The logical pluralist in me certainly thinks so. Nonetheless, I will set aside these issues as outside the scope of the current discussion.

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We should be careful, of course, to avoid the subject-matter merely being comprised of some set of acceptable inferences, as this threatens circularity.

The account of informal proofs as depending on content as well as their form does lose us some nice features that a Formalist-Reductionist account would have. For example, if all of our proofs could be reduced to formal proofs alone, then the required logic could remain topic-neutral, which now will be explicitly given up. Likewise, Larvor points out that “we have to abandon the hope of establishing a general test for validity” (Larvor 2012, p. 723). We shouldn’t be unhappy to see these go, however, given that they are so closely linked to formality which we have good reason to reject as the right account of informal proofs (see the first half of this thesis). Rather, it will become clear that these features can have no general place in the new, more dynamic approach to mathematical proof conceived of in terms of content and action.

The second main proposal by Larvor is a switch to emphasising the ac- tivities involved in inferring, arguing and proving. Above, we saw a number of mentions of the acceptable moves, steps and inferences in some given domain. Regarding these, Larvor says:

If we think of an argument as a sequence of propositions con- nected by logical relations, it is hard to see how the content of the argument can play a role in the step from one propo- sition to the next. This is in part because a classically trained philosophical imagination is dominated by general logic, but also because orthodox philosophical education urges us to forget that the movement from one line of a proof to the next is anaction. (Larvor 2012, p. 721)

Larvor argues that we should recognise the purely propositional framework as being too limited to properly account for actual arguments found in math- ematical practice and mathematical proofs. The point is not merely that we should recognise the actions involved in moving between propositions, but rather that adopting such a focus reveals that the objects of our actions actually form a much broader class than just propositions.

The liberating insight is to notice that in making arguments, we act on all sorts of items in addition to propositions and well-formed formulae. Sometimes, we act inferentially on non- propositional representations of the subject-matter such as dia- grams, notational expressions, physical models, mental models

and computer models. (Larvor 2012, p. 721)

More specifically to informal proofs, the kinds of steps found in mathematics are not limited to actions on propositional contents, something Larvor illus- trates with a series of examples from diverse areas of mathematics. Indeed, it may even be that these actions do not have objects at all, such as if the subject-matter is the manifestation of the action. Larvor’s example of the last point is the demonstration by a gymnast that some complex gymnastics move is possible by performing the move.

The framework being proposed by Larvor, then, is to see proofs as sys- tems of inferential actions. This is far removed from the alternative, tra- ditional view of proofs as abstract objects made up of sequences of propo- sitions.18 Inferential actions are just those actions which can be used in arguments and, in the mathematical case, proofs. Of course, as described above, the inferential actions acceptable for some particular proof depends on the domain the proof is in.

There is a strong dose of Lakatos and Kneebone in this conception of inferences as found in actual proofs. We can view the key point as being that we should switch from a static conception of proofs to a dynamic one. While the static conception is primarily concerned with the stops along the way and the stepping-stones through the proof, in contrast, the dynamic view is concerned with the movement through the proof and the actual steps being made, as it is these which ultimately take us through the proof to establish a theorem. That isn’t to say that the places we stop aren’t important. The full picture that should emerge of informal proof will be one which takes account of how the non-propositional actions found in the proof relate to the propositional content of that proof as in the Wigginsian anti-intellectualist view of Ryle. This aligns with the approach to practical and propositional knowledge being argued for in this chapter.

Again, the move to the action-oriented perspective gives up on cer- tain desirable features, especially when combined with validity as content- dependent. Primarily, unlike in formal logic, there is now no general test

18_{Proofs and arguments as abstract objects is not restricted to the formal proofs and}

arguments. For example, Leitgeb takes informal proofs to be abstract objects: [...] we regard mathematical proofs per se as abstract entities which are independent of any material instantiation. (Leitgeb 2009, p. 266)

Similarly, (Simard Smith & Moldovan 2011) treats arguments as abstract objects (al- though abstract objects which can come into existence and disappear again).

for validity because there is no full and final list of inferential actions. Even when limited to some domain, if that domain is anything beyond the simplest cases then it will simply not be possible to fully settle all of the actions that might be permissible within that domain. Although this is certainly a cost of the view, as Larvor points out, it is a rather mild one. This is so because it is something that we have been lead to believe by the Formalist-Reductionist tradition that we will get, yet isn’t really something we should be expecting once we pay attention to proof in practice. Indeed the open-ended nature of mathematics and mathematical methods is vitally important to its growth and development, as we saw in the previous chapter.

Let us briefly mention the place of rigour as it is sketched in Larvor’s view. Larvor says that

[F]or every kind of inferential action, there must be a correspond- ing means of control, to ensure rigour. Sometimes these controls are simple rules like ‘do not divide by zero’. In other cases, these controls may be the fruit of mathematical research [...] Demon- strating rigour involves making the controls on inferential acts explicit, which is why some diagrams disappear from the final published version of a mathematical argument. The problem is not with diagrams as such, but rather that the actions performed on these diagrams in this piece of work do not have established, agreed controls. (Larvor 2012, p. 728)

Such controls are important— mathematicians should be careful not to di- vide by zero or abuse diagrams and infinite series. Larvor is right that it is often a fruitful project to make these explicit and that this is connected to mathematical rigour. However, we might be sceptical that rigour is fully accounted for by such corresponding controls, for I take correct and rigor- ous proving to be connected to practical knowledge and it has been argued that this is not fully enumerable in terms of explicit rules or principles in any reasonable sense. Just as in Ryle’s point that there may be regulative propositions, rules and maxims which apply to practical knowledge, but these cannot be the whole of what knowledge-how amounts to, nor should we expect there to be a particular list of controls which ensure rigour. As such, we should not expect to be able to demonstrate rigour in the way described either, unless it reduces to the Formalist-Reductionist position,

which is not what is intended. If not, though, we need to answer the further question of how we will ever manage to be confident that there is not some further ‘hidden’ rule that we violate in a given proof?

We can also wonder if such controls will form a unified class at all, or whether there is a range of different principles, between those necessary for rigorous proving and those which are merely good proof etiquette. For ex- ample, if I were to switch languages (from English to Japanese, to Afrikaans etc.) between each line of a proof, is this a lack of rigour or just poor style? Just like the open texture of mathematical concepts, for any given math- ematical domain it might well be that there is a never-ending horizon of ways to mathematically misbehave. The way these are avoided is not about implicit rules, but about learning how to behave well. This is not to say that there are no such rules; in line with the arguments from Ryle we can extract them from practice and describe them in exactly the same way that we can identify logics which our practices cohere with. We certainly have a great deal of rules and heuristics taught in classrooms and lecture halls, for example. The point is just that there is something more than this to rigour. In the next chapter I will argue that the correct account of mathematical rigour should connect it to intellectual,mathematical virtues.