ANTECEDENTES INVESTIGATIVOS

The Strong Proposal to see mathematical knowledge as best explained through a virtue epistemological lens can provide a better alternative to the Formalist- Reductionist account of proofs. Rather than being hindered and undermined by the issues we have seen raised against the Formalist-Reductionist account throughout the thesis, the virtue epistemological view on proofs I will offer positively thrives on them, as we shall soon see.

The first move that we make on the virtue approach needs us to ob- serve that in the virtue epistemology literature the key to knowledge is

virtuous intellectual activity. It is through virtuous acts, acts in which we exhibit or manifest relevant virtues, that we gain knowledge. The Formalist- Reductionist approach, meanwhile, emphasises proofs as construed as ob- jects, where we can then study these objects to discover what makes them good/correct/rigorous/etc. But then the account fares poorly with respect to our proving practices, having tried to abstract away from their material instantiation. In contrast to the proofs-as-objects view, then, the virtue ac- count should focus instead on proving as an activity, as argued for in the previous chapter.

Let us now proceed to locate the role of virtues in this account of proofs and the knowledge it secures. The virtue epistemological view is that in

order for the proving activities to secure our mathematical knowledge, they need to be virtuous in the appropriate way. Once again, how the details of this are filled out will depend on the particular virtue epistemology one favours.

Starting with the reliabilist, for someone producing a proof to have se- cured mathematical knowledge therewith, the proving must be completed with the relevant skills and competences. The mathematician needs to cor- rectly deploy the particular mathematical skills involved in whatever the proof is, rightly observing any limitations or restrictions on the domains of application. Furthermore, these skills need to be tied together to form a coherent whole, which delivers the final theorem as the result of the manoeu- vres combined correctly, that is, avoiding errors and mistakes, which will be the result of general level competences. Similarly, in the case of checking or learning from a previous proof, the proof on the page (or wherever it may be) acts as a guide or recipe as to how one should carry out the actions of the proof, as described in chapter 4. Still, the person doing the checking must accurately follow the techniques being presented and see how one step follows the last in order to come to know the ultimate solution. Importantly, following steps in this sense does not need to be filled out as following the un- derlying formal moves, but instead is about the fitting together of the steps in the overall reasoning pattern and recognising what follows from what in the moves that are being made, moves which can certainly be informal in the operative sense.

Virtue responsibilists, on the other hand, require that the mathemati- cian comes to know the proved theorem through acts of mathematical virtue, which is to say that their proving activities must be virtuous and free from vices. In particular, they gain knowledge through proving if the activity of this instantiates the necessary virtue of mathematical rigour. The nature of rigour is a major question of philosophy of mathematics, one which has not been done justice by the Formalist-Reductionist answer, so I will dis- cuss this in greater depth in the following section as something additional that the Strong Virtue Proposal can offer besides the expected account of mathematical knowledge. Importantly, following the Zagzebski framework, the virtue of rigour will have an epistemic motivational component, usually aimed at establishing the truth of the theorem, or making cognitive contact with mathematical reality (whatever form that takes), and also a success

component of actually doing so. Once again, this builds in the fact that our right reasoning must track the contours and landscape of mathematics.

Let us illustrate these both using the Stallings example we began the chapter with, where these virtues go astray to lead to a failed proof. Re- call that Stallings’s proof of the Poincar´e Conjecture failed because it relies on deploying a theorem in a case for which it doesn’t hold. The mathe- matical reliabilist account of what has gone on here is that what Stallings describes as his “psychological problem, a blindness, an excitement, an in- hibition of reasoning by an underlying fear of being wrong” (Stallings 1966, p. 88) represents a failure of his usual skills and competence at putting together a proof to form a complete argument, check for errors and, in par- ticular, observe the domain of application of theorems being used. The talk of developing “[t]echniques leading to the abandonment of such inhi- bitions” (Stallings 1966, p. 88) can be taken seriously; the development of such skills is paramount in securing correct proofs and further mathe- matical knowledge. For the mathematical responsibilist, the focus will be on the misapplication of theorem 0 as a failure on his part to be rigorous in his proving, rigour being a mathematical character trait to be discussed shortly. Additionally, the responsibilist would also be more interested in the idea that developing the techniques for avoiding this is something desirable to “every honest mathematician” (Stallings 1966, p. 88) as the intellectual honesty will work alongside rigour to ensure that our motivations in our mathematical proving are of the correct sort, rather than simply being di- rected at fame and fortune.10 For both approaches, the failure to enact the relevant virtues in the creation of the proof are how the proof came to be mistaken and its reasoning flawed, with the upshot that the activities did not deliver mathematical knowledge to Stallings of the truth of the Poincar´e Conjecture.

What has been given here is a description of how proving as an activity can deliver or fail to deliver mathematical knowledge. However, the chal- lenge on which I began was to account for proofs, rather than the activities surrounding creating and verifying them. In the previous chapter, I put forward a re-orientation of priority, where we focus on proving activities first and see proofs as objects or arguments as of secondary concern. Proofs

10_{Of course, this does not mean that this cannot be a part of your motivation, just}

don’t operate in a vacuum was a slogan form of the idea that a proof con- sidered in some idealised sense will miss out on the crucial connection to the provers who create, know, understand and employ them, a connection I have claimed is fundamental to the mathematical knowledge that proofs are aimed at. Nevertheless, there is an emerging worry here which must be addressed: that prioritising the individual activities appears to take us too far in an individualistic direction. The fact is that we do have a seemingly robust notion of what is sufficient for a proof, a set of standards taught in classrooms and lecture halls which is enforced by teachers and even the referees for journals. One might think that it cannot be simply down to the individual whether a proof is enough to secure knowledge, as this seems hopelessly subjective with respect to mathematics, which should be held up as objective.

In response, though, we can observe that this does not really cut to the core of the issue and that this instead merely reintroduces certain Tra- ditionalist attitudes to mathematical proof. For one thing, it seems that this response seeks to re-idealise proofs as something which are ‘out there’ to discover, thereby conflating the contours, structures and relationships of mathematics, on one hand, with the proofs themselves on the other. Cru- cially, proponents of the Strong Virtue Proposal can and should accept that there are many operative canons in mathematical practice and that proof is standardised in numerous ways. None of this poses a difficulty, however. The point is that the standards are set for how to best structure and commu- nicate proofs, thereby also helping to inculcate mathematical reasoning and problem-solving into students by demonstrating how to set out reasoning in clear and cogent ways. That each individual needs to go about actually carrying out the reasoning in order to gain the specific type of knowledge associated with proving (as opposed, say, to testimonial knowledge of its correctness) does not necessarily impede the objectivity of the mathemat- ics at stake. To return to the map metaphor: we can agree that there are important map-making conventions, while asserting that properly knowing the route it describes involves traversing it.

Under the current attitude, we can also do better than the Traditionalist in discussing the conventions and standards surrounding proving. For exam- ple, the claim that there are well-guarded standards of mathematical proofs must come with some major qualifications. The fact is that the demands

such conventions place on us vary from context to context, with more details demanded for proofs for students and less for discussions between colleagues etc. These differences seem to concern the granularity of the proof, or how much can be assumed on the part of the reader. For instance, in Pettigrew’s review (Pettigrew 2016) of (Burgess 2015) he points out that informal proofs must communicate the key ideas which deliver the truth of the theorem, and that it is this rather than the mere convincing which is important for a proof to be successful. For the Traditionalist, such a statement would fit poorly with the constant standards of rigour which are assumed, but for the Virtue approach this is only natural since virtues can be assessed across contexts and situations. Besides the virtue of rigour which can be displayed in prov- ing, many other virtues are relevant to mathematical work, including virtues (in whatever sense they are taken) which pertain to communicating and col- laborating. It may well be that very coarse proofs which only include the main ideas might not fully display whether the thinking underlying them is rigorous or not, in that they leave substantial gaps in between these main ideas, but could still be sufficient to communicate how to go about rigorously proving something if we address them to the right people. The point is that a virtue approach embraces also the diversity of purposes for which we em- ploy proofs, as set out at the start of the introduction to this thesis. While one such purpose is to fully set out how to deduce some conclusion, another may be merely to communicate how this is done to a fellow researcher who does not need the full explanation to arrive at mathematical knowledge, by virtue of their existing knowledge and abilities. The virtue picture does give us a way to account for how these different facets come together and what to say when they come apart, in particular that divergent purposes might need to be assessed with respect to different virtues.

Ultimately, I take the point here to be that communal standards re- flect something about the way in which we systematise our communication of mathematical ideas, approaches and proofs. This does diverge from the question of how proofs and proving relate to mathematical knowledge, but has been constantly conflated in Traditionalist and Formalist-Reductionist approaches. Let us now proceed to see what can be said concerning mathe- matical rigour from the virtue perspective.