ENCUESTA A LOS TRABAJADORES

Proofs play numerous roles in our mathematical practices and serve many different functions, but one I am here primarily interested in is their role in mathematical epistemology. Previous considerations of mathematical knowledge seem to have paid little attention to the idea that the knowl- edge we get from proofs is arrived at by the activities of proving. Even the L¨owe & M¨uller papers discussed above, which seem to be going in the right direction, start from an idea of merely having access to a proof and finish on a modalised notion which grants us knowledge of everything we are skilled enough to do with respect to a context, without requiring us to actually do the work of obtaining the knowledge we have. Larvor is correct to empha- sise that the movement through a proof involves inferential actions, but his focus lies elsewhere and the paper does not make explicit the impacts this has on mathematical epistemology.

The important idea is that it is the activity of proving which is of pri- mary epistemological significance in mathematics, withproofs themselves of

secondary importance. Outside of philosophy this has been picked up on in more popular reflections on mathematics, such as by Marcus du Sautoy in (du Sautoy 2015), as quoted at the start of this chapter. He sees proofs as narratives, describing the journey across the mathematical terrain from familiar and well-trodden starting points to far-off realms. He continues as follows:

Within the boundaries of the familiar land of the Shire are the axioms of mathematics, the self-evident truths about numbers, together with those propositions that have already been proved. This is the setting for the beginning of the quest. The journey from this home territory is bound by the rules of mathematical deduction, like the legitimate moves of a chess piece, prescribing the steps you are permitted to take through this world. At times you arrive at what looks like an impasse and need to take that characteristic lateral step, moving sideways or even backwards to find a way around. Sometimes you need to wait for new mathematical characters like imaginary numbers or the calculus to be created so you can continue your journey. (du Sautoy 2015)19

Reading this, there is a touch of formalistic thinking in the further analogy to moves in chess20 _{which we wouldn’t want to take too seriously, but the} notion of a journey fits very well with the thought that we should emphasise the activity of proving. The quote is, of course, very reminiscent of the well-known picture from G. H. Hardy:

I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can. There are some peaks which he can distinguish easily, while others are less clear. He sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. B is now fixed in his vision, and from

19

I am grateful to Ursula Martin for pointing me to this article.

this point he can proceed to further discoveries. In other cases perhaps he can distinguish a ridge which vanishes in the distance, and conjectures that it leads to a peak in the clouds or below the horizon. But when he sees a peak he believes that it is there simply because he sees it. If he wishes someone else to see it, he points to it, either directly or through the chain of summits which led him to recognise it himself. When his pupil also sees it, the research, the argument, theproof is finished. (Hardy 1929, p. 18)

The point of these rather long quotes is that the metaphor of mathematics as a huge landscape has been drawn on before. However, du Sautoy’s way of speaking is preferable to Hardy’s for now, as it brings out the active nature of proving rather than the more passive language of the ‘observer’. The claim I am making is that if we want to attain knowledge, this requires finding and following the path to get there. We can think of proofs, via a similar metaphor, as maps or directions providing us with a guide as to how to get from one place to another, from A to B. While the activity of proving is about traversing the mathematical landscape, a proof provides a record of the series of actions required to reach a new mathematical location and is used to communicate what the discoverer of the proof went through to others who wish to follow the same road and gain the same mathematical knowledge.21

Indeed, there is clear linguistic evidence for such an idea in the fact that much of the standard terminology in informal proofs is imperatival.

21_{I have recently discovered, thanks to Josh Habgood-Coote, that Ryle actually uses}

the same metaphor for mathematical discovery:

[...] the pioneering path-finder, Pythagoras say, has no tracks to follow; and any particular sequence of paces that he tentatively takes through the jungle may soon have to be marked by him as leading only into swamps or thickets. All the same, it may be, though it need not be, that in a day’s time or a year’s time he will have made a track along which he can now guide docile companions safely and easily right through the jungle. How does he achieve this? Not by following tracks, since there are none to follow. Not by sitting down and wringing his hands. But by walking over ground where tracks certainly do not exist, but where, with luck, assiduity and judgement, tracks might and so perhapswill exist. All his walkings are experimental walkings on hypothetical tracks or candidate-tracks or could-be tracks, or tracks on appro; and it is by so walking that, in the end, while of course he finds lots and lots of impasses, he also finds (if he does find), a viable track. (Ryle 1971, p. 224)

Common terms are ‘let’, ‘assume’, ‘suppose’, ‘define’, ‘construct’, ‘observe’, ‘consider’, ‘reduce’, ‘rearrange’, ‘note’ and many more.22 Once again, we find ourselves with a point familiar from Ryle:

We certainly can, in respect of many practices, like fishing, cook- ing and reasoning, extract principles from their applications by people who know how to fish, cook and reason. Hence Izaak Walton, Mrs. Beeton and Aristotle. But when we try to express these principles we find that they cannot easily be put in the indicative mood. They fall automatically into the imperative mood. (Ryle 1946, pp. 11–12)

A proof thus tells the reader what to do in order to prove some theorem, and thereby makes one important role of proofs to guide us through the inferen- tial actions needed to get to a certain place. An equally good analogy, then, would have been recipes in cookbooks: the recipe itself is only important in so far as it directs you how to make the cake in question. While we can talk about better or worse recipes, this is derivative on how well it guides us through our baking activities.23 In (Robinson 1991), Robinson describes proofs in a way similar to du Sautoy’s narrative idea:

[A] kind of meaningful narrative [...] more like a story, or even a drama, conveyed to us in language calling on our semantic and intuitive understanding. [...] To follow an informal proof as it unfolds in time is to understand the story as it develops. (Robinson 1991, p. 269)

Now certainly this seems right in certain respects, but it appears to suggest that we are passive observers to the unfolding drama with our understanding just being used to follow the action from afar. On the contrary, I take understanding a proof to involve being part of the action. The proof tells us which actions to take; the mathematician acts them out. Proofs thus don’t operate in a vacuum, securing their targets in the abstract, but rather they are secondary to the mathematical activities they guide us through,

22

This should be familiar to anyone who has looked at mathematical proofs, but I invite anyone sceptical of this to open up a few recent pure mathematics articles on the ArXiv and check for themselves. An interesting study to carry out in the future would be to do a proper analysis of some body of real proofs.

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Though the baking analogy works slightly less well because we also judge recipes for the tastiness of the baked goods they produce.

activities which are themselves embedded in a practical context and carried out by agents.

To conclude, by seeing proofs from thisaction-centred perspective we can thus say something new about the relationship between proofs and math- ematical knowledge. Gaining knowledge of mathematics from proofs is ac- tually done through the activities of proving: blazing a new trail through the mathematical landscape or following the paths that others have set, by following the instructions they have given us in their proofs. The epistemic importance thus lies primarily with the activities and actions, not the proof itself or its mere existence. The Formalist-Reductionist project misses out on this crucial idea, according the primary importance to proofs themselves, or worse still to unaccessed or inaccessible formal proofs underlying them, thereby failing to correctly explain the epistemic role of proofs. We, on the contrary, are in a place to explain how mathematical knowledge actually connects to proofs and proving activities. This will be the topic of the next chapter.