Abstract
In (Avigad, 2020), Jeremy Avigad makes a novel and insightful argument, which he presents as part of a defence of the ‘Standard View’ about the relationship between informal mathematical proofs (that is, the proofs that mathematicians write for each other and publish in mathematics journals, which may in spite of their ‘informal’ label be rather more formal than other kinds of scientific communication) and their corresponding formal derivations (‘formal’ in the sense of computer science and mathematical logic). His argument considers the various strategies by means of which mathematicians can write informal proofs that meet mathematical standards of rigour, in spite of the prodigious length, complexity and conceptual difficulty that some proofs exhibit. He takes it that showing that and how such strategies work is a necessary part of any defence of the Standard View.
In this paper, I argue for two claims. The first is that Avigad’s list of strategies is no threat to critics of the Standard View. On the contrary, this observational core of heuristic advice in Avigad’s paper is agnostic between rival accounts of mathematical correctness. The second is that that Avigad’s project of accounting for the relation between formal and informal proofs requires an answer to a prior question: what sort of thing is an informal proof? His paper havers between two answers. One is that informal proofs are ultimately syntactic items that differ from formal derivations only in completeness and use of abbreviations. The other is that informal proofs are not purely syntactic items, and therefore the translation of an informal proof into a derivation is not a routine procedure but rather a creative act. Since the ‘syntactic’ reading of informal proofs reduces the Standard View to triviality, makes a mystery of the valuable observational core of his paper, and underestimates the value of the achievements of mathematical logic, he should choose some version of the second option.
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Notes
Bourbaki (1949).
Poincaré (1908). On his conception of proof, “Verification differs from proof precisely because it is analytical,… It leads to nothing because the conclusion is nothing but the premisses translated into another language. A real proof, on the other hand, is fruitful because the conclusion is in a sense more general than the premisses.” (1902, p. 396 in Benacerraf & Putnam). For a nuanced discussion of Poincaré’s views on formal logic, see McLarty (1997).
“The question where mathematical exactness does exist, is answered differently by the two sides; the intuitionist says: in the human intellect, the formalist says: on paper.” Brouwer (1913) p. 83.
Typically, there are lots of derivations that might formalise a given proof, and sometimes they vary sufficiently to make trouble for Feferman’s claim that the corresponding derivation captures the logical structure of the informal proof. See Tanswell (2015) for cases.
Azzouni modified his view considerably in his (2009) and (2017). Nevertheless, his phrase ‘derivation-indicator’ has taken on a life beyond his writing. Since Avigad’s most recent work is the focus of the present paper, it is worth noting that Azzouni lists Avigad among the acknowledgements of his (2009) paper.
Azzouni, 2009, p. 25. Italics in original.
Of course, beliefs about non-existent formal derivations can be causally effective, and Azzouni makes moves in that direction in his (2017). There, he attempts to rescue his claim—that in reading an informal proof, mathematicians are unconsciously recognising algorithmic processes—by suggesting that the objects of the algorithms need not be linguistic strings. This thought is already in the literature, e.g., De Toffoli (2017), Giardino (2017), Larvor (2012), and Manders (2008) except that Azzouni re-presents manipulations as algorithmic processes. That may work for Azzouni, but it comes at the cost of separating his thesis (reading mathematical proofs is algorithm-recognition) from the idea that informal proofs are always-already formed in something like first-order logic with ZFC. He knows this, and makes moves to reconnect the two claims in the later sections of his paper.
This point has been raised in a helpfully clear way by Zoe Ashton (PhD thesis, forthcoming). It is not a topic of the present paper, but it is easy to see where one might start. Formalisation does all sorts of work in mathematics aside from securing proofs, such as sharpening and deepening ideas and preparing them for communication (see Hamkins pp. 162-3). It is distinctive of modern mathematics (roughly, Hilbert and after). Moreover, proof theory (the branch of formal logic) supplies mathematical models of mathematical reasoning. One might say that proof theory is what mathematics has to say about proof. It is not then surprising that it is also what many mathematicians have to say about proof. This is Hersh’s line (1997 p. 154): the mathematician who insists on the Standard View is like the economist who forgets that his model of the economy is only a model. Hersh calls for the model to be tested. Perhaps current work on digital proof assistants is an answer to that call.
A philosopher in the Rav tendency might argue thus: mathematicians formalise, but the art of it is tacit knowledge. It is not a summatively assessed part of mathematical training (see Tatton-Brown (2020) for an account of how it is learned). Mathematicians who have entirely internalised the art of formalisation may feel that they’re doing almost nothing, something ‘routine’, rather as a virtuoso jazz musician might insist that they’re not thinking about music theory or technique while improvising.
Some anecdotal corroboration for some of these points may be found here: https://mathoverflow.net/questions/338607/why-doesnt-mathematics-collapse-even-though-humans-quite-often-make-mistakes-in.
I will indulge in a joke about the creative tension between 3 and 5. There are two sorts of mathematicians: the ones who, when presented with a new mathematical structure ask, ‘What is an example of this?’ and the ones who ask, ‘What is this an example of?’
I owe this observation to a comment by Dirk Schlimm at the 2020 meeting of the Association for the Philosophy of Mathematical Practice, during the Q&A of Avigad’s presentation of his paper.
Avigad says that, ‘Hamami’s model is essentially correct…’ (2020, p. 15). ‘Essentially’ here warns us not to hold Avigad responsible for every nuance of Hamami’s elaboration of the Standard View.
From the abstract. Italics in original.
Unpublished paper. Though this paper is unpublished, it is a developed piece of work and benefits from two decades of refinement. We may therefore treat it as a reasonably reliable expression of Manders’ view.
Though it can sometimes feel that way. Here is Alain Connes, ‘…when a mathematician works, he is in fact reflecting up on a certain field, in which he encounters mathematical beings, and ends up playing with them, until they become familiar to him… After a time… either we get nowhere,… or else we manage to get hold of some result. Then begins the onerous task—the obligation to write up for the benefit of the mathematical community a polished article that is as compelling as possible.’ (Connes et al., 2001, p. 24f). Connes insists that these are two wholly distinct types of activity. Nevertheless, one supplies materials for the other—untranslated.
Imagine pushing one potato surface into the other, so that they intersect. The intersection is the required common loop. Proof taken from Tim Gowers’ Twitterfeed.
Personal communication at the APMP conference in Zurich, 2020.
For a spectacular recent example, see Castelvecchi (2021).
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Acknowledgements
I am grateful for valuable comments to two anonymous reviewers and to audience members at a conference in January 2021 organised by Silvia De Toffoli.
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Larvor, B.P. On the unreasonable reliability of mathematical inference. Synthese 200, 332 (2022). https://doi.org/10.1007/s11229-022-03812-w
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DOI: https://doi.org/10.1007/s11229-022-03812-w