Abstract
This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.
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Notes
See for example Manin (1981), Van Bendegem (1988, 1990), who uses this thought as a motivation for philosophy of mathematical practice; Hersh (1997), Rav (1999) on the mathematical interest of topic-specific proof ideas and (2007), Robinson (2000) to make room for the explanatory function of some proofs; Thurston (2006: p. 48), Auslander (2008), Pelc (2009) to argue the irrelevance of derivations to mathematicians’ confidence in theorems; Hales (2012: p. x).
Lakatos (1976) predates the formation of the philosophy of mathematical practice as a field. Azzouni (2004, 2009, 2013), Detlefsen (2008), Buldt et al. (2008), Goethe and Friend (2010), Larvor (2012a, b, 2016), Leitgeb (2009) seeking to rehabilitate Gödel’s allusions to mathematical perception, Macbeth (2015) and Tanswell (2015) all problematise the logical relation between derivations and proofs. For Avigad (2008: p. 306), the difference between derivations and real proofs is the background for reflection on the growing use of computers in mathematics.
Hardy (1929): p. 18.
This broadening of the conception of logic has some affinity with the heterogeneous logic project of Barwise and Etchemendy, as Shin explained it, “... the heterogeneous logic project aims to expand the territory of logic... Logic is not inherently tied up with sentential representation only.” (Shin (2004: p. 93).
Feferman, in the (2012) article already quoted, argues that diagrams persist in mathematics because they are essential to human understanding of proofs. Suppose this is true; this suggests the possibility of giving an account of the argument-structure of the picture-proofs that he considers without analysing away the inferential role of the pictures. Such an account would contradict Feferman’s position, as quoted here.
This remains true in the case computer proofs, so long as by ‘writer’ we mean the programmer rather than the machine. Note, the reader and author of a proof need not engage in precisely the same activities because they may take different routes from one stage to another. One might have to work out in detail some sub-result that the other can see immediately as a special case of some more powerful lemma already known, for example. (I came to appreciate this point after hearing Yacin Hamami make it on the occasion of his doctoral defence.)
As Pollard put it in his review of Hersh (2014), “The real question is how the arguments [in mathematical proofs] manage to present the experts with a mathematical grade of evidence.” (2014: p. 273).
Recall Wittgenstein’s remark that “mathematics is a MOTLEY [BUNTES Gemisch] of techniques of proof”. (RFM III 46:176). Given this variety, we should not insist on seeking a single model to account for all mathematical inferences.
See Carter (2010).
(2008a): p. 69. ‘Recent’ here means ‘recent in the mid-1990s’, the time of writing.
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.
I have elaborated and defended the notion of inferential actions in Larvor (2012a). An inferential action is anything one might do that contributes to the logic of an argument. Such actions include speech-acts, manipulations of matter (such as physical experiments), taking of measurements, manipulation of physical or digital models, calculations, operations on propositions (such as applications of logical rules like modus ponens), bodily performances and in mathematics, a host of topic-specific moves in the sense of Rav (1999). Proofs are full of imperatives that instruct the reader to build, change, move, identify or otherise re-order all sorts of objects, including but not only, propositions. What matters is that such actions be governed by (explicit or implicit) norms. If the aim of logic is to understand the difference between valid and invalid inferences, then these actions fall within its scope, even though we then lose any prospect of a single unifying theory of logic. Since formal mathematical logic already includes a multitude of systems with varying symbolisms and logical strengths, with extensions designed for the analysis of modal, temporal and deontic arguments, this is no great loss.
With the exception of Elements I.4, where a triangle is rigidly displaced. This is well known to commentators (see, e.g. Ferreirós 2016 p. 130n20, or Russell 1902, who calls superposition “a tissue of nonsense”). Euclid seems to have wished to avoid superposition wherever possible. In any case, superposition is still a highly disciplined move, even if it does not fit the constraints of Euclid’s overall practice.
Literally, ‘translator, traitor’, meaning that all translation traduces (except that the Italian original has a play on words that cannot be reproduced in English).
More precisely, to take the case of I.1, that the circles intersect is co-exact information because it is unaffected by bad drawing—however ovoid or tomato-shaped your circles are, in drawing the second one, you have to start inside the first, go outside the first in order to pass the other side of the centre of the second circle, and then re-enter the first circle again. This robustness is what rigour means in this context. (Manders 2008a: p. 69)
For the historical context of this point, see Ferreirós (2016) chapter eight, and most especially p. 213.
See Starikova (2010) for a case study on this point.
Interestingly, Alan Weir, developing a defence of formalism against criticism based on the difference between real proofs and proof-theoretic derivations, makes a similar point, “I did not require that the formal rules of a derivation be what we would recognise as logical rules.” (2016: p. 30). He then lists all manner of rules for actions that we might count as inferential: “basic rules for differentiation, power rules, product rules, quotient rules, the chain rule and so on.” Both he and we note that automation of proofs need not be, and probably cannot be, the encoding of proofs in some version of formal logic. Rather, what is automated is a much wider class of inferential actions, including topic-specific actions on objects other than propositions (or sentences).
I make this argument in greater detail in Larvor (2012a).
I am grateful to Robert Thomas for this distinction, and for pointing out my arbitrary restriction to two-dimensional representations.
I am grateful to an anonymous referee for this way of putting the point.
Silvia De Toffoli, in an unpublished presentation, has taken some steps towards classifying diagrams that can support inferences. Her principal criterion is the Jordan curve theorem. On the significance of the Jordan Curve theorem for reasoning with pictures, see also Corfield (2003: p. 257).
This criterion was originally inspired by some remarks in Toffoli and Giardino (2014) about the connections between knot diagrams and other notations.
‘Euclidean’ in a modern topology textbook does not mean, of couse, what it means when we discuss Euclid’s Elements.
And Starikova (2010). Her work on geometric group theory is in the same line but raises problems that lie beyond the scope of this paper.
A ‘knot’ here is a closed loop, which may be tangled, such as the trefoil knot. None of the useful knots known to sailors and fishermen are ‘knots’ for this purpose.
That is, its ambient isotopy class.
See Giardino (2017) for a further elaboration of this notion.
See Corfield (2003) chapter 10.4. I am grateful to an anonymous referee for drawing this to my attention.
This point, about the relationship between knot-diagrams and other, more discursive sorts of representation, was the the inspiration for our criterion (c).
As we saw with Euclid, another way of examining the role of intuition in a proof is to try to code it into a computer. There is at present no known way of doing this with Alexander’s proof. Movies are another option. Some of the illustrations in De Toffoli and Giardino (2016) are stills from an animation by Ester Dalvit.
Tanswell (2015) makes a similar argument, that informal proofs underdetermine the details of their formal reconstructions so radically that the former cannot be meaningfully identified with the latter.
(p. 271). Leitgeb offers no empirical grounds for these claims about what mathematicians do or do not know. At present it is normal in the philosophy of mathematical practice to make remarks of this sort on the basis of one’s standing as a mathematician or acquaintance with mathematicians. Therefore, this is not a complaint about Leitgeb, but rather a point about the state of the sub-field. Besides, other published versions of this claim (by, for example, Kreisel) have not provoked rebukes or rebuttals from other mathematicians.
Accounts of meaning that ground the intelligibility of speech and thought in social practice (such as Wittgenstein’s anti-private language argument or Vygotsky’s social-developmental theory) might suggest stronger versions of point (a).
Indeed, Feferman had no motive to provide such an analysis, because he thought that the underlying logical structure of every good proof is given by a derivation. His chief argument for this seems to be that otherwise, formal logic would lose its importance for philosophy of mathematics (2012:384–385).
Larvor (2012a).
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I am grateful to conference audiences in Oxford, Paris, Nancy and Brussels for their patience and penetrating questions. Much of the work on this paper took place during my year as a visiting professor at the Vrije Universiteit Brussel, and I am grateful to the directors and participants of the Strategic Research Project on Logic and Philosophy of Mathematical Practices at VUB for their material and intellectual support. Valeria Giordino, Silvia de Toffoli, Irina Starikova and three anonymous referees read drafts and made helpful comments.
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Larvor, B. From Euclidean geometry to knots and nets. Synthese 196, 2715–2736 (2019). https://doi.org/10.1007/s11229-017-1558-x
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DOI: https://doi.org/10.1007/s11229-017-1558-x