The beauty (?) of mathematical proofs -- explanatory proofs

By Catarina Dutilh Novaes

This is the fourth installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is here; Part II is here; Part III is here). In this post I present a brief survey of the debates in the literature on what it means for a mathematical proof to be explanatory.

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Quite a bit has been said on explanation and mathematical proofs in recent decades (Mancosu & Pincock 2012). Although the topic itself has an old and distinguished pedigree (it was extensively discussed by ancient authors such as Aristotle and Proclus, as well as by Renaissance and early modern authors – Mancosu 2011, section 5), in recent decades the debate was (re-)ignited by the work of Steiner in the late 19070s, thus generating a wealth of discussions. This brief overview could not possibly do justice to the richness of this material, so what follows is a selection of themes particularly pertinent for the present purposes.

The issue of what makes scientific theories or arguments more generally explanatory is again a question as old as philosophy itself; indeed, it is of the main issues discussed in Aristotle’s theory of science (in particular in the Posterior Analytics). The traditional, Aristotelian account has it that a scientific explanation is truly explanatory iff it accurately tracks the causal relations underlying the phenomena that it seeks to explain. To mention a worn-out but still useful example: the fact that it is 25 degrees C outside and the fact that a well-functioning thermometer indicates ’25 C’ (typically) occur simultaneously, but an explanation of the former phenomenon in terms of the latter gets the causal order the wrong way round: it is the outside temperature of 25 degrees C that causes thermometers to indicate ’25 C’, not the converse.

In the 20^th century, the issue regained prominence, at first with Hempel’s (1965) formulation of his famous Deductive-Nomological model of scientific explanation. In the spirit of the logical positivistic rejection of all things metaphysical, Hempel’s goal was to offer an account of scientific explanation that would do away with traditional but dubious (i.e. metaphysical) concepts such as causation. Much criticism has been voiced against Hempel’s model on different grounds, and one line of attack, espoused in particular by Salmon (1984), emphasized the unsuitability of doing away with causation altogether.

When it comes to mathematics, the question them becomes: are mathematical proofs explanatory in the same way as scientific theories are? It is in no way obvious that a causal story can be told for mathematical proofs. Does it make sense to say that some mathematical truths can cause some other mathematical truths? For this to be the case, one would presumably have to accept not only the independent existence of mathematical entities, but also the idea that they can causally influence each other. Now, while this is not as such an incoherent position (and seems to be something that a full-blown Platonist such as Hardy might be happy to endorse), it comes with heavy metaphysical as well as epistemological (as per Benacerraff’s challenge) costs.

And thus, most authors involved in these debates seem to recognize that a full-blown causal story is unlikely to do a good job in the case of mathematical proofs. They then endeavor to look for different accounts of the explanatoriness of mathematical proofs, which may eventually still be reconcilable with broader conceptions of scientific explanation, or, if not, would lead to mathematical exceptionalism – i.e. the idea that explanation in mathematics is a different beast altogether. (For the present purposes, this specific issue will be set aside.)

In his seminal work in the late 1970s, Steiner introduced the notion of a characterizing property as what distinguishes explanatory from non-explanatory proofs: “an explanatory proof makes reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property” (Steiner 1978, 143). The notion of dependence clearly seeks to capture a non-causal analogue of the notion of causation in scientific explanation more generally. As described by Mancosu & Pincock (2012, 16), “Steiner argues that these dependence relations require both that the entity or structure be uniquely picked out by some characterizing property, and that the explanatory proof be part of a family of proofs where this property is varied.”

Notice that the idea of a ‘family of proofs’ seems related to Hardy’s generality requirement that the core idea of a proof be present in other relevant proofs (see here for Hardy's criteria). In turn, the idea that an explanatory proof makes it evident that the result depends on the characterizing property resembles Hardy’s notion of inevitability, and arguably brings along economy/simplicity as well.

Steiner’s account has been criticized for a number of reasons, but it is fair to say that much of the debate in recent decades is still tied to how he framed the issue. As pointed out by Mancosu (2011), Steiner’s is a local account, i.e. explanatoriness is a local property of a given proof. In contrast, Philip Kitcher’s (1989) approach may be described as global/holistic in that the explanatoriness of proofs is viewed in the broader context of (mathematical) knowledge as a whole. For Kitcher, the key notion is that of unification: “an explanatory proof in pure mathematics is one that is part of a small collection of argument patterns that allows the derivation of the mathematical claims that we accept.” (Mancosu & Pincock 2012, 15) We are now reminded of Hardy’s global properties of seriousness and depth.

A presupposition running through the philosophical literature is that there really is a meaningful distinction between proofs that are explanatory and proofs that are not.  ((Resnik & Kushner 1987) is one of the few exceptions.) The distinction is often formulated in terms of a distinction between proofs that merely establish that the conclusion is the case (that is, if the premises are the case) and proofs establishing why the conclusion is the case; the former merely demonstrate, while the latter explain (Colyvan 2102, 76).[1] (Recall that, as famously claimed by van Fraassen (1980), explanations are answers o ‘why’ questions.) The point is obviously that explanatoriness is (presumably) a desirable property in a proof, and all things being equal, mathematicians (should) prefer explanatory proofs over non-explanatory ones. However, a number of questions immediately arise: is this distinction absolute, or is it context-dependent? Is it a sharp distinction, or is it one allowing for degrees such that explanatoriness becomes a comparative notion? (A proof is not explanatory an sich, but more explanatory than other proofs, as well as potentially less explanatory than others.)

Not coincidentally, the same questions can be raised concerning the purported beauty of proofs. Is a beautiful proof beautiful in an absolute sense, i.e. is beauty an objective, absolute property of certain proofs? Or can a proof be viewed as beautiful by one person but as not particularly beautiful by another person, while both judgments are legitimate and ‘correct’ in some sense or another? Is beauty best viewed as a sharp category, or should it be viewed as a comparative notion? In both cases, what is being tracked is the phenomenon of mathematicians displaying predilections for certain (kinds of) proofs over others, which suggests that more is at stake than simply establishing the truth of a theorem. Perhaps the proofs that mathematicians tend to prefer are more beautiful than others, or else they are more explanatory than others, and if this is so, we may wonder what it is that these preferred proofs have in common.

FULL SERIES: 

Part I is here

Part II is here

Part III is here

Part IV is here

Part V is here

Part VI is here

Part VII is here
Part VIII is here

References

Colyvan, M.,2012, Introduction to the Philosophy of Mathematics. Cambridge: CUP.

Hempel, C., 1965, Aspects of Scientific Explanation and Other Essays in the Philosophy of Science, New York: Free Press.

Kitcher, P., 1989, ‘Explanatory Unification and the Causal Structure of the World’, in Scientific Explanation, P. Kitcher and W. Salmon, 410–505. Minneapolis: University of Minnesota Press.

Mancosu, P., 2011, 'Explanation in mathematics'. In E. Zalta (ed.), Stanford Encyclopedia of Philosophy

Mancosu, P. & Pincock, C., 2012, 'Mathematical explanation'. Oxford Bibliographies.

Resnik, M., and D. Kushner, 1987, “Explanation, Independence, and Realism in Mathematics”, British Journal for the Philosophy of Science, 38: 141–158.

Salmon, W., 1984, Scientific Explanation and the Causal Structure of the World, Princeton: Princeton University Press.

Steiner, M., 1978, “Mathematical Explanation”, Philosophical Studies, 34: 135–151.

Van Fraassen, B., 1980, The Scientific Image. Oxford: Oxford University Press

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[1] Recall Rota’s description of a beautiful proof as “when it leads us to perceive the actual, not the logical inevitability of the statement that is being proved”.