Monday, 26 October 2015

Podcast: Was medieval logic "formal"?

By Catarina Dutilh Novaes

Another instance of some shameless self-promotion... Here is a podcast with an interview with me by the ever-wonderful Peter Adamson -- the host of the fabulous podcast series History of Philosophy without any Gaps -- on Latin medieval logic, more specifically the senses in which medieval logic can (or cannot) be said to be formal -- both according to contemporary notions of formality and medieval ones. Hope some of you will enjoy it!

Saturday, 17 October 2015

Talk: Lessons from the Language(s) of Fiction

Back in January, I posted some reflections on what fictional languages can tell us about what meaning can and cannot be, here and here. Those thoughts eventually became a paper jointly written with one of my students, Phoebe Chan, which is forthcoming in Res Philosophica next April, "Against Truth-Conditional Theories of Meaning: Three Lessons from the Language(s) of Fiction".

For those who are interested in these topics, I gave a talk based on this paper at the Durham Arts & Humanities Society last Thursday evening. The talk was recorded, and is available to listen to on Soundcloud, for a few months at least.


© Sara L. Uckelman, 2015.

Wednesday, 14 October 2015

The beauty (?) of mathematical proofs -- Functional and non-functional beauty

By Catarina Dutilh Novaes

This is the seventh installment  of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is herePart II is herePart III is herePart IV is herePart V is here; Part VI is here). I now turn to beauty properly speaking, and discuss ways in which mathematical proofs are beautiful both in a functional and in a non-functional way.

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Prima facie, the concept of functional beauty is strikingly simple: a thing is beautiful insofar as it performs its function(s) well. It seems clear that, generally speaking, for something to fulfill its function is a good thing: normally, it will be useful and advantageous (e.g. it typically enhances fitness for organisms). So it is not surprising that function and beauty should become closely associated. As detailed in (Parsons & Carlson 2009), to date the most comprehensive study of this concept, functional beauty has a venerable pedigree, dating back to classical Greek philosophy (Aristotle in particular, which is not surprising given his interest in function and teleology), and having been particularly popular in the 18th century. As famously noted by Hume:

This observation extends to tables, chairs, scritoires, chimneys, coaches, saddles, ploughs, and indeed to every work of art; it being a universal rule, that their beauty is chiefly deriv’d from their utility, and from their fitness for that purpose, to which they are destin’d. (Hume 1960, 364)

But of course, much complexity lies behind the concept of function itself, which is what is doing all the work. What determines the function(s) of an object, artifact or organism? The concept of function occupies a prominent role in biology, in fact since Aristotle but with renewed strength since the advent of evolutionary biology. (Indeed, Parsons and Carlson (2009) rely extensively on work on function within philosophy of biology, e.g. Godfrey-Smith’s work.) Here however we should focus on artifacts, given that the goal is to increase our understanding of the (putative) beauty of mathematical proofs, which, despite a potentially problematic ontological status (more on which shortly), come closer to artifacts than to organisms or natural objects such as e.g. planets or rocks. Parsons and Carlson (2009, 75) offer the following definition of the (proper) function of an artifact:

An artifact has proper function if and only if it currently exists because, in recent past, its ancestors were successful in meeting some need or want in the marketplace because they performed that function, leading to the manufacture and distribution of that artifact.

Monday, 12 October 2015

The beauty (?) of mathematical proofs -- A proof is and is not a dialogue

By Catarina Dutilh Novaes

This is the sixth installment (two more to come!) of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is herePart II is herePart III is here;Part IV is here; Part V is here). After having introduced the dialogical conception of proofs in the previous post, in this post I explain why proofs do not appear to be dialogues, and what the prospects are for an absolute notion of the explanatoriness of proofs.

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At this point, the reader may be wondering: this is all very well, but obviously deductive proofs are not really dialogues! They are typically presented in writing rather than produced orally (though of course they can also be presented orally, for example in the context of teaching), and if at all, there is only one ‘voice’ we hear, that of Prover. So at best, they must be viewed as monologues. My answer to this objection is that Skeptic may have been ‘silenced’, but he is still alive and well insofar as the deductive method has internalized the role of Skeptic by making it constitutive of the deductive method as such. Recall that the job of Skeptic is to look for counterexamples and to make sure the argumentation is perspicuous. This in turn corresponds to the requirement that each inferential step in a proof must be necessarily truth preserving (and so immune to counterexamples), and that a proof must have the right level of granularity, i.e. it must be sufficiently detailed for the intended audience, in order to achieve its explanatory purpose.

Let us discuss in more detail the phenomenon of different levels of granularity in mathematical proofs, as it is directly related to the issue of explanatoriness. It is well known that the level of detail with which the different steps in a proof are spelled out will vary according to the context: for example, in professional journals, proofs are more often than not no more than proof sketches, where the key ideas are presented. The presupposition is that the intended audience, namely professional mathematicians working on similar topics, would be able to reconstruct the details of the proof should they feel the need to do so (e.g. if they somehow doubt the results). In contrast, in the context of textbooks or in classroom situations, proofs tend to be presented in much more detail, precisely because the intended audience is not expected to have the level of expertise required to reconstruct the proof from a proof-sketch. What is more, the intended audience is in the process of learning the game of formulating and understanding mathematical proofs, and so proofs where each step is clearly spelled out is what is required. Furthermore, different areas within mathematics tend to have different standards of rigor for proofs, again in function of the intended audience.

What the phenomenon of different levels of granularity suggests when it comes to the explanatoriness of proofs is that, for a proof to be explanatory for its intended audience, the right level of granularity must be adopted.[1] If a proof is to be explanatory in the sense of making “something that is initially puzzling less puzzling; an explanation reduces mystery” (Colyvan 2012, 76), the decrease of puzzlement is at least in first instance inherently tied to the agent to whom something should become less puzzling.

Friday, 9 October 2015

The beauty (?) of mathematical proofs -- explanatory persuasion as the function of proofs

By Catarina Dutilh Novaes

This is the fifth installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is herePart II is herePart III is here; Part IV is here). In this post I bring in my dialogical conception of proofs (did you really think you'd be spared of it this time, dear reader?) to spell out what I take to be one of the main functions of mathematical proofs: to produce explanatory persuasion.

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Framing the issue in these terms allows for the formulation of two different approaches to the matter: explanatoriness as an objective, absolute property of the proofs themselves; or as a property that is variously attributed to proofs first and foremost based on pragmatic reasons, which means that such judgments may by and large be context-dependent and agent-dependent. (A third approach may be described as ‘nihilist’: explanation is simply not a useful concept when it comes to understanding the mathematical notion of proof.) Some of those instantiating the first approach are Steiner (1978) and Colyvan (2010); some of those instantiating the second one are Heinzmann (2006) and Paseau (2011). (It is important to bear in mind that the discussion here pertains to so-called ‘informal’ deductive proofs (such as proofs presented in mathematical journals or textbooks), not to proofs within specific formal systems.)

For reasons which will soon become apparent, the present analysis sides resolutely with so-called pragmatic approaches: the notion of explanation is in fact useful to explain the practices of mathematicians with respect to proofs, in particular the phenomenon of proof predilection, but it should not be conceived as an absolute, objective, human-independent property of proofs. One important prediction of this approach is that mathematicians will not converge in their judgments on the explanatoriness of a proof, given that these judgments will depend on contexts and agents (more on this in the final section of the paper).

Perhaps the conceptual core of pragmatic approaches to the explanatoriness of a mathematical proof is the idea that explanation is a triadic concept, involving the producer of the explanation, the explanation itself (the proof), and the receiver of the explanation. The idea is that explanation is always addressed at a potential audience; one explains something to someone else (or to oneself, in the limit).[1] And so, a functional perspective is called for: what is the function (or what are the functions) of a proof? What is it good for? Why do mathematicians bother producing proofs at all? While these questions are typically left aside by mathematicians and philosophers of mathematics, they have been raised and addressed by authors such as Hersh (1993), Rav (1999), and Dawson (2006).

One promising vantage point to address these questions is the historical development of deductive proof in ancient Greek mathematics,[2] and on this topic the most authoritative study remains (Netz 1999). Netz[3] emphasizes the importance of orality and dialogue for the emergence of classical, ‘Euclidean’ mathematics in ancient Greece:

Thursday, 8 October 2015

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 herePart 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 20th 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.

Wednesday, 7 October 2015

The beauty (?) of mathematical proofs -- beauty and explanatoriness

By Catarina Dutilh Novaes

This is the third installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is here; Part II is here). In this post I start drawing connections (later to be discussed in more detail) between beauty and explanatoriness.

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A hypothesis to be investigated in more detail in what follows is that there seems to be an intimate connection between attributions of beauty to mathematical proofs and the idea that mathematical proofs should be explanatory. Indeed, the reductive account of Rota in terms of enlightenment immediately brings to mind the ideal of explanatoriness:

We acknowledge a theorem's beauty when we see how the theorem "fits" in its place, how it sheds light around itself, like a Lichtung, a clearing in the woods. We say that a proof is beautiful when such a proof finally gives away the secret of the theorem, when it leads us to perceive the actual, not the logical inevitability of the statement that is being proved. (Rota 1997, 182).

It is not surprising that there should be such a connection for non-literal, reductive accounts such as Rota’s; explanatoriness is a very plausible candidate as the epistemic property that is actually being tracked by these apparently aesthetic judgments. However, the connection is arguably present both in reductive and in non-reductive accounts. Indeed, it is striking to notice that many of the properties that Hardy (1940) attributes to beautiful proofs are in fact properties typically associated with explanatoriness in the literature (to be discussed in an upcoming post). According to Hardy, a beautiful mathematical proof is:

  • ·      Serious: connected to other mathematical ideas
  • ·      General: idea used in proofs of different kinds
  • ·      Deep: pertaining to deeper ‘strata’ of mathematical ideas
  • ·      Unexpected: argument takes a surprising form
  • ·      Inevitable: there is no escape from the conclusion
  • ·      Economical (simple): no complications of detail, one line of attack

Tuesday, 6 October 2015

The beauty (?) of mathematical proofs - methodological considerations

By Catarina Dutilh Novaes

This is the second installment in my series of posts on the beauty, function, and explanatoriness of mathematical proofs (Part I is here). I here discuss methodological issues on how to adjudicate the 'dispute' between the reductive and the literal accounts of the beauty of proofs, discussed in Part I.

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But what could possibly count as evidence to adjudicate the ‘dispute’ between the literal/non-reductive camp and the non-literal/reductive camp? We are now confronted with a rather serious methodological challenge, namely that of determining what counts as ‘data’ on this issue (and potentially other issues in the philosophy of mathematics). Both sides seem to have compelling arguments, but it is not clear that a top-down approach with conceptual, philosophical argumentation alone will be sufficient.[1] However, it seems that merely anecdotal evidence (“I am a mathematician and I use aesthetic terminology in a literal (or non-literal) sense”) will not suffice either. Firstly, there are of course limits to self-reflective knowledge. Secondly, what is to rule out that some mathematicians use aesthetic terminology as proxy for epistemic properties, while others use the terminology in a literal sense instead? It is not clear that a uniform account is what we are looking for.[2] Moreover, it may be a case of an is-ought gap: perhaps mathematicians do use aesthetic vocabulary in a particular way (either literal or non-literal), but should they use this vocabulary in this way and not in another way?

Ultimately, the question is: what is the explanandum in a philosophical account of the (presumed) aesthetic dimension of mathematical proofs? Are we (merely) offering an account of the aesthetic judgments of mathematicians? (Something that might be conceived as belonging to the sociology rather than the philosophy of mathematics.)[3] Or are we dealing with a crucial component of mathematical practice, one that fundamentally influences how mathematicians go about? Or perhaps the goal is to explain (purported) human-independent properties of proofs such as beauty and ugliness? What will count as data in this investigation will depend on what the theorist thinks is being investigated in the first place.[4]

Monday, 5 October 2015

The beauty (?) of mathematical proofs - reductive vs. literal approaches

By Catarina Dutilh Novaes

I am currently working on a paper provisionally entitled 'Beauty, function, and explanation in mathematical proofs', and so this week I will post what I have so far as a series of blog posts. Here I start with a discussion on the current literature on the presumed beauty of some mathematical proofs. As always, comments very welcome!

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It is well known that mathematicians often employ aesthetic adjectives to describe mathematical entities, mathematical proofs in particular. Poincaré famously claimed that mathematical beauty is “a real aesthetic feeling that all true mathematicians recognize.” In a similar vein, Hardy remarked that “there is no permanent place in the world for ugly mathematics.” Indeed, in A Mathematician’s Apology Hardy offers a detailed discussion of what makes a mathematical proof beautiful in his view. More recently, corpus analysis of the laudatory texts on the occasion of mathematical prizes shows that they are filled with aesthetic terminology (Holden & Piene 2009, 2013). But it is not all about beauty; certain kinds of proofs that still encounter resistance among mathematicians, such as computer-assisted proofs or probabilistic proofs, are sometimes described as  ‘ugly’ (Montaño 2012). Indeed, mathematicians seem to often use aesthetic vocabulary to indicate their preferences for some proofs over others.
What exactly is going on? Even if we keep in mind that, in colloquial language, it is quite common to use aesthetic terminology in a rather loose sense (‘he has a beautiful mind’; ‘things got quite ugly at that point’), the robustness of uses of this terminology among mathematicians seems to call for a philosophical explanation. What are these judgments tracking? Are these judgments really tracking aesthetic properties of mathematical proofs? Or are these aesthetic terms being used as proxy for some other, non-aesthetic property or properties? Is it really the case that “all true mathematicians” recognize mathematical beauty when they see it? Do they indeed converge in their attributions of beauty (or ugliness) to mathematical proofs? And even assuming that there is a truly aesthetic dimension in these judgments, is beauty a property of the proofs themselves, or is it rather something ‘in the eyes of the beholder’? These and other issues are some of the explanatory challenges for the philosopher of mathematics seeking to understand why mathematicians systematically employ aesthetic terminology to talk about mathematical proofs (as well as other mathematical objects and entities).