Thursday, 30 June 2016

Probabilistic and logical approaches in formal epistemology - Interview in The Reasoner

The latest issue of The Reasoner has an interview with my colleagues Jan-Willem Romeijn and Barteld Kooi by Rohan French and myself. The topic is probabilistic and logical approaches in formal epistemology. Go check it out!

Tuesday, 17 May 2016

CFA: Foundations of Mathematical Structuralism

CFA: Foundations of Mathematical Structuralism

12-14 October 2016, Munich Center for Mathematical Philosophy, LMU Munich

In the course of the last century, different general frameworks for the foundations of mathematics have been investigated. The orthodox approach to foundations interprets mathematics in the universe of sets. More recently, however, there have been other developments that call into question the whole method of set theory as a foundational discipline. Category-theoretic methods that focus on structural relationships and structure-preserving mappings between mathematical objects, rather than on the objects themselves, have been in play since the early 1960s. But in the last few years they have found clarification and expression through the development of homotopy type theory. This represents a fascinating development in the philosophy of mathematics, where category-theoretic structural methods are combined with type theory to produce a foundation that accounts for the structural aspects of mathematical practice. We are now at a point where the notion of mathematical structure can be elucidated more clearly and its role in the foundations of mathematics can be explored more fruitfully.

The main objective of the conference is to reevaluate the different perspectives on mathematical structuralism in the foundations of mathematics and in mathematical practice. To do this, the conference will explore the following research questions: Does mathematical structuralism offer a philosophically viable foundation for modern mathematics? What role do key notions such as structural abstraction, invariance, dependence, or structural identity play in the different theories of structuralism? To what degree does mathematical structuralism as a philosophical position describe actual mathematical practice? Does category theory or homotopy type theory provide a fully structural account for mathematics?

Confirmed Speakers:

Prof. Steve Awodey (Carnegie Mellon University)
Dr. Jessica Carter (University of Southern Denmark)
Prof. Gerhard Heinzmann (Université de Lorraine)
Prof. Geoffrey Hellman (University of Minnesota)
Prof. James Ladyman (University of Bristol)
Prof. Elaine Landry (UC Davis)
Prof. Hannes Leitgeb (LMU Munich)
Dr. Mary Leng (University of York)
Prof. Øystein Linnebo (University of Oslo)
Prof. Erich Reck (UC Riverside)

Call for Abstracts:

We invite the submission of abstracts on topics related to mathematical structuralism for presentation at the conference. Abstracts should include a title, a brief abstract (up to 100 words), and a full abstract (up to 1000 words), blinded for peer review. Authors should send their abstracts (in pdf format), together with their name, institutional affiliation and current position We will select up to five submissions for presentation at the conference. The conference language is English.

Dates and Deadlines:

Submission deadline: 30 June, 2016
Notification of acceptance: 31 July, 2016
Registration deadline: 1 October, 2016
Conference: 12 - 14 October, 2016

For further details on the conference, please visit:

Sunday, 24 April 2016

CFA: workshop on argument strength

When: December 1-2, 2016
Where: Institute of Philosophy II, Ruhr-University Bochum

Arguments vary in strength. The strength of an argument is affected by e.g. the plausibility of its premises, the nature of the link between its premises and conclusion, and the prior acceptability of the conclusion.

The aim of this workshop is to bring together experts from the fields of artificial intelligence, philosophy, logic, and argumentation theory to discuss questions related to the strength of arguments. Such questions include:

-Which factors influence the strength of an argument?
-What are the pros and cons of different formal representations of argument strength?
-How to formally model qualifiers on the conclusions of arguments?
-How does argument strength propagate when inferences are chained?
-How do arguments accrue?
-Can weaker arguments defeat and/or defend stronger arguments?
-When do more specific arguments defeat more general arguments and vice versa?
-How do formal and informal approaches to argument strength relate?
-How do preferences assigned to premises influence the evaluation of arguments?

Keynote speakers:
Gerhard Brewka (University of Leipzig)
Gabriele Kern-Isberner (TU Dortmund)
Beishui Liao (Zhejiang University)
Henry Prakken (Utrecht University)

Leon Van Der Torre (University of Luxembourg)

Abstract submission:
Authors are invited to submit an abstract (500-1000 words) related to the above or any other questions on the topic of argument strength to by August 1, 2016.

Important dates:
submission deadline: August 1, 2016
notifications: September 1, 2016
workshop: December 1-2, 2016

Organizing committee:
Mathieu Beirlaen
AnneMarie Borg
Jesse Heyninck
Dunja Šešelja
Christian Straßer

Friday, 18 March 2016

Five Years MCMP: Quo Vadis, Mathematical Philosophy?

The Munich Center for Mathematical Philosophy invites participation to the following event:

Five Years MCMP: Quo Vadis, Mathematical Philosophy?

MCMP, LMU Munich
June 2-4, 2016

On the one hand, the workshop will celebrate the five years of existence of the Munich Center for Mathematical Philosophy (MCMP). On the other hand, and much more importantly, the workshop will be devoted to the question of where we should be heading in the future: what next, mathematical philosophy?

The workshop will consist of:

— a brief look back at five years MCMP;
— 16 short talks by young mathematical philosophers;
— three evening lectures on the logical empiricist background to mathematical philosophy;
— three general discussion sessions;
— and an "Ideas Session" in which the participants will be asked to contribute new ideas for the application of logical and mathematical methods to philosophical problems and questions.

Registration deadline: May 1st 2016

Prof. Dr. Hannes Leitgeb
Prof. Dr. Stephan Hartmann

Monday, 29 February 2016

Rationality Summer School: Call for applications

Call for applications: International Rationality Summer Institute 2016

40 full stipends

We invite applications for the first International Rationality Summer Institute (IRSI), which will take place from September 4-16, 2016, in Aurich (Germany). The topic of the Summer Institute is human rationality from a psychological, philosophical, and cognitive (neuro)science perspective.

Topics of the courses are: Rationality and normativity, Norms vs. evidence in reasoning research, Rational belief change, Inductive reasoning, Causal cognition, Probabilistic reasoning and argumentation, Language and reasoning, Mental models and rationality, Probabilities and conditionals, Bounded rationality, Neural bases of reasoning, Development of reasoning, Logical and probabilistic approaches to rationality, Intuition and analytic thinking, Scientific objectivity and inductive inference.

Faculty members are: John Broome, Vincenzo Crupi, Igor Douven, Aidan Feeney, York Hagmayer, Stephan Hartmann, Konstantinos Katsikopoulos, Martin Monti, David Over, Arthur Paul Pedersen, Jérôme Prado, Eva Rafetseder, Marco Ragni, Hans Rott, Jan Sprenger, Jakub Szymanik, and Valerie Thompson. In addition to the courses, we will have two keynote speakers: Gerd Gigerenzer and Johan van Benthem.

We invite applications by doctoral students and early-stage postdocs interested in human rationality and with a background in psychology, philosophy, cognitive (neuro)science, or related fields. Advanced Master’s students with a Bachelor’s degree in one of the disciplines and with an outstanding interest in the topic are also encouraged to apply.

The IRSI is generously funded by the Volkswagen Stiftung. Successful applicants will get a full stipend that covers the participation fee, board and lodging, and the reimbursement of traveling costs.

Applications close on April 15, 2016

The IRSI is organized by Markus Knauff, Patricia Garrido-Vásquez, and Marco Ragni (Giessen). Advisory board: Ralph Hertwig (Berlin), Gabriele Kern-Isberner (Dortmund), Gerhard Schurz (Düsseldorf), Wolfgang Spohn (Konstanz), and Michael Waldmann (Göttingen).

Please find more information on the Summer Institute and on how to apply at For inquiries, please send an e-mail to

Friday, 26 February 2016

Swamplandia 2016 - schedule and abstracts

Submission deadline for Swamplandia 2016 is approaching. Meanwhile, tentative schedule with keynote speakers' titles and abstracts is available online. Here.

Wednesday, 10 February 2016

On the adoption of logical principles

Two weeks ago I had the pleasure of attending a one-day workshop on The Nature of Logic organized by the University of York. The focus of the day was Saul Krikpe's unpublished works on the 'adoption problem', an interpretation of Lewis Carroll's "What the Tortoise Said to Achilles". "What the Tortoise Said to Achilles" is probably my favorite piece of philosophy, ever; York is a day-trip away from Durham; and it was a chance to hear Kripke speak in the flesh, all three reasons to expect a very interesting and enjoyable day, and the workshop did not disappoint.

The talks were all thought-provoking, but it was the first, by Romina Padró, that set the stage for the day and also triggered the thoughts that I want to try to articulate here today. Padró recently completed her dissertation on What the Tortoise Said to Kripke: the Adoption Problem and the Epistemology of Logic. The "Adoption Problem" is detailed in S. 2.2, but the basic issue of this: Suppose you are confronted with someone, call him Harry, who has "no notion of the principles in question [modus ponens and universal instantiation] and has never inferred in accordance with them" (p. 31). Surely Harry has an impoverished reasoning ability and it would be useful to introduce him to these logical principles, such that he accepts them and can henceforth go on to reason according to them. This is the adoption of a logical principle:

By 'adopt' here we mean that the subject, Harry in this case, picks up a way of inferring according to, say, UI, something he wasn't able to do before, on the basis of the acceptance of the corresponding logical principle (p. 31, emphasis in the original).

The adoption problem is then whether such principles as MP and UI can be adopted. Padró's talk at the workshop was directed at arguing that they cannot: That in order to apply MP after it has been accepted, one must already be able to appeal to a notion of modus ponens. This is precisely what the Tortoise is pointing out to Achilles in Carroll's classic piece.

I remain unconvinced by Padró's argument, in part because it seems to me that Harry can accept a principle without applying it, and that once he has accepted it, he can then go on to apply it -- if he cannot apply it, then I would argue he hasn't in fact accepted it, contrary to assumption. But I will leave this point aside, and assume that there are some principles which cannot be adopted, and that MP and UI are, if anything are, prime candidates for such principles. The questions that I had -- and they are only questions, I don't have any idea how one would go about answering them, which is part of why I'm writing this, in case the collective power of the internet is smarter than me (it almost certainly is) -- stem from generalising the issue.

Padró's talk focussed on whether or not MP and UI are adoptable, and mentioned briefly other logical principles that may be similar, such as &I and &E, as well as some that likely can be adopted, such as disjunctive syllogism. This raises a general methodological point: How does one determine if a principle is adoptable? If every logical principle is adoptable, then we have no problem; if no logical principle is adoptable, then we have no problem. But if some are and some are not, then it would be useful to have a principled way of identifying them, preferably in advance. The argument for MP and UI is that in order to apply them, one must invoke the principles themselves:

If someone who never inferred in accordance with MPP were to be told that "For any A and B, if A then B, and A, then B," the subject wouldn’t be in a better position to perform a MPP inference. For the principle to be of use with any particular inference, she will need to infer in accordance with the MPP pattern that she does not use in the first place: in any particular case, she will only get to B from her premises by performing a MPP inference on the instantiation of 'For all A and B, if A, and if A then B, then B,' but that is exactly what she couldn't do to begin with (p. 36).

It seems then that one could argue that &I and &E cannot be adopted, since one must already have a concept of conjunction in order to introduce or eliminate conjunctions. But surely this is a matter of how the rule is formulated: With sufficient cleverness, I'm sure I could define &I and &E in a way that doesn't use 'and' at all, but only 'or' and 'not'. Would the principle then be adoptable, because it is formulated without appeal to the notion it is purporting to introduce?

If the answer is yes, then it immediately raises this question: If whether a principle can be adopted depends on how it is formulated, how do we know that MP and UI cannot be reformulated in a way that doesn't invoke them? For example, surely one could formulate MP in such a way that all Harry needs to know is disjunction and negation. If one wishes to maintain that MP-formulated-with-conditionals is not adoptable while MP-formulated-with-disjunction-and-negation is, then there is good reason to think that one must maintain that these are distinct logical principles. In that case, we're left with what I suspect is an extremely difficult question to answer: What are the identity conditions of logical principles?

At this point, I have no good intuitions about how to begin answering these questions.

© 2016 Sara L. Uckelman.

Saturday, 30 January 2016

Meta-arithmetic and philosophy CFP (Swamplandia 2016)

Swamplandia 2016
Meta-arithmetical results and their philosophical meaning
Ghent, May 30 - June 1, 2016

Logicians and mathematicians devoted considerable effort to investigate the properties and limitations of arithmetical theories. Unfortunately, philosophical motivations and implications of some of these results are either not known or not clear. The main aim of the workshop is to present philosophically relevant meta-arithmetical results and discuss their philosophical implications in more depth. The workshop is focused on, but not restricted to formal theories of truth, theories of provability in arithmetic, logic of provability and philosophically relevant results about complexity or computability. Keynote speakers will deliver invited lectures and give extended tutorials. The title of the workshop comes from the fact that philosophical approaches to mathematical results are rather tricky.

Keynote speakers
Diderik Batens (Ghent University)
Cezary Cieśliński (University of Warsaw)
Jeffrey Ketland (University of Oxford)
Lavinia Picollo (Munich Center for Mathematical Philosophy)
Saeed Salehi (University of Tabriz)
Peter Verdée (Université Catholique de Louvain)
Albert Visser (Utrecht University)

We welcome submissions of papers that strike a balance between technical developments and philosophical discussion. If you’re interested in presenting at the workshop, please send your extended abstract (1000-1500 words) prepared for double-blind review in PDF format to

by March 1, 2016. Authors of accepted papers will have 30-45 minutes to present their work.

A Studia Logica volume on the philosophical aspects of meta-arithmetical and set-theoretic results will be edited by the organizers. Participants are welcome to submit papers for the volume some time after the conference. Details TBA.

Presentation abstract submission: March 1, 2016
Acceptance notification: April 15, 2016
Fee payment deadline:  May 1, 2016
Workshop: May 30, 2016 - June 1, 2016

Faculty: 60 EUR
Students: 40 EUR
Late fee: 30 EUR + basic fee
If your attendance will not be covered by any grant or if you are a student with financial difficulties, please include a statement saying so at the end of your extended abstract, so we can consider you for a conference fee waiver.

Organizers: Rafal Urbaniak, Pawel Pawlowski and Erik Weber

Wednesday, 27 January 2016

Two doctoral fellowships at the MCMP

You would like to write a PhD thesis at the Munich Center for Mathematical Philosophy (MCMP) on paradoxes of truth and/or vagueness, and on the metaphilosophical question about how to handle diverging solutions to such paradoxes? Great! Then please consider applying for one of the

*** Two Doctoral Fellowships at the MCMP ***

which we are advertising right now (as part of the European Training Network DIAPHORA that includes philosophers from Barcelona, Munich, Neuchatel, Stirling, Stockholm, Edinburgh, Paris).

More information can be found at:

Tuesday, 8 December 2015

Why I don't care what possible worlds are

This afternoon, I lectured to my 2nd year students on Lewis and Stalnaker on possible worlds (with a bit of Kripke thrown in since we'd done the 1st lecture of Naming and Necessity two weeks ago). I included these two papers in the syllabus for the same reason I did last year -- because they are pieces of work of historical importance for their role in the debate on realism w.r.t. possible worlds. And like last year, I found both pieces difficult to lecture on, not because they are especially difficult, or especially problematic, but because, as a modal logician, I simply don't care. Resolving this debate -- whether possible worlds really are "out there" like Lewis thinks or whether they're more of a pragmatic tool as Stalnaker thinks -- will not change my practice one whit.

I try not to let my students know that I feel this way (I try to keep my philosophical "politics" out of the classroom -- except when the opportunity to rant on why I think "not philosophical enough" is a horrible criticism, but that is not apropos here), but I do try to let them know that there is more to the issue than resolving the debate, there is the question of whether the debate needs to be resolved before modal logicians can go about their business with impunity. Last year I put it as an essay question, but I don't remember if anyone took it up. This year, in yesterday's tutorial I divided the group into two and randomly told one "You prepare a case in favor of realism", and the other group "You guys get anti-realism", and during the ensuing discussion, I heard someone sort of whisper to someone else "Does it matter?", which I thought an appropriate to revisit the issue. We discussed it some in the tutorial, with one person feeling quite strongly that if one didn't properly settle the 'foundational' issues, then there would be no guarantee that the modal logician wouldn't one day be led astray. At the end of lecture today I posted two questions hoping to get people's gut feelings -- who thinks Lewis is right, who thinks he's not, and who thinks the question has to be resolved, and who thinks it doesn't. As expected, I got roughly equal hands for each, and was lucky enough to have two people willing to articulate their gut feelings. One (on the side of "yes, we do") argued from the basis of metaphysical possibility: If we're going to use possible worlds for analysing metaphysical possibility, we're sure going to want to know if they are metaphysically possible! The other said that you might need to look at reality to determine which axioms you adopted, but after that, it shouldn't matter what possible worlds in fact are when you start using them as a tool in modal reasoning.

All of this set me up to spend some more time thinking this afternoon about why I don't care. It's a rather scientific, rather than philosophical, position to take -- scientists don't care what the "real nature" of particles are (well, except for the foundationalists, i.e., the physicists), mathematicians don't care what numbers "really" are, modal logicians don't care what possible worlds "really" are, etc. The foundational issue raised in the tutorial yesterday gave rise to an apt comparison with mathematics: Mathematicians don't really care about what numbers are, because whatever they are, they sure work really really well, and by now it seems highly unlikely that we could discover something about what numbers are that would cast the results that we've derived using them into doubt. Modal logic isn't in quite the same position with respect to possible worlds, but it seems similar.

I also thought about what a situation in which it mattered what possible worlds were, metaphysically, would look like -- in what sort of situation would the metaphysical nature of possible worlds make a difference? Well, when discussing metaphysical possibility/necessity, as noted above. I happen to find that concept a highly dubious one (on extra-logical grounds), so I'm happy to simply put up my hands and say "that is not a modal concept I am interested in explicating". But as I tried to come up with concrete scenarios in which modal logic is applied, rather than simply theorized about, in each of these cases, the notion of possible world was interpreted as something quite concrete: For example, states of a computer programme. Then I thought about the other student's comment about needing to hash out what the right axioms were, and that possibly being when it was necessary to know something about the metaphysical status of possible worlds. But what is it that axioms specify? Do they specify anything about the worlds themselves? No: What modal axioms do is specify how the worlds are related to each other, and these axioms will hold (or not) in virtue of the relations between the worlds -- whatever the worlds may be. They may be Lewisian possible worlds, they may be states of a computer, they may be moments in time, they may be pebbles, they may be fruitcakes. The axioms -- that which really is the meat of modal reasoning -- are all about how the worlds are related to each other, and not about how the worlds are composed [1].

And that is at least part of the reason why I, as a modal logician, don't really care about what possible worlds are.

[1] At this point, I realize that everything that I've been saying is about propositional modal logic, and that if what you're interested in is quantified modal logic, then you might object that how the worlds are composed, i.e., what objects are in them and what properties those objects have, is of crucial importance, AND that the axioms adopted will have consequences for the internal composition, e.g., whether the Barcan or Converse Barcan formulas are axioms. To which I would reply: Hmmm, this is very interesting, I will have to think on the case of quantified modal logic further.

© 2015, Sara L. Uckelman

Monday, 2 November 2015

The beauty (?) of mathematical proofs -- empirical predictions

By Catarina Dutilh Novaes

This is the final post in my series on beauty, function, and explanation in mathematical proofs (Part I is herePart II is herePart III is herePart IV is herePart V is herePart VI is here; Part VII is here). Here I tease out some empirical predictions of the account developed in the previous posts, according to which beauty and explanatoriness will largely (though not entirely) coincide in mathematical proofs. I also comment on how the account, based on a dialogical conception of mathematical proofs, could be made more palatable for those who would prefer a non-relative, absolute analysis of beauty and explanatoriness.


To summarize, the present account defends the thesis that when mathematicians employ aesthetic vocabulary to describe proofs, both positively (‘beautiful’, ‘elegant’) and negatively (‘ugly’, ‘clumsy’), they are by and large (though not exclusively) tracking the epistemic property of explanatoriness (or lack thereof) of a proof. Up to this point, the account is compatible with both subjective (agent-relative) and objective understandings of beauty and explanation, so long as the two dimensions go together (i.e. both understood as either subjective or as objective). However, on the basis of a dialogical conception of mathematical proofs, I’ve also argued that both explanation and beauty are essentially relative notions with respect to proofs: an explanation is not explanatory an sich, but rather explanatory for its intended audience; and if a proof is deemed beautiful to the extent that it fulfills this explanatory function, then beauty too emerges as a relative notion.

I’ve also suggested ways in which the present account can be made more palatable for those who strongly prefer objective accounts of explanatoriness and beauty. By maximally expanding the range of Skeptics who will deem a proof explanatory – and so aiming towards the notion of a universal audience – in the limit (idealized) case a proof may be deemed explanatory by all (i.e. those who have the required expertise to understand it in the first place). On this conception then, a proof may also be understood to be beautiful in an absolute sense, i.e. insofar it fulfills its explanatory function towards any potential (suitably qualified) audience. The conception of beauty as fit defended by Raman-Sundström (2012), which relies on an objectively conceived notion of fit,[1] may be viewed as an example of such an account, and indeed her description of fit bears a number of similarities with concepts typically associated with explanatoriness.[2]

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.


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.


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.


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.


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.


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.


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!


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).