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).

Monday, 28 September 2015

Easy as 1, 2, 3 ? -- Wittgenstein on counting

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

“A B C
It's easy as, 1 2 3
As simple as, do re mi
A B C, 1 2 3
Baby, you and me girl”

45 years ago, Michael Jackson and his troupe of brothers famously claimed that counting is easy peasy. But how easy is it really? (We’ll leave aside the matter of the simplicity of A B C and do re mi for present purposes!)

Counting and basic arithmetic operations are often viewed as paradigmatic cases of ‘easy’ mental operations. It might seem that we are all ‘born’ with the innate ability for basic arithmetic, given that we all seem to engage in the practice of counting effortlessly. However, as anyone who has cared for very young children knows, teaching a child how to count is typically a process requiring relentless training. The child may well know how to recite the order of numbers (‘one, two, three…’), but from that to associating each of them to specific quantities is a big step. Even when they start getting the hang of it, they typically do well with small quantities (say, up to 3), but things get mixed up when it comes to counting more items. For example, they need to resist the urge to point at the same item more than once in the counting process, something that is in no way straightforward!

The later Wittgenstein was acutely aware of how much training is involved in mastering the practice of counting and basic arithmetic operations. (Recall that he was a schoolteacher for many years in the 1920s!) Indeed, counting and adding objects can be described as a specific and rather peculiar language game which must be learned by training, and which raises all kinds of philosophical questions pertaining to what it is exactly that we are doing when we count things. Perhaps my favorite passage in the whole of the Remarks on the Foundations of Mathematics is #37 in part I:

Monday, 7 September 2015

Cambridge Companion to Medieval Logic - Table of Contents

By Catarina Dutilh Novaes

(This post can be safely classified as an instance of shameless self-promotion, but here we go anyway...) Last week Stephen Read and I delivered the full manuscript of the forthcoming Cambridge Companion to Medieval Logic to Cambridge University Press. We still need to go through the whole production process (including indexing), but at this point it is safe to assume the volume will appear somewhere in 2016. We've been working on this volume for nearly 3 years, and so we are suitably thrilled to be nearing completion!

Many people asked me about the Table of Contents for the volume, and so I figured I might as well make it public -- now that we know there will not be any changes to chapters and/or contributors. Here it is:

0   Introduction – Catarina Dutilh Novaes and Stephen Read      

PART I: Periods and traditions

1   The Legacy of Ancient Logic in the Middle Ages – Julie Brumberg-Chaumont         
2   Arabic Logic up to Avicenna – Ahmad Hasnawi and Wilfrid Hodges  
3   Arabic Logic after Avicenna – Khaled El-Rouayheb      
4   Latin Logic up to 1200 – Ian Wilks          
5   Logic in the Latin Thirteenth Century – Sara L. Uckelman and Henrik Lagerlund   
6   Logic in the Latin West in the Fourteenth Century – Stephen Read  
7   The Post-Medieval Period – E. Jennifer Ashworth   

PART II: Themes
8   Logica Vetus – Margaret Cameron           
9   Supposition and properties of terms – Christoph Kann          
10 Propositions: Their meaning and truth – Laurent Cesalli        
11 Sophisms and Insolubles – Mikko Yrjönsuuri and Elizabeth Coppock           
12 The Syllogism and its Transformations – Paul Thom    
13 Consequence – Gyula Klima          
14 The Logic of Modality – Riccardo Strobino and Paul Thom      
15 Obligationes – Catarina Dutilh Novaes and Sara L. Uckelman 

Bridges 2 – Workshop at Rutgers

The Rutgers Philosophy Department and the Rutgers Center for Cognitive Science will be hosting a workshop (on Logic, Language, Epistemology, and Philosophy of Science) September 18-20. The workshop will bring together scholars from the NYC area, Amsterdam (the Institute for Logic, Language, and Computation), and Munich (the Munich Center for Mathematical Philosophy). The schedule for the workshop is posted on the Bridges 2 webpage:

The event is open to the public.

Wednesday, 26 August 2015

Formal Methods in Philosophy: a Brief Introduction (Part II)

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

This is the second and final part of my 'brief introduction' to formal methods in philosophy to appear in the forthcoming Bloomsbury Philosophical Methodology Reader, being edited by Joachim Horvath. (Part I is here.) In this part I present in more detail the four papers included in the formal methods section, namely Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. 


Some of the pioneers in formal/mathematical approaches to philosophical questions had a number of interesting things to say on the issue of what counts as an adequate formalization, in particular Tarski and Carnap – hence the inclusion of pieces by each of them in the present volume. Indeed, both in his paper on truth and in his paper on logical consequence (in the 1930s), Tarski started out with an informal notion and then sought to develop an appropriate formal account of it. In the case of truth, the starting point was the correspondence conception of truth, which he claimed dated back to Aristotle. In the case of logical consequence, he was somewhat less precise and referred to the ‘common’ or ‘everyday’ notion of logical consequence.

These two conceptual starting points allowed Tarski to formulate what he described as ‘conditions of material adequacy’ for the formal accounts. He also formulated criteria of formal correctness, which pertain to the internal exactness of the formal theory. In the case of truth, the basic condition of material adequacy was the famous T-schema; in the case of logical consequence, the properties of necessary truth-preservation and of validity-preserving schematic substitution. Unsurprisingly, the formal theories he then went on to develop both passed the test of material adequacy he had formulated himself. But there is nothing particularly ad hoc about this, since the conceptual core of the notions he was after was presumably captured in these conditions, which thus could serve as conceptual ‘guides’ for the formulation of the formal theories.

Friday, 21 August 2015

Formal Methods in Philosophy: a Brief Introduction (Part I)

By Catarina Dutilh Novaes
(Cross-posted in NewAPPS)

There is a Bloomsbury Philosophical Methodology Reader in the making, being edited by Joachim Horvath (Cologne). Joachim asked me to edit the section on formal methods, which will contain four papers: Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. It will also contain a brief introduction to the topic by me, which I will post in two installments. Here is part I: comments welcome!


Since the inception of (Western) philosophy in ancient Greece, methods of regimentation and formalization, broadly understood, have been important items in the philosopher’s toolkit (Hodges 2009). The development of syllogistic logic by Aristotle and its extensive use in centuries of philosophical tradition as a formal tool for the analysis of arguments may be viewed as the first systematic application of formal methods to philosophical questions. In medieval times, philosophers and logicians relied extensively on logical tools other than syllogistic (which remained pervasive though) in their philosophical analyses (e.g. medieval theories of supposition, which come quite close to what is now known as formal semantics). But the level of sophistication and pervasiveness of formal tools in philosophy has increased significantly since the second half of the 19th century. (Frege is probably the first name that comes to mind in this context.)

It is commonly held that reliance on formal methods is one of the hallmarks of analytic philosophy, in contrast with other philosophical traditions. Indeed, the birth of analytic philosophy at the turn of the 20th century was marked in particular by Russell’s methodological decision to treat philosophical questions with the then-novel formal, logical tools developed for axiomatizations of mathematics (by Frege, Peano, Dedekind etc. – see (Awodey & Reck 2002) for an overview of these developments), for example in his influential ‘On denoting’ (1905). (Notice though that, from the start, there is an equally influential strand within analytic philosophy focusing on common sense and conceptual analysis, represented by Moore – see (Dutilh Novaes & Geerdink forthcoming).) This tradition was then continued by, among others, the philosophers of the Vienna Circle, who conceived of philosophical inquiry as closely related to the natural and exact sciences in terms of methods. Tarski, Carnap, Quine, Barcan Marcus, Kripke, and Putnam are some of those who have applied formal techniques to philosophical questions. Recently, there has been renewed interest in the use of formal, mathematical tools to treat philosophical questions, in particular with the use of probabilistic, Bayesian methods (e.g. formal epistemology). (See (Papineau 2012) for an overview of the main formal frameworks used for philosophical inquiry.)

Tuesday, 18 August 2015

Book review: John P. Burgess' Rigor and Structure (OUP)

Rigor and Structure, Burgess tells us in the preface, was originally intended to provide for mathematical structuralism the sort of survey that A Subject with No Object (Burgess & Rosen, 1999) provided for nominalism. However, the book that Burgess has ended up writing is importantly different from his earlier work with Rosen. In large part, this is because, for Burgess, not only is mathematical structuralism true --- whereas he took nominalism to be false --- but moreover it is a ''trivial truism'', at least as a description of modern mathematics from the beginning of the twentieth century onwards (Burgess, 2015, 111). Thus, instead of providing philosophical arguments in favour of mathematical structuralism, Burgess instead devotes the first half of the book (Chapters 1 and 2) to providing an historical account of how mathematics developed into the modern discipline of which mathematical structuralism is so obviously a true description. And this is where the other component of the title enters the story. For it is Burgess' contention that modern structuralist mathematics --- which he explores in the second half of the book, that is, in Chapters 3 and 4 --- is an inevitable consequence of the long quest for rigor, which began, so far as we know, with Euclid's Elements, and was completed by work in the nineteenth and early twentieth century that led to the arithmetization of analysis, the axiomatization or arithmetic, analysis, and geometry, the formulation of non-Euclidean geometries, and the founding of modern algebraic theories, such as group theory.

Thus, in the first two chapters of Rigor and Structure, Burgess asks two questions: What is mathematical rigor? Why did mathematicians strive so hard to achieve it throughout the period just described? To answer the first question, Burgess turns initially to the pronouncements of mathematicians themselves, but he finds little that is precise enough to satisfy a philosopher there. So he turns next to Aristotle and, looking to the Posterior Analytics, extracts the following suggestion:

Mathematical rigor requires that:
  • ''every new proposition must be deduced from previously established propositions'';
  •  ''every new notion must be defined in terms of previously explained notions'';
  • there are primitive notions from which the chain of definitions begins;
  • there are primitive postulates from which the chain of deductions begins;
  • ''the meaning of the primitives and the truth of the postulates must be evident''.
(Burgess, 2015, 6-7)

Monday, 27 July 2015

Women in Logic: two new initiatives

For those who haven't yet come across these, I have two new initiatives relating to women in logic to advertise:

  • Women in Logic group on Facebook: "A group for women in Logic, philosophical, mathematical or computational. or any other kind of formal logic that you care about." Membership is not restricted to women.
  • Female Professors of Logic, an editable google spreadsheet. One outcome of this will be to give a list of people who should have wikipedia pages if they don't already.

Please share widely and contribute as you can.

© 2015 Sara L. Uckelman

Tuesday, 21 July 2015

Reductio arguments from a dialogical perspective: final considerations

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the final post in my series on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, here is Part IV, and here is Part V. I now return to the issues raised in the earlier posts equipped with the dialogical account of deduction, and of reductio ad absurdum in particular.


A general dialogical schema for reductio ad absurdum, following Proclus’ description but inspired by the Socratic elenchus, might look like this:
  1. Interlocutor 1 commits to A (either prompted by a question from interlocutor 2, or spontaneously), which corresponds to assuming the initial hypothesis.
  2. Interlocutor 2 leads the initial hypothesis to absurdity, typically by relying on additional discursive commitments of 1 (which may be elicited by 2 through questions).
  3. Interlocutor 2 concludes ~A.

The main difference between the monological and the dialogical versions of a reductio is thus that in the latter there is a kind of division of labor that is absent from the former (as noted above). The agent making the initial assumption is not the same agent who will lead it to absurdity, and then conclude its contradictory. And so, the perceived pragmatic awkwardness of making an assumption precisely with the goal of ‘destroying’ it seems to vanish. Moreover, the adversarial component provides a compelling rationale for the general idea of ‘destroying’ the initial hypothesis; indeed, while the adversarial component is present in all deductive arguments (in particular given the requirement of necessary truth preservation, as argued above), it is even more pronounced in the case of reductio arguments, that is the procedure whereby someone’s discursive commitments are shown to be collectively incoherent since they lead to absurdity. There remains the question of why interlocutor 1 would want to engage in the dialogue at all, but presumably she simply wishes to voice a discursive commitment to A. From there on, the wheel begins to spin, mostly through 2’s actions.

Monday, 20 July 2015

Conference on Belief, Rationality, and Action over Time

University of Wisconsin-Madison, September 5-7The goal is to get action theorists and epistemologists (especially formal epistemologists) together to think about topics related to diachronic rationality and belief.  All are welcome, but attendees are expected to have read the papers beforehand.  Register for free here.

Organizers:  Mike Titelbaum, Sergio Tenenbaum, Chrisoula Andreou, and Sarah Paul
Funded by the Canadian Journal of Philosophy, the University of Wisconsin, and a gift from Rodney J. Blackman. 

Friday, 17 July 2015

Dialectical refutations and reductio ad absurdum

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the fifth installment of my series of posts on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, and here is Part IV. In this post I discuss a closely related argumentative strategy, namely dialectical refutation, and argue that it can be viewed as a genealogical ancestor of reductio ad absurdum.


Those familiar with Plato’s Socratic dialogues will undoubtedly recall the numerous instances in which Socrates, by means of questions, elicits a number of discursive commitments from his interlocutors, only to go on to show that, taken collectively, these commitments are incoherent. This is the procedure known as an elenchus, or dialectical refutation.

The ultimate purpose of such a refutation may range from ridiculing the opponent to nobler didactic goals. The etymology of elenchus is related to shame, and indeed at least in some cases it seems that Socrates is out to shame the interlocutor by exposing the incoherence of their beliefs taken collectively (for example, so as to exhort them to positive action, as argued in (Brickhouse & Smith 1991)). However, as noted by Socrates himself in the Gorgias (470c7-10), refuting is also what friends do to each other, a process whereby someone rids a friend of nonsense. An elenchus can also have pedagogical purposes, in interactions between masters and pupils.

There has been much discussion in the secondary literature on what exactly an elenchus is, as well as on whether there is a sufficiently coherent core of properties for what counts as an elenchus, beyond a motley of vaguely related argumentative strategies deployed by Socrates (Carpenter & Polansky 2002). (A useful recent overview is (Wolfsdorf 2013); see also (Scott 2002).) For our purposes, it will be useful to take as our starting point the description of the ‘Socratic method’ in an influential article by G. Vlastos (1983) (a much shorter version of the same argument is to be found in (Vlastos 1982), and I'll be referring to the shorter version). Vlastos distinguishes two kinds of elenchi, the indirect elenchus and the standard elenchus:

Thursday, 16 July 2015

A precis of the dialogical account of deduction

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the fourth installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, here is Part II, and here is Part III. In this post I offer a précis of the dialogical account of deduction which I have been developing over the last years, which will then allow me to return to the issue of reductio arguments equipped with a new perspective in the next installments. I have presented the basics of this conception in previous posts, but some details of the account have changed, and so it seems like a good idea to spell it out again.


In this post, I present a brief account of the general dialogical conception of deduction that I endorse. Its relevance for the present purposes is to show that a dialogical conception of reductio ad absurdum arguments is not in any way ad-hoc; indeed, the claim is that this conception applies to deductive arguments in general, and thus a fortiori to reductio arguments. (But I will argue later on that the dialogical component is even more pronounced in reductio arguments than in other deductive arguments.)

Let us start with what can be described as functionalist questions pertaining to deductive arguments and deductive proofs. What is the point of deductive proofs? What are they good for? Why do mathematicians bother producing mathematical proofs at all? While these questions are typically ignored by mathematicians, they have been raised and addressed by so-called ‘maverick’ philosophers of mathematics, such as Hersh (1993) and Rav (1999). One promising vantage point to address these questions is the historical development of deductive proof in ancient Greek mathematics,[1] and on this topic the most authoritative study remains (Netz 1999). Netz emphasizes the importance of orality and dialogue for the emergence of classical, ‘Euclidean’ mathematics in ancient Greece:

Greek mathematics reflects the importance of persuasion. It reflects the role of orality, in the use of formulae, in the structure of proofs… But this orality is regimented into a written form, where vocabulary is limited, presentations follow a relatively rigid pattern… It is at once oral and written… (Netz 1999, 297/8)

Wednesday, 15 July 2015

Problems with reductio proofs: "jumping to conclusions"

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the third installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, and here is Part II. In this post I discuss issues pertaining specifically to the last step in a reductio argument, namely that of going from reaching absurdity to concluding the contradictory of the initial hypothesis.


One worry we may have concerning reductio arguments is what could be described as ‘the culprit problem’. This is not a worry clearly formulated in the protocols previously described, but one which has been raised a number of times when I presented this material to different audiences. The basic problem is: we start with the initial assumption, which we intend to prove to be false, but along the way we avail ourselves to auxiliary hypotheses/premises. Now, it is the conjunction of all these premises and hypotheses that lead to absurdity, and it is not immediately clear whether we can single out one of them as the culprit to be rejected. For all we know, others may be to blame, and so there seems to be some arbitrariness involved in singling out one specific ingredient as responsible for things turning sour.

To be sure, in most practical cases this will not be a real concern; typically, the auxiliary premises we avail ourselves to are statements on which we have a high degree of epistemic confidence (for example, because they have been established by proofs that we recognize as correct). But it remains of philosophical significance that absurdity typically arises from the interaction between numerous elements, any of which can, in theory at least, be held to be responsible for the absurdity. A reductio argument, however, relies on the somewhat contentious assumption that we can isolate the culprit.

However, culprit considerations do not seem to be what motivates Fabio’s dramatic description of this last step as “an act of faith that I must do, a sacrifice I make”. Why is this step problematic then? Well, in first instance, what is established by leading the initial hypothesis to absurdity is that it is a bad idea to maintain this hypothesis (assuming that it can be reliably singled out as the culprit, e.g. if the auxiliary premises are beyond doubt). How does one go from it being a bad idea to maintain the hypothesis to it being a good idea to maintain its contradictory?

Tuesday, 14 July 2015

Problems with reductio proofs: assuming the impossible

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is a series of posts with sections of the paper on reductio ad absurdum from a dialogical perspective that I am working on right now. This is Part II, here is Part I. In this post I discuss issues in connection with the first step in a reductio argument, that of assuming the impossible.


We can think of a reductio ad absurdum as having three main components, following Proclus’ description:

(i) Assuming the initial hypothesis.
(ii) Leading the hypothesis to absurdity.
(iii) Concluding the contradictory of the initial hypothesis.

I discuss two problems pertaining to (i) in this post, and two problems pertaining to (iii) in the next post. (ii) is not itself unproblematic, and we have seen for example that Maria worries whether the ‘usual’ rules for reasoning still apply once we’ve entered the impossible world established by (i). Moreover, the problematic status of (i) arises to a great extent from its perceived pragmatic conflict with (ii). But the focus will be on issues arising in connection with (i) and (iii).

A reductio proof starts with the assumption of precisely that which we want to prove is impossible (or false). As we’ve seen, this seems to create a feeling of cognitive dissonance in (some) reasoners: “I do not know what is true and what I pretend [to be] true.” (Maria) This may seem surprising at first sight: don’t we all regularly reason on the basis of false propositions, such as in counterfactual reasoning? (“If I had eaten a proper meal earlier today, I wouldn’t be so damn hungry now!”) However, as a matter of fact, there is considerable empirical evidence suggesting that dissociating one’s beliefs from reasoning is a very complex task, cognitively speaking (to ‘pretend that something is true’, in Maria’s terms). The belief bias literature, for example, has amply demonstrated the effect of belief on reasoning, even when participants are told to focus only on the connections between premises and conclusions. Moreover, empirical studies of reasoning behavior among adults with low to no schooling show their reluctance to reason with premises of which they have no knowledge (Harris 2000; Dutilh Novaes 2013). From this perspective, reasoning on the basis of hypotheses or suppositions may well be something that requires some sort of training (e.g. schooling) to be mastered.

Monday, 13 July 2015

Problems with reductio proofs: cognitive aspects

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

As some readers may recall, I ran a couple of posts on reductio proofs from a dialogical perspective quite some time ago (here and here). I am now *finally* writing the paper where I systematize the account. In the coming days I'll be posting sections of the paper; as always, feedback is most welcome! The first part will focus on what seem to be the cognitive challenges that reasoners face when formulating reductio arguments.


For philosophers and mathematicians having been suitably ‘indoctrinated’ in the relevant methodologies, the issues pertaining to reductio ad absurdum arguments may not become immediately apparent, given their familiarity with the technique. And so, to get a sense of what is problematic about these arguments, let us start with a somewhat dramatic but in fact quite accurate account of what we could describe as the ‘phenomenology’ of producing a reductio argument, in the words of math education researcher U. Leron:

We begin the proof with a declaration that we are about to enter a false, impossible world, and all our subsequent efforts are directed towards ‘destroying’ this world, proving it is indeed false and impossible. (Leron 1985, 323)

In other words, we are first required to postulate this impossible world (which we know to be impossible, given that our very goal is to refute the initial hypothesis), and then required to show that this impossible world is indeed impossible. The first step already raises a number of issues (to be discussed shortly), but the tension between the two main steps (postulating a world, as it were, and then proceeding towards destroying it) is perhaps even more striking. As it so happens, these are not the only two issues that arise once one starts digging deeper.

To obtain a better grasp of the puzzling nature of reductio arguments, let us start with a discussion of why these arguments appear to be cognitively demanding – that is, if we are to believe findings in the math education literature as well as anecdotal evidence (e.g. of those with experience teaching the technique to students). This will offer a suitable framework to formulate further issues later on.