Tuesday, 17 January 2017

The Principal Principle does not imply the Principle of Indifference

Recently, James Hawthorne, Jürgen Landes, Christian Wallmann, and Jon Williamson published a paper in the British Journal of Philosophy of Science in which they claim that the Principal Principle entails the Principle of Indifference -- indeed, the paper is called 'The Principal Principle implies the Principle of Indifference'. In this post, I argue that it does not.

All Bayesian epistemologists agree on two claims. The first, which we might call Precise Credences, says that an agent's doxastic state at a given time $t$ in her epistemic life can be represented by a single credence function $P_t$, which assigns to each proposition $A$ about which she has an opinion a precise numerical value $P_t(A)$ that is at least 0 and at most 1. $P_t(A)$ is the agent's credence in $A$ at $t$. It measures how strongly she believes $A$ at $t$, or how confident she is at $t$ that $A$ is true. The second point of agreement, which is typically known as Probabilism, says that an agent's credence function at a given time should be a probability function: that is, for all times $t$, $P_t(\top) = 1$ for any tautology $\top$, $P_t(\bot) = 0$ for any contradiction $\bot$, and $P_t(A \vee B) = P_t(A) + P_t(B) - P_t(AB)$ for any propositions $A$ and $B$.

So Precise Credences and Probabilism form the core of Bayesian epistemology. But, beyond these two norms, there is little agreement between its adherents. Bayesian epistemologists disagree along (at least) two dimensions. First, they disagree about the correct norms concerning updating on evidence learned with certainty --- some say they are diachronic norms concerning how an agent should in fact update; others say that there are only synchronic norms concerning how an agent should plan to update; and others think there are no norms concerning updating at all. Second, they disagree about the stringency of the synchronic norms that don't concern updating. Our concern here is with the latter. Some candidates norms of this sort: the Principal Principle, which says how an agent's credences in propositions concerning the objective chances should relate to her credences in other propositions (Lewis 1980); the Reflection Principle, which says how an agent's current credences in propositions concerning her future credences should relate to her current credences in other propositions (van Fraassen 1984, Briggs 2009); and the Principle of Indifference, which says, roughly, that an agent with no evidence should divide her credences equally over all possibilities (Keynes 1921, Carnap 1950, Jaynes 2003, Williamson 2010, Pettigrew 2014). Those we might call Radical Subjective Bayesians adhere to Precise Credences and Probabilism, but reject the Principal Principle, the Reflection Principle, and the Principle of Indifference. Those we might call Moderate Subjective Bayesians adhere to Precise Credences, Probabilism, and the Principal Principle (and also, quite often, the Reflection Principle), but they reject the Principle of Indifference. And the Objective Bayesians accept all of the principles.

In a recent paper, Hawthorne et al. (2015) (henceforth, HLWW) argue that Moderate Subjective Bayesianism is an inconsistent position, because the Principal Principle (and, indeed the Reflection Principle) entails the Principle of Indifference. Thus, it is inconsistent to accept the former and reject the latter. We must either reject the Principal Principle, as the Radical Subjective Bayesian does, or accept it together with the Principle of Indifference, as the Objective Bayesian does.

Notoriously, as Lewis originally stated it, the Principal Principle includes an admissibility condition (266-7, Lewis 1980). Equally notoriously, Lewis did not provide a precise account of this condition, thereby leaving his formulation of the principle similarly imprecise. HLWW do not give a precise account either. But they do appeal to two principles that they take to follow intuitively from the Principal Principle. And from these two principles, together with the Principal Principle itself, they derive what they take to be an instance of the Principle of Indifference. The first principle to which they appeal --- their Condition 1 --- is in fact provable, as they note. The second --- their Condition 2 --- is not. Indeed, as we will see, on the correct understanding of admissibility, it is false. Thus, the HLWW argument fails. What's more, its conclusion is not true. It is possible to satisfy the Principal Principle without satisfying the Principle of Indifference, as we will see below. Moderate Subjective Bayesianism is a coherent position.

Introducing the Principal Principle

We begin by introducing the Principal Principle. To aid our statement, let me introduce a piece of notation. Given a proposition $A$ and a real number $0 \leq x \leq 1$, let $C^A_x$ be the following proposition: The current objective chance of $A$ is $x$. And we will let $P_0$ be the credence function of our agent at the very beginning of her epistemic life --- when she is, as Lewis would say, a superbaby; that is, she is not yet in receipt of any evidence. Then, as Lewis originally formulates the Principal Principle, it says this:

Lewis' Principal Principle Suppose $A$, $E$ are propositions and $0 \leq x \leq 1$. Then it should be the case that $$P_0(A | C^A_xE) = x $$providing (i) $P_0(C^A_xE) > 0$, and (ii) $E$ is admissible for $A$.

In this version, the principle applies to an agent only at the beginning of her epistemic life; it governs her initial credence function. In this situation, the principle says, her credence in a proposition $A$ conditional on the conjunction of some proposition $E$ and a chance proposition that says that the chance of $A$ is $x$ should be $x$, providing the conditional probability is well-defined and $E$ is admissible for $A$.

The motivation for the admissibility condition is this. Suppose $E$ entails $A$. Then we surely don't want to demand that $P_0(A | C^A_xE) = x$. After all, if $x < 1$, then such a demand would conflict with Probabilism, since it is a consequence of Probabilism that, if $E$ entails $A$, then $P_0(A | C^A_xE) = 1$. Thus, we must at least restrict the Principal Principle so that it does not apply when $E$ entails $A$. But there are other cases in which the Principal Principle should not be imposed, even if such an application would not be outright inconsistent with other norms such as Probabilism. For instance, suppose that $E$ entails that the chance of $A$ at some time in the future is $x' \neq x$. Then, again, we don't want to require that $P_0(A | C^A_xE) = x$. The moral is this: if $E$ contains information about $A$ that overrides the information that the current chance of $A$ gives about $A$, then it is inadmissible. Clearly any proposition that logically entails $A$ provides information that overrides the current chance information about $A$; and so does a proposition that entails something about the future chance of $A$. So much for propositions that are inadmissible. Are there any we can be sure are admissible? According to Lewis, there are, namely, propositions solely concerning the past or the present. Thus, Lewis does not give a precise account of admissibility: he gives a heuristic --- $E$ is admissible for $A$ if $E$ does not provide information about $A$ that overrides the information contained in propositions about the current chance of $A$ --- and he gives examples of propositions that do and do not provide such information --- I've recalled some of Lewis' examples here.

Now, as Lewis himself noted, the Principal Principle has implausible consequences when the chances are self-undermining --- that is, when the chances assign a positive probability to outcomes in which the chances are different. This happens, for instance, for Lewis' own favoured account of chance, the Humean account or Best System Analysis. This lead to reformulations of the Principal Principle, such as Thau's and Hall's New Principle (Lewis 1994, Thau 1994, Hall 1994) and Ismael's General Recipe  (Ismael 2008). HLWW say nothing explicitly  about whether or not chances are self-undermining. But, since they are interested in investigating the Principal Principle and not the New Principle or the General Recipe,  I take them to assume that chances are not self-undermining. I will do likewise.

The HLWW argument

However imprecise Lewis' account of admissibility is, HLWW take it to be precise enough to allow us to be confident of the following principles:

Condition 1  If
(1a) $E$ is admissible for $A$, and
(1b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,
(1c) $EF$ is admissible for $A$.

Now, HLWW propose to make (1b) precise as follows: $$P_0(A | FC^A_xE) = P_0(A | C^A_xE)$$ That is, $C^A_xE$ contains no information that renders $F$ relevant to $A$ just in case $C^A_xE$ renders $A$ probabilistically independent of $F$. With that explication in hand, Condition 1 now actually follows logically from Lewis' Principal Principle, as HLWW note. After all, by (1a) and Lewis' Principal Principle, $P_0(A | C^A_xE) = x$. And, by the explication of (1b), $P_0(A | C^A_xE) = P_0(A | FC^A_xE)$. Daisychaining these identities together, we have $P_0(A | FC^A_xE) = x$, which is (1c).

Condition 2 If
(2a) $E$ is admissible for $A$, and
(2b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,
(2c) $E(A \leftrightarrow F)$ is admissible for $A$.

This is not provable. Indeed, as we will see below, it is false. Nonetheless, together with Lewis' Principal Principle, Conditions 1 and 2 entail a constraint on an agent's credence function that HLWW take to be the constraint imposed by the Principle of Indifference.

Proposition 1 Suppose Lewis' Principal Principle together with Conditions 1 and 2 hold. And suppose that there are propositions $A$, $E$, and $F$ and $0 < x < 1$ such that $E$ is admissible for $A$. Suppose further that $F$ is atomic and contingent. Then

(i) If $C^A_xE$ contains no information that renders $F$ relevant to $A$, then the following is required of the agent's initial credence function: $P_0(F | C^A_xE) = 0.5.$

(ii) If $C^A_xE$ contains no information whatsoever about $F$ (so that $P_0(F | C^A_xE) = P_0(F)$), then the following is required of the agent's initial credence function: $P_0(F) = 0.5$

HLWW take Proposition 1 to show that the Principle of Indifference follows from the Principal Principle. After all, Condition 1 is simply a theorem. And they take Condition 2 to be a consequence of the Principal Principle, given the correct understanding of admissibility. So if you assume the Principal Principle, you get all of the hypotheses of the theorem. However, as we will see in the next two sections, Condition 2 is in fact false.

Levi's Principal Principle and Levi-Admissibility

Above, we stated the Principal Principle as follows:

Lewis' Principal Principle $P_0(A | C^A_xE) = x$, providing (i) $P_0(C^A_xE) > 0$,  and (ii) $E$ is admissible for $A$.

Now suppose we make the following assumption about admissibility:

Current Chance Admissibility Propositions about the current objective chances are admissible.

Thus, for instance, $P_0(A | C^A_xC^B_y) = x$, providing $P_0(C^A_xC^B_y) > 0$, which also ensures that $C^A_x$ and $C^B_y$ are compatible.

Now suppose that, if $ch$ is a probability function defined over all the propositions about which the agent has an opinion, $C_{ch}$ is the proposition that says that the objective chances are given by $ch$. Then it follows from the Principal Principle and Current Chance Admissibility that $P_0(A | C_{ch}) = ch(A)$. But it also follows from this that:

Levi's Principal Principle (Bodgan 1984, Pettigrew 2012) $P_0(A | C_{ch}E) = ch(A | E)$, providing $P_0(C_{ch}E), ch(E) > 0$.

This is a version of the Principal Principle that makes no mention of admissibility. From it, something close to Lewis' Principal Principle follows: If $P_0(C^{A|E}_x E) > 0$, then $$P_0(A | C^{A|E}_x E) = x$$ where $C^{A|E}_x$ is the proposition: The current objective chance of $A$ conditional on $E$ is $x$. What's more, while Levi's version does not mention admissibility, since it applies equally when the proposition $E$ is not admissible, it does suggest a precise account of admissibility. And it is possible to show that, if we take the version of Lewis' Principal Principle that results from understanding admissibility in this way, it is a consequence of Levi's Principal Principle.

Levi-Admissibility $A$ is Levi-admissible for $E$ if, for all possible chance functions $ch$, $ch(A | E) = ch(A)$.

That is, on this account $A$ is admissible for $E$ if every chance function renders $A$ and $E$ stochastically independent. Three points are worthy of note:
  1. All propositions providing future information about the chance of $A$ or information about the truth value of $A$ are Levi-inadmissible, since $A$ will be stochastically dependent on such propositions according to all possible current chance functions. So this account of admissibility agrees with the examples of clearly inadmissible propositions that we gave above.
  2. All propositions solely about the past are Levi-admissible, since all such propositions will now be true or false and will be assigned chance 1 or 0 accordingly by all possible current chance functions. So this account of admissibility agrees with the examples of clearly admissible propositions that we gave above.
  3. If $A$ is Levi-admissible for $E$, then $P_0(A | C^A_xE) = P_0(A | C^{A|E}_xE ) = x$. That is, Lewis' Principal Principle follows from Levi's version if we understand Lewis' notion of admissibility as Levi-admissibility.
Taken together, (1), (2), and (3) entail that Levi-admissibility has all of the features that Lewis wished admissibility to have.

Now, although Levi's account of admissibility recovers Lewis' examples, it might seem to be too demanding. Suppose, for instance, that $A$ is a proposition concerning the toss of a coin in Quito --- it says that it will lands heads --- while $E$ is a proposition concerning tomorrow's weather in Addis Ababa --- it says that it will rain. Then, intuitively, $E$ is admissible for $A$. But $E$ is not Levi-admissible for $A$. After all, we are considering an agent at the beginning of her epistemic life. And so there are certainly possible chance functions --- probability functions that, for all she knows, give the objective chances --- that do not render $E$ and $A$ stochastically independent.

However, in fact, on closer inspection, the Levi-admissibility verdict is exactly right. Consider my credence in $A$ conditional on $E$ and the chance hypothesis $C^A_{0.5}$, which says that the coin in Quito is fair and so the unconditional chance of $A$ is 0.5. Amongst the chance functions that are epistemically possible for me, some make $E$ irrelevant to $A$, some make it positively relevant to $A$ and some make it negatively relevant to $A$. Indeed, we might suppose that the possible chances of $A$ conditional on $E$ run the full gamut of values from 0 to 1. In that case, surely we don't want to say that $E$ is admissible for $A$ and thereby impose, via the Principal Principle, the demand that our agent's credence in $A$ conditional on $E$ and $C^A_{0.5}$ is 0.5. After all, if I choose to place most of my prior credence on the chance hypotheses on which $E$ is positively relevant to $A$, then my credence in $A$ conditional on $E$ and $C^A_{0.5}$ should not be 0.5 --- it should be something greater than 0.5. If I choose to place most of my prior credence on the chance hypotheses on which $E$ is negatively relevant to $A$, then my credence in $A$ conditional on $E$ and $C^A_{0.5}$ should not be 0.5 --- it should be something less than 0.5. Of course, we might think that it is irrational for our agent, a superbaby with no evidence one way or the other, to favour the positive relevance hypotheses over those that posit neutral relevance and negative relevance. We might think that she should spread her credences equally over all of the possibilities, in which case their effects will cancel out, and her credence in $A$ conditional on $E$ and $C^A_{0.5}$ will indeed be 0.5. But of course to do this is to assume the Principle of Indifference and beg the question.

The failure of Condition 2

With this precise account of admissibility in hand, we can now test to see whether or not it vindicates Condition 2 --- recall, HLWW claim that this is a consequence of the Principal Principle. As we saw above, Condition 2 runs as follows:

Condition 2 If
(2a) $E$ is admissible for $A$, and
(2b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,
(2c) $E(A \leftrightarrow F)$ is admissible for $A$.

Now suppose that Lewis' Principal Principle is true, and assume that admissibility means Levi-admissibility. Then this is equivalent to:

Condition 2$^*$ If $ch$ is a possible chance function, and
(2a$^*$) $ch(A | E) = ch(A)$, and
(2b$^*$) $ch(A | FE) = ch(A | E)$,
(2c$^*$) $ch(A | E(A \leftrightarrow F)) = ch(A)$.

However, this is false. Indeed, we can show the following:

Proposition 2 For any value $0 \leq y \leq 1$, there is a chance function $ch$ such that (2a$^*$) and (2b$^*$) hold, but $$ch(A | E(A \leftrightarrow F)) = y$$

Thus, (2a$^*$) and (2b$^*$) impose no constraints whatsoever on the chance of $A$ conditional on $E(A \leftrightarrow F)$.

Thus, it is possible that $E$ is Levi-admissible for $A$ and that $C^A_xE$ carries no information whatsoever about $F$, and yet $E(A \leftrightarrow F)$ is not Levi-admissible for $A$. Thus, Condition 2 is false and the HLWW argument fails.

Levi's Principal Principle and the Principle of Indifference

Of course, the failure of an argument does not entail the falsity of its conclusion. It might yet be the case that the Principal Principle entails the Principle of Indifference, even if the HLWW argument does not show that. But in fact we can show that this is not true. To see this, we note a sufficient condition for satisfying Levi's Principal Principle:

Proposition 3 Suppose $C$ is the set of all possible chance functions. Then, if $P_0$ is in the convex hull of $C$, then $P_0(A | C_{ch} E) = ch(A | E)$.

Now, if Levi's Principal Principle entails the Principle of Indifference, and the Principle of Indifference entails that every atomic proposition has probability 0.5, then it follows that every member of the convex hull of the set of possible chance functions must assign probability 0.5 to every atomic proposition. But it is easy to see that this is not true. Let $F$ be the atomic proposition that says that a sample of uranium will decay at some point in the next hour. In the absence of evidence, the possible chances of $F$ range over the full unit interval from 0 to 1. Thus, there are members of the convex hull of the set of possible chance functions that assign probabilities other than 0.5 to $F$. And, by Proposition 3, these members will satisfy Levi's Principal Principle.

Applying Levi's Principal Principle

A possible objection: Levi's Principal Principle is all well and good in theory, but it is not applicable. Suppose we are interested in a proposition $A$; and we have collected evidence $E$. How might we apply Levi's Principal Principle in order to set our credence in $A$? In the case of Lewis' version of the principle, we need only know the chance of $A$ and the fact that $E$ is admissible for $A$, and we often know both of  these. But, in order to apply Levi's version, we must know the chance of $A$ conditional on our evidence $E$. And, at least for large and varied bodies of evidence, we never know this. Or so the objection goes.

But the objection fails. In fact, Levi's Principal Principle may be applied in those cases. You don't have to know the chance of $A$ conditional on $E$ in order to set your credence in $A$ when you have evidence $E$. You simply have to have opinions about the different possible values that that conditional chance might take. You then apply Levi's Principal Principle, together with the Law of Total Probability, which jointly entail that your credence in $A$ given $E$ should be your expectation of the chance of $A$ given $E$. Of course, neither Levi's Principal Principle nor the Law of Total Probability will tell you how to set your credences in the different possible values that the conditional chance of $A$ given $E$ might take. But that's not a problem for the Moderate Subjective Bayesian, who doesn't expect her evidence to pin down a unique credal response. Only the Objective Bayesian would expect that. You pick your probability distribution over those possible conditional chance values and Levi's Principal Principle does the rest via the Law of Total Probability.


The HLWW argument purports to show that the Principal Principle entails the Principle of Indifference. But it fails because, on the correct understanding of admissibility, Condition 2 is not a consequence of the Principal Principle; and indeed it is false. What's more, we can see that there are credence functions that satisfy the correct version of the Principal Principle --- namely, Levi's Principal Principle --- that do not satisfy the Principle of Indifference. The logical space is therefore safe once again for Moderate Subjective Bayesians, that is, those who accept Precise Credences, Probabilism, the Principal Principle (and perhaps the Reflection Principle), but who deny the Principle of Indifference.


  • Bogdan, R. (Ed.) (1984). Henry E. Kyburg, Jr. and Isaac Levi. Dordrecht: Reidel.
  • Briggs, R. (2009). Distorted Reflection. Philosophical Review, 118(1), 59–85.
  • Carnap, R. (1950). Logical Foundations of Probability. Chicago: University of Chicago Press.
  • Hall, N. (1994). Correcting the Guide to Objective Chance. Mind, 103, 505–518.
  • Hawthorne, J., Landes, J., Wallman, C., & Williamson, J. (2015). The Principal Principle Implies the Principle of Indifference. The British Journal for the Philosophy of Science
  • Ismael, J. (2008). Raid! Dissolving the Big, Bad Bug. Noûs, 42(2), 292–307.
  • Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge, UK: Cambridge University Press.
  • Keynes, J. M. (1921). A Treatise on Probability. London: Macmillan.
  • Lewis, D. (1980). A Subjectivist’s Guide to Objective Chance. In R. C. Jeffrey (Ed.) Studies in Inductive Logic and Probability, vol. II. Berkeley: University of California Press.
  • Lewis, D. (1994). Humean Supervenience Debugged. Mind, 103, 473–490.
  • Pettigrew, R. (2012). Accuracy, Chance, and the Principal Principle. Philosophical
    121(2), 241–275.
  • Pettigrew, R. (2014). Accuracy, Risk, and the Principle of Indifference. Philosophy
    and Phenomenological Research
  • Thau, M. (1994). Undermining and Admissibility. Mind, 103, 491–504.
  • van Fraassen, B. C. (1984). Belief and the Will. Journal of Philosophy, 81, 235–56.
  • Williamson, J. (2010). In Defence of Objective Bayesianism. Oxford: Oxford University Press.

Tuesday, 20 December 2016

Assistant professorship in mathematical philosophy, University of Gdansk

Assistant Professorship
 (“adiunkt” in Polish terminology) in the Chair of Logic, Philosophy of Science and Epistemology is available at the Department of Philosophy, Sociology and Journalism, University of Gdansk, Poland. The position is to start sometime between  July 1 and September 1, 2017, for a fixed period of time with the possibility of extension. Decisions about the exact beginning date of the contract and the number of years will be made during the hiring process. No knowledge of Polish is required.

Details available here.

Sunday, 18 December 2016

Call for submissions: PhDs in Logic IX, Bochum, 2nd - 4th May 2017

PhDs in Logic is an annual graduate conference organised by local graduate students. This interdisciplinary conference welcomes contributions to various topics in mathematical logic, philosophical logic, and logic in computer science. It involves tutorials by established researchers as well as short (20 minutes) presentations by PhD students, master students and first-year postdocs on their research.
We are happy to announce that the ninth edition of PhDs in Logic will take place at the Ruhr University Bochum, Germany, during 2nd - 4th May 2017.

Confirmed tutorial speakers are :
Petr Cintula (Czech Academy of Sciences)
María Manzano (University of Salamanca)
João Marcos (University of Natal)
Gabriella Pigozzi (Paris Dauphine University)
Christian Straßer (Ruhr-University Bochum)
Heinrich Wansing (Ruhr-University Bochum)

Abstract submission:
PhD students, master students and first-year postdocs in logic from disciplines, that include but are not limited to philosophy, mathematics and computer science are invited to submit an extended abstract on their research. Submitted abstracts should be between 2 and 3 pages, including the relevant references. Each abstract will be anonymously reviewed by the scientific committee. Accepted abstracts will be presented by their authors in a 20-minute presentation during the conference. The deadline for abstract submission is 2nd February 2017. Please submit your blinded abstract via: https://easychair.org/conferences/?conf=phdsinlogic9

For more information please see:

Local organisers:

Christopher Badura, AnneMarie Borg, Jesse Heyninck and Daniel Skurt

Thursday, 27 October 2016

Assistant Professorship at the MCMP

Ludwig-Maximilians-University Munich is seeking applications for one

Assistant Professorship position in Logic and Philosophy of Language
(for three years, with the possibility of extension)

at the Chair of Logic and Philosophy of Language (Professor Hannes Leitgeb) and the Munich Center for Mathematical Philosophy (MCMP) at the Faculty of Philosophy, Philosophy of Science, and Study of Religion. The position, which is to start on April 1st 2017, is for three years with the possibility of extension.

The appointee will be expected (i) to do philosophical research, especially in logic and philosophy of language, (ii) to teach five hours a week in areas relevant to the chair, and (iii) to participate in the administrative work of the MCMP.

The successful candidate will have a PhD in philosophy or logic, will have teaching experience in philosophy and logic, and will have carried out research in logic and related areas (such as philosophy of logic, philosophy of language, philosophy of mathematics, formal epistemology).

Women are currently underrepresented in the Faculty, therefore we particularly welcome applications for this post from suitably qualified female candidates. Furthermore, given equal qualification, severely physically challenged individuals will be preferred.

Applications (including CV, certificates, list of publications), a description of planned research projects (1000-1500 words), and letters of reference of two referees should be sent either by email (ideally all requested documents in just one PDF document) or by mail to

Ludwig-Maximilians-Universität München
Faculty of Philosophy, Philosophy of Science and Study of Religion
Chair of Logic and Philosophy of Language / MCMP
Geschwister-Scholl-Platz 1
80539 München
E-Mail: office.leitgeb@lrz.uni-muenchen.de


December 1st, 2016.

If possible at all, we very much prefer applications by email.

Contact for informal inquiries: office.leitgeb@lrz.uni-muenchen.de

More information about the MCMP can be found at http://www.mcmp.philosophie.uni-muenchen.de/index.html.

The German description of the position is to be found at http://www.uni-muenchen.de/aktuelles/stellenangebote/wissenschaft/20161017140416.html.


Wednesday, 12 October 2016

Entia et Nomina 2017 CFP

The “Entia et Nomina” series features English language workshops for researchers in formally oriented philosophy, in particular in logic, philosophy of science, formal epistemology and philosophy of language. The aim of the workshop is to foster cooperation among philosophers with a formal bent. Previous editions took place at Gdansk University, Ghent University (as part of the Trends in Logic series), Jagiellonian University, and Warsaw University. The sixth conference in the series will take place in Palolem, Goa, India, on 29 January - 5 February 2017. Invited speakers confirmed so far include:

Krzysztof Posłajko (Jagiellonian University)
Katarzyna Kijania-Placek (Jagiellonian University)
Tomasz Placek (Jagiellonian University)
Nina Gierasimczuk (Danish Technical University)
Cezary Cieślinski (Warsaw University)
Marcello Dibello (City University of New York)

Authors of contributed papers are requested to submit short (up to 2 normalized pages) and extended (up to 6 pages) abstracts, prepared for blind-review, in PDF format, by 30.10.2016. Decisions about acceptance will be communicated by 20.11.2016.

Authors of accepted papers will have 40 minutes to present their work. Each paper will be followed by a 10 minute commentary prepared beforehand by another participant. Accepted participants might also be asked to comment on at least one talk. Commentaries will be followed by 10-15 minutes of discussion. Applications can be made also for the role of commentator only, in which case only a short CV is requested. We aim to make the short versions of accepted papers available to the participants ahead of the conference.

Please send your abstracts, questions and any inquiries to both Rafal Urbaniak <rfl.urbaniak@gmail.com> and Juliusz Doboszewski <jdoboszewski@gmail.com>.

Tuesday, 4 October 2016

CFA: The Fifth Reasoning Club Conference

Call for Abstracts: The Fifth Reasoning Club Conference
University of 
Torino, 18-19 May 2017

Keynote speakers: 
Branden FITELSON (Northeastern University, Boston)
Jeanne PEIJNENBURG (University of Groningen)
Katya TENTORI (University of Trento)
Paul EGRÉ (Institut Jean Nicod, Paris)
Please visit http://www.llc.unito.it/notizie/fifth-reasoning-club-meeting-llc-2017 for further information.

Submissions for the Fifth Reasoning Club Conference are now open. All PhD candidates and early career researchers with interests in reasoning and inference, broadly construed, are encouraged to submit an abstract of up to 500 words (prepared for blind review) via Easy Chair at https://easychair.org/conferences/?conf=rcc17
We especially welcome members of groups that are underrepresented in philosophy to submit. We are committed to promoting diversity in our final programme.

The deadline for submissions is 1 February 2017. The final decision on submissions will be made by 15 March 2017. 

Grants will be available to help cover travel costs for contributed speakers. To apply for a travel grant, please send a CV and a short travel budget estimate in a single pdf file by 1 February 2017 to reasoningclubconference2017@gmail.com.

For any queries please contact Vincenzo Crupi or Jason Konek.

The Reasoning Club is a network of institutes, centres, departments, and groups addressing research topics connected to reasoning, inference, and methodology broadly construed. It issues the monthly gazette The Reasoner.

Earlier editions of the meeting were held in Brussels, Pisa, Kent, and Manchester

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 tomathematicalstructuralism2016@lrz.uni-muenchen.de. 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: http://www.mathematicalstructuralism2016.philosophie.uni-muenchen.de/index.html

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 argumentstrength2016@gmail.com 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 http://www.irsi2016.de. For inquiries, please send an e-mail to info@irsi2016.de.

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.