Friday, 28 March 2014

Counting Infinities

(Cross-posted at NewAPPS)

In his Two New Sciences (1638), Galileo presents a puzzle about infinite collections of numbers that became known as ‘Galileo’s paradox’. Written in the form of a dialogue, the interlocutors in the text observe that there are many more positive integers than there are perfect squares, but that every positive integer is the root of a given square. And so, there is a one-to-one correspondence between the positive integers and the perfect squares, and thus we may conclude that there are as many positive integers as there are perfect squares. And yet, the initial assumption was that there are more positive integers than perfect squares, as every perfect square is a positive integer but not vice-versa; in other words, the collection of the perfect squares is strictly contained in the collection of the positive integers. How can they be of the same size then?

Galileo’s conclusion is that principles and concepts pertaining to the size of finite collections cannot be simply transposed, mutatis mutandis, to cases of infinity: “the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.” With respect to finite collections, two uncontroversial principles hold:

Part-whole: a collection A that is strictly contained in a collection B has a strictly smaller size than B.

One-to-one: two collections for which there exists a one-to-one correspondence between their elements are of the same size.

What Galileo’s paradox shows is that, when moving to infinite cases, these two principles clash with each other, and thus that at least one of them has to go. In other words, we simply cannot transpose these two basic intuitions pertaining to counting finite collections to the case of infinite collections. As is well known, Cantor chose to keep One-to-one at the expenses of Part-whole, famously concluding that all countable infinite collections are of the same size (in his terms, have the same cardinality); this is still the reigning orthodoxy.

In recent years, an alternative approach to measuring infinite sets is being developed by the mathematicians Vieri Benci (who initiated the project) Mauro Di Nasso, and Marco Forti. It is also being further explored by a number of people – including logicians/philosophers such as Paolo Mancosu, Leon Horsten and my colleague Sylvia Wenmackers. This framework is known as the theory of numerosities, and has a number of theoretical as well as more practical interesting features. The basic idea is to prioritize Part-whole over One-to-one; this is accomplished in the following way (Mancosu 2009, p. 631):

Informally the approach consists in finding a measure of size for countable sets (including thus all subsets of the natural numbers) that satisfies [Part-whole]. The new ‘numbers’ will be called ‘numerosities’ and will satisfy some intuitive principles such as the following: the numerosity of the union of two disjoint sets is equal to the sum of the numerosities.
Basically, what the theory of numerosities does is to introduce different units, so that on these new units infinite sets comes out as finite. (In other words, it is a clever way to turn infinite sets into finite sets. Sounds suspicious? Hum…) In practice, the result is a very robust, sophisticated mathematical theory, which turns the idea of measuring infinite sets upside down.

The philosophical implications of the theory of numerosities for the philosophy of mathematics are far-reaching, and some of them have been discussed in detail in (Mancosu 2009). Philosophically, the mere fact that there is a coherent, theoretically robust alternative to Cantorian orthodoxy raises all kinds of questions pertaining to our ability to ascertain what numbers ‘really’ are (that is, if there are such things indeed). It is not surprising that Gödel, an avowed Platonist, considered the Cantorian notion of infinite number to be inevitable: there can be only one correct account of what infinite numbers really are. As Mancosu points out, now that there is a rigorously formulated mathematical theory that forsakes One-to-one in favor of Part-whole, it is far from obvious that the Cantorian road is the inevitable one.

As mathematical theories, Cantor’s theory of infinite numbers and the theory of numerosities may co-exist in peace, just as Euclidean and non-Euclidean geometries live peacefully together (admittedly, after a rough start in the 19th century). But philosophically, we may well see them as competitors, only one of which can be the ‘right’ theory about infinite numbers. But what could possibly count as evidence to adjudicate the dispute?

One motivation to abandon Cantorian orthodoxy might be that it fails to provide a satisfactory framework to discuss certain issues. For example, Wenmackers and Horsten (2013) adopt the alternative approach to treat certain foundational issues that arise with respect to probability distributions in infinite domains. It is quite possible that other questions and areas where the concept of infinity figures prominently can receive a more suitable treatment with the theory of numerosities, in the sense that oddities that arise by adopting Cantorian orthodoxy can be dissipated.

On a purely conceptual, foundational level, the dispute might be viewed as one between Part-whole and One-to-one, as to which of the two is the most fundamental principle when it comes to counting finite collections – which would then be generalized to the infinite cases. They are both eminently plausible, and this is why Cantor’s solution, while now widely accepted, remains somewhat counterintuitive (as anyone having taught this material to students surely knows). Thus, it is hard to see what could possibly count as evidence against one or the other


Now, after having thought a bit about this material (prompted by two wonderful talks by Wenmackers and Mancosu in Groningen yesterday), and somewhat to my surprise, I find myself having a lot of sympathy for Galileo’s original response. Maybe what holds for counting finite collections simply does not hold for measuring infinite collections. And if this is the case, our intuitions concerning the finite cases, and in particular the plausibility of both Part-whole and One-to-one, simply have no bearing on what a theory of counting infinite collections should be like. There may well be other reasons to prefer the numerosities approach over Cantor’s approach (or vice-versa), but I submit that turning to the idea of counting finite collections is not going to provide relevant material for the dispute in the infinite cases. In fact, from this point of view, an entirely different way of measuring infinite collections, where neither Part-whole nor One-to-one holds, is at least in principle conceivable. In what way the term ‘counting’ would then still apply might be a matter of contention, but perhaps counting infinities is a totally different ball game after all.

Thursday, 27 March 2014

CFP: *Extended Deadline* Symposium on the Foundations of Mathematics, Kurt Gödel Research Center, University of Vienna, 7-8 July 2014.

Date and Venue: 7-8 July 2014 - Kurt Gödel Research Center, Vienna

Confirmed Speakers:
  • Sy-David Friedman (Kurt Gödel Research Center for Mathematical Logic)
  • Hannes Leitgeb (Munich Center for Mathematical Philosophy)
Call for Papers: We welcome submissions from scholars (in particular, young scholars, i.e. early career researchers or post-graduate students) on any area of the foundations of mathematics (broadly construed). Particularly desired are submissions that address the role of set theory in the foundations of mathematics, or the foundations of set theory (universe/multiverse dichotomy, new axioms, etc.) and related ontological and epistemological issues. Applicants should prepare an extended abstract (maximum 1,500 words) for blind review, and send it to sotfom [at] gmail [dot] com. The successful applicants will be invited to give a talk at the conference and will be refunded the cost of accommodation in Vienna for two days (7-8 July).

*New* Submission Deadline: 15 April 2014
Notification of Acceptance: 30 April 2014

Set theory is taken to serve as a foundation for mathematics. But it is well-known that there are set-theoretic statements that cannot be settled by the standard axioms of set theory. The Zermelo-Fraenkel axioms, with the Axiom of Choice (ZFC), are incomplete. The primary goal of this symposium is to explore the different approaches that one can take to the phenomenon of incompleteness. One option is to maintain the traditional “universe” view and hold that there is a single, objective, determinate domain of sets. Accordingly, there is a single correct conception of set, and mathematical statements have a determinate meaning and truth-value according to this conception. We should therefore seek new axioms of set theory to extend the ZFC axioms and minimize incompleteness. It is then crucial to determine what justifies some new axioms over others. Alternatively, one can argue that there are multiple conceptions of set, depending on how one settles particular undecided statements. These different conceptions give rise to parallel set-theoretic universes, collectively known as the “multiverse”. What mathematical statements are true can then shift from one universe to the next. From within the multiverse view, however, one could argue that some universes are more preferable than others. These different approaches to incompleteness have wider consequences for the concepts of meaning and truth in mathematics and beyond. The conference will address these foundational issues at the intersection of philosophy and mathematics. The primary goal of the conference is to showcase contemporary philosophical research on different approaches to the incompleteness phenomenon. To accomplish this, the conference has the following general aims and objectives: (1) To bring to a wider philosophical audience the different approaches that one can take to the set-theoretic foundations of mathematics. (2) To elucidate the pressing issues of meaning and truth that turn on these different approaches. (3) To address philosophical questions concerning the need for a foundation of mathematics, and whether or not set theory can provide the necessary foundation.

Scientific Committee: Philip Welch (University of Bristol), Sy-David Friedman (Kurt Gödel Research Center), Ian Rumfitt (University of Birmigham), John Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Gödel Research Center), Neil Barton (Birkbeck College), Chris Scambler (Birkbeck College), Jonathan Payne (Institute of Philosophy), Andrea Sereni (Università Vita-Salute S. Raffaele), Giorgio Venturi (Université de Paris VII, “Denis Diderot” - Scuola Normale Superiore)

Organisers: Sy-David Friedman (Kurt Gödel Research Center), John Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Gödel Research Center), Neil Barton (Birkbeck College), Carolin Antos (Kurt Gödel Research Center)

Conference Website: sotfom [dot] wordpress [dot] com

Further Inquiries: please contact Claudio Ternullo (ternulc7 [at] univie [dot] ac [dot] at) Neil Barton (bartonna [at] gmail [dot] com) John Wigglesworth (jmwigglesworth [at] gmail [dot] com)

Tuesday, 25 March 2014

Buchak on risk and rationality II: the virtues of expected utility theory

In a previous post, I gave an overview of the alternative to expected utility theory that Lara Buchak formulates and defends in her excellent new book, Risk and Rationality (Buchak 2013).  Buchak dubs the alternative risk-weighted expected utility theory.  It permits agents to have risk-sensitive attitudes.  In this post and the next one, I wish to argue that risk-weighted expected utility theory is right about the constraints that rationality places on our external attitudes, but wrong about the way our internal attitudes ought to combine to determine those external attitudes (for the internal/external attitude terminology, as well as other terminology in this post, please see the previous post):  that is, I agree with the axioms Buchak demands our preferences must satisfy, but I disagree with the way she combines probabilities, utilities, and risk attitudes to determine those preferences.  I wish to argue that, in fact, we ought to combine our internal attitudes in exactly the way that expected utility theory suggests.  In order to maintain both of these positions, I will have to redescribe the outcomes to which we assign utilities.  I do this in the next post.  In this post, I want to argue that all the effort that we will go to in order to effect this redescription is worth it.  That is, I want to argue that there are good reasons for thinking that an agent's internal attitudes ought to be combined to give her external attitudes in exactly the way prescribed by expected utility theory. (These three posts will together provide the basis for my commentary on Buchak's book at the Pacific APA this April.)

Friday, 21 March 2014

Logical foundations for mathematics? The first-order vs. second-order ‘dichotomy’? (Part IV of 'Axiomatizations of arithmetic...')

(It took me much longer than I had anticipated to get back to this paper, but here is the final part of my paper on axiomatizations of arithmetic and the first-order/second-order divide. Part I is here; Part II is here; Part III is here. As always, comments are welcome!)

3. Logical foundations for mathematics? The first-order vs. second-order ‘dichotomy’?

Given the (apparent) impossibility of tackling the descriptive and deductive projects at once with one and the same underlying logical system – what Tennant (2000) describes as ‘the impossibility of monomathematics’ – what should we conclude about the general project of using logic to investigate the foundations of mathematics? And what should we conclude about the first-order vs. second-order divide? I will discuss each of these two questions in turn.

If the picture sketched in the previous sections is one of partial failure, it can equally well be seen as a picture of partial success. Indeed, a number of first-order mathematical theories can be made to be categorical with suitable second-order extensions (Read 1997). And thus, as argued by Read, there is a sense in which the completeness project of the early days of formal axiomatics has been achieved (despite Gödel’s results), namely in the descriptive sense countenanced by Dedekind and others.  

Moreover, categoricity failure must not be viewed as a complete disaster, if one bears in mind Shapiro’s (1997) useful distinction between algebraic and nonalgebraic theories:
Roughly, non-algebraic theories are theories which appear at first sight to be about a unique model: the intended model of the theory. We have seen examples of such theories: arithmetic, mathematical analysis… Algebraic theories, in contrast, do not carry a prima facie claim to be about a unique model. Examples are group theory, topology, graph theory… (Horsten 2012, section 4.2)
In this vein, proofs of (non-)categoricity can be viewed as a means of classifying algebraic and non-algebraic theories (Meadows 2013). This means that the descriptive (non-algebraic) project of picking out a previously chosen mathematical structure and describing it in logical terms has developed into the more general descriptive project of studying theories and groups of theories not only insofar as they instantiate unique structures (i.e. non-algebraic as well as algebraic versions of the descriptive project).

On the deductive side, things may seem less rosy at first sight. In a sense, first-order logic is not only descriptively inadequate: it is also deductively inadequate, given the impossibility of a deductively complete first-order theory of the natural numbers, and the fact that first-order logic itself is undecidable (though complete). It does have a better behaved underlying notion of logical consequence when compared to second-order logic, but it still falls short of delivering the deductive power that e.g. Frege or Hilbert would have hoped for. In short, first-order logic might be described as being ‘neither here nor there’.

However, if one looks beyond the confines of first-order or second-order logic, developments in automated theorem proving suggest that the deductive use as described by Hintikka is still alive and kicking. Sure enough, there is always the question of whether a given mathematical theorem, formulated in ‘ordinary’ mathematical language, is properly ‘translated’ into the language used by the theorem-proving program. But automated theorem proving is in many senses a compelling instantiation of Frege’s idea of putting chains of reasoning to test.

Recently, the new research program of homotopy type-theory promises to bring in a whole new perspective to the foundations of mathematics. In particular, its base logic, Martin-Löf’s constructive type-theory, is known to enjoy very favorable computational properties, and the focus on homotopy theory brings in a clear descriptive component. It is too early to tell whether homotopy type-theory will indeed change the terms of the game (as its proponents claim), but it does seem to offer new prospects for the possibility of unifying the descriptive perspective and the deductive perspective.

In sum, what we observe currently is not a complete demise of the original descriptive and deductive projects of pioneers such Frege and Dedekind, but rather a transformation of these projects into more encompassing, more general projects.

As for the first-order vs. second-order divide, it may be instructive to look in more detail into the idea of second-order extensions of first-order theories, specifically with respect to arithmetic. Some of these proposals can be described as ‘optimization projects’ that seek to incorporate the least amount of second-order vocabulary so as to ensure categoricity, while producing a deductively well-behaved theory. In other words, the goal of an optimal tradeoff between expressiveness and tractability may not be entirely unreasonable after all.

One such example is the framework of ‘ancestral logic’ (Avron 2003, Cohen 2010). Smith (2008) argues on plausible conceptual grounds that our basic intuitive grasp of arithmetic surely does not require the whole second-order conceptual apparatus, but only the concept of the ancestral of a relation, or the idea of transitive closure under iterable operations (my parents had parents, who in turn had parents, who themselves had parents, and so on). Another way to arrive at a similar conclusion is to appreciate that what is needed to establish categoricity by extending a first-order theory is nothing more than the expressive power required to formulate the induction schema, or equivalently the last, second-order axiom in the Dedekind/Peano axiomatization (the one needed to exclude ‘alien intruders’). Here again, the concept of the ancestral of a relation is a plausible candidate (Smith 2008, section 3; Cohen 2010, section 5.3).

Extensions of first-order logic with the concept of the ancestral yield a number of interesting systems (Smith 2008, section 4; Cohen 2010, chapter 5). These systems, while not being fully axiomatizable (Smith 2008, section 4), enjoy a number of favorable proof-theoretical properties (Cohen 2010, chapter 5). Indeed, they are vastly ‘better behaved’ from a deductive point of view than full-blown second-order logic – and of course, they are categorical.

Significant for our purposes is the status of the notion of the ancestral, straddled between first-order and second-order logic. Smith argues that the fact that this notion can be defined in second-order terms does not necessarily mean that it is an essentially higher-order notion:

In sum, the claim is that the child who moves from a grasp of a relation to a grasp of the ancestral of that relation need not thereby manifest an understanding of second-order quantification interpreted as quantification over arbitrary sets. It seems, rather, that she has attained a distinct conceptual level here, something whose grasp requires going beyond a grasp of the fundamental logical constructions regimented in first-order logic, but which doesn’t takes as far as an understanding of full second-order quantification. (Smith 2008)

What this suggests is that the first-order vs. second-order divide itself may be too coarse to describe adequately the conceptual building blocks of arithmetic. It is clear that purely first-order vocabulary will not yield categoricity, but it would be misguided to view the move to full-blown second-order logic as the next ‘natural’ step. In effect, as argued by Smith, the concept of the ancestral of a relation is essentially neither first-order nor second-order, properly speaking. So maybe the problem lies precisely in the coarse first-order vs. second-order dichotomy when it comes to the key concepts at the foundations of arithmetic (such as the concept of the ancestral, or Dedekind’s notion of chains). We may need different, intermediate categories to classify and analyze these concepts more accurately.


4. Conclusions

My starting point was the observation that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. I then turned to Hintikka’s distinction between descriptive and deductive approaches for the foundations of mathematics. Both approaches were represented in the early days of formal axiomatics at the end of the 19th century, but the descriptive approach was undoubtedly the predominant one; Frege was then the sole representative of the deductive approach.

Given the (apparent?) impossibility of combining both approaches in virtue of the orthogonal desiderata of expressiveness and tractability, one might conclude (as Tennant (2000) seems to argue) that the project of providing logical foundations for mathematics itself is misguided from the start. But I have argued that a story of partial failure is also a story of partial success, and that both projects (descriptive and deductive) remain fruitful and vibrant. I have also argued that an investigation of the conceptual foundations of arithmetic seems to suggest that the first-order vs. second-order dichotomy is in fact too coarse, as some key concepts (such as the concept of the ancestral of a relation) seem to inhabit a ‘limbo’ between the two realms.

One of the main conclusions I wish to draw from these observations is that there is no such thing as a unique project for the foundations of mathematics. Here we focused on two distinct projects, descriptive and deductive, but there may well be others. While it may seem that these two perspectives are incompatible, there is both the possibility of ‘optimization projects’, i.e. the search for the best trade-off between expressive and deductive power (e.g. ancestral arithmetic), and the possibility that an entirely new approach (maybe homotopy type-theory?) may even dissolve the apparent impossibility of fully engaging in both projects at once. It is perhaps due to an excessive focus on the first-order vs. second-order divide that we came to think that the two projects are incompatible.

At any rate, the choice of formalism/logical framework will depend on the exact goals of the formalization/axiomatization. Here, the focus has been on the expressiveness-tractability axis, but there may well be other relevant parameters. Now, if we acknowledge that there may be more than one legitimate theoretical goal when approaching mathematics with logical tools (and here we discussed two, prima facie equally legitimate approaches: descriptive and deductive), then there is no reason why there should be a unique, most appropriate logical framework for the foundations of mathematics. The picture that emerges is of a multifaceted, pluralistic enterprise, not of a uniquely defined project, and thus one allowing for multiple, equally legitimate perspectives and underlying theoretical frameworks. A plurality of goals suggests a form of logical pluralism, and thus, perhaps there is no real ‘dispute’ between first-order and second-order logic in this domain.


Thursday, 13 March 2014

Summer school: Proof, Truth, Computation (deadline approaching!)

Summer School "Proof, Truth, Computation" (PTC 2014)

20-25 July 2014, Chiemsee, Germany

Call for applications by young researchers. Deadline: 17 March 2014.

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This is to invite young researchers (PhD students and post-docs) to apply for the upcoming summer school

"Proof, Truth, Computation. Modern Foundations of Mathematics and Contemporary Philosophy".

Application by female scientists is particularly encouraged.

The event will take place from 21st to 25th July 2014 (arrival 20th July afternoon, departure 25th July after noon) in the Benedictine nunnery Frauenwoerth on the Fraueninsel in Chiemsee between Munich and Salzburg:


The Volkswagen Foundation will kindly sponsor this event:


To get an idea of this summer school, especially of its interdisciplinary character, please see the material provided at end of this message. Junior participants will be particularly expected to contribute to the questions and answers sessions and to the round table discussions.

Important dates:

Deadline for application: 17th March 2014
Notification of acceptance: 24st March 2014
Communication of precise air fare (if applicable): 31st March 2014

Applications are to be sent, in a single PDF document, by email, to


PhD students need to send a CV of at most 2 pages, a brief letter of motivation and one letter of reference. Postdocs only need to send a CV of at most 2 pages. All applicants need to tell whether they also apply for funding and, if so, to which extent. Only a limited amount of funding is available. Applicants for funding are expected to stay for the whole week, and to tell the extent to which they can be funded by other sources.

If your application for funding is successful, then you will be offered reimbursement of the travel and lodging expenses that you cannot cover from other sources. This will require that you choose the cheapest travel option, and that you book your trip by 31st March 2014 in case of flights and by the earliest possible date in case of long-distance trains. We hope to be able to contribute partially to your subsistence expenses (meals).

Meals (breakfast, lunch and dinner - the latter two excluding drinks) for four and a half days will be EUR 180. PhD students and postdocs are expected to share double rooms, for EUR 125 each person for the whole week (5 nights).

Organising committee:

Hannes Leitgeb <hannes.leitgeb@lmu.de>
Iosif Petrakis <petrakis@math.lmu.de>
Peter Schuster <pschust@maths.leeds.ac.uk>
Helmut Schwichtenberg <schwicht@math.lmu.de>

Enquiries are to be directed to:


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Main Topics

Truth Theories
Predicativity
Constructivity
Proof Theory
Formal Epistemology
Set-theoretic Truth

Further topics will include:

Homotopy, Types and Univalence
Program Extraction from Proofs
Coalgebraic and Categorical Semantics
Minimal Type Theory

Speakers and disputants (preliminary list)

Tatiana Arrigoni
Steve Awodey
Marco Benini
Ulrich Berger
Andrea Cantini
Thierry Coquand (to be confirmed)
Laura Crosilla
Branden Fitelson
Sy Friedman (to be confirmed)
Volker Halbach
Hajime Ishihara
Gerhard Jaeger
Peter Koellner
Hannes Leitgeb
Maria Emilia Maietti
David Makinson
Yiannis Moschovakis
Sara Negri
Erik Palmgren
Dirk Pattinson
Jan von Plato
Dieter Probst
Joan Rand-Moschovakis
Michael Rathjen
Giuseppe Rosolini (to be confirmed)
Giovanni Sambin
Monika Seisenberger
Philip Welch
Andreas Weiermann

Aims and Scope

Mathematical methods are about to shape some branches of contemporary philosophy just as they have formed most of the natural and many of the social sciences. The thread of the school we propose is to mirror this development, known as mathematical philosophy or formal epistemology; to highlight the challenges that arise from it; and to display its repercussions in mathematics. As for theoretical computer science, a quite comparable spin-off of mathematics, the principal counterpart within mathematics is mathematical logic.

Since many of the objects of study lie beyond the typical commitment of contemporary mathematics, it is decisive to include non-classical issues such as predicativity and constructivity. Proof theory does indeed play a pivotal role: as the area of mathematical logic that is closest to the understanding of logic as the science of formal languages and reasoning, it is predestined for interaction both with philosophical and computer science logic.

A hot topic that crosses over wide ranges of the school, and is most prominently represented within, is whether axiomatic theories of truth and of related notions, such as provability and knowledge, are possible at all in the stress field between syntax and semantics. Rational belief and rational choice, epistemic issues of principal philosophical relevance, are put under mathematical scrutiny by applying probabilism: that is, the thesis that a rational agent's degrees of belief should conform to the axioms of probability theory.