MCMP goes Coursera!

LMU Munich explores new forms of collaborative learning – by making use of Massive Open Online Courses (MOOCs). The MCMP takes part in creating the first LMU online courses offered on Coursera, a leading platform for online education. Coursera's classes may be accessed by anyone anywhere, at no charge and without having to meet any special entrance requirements. The MOOCs will consist of video lectures, machine-graded quizzes, collaborative online learning forums, reading lists and seminar assignments.

The first MCMP online course, "Introduction to Mathematical Philosophy" by Hannes Leitgeb and Stephan Hartmann, will start in July 2013 and last 8 weeks. Learn more about the course and how to enroll on Coursera on the MCMP website!

Post-Docs in Philosophy of Canonical Quantum Gravity

Call for postdoctoral positions for working on the ERC (European Research Project) project

Philosophy of Canonical Quantum Gravity
Principal Investigator: Gabriel Catren

The European Research Council (ERC) project Philosophy of Canonical Quantum Gravity announces one-year post-doctoral positions, with the possibility of an extension depending on the final evaluation. The profile of the searched candidate is that of a candidate with a strong background in pure mathematics and mathematical physics interested in the conceptual and mathematical foundations of theoretical physics.

Profile of the position: The chosen candidate is expected to work on one of the following areas:

Philosophy of spacetime and gravitational theories. 

Philosophy of quantum mechanics in the light of symplectic geometry, geometric quantization and the theory of constrained Hamiltonian systems. 

Philosophy of quantum gravity.

Scientific environement: Laboratoire SPHERE (UMR 7219), Centre National de la Recherche Scientifique (CNRS) – Université Paris Diderot, Paris, France. Eligibility requirements: Eligible candidates must hold a PhD or equivalent in philosophy, physics or mathematics.

Application Procedure: Applicants are invited to submit a dossier by email (in English or French) including:

• a cover letter explaining the candidate’s research interests and her/his skills for contributing to an interdisciplinary project at the intersection between philosophy, physics, and mathematics. 

• a curriculum vitae including a publication list; 

• university certification of the last candidates’ degree. 

• a copy of PhD thesis and a sample of other relevant writings, published or not; 

• the name and electronic address of one knowledgeable scholar who could provide a recommendation letter if necessary.

Scholars of all nationalities are welcome to apply.

For further information, please contact Gabriel Catren: gabrielcatren@gmail.com. 

The dossier must be addressed to the following electronic address: gabrielcatren@gmail.com.

Application deadline: Open until filled.

The logic and psychology of confirmation and information search

 

Workshop

The logic and psychology of confirmation

                                                and information search

Munich, 26 February 2013

The workshop is the mid-term event within a three-year research project funded by the DFG on Models of information search: A theoretical and empirical synthesis. The project is part of the wider DFG Priority Program New frameworks of rationality (SSP 1516), launched in 2010 to promote the collaboration between psychologists, philosophers and other related research communities towards an integrated approach to the study of human rationality. The meeting is free to attend and will take place in the Statistics Seminar Room, Ludwigstrasse 31/33, first floor.

    SCHEDULE

              9:30         Welcome and coffee

             10:00         Björn Meder and Jonathan Nelson (MPI Berlin)

                               Is people’s information search behavior sensitive

                               to different reward structures?

             10:30         Vincenzo Crupi (University of Turin, MCMP)

                               Pathways between Bayesian confirmation theories

    and models of information search

             11:00         Coffee break

             11:30         Laura Martignon (Ludwigsburg)

                               On mermaids and princesses

12:00         Katya Tentori (Trento)

                  Probability vs. confirmation judgments: A comparison

   in accuracy and test-retest reliability                              

12:30         Gustavo Cevolani (Bologna)

                               Similarity, verisimilitude, and information


 

What does it mean to be irrational?

This has been circulating today on Facebook, rather excellent!

Review of Mancosu's 'The Adventure of Reason'

I've been asked to review Paolo Mancosu's book The Adventure of Reason -- Interplay between Philosophy of Mathematics and Mathematical Logic, 1900-1940 (OUP, 2010) for Mind. What follows is a draft of my review, so comments and suggestions are most welcome!
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The first half of the 20^th century witnessed the birth and flourishing of a radically novel subdiscipline: mathematical logic. While mathematics and logic had been on friendly terms at least since the 17^th century (Mugnai 2011), and while there were 19^th century precursors to the idea of applying mathematics to logic (e.g. Boole) and logic to mathematics (e.g. Frege), it is only in the first half of the 20^th century that mathematical logic became a fully mature subdiscipline/research program.

The Adventure of Reason offers a compelling narrative of this exciting chapter of the history of logic and mathematics. Paolo Mancosu is one of the world’s leading experts on these developments, and while many others have made important contributions to the topic, it is fair to say that Mancosu has gone a step beyond with his painstaking work on sources other than the canonical, printed versions of articles and books. He has done extensive research at a number of archives both in Europe and North America, examining documents such as letters, minutes of meetings, informal reports, unpublished transcriptions of lectures – in short, unpublished material of various sorts. This approach has enabled him to produce new insights and at times to provide novel answers to interpretive open questions concerning some of the towering figures in this tradition (such as the controversy on whether Tarski did or did not accept domain variation in his definition of logical consequence – chapter 16).

The book essentially consists of re-prints of several of Mancosu’s articles (some of them co-authored pieces) on a wide range of topics and authors, but all within the framework of the emergence of mathematical logic in the 20^th century and its tight connections with philosophical discussions on the nature of mathematics (as the subtitle of the book indicates). It is divided into five parts: Part I is composed of a long chapter surveying the developments from 1900 to 1935; Part II focuses on the foundations of mathematics program, in particular Russell, Hilbert, Bernays and Gödel; Part III is less ‘mainstream’ and is dedicated to the contacts between phenomenology and the philosophy of the exact sciences, pioneered by Husserl and pursued by less well-known figures such as Weyl, Becker and Mahnke; Part IV is rather short and concerns Tarski and Quine on nominalism; Part V focuses on Tarski and the Vienna Circle on truth and logical consequence, and includes a hitherto unpublished lecture by Tarski on completeness and categoricity (more on which below).

Among the many innovations introduced by mathematical logicians in the first half of the 20^th century, the emergence of formal systems stands out: these are collections of axioms and rules of transformation, typically formulated in a specially designed language (a formal language), which would allow for the precise investigation of whole portions of mathematics by means of axiomatizations/formalizations. Starting with the system presented in Whitehead and Russell’s 1910 Principia Mathematica (which in turn was heavily inspired by Frege’s Begriffsschrift and by the notation introduced by Peano), there was a crescendo of enthusiasm and optimism concerning what could be accomplished with this novel tool, which then culminated in Hilbert’s program in the 1920s (the topic of Part II). The basic idea was that meta-mathematical questions such as the consistency of arithmetic would become mathematical questions once adequately formulated within a convenient formal system, and could thus be investigated and settled by purely mathematical means.

At first, it seemed that these powerful tools, formal systems, offered almost limitless possibilities for the investigation of such foundational issues. (Hilbert: “We must know. We will know.”) But it also became increasingly clear that the desired meta-properties of these systems, in particular in view of the goal of describing portions of mathematics completely (in different senses of ‘completely’ – see chapter 1 of the book and Awodey and Reck 2002), could not be taken for granted. A seemingly well-designed, plausible collection of axioms, say the axioms of Peano arithmetic, could still fail to deliver a complete description of the mathematical theory in question. So the properties of these very systems also had to be investigated in a systematic way – the meta-meta-level of investigation, so to speak. Now, once thoroughly formalized, formal systems become mathematical objects themselves, thus allowing for the application of the same general methodology. So one remarkable feature of mathematical logic in this period is that it set out to establish its own limits by means of the very methodology it had inaugurated.

As is well known, the most astonishing limitative results were proved by Gödel, who in the early 1930s showed (the First Incompleteness Theorem) that any consistent system containing arithmetic (in a suitable sense of ‘containing’) is inherently incomplete in that there is a sentence which can neither be proved nor refuted by the system, though true according to the standard model of arithmetic. The Second Incompleteness Theorem established that Hilbert’s dream of proving the consistency of arithmetic within arithmetic itself could not be realized. And while it became generally accepted that Gödel’s results represented a fatal blow to Hilbert’s program, this by no means meant the end of mathematical logic as a research program. In fact, it seemed to count as a victory rather than as a failure, as the methodology was thus shown to be able to investigate and delineate its own limits. The emergence of model-theory with Tarski’s work on truth and logical consequence (Part V) was also a response to Gödel’s results and inaugurated a new approach in mathematical logic, alongside the proof-theoretical approach originally envisaged by Hilbert and collaborators.

Ultimately (and as noted by M. Potter in his review of the book in Philosophia Mathematica, 2012), Mancosu does not (and does not intend to) offer a radically novel picture of the development of mathematical logic in the first half of the 20^th century. The broad lines of his narrative are very much the ones largely agreed-upon in the literature. But starting from the general picture, he fills in the gaps, thus offering a very detailed account of these developments. Moreover, Mancosu is to be praised for bringing to the fore the contribution of less well-known figures such as Becker, Behmann and others, and for his keen eye for how these developments unfolded historically. His is a narrative focusing on actors and processes, not only on the results of their endeavors (theories, theorems etc.). (One of my favorite chapters in the volume is his piece on the immediate reception of Gödel’s incompleteness theorems (chapter 7), which describes how, after initial shock and some skepticism, consensus emerged concerning the correctness of the results and their wide-ranging implications.)

One may wonder what justifies the publication of a collection of previously published papers, which moreover have not been substantially altered for their inclusion in this volume. Now, the book does contain updated bibliographical lists of works on the topics that have appeared since the original publication of the articles. It also includes helpful summaries for each of its five parts, which thus make manifest that the different individual articles are all part of a larger, general project. At any rate, the fact that these articles are all conveniently assembled in just one volume represents a useful resource for all those interested in these topics.

It must be said, however, that the book is not exactly a page-turner, which may well be due to the fact that, in the end, it remains a collection of articles rather than a book conceived as such. In fact, it could (but need not be) viewed as a reference work, with its impressive collection of ‘facts and figures’ (accompanied by a rather detailed index of names, but alas no index of terms). It is for the most part also a very useful pedagogical resource, where some difficult concepts and results are explained in an accessible way (especially in the long survey in chapter 1).

To close this review, let me briefly discuss the two thus far unpublished chapters of the book, namely a transcription of Tarski’s lecture “On the Completeness and Categoricity of Deductive Systems”, and Mancosu’s own discussion of the lecture and its content (chapters 18 and 17, respectively, the last two chapters of the book). Completeness and categoricity are both crucial desiderata for formal systems, and are in a sense each other’s duals; while completeness ensures that all the relevant facts about a given portion of mathematics can be captured by a deductive system, categoricity ensures that only these relevant facts are captured by the system, i.e. that it is not also a description of structures falling outside the targeted mathematical domain/theory (‘alien intruders’, in Dedekind’s terms). When completeness fails, not enough fish are caught; when categoricity fails, too many fish are caught.

In his lecture, Tarski remarks that, in view of Gödel’s results, absolute completeness is a rare phenomenon; there are few deductive theories that can be complete (i.e. only those not sufficiently expressive so as not to allow for a formalization of arithmetic). He then introduces two weaker concepts of completeness, namely relative completeness and semantic completeness (the interested reader will have to consult the text itself for details, as limitations of space prevent me from offering further technical elaborations), and asks himself what methods could be used to show a given system to be relatively/semantically complete. Lest one should despair, Tarski announces the grand news: “We shall see, namely, that the concepts of relative and semantical completeness are closely related to the concept of categoricity (due to Veblen) and the investigation of the latter concept does not require in general any special and subtle methodological investigations.” (p. 490) Tarski then proves two theorems relating (relative and semantic) categoricity to (relative and semantic) completeness, and in view of the abundance of categorical systems, he concludes: “in opposition to absolute completeness, relative or semantical completeness occurs as a common phenomenon.” (p. 492) We thus have yet another display of Tarski’s uncanny ability to convey difficult technical concepts in an accessible way, while at the same time drawing sophisticated general conclusions from the results he discusses. A great way to end the book, and accordingly an appropriate way to end this review.

Amsterdam Workshop on Truth (13-15 March 2013)

The Amsterdam Workshop on Truth is organised by the Institute for Logic, Language, and Computation of the University of Amsterdam.

The workshop will take place from Wednesday 13th to Friday 15th of March 2013.

The workshop is intended to serve as a meeting point for researchers working on the philosophy of truth in order to discuss latest results and work in progress. It will address a wide range of truth-related topics and it is open to more formal or less formal approaches.

The following speakers have confirmed participation:

Stefan Wintein, Philip Welch, Albert Visser, Giulia Terzian, Johannes Stern, Jönne Speck, Sonja Smets, Georg Schiemer, Robert van Rooij, Carlo Nicolai, Iris Loeb, Øystein Linnebo, Graham Leigh, Jeffrey Ketland, Leon Horsten, Volker Halbach, Nina Gierasimczuk, Martin Fischer, Theodora Achourioti. 

Attendance is free of charge, however, registration is required.
The deadline for registering is March the 3rd.

The workshop will start on Wednesday at 12:00 and end on Friday at 15:00.

For more information, visit the workshop website (www.illc.uva.nl/truth/truth13) or contact the organisers Dora (t.achourioti AT uva.nl) and Cian (chartiec AT tcd.ie).

MCMP website relaunched!

We just relaunched our MCMP website with lots of little improvements and new contents! You might already have checked out our new front page or had a look at everybody's updated profile pages. We also added a video search function to the media page - still an experiment but one more way to explore the MCMP's topics.

As always, we are looking forward to getting your feedback on the new page. And of course we are working on implementing further improvements, stay tuned! Thanks for linking to www.lmu.de/mcmp from your own page and for your "like" on facebook here: www.facebook.com/lmu.mcmp!

Mathematics as "Pretense"?

A rather fashionable "nominalistic" view in the last decade in relation to mathematics is a view, or perhaps bundle of views (also sometimes called "fictionalism"), that mathematical reasoning is a kind of "pretense". Rather than thinking it to be the case that 3 + 5 = 8 or that there are infinitely many prime numbers, we somehow merely play a game of pretense with the sentences "3 + 5 = 8" and "there are infinitely many prime numbers". There are echoes here of old-fashioned game formalism, if-thenism and deductivism. And so it is that some philosophers have come to compare mathematics---a fundamental component of human knowledge and science---with Santa Claus.

My response to this kind of pretense theory is usually along the following lines. At the moment, no actual scientist has used the Santa Claus story to compute the Lamb shift, analyse quark confinement or perturbations in binary star systems and so on. So, I would be very interested to know why not!

The basic claims of such pretense theories are in some sense connected to the meanings of mathematical sentences; but they are neither abstract metaphysical claims (e.g., "mathematical facts are necessities which trivially supervene on all contingencies"), or rather abstract epistemological claims (e.g., normative claims, concerning rationality, "oughts", and so on). What is interesting about such claims is that they seem to have some empirical content at the level of cognitive psychology, and therefore may be subjected to empirical investigation.

In an interesting forthcoming article, "Pretense, Mathematics, and Cognitive Neuroscience" (BJPS 2013), Jonathan Tallant argues:

Abstract
A pretense theory of a given discourse is a theory that claims that we do not believe or assert the propositions expressed by the sentences we token (speak, write, and so on) when taking part in that discourse. Instead, according to pretense theory, we are speaking from within a pretense. According to pretense theories of mathematics, we engage with mathematics as we do a pretense. We do not use mathematical language to make claims that express propositions and, thus, we do not use mathematical discourse to make claims that are either true or false. In this paper I make use of recent findings from cognitive neuroscience and developmental science to suggest that pretense theories of mathematics fail.

States

In physics, one considers various quantities and functions, usually defined on spacetime, or on some assembly of particles. The quantities and functions are (usually) mixed mathematical entities, with abstract values. Laws of physics are then propositions which have semantic content: they say that these functions are related in some way -- usually via a differential equation. An important kind of value is a physical state. We have some sort of system $S$ (say a point particle, or a rigid body, or an assembly of particles, or a region of gas, etc.), and we think that it can "be" in some range of possible physical states.

I mention this only because it seems to me that there is a persistent mistake in some recent literature about applied mathematics and indispensability, in assuming that physical states are "concrete" entities, presumably of a spectacularly peculiar sort. But they are not "concrete". "Physical" does not imply "concrete". For example, a wavefunction $\Psi$ is physical, but is not a concrete entity: it's a function to $\mathbb{C}$. The wavefunction is the physical state.

For example, for an $N$-particle system in classical mechanics, the physical states are $N$-tuples of ordered pairs,

$((\mathbf{r}_i, \mathbf{v}_i) \mid i \in \{1,\dots, N\})$,

where $\mathbf{r}_i$ is the location of particle $i$, and $\mathbf{v}_i$ is the instantaneous velocity of particle $i$. Although certain kinds of equivalences may have to be taken into account, the points in the state space are the physical states. Taken together, they form a structure, which in a sense is "like" the manifold $\mathbb{R}^{6N}$, because the locations and velocities can be co-ordinatized as triples of reals (i.e., given a co-ordinate chart $\phi$ on space, and a point $\mathbf{r}$ in space, we have $\phi(\mathbf{r}) = (x, y, z) \in \mathbb{R}^3$). In classical mechanics, when one moves to the phase space, the structure is called a symplectic manifold.

Consider a 3-particle system, and consider the state

$\alpha := ((\mathbf{r}_1, \mathbf{v}_1),(\mathbf{r}_2, \mathbf{v}_2),(\mathbf{r}_3, \mathbf{v}_3))$.

This state $\alpha$ is not a concrete thing. What, for example, is the speed of this state $\alpha$? Its location? Such questions are absurd. A sequence $\alpha$ of positions and velocities is not a concrete entity (like, perhaps, the point particle itself). And despite the claims of some fictionalists, a physical state is also not like Santa Claus or Jane Eyre, because physical states are what are our scientific theories are about, quite unlike Santa and Jane Eyre.