Mathematics and Scientific Representation, Claim 2: Individuate Contributions via Content

With the move to Missouri complete and my book safely in press, there is now an opportunity to continue blogging through the key claims of the book. The first key claim is that mathematics contributes many things to the success of science. Next comes

2. These contributions can be individuated in terms of the contents of mathematical scienti c representations.

In principle, we might divide up the contributions which mathematics makes in any number of ways. These include historically, by scientific discipline, by the most successful and by mathematical technique. But it seemed most productive to consider different sorts of scientific representations, where the 'sorts' were isolated by the content of that representation. The hope is then that representations are available which are more or less pure instances of this or that kind of representation. And, for the successful representations, we can then look at the mathematics and see how it is contributing to the success of that sort of representation.

Approached in this way, there is a crucial difference between two ways in which some parts of mathematics might relate to a given representation. Some mathematics will be intimately part of the mathematics, in the following sense: the content of the representation is that some mathematical structure bears some structural relation to a target system, under some interpretation. So, the mathematics related directly to that structure is what I call intrinsic to the representation. This has immediate consequences for grasping the content of the representation, which I think of as given by the accuracy conditions for that representation. To understand the representation, one must believe that the claims of the intrinsic mathematics are true.

On the other hand, there may be additional mathematical claims which are extrinsic to the representation, but which are still relevant to the success of that representation. As a central example, consider the case where some stronger mathematical theory is deployed to solve the equations given in some weaker mathematical theory. It is wrong to say that an understanding of the representation requires a belief in the stronger mathematical claims, but these claims are still relevant to a consideration of the ultimate success of the representation. More generally, the mathematical relationships between representations may go beyond the intrinsic mathematics of the representations, but these relationships can still be central to an account of the success of the representations.

New Group: Association for the Philosophy of Mathematical Practice

On the weekend I was fortunate to attend the first meeting of a new group: the Association for the Philosophy of Mathematical Practice. This group aims to promote a somewhat different approach to the philosophy of mathematics than what has become standard. As the statement of purpose of the group puts it,

Over the last few years approaches to the philosophy of mathematics that focus on mathematical practice have been thriving. Such approaches include the study of a wide variety of issues concerned with the way mathematics is done, evaluated, and applied, and in addition, or in connection therewith, with historical episodes or traditions, applications, educational problems, cognitive questions, etc. We suggest using the label “philosophy of mathematical practice” as a general term for this gamut of approaches, open to interdisciplinary work.

In 2009, a group of researchers in this field gathered to promote the creation of the Association for the Philosophy of Mathematical Practice, APMP. This association aims to become a common forum that will stimulate research in philosophy of mathematics related to mathematical activity, past and present, and foster joint actions.

The goals of APMP are:

● to foster the philosophy of mathematical practice, that is, a broad outward-looking approach to the philosophy of mathematics which engages with mathematics in practice (including issues in history of mathematics, the applications of mathematics, cognitive science, etc.);
● to gather a group of interested people that forms a coherent community, and makes us more visible to the wider communities of, e.g., philosophers of science, historians of mathematics, mathematics educationalists, etc.;
● to stimulate research in philosophy of mathematics related to mathematics in practice, and enhance our opportunities for developing research projects;
● to facilitate the exchange of information among us in all kinds of ways, and stimulate common projects, meetings, etc.

The quality of the papers of the conference was quite high and showed, I hope, the potential for fruitful cooperation in the philosophy of mathematics between philosophers, historians, cognitive scientists, sociologists, etc.

Anyone interested in joining the association can contact me or one of the organizers listed on the site above. The next planned meeting is scheduled to occur at the Nancy Congress in July. I will post more details as they become available!

New Journal: Journal for the History of Analytical Philosophy

This week marks the official launch of the new Journal for the History of Analytical Philosophy. I think it is a very exciting opportunity for scholars working in this field. I would emphasize the open access character of the journal. All articles will be freely available in electronic form. The hope is that the journal can provide a forum for rigorous scholarship for the broadly conceived history of analytic philosophy.

As the mission statement of the journal indicates:

JHAP aims to promote research in and discussion of the history of analytical philosophy. ‘Analytical’ is understood broadly and we aim to cover the complete history of analytical philosophy, including the most recent one. JHAP takes the history of analytical philosophy to be part of analytical philosophy. Accordingly, it publishes historical research that interacts with the ongoing concerns of analytical philosophy and with the history of other twentieth century philosophical traditions. In addition to research articles, JHAP publishes discussion notes and reviews.

This goes some way to addressing Leiter's recent skeptical remark that "I trust they will publish articles that also explain how what used to be an actual movement in philosophy ceased to exist!" I would suggest that one of the issues worth discussing in the journal itself is the sort of position that Leiter alludes to here. But of course I also hope that more ordinary scholarship directed at questions in the history of analytic philosophers, and their relations to other philosophers, can be addressed.

The editorial team is

Editor in Chief
Mark Textor, King's College London, UK

Associate Editors
Juliet Floyd, Boston University, US
Greg Frost-Arnold, University of Nevada, Las Vegas, US
Sandra Lapointe, Kansas State University, US
Douglas Patterson, Kansas State University, US
Chris Pincock, Purdue University, US
Richard Zach, University of Calgary, CAN

Assistant Editor
Ryan Hickerson, Western Oregon University, US

Review Editor
Mirja Hartimo, University of Helsinki, FI

and the advisory board is

Steve Awodey, Carnegie Mellon University, US
Michael Beaney, University of York, UK
Arianna Betti, Free University of Amsterdam, NL
Patricia Blanchette, University of Notre Dame, US
Richard Creath, Arizona State University, US
Michael Friedman, Stanford University
Leila Haaparanta, University of Tempere, FI
Tom Hurka, University of Toronto, CAN
Peter Hylton, University of Illinois, Chicago, US
Bernard Linsky, University of Alberta, CAN
Ulrich Majer, University of Göttigen, D
Paolo Mancosu, University of California, Berkeley, US
Volker Peckhaus, University of Paderborn, D
Eva Picardi, University of Bologna, IT
Ian Proops, University of Texas, Austin, US
Erich Reck, University of California, Riverside
Alan Richardson, University of British Columbia, CAN
Thomas Ricketts, Pittsburgh University, US
Peter Simons, Trinity College Dublin, IRE
Thomas Uebel, University of Mancherster, UK
Joan Weiner, Indiana University, Bloomington, US
Jan Wolenski, Jagiellonian University, PL

Priest on Dialetheism in the NYT

Many readers of this blog have surely already seen this, but for the rest, be sure to check out Graham Priest's wonderfully accessible take on "Paradoxical Truth" in the much criticized NYT philosophy blog.

Philosophy Viral Video?

Has Roy Cook produced the first viral video for philosophers? You decide.

Mathematics and Scientific Representation, Claim 1: Many Contributions

In the next few weeks, I hope to go through 12 of the key claims which I try to defend in my book manuscript. At its most general, the topic of the book is how mathematics helps in science. I assume to start that science is quite successful. This success is not limited to its ability to generate consensus amongst its practitioners, but extends to its predictions and contributions to technological innovations. I more or less assume some kind of scientific realism, then, although exactly how realist we should be is part of the discussion of the book.

So, what does mathematics contribute to the success of science? I argue that

1. A promising way to make sense of the way in which mathematics contributes to the success of science is by distinguishing several diff erent contributions.

Many philosophers seem to think that there is one thing which mathematics does. Perhaps the most influential view along these lines goes back (at least) to Wittgenstein's Tractatus:

In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics. (6.21)

But this seems too narrow. Mathematics makes any number of contributions to the success of science, and there is no straightforward way to reduce them all to a single kind.

The problems with Wittgenstein's approach are obvious. In many cases, we have no clue what the non-mathematical inputs or outputs are supposed to be. We start with mathematical descriptions and we end with equally mathematical descriptions. Either there is something defective in scientific practice, or Wittgenstein's approach is wrong. Beyond this sort of inferential or deductive contribution, there must be other kinds of contributions. But how are we to enumerate these contributions, and is there anything to be said about what they might have in common?

Workshop: The Role of Mathematics in Science

Readers of this blog in the Toronto area may want to check out a workshop this Friday at the University of Toronto, IHPST. It is on the role of mathematics in science and the speakers are me, Margaret Morrison (Toronto), Steven French (Leeds), Alex Koo (Toronto) and Alan Baker (Swarthmore). The program is online here.