Mathematics and philosophy

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Mathematics and philosophy

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1richardbsmith
Mrz. 13, 2010, 10:15 pm

Some of the greatest philosophers have had strong backgrounds in mathematics, as well as some in the sciences.

Do philosophers today study mathematics as part of their profession?

2johnnylogic
Mrz. 14, 2010, 2:17 pm

This is a subject near and dear to me. Many philosophers in the analytic tradition study mathematics as part of their profession. Particularly those in particular philosophical sub-disciplines:logic (naturally), epistemology, philosophy of science, and philosophy of mathematics (naturally). Nearly all areas can be enhanced by such study, such as metaphysics (esp. mereology), political philosophy (esp. models of distributive justice), philosophy of economics (esp. methodological and econometric problems), for instance. However, college curricula generally do not focus heavily on mathematics in philosophy programs, with the exception of the study of logics and set theory.

Are you interested in any particular branch of mathematics or philosophy?

3richardbsmith
Mrz. 14, 2010, 2:41 pm

I am sitting here right now, beating my head on beginner statistics. (Self study with some texts.) My very undisciplined studies took me from calculus, back to algebra, back to trigonometry, to group theory, to graphing transformations, back to exponent and logarithms (I never really got logarithms in previous exposures), and now to statistics.

(Running into regressions pushed me into statistics.)

My question though is from a curiosity about philosophy, and its relationship with science and mathematics. Discussions frequently focus on ethics or metaphysics, but some of the great philosophers made major contributions to the harder sciences and to mathematics.

I wondered if mathematics remains important to the discipline, or if philosophy is moving away from those fields. Like you mention with a less focus on science and math in philosophy curricula.

Do other areas such as evolution, genetics, particle physics, cosmology, and other sciences enter into the discipline?

4bjza
Mrz. 15, 2010, 12:49 pm

Re: "Discussions frequently focus on ethics or metaphysics..."

Are we talking about philosophical discussions in forums like this, bookstore shelves, or at professional conferences and in professional journals? If the first two, we'd also have marvel at the apparent wider interest in Plato and Aristotle than all 20th century philosophers combined.

If the last option, I'd need to see some numbers before I accepted that. ;)

5richardbsmith
Bearbeitet: Mrz. 15, 2010, 3:45 pm

bjza,

I am far from a professional or especially well read in the field. It was just a question that has been on the brain - the impact of contemporary science and math on philosophical inquiries.

Your point about Plato and Aristotle is good, though surely Plato has the bigger share of the two.

I take your comment, and the response from johnnylogic, to indicate that at least professional discussions typically extend to more than arguing opinions on metaphysics and ethics, and that mathematics remains important to philosophy.

Thanks.

6johnnylogic
Bearbeitet: Mrz. 15, 2010, 10:02 pm

richardbsmith,

bjza points out some good distinctions-- the professional community is quite different from popular conceptions of philosophy. Professional discussions are highly critical and full of standard cases and subtle distinctions that go unnoticed in most conversations outside the field.

If you are interested in seeing firsthand the level of interest and significance of maths and sciences to philosophy, I urge you to first read some introductory philosophy of science, epistemology (esp. formal), and philosophy of mathematics books, and browse the Stanford Encyclopedia of Philosophy. Here are a few good books along these lines:

*Philosophy of Science: The Central Issues
*Representing and Intervening : Introductory Topics in the Philosophy of Natural Science
*Theory and Reality : An Introduction to the Philosophy of Science
*Uncertain Inference
*Thinking Things Through: An Introduction to Philosophical Issues and Achievements
*Thinking about Mathematics: The Philosophy of Mathematics

When you are comfortable with the basics, you can go over to Science, Logic, and Mathematics and Formal Epistemology sections of PhilPapers and see professional-level scientifically/mathematically oriented philosophy in action. In short, most sub-disciplines of science and mathematics is of interest to some set of philosophers.

7richardbsmith
Mrz. 15, 2010, 10:27 pm

Thanks so much for the references. And especially the link to PhilPapers. I had not known of that resource.

8NoLongerAtEase
Mrz. 16, 2010, 12:39 am

Richard-

I can only answer with an affirmative: yes! Reflection on mathematics still animates philosophers as much as ever! That said, we are not usually trained in mathematics in any professional way. I took graduate courses in set theory and mathematical logic (most good philosophers have worked their way through Godel's proof at least once) but these courses were not *required*.

Simply put, professional training in philosophy is narrowly focused. Unless one is writing a dissertation *in* logic or the philosophy of physics, one typically doesn't take mathematics courses, though certainly anyone that wants to is encouraged so long as the course content has some overlap with their research.

Still though we're not proving many theorems, philosphers still spend lots of time thinking *about* mathematics; the philosophy of mathematics is a burgeoning sub-field and, to my mind, is one of the places where the most important issues in general metaphysics and epistemology come into clear focus.

In addition to the general philosophy of science resources mentioned above, if you want to get a sense of what philosophers of mathematics think about I would direct you to James Robert Brown's "Introduction to the Philosophy of Mathematics" published by Routledge.

You should probably also have a look at Paul Benecerraf's seminal essays "What Numbers Could Not Be" and "Mathematical Truth". The latter is, to my mind, the most important paper in the philosophy of mathematics over the last four or five decades. Both should be available through Jstor. If you do not have Jstor access, "Mathematical Truth" is available in anthology format. See, for example, W.D. Hart's 'Philosophy of Mathematics' published by Oxford.

9bjza
Mrz. 18, 2010, 10:56 pm

8> Really enjoyed "What Numbers Could Not Be." Will be reading more Benecerraf, I think.

10indregard
Mrz. 31, 2010, 2:30 pm

You might also be interested in Alain Badiou, who has brought mathematics back on the agenda in what is perhaps still called "continental philosophy". His two great works, "Being and Event" and "Logics of Worlds" both deal heavily with mathematics, while "Number and Numbers" is about, well, the philosophy of number theory.

11richardbsmith
Mrz. 31, 2010, 8:46 pm

Thanks for the suggestions. Any thoughts or comments on Turing and John von Neumann?

12johnnylogic
Bearbeitet: Mrz. 31, 2010, 9:22 pm

Both Turing and John von Neumann were mathematical luminaries, and two of my intellectual heroes.

Turing's work cryptography, computability and mathematical biology See Hodges' definitive biography, Alan Turing: The Enigma for his tragic story.

Von Neumann seemed to work on nearly every major project in the first half of last century: he axiomatized set theory and quantum physics, co-created the foundations for game theory, worked on the Manhattan Project, and contributed to several fundamental pieces of computer science (from his standard architecture for stored programs and cpu, to Monte Carlo simulation, to the theory of logical automata). Formidable! John von Neumann and Norbert Wiener is a good biography of both von Neumann and the founder of cybernetics, Norbert Wiener.

13richardbsmith
Mrz. 31, 2010, 9:38 pm

Logical automata is what I am getting hints of, well above my math level. But it is intriguing that computational logic may precisely describe the workings of everything.

Ideas of discreteness and infinitesimalness even in space and time.

It is overwhelming to consider what great minds have worked and are working on discovery. What will the next decades produce?

14kukulaj
Bearbeitet: Apr. 9, 2010, 9:13 am

I don't know much about what professional philosophers do, but I like to scratch my head over many of the books they write!

I have worked for many years developing software used by engineers to check the correctness of digital circuit designs. On the one hand, this is right in the thick of the mathematics of automata. On the other hand, one runs smack into some nasty issues about what it means for a design to be correct, and what it means to prove that a design is correct. Mechanizing Proof by Donald MacKenzie is a delightful book that covers much of the fun.

The automata I have been working with have been finite state machines. Once you jump over to infinite state machines, in particular to Turing machines, life gets far more interesting. In particular, the semantics of programming languages is just amazingly tangled. Practical Foundations of Mathematics by Paul Taylor looks like a really great starting point.

Maybe Ludwig Wittgenstein's Tractatus Logico-Philosophicus was the ultimate proposal for unifying philosophy and mathematics. The marriage didn't last too long, though. A big fork in the road was Goedel's proof of the undecidability of some propositions in number theory. Plus with books like J. L. Austin's How to Do Things With Words it seems to have gotten clearer that mathematics doesn't so easily address many of the affairs of life.

15richardbsmith
Apr. 9, 2010, 9:08 am

I am trying to work through Roger Penrose Shadows of the Mind now.

16richardbsmith
Apr. 9, 2010, 5:46 pm

modalursine has started a topic on Pro and Con which is somewhat touching on this topic. I would sure love to see some of your inputs on his topic about AI and consciousness.

17Landric
Mai 24, 2010, 9:40 pm

I did, and I still do.

18bjza
Mai 29, 2010, 10:58 pm

In the realm of work framed more for mathematicians than philosophers, I've just recently read Gödel's Proof and The Mathematical Experience. I'd recommend either for someone who has a mathematical background (or at least isn't afraid of maths) and is interested in the philosophy of mathematics. The first is an introduction to the sort of issues which made mathematics an important topic in philosophy. The second shows how mathematics continues to open new philosophical issues, despite widespread beliefs in its purity, self-evidentiality, or practicality.

And if you need something light to cleanse the palette, there's always Look Around You's module on maths: http://www.youtube.com/watch?v=Pj2NOTanzWI

19richardbsmith
Mai 31, 2010, 9:11 am

I have been looking recently at The Mathematical Experience to purchase. Thanks for the recommendations.

20Mr.Durick
Mai 31, 2010, 5:45 pm

It is a long time since I read The Mathematical Experience, but I count it as one of the memorable readings of my life. I remember The World of Mathematics fondly also and probably got as much out of it, but the former seemed to change my understanding whereas the latter instead informed it.

Robert