New Draft: Mathematics, Science and Confirmation Theory

Here is the latest version of my paper from the PSA. As noted earlier, the goal of the session was to establish some links between philosophy of mathematics and philosophy of science. My aim was to make the connection through confirmation, although all I have done so far in this paper is raised the issue in what is hopefully a useful and novel way. This part of an ongoing project, so comments are certainly welcome!

Mathematics, Science and Confirmation
Abstract: This paper begins by distinguishing intrinsic and extrinsic contributions of mathematics to scientific representation. This leads to two investigations into how these different sorts of contributions relate to confirmation. I present a way of accommodating both contributions that complicates the traditional assumptions of confirmation theory. In particular, I argue that subjective Bayesianism does best accounting for extrinsic contributions, while objective Bayesianism is more promising for intrinsic contributions.

Richardson on Carus on Carnap

Richardson has a review in NDPR of Carus' recent book on Carnap. It is fairly sympathetic, but I think it strikes the right note of skepticism about Carus' attempts to extract an Enlightenment project from Carnap's work that will not only rescue some notion of explication in the service of clarifying scientific knowledge, but will also relate scientific knowledge to normative disputes in ethics and politics. As I read the book, Carus does an excellent job clarifying Carnap's moves towards a defensible picture of explication, but the link to values is still hard to make out. As Richardson puts it,

Carus's book leaves, that is to say, more to be done to specify and implement the project he announces. One can only hope that he continues to work in this vein and to inspire others to do so also. I am not convinced that what is at stake in interpreting Carnap's philosophy is ultimately our Western way of life, but, given the well-known social projects of the Vienna Circle, it would not be surprising if some aspects of interpreting Carnap's project aided in our philosophical understanding of our own social projects. I hope this review has given some indication of the multiple levels on which Carus's book is worth engaging philosophically. The book will be central to the continuing detailed scholarly discussions of Carnap's philosophy. More than this, it will, I hope, help raise to consciousness several larger issues regarding the social import of key projects within analytic philosophy.

Nahin on Torricelli’s Funnel

On the beach I finally got a chance to start (if not finish) Nahin’s When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. So far it is an engrossing survey of work on maximum/minimum problems that leads one gently into the mathematical intricacies of the history without being overly technical.

One of the first examples that Nahin discusses is Torricelli’s Funnel (also known as Gabriel’s Horn). One can think of it as the result of rotating the "first quadrant branch" (positive x, positive y) of the hyperbola xy = 1 around the x-axis. If we consider its surface, then we can show that the total surface is greater than any finite number. But if we consider its volume, then we can show that it is finite. For this case, the volume is pi. While I had come across this example before in Mancosu’s book, Nahin raises a paradox which I think should be more widely known for those working on mereology and "intuitions" in philosophy. As Nahin puts it, if the surface is infinite, then we cannot paint the funnel with a bucket of paint, no matter how large the bucket is. But if the volume is finite, we can paint the inside of the funnel by filling the funnel with paint! Do we have a proof that this figure is impossible?

No! Nahin points out that we are tacitly working with two different notions of paint: "mathematical paint" and "real paint". If we consider real paint, it is composed of molecules of some finite size. So, if we pour the real paint into the funnel, then it will not coat the inside of the funnel because at some distance its molecules will no longer fit. But if we consider "mathematical paint", then a finite amount can coat the outside of the funnel, or the inside of the funnel, because a finite volume can be spread out to any degree over a surface if we place no limit on the thickness of the coating. This elegant example shows just how liable we are to make mistakes when we consider these sorts of mereological questions if we fail to pay attention to the mathematical subtleties.

A Course in the History of Analytic Philosophy

Following up the previous post, here is the list of lectures that I gave here in Taiwan, with the readings for each lecture. I had 22 sessions, with an hour and a half per session, but pressed into four weeks. I ended up with only 18 lectures, with some sessions having more reading than others. An introductory course in deductive logic was presupposed.

1 What is analytic philosophy? What is the history of analytic
philosophy?
Glock, What is Analytic Philosophy?, pp. 21-48.

2 Kant and Mill
(i) Kant, Prolegomena to Any Future Metaphysics (1783), Preamble
and First Part. (ii) Mill, A System of Logic (1843), Book I, Ch. 3, sections 6-9, Ch. 5 & Book II, Ch. 6.

3 Frege, Foundations: Project & Critical Phase
Foundations, Introduction, sections 1-44

4 Frege, Foundations: Constructive Phase
Foundations, sections 45-69

5 Frege, Foundations: Implications
Foundations, sections 70-109

6 Frege, Two later papers
"Sense and Reference", "The Thought"

7 Moore and Russell
Moore, "Refutation of Idealism"

8 Russell on Denoting
Russell, "On Denoting"

9 Russell, Problems: Perception
Problems of Philosophy, ch. 1-4

10 Russell, Problems: Universals
Problems of Philosophy, ch. 5-10

11 Russell, Problems: Judgment
Problems of Philosophy, ch. 11-15

12 Wittgenstein, Tractatus: Metaphysics
1-2.063

13 Wittgenstein, Tractatus: Picturing
2.1-4.28

14 Wittgenstein, Tractatus: Logic
4.3-5.5571

15 Wittgenstein, Tractatus: Nonsense
5.6-7

16 The Vienna Circle
Neurath, Carnap, Hahn, "The Scientific World Conception: The Vienna Circle", Schlick, "The Turning Point in Philosophy", Carnap, "Elimination
of Metaphysics"

17 Protocol Sentences
Neurath, "Physicalism", "Protocol Sentences", Carnap, "Protocol
Sentences"

18 Carnap & Quine
Carnap, "Empiricism, Semantics and Ontology", Quine, "Two Dog-
mas of Empiricism"

Ideally there would be two more lectures: (i) one after 17 filling out the second phase of the protocol sentence debate with Schlick's "Foundation of Knowledge" and some later Neurath papers "Radical Physicalism and the 'Real World'" and "Unity of Science as a Task" and (ii) a final lecture bringing together some of the lessons for the history of analytic philosophy and noting some later developments with Quine and post-Quine. While this is a lot for one semester, for fifteen weeks I think it is a good balance of coverage of material and detailed discussion.

Teaching the History of Analytic Philosophy

Rather than worrying about the nature of analytic philosophy or taking a poll on who the most important philosophers are, I wanted to raise the issue of how we should structure an introductory course on the history of analytic philosophy. It seems to me that history of analytic has reached a kind of maturity that we associate with other areas of history like modern or Kant. With these topics, there are a few standard ways to organize an introductory course. There are also some companion introductory books that can be used to supplement readings of the classic primary texts.

But in the history of analytic philosophy, we don't really have either a standard syllabus or adequate companion books. Over the years several people have asked me how I teach my history of analytic classes. Unfortunately, I have tried out several different ways of organizing a Frege-Russell-Wittgenstein course and have always had trouble with the Russell part. My latest experience in Taiwan has convinced me that we should have a few basic goals when organizing such a course.

First, as with any history of philosophy course, the readings should be mostly drawn from the primary texts of the philosophers themselves. These readings should cover several different philosophers and span various areas of philosophy. It is not that useful, I would argue, to just focus on philosophy of language, for example, or just philosophy of mathematics. Picking just one area of philosophy would give a misleading impression of early analytic philosophy.

Second, there should be some attempt to relate the readings together into some kind of sustained narrative. One problem with some introductory books out there right now is that they cover just one philosopher. So, they miss the important interactions and disputes between philosophers that are crucial to the development of analytic philosophy. The narrative need not be some kind of continuous advancement of understanding, but could just as easily include several false starts or strange innovations on a given issue.

Third, students should learn not only the material covered, but also come to appreciate its remoteness from many of our contemporary ways of doing philosophy. As I put this point in my most recent course, early analytic philosophy was done in a different context, where a context includes a choice of problems, methods and standard positions. So, we should acquaint our students with this different context and help them to see how radical a transformation was effected by people like Frege, Russell and Wittgenstein.

On this last point, I can imagine the reply

Aren't we analytic philosophers, after all? If we are analytic philosophers, then surely we share more or less the same philosophical context of earlier analytic philosophers. So, the sorts of misunderstandings that result from variation in philosophical context simply cannot arise when we look back at these writings.

I want to suggest that this sort of response is misguided. What it ignores is that in its early stages analytic philosophy involved a radical change in the way philosophy was done. If this is right, then these early analytic philosophers were operating in a very different philosophical context from the one we enjoy now. This is for the simple reason that their philosophical contributions changed the philosophical context in many respects. These include all the features of a context: what problems are important, which methods are appropriate and which answers are viable. Many more traditional aspects of philosophy were thrown out and many new problems and techniques were imported into philosophy. The revolutionary character of early analytic philosophy, then, means that we must be careful in our approach to these writings, perhaps even more careful than when reading Aristotle or Aquinas where the difference in philosophical context is obvious and uncontested.

In future posts I hope to flesh out more how this sort of course would proceed, but for now I would be interested in hearing from teachers and students what their thoughts are on this sort of issue.

New Book: Glock, What is Analytic Philosophy?

As part of my course here in Taiwan I have been trying to read Glock's 2008 book What is Analytic Philosophy? (reviewed here). I am only about half-way through, but it is already one of the most encouraging contributions to the field in recent years. To start, Glock shows an in-depth knowledge of the many different aspects of analytic philosophy, from its earliest stages to its contemporary manifestations in Europe. Another positive is the level at which the book is written. While some exposure to analytic philosophy is necessary to appreciate his main points, most of the discussion would be accessible to a detemined undergraduate or a beginning graduate student.

Glock's main claim is set out in his "Introduction":

According to the [historical conception], analytic philosophy is first and foremost a historical sequence of individuals and schools that influenced, and engaged in debate with, each other, without sharing any single doctrine, problem, method or style ... [But] a purely historical conception ignores the fact that philosophers can be more or less analytic on grounds other than historical ties. These worries can be laid to rest if we acknowledge that analytic philosophy is a tradition held together not just by relations of influence, but also by overlapping similarities (pp. 19-20).

So, Glock offers a hybrid account of what analytic philosophy is. He combines both historical influence with similarities in philosophical commitments.

As I continue to read the book I will be interested to see if Glock addresses a tension that I see in many attempts to characterize analytic philosophy. On the one hand, we want to understand why analytic philosophy developed at the time and place that it did. On the other hand, if we are sympathetic to analytic philosophy, we also want to explain what is good or best about it compared to other developments in philosophy. Both desires can be easily combined if there are conclusive philosophical arguments for certain distinctive views of analytic philosophy, and these arguments were presented by the early analytic philosophers. But, in my experience at least, these things are very hard to find. As a result, many historians feel forced to choose either a purely causal reconstruction of historical developments or else a timeless reconstruction of philosophical arguments. Some third alternative is clearly needed.

Back to Blogging!

Apologies to the few regular readers of this blog for my lack of posts through May. I finished up my sabbatical visit at the Center for the Philosophy of Science in Pittsburgh at the beginning of May. I highly recommend it for anybody at any rank whose work intersects with the philosophy of science. Among other things, John Norton organized a reading group for the visiting fellows where we read each other's work in progress. While this caused some disturbing flashbacks to graduate school for me, the whole group was great and I certainly learned a lot about different areas of philosophy of science. PhilSci Archive now has a page where some of the work done by fellows this year can be seen. Everyone should consider applying for 2010-2011 -- the deadline for applications is in December.

After Pittsburgh I travelled to Taiwan where I am teaching an intensive four-week summer course on the history of analytic philosophy at Soochow University. This has been a very enlightening experience, and also very time consuming as Michael Mi, my host, asked me to circulate my lecture notes to the students to help them follow along. We are now in the last week of the class, so some of my energies can turn back to blogging! I plan on posting more about this class and the issues it raises for teaching the history of analytic soon.