New Book: Weyl, Mind and Nature: Selected Writings

Princeton University Press has recently reissued Weyl's classic Philosophy of Mathematics and Natural Science along with a collection of essays by Weyl. Brandon Fogel reviews the latter in NDPR, noting

Peter Pesic's collection is the latest, and to date most significant, salvo in the effort to bring long overdue attention to Weyl's philosophical ideas, particularly those regarding science and its integration with mathematics.

While Weyl's philosophical views come across as a bit strange, he surely marks one of the most serious attempts to integrate a phenomenological starting point with a genuine understanding of our scientific knowledge.

Mathematics and Scientific Representation: Summary and Chapter 1

More than a year ago I posted a fairly vague description of a book project on the ways in which mathematics contributes to the success of science. I have made some progress on bringing together this material and thought it would be useful to post a summary of the chapters of the book along with an introductory chapter where I give an overview of the main conclusions of the book. Hopefully this is useful to other people working on similar projects. Critical suggestions for what is missing or who else is doing similar stuff is of course welcome!

Update (Feb. 17, 2011): The link to chapter 1 has been replaced with the final version. The link to the summary has been removed.

The Polymath Project

Gowers and Nielsen offer in the current issue of Nature a report on the online collaboration in mathematics known as the Polymath Project. It is hard to know what to make of it all without delving into the details and trying to understand if there is anything special about this problem which lends itself to collaboration. But two passages jump out for the philosopher:

This theorem was already known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights.

The working record of the Polymath Project is a remarkable resource for students of mathematics and for historians and philosophers of science. For the first time one can see on full display a complete account of how a serious mathematical result was discovered. It shows vividly how ideas grow, change, improve and are discarded, and how advances in understanding may come not in a single giant leap, but through the aggregation and refinement of many smaller insights. It shows the persistence required to solve a difficult problem, often in the face of considerable uncertainty, and how even the best mathematicians can make basic mistakes and pursue many failed ideas. There are ups, downs and real tension as the participants close in on a solution. Who would have guessed that the working record of a mathematical project would read like a thriller?

At over 150 000 words, these records should keep some philosopher busy for a while!

Nobel Prize for Efficient Markets Hypothesis?

One of the core ideas driving the derivation of the Black-Scholes model is the efficient markets hypothesis. Exactly what this comes to is hopefully something I'll post on next week. But for now I'll pass on this from NPR's Marketplace:

Kai Ryssdal's final note.

Not so much news as a commentary on the state of the economic profession. The Nobel Prize in economics comes out Monday morning. I obviously have no idea who's going to win, but the markets think they do. The betting line at Ladbrokes, in London, has Eugene Fama of the University of Chicago as a 2-to-1 favorite.

That's all well and good except for this: Fama's best known for something called the Efficient Markets Theory. That the markets are, in essence, always right. I dunno, I'd say that's a tough sell after the year and a half we've just had. More to come on Monday.

Mathematics, Financial Economics and Failure

In a recent post I noted Krugman's point about economics being seduced by attractive mathematics. Since then there have been many debates out there in the blogosphere about the failures of financial economics, but little discussion of the details of any particular case. I want to start that here with a summary of how the most famous model in financial economics is derived. This is the Black-Scholes model, given as (*) below. It expresses the correct price V for an option as a function of the current price of the underlying stock S and the time t.

My derivation follows Almgren, R. (2002). Financial derivatives and partial differential equations. American Mathematical Monthly, 109: 1-12, 2002.

In my next post I aim to discuss the idealizations deployed here and how reasonable they make it to apply (*) in actual trading strategies.

A Derivation of the Black-Scholes Model

A (call) option gives the owner the right to buy some underlying asset like a stock at a fixed price K at some time T. Clearly some of the factors relevant to the fair price of the option now are the difference between the current price of the stock S and K as well as the length of time between now and time T when the option can be exercised. Suppose, for instance, that a stock is trading at 100$ and the option gives its owner the right to buy the stock at 90$. Then if the option can be exercised at that moment, the option is worth 10$. But if it is six months or a year until the option can be exercised, what is a fair price to pay for the 90$ option? It seems like a completely intractable problem that could depend on any number of factors including features specific to that asset as well as an investor's tolerance for risk. The genius of the Black-Scholes approach is to show how certain idealizing assumptions allow the option to be priced at V given only the current stock price S, a measure of the volatility of the stock price σ , the prevailing interest rate r and the length of time between now and time T when the option can be exercised. The only unknown parameter here is σ , the volatility of the stock price, but even this can be estimated by looking at the past behavior of the stock or similar stocks. Using the value V computed using this equation a trader can execute what appears to be a completely risk-free hedge. This involves either buying the option and selling the stock or selling the option and buying the stock. This position is apparently risk-free because the direction of the stock price is not part of the model, and so the trader need not take a stand on whether the stock price will go up or down.

The basic assumption underlying the derivation of (*) is that markets are efficient so that ``successive price changes may be considered as uncorrelated random variables" (Almgren, p. 1). The time-interval between now and the time T when the option can be exercised is first divided into N -many time-steps. We can then deploy a lognormal model of the change in price δ S_j at time-step j :

δ S_j = a δ t + σ S ξ_j

The ξ_j are random variables whose mean is zero and whose variance is 1 (Almgren, p. 5). Our model reflects the assumption that the percentage size of the random changes in S remains the same as S fluctuates over time (Almgren, p. 8). The parameter a indicates the overall ``drift" in the price of the stock, but it drops out in the course of the derivation.

Given that V is a function of both S and t we can approximate a change in V for a small time-step &delta t using a series expansion known as a Taylor series

δ V = V_t δ t + V_s δ S + 1/2 V_{SS} δ S^2

where additional higher-order terms are dropped. Given an interest rate of r for the assets held as cash, the corresponding change in the value of the replicating portfolio Π = DS+C of D stocks and C in cash is

δ Π = Dδ S + r C δ t

The last two equations allow us to easily represent the change in the value of a difference portfolio which buys the option and offers the replicating portfolio for sale. The change in value is

δ(V-Π)=(V_t - rC)δ t + (V_S - D)δ S + 1/2 V_{SS} δ S^2

The δ S term reflects the random fluctuations of the stock price and if it could not be dealt with we could not derive a useful equation for V . But fortunately the δ S term can be eliminated if we assume that at each time-step the investor can adjust the number of shares held so that

D=V_S

Then we get

δ(V-Π)=(V_t - rC)δ t + 1/2 V_{SS} δ S^2

The δ S^2 remains problematic for a given time-step, but we can find it for the sum of all the time-steps using our lognormal model. This permits us to simplify the equation so that, over the whole time interval Δ t ,

Δ(V-&Pi) = (V_t - rC + 1/2 σ^2 S^2 V_{SS})Δ t

Strictly speaking, we are here applying a result known as Ito's Lemma.

What is somewhat surprising is that we have found the net change in the value of the difference portfolio in a way that has dropped any reference to the random fluctuations of the stock price S . This allows us to deploy the efficient market hypothesis again and assume that Δ(V-Π) is identical to the result of investing V-Π in a risk-free bank account with interest rate r . That is,

Δ(V-Π) = r (V-Π)Δ t

But given that V-Π = V - DS - C and D = V_S , we can simplify the right-hand side of this equation to

(rV - rV_S S - rC)Δ t

Given our previous equation for the left-hand side, we get

(*) V_t + 1/2 σ^2 S^2 V_{SS} + rSV_S - rV = 0

after all terms are brought to the left-hand side.

Critical Notice of Mark Wilson's Wandering Significance

I have posted a long critical notice of Mark Wilson's amazing book Wandering Significance: An Essay on Conceptual Behavior. It will eventually appear in Philosophia Mathematica. My impression is that even though the book came out in 2006 and is now available in paperback, it has not really had the impact it should in debates about models and idealization. I think this is partly because the book addresses broad questions about concepts that don't often arise in philosophy of science or philosophy of mathematics. But if you start to read the book, it becomes immediately clear how important examples from science and mathematics are to Wilson's views of conceptual evaluation. So, I hope my review will help philosophers of science and mathematics see the importance of the book and the challenges it raises.

Cole's Practice-Dependent Realism and Creativity in Mathematics

Julian Cole's "Creativity, Freedom and Authority: A New Perspective on the Metaphysics of Mathematics" is now available via the Australasian Journal of Philosophy. Cole develops what seems to me to be the most careful version of a social constructivist metaphysics for mathematics. Basically the idea is that the activities of mathematics constitute the mathematical entities as abstract entities. This makes it coherent for Cole to insist that the entities have many of the traditional features of abstract objects such as being outside space and time and lacking causal relations. Crucially for the causal point, even though the mathematicians constitute the mathematical entities, they do not cause them to exist.

One consideration in favor of his view that Cole emphasizes is the creativity that mathematicians have to posit new entities. Qua mathematician, he notes "the freedom I felt I had to introduce a new mathematical theory whose variables ranged over any mathematical entities I wished, provided it served a legitimate mathematical purpose" (p. 589). Other mathematicians have of course said similar things, from Cantor's claim that "the essence of mathematics lies precisely in its freedom" (noted by Linnebo in his essay in this volume) and Hilbert's conception of axioms in his debate with Frege.

I have two worries with this starting point. First, is it so clear that mathematicians really have this freedom? The history of mathematics seems filled with controversies about new objects or new mathematical techniques that seem to presuppose the existence problematic objects. Second, even if mathematicians have a certain kind of freedom to posit new objects, how do we determine that this freedom is independent of prior metaphysical commitments? One option for the traditional platonist or the ante rem structuralist is to insist that mathematicians are now free to posit new objects only because it is highly likely that these new objects can find a place in their background set theory or theory of structures. This of course would not settle the issue against practice-dependent realism, but it gives the realist a strategy to accommodate the same data.