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.

Mandelbrot (1924-2010)

The New York Times obituary gives a useful overview of his career and contributions to applications.

Okasha Takes On The Inclusive Fitness Controversy

In a helpful commentary in the current issue of Nature Samir Okasha summarizes the recent dispute about inclusive fitness. In an article from earlier this year E. O. Wilson and two collaborators argued that inclusive fitness (or kin selection) was dispensable from an explanation of altruistic behavior. For my purposes what is most interesting about this debate is that the Wilson argument depends on an alternative mathematical treatment which seems to get rid of the need for anything to track inclusive fitness. As a result, inclusive fitness is seen merely as a book-keeping device with no further explanatory significance.

Okasha suggests that the dispute is overblown and that each of the competing camps should recognize that a divergence in mathematical treatment need not signal any underlying disagreement. As he puts it at one point

Much of the current antagonism could easily be resolved — for example, by researchers situating their work clearly in relation to existing literature; using existing terminology, conceptual frameworks and taxonomic schemes unless there is good reason to invent new ones; and avoiding unjustified claims of novelty or of the superiority of one perspective over another.

It is strange that such basic good practice is being flouted. The existence of equivalent formulations of a theory, or of alternative modelling approaches, does not usually lead to rival camps in science. The Lagrangian and Hamiltonian formulations of classical mechanics, for example, or the wave and matrix formulations of quantum mechanics, tend to be useful for tackling different problems, and physicists switch freely between them.

This point is right as far as it goes, but my impression is that some biologists and philosophers of biology over-interpret the concept of fitness. If Wilson et. al. are correct, then there is simply no need to believe that inclusive fitness tracks any real feature of biological systems. And this interpretative result would be significant for our understanding of altruism and natural selection more generally.

Mathematics and Scientific Representation Update

As readers of this blog are already aware, for some time now I have been working on a book called Mathematics and Scientific Representation which aims to say something useful about how mathematics helps in science. This is a project which combines elements of the philosophy of mathematics with the philosophy of science and so will hopefully be of interest to both communities.

I have recently assembled the chapters into what is hopefully their final form. An outline of the project is available in a revised chapter 1. In the next few weeks, I hope to blog through the 12 key claims which I present at the end of this chapter. Comments and links to other ways of exploring these issues are especially welcome.

More on Epimenides

As Jonathan Livengood helpfully pointed out in a comment on my last post, Bayle links Epimenides to the semantic paradoxes in his Dictionary entry for Euclid (1740, although perhaps also in an earlier edition). A modern source cited by Bayle is Gassendi, and when you track down that reference you indeed find a link between Epimenides and the semantic paradoxes:




This comes from p. 40 of the first volume of the Opera Omnia from 1658, which has been scanned and posted online here. I don't know Latin, but the reference to this case as "celebre" suggests that Gassendi does not take himself to be making a new connection.

Who First Linked Epimenides to the Semantic Paradoxes?

As part of a philosophy of logic seminar on theories of truth I have developed an amateur interest in the history of discussions of logical and semantic paradoxes. As is well known, the Liar paradox can be traced to Epimenides and appears in the New Testament:

It was one of them, their very own prophet, who said, 'Cretans are always liars, vicious brutes, lazy gluttons.' That testimony is true. (Titus 1: 10-13, NRSV)

Russell makes allusions to this passage several times, including in "Mathematical Logic as Based on a Theory of Types" (see here.)

Given the discussion of these sorts of paradoxes in by medieval logicians, I was surprised to find this passage in Spade's article on Insolubles in the Stanford Encyclopedia:

One initially plausible stimulus for the medieval discussions would appear to be the Epistle to Titus 1:12: "One of themselves, even a prophet of their own, said, The Cretians [= Cretans] are always liars, evil beasts, slow bellies." The Cretan in question is traditionally said to have been Epimenides. For this reason, the Liar Paradox is nowadays sometimes referred to as the “Epimenides." Yet, blatant as the paradox is here, and authoritative as the Epistle was taken to be, not a single medieval author is known to have discussed or even acknowledged the logical and semantic problems this text poses. When medieval authors discuss the passage at all, for instance in Scriptural commentaries, they seem to be concerned only with why St. Paul should be quoting pagan sources.[5] It is not known who was the first to link this text with the Liar Paradox.

So, was Russell the first to make this link, or was he merely drawing on other sources?

My first thought was that Hegel or some other post-Kantian must have made the link, and Russell is merely repeating it. Through the power of Google Books I was able to find a passage in the English translation of Lotze's Logic:

One dilemma nicknamed Pseudomenos dates from Epimenides, who being a Cretan himself asserted that every Cretan lies as soon as he opens his lips. If what he asserted is true, he himself lied, in which case what he said must have been false; but if it false it is still possible that the Cretans do not always lie but lie sometimes, and that Epimenides himself actually lied on this occasion in making the universal assertion. In this case there will be no incongruity between the fact asserted and the fact that it is asserted, and a way out of the dilemma is open to us (Book II, Chapter IV).

This translation dates from 1884 and seems to be from the second edition of the Logic from 1880. I have not checked the German or the first edition.

It seems likely that Russell read Lotze's Logic, either in this very translation or the original German, as he notes Lotze's Metaphysik in his readings from 1897 and of course discusses Lotze's views on geometry in the fellowship essay. Still, it seems unlikely to me that Lotze was the first person to make the link. Any other candidates or evidence to consider?