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Philosophy's Loss of Logic to Mathematics

Woosuk Park
Publication Date: 
Number of Pages: 
Studies in Applied Philosophy, Epistemology and Rational Ethics
[Reviewed by
Michael Berg
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I was an undergraduate in the 1970s, when the world was very different; but that is neither here nor there, and it’s no use crying over spilled milk in the company of other curmudgeons. In any event, in addition to my mathematics major, I did a minor in philosophy. This ecumenism was in large part due to the influence of a wonderful junior professor, a student of Robert Solovay, the late Telis K. Menas, who played pied piper to a whole parade of undergraduate rats, myself among them. As it turned out, I stayed the course, so to speak, and even did some mathematical logic (as an avocation) in graduate school. To this day, it fascinates me, and my fellow traveler exploits in philosophy still resonate these many years later. That said, in my opinion the book under review addresses a set of truly fascinating themes, and critical ones, for philosophers interested in mathematics, mathematicians interested in philosophy, and historians of these disciplines.

Right off the bat, the title of the book conveys a historical reality that is well worth coming to terms with on its own terms. Why indeed — and how — did logic, classically the province of, e.g., Aristotle, come to be considered part of mathematics? I suppose the easiest answer to this question, or at least a first pass at the answer, would be that it’s all about axiomatization. In other words, once rules of the game of doing logic (one thinks of Boole’s The Laws of Thought) are written down, and especially once they are encoded in a uniform language, they become fodder for mathematics in the modern sense. Arguably, the Greeks actually saw this already: Euclid, after all is the father of the axiomatic method. But in the modern way of thinking (whatever that means), we are driving at the notion of a theory as a more or less well-defined thing. It should have objects it concerns itself with and rules for playing the game, and so forth.

Accordingly, what Aristotle concerned himself with is now largely subsumed by first order logic, or first order sentential calculus. We still find such things as his tertium non datur, i.e. the law of the excluded middle, but it now looks like \(\forall P: \lnot[P\land (\lnot P)]\), and we do get modus ponens and modus Tollens, of course, and a lot more, all very classical stuff, but it’s clearly more mathematically spiced than it used to be.

So, if there is really nothing new under the sun, i.e. if all we’re doing is to rewrite what “the philosopher” (to take the phrase of St. Thomas Aquinas) already said in more and more fancy language ca. 360 BC, what’s the point of it all? What business does mathematics have making this play for logic?

Well, we now have to look toward the 19th century for an answer, with the advent and work of Gottlob Frege. The book under review starts by highlighting how Frege went beyond Aristotle (and two millennia of custom and tradition) via and analysis of Frege’s distinction between the notions of “falling under” and “subordinations” (cf. the book’s second chapter). Park quotes somewhat extensively from Frege’s 1892 paper “On Concept and Object,” to wit the following (loc. cit. p. 11):

I [Frege] call the concepts under which an object falls its properties … If the object \(\Gamma\) has the properties \(\Phi\), \(X\), and \(\Psi\), I may combine them into \(\Omega\); so that is the same thing if I say that \(\Gamma\) has the property \(\Omega\), or that it has the properties \(\Phi\), \(X\), and \(\Psi\). I then call \(\Phi\), \(X\), and \(\Psi\) marks of the concept \(\Omega\), and at the same time properties of \(\Gamma\). It is clear that the relations of \(\Phi\) to \(\Gamma\) and to \(\Omega\) are quite different …\(\Gamma\) falls under the concept \(\Phi\); but \(\Omega\), which is itself a concept, cannot fall under the first-level concept \(\Phi\); only to a second-level concept could it stand in a similar relation. \(\Omega\) is, on the other hand, subordinate to \(\Phi\).

This fragment illustrates, then, a particularly modern mathematical theme, and one which we really run up against more often than we might realize, i.e. the business of introducing a hierarchy of how we construct and communicate qualified and quantified statements. It is in this realm that we encounter the most dramatic “modern” paradox, namely, Russell’s, which we all foist on our rookies in transition courses to upper division pure mathematics: the “set” of all sets that are not elements of themselves is not a set, since it defies effective membership criteria — it is a member of itself if and only if it isn’t a member of itself. A philosopher would argue that we really have the liar’s paradox all over again, but mathematics, with its mandate to axiomatize, takes this as much more: an opportunity to delineate safe ground for set theory (e.g.) along the lines of Russell’s theory of types. In this way, a set and its elements cannot be of the same type, or Frege might say order, and we are able to navigate safely, away from these paradoxes, if only by \(\varepsilon/3\).

Well, let’s get down to brass tacks, as far as Park’s book is concerned, now that we have discussed this initial “tension” between Aristotle and Frege. The book is laid out along historical lines in that it has four parts titled, respectively, “The Fregean Legacy,” “The Hilbert School,” “Goedel and Tarski,” and “Back to Aristotle.” Such a taxonomy in historical terms is certainly the right way to go for what Park has in mind, which is exemplified by the following (cf. Park, p.1):

Largely due to the foundational approaches in the first half of the twentieth century … we do not fully understand the problems of the application of mathematics … I want to hint at the urgent need to reconsider the Aristotelian position in logic and mathematics, which disappeared almost completely from the scene, without good reason, in the early twentieth century …

Accordingly, the book offers something of a sed contra to Frege, Hilbert, Goedel, and Tarski, with one of the main players in Park’s endgame being the Jesuit mathematician Giuseppe Biancani (1566–1624), who introduced the notion of scientiae mediae in connection with the peculiar status of mathematics vis a vis physics (cf. p. 217 of the book under review). In this regard Park states (cf. p. 219) that

philosophy of mathematics in the twentieth century largely ignored the problems of applying mathematics. Only quite recently, we began to realize the significance of the philosophical issues involved in the application of mathematics in all the different individual sciences …

and he proposes to inspire his readers to join him in this enterprise of re-examining modern (mathematical) logic with a critical philosophical eye.

Beyond that, the book provides a very interesting and accessible treatment of some of the relevant work of the mathematicians and logicians already mentioned, as well as a philosopher’s analysis of classical problems abutting to logic, e.g. certain ontological themes addressed by Duns Scotus (1266–1308). It is also noteworthy that Park devotes a chapter in Part III to Goedel’s famous formulation of the ontological argument for the existence of God (originally presented by St. Anselm). The chapter ends with an appearance by St. Augustine.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in the publisher's webpage.