Continuing: The Modal Logical Interpretation and "Equivalent Descriptions”

In 1976, when I delivered the John Locke Lectures at Oxford, I often spent time with Peter Strawson, and one day at lunch he made a remark I have never been able to forget. He said, "Surely half the pleasure of life is sardonic comment on the passing show".  This blog is devoted to comments, not all of them sardonic, on the passing philosophical show.

Hilary Putnam

Although Rudolf Carnap introduced the notion of the idea of rational reconstruction in his great work on epistemology[1], for our purposed it can be better illustrated with the story of “imaginary” numbers. As Menachem Fisch has described, British algebraist were tormented for nearly a century by the question of the “reality” of what we now call the “complex numbers”[2]. Yet, in the end, even introductory textbooks in analysis often tell us that we can stipulate that a complex number is simply an ordered pair of members of R, the field of real numbers. And real numbers can be identified with Dedekind cuts on the field Q of rational numbers, and rationals themselves can be identified with ordered pairs of members of Z, the ring of the positive and negative integers together with zero, or rather with equivalence classes of such ordered pairs.[3] And what are the “integers" of which Z is composed? We can stipulate, for example, that they are ordered pairs consisting of a natural number and one of three objects (say, the null set Ø, its singleton {Ø}, , and zero), and multiplication and addition can be defined in accordance with the rules that the product of two numbers of unlike sign (i.e. of a positive number and a negative number) is negative, and the product of two numbers of like sign is positive (e.g., <Ø,2> .<{Ø},3> =  <{Ø},6>).  And what are the natural numbers? Well, von Neumann taught us that we can stipulate that they are Ø, {Ø}, {Ø,{Ø},{Ø, {Ø}},…[4] And just as we stipulate definitions for multiplication and addition of members of Z (i.e. for addition and multiplication of the ordered pairs with which they were identified) so that the usual rules for multiplication and addition of “arbitrary integers” hold, so stipulate definitions for addition and multiplication of complex numbers (i.e. for addition and multiplication of the ordered pairs with which they were identified) so that the usual rules for the multiplication and addition of “complex numbers” hold. Of course, it is necessary to prove that all these stipulations are consistent, and that the distributive, commutative, associative, etc., laws are all forthcoming, but that is straightforward mathematical work. And voila! a century of worry by some of the greatest algebraists in the world over the “reality” of, e.g., the square root of minus one is passé. i =_df <0,1>, 1(the complex number)=_df<1,0> [the ordered pair of two real numbers], and “i . i = -1” becomes “<0,1> . <0,1> = <-1,0>” — and we define ‘multiplication’ of these particular ordered pairs so that this follows immediately from the definition. I repeat: an ontological worry about the “existence” of the complex numbers (and particularly about the existence of such a strange thing as a “square root of minus one”) is replaced by a mathematical problem—and not that difficult a one—of establishing the consistency and the logical consequences of a set of  stipulations.

Of course, these stipulations have the strange consequence that there are now five “ones”: one the “natural number” (e.g., {Ø} if we adopt von Neumann’s system); one the member of “the ring of integers” Z; one the rational number (the equivalence class to which 1/1, 2/2, ….and

-1/-1, -2/-2,… all belong); one the real number (the Dedekind cut whose left member A is the set of rational numbers less that the “one” of Z); and one the complex number! And what do mathematicians do about that? Why they simply ignore it!

Note that Benacerraf’s problem could have been raised here, but wasn’t. One could have said that these definitions have many alternatives which would work just as well (which is of course true), so how can it be that any one of them is “really right”? What the complex numbers really are has not been answered. But no one supposed, after the century of torment that Fisch describes so well, that there was such a thing as “what the complex numbers really are”. Dedekind did suppose that there was such a thing as what the integers really are, namely “a free creation of the human mind”[5] and Kronecker famously said that “God made the natural numbers. Everything else is the work of man”, but basically there was no problem.. But once the idea of treating complex numbers as appropriate logical constructions as taken hold—together with the idea that this could be done in different ways—and once Whitehead and Russell had used the same technique to build up, successively, the ring Z, the field Q, the field R, and the field of complex numbers C, starting with their own construction of the natural numbers (of each type from the second up) as sets of sets, nothing but sets was left as a basis[6]. Reference to the natural numbers too dissolves into reference to any infinite sequence of sets you choose. But why did Benacerraf worry about that fact, whereas Quine, for example, did not?

Perhaps for the simple reason that Quine, who felt forced, as a self-described reluctant Platonist[7] to simply “acquiesce” in the existence of sets, was simultaneously (if strangely) a complete skeptic about reference, and thus could never take seriously the problem of how we can refer to sets if sets are causally inert entities. In contrast, Benacerraf was a realist about reference. This is what Wagner refers to when he writes, “but Benacerraf's route would appear to turn on very delicate, tendentious formulations regarding causation and its role in justification.” While this may be right as a description of the paper to which Wagner refers, I suspect that there was much more behind Benacceraf’s raising the problem of reference to mathematical entities in the way he did. I will come back to this suspicion of mine in a moment. But right now I want to note the following: if, for the time being, we are willing to take reference to sets for granted, then the example of what has become the standard way of introducing the natural numbers (e.g., von Neumann), the ring of integers Z, the rationals Q, the reals R, and the complex numbers C, shows how a rational reconstruction can “defuse” a metaphysical problem, not by showing that there is one right way to think about the issue, but by showing a number of ways we could have decided to think and talk that would work equally well.  And this applies not only to ontological issues, although such are our concern here, but to rational reconstruction in general; it is not important that the theory of truth can be formalized a là Tarksi or a là Kripke[8]; what is important is that our concept of truth can be rendered non-contradictory.

Bringing this back to Wagner: one could have asked the mathematicians who decided to “identify” complex numbers with ordered pairs of reals, “Are you making the semantic claim that, e.g., “3+5i” means <3,5>? What possible semantic theory can support that? Why don’t you say that the square root of minus one is a fiction, and we can now live without it? One could have asked Frege, are you saying that, for example, “two” means the property of being an extension (i.e., a set) that can be put in one to one correspondence with the integers zero, one? What possible semantic theory can support that? Why don’t you say that arithmetic is a fiction, and we can now live without one, two, three,….etc.? One could have asked Tarski, aren’t you saying that the idea that there is such a thing as “truth” was a fiction and we can now live without it? In sum, that a concept needs to be replaced by a less problematic one, and that this can be done in more than one way, does not mean that the original concept was a fiction. I am not an eliminationist with respect to arithmetic, and I am not a semantic-modalist either; I am proposing a rational reconstruction.

But, granted that some rational reconstruction is called for here (and a great deal of it has taken place in mathematics itself since the nineteenth century), why can’t we stop with Quine? Why not just take sets as basic, and accept it that the work I described above of providing satisfactory definitions of Z, Q, R, and C has done the “housecleaning” work that was sorely needed? The answer is that a rational reconstruction is meant to defuse a paradox. Defining a problematic concept in terms of equally problematic concepts isn’t rational reconstruction. The work of rational reconstruction done by the 19^th century mathematicians and their 20^th century successors was not designed to resolve Benacerraf’s problem. It put the theory of real and complex variables on a firm footing, and that was a great achievement. But that is not our task here.

I said above that “I suspect that there was much more behind Benacceraf’s raising the problem of reference to mathematical entities in the way he did”. What I have in mind is this: Benacerraf is a Frege scholar, and he knows that the notion of “set” was quite unclear as late as the beginning of the twentieth century. I don’t have in mind simply the Russell paradox; I have in mind that the question of whether sets are simply the extensions of (possible) predicates haunted the whole late-nineteenth early-twentieth century discussion. Today that idea has been rejected (in part because possible predicates, or “properties” seems more problematic than sets, and in part because another notion, the so called notion of a “random” set, or an “arbitrary” collection” has come to seem more suitable for mathematics). But if the natural numbers seemed to be “the work of God”[9], set theory seems too recent (and two recently problematic) an invention to have such a sanctified metaphysical status. And aside from the fact that “set” is somewhat of a neologism, the fact is that sets too can be identified with other mathematical entities; in fact functions would seem to be a natural choice. Should we just say that here too there are simply “alternative rational reconstructions”? It is true that Benacerraf himself only speaks of the problem of the arbitrariness involved in identifying numbers with the members of any particular omega-sequence; I hope he will not mind if, when his problem is extended to all  mathematical entities, as illustrated by the fact that sets themselves can be identified with functions and vice versa, I henceforth speak of Benacerraf’s Paradox.

 If, as I believe, Benacerraf’s Paradox shows that the notion of sets as objects and arbitrary functions as objects are less than fully clear; if we don’t, in fact, know what it means to be a “Platonist about  sets or functions” (especially if, as Wagner explicitly does, we reject the idea of equivalent descriptions aka “conceptual relativity”!), how can showing that one could take either as basic and treat the other as a construction help? Granted, that I could think of functions as “real” and sets as different sorts of functions (and say, truly, that this can be done in more than one way, as far as mathematics is concerned), and granted that I could think of sets as “real” and functions as different sorts of sets (and say, truly, that this can be done in more than one way, as far as mathematics is concerned), how can that satisfy my desire to be clear about what I am doing when I do one or the other? Quine tells me to be a “sectarian”, and choose one and reject the other, but perhaps change my choice from time to time for some sort of enlightenment[10] and Wittgensteinians will say that my worry is “metaphysical”—but of course it is! What I am seeking is the right metaphysics.

---

[1] Rudolf Carnap, Logische Aufbau der Welt (Berlin-Schlachtensee: Weltkreis Verlag, 1928), 138 ff. Carnap wrote “rationale Nachkonstruktion”; in the English translation, The Logical Structure of the World (Berkeley: University of California Press, 1967), p. 220, this is translated as “rational reconstruction”.  “Rational reconstruction” was also used by Hans Reichenbach in Experience and Prediction (Chicago: University of Chicago Press, 1938), who attributed the term to Carnap (footnote 1, p. 5).

[2] Menachem Fisch, "The Emergency Which has Arrived: The Problematic History of 19th Century British Algebra - A Programmatic Outline", The British Journal for the History of Science, 27: 247-276, 1994.

[3] One chooses equivalence classes and not simply ordered pairs consisting of the numerator and the denominator so that 3/7 and 6/14 will turn out to be the same rational number.

[4] I.e.,  0 = Ø, 1 = {0}, 2 = {0,1}, 3 = {0,1,2} … each natural number, starting with zero, is the set of all smaller natural numbers!

[5] REF http://www.math.uwaterloo.ca/~snburris/htdocs/scav/dedek/dedek.html

[6] I am describing the theory of types as Ramsey simplified it here, not as Whitehead and Russell presented it.

[7] In Theories and Things (Cambridge, MA: Harvard University Press, 1990), p. 100, Quine famously described himself as a reluctant Platonist (“I have felt that if I must come to terms with Platonism, the least I can do is keep it extensional”).

[8] Saul Kripke, "Outline of a Theory of Truth", Journal of Philosophy 72 (1975): 690–716.

[9] Leopold Kronecker famously said that “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”. Quoted in Weber, H. "Leopold Kronecker", Mathematische Annalen (Springer Berlin / Heidelberg) 43 (1893): 1–25.

[10] Quine wrote“[The sectarian] is as free as the ecumenist to oscillate between [empirically equivalent but incompatible] theories for the sake of added perspective (sic) from which to triangulate on problems. In his sectarian way, he does deem the one the true and the alien terms in the other as meaningless, but only so long as he is entertaining the one theory rather than the other. He can readily shift the shoe to the other foot.” Pursuit of Truth (Cambridge, MA: Harvard University Press: 1990), 100.