Tuesday, December 30, 2014

Your questions about my “Husserl and Twin Earth” Post
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

In my previous post, I reported that Dagfinn Føllesdal informed me that Husserl had the Twin Earth thought experiment in mind in 1911.
Several comments have come in, including requests for more detail about what Husserl had in mind. Dagfinn’s analysis of Husserl’s thoughts about this will appear in Dagfinn Føllesdal, “Husserl and Putnam on Twin Earth”, in Themes from Putnam, Lauener Library of Analytical Philosophy, ed. M. Frauchiger (Berlin: Ontos). I don’t want to comment on the Husserl passage myself until after his paper appears.

Saturday, December 13, 2014

Mathematical "existence"
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

The preceding three posts have been fairly technical and full of details. So that the wood won’t be lost for the trees, I want to give a fairly short summary of the main points. Here they are:

(1) An interpretation of mathematics must be compatible with a scientific realist understanding of physics. From my first publications on the subject in the 1960s I have insisted that it is not enough that the theorems of pure mathematics used in physics come out true under one’s interpretation of mathematics—even some antirealist interpretations arguably meet that constraint—the content of the “mixed statements” of science (empirical statements that contain some mathematical terms and some empirical terms) must be interpretable in a “realist”, that is to say, a non-verificationist (or not simply “operationalist”) manner.  I believe many proposed interpretations fail that test. (Brouwer’s Intuitionism was my example in “What is Mathematical Truth”).
(2) Both objectualist interpretations (interpretations under which mathematics presupposes the mind-independent existence of sets as “intangible objects”) and potentialist/structuralist interpretations (interpretations under which mathematics only presupposes the possible existence of stuctures with the properties ascribed to sets, such as the “modal logical interpretations”) can meet the foregoing constraint.
(3) But the actual existence of sets as “intangible objects” suffers not only from familiar epistemological problems, but from a generalization of a problem first pointed out by Paul Benacerraf, “Benacerraf’s Paradox”, namely many identities (or proposed indenties) between different categories of mathematical “objects” seem undefined—e.g. are sets a kind of function or are functions a sort of set? For me, this tips the scales decisively in favor or potentialism/structuralism.
(4) In “Mathematics Without Foundations”, where I first proposed the modal logical interpretation), I claimed that conceptualism and potentialism are “equivalent descriptions”. In the three prededing posts I have retracted that claim.  But I don’t agree with Steven Wagner that rejecting objectualism requires one to say that sets, functions, numbers, etc., are fictions, and that the mathematics  student on the street is making a mistake when she says that there is a prime number between 17 and 34.  I now defend the view that potentialism is a rational reconstruction of our talk of “existence” in mathematics. Rational reconstruction does not “deny the existence” of sets (or, to use the example I used in the last post), of “a square root of minus one”; it provides a way of construing such talk that avoids the paradoxes.

Friday, December 12, 2014

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+5imeans <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 19th century mathematicians and their 20th 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.

Thursday, December 11, 2014

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

In a forthcoming paper[1], Steven Wagner raises a dilemma (actually, a trilemma) for the view I defended in “Mathematics without Foundations”. There, and in one place in my previous post, I described the modal logical interpretation (an interpretation on which quantification over mathematical objects is interpreted as talk of possible existence of structures[2])  and the objectualist interpretation, on which mathematical objects are taken to be genuine immaterial entities–as “equivalent descriptions”. Wagner asks what ”equivalent” means here, and argues (rightly) that mathematical equivalence (interdeducibility) is too weak, and that there are problems with interpreting my “equivalent” as “semantically equivalent” (which he thinks is what I meant). And if my talk of equivalence really disguised the claim that saying that a number or a set exists is talking about a fiction, and the modal logical interpretation is a way of eliminating those fictions from our discourse, I should not have spoken of equivalent descriptions.
Well, I agree that I should not have spoken of equivalent descriptions. But neither am I willing to describe myself as an eliminationist. This post and the posts that will follow will explain the alternative I see to accept Wagner’s trilemma.[3]
Escaping the trilemma
In fact, semantic equivalence is not what I have ever claimed. I have long written papers[4] defending the idea that two theories that have incompatible ontologies or make incompatible claims if we take them at “face value”, are, nevertheless, sometimes “equivalent descriptions”, and I have frequently said that this is not a claim about “translation practice”, i.e. about semantics. 
An example I have used for many years is the following: imagine a situation in which there are exactly three billiard balls on a certain table and no other objects (e.g. the atoms, etc., of which the billiard balls consist do not count as “objects” in that context). Consider the two descriptions, “There are only seven objects on that table: three billiard balls, and four mereological sums containing more than one billiard ball” and “There are only three objects on the table, but there are seven sets of individuals that can be formed of those objects.” What it means to be a realist who recognizes conceptual relativity with respect to this case is to believe that there is an aspect of reality which is independent of what we think at the moment (although we could, of course, change that reality by adding or subtracting objects from the table), which is correctly describable either way.
I do not think that the two sentences I just used as examples have the same meaning by any reasonable and non-tendentious standards of sameness of meaning. For one thing, the first sentence implies the existence of mereological sums, and the second implies the existence of sets, and I do not see that someone who accepts the first sentence is committed to the ontology of sets at all, or that someone who accepts the second is committed to the ontology of mereological sums at all. These are not synonymous sentences. They are not “semantically equivalent”. And similarly, an arithmetic sentence, or a sentence of set theory, is not synonymous with its image under the sort of mapping of mathematical sentences onto modal logical sentences I proposed in “Mathematics without Foundations”. These pairs of sentences are not semantically equivalent. But, I claimed, they are “equivalent descriptions”. But what was the criterion?
The example of the three balls was an artificial one. In my closing lecture to the conference in honor of my 80th birthday in Dublin[5], I said the following about a genuine scientific example (a case of quantum-mechanical “duality”):

My own notion of “conceptual relativity” (which I originally called “cognitive equivalence” is beautifully illustrated by [the duality example]. The different “representations” are perfectly intertranslatable; it is just that the translations don’t preserve “ontology”.
What do they preserve?  Well, they don’t merely preserve macro-observables. They also preserve explanations. An explanation of a phenomenon goes over into another perfectly good explanation of the same phenomenon under these translations.
But who’s to say what is a phenomenon? And who’s to say what is a perfectly good explanation?  My answer has always been: physicists are; not linguists and not philosophers.”

 —And I gave a similar explanation in my first paper on this sort of equivalence in 1978.
But now, I admit, a problem like Wagner’s “dilemma” does arise. Of course, if we are to apply this criterion to mathematics, we shall have to say that it is mathematicians and not physicists who should be the ones to say. But now a number of problems arise, problems that make me think I should not have tried to export the notion of “equivalent descriptions” from empirical science to the present case at all. Mathematics, after all, is not about “phenomena”, but about proofs, ways of conceiving of mathematical problems, mathematical approaches, and much more. And it does not seem reasonable to think that the mapping Hellman and I proposed of mathematical assertions onto modal-logical assertions preserves  these.  It is not just that a proof and its formalization in modal logical terms aren’t “semantically” equivalent; if the criterion is supposed to be that the mathematician would regard them as the same, the objection immediately arises that the modal-logical version, unlike the quantum-mechanical representations I mentioned in Dublin, is not one mathematicians are even aware of. (And if they were, I doubt very much that they would regard them as equivalent, except in the sense of deducible from each other, which is clearly insufficient.) The pragmatic criterion of equivalence I proposed for physics has no obvious analogue here. So, though not for Wagner’s reasons, I do see a real difficulty with what he calls “translational semantic modalism”.

The alternative I now propose is this: the interpretation of arithmetic and set theory as modal statements is neither a piece of straightforward semantics nor a substitute for something we have come to reject as false (as if numbers and sets were “fictional entities”). The modal logical interpretation is a rational reconstruction.
(to be continued)

[1] Steven Wagner, “Modal and Objectual”, forthcoming in  The Philosophy of Hilary Putnam (Chicago: Open Court, The Library of Living Philosophers, 2015).
[2] E.g., the graphs described in the previous post.
[3] What follows is adapted from some paragraphs in my Reply to the paper cited in note 1, which will appear in the same volume.
[4]  My first paper about cognitive equivalence of theories which are incompatible if simply conjoined was “Equivalenza,” Trans. P. Odifreddi. Enciclopedia, vol. 5 (Torino, Italy: Giulio Einaudi Editore, 1978), 547-564; English version published as “Equivalence” in Realism and Reason, 26-45. After that came my 1987 Carus Lectures, The Many Faces of Realism, in which I used the mereological sums example for the first time; “Reply to Jennifer Case.” Revue Internationale de Philosophie 55.4 (December 2001): 431-438; Lecture 2 in Ethics Without Ontology (2004); “Sosa on Internal Realism and Conceptual Relativity” in J. Greco (ed), Sosa and His Critics (Oxford: Blackwells, 2008);
 and probably other papers.
[5] “From Quantum Mechanics to Ethics and Back Again,” in my Philosophy in the Age of Science, 51-71.