Download 9. Indispensability arguments in the philosophy of mathematics

9.
INDISPENSABILITY ARGUMENTS IN THE PHILOSOPHY OF
MATHEMATICS
If one consults the Stanford Encyclopedia of Philosophy on the topic
“Indispensability Arguments in the Philosophy of Mathematics,”i one finds (as part of a
moderately lengthy entry written by Mark Colyvan) the following paragraph:
From the rather remarkable but seemingly uncontroversial fact that mathematics is
indispensable to science, some philosophers have drawn serious metaphysical
conclusions. In particular, Quine…ii and Putnam…iii have argued that the
indispensability of mathematics to empirical science gives us good reason to
believe in the existence of mathematical entities. According to this line of
argument, reference to (or quantification over) mathematical entities such as sets,
numbers, functions and such is indispensable to our best scientific theories, and so
we ought to be committed to the existence of these mathematical entities. To do
otherwise is to be guilty of what Putnam has called “intellectual dishonesty.”iv
Moreover, mathematical entities are seen to be on an epistemic par with the other
theoretical entities of science, since belief in the existence of the former is justified
by the same evidence that confirms the theory as a whole (and hence belief in the
latter). This argument is known as the Quine-Putnam indispensability argument for
mathematical realism. There are other indispensability arguments, but this one is by
far the most influential, and so in what follows I’ll concentrate on it.
From my point of view, Colyvan’s description of my argument(s) is far from right.
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The fact is that in “What is Mathematical Truth” I argued that the internal success and
coherence of mathematics is evidence that it is true under some interpretation, and that its
indispensability for physics is evidence that it is true under a realist interpretationthe
antirealist interpretation I considered there was Intuitionism. This is a distinction that
Quine nowhere draws. It is true that in Philosophy of Logic,v I argued that at least some
set theory is indispensable in physics as well as logic (Quine had a very different view on
the relations of set theory and logic, by the way), but both “What Is Mathematical
Truth?” and “Mathematics Without Foundations” were published in Mathematics, Matter
and Method together with “Philosophy of Logic,” and in both of those papers I said that
set theory did not have to be interpreted Platonistically. I said that modal-logical
mathematics (i.e. mathematics which takes mathematical possibility as primitive and not
abstract entities of any kindvi) and mathematics which takes sets as primitive are
“equivalent descriptions.” In fact, in “What Is Mathematical Truth?”vii I said, “the main
burden of this paper is that one does not have to ‘buy’ Platonist epistemology to be a
realist in the philosophy of mathematics. The modal logical picture shows that one
doesn’t have to ‘buy’ Platonist ontology either.” Obviously, a careful reader of
Mathematics, Matter and Method would have had to know that I was in no way giving an
argument for realism about sets as opposed to realism about modalities.
In sum, my “indispensability” argument was an argument for the objectivity of
mathematics in a realist sense—i.e. for the idea that mathematical truth must not be
identified with provability. Quine’s indispensability argument was an argument for
“reluctant Platonism,” which he himself characterized as accepting the existence of
“intangible objects”viii (numbers and sets). The difference in our attitudes goes back at
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least to 1975.
In addition, there is a premise in Colyvan’s formalization of the supposed “QuinePutnam indispensability argument” the “and only” part of which I have never subscribed
to in my life, namely:
“(P1) We ought to have ontological commitment to all and only the entities that are
indispensable to our best scientific theories.”
Nevertheless, there was a common premise in my argument and Quine’s, even if
the conclusions of those arguments were not the same. That premise was “scientific
realism,” by which I meant the rejection of operationalism and kindred forms of
“instrumentalism.” I believed (and in a senseix Quine also believed) that fundamental
physical theories are intended to tell the truth about physical reality, and not merely to
imply true observation sentences.
Objections to indispensability arguments
A common objection to arguments from indispensability for physics to realism with
respect to mathematics (I shall consider more sophisticated objections shortly) is that we
do not yet have, and may indeed never have, the “true” physical theory; my response is
that, at least when it comes to the theories that scientists regard as most fundamental
(today that would certainly include quantum field theories) we should regard all of the
rival theories as candidates for truth or approximate truth, and that any philosophy of
mathematics that would be inconsistent with so regarding them should be rejected. This
“indispensability argument” has continued to appear in my work right up to the present
day.x
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I believe that the most serious objections to my “indispensability arguments”
depend, albeit in very different ways, on considering nominalist alternatives to present
day theoretical physics. The two most important such objections, I believe, are due to
Hartry Field and to Gideon Rosen and I will consider them in the next two sections. [In
the Appendix, I will argue briefly that if my arguments are accepted, not only
intuitionism and nominalism need to be rejected as philosophies of mathematics, but
also—and I know this will be controversial—the idea that “predicative” mathematics is
all that science needs.] But since I began by quoting Mark Colyvan, in the rest of the
present section I shall discuss the objections that he mentions in his entry in The Stanford
Encyclopedia of Philosophy. I shall discuss them, of course, on the assumption that they
are supposed to be objections to my arguments. It may be that Colyvan only intended
them to apply to Quine’s argument. (I suspect that while Quine certainly accepted
something like (P1), the idea that he accepted or used the idea of “confirmation” is wildly
off the mark, and several of the objections depend on that idea. But Quine-interpretation
is not the purpose of this essay.)
The objections in question are:
(1)
“[T]he debate continues in terms of indispensability, so we would be well
served to clarify this term. The first thing to note is that ‘dispensability” is not the same
as ‘eliminability.’ If this were not so, every entity would be dispensable (due to a theorem
of Craig).”
(2)
“These issues naturally prompt the question of how much mathematics is
indispensable (and hence how much mathematics carries ontological commitment).”
(3)
“Confirmational holism is the view that theories are confirmed or
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disconfirmed as wholes.xi So, if a theory is confirmed by empirical findings, the whole
theory is confirmed. In particular, whatever mathematics is made use of in the theory is
also confirmed.xii Furthermore, as Putnam stressed in “What is Mathematical Truth,” it is
the same evidence that is appealed to in justifying belief in the mathematical components
of the theory that is appealed to in justifying the empirical portion of the theoryxiii (if
indeed the empirical can be separated from the mathematical at all). Naturalism and
holism taken together then justify (P1). Roughly, naturalism gives us the ‘only’ and
holism gives us the ‘all’ in (P1).”
(4)
“There have been many objections to the indispensability argument, for
example in Charles Parsons’ ‘Mathematical Intuition,’xiv concern that the obviousness of
basic mathematical statements is left unaccounted for by the Quinean picture and Philip
Kitcher’s The Nature of Mathematical Knowledgexv worry that the indispensability
argument doesn’t explain why mathematics is indispensable to science. The objections
that have received the most attention, however, are those due to Hartry Field, Penelope
Maddy and Elliott Sober. In particular, Field’s nominalisation program has dominated
recent discussions of the ontology of mathematics.”
(5)
“Maddy’s first objection to the indispensability argument is that the actual
attitudes of working scientists towards the components of well-confirmed theories vary
from belief, through tolerance, to outright rejection”.xvi The point is that naturalism
counsels us to respect the methods of working scientists, and yet holism is apparently
telling us that working scientists ought not to have such differential support to the entities
in their theories. Maddy suggests that we should side with naturalism and not holism
here. Thus we should endorse the attitudes of working scientists who apparently do not
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believe in all the entities posited by our best theories. We should thus reject (P1).
(6)
“Thus if mathematics is confirmed along with our best empirical
hypotheses (as indispensability theory claims), there must be mathematics-free
competitors. But Sober points out that all scientific theories employ a common
mathematical core. Thus, since there are no competing hypotheses, it is a mistake to think
that mathematics receives confirmational support from empirical evidence in the way
other scientific hypotheses do.”
(7)
“The two most important arguments against mathematical realism are the
epistemological problem for Platonism—how do we come by knowledge of causally inert
mathematical entities?xvii—and the indeterminacy problem for the reduction of numbers
to sets—if numbers are sets, which sets are they?”xviii
Although this is a large budget of objections, I can reply to them fairly quickly
because they all assume that my “indispensability argument” (as opposed to the fictitious
“Quine-Putnam Indispensability argument”) depends on P1 and on “confirmational
holism,” and neither supposition is correct. In addition, they all assume that my
indispensability argument was an argument for “Platonism” in Quine’s sense, and, as I
have already pointed out, this was not the case.xix Here follows a brief word about each of
them:
Re (1): [“The first thing to note is that “dispensability” is not the same as
“eliminability’. If this were not so, every entity would be dispensable (due to a theorem
of Craig).”] The Craig theorem only shows that is logically possible to replace a theory T
which contains terms for unobservable entities (mathematical or non-mathematical) with
a theory—in effect, a Turing Machine—which recursively enumerates exactly the
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theorems of the first theory which contain only “observation terms”xx. Such an
“elimination” is only acceptable to an instrumentalist who thinks of theories as mere
prediction machines. I repeat that my “indispensability arguments” were intended to
show that it is incoherent to attempt to be a scientific realist with respect to physics (that
is, a realist with respect to the unobservable entities postulated by physics) and a
verificationist, intuitionist, nominalist, or what-have-you, with respect to mathematics.
They were not arguments against instrumentalism, although I have published a number of
such arguments in my time.xxi
As for clarifying the notion of “indispensability”: it is enough for my purposes that
all the different versions of, say, present day quantum field theory so far proposed
presuppose enough mathematics to develop the theory of unitary transformations of the
state-vectors in certain abstract spaces (Hilbert spaces and Fock spaces).
Re (2): [“These issues naturally prompt the question of how much mathematics is
indispensable (and hence how much mathematics carries ontological commitment).”]
Well, I reject “and hence” root and branch. I have never said that only what is
indispensable to natural science carries “ontological commitment” (if you insist on
speaking that way). Nor do I think that only as much mathematics as is used in physics
“carries ontological commitment,” in the sense of being true under a realist interpretation.
In my view, once we have agreed that number theory and the real and complex analysis,
differential geometry, etc. that are used in physics are neither to be rejected as “fiction”
nor reinterpreted in a verificationist way, then I think it would be absurd to say, “Well, I
accept “ontological commitment,” in that sense, to sets of integers and even sets of reals,
but not to sets of sets of reals.” I don’t think that “set” is a notion that it makes sense to
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interpret in a realist way up to a certain rank  and a non-realist way at rank +1. But I
agree with Colyvan that the question how much mathematics is indispensable in physics
is an interesting and difficult one.
Re (3) [My alleged “confirmational holism”]: I have never claimed that
mathematics is “confirmed” by its applications in physics (although I argued in “What is
Mathematical Truth” that there is a sort of quasi-empirical confirmation of mathematical
conjectures within mathematics itself) What I claimed in “What is Mathematical Truth”
(and subsequently) is, I repeat, that a prima facie attractive positionrealism with
respect to the theoretical entities postulated by physics, combined with antirealism with
respect to mathematical entities and/or modalitiesdoesn’t work.
Re (4): [“[I]f a theory is confirmed by empirical findings, the whole theory is
confirmed. In particular, whatever mathematics is made use of in the theory is also
confirmed.”] I agree with “Parson’s concern that the obviousness of basic mathematical
statements is left unaccounted for by the Quinean picture.” That’s why I don’t think it at
all plausible to think that numbers are “intangible objects” whose existence we “confirm”
in the same way that we confirm the existence of, say, mesons. My argument was never
intended to be an “epistemology of mathematics.” If anything, it is a constraint on
epistemologies of mathematics from a scientific realist standpoint. (Recall that in “What
is Mathematical Truth” I said that there are reasons internal to mathematics itself for
concluding that mathematics is “true under some interpretation”).
Re (5): [“Maddy suggests that we should side with naturalism and not holism here.
Thus we should endorse the attitudes of working scientists who apparently do not believe
in all the entities posited by our best theories.”] I agree with Maddy that the actual
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attitudes of working scientists towards the components of well-confirmed theories vary
from belief, through tolerance, to outright rejection. But this does not invalidate my
“indispensability arguments,” because they were never offered as an account of which
“ontological commitments” are “confirmed.”xxii For my purposes, it suffices to point out
that there is no serious quantum field theorist who does not believe something like the
following: (1) there are such things as quantum mechanical “states” of physical systems.
Those states determine the probabilities with which various physical interactions have
various observable effects. (2) The “time-evolution” of physical systems (that is, the way
in which those “states” change over time) is governed (except possibly at times of a
quantum mechanical “collapse”xxiii) by the Dirac equation, or by an equation (possibly a
non-linear one) to which the Dirac equation is typically an extremely good
approximation. (3) In virtue of (1) and (2), present day quantum field theories (with their
various “representations”) are legitimately believed to be approximately correct
descriptions of the relevant physical systems. If one understands this approximate
correctness in a realist way, then an argument against “verificationism” in the philosophy
of mathematics, which I publishedxxiv in criticism of an unpublished remark of
Wittgenstein’sxxv still seems pretty good to me. In brief, the argument is that when a
mathematical physicist accepts a theory according to which the time-evolution of a
physical system obeys a differential equation or system of equations, she is committed
the claim that those equations have solutions (in real numbers, or, in certain cases, in
complex numbers) for each real value of the time parameter t. Suppose, now, that the
statement “the solution to such and such equations for the given value of t is in the
interval between r1 and r2” is true. If the physicist makes the mistake of thinking that
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(mathematical) truth is the same as provability by human beings, then she must construe
this statement as implying that it is possible to calculate the solution to the equations in
question for the specified value of t, and the solution will be found to lie in the interval
between r1 and r2. But, given present knowledgexxvi, this is something it is unsafe to
commit oneself to, even if the value of the magnitude in question does lie in that interval.
I repeat, my indispensability arguments address the question whether antirealism
with respect to mathematics is compatible with realism with respect to physics. That
scientists do not take seriously all of the “components of well confirmed theories” that
they accept does not refute my argument, unless what they “do not take seriously”
includes the whole mathematical apparatus of fundamental physics. And obviously it
doesn’t.
Re (6): [“Sober points out that all scientific theories employ a common
mathematical core. Thus, since there are no competing hypotheses, it is a mistake to think
that mathematics receives confirmational support from empirical evidence in the way
other scientific hypotheses do.”] I completely agree with Elliot Sober that it is a mistake
to think that mathematics receives confirmational support from empirical evidence in the
way other scientific hypotheses do.
Re (7): [“The two most important arguments against mathematical realism are the
epistemological problem for Platonism—how do we come by knowledge of causally inert
mathematical entities? and the indeterminacy problem for the reduction of numbers to
sets—if numbers are sets, which sets are they?”]. This objection illustrates both the way
in which it is assumed that any defense of realism in the philosophy of mathematics must
be a defense of “Platonism,” and the way in which the idea that there are “equivalent
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descriptions” in mathematicsan idea which, as mentioned earlier, was already defended
in “What is Mathematical Truth?”has been ignored for over thirty years. For my most
recent defense of that idea, see my Ethics without Ontology, especially chapters 2 and 3.
There I argue that the decision to identify the natural numbers with one omega-sequence
of sets rather than with another is simply a matter of conventionyet another instance of
“equivalent descriptions.”
Hartry Field’s Position
Hartry Field understands very well what my arguments were, and he attempts to
meet them in their own terms. What he tries to do is show that mathematical physics can
be restated in a way that avoids all quantification over abstract entities. For this purpose,
as is well known, he assumes that points (in space or space-time) are concrete particulars.
Although this assumption has been attacked, it does not seem illegitimate to me (whether
the assumption is correct as a matter of contingent physical fact is another question).
Then, rather than construe such a statement as:
(S1) The charge of O is r.
as asserting that a substitution instance of the open sentence “the charge of 1 is 2”
is truenamely, a substitution instance in which “1” is replaced by the name of a
physical object and “2”is replaced by the name of an abstract entity, a real
numberField will reformulate (S1) thus:
(S2) Charge (O, G)
where “Charge (x, y) is a two-place predicate which replaces the real number-
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valued function charge(x) of classical physics, and G is a suitable geometric object (e.g. a
pair of line-segments the ratio of whose lengths is r). Line-segments, are, of course,
construed as mereological sums (not sets!) of points, and a pair of line-segments is just
the mereological sum of the two line-segments. (I have not followed Field’s own
construction exactly, but this will give the philosophical “flavor” of that construction.) In
this way, all quantifications over sets and numbers become quantifications over
mereological sums of points, which are taken to be nominalistically acceptable
particulars. So sets and numbers are not indispensable in physics after all, as both Quine
and I assumed. Voila!
Well, I agree that, assuming the nominalistic acceptability of such geometrical
objects as line-segments and pairs of line-segments, and such properties/relations of linesegments as congruence and equal-ratios of lengths (in the case of two pairs of linesegments), Field has shown that much or perhaps even all of classical physics can avoid
any use of set theory at all. (A somewhat improved and clarified version of Field’s
mathematical claims is presented by John Burgess and Gideon Rosen in their book A
Subject with No Object.xxvii) My objection to Field’s proposed solution is that while it
may work for classical physics, it is not sufficiently robust; it probably cannot be adapted
to the post-classical physical theories that are today the live candidates for quantum
theories of gravity. Here I will have to be moderately technical.
Very roughly, my objection is that all theories of quantum gravity assume that
space-times can be superimposed. That means that space-time does not have a
determinate metric at all, and hence one cannot speak of determinate “straight lines”
(geodesics) at all, or even of curve-segments with equal or unequal lengths (or of two
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pairs of curve-segments whose lengths have the same ratio) unless future physics
abandons the idea that space-time itself is quantum mechanical (a point implicitly made
by Wheeler, Thorne and Misner in their classic book Gravitation.xxviii They famously
wrote that space-time is a “foam of worm-holes” at the quantum scale.)
In sum, if the theories of quantum gravity are right about this much: that our world
exists in a superposition of different space-times, then if someone says “consider a line
on a space-like hypersurface ,” what they say is literally meaningless: there is no such
thing as “one of the lines on the surface” (in fact there is no such thing as “the surface”!).
There is only a superposition of space-times.
If we go back to the sentence (S2) above:
(S2) Charge (O, G)
the problem is that the geometric object (the pair of line-segments) G is not an
object that makes any sense in quantum gravitation.
As everyone knows, we certainly do not know what a satisfactory theory of
quantum gravity will look like in any detail. But for the reasons just given, I don’t see
how, for example, string theory or quantum loop gravity could be “nominalized” à la
Field, nor how any other theory which lacks a classical space-time could be. For this
reason, I am skeptical that Field’s proposal can work. To reject theories that need more
set theory than Field allows would require us to ban precisely the theories or theorysketches that cutting edge physicists are working on.
A historical remark: that the whole mathematics of physics, including the calculus,
was just a part of geometry is not a new idea. This is something that Leibniz and Kant
regarded as obvious (in Kant’s case, because that is how Newton understood the
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calculus). In a sense, Field’s proposal is an attempted revival of the seventeenth and
eighteenth century view (in a more sophisticated form, of course). When I say the
proposal is not “robust,” what I mean is that it depends essentially on features of that
classical view that are no longer accepted, and that are not likely be features of future
fundamental physics.
Gideon Rosen’s Position
Gideon Rosen is not a nominalist, but he believes that neither the arguments for nor
the arguments against nominalism are rationally compelling. In a deep and beautiful
paper titled Nominalism, Naturalism, Epistemic Relativismxxix, he argues that both
nominalism and anti-nominalism are rationally permissible. But like both myself and
Hartry Field he believes that “it would be patently unreasonable for an informed citizen
of the modern world simply to reject modern science,” and so he goes on to say,xxx
This means that any case for the permissibility of the suspension of judgment about
the abstract must show how nominalism of this sort is compatible with taking
science seriously as a source of information about concrete nature. This is of course
the point of the familiar “nominalization” programs. When these programs are not
pitched as dubious hermeneutic proposals, they are best understood as attempts to
reconcile doubts about the literal truth of extant science with a policy of accepting
Platonistic theories as instruments for description and explanation. I shall not
discuss these programs in detail here. Instead I shall describe what I regard a simple
trick for nominalizing at a stroke any theory whatsoever. Given any theory T
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formulated in the usual mathematical terms, the trick returns a theory that does
without abstract objects, but which is nonetheless in a certain sense equivalent to
the original.xxxi
Rosen believes that “we” (and his “we” certainly includes scientific realists of my
ilk) “indulge in Platonistic discourse;xxxii we often believe what we say; we harbor no
secret reservations; and all this is rationally permissible by our lights. Indeed [Rosen
continues] let us make the further assumption that so far as we can see, the cumulative
scientific case for certain abstract entities is so utterly compelling that it would
unreasonably cautious to demur.”xxxiii
Rosen illustrates (what he argues to be) the possibility of an equally rational
nominalistic version of scienceone that might have been our version had “cultural
history had gone somewhat differently”xxxivby imagining a world called “Bedrock”:xxxv
That is how we are, but in Bedrock things are differentBedrocker science and
mathematics are indiscernible from ours. Their textbooks, their advanced teaching,
the transcripts of their laboratory conversations, and the rest are all thoroughly
Platonistic in the sense that the sentences they contain imply the existence of
abstract entities.
The difference is that in Bedrock they have reservations about this aspect of what
they saycarefully considered and fully articulate reservations. Bedrockers are
encouraged from early childhood on to suspect that only concrete things exist, and
that discourse about the abstract is at best a useful fiction. If you ask them how they
square this with what they say when they’re doing science, they smile as if they’ve
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heard the question a thousand times and deliver themselves of a speech along the
following lines.
(Because the speech is very long, I shall “set the stage” in my own words.)
Consider the following sentence:
(1) The number of Martian moons = 2
Now, define the concrete core of a world W to be the largest wholly concrete part
of W: the mereological sum of all the concrete objects that exist in W.
Let me now quote the Bedrockers’ words:xxxvi
Now suppose ours is a numberless world and that (1) is therefore false. If we were
concerned to speak the truth, we would never countenance its assertion. But the fact
is, we are rarely concerned to speak the truth. Our unhedged assertoric utterances
normally aspire to a weaker condition we call nominalistic adequacy. S is
nominalistically adequate iff the concrete core of the actual world is an exact
intrinsic duplicate of the concrete core of some world at which S is truethat is,
just in case things are in all concrete respects as if S were true.
The “trick” to which Rosen referred, the one that, given a theory that quantifies
over numbers and sets is supposed to return a theory that does without abstract entities,
but which is nonetheless in a certain sense equivalent to the original, is to do what the
Bedrockers do: replace the claim that, to use Rosen’s own example, quantum
electrodynamics is true, with the claim that quantum electrodynamics is nominalistically
adequate.
Have the Bedrockers really found a way to exhibit a “rationally permissible”
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substitute for any physical theory that quantifies over numbers and sets, including
quantum electrodynamics, as Rosen claims? I shall suggest two reasons in favor of a
negative answer to this question.
First, a reason for doubting that the Bedrocker’s “trick” has been really described
in a way that makes sense.
Both the notions on which the “trick” dependsnamely, “concrete object” and
“exact intrinsic duplicate”are difficult and perhaps impossible to make sense of in what
we take to be our best theoretical physics. Already in relativistic electromagnetic theory a
problem arises: are fields “concrete objects” or not? They have mass and carry energy
and momentum so the answer would seem to be “yes.” But what nominalistically
acceptable predicates can we use to describe them? Normally one describes them by
specifying the components of the field-vectors at points in space. If one undertakes to do
this nominalistically, then one undertakes to carry out a program like Hartry Field’sbut
the Bedrocker’s “trick” was supposed to bypass the need to do that.
As for “intrinsic duplicate”: if an intrinsic duplicate of, say, my brain (or an
electromagnetic field) is required to have all the properties of my brainincluding
properties described in the language of mathematical physicswhy should we agree that
a nominalist is entitled to any such notion? And if an intrinsic duplicate is only required
to “match” the possible entity it is a duplicate of with respect to nominalistically
acceptable predicates (“concrete respects”), then which predicates are they?
If the problem was bad in classical physics, it may well be hopeless in quantum
field theories, whether or not they include gravitation. Here is a dilemma I propose for
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Gideon Rosen to consider: are electrons “concrete” or not?
Assume Rosen answers “yes.” Then the following problem arises: quantum
physicists have long rejected the idea that particles in states which are “indeterminate” in
one or another respect (particles in a superposition of states in which a property has
different values) “really” possess a definite value of the indeterminate property only we
can’t know what it is; that way of looking at quantum mechanical indeterminacy literally
makes no sense in contemporary quantum mechanics, and in field theories in particular.
But among the properties that enter into superpositions are (1) the number of electrons in
the system; and (2) the number of protons in the system.
So suppose we have a state that is a superposition of a state in which the system
consists of N electrons and one in which it consists of M protons. (Such states are
perfectly possible.) Then if someone says “consider one of the electrons in the system,”
what they say is literally meaningless: there is no such thing as “one of the electrons in
the system,” there is only a system in a superposed state. There are definite expectation
values for different measurements, including measurements designed to find electrons
and measurements designed to find protons, and we can say what these expectation
values are, but before the measurement interaction there isn’t a definite number of
“electrons” and “protons” there. This means that strictly speaking there aren’t such
“objects” as electrons, in the “ontology” of the theory; what there are are quantized fields,
and talk of “electrons” is a useful way of talking of, for example, certain components of
vectors in the Fock space that we use to describe those fields. (Note: I am not saying
“electrons don’t exist;” I am saying that electron-talk is not object-talk).xxxvii If, in spite of
this, Rosen (or better, the Bedrocker) claims that electrons are “concrete,” what
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nominalistically acceptable predicates does he use to describe them? And how does he
formalize such a sentence as “the number of electrons in the box is indeterminate, but the
state is 1/2(two electrons in the box) + 1/2 (three electrons in the box)”? If the answer
is “We Bedrocker’s don’t recognize any such statement as truth-apt, but only as
nominalistically adequate,” then can the Bedrocker also explain what it means to say that
a” concrete” system can be an “intrinsic duplicate” of a system in a possible world in
which there are numbers, and in which that sentence is true? Again we face the question,
doesn’t the notion of an “intrinsic duplicate” presuppose some notion of a
nominalistically acceptable description of a “concrete object” and its “concrete respects”?
Now assume that Rosen answers “no,” electrons are not “concrete.” Short of
reverting to instrumentalism, what then are the “concrete objects” that quantum field
theory describes? Fields? But fields too can be superimposed.
I suspect that the “trick” Rosen describes depends, in fact, on assuming that all
possible things are good old fashioned “concrete objects,” which we know how to
describe in nominalistic language, or good old fashioned “abstract entities,” which are not
in spacetime, and which are causally inert. But the entities of modern physics are neither
of the above.
Second, a reason for doubting that the Bedrocker’s “trick” produces inferences
that the Bedrocker’s themselves have any reason to see as justified.
In “Explanation and Reference,”xxxviii a paper I wrote back when I was still battling
Logical Positivism, I gave an argument that I think can be easily adapted against
Bedrockism. Here is the original argument (pp. 210-211):
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When a realistically minded scientistaccepts a theory, he accepts it as true
(or probably true, or approximately-true, or probably approximately-true). Since he
also accepts logic, he knows that certain moves preserve truth. For example, if he
accepts a theory T1 as true and he accepts a theory T2 as true, then he knows that
T1˄T2 is also true, by logic, and so he accepts T1˄T2. If we talk about probability,
we have to say that if T1 is very highly probably true and T2 is is very highly
probably true than the conjunction T1˄T2 is also highly probable (though not as
highly as the conjuncts separately), provided that T1 is not negatively relevant to T2
i.e. provided that T2 is not only highly probable on the evidence, but also no less
probable on the assumption of T1 (this is a judgment that must be made on the basis
of what T1 says and background knowledge, of course). If we talk about
approximate-truth, then we have to say that the approximations probably involved
in T1 and T2 need to be compatible for us to pass from the approximate-truth of T1
and T2 to the approximate-truth of their conjunction. None of these matters is at all
deep, from a realist point of view. But even if we confine ourselves to the simplest
case, the case in which we can neglect the chances of error and the presence of
approximations, and treat the acceptance of T1 and T2 as simply the acceptance of
them as true, I want to suggest that the move from this acceptance to the acceptance
of their conjunction is one to which one is not entitled on positivist philosophy of
science. One of the simplest moves that scientists daily make, a move they make as
a matter of propositional logic, a move which is central if scientific inquiry is to
have any cumulative character at all, is arbitrary if positivist philosophy of science
is right.
20
As I went on to explain:xxxix
The difficulty is very simple. Acceptance of T1, for a positivist, means acceptance
of T1 as leading to successful predictions (i.e. all observation sentences which are
theorems of T1 are true). Similarly, the acceptance of T2 means acceptance of T2
as leading to successful predictions. But from the fact that T1 leads to successful
predictions and the fact that T2 leads to successful predictions it does not follow at
all that the conjunction T1˄T2 leads to successful predictions. The difficulty, in a
nutshell, is that the predicate which plays the role of truththe predicate “leads to
successful predictions”does not have the properties of truth. The positivist may
teach in his philosophy class that acceptance of a scientific theory is acceptance of
it as “simple and leading to true predictions,” and then go out and do scienceby
verifying theories T1 and T2, conjoining theories which have previously been
verified, etc.but then there is just as great a discrepancy between what he teaches
in his philosophy seminar and his practice as there was between Berkeley’s
teaching that the world consisted of spirits and their ideas and continuing in
practice to daily rely on the material object conceptual system.
The relevance to Bedrock is this: the Bedrockers say “our unhedged assertoric
utterances normally aspire to a weaker condition [than truth] we call nominalistic
adequacy.” But the predicate “is nominalistically adequate” no more has the properties of
truth than the positivists’ “leads to true predictions” (or “is simple and leads to true
predictions”) does. Whenever their scientists employ one of the simplest truth-functional
inferences there is, conjunction-introduction, they make a move to which they are not
21
entitled. If they really “aspire to” nominalistic adequacy and not truth, then why do they
constantly make inferences which lead from true premises to true conclusions, but not, in
general, from nominalistically adequate premises to nominalistically adequate
conclusions (assuming the difficulties I have raised about even defining ““nominalistic
adequacy” can be met)?
Third, a more traditional sort of objection.
Suppose we define a correct explanation to be which, in addition to being
acceptable as an explanation by the standards both we and the Bedrockers are assumed to
accept, possesses true premises (and, of course, a true conclusion). A question I should
like to put to the Bedrockers is this: “You tell me that in science you don’t aspire to what
I just called correct explanations, you say that you only have the “weaker” aspiration of
finding nominalistically adequate explanations. Moreover, all of the major explanations
you and we accept are fictions by your lights. Very well then, do you have any idea what
the correct explanation of, say, the orbits of the planets is, given that the one taught in
your text books has false premises?”
What are the Bedrockers supposed to say? The notion of “correct explanation”
doesn’t involve any concepts they don’t have. So why aren’t they interested in correct
explanations? Unlike Rosen, I think that disinterest in correct explanation is prima facie a
rational shortcoming, and nothing in his brilliant philosophical fiction about Bedrock
exonerates the Bedrockers from criticism here. To quote one of my former selves, in the
light of the last two objections, I am inclined to say that the Bedrocker’s epistemology
makes the success of science a miracle.xl
22
Appendix
As I promised in the foregoing essay, in this Appendix I will argue briefly that if
my arguments are accepted, not only intuitionism and nominalism need to be rejected as
philosophies of mathematics, but also the idea that “predicative” mathematics is all that
science needs.
Recall that I began by saying that my answer to the objection that we do not yet
have the “true” physical theory is that we should regard each of the rival theories as a
candidate for truth or approximate truth, and that any philosophy of mathematics that
would be inconsistent with so regarding them should be rejected. (This is what I mean by
“scientific realism.”)
An example of such a view is Wittgenstein’s view (which he did not publish) that
the notion of a real number is vague (“there is no set of irrational numbers”).xli Classical
physics assumes that there is a point corresponding to every triple of real numbers
[alternatively, if you prefer an ontology of regions, a sphere the coordinates of whose
center are any given triple of real numbers]. If we accepted that there is no determinate
totality of real numbers, as Wittgenstein urged, then this important claimthe claim that
the geometrical continuum is isomorphic to the analytic continuumwould become
inexpressible.
Of course it is often said that the continuum is only an “idealization.” This may
indeed turn out to be true for physical reasonsindeed, in the preceding essay I argued
that it has, in a sense, turned out to be true not because spacetime has turned out to be
23
discrete, but because we live in a world which features a superposition of
spacetimesbut a scientific realist should insist that the issue not be prejudged by
adopting a finitist philosophy of mathematics. Philosophers should not be in the business
of telling physicists what space-time “really” is, or what it is “meaningless” to say it is.
Let us turn now to the claim that predicative set theory is adequate to the needs of
physics. I would not dare to challenge Solomon Feferman’s claim (in, e.g. the papers
collected in In the Light of Logicxlii to have shown that the theorems that are needed in the
“applications” of physical science can all be derived within predicative mathematics.
What I am skeptical about is whether the predicative mathematician can answer the
difficulty that I just raised as an objection to Wittgenstein’s finitism. The statement “there
is a point corresponding to every triple of real numbers” [alternatively, “there is a sphere
the coordinates of whose center are any given triple of real numbers”] is not, as far as I
can see, expressible without quantifying over all triples of real numberswhich is just
what predicative analysis forbids!
If I understand Feferman correctly, he would recommend that the physicist replace
the statement I just mentioned with a schema to the effect that “there is a point
corresponding to every triple of real numbers of order ,” where  is a “dummy letter”
for any predicatively attainable ordinal. In other words, the statement there is a point
corresponding to every triple of real numbers of order “ might be asserted with what in
the context of Principia Mathematics has long been called “typical ambiguity”except
that the ambiguity here would be one of “orders” of predicativity, not types. Indeed, in
the context of set theories which have either types or orders of predicativity or both, the
device of “typical ambiguity” is familiar from the time of Principia Mathematica. But it
24
is a highly unnatural device in the context of an empirical theory. What could the
justification for such a schema be, except that one believes (but thinks for philosophical
reasons that one is not allowed to say) that every triple of real numbers is the coordinate
triple of a point in space [or the center of a unit sphere in space]?xliii
Of course one could (and many do) wax skeptical, not on physical but on
supposedly “conceptual” grounds, about whether physics “really needs” to talk about
every point in space or every region in space; but I have difficulty in seeing the difference
between such skepticism and outright “instrumentalism.” Indeed, if adequacy for
“applications” just means adequacy for testable predictions, then the view that that is all
that physics is after is precisely the denial of what I called “scientific realism.”
But what of the quantum theories of gravitation that I spoke about in the preceding
essay? It is true that here we lose the notion of points and spheres as determinate objects
in physical spacetime, but if anything it is even easier to show that the world view of
quantum mechanics needs all the real numbersand, if fact, we don’t have to talk about
quantum gravity at all to this. The world view of quantum mechanics is that there is a
definite totality of possible “states” of a physical systemthese are what the “statevectors” represent. And if it is even physically possible that future time is infinite, then it
is easy to show that there are as many distinct states of a very simple system as there are
real numbers (I will spare you the details).
9. Indispensability arguments in the philosophy of mathematics
i
Mark Colyvan, “Indispensability Arguments in the Philosophy of Mathematics,” in E.N.
Zalta, ed., The Stanford Encyclopedia of Philosophy (Fall 2004 Edition), URL =
25
<http://Plato.stanford.edu/archives/fall2004/entries/mathphil-indis/>. Colyvan is also the
author of The Indispensability of Mathematics (Oxford: Oxford University Press, 2001).
ii
The author of this entry, Mark Colyvan, is referring to W.V. Quine, “Carnap and
Logical Truth,” reprinted in The Ways of Paradox and Other Essays, revised edition
(Cambridge, Mass.: Harvard University Press, 1976), 107-132 and in Paul Benacerraf
and Hilary Putnam, eds., Reading in the Philosophy of Mathematics (Cambridge:
Cambridge University Press,1983), 355-376; W.V. Quine, “On What There Is,” Review
of Metaphysics, 2 (1948): 21-38; reprinted in From a Logical Point of View (, Cambridge,
Mass.: Harvard University Press, 19802), 1-19; W.V. Quine, “Two Dogmas of
Empiricism,” Philosophical Review, 60, 1 (January 1951): 20-43; reprinted in his From a
Logical Point of View (Cambridge, Mass.: Harvard University Press, 1961), 20-46; W. V.
Quine, “Things and Their Place in Theories,” in Theories and Things (Cambridge, Mass.:
Harvard University Press, 1981), 1-23; W.V. Quine, “Success and Limits of
Mathematization,” in Theories and Things (Cambridge, Mass.: Harvard University Press,
1981), 148-155.
iii
Colyvan is referring to Hilary Putnam, “What is Mathematical Truth,” Historia
Mathematica, 2 (1975): 529-543; reprinted in Philosophical Papers, vol. 1: Mathematics,
Matter and Method (Cambridge: Cambridge University Press, 19792), 60-78; and Hilary
Putnam, “Philosophy of Logic” (1979); reprinted in the same volume, 323-357.
iv
Putnam, “Philosophy of Logic,” 347.
v
Philosophy of Logic (New York: Harper and Row, 1971).
26
I proposed such an interpretation of mathematics in “Mathematics Without
vi
Foundations,” cited in a previous note. For a detailed development, see Geoffrey
Hellman, Mathematics Without Numbers (Oxford: Oxford University Press, 1989).
vii
“What is Mathematical Truth?,” 72.
viii
E.g., Quine wrote “[The words ‘five’ and ‘twelve’] name two intangible objects,
numbers, which are sizes of sets of apples and the like.” Theories and Things
(Cambridge, Mass.: Harvard University Press, 1990), 149.
For the reasons that I say “in a sense” see my “The Greatest Logical Positivist,” in
ix
Realism with a Human Face, ed. by James Conant (Cambridge, Mass.: Harvard
University Press, 1990), 268-277.
x
See chapters 25 and 26 here. Both of these chapters are critical of some of
Wittgenstein’s unpublished remarks on mathematics.
xi
Quine, “Two Dogmas of Empiricism,” 41. What Quine actually wrote is that “Our
statements about the external world face the tribunal of sense experience not individually
but only as a corporate body.” This doesn’t mention (1) “theories” or (2) “confirmation.”
When asked how exactly we go about judging statements about the external world in the
light of experience, Quine’s famous advice was to “settle for psychology.” Quine was
endorsing a Duhemian thesis in “Two Dogmas,” not propounding a claim about the logic
of “confirmation.”
xii
In support of this Colyvan cites Quine, “Carnap and Logical Truth,” 120-122.
xiii
No such claim appears in “What is Mathematical Truth.”
xiv
Proceedings of the Aristotelian Society, 80 (1979-1980): 145-168.
xv
(New York: Oxford University Press, 1984).
27
Penelope Maddy, “Indispensability and Practice,” Journal of Philosophy, 89, 6 (1992):
xvi
275-289; quotation from p. 280.
Paul Benacerraf, “Mathematical Truth,” Journal of Philosophy, 70 (1973): 661-679;
xvii
reprinted in Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics:
Selected Readings (Cambridge: Cambridge University Press 19832), 403-420 and in
W.D. Hart, ed., The Philosophy of Mathematics (Oxford: Oxford University Press, 1996),
14-30.
xviii
Paul Benacerraf, “What Numbers Could Not Be,” The Philosophical Review, 74
(1965): 47-73; reprinted in Benacerraf and Putnam, eds., Philosophy of Mathematics,
272-294.
For a detailed discussion of Quine’s ontological views see my Ethics without Ontology
xix
(Cambridge, Mass.: Harvard University Press, 2004), especially chapters 2, 3, and 4.
xx
See “Craig’s Theorem” in my Mathematics, Matter and Method, 228-236.
xxi
One such argumentthe one in “Explanation and Reference”is quoted later in this
essay.
xxii
I criticize Quine’s “ontological commitment” talk in Chapter 4 of Ethics without
Ontology.
xxiii
See my, “A Philosopher Looks at Quantum Mechanics (Again),” British Journal for
the Philosophy of Science, 56, 4 (Dec. 2005): 615-634; reprinted here as chapter 5.
xxiv
See chapter 24 here. That argument can also be found in full in chapter 10 of this
volume, in the section titled “Where Wittgenstein Went Wrong.”
xxv
Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, ed. by G.H. von
Wright, Rush Rhees, G.E.M. Anscombe (Oxford: Blackwell, 1978), part V, §34.
28
xxvi
For the reasons that, given present knowledge, the truth of such a statement probably
does not in all cases imply its provability by a calculation human beings could actually
carry out, see Chapter 10 of this volume, the section titled “Where Wittgenstein Went
Wrong.”
xxvii
John Burgess and Gideon Rosen, A Subject with No Object Object; Strategies for
Nominalistic Interpretation of Mathematics (Oxford: Oxford University Press, 1997).
xxviii
C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (San Francisco: W. H.
Freeman, 1972).
Gideon Rosen, “Nominalism, Naturalism, Epistemic Relativism,” Philosophical
xxix
Perspectives, 15, Metaphysics (2001): 69-91.
xxx
Ibid., 73.
xxxi
However, Rosen is not himself a nominalist. He adds (pp. 73-4), “Before I describe
the trick, I should emphasize that I do not endorse the novel theories it delivers. The
nominalistic version of, say, quantum electrodynamics is in some sense a different theory
from standard QED. But it is not a better, more acceptable, theory. The nominalistic
version is of interest, not as an account of what science actually says, and certainly not as
an account of what it ought to be saying, but rather as an account of what science might
have said, and might have been justified in saying had cultural history gone somewhat
differently.”
xxxii
“Platonistic discourse” is Rosen’s term for discourse that quantifies over numbers
and sets.
xxxiii
Rosen, “Nominalism, Naturalism, Epistemic Relativism,” 74.
xxxiv
Ibid., 73.
29
xxxv
Ibid., 74.
xxxvi
Ibid., 75.
xxxvii
This argument now [April 5, 2011] seems bad to me, because it totally ignores the
Bohm and GRW interpretations that I myself argue we should take serious in chapters 5
and 6. In those interpretations particles do have positions, or (in the case of GRW) at
certain moments (when there is a “collapse”) they do. A better argument would have
been to point out that to “nominalize” quantum mechanics one would have to nominalize
the mathematical notion of probability, because an “intrinsic duplicate” of a quantum
mechanical world obviously has to have the same probabilities of nominalisticallydescribable occurrences (trajectories, in the case of a Bohmian world (if they can be
described nominalistically), and “flashes” in the case of a GRW world with a
“flash”ontology; but how can the Bederocker say that without talking mathematics? I
should also have pointed out that Rosen’s interpretation is not nominalistic in my sense; it
is, rather, a somewhat vague modal logical interpretation.
xxxviii
“Explanation and Reference” was first published in Glenn Pearce and Patrick
Maynard, eds., Conceptual Change (Dordrecht: Reidel, 1973), and reprinted in Mind,
Language and Reality, 196-214. I am quoting from the latter.
xxxix
xl
Ibid., 210-211.
In “What is Mathematical Truth” I wrote “The positive argument for realism is that it is
the only philosophy that doesn’t make the success of science a miracle” (p. 73). As an
argument against a sophisticated antirealism such as Dummett’s, or Crispin Wright’s, or
my own former “internal realism,” this was not a good argumentindeed, as I pointed
out in Reason, Truth and History (Cambridge, Mass.: Cambridge University Press, 1981),
30
a sophisticated antirealist need not deny that “terms in a mature science typically refer
and theories accepted in a mature science are typically approximately true”this was my
characterization of “scientific realism.” But as an argument against positions such as
logical positivism, van Fraassen’s “constructive empiricism,” and Bedrocker nominalism
I still believe it.
xli
Wittgenstein wrote, “It might be said, besides the rational points there are diverse
systems of irrational points to be found in the number line. There is no system of
-system, no ‘set of irrational numbers’ of higherorder infinity” (Ludwig Wittgenstein, Remarks on the Foundations of Mathematics,
edited by G.H. von Wright, R. Rhees and G.E.M. Anscombe [Cambridge, MA: MIT
Press, 1991], §33). In the 1956 edition, this material was in (what was then) Appendix II
of Part I. The present Part II includes a few remarks that were left out of the 1956 edition,
and, in addition, “the arrangement of sentences and paragraphs into numbered Remarks
corresponds to the original text (which was not wholly the case in the 1956 edition).”
(The “Editors Preface to the Revised Edition”, 31). The numbers of the Remarks are,
therefore, different from those in the 1956 edition, as is the order of some of the
Remarks.
xlii
Solomon Feferman, In the Light of Logic (Oxford: Oxford University Press, 1998).
xliii
One might also ask how a physicist could be expected to discover such a schema? By
first forming the classical hypothesis, and then weakening it to fit the predicative
straightjacket?
31