A Poorly Structured, Slightly Rushed Essay on the "Indispensability Argument"

1.      Outline the Quine-Putnam indispensability argument and critically evaluate one line of attack on this argument.

The “Quine-Putnam indispensability argument” is the name given to the simplified synthesis of the similar pro-‘realist’ arguments made in the 1970s by two of the most important figures in 20^th Century philosophy – the great Harvard logician, Willard van Orman Quine, and the great Harvard philosopher-chameleon, Hilary Whitehall Putnam (1928-2016). It is perhaps the most important and widely discussed argument in the contemporary philosophy of mathematics.

In his Stanford Encyclopedia article, “Indispensability arguments in the Philosophy of Mathematics”, Mark Colyvan gives what is probably the most concise version of the argument:

Premise 1: We ought to have ontological commitment to all and only those entities indispensable to our best scientific theories.

Premise 2: Mathematical entities are indispensable to our best scientific theories.

Conclusion: We ought to have ontological commitment to mathematical entities (Colyvan, 2015 (1998), Section 1).[1]

In this essay, it is this first premise that I will be focussing on. My examination of this “Quinean Ontic Thesis” (as Colyvan calls it (1998)) will mainly centre on one particular line of attack that has been visited upon it: Penelope Maddy’s arguments from mathematical and scientific practice. I will argue that Maddy ultimately succeeds in her argument that Quine’s “confirmational holism” is incompatible with naturalism (even if my own views diverge from hers in several important ways).^[2] In mounting this case, I will be simultaneously evaluating what is probably the major response to Maddy’s argument: Mark Colyvan’s strong defence of indispensability (1998, 2001).  Ultimately, I will diverge from both Colyvan and Maddy: I will argue that Maddy’s attack does succeed in doing serious damage to the first premise, and yet I will claim that the truth-oriented (rather than ontological) versions of the indispensability argument, like Putnam’s version of the argument or Resnik’s “Pragmatic indispensability argument”, are not touched by this line of attack at all. I will thus conclude that science does give us good reason to believe that maths possess some kind of truth, but that the specifically ontological question is underdetermined by indispensability considerations.

In her 1992 paper “Indispensability and Practice”, Penelope Maddy mounts what I believe to be an ingenious case against Quine’s ontological holism.  Maddy argues that if one takes seriously the kind of commitment to “naturalism” that Quine espoused – “the claim that a philosopher can criticize scientific practice, but only on scientific grounds […] for good scientific reasons” (Maddy, 1992: 276) – Quine’s confirmational holism is itself untenable, since its overly idealised, philosophy-first rigidity ignores the way in which mathematics and science actually work.

Maddy first tries to undermine Quine’s reductionist ontology by highlighting the way in which his Ontic Thesis clashes with the behaviour of mathematicians. Although it may seem moot to draw on mathematical practice (given that Quine’s naturalism did not apply to mathematical practice), I think Maddy shows that Quine’s holism has sufficiently bizarre implications for mathematics to raise serious doubts about its reasonableness. Maddy’s basic argument in this section is that the ontological parsimony that Quine’s doctrine requires (an instance of his notorious proclivity for “desert landscapes”) forces Quine to make demarcations of mathematical ‘reality’ where mathematicians see none. Though it is, of course, necessary for any ontological realist in mathematics to make demarcations of some kind or another (one can’t say that any object imagined by a mathematician exists), I think Maddy is successful in showing the unique strangeness of Quine’s partitioning.

Maddy’s best example of the odd mathematical consequences of Quine’s doctrine comes from a statement Quine made in reply to his fellow philosopher of mathematics, Charles Parsons: “I recognize indenumerable infinites only because they are forced on me by the simplest known systematizations of more welcome matters. Magnitudes in excess of such demands, e.g. M,, or inaccessible numbers, I look upon only as mathematical recreation and without ontological rights” (Quine, 1984, 783). As Maddy points out, mathematicians working in set theory do not regard Quine’s dividing point as in any way special – in fact, they would probably think the demarcation quite bizarre.

More broadly, Maddy argues that Quine’s system simply fails to accommodate a basic fact about mathematical practice: that pretty much all mathematicians are, as Shapiro would put it (1998), “working realists” who take no notice of any considerations resembling Quine’s.[3] As Maddy herself puts it, “Mathematicians believe the theorems of number theory and analysis not to the extent that they are useful in applications but insofar as they are provable from the appropriate axioms” (Maddy, 1992: 279).

As I will argue next paragraph, I believe Maddy’s arguments are quite strong here – but I think it’s also worth bearing in mind that Quine himself would not have been convinced by them. As a point of fact, the whole basis for Quine’s confirmational holism was his strictly scientific/empirical naturalism: the belief that science is the only arbiter of ontology, and that scientific theories are the only possible foundation for a rigorous ontology.[4] It is therefore evident that Quine would not have felt obliged to expand his conception of naturalism merely to accommodate in his ontology something as wishy-washy and unempirical as the attitudes of mathematicians. Of course, Maddy wants to say that Quine’s naturalism was polluted by confirmational holism: a true naturalist, in her view, wouldn’t privilege a philosophy-first ontological doctrine over the actual practice of mathematicians and scientists (Maddy, 1992). But Maddy’s problem is that – at least with regard to Quine’s views – she is making a circular argument. Her defence of the importance of mathematical practice relies directly on claims about the “success” of mathematics as a discipline in its own right, and yet Quine saw the indispensability of mathematics in science as the singular criterion for judging mathematical success.

Since I am more open to a pluralistic naturalism, however, I ultimately side more with Maddy. Like Maddy, I believe it is correct to say that mathematics has been successful as a discipline in its own right. My reason for this is the same one that Putnam gives in his seminal paper “What is Mathematical Truth?”: the remarkable “consistency and fertility of classical mathematics”, which is explicable only if the abstract ‘territory’ which mathematicians explore has at least some objective contours (Putnam, 1972: 73). It is for this reason that I also share Maddy’s belief that we cannot ignore mathematicians’ own views on what counts as legitimate mathematics and what does not.^[5] Most importantly, I believe, like Maddy, that there is something sufficiently strange about Quine’s demarcation of mathematics to worry that there is something wrong with Quine’s ontological approach. (There is also the problem that Eliot Sober (1994) raises: mathematics is not itself empirical but a priori, and the truth of mathematical theories does not rest or fall on results in science.) Ultimately, all of this makes me think that in judging mathematics on an empirical basis, Quine is making some kind of mistake.

As I mentioned earlier, Maddy also uses examples from scientific practice to repudiate the Quine Ontic Thesis. I believe that Maddy’s scientific-practice objections to the Quinean program are even more powerful than the mathematical-practice objections, since I think these really do succeed in showing that Quine’s commitment confirmational holism leads him to violate his more basic commitment to naturalism (at least in part).

Maddy makes her case for the non-naturalism of confirmational holism through a series of examples from scientific practice – both historical and current. Her first historical recruit is the attitude of late 19^th Century scientists towards atoms. Maddy briefly describes the history: the postulated entity known as the ‘atom’ had become indispensable to the field of chemistry from about 1860 – enabling increasingly powerful predictions and calculations – and yet a large section of the late-19^th Century scientific community doubted that this entity really existed (Maddy, 1992).  Since atoms were invisible and mysterious, and since they played no role in the physics of the day (widely regarded to be more solid than the chemistry), many of the most eminent scientists of the day – including men like Ostwald and Poincaré – believed that atoms were just useful calculating devices (Colyvan, 1998: 42). The moral Maddy derives from this historical period is not hard to figure out: since confirmational holism clashes so violently with the behaviour of these scientists, confirmational holism can’t really be a ‘naturalistic’ principle.

It’s worth emphasising that Maddy’s ultimate claim here is not the simpleminded inference that we should accept any ontological view expressed by a few important scientists. Instead, she claims simply that Quine’s purist system of ontology cannot be regarded as the one measure of rationality – especially not if we claim to be scientific naturalists. In her own words, a genuine naturalist “must allow a distinction to be drawn between parts of a theory that are true and parts that are merely useful” (Maddy, 1992: 281).

As before, it is worth nothing that Quine himself would probably not have been swayed by this argument. Thankfully, we can get some sense of how Quine might have replied in the response of Quine’s mouthpiece, Mark Colyvan, in his 1998 paper “In Defence of Indispensability”. Perhaps the most noteworthy thing about Colyvan’s argument in this paper is how much ground he concedes to Maddy. Indeed, Colyvan essentially concedes Maddy’s central argument: that confirmational holism is often at odds with strict scientific naturalism. Colyvan’s main counter-claim, however, is that Maddy’s strict scientific naturalism is excessive, and does not reflect Quine’s own naturalism. This allows Colyvan to say that the contradiction Maddy supposedly identifies between confirmational holism and naturalism is really a problem she has created for herself (Colyvan, 1998: 46). On Colyvan’s view (and, he believes, Quine’s), accepting naturalism and disavowing “first philosophy” does not entail disavowing all normative philosophical systems.

When it comes to Maddy’s atomic example, Colyvan is again quite concessionary. Instead of coming to a conclusion as to what a good Quinean naturalist should say about the attitude of scientists like Poincare and Ostwald towards atoms, he merely makes three suggestions: i) that they “were making a mistake” (Colyvan’s own preference); ii) that “the controversy over atomic theory at the time gives us good reason to think that prior to 1913 chemistry/atomic theory was in a crisis period and thus the Quinean could suspend judgment on the ontological commitments of the theory”; or iii) that “given the evidence at the time it would be unwise to give total commitment to either the existence or the non-existence of atoms—some degree of belief strictly between zero and one would be appropriate” (Colyvan, 1998: 49-50). Colyvan thinks the existence of these options should be sufficient to dispel Maddy’s reservations (Colyvan, 1998: 50).

In my view, Colyvan’s response is really too concessionary here. In fact, I suspect Maddy herself would agree with the last option. This is a problem, I think, because it means that Colyvan is diluting confirmational holism to the point that it’s almost not a substantive doctrine at all (it’s almost verging on commonsense). If Quine didn’t really believe we should be ontologically committed to all and only those entities indispensable to our best scientific theories – if he instead believed that we should be ontologically committed to all and only those entities indispensable to our best scientific theories proportional to various other considerations, such as aesthetic ones – then how could he have made decisive judgments on ontological questions? Surely the point of confirmational holism is to make ontological questions clear-cut and precise – to act as a kind of ontological guillotine. If it is made vague, then it is (presumably) made useless.   

The other way in which Maddy uses scientific practice in “Indispensability and Practice” to make her case for the irreconcilability of naturalism and confirmational holism is by giving examples of applications of mathematics that are obviously not deserving of ontological commitment (on account of the obviously false assumptions they employ). Maddy’s main example of this is the average “freshman physics text”, which is “littered with applications of mathematics that are expressly understood not to be literally true: e.g., the analysis of water waves by assuming the water to be infinitely deep or the treatment of matter as continuous in fluid dynamics or the representation of energy as a continuously varying quantity” (Maddy, 1992: 281). The indispensability of such unrealistic mathematical tools is not in dispute, and yet only an insane person would think that that fact is sufficient to ontologically commit us to the mathematical objects required by these mathematical tools.

I think Maddy’s attack here succeeds in inflicting yet more damage on Quine’s confirmational holism, though I also believe that Maddy misses an opportunity in this passage to invoke the plentiful applications of unrealistic mathematics in economics – a field which did (surprisingly) fall under Quine’s very broad conception of science (Quine, 1995: 49). As anyone who has read a freshman economics text realises, neoclassical economics uses highly simplified, toy models of the real world, and typically eschews dynamic modelling – instead making totally unrealistic assumptions about tendencies towards “equilibrium”. And just as with physics, nobody would say that we should be ontologically committed to the mathematical entities needed for these models.

The way Colyvan replies to this latter class of Maddy examples in “In Defence of Indispensability” is, again, not to deny their validity, but simply to claim that scientists often do worry about the ontology of the mathematics they are using – thus proving that it is certainly not anything goes. Colyvan’s example of useful maths that was viewed with suspicion by scientists is the Dirac delta function. This strange mathematical object proved very useful in quantum mechanics, and yet, as Colyvan notes, it “attracted much criticism” (Colyvan, 1998: 52).

It’s worth nothing that Colyvan is once again making massive concessions here. He has strayed so far from the literal Quine Ontic Thesis and the literal doctrine of confirmational holism that he is looking for evidence of scientists not taking confirmational holism too seriously (by being suspicious of the ontology of indispensable mathematics) in order to show that some enervated version of confirmational holism is still possible. It’s a strange position, in my opinion.

All in all, my conclusion about Maddy’s attack on the first premise of the Quinean indispensability argument is that it is broadly successful. As I’ve made clear, I think Maddy’s success is even evident in Mark Colyvan’s attempt to defend the premise, since Colyvan’s defence involves diluting the doctrine of confirmational holism to such an extent that it basically loses all meaning. Though I haven’t mentioned this philosopher so far, I also think that Maddy’s critique is boosted by some of the observations of Stephen Yablo (1998) about the ineliminably metaphorical nature of a lot of the language in our best scientific theories. As Yablo notes, Quine never adequately acknowledged the importance of figurative language in our scientific discourse, and this is reflected in the rigidity of the doctrine of confirmational holism.[6]

As I suggested in my introduction, I certainly don’t think that there isn’t some salvageable form of the indispensability argument. Indeed, I think the truth-oriented versions of the argument are totally unassailable. One example of such an argument is Michael Resnik’s “Pragmatic Indispensability Argument”, laid out in his book Mathematics as a Science of Patterns. Resnik’s formulation doesn’t rely on confirmational holism, and thus doesn’t have a premise nearly as strong as the Quine Ontic Thesis. Instead, its two key premises are that “We are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true” and “We are justified in using science to explain and predict” (Resnik, 1997: 47). The idea is that science’s success only makes sense if the mathematics used in it is largely or wholly true.

I personally see no way of denying this argument. As I made clear in footnote 1, Putnam also made a similar truth-oriented argument, and was generally less rigid about ontology than Quine, holding no such doctrine as confirmational holism. This makes his indispensability arguments  more attractive, in my view.

Bibliography

Colyvan, Mark (1998). “In Defence of Indispensability”, Philosophia Mathematica, 6 (1): 39-62.

(2001). The Indispensability of Mathematics, Oxford University Press.

Maddy, Penelope (1992). “Indispensability and practice”, Journal of Philosophy, 89 (6): 275-289.

Putnam, Hilary (1975). “What is Mathematical Truth?”, in Mathematics, Matter and Method, Cambridge University Press 60--78.

Quine, Willard van Orman (1961). “On What There Is”, in Tim Crane & Katalin Farkas (eds.), From a Logical Point of View, Harvard University Press, 21-38.

(1995). From Stimulus to Science, Harvard University Press, Cambridge, Mass.

Resnik, Michael (1997). Mathematics as a Science of Patterns, Oxford University Press.

Shapiro, Stewart (1996). The Philosophy of Mathematics: Structure and Ontology, Oxford University Press.

Yablo, Stephen (1998). “Does Ontology Rest on a Mistake?”, Aristotelian Society Supplementary Volume, 72 (1): 229 - 283.

---

[1] It should be noted that this formulation is very Quine-biased, and doesn’t very well capture Putnam’s indispensability arguments. In fact, Putnam criticises Colyvan’s formulation directly in chapter 9 of the 2012 book Philosophy in an Age of Science: “From my point of view, Colyvan’s description of my argument(s) is far from right” (Putnam, 2014: 1).  Putnam’s objections to the formulation are several. Firstly, it doesn’t capture the important distinctions between his and Quine’s views: “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” […] “a distinction that Quine nowhere draws” (Putnam, 2014: 2).  Secondly, Putnam points out that he never argued for anything as straightforward as “ontological commitment” to objects or entities (and was never a “Platonist” per se): in both “What Is Mathematical Truth?” and “Mathematics Without Foundations”, he “said that set theory did not have to be interpreted Platonistically” and that “modal-logical mathematics (i.e. mathematics which takes mathematical possibility as primitive and not abstract entities of any kind) and mathematics which takes sets as primitive are “equivalent descriptions”” (Putnam, 2014: 2). Whereas Putnam’s “indispensability” argument was – as he puts it – “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” (numbers and sets)” (Putnam, 2014: 2). Finally, Putnam never subscribed to the “and only” part of the first premise (Putnam, 2014: 3).

Despite all these problems with the formulation, it is Colyvan’s exposition of the argument that I will be attacking in my essay. One good reason for sticking with this Quine-biased formulation is that most of the literature on the ‘Quine-Putnam indispensability argument’ is also Quine-biased.

[2] ‘Confirmational holism’ is a slightly misleading name for Quine’s mechanical system of ontology. In essence, it refers to Quine’s claim, famously expressed in his essay “On What There Is”, that a scientific theory  is “committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true” (Quine, 1943: 33).

The key feature of this doctrine is its rigidity. According to Quine, if a new grand theory of physics were to come along that bettered our current Relativity/quantum mechanics synthesis in terms of explanatory accuracy, predictive power, ontological parsimony (and so on), we should immediately abandon all ontological commitments to entities that aren’t necessary for the truth of the affirmations of the new theory and immediately adopt ontological commitments to all and only the entities that are. (It’s clear why I called it a “mechanical system of ontology”.)

[3] Hilbert was unusual (and his program failed anyway).

[4] We will later see that Colyvan, a neo-Quinean, also defends this view (1998, 2001).

[5] Note, however, that I am not endorsing Maddy’s own ontology. I have as yet said nothing about whether any mathematical “objects” “exist”. 

[6] As Yablo writes in “Does Ontology Rest on a Mistake?”, “Intentional attributions, subjunctive conditionals, and so on are said [by Quine] to have ‘no place in an austere canonical notation for science’ suitable for ‘limning the true and ultimate structure of reality’”.