Sunday, 20 January 2013

The Quine-Putnam Indispensability Argument

Many years ago I finished my PhD, entitled "The Mathematicization of Nature" (1998, LSE), in which I discussed the applicability of mathematics, the Quine-Putnam indispensability argument and considered a number of nominalist responses to it, in the end rejecting them all. The monograph Burgess & Rosen 1997, A Subject with No Object, had appeared a year earlier. At the time, I'd considered the issue definitively settled. And so I decided not to bother publishing anything in the area, as it would be pointless. (I did publish Ch. 5, which was about truth theories and deflationism.)

Jeez was I wrong! In the last fourteen years, the debate about the indispensability argument has continued, taking off in many different directions. And I'm pretty baffled at the whole thing. Even the formulation of the Indispensability Argument often given is incorrect, as far as I can see. So, here is mine, and I think it is reasonably faithful to the intentions of both Quine and Putnam.

1. Nominalism

Nominalism (in mathematics) is the claim that there are no numbers, sets, functions, and so on. (In addition, nominalism normally implies also that there are no syntactical types: i.e., finite sequences of symbols. Consequently there is a problem for nominalism at the level of syntax, a problem discussed long ago by Quine & Goodman 1947, "Steps Toward a Constructive Nominalism".) In particular, there are no mixed sets and no mixed functions. A mixed set is a set of non-mathematical entities, and a mixed function is a function whose domain or range includes some non-mathematical entities.

However, modern science is up-to-its-neck in mixed sets and functions. All the various quantities invoked in science are mixed functions. Laws of nature express properties of such mixed functions, and express relations between them. A differential equation in physics usually expresses some property of some mixed function(s). For example, it might say that a function defined on time instants has a certain property.

2. The Quine-Putnam Indispensability Argument 

Quine and Putnam both gave versions of an argument, which I formulate like this:
The Quine-Putnam Indispensability Argument
(1) Mathematicized theories are inconsistent with nominalism.
(2) Our best scientific theories are mathematicized.
(C) So, if one accepts our best scientific theories, one must reject nominalism. 
(The name "Quine-Putnam Indispensability Argument" derives, I believe, from Hartry Field.)

The argument for the first premise (1) is based on the following kind of example. Maxwell’s Laws include the mathematicized law:
At any spacetime point $p$, $(\underline{\nabla} \cdot \underline{B})(p) = 0$.
This is often abbreviated "$(\underline{\nabla} \cdot \underline{B}) = 0$", but it is clear that quantification over spacetime points is implicitly intended.

Since $\underline{B}$ is a vector field on spacetime, it is a mixed function, whose domain is spacetime, and whose range is some vector space (one that is isomorphic to $\mathbb{R}^3$). If nominalism is true, it follows that $\underline{B}$ does not exist, and therefore that Maxwell's Law, "$(\underline{\nabla} \cdot \underline{B}) = 0$", is false. (A slightly fancier version of this would refer instead to the electromagnetic field tensor $F_{ab}$, whose components unify the $\underline{B}$-field and the $\underline{E}$-field; but the considerations are more or less the same.) In general, if nominalism is true, then any such mathematicized theory is false. This establishes (1).

If this is right, then we have a major worry: this shows that a certain philosophical theory (nominalism) contradicts science. This is probably the central reason I am suspicious  of nominalism.

The argument for the second premise (2) requires one to compare our working mathematicized theories (Maxwell’s theory; Schroedinger equation; Einstein’s field equations; Yang-Mills gauge theories, etc.) with proposed nominalistic replacements. Having done this, one then concludes that either there are insuperable technical obstacles to the nominalization of such theories; or, though there may be, for certain mathematicized theories, nominalized replacements, even so, the mathematicized original is always a scientifically better theory, by scientific standards. (This is the sort of point emphasized by John Burgess, who semi-hemi-demi-jokingly suggested that nominalists might submit articles with their replacement theories to The Physical Review.)

So, our best scientific theories are mathematicized and are inconsistent with nominalism. Hence, if one accepts such theories, one must reject nominalism. This conclusion is epistemic only in a conditional sense. It simply says that one cannot have one’s cake and eat it. One cannot be a nominalist and a scientific realist.

3. Responses 

3.1 Rejecting (1): The rough idea is that mathematicized theories are consistent with nominalism. So, such theories may be true even though there are no mathematical entities. So, the magnetic field $\underline{B}$ doesn’t exist, but, even so, Maxwell’s Laws are true. This kind of view is advocated by Jody Azzouni (2004, Deflating Existential Consequence: A Case for Nominalism), but I'm not sure I quite understand it.

3.2 Rejecting (2): Our working scientific theories can be nominalized, and such theories are epistemically better. The betterness consists in the advantage that issues from the elimination of mathematicalia. This is essentially Hartry Field’s approach (Field 1980, Science Without Numbers).

3.3 Accepting, but living with, the conclusion: a nominalist might accept the Quine-Putnam argument, conceding the premises, but insist that one may “accept” mathematicized scientific theories in a weaker sense, which involves only accepting their nominalistic content. This is essentially Mary Leng’s and Joseph Melia's approach (Leng 2010, Mathematics and Reality; and Melia 2000, "Weaseling Aaway the Indispensability Argument" (Mind)).


  1. Thanks Jeff for your concise and clear summary. Could I ask, not having read Field, is the 'advantage that issues from the elimination of mathematicalia" basically that science then becomes nominalism friendly? and is the appeal of nominalism for the would be scientific realist partly explained by the worry that realism about abstracta and mathematicalia seem to make a thoroughgoing materialism impossible?

    apologies for any naiveté, not a professional philosopher and all that.

  2. Ad 3.1, I think the basic idea in it's negative form is simple enough. It is just a rejection of the Quinean criterion of ontological commitment. One can quantify-over without being-committed-to. This naturally raises a lot of questions about what the criteria for commitment *are*. I think Azzouni's answer is unsatisfying in this respect, but one can imagine how this project might go, e.g. it might have something to do with believer/speaker intentions.

  3. Joseph,

    "is the 'advantage that issues from the elimination of mathematicalia" basically that science then becomes nominalism friendly? and is the appeal of nominalism for the would be scientific realist partly explained by the worry that realism about abstracta and mathematicalia seem to make a thoroughgoing materialism impossible?"

    The advantage is ontological parsimony; so a nominalized scientific theory doesn't require the existence of abstract entities (like vector fields, or vectors). So one can defuse the epistemological problem of "access": how we "know" about abstract entities, given that they're non-causal.
    Field also highlights another advantage, namely that a nominalized theory (of the kind he gives in his 1980 monograph) explains the conventional role played by the mathematical aspects of usual scientific theories. Roughly, the mathematics is only increasing the conceptual simplicity of theories (which is why it is useful), and not contributing to its genuine physical content.


    1. Here is a contention and agreement. One extension of Quine's argument is that all forms of abstracta reduce to 'applications' which only have validity through pragmatic reference. Then not only does math get caught up in a vast contingency of conflating validity with usefulness or vice versa (a kind of Jacob's Ladder problem), but there is an appealing argument for the universalism of applications that may open mathematics to what you call 'mixed functions'.

      For example, if a set is not a universal set in terms of its contents, what is it saying about its usefulness? Although this may reduce adequately to a claim that a function is an application, it would do not not make assumptions about what this means---since 'application' suddenly may mean 'mathematics' to the mathemician---it does not mitigate arguments that 'other applications' could be equally useful. Perhaps this amounts to the claim that mathematicians are attracted to an 'illusory' usefulness much in the way that statisticians sometimes become poor economists.

      I find it appealing that mathematics may be 'just one form' of usefulness, and I think there is no implicit problem in widening the field of potential quasi-mathematical applications. The question is really one of standardization, once it is accepted---I think it is obvious to accept---that mathematics is a form of nominalism. And amongst other questions is the question of whether math has been 'synched' to real cognitive processes, or instead merely taps into strengths and weaknesses, proving things that are already true about the mind, yet remain trivial.

  4. Colin,

    Yes, in a nutshell, that sounds right.
    But I think Azzouni formulates matters *epistemically* (in terms of beliefs, etc.), whereas Quine's analysis is semantic, and concerns the existential implications of sentences. That is, if a sentence (e.g., a natural language sentence) $S$ is regimented as $\exists x Fx$, then $S$ implies that there are $F$s.
    There is wriggle room here at the "regimentation" stage; so Quine gives lots of examples of eliminating apparent existential implications in Word & Object.
    But not so much wriggle room, I think, at the semantic implication stage, because it's hard to see what $\exists x Fx$ could mean except that there are $F$. For the semantic clause defining $\exists x Fx$ is: it's true iff there are $F$s. I suppose one could have say a free logic, or a number of different quantifiers in operation, with inner and outer domains and whatnot. This is a sort of Graham Priest direction. But I think this isn't what Azzouni has in mind, and it's connected to speaker intentions and beliefs, as you say.

    So, as far as I understand it, Azzouni notion of "ontological commitment" is an epistemic notion, rather than Quine's semantic one.


    1. I think Quine's argument is syntactic, while his style is semantic.

  5. To be clear, my view of nominalism DOES support the view that nominalism can make statements, or prove entities, albeit these are not entities in the sense of mathematics. For example, the categorical-deductive sets {A-B:C-D, A-D:C-B} can refer to any appropriate object in space, although with some level of metaphoricalization, or at least some statement about its intended functionality.

    More about this in my book, The Dimensional Philosopher's Toolkit (2013, 2014), not to be confused with Baggini and Fosl's classic.

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