If Quine-Putnam's argument is (following the SEP):

  • (P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
  • (P2) Mathematical entities are indispensable to our best scientific theories.
  • (C) We ought to have ontological commitment to mathematical entities.

I am reading this as a a transfer argument, where the supposed existence of the entities indispensable to our scientific theories is transferred to the mathematical objects used in those scientific theories.

Assuming this is tenable, is there a possible attack on premise (P1) along the lines that our best scientific theories do not necessarily involve any ontological commitment, so that no ontological commitment would be required of mathematics even if they were indispensable to our best scientific theories?

To restrict the discussion to physics, physics could be argued to propose only descriptive models that "work" more or less well in some range of parameters but do not necessarily carry a commitment to the existence of what they model. For example, "elementary particles" as point particles of matter are now superseded by "fields", just because the fields picture has more explanatory power. But if physicists needed to abandon the fields picture in favor of another model that would be a more accurate description, they would presumably do so, rather than cling to "particles" or "fields" just because they would have made some sort of "ontological commitment" through these models (it is actually more complicated, Newtonian gravity is useful in some range of parameters, while Einstein's SR and GR are useful in other ranges of parameters, there is a "classical limit" to QM, etc etc.)

So, if physics itself was seen to not actually make any kind of "ontological commitment" but instead be concerned with the descriptive power of models only, there would be no "ontological commitment" in physics to transfer to mathematics, even if mathematics were somehow deemed "indispensable" to physics.

As this kind of attack on (P1) been advanced by any philosopher?

Addendum: a variant on this attack could possibly be conceived by asking what happened to the ontological commitment to obsolete objects. If physics used some mathematical object X in theory A that was then superseded by theory B which no longer used mathematical object X, would the ontological commitment conferred to mathematical object X by its indispensability in physical theory A have to be retracted?

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    P1 is a formulation of scientific realism. For an anti-realist, there is no point engaging with the indispensability argument at all: if one does not believe that even physics describes reality (at least, partially, approximately and fallibly) why bother with mathematics. To those who do engage, the target is P2.
    – Conifold
    Apr 22 at 23:43
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    Ontological commitment is not some permanent attachment to a theory. It just means a realist accepts as existing those things that their best theories quantify over. If your best theories quantify over electrons then electrons exist. If your best theories quantify over numbers then numbers exist. If you exchange one theory for a better one, your ontological commitment changes with it.
    – Bumble
    Apr 23 at 0:01
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    My take on this is that P1 is so obviously a matter of belief & choice at best that it isn't even worth much bothering. Lots of philosphers won't buy into it (e.g., Bas van Fraassen, Karl Pearson, Wittgenstein, Thomas Kuhn, Nancy Cartwright,...), but it is somewhat harder to find straight on attacks, as people frankly deal with more interesting things (said of course somewhat polemically, but ultimately this is how I see it). Apr 23 at 12:33
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    Different theories do indeed posit different objects and have differing ontologies. In fact, Quine is content to allow for Ontological Relativity (he has a paper of that name). We cannot be certain of a particular ontology, nor do we know we will converge to true objects. We are just sailing along in Neurath's boat progressing in the direction of better theories. It's the best we can do.
    – Bumble
    Apr 23 at 14:32
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    @JD Sure, then maybe we should just replace "ontological commitment" by "ontological fantasy of the day" in the philosophy books.
    – Frank
    Apr 23 at 23:45

4 Answers 4


This popular 'indispensability argument' is completely fallacious. For instance, it claims that "reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories". This is plainly false. It is well-known to all real experts in logic that very weak subsystems of second-order arithmetic suffice for all empirically backed real-world applications of mathematics to date.

In particular, our best scientific theories only rely on mathematics that can be carried out in such weak systems, apparently not needing reasoning about anything beyond arithmetical sets, and this can be handled easily by ACA0 or at most ACA (i.e. ACA0 plus full induction). So the popular notion that set theory is indispensable to science is just bogus.

More specifically, "mathematical entities" is ill-defined until you specify a foundational system and provide some kind of real-world or ontological interpretation, which Putnam utterly failed to do. In fact, ACA can be interpreted to have nothing to do with sets at all. On the surface, ACA is about ℕ and its arithmetical subsets. However, that is mere appearance. We could use the obvious interpretation of ℕ in terms of finite binary encodings in some specific physical medium with the appropriate operations on them, and we could interpret those 'subsets' as simply arithmetical formulae with one free variable. For any such set S = { x : x∈ℕ ∧ Q(x) }, the truth-value of "k∈S" for k∈ℕ is simply the truth-value of "Q(k)", which is already well-defined once you believe the meaningfulness of PA.

Therefore even if we accept (P1), if you think carefully you will see that (P2) is utterly meaningless. What is defensible is:

(P3) We can reasonably have ontological commitment to existence of a model of a foundational system whose theorems are indispensable to our best scientific theories.

(P4) The theorems of ACA0 are indispensable to our best scientific theories.

Conclusion: We can reasonably have ontological commitment to the existence of a model of ACA0. No problem. Almost all real logicians believe this.

  • The argument uses "mathematical entities" so as to leave open which foundational system is at play. So its seems a bit heavy handed to criticize Putnam for "failing" to specify a foundational system. Quine himself uses the argument to justify transfinite set theory, yes, but contemporary defenders of the argument need not commit themselves to this ( as you yourself note)
    – emesupap
    Apr 23 at 20:47
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    @Papuseme: I read this answer as a form of nominalism, in which it is argued that we only need to bother dealing with very simple mathematical objects (the natural numbers), and can therefore sidestep many of the traditional objections to nominalism. In that interpretation, the idea is not necessarily that Putnam is wrong, but rather that Putnam is trivial.
    – Kevin
    Apr 24 at 6:11

P1, P2 and C are all open to challenge. Take the electron. The term is really shorthand for 'that which causes certain types of detectable effects'. We can be committed to the existence of the effects, and we can be committed to their seeming to have a common cause, but the history of science is littered with ideas such as phlogiston, the ether, fluxions and so on that would once have been considered indispensable to the then best theories. It might be that in n years the idea of an electron will join all the other outdated terms. When you say that mathematical entities are indispensable, one might argue that they are just stand-ins representing the attributes of the 'real' entities, so they do not count as objects which demand an ontological commitment in themselves. So, the argument can be paraphrased another way:

Our theories assume the existence of entities whose fundamental nature is questionable.

We use mathematical symbols to represent the apparent properties of those entities.

Ergo we should be committed to the existence of mathematical objects.

I struggle to think of a less appropriate use of the word 'ergo'.


These terms are broad brushes on articulate positions, but...

if physics itself was seen to not actually make any kind of "ontological commitment" but instead be concerned with the descriptive power of models only

Ontological commitment as Quine entails it involves quantification over real things. Thus, for realists, physical theories are themselves committed to real entities. If one abandons that premise and instead moves away from realism to see physical theories are useful tools, fictions, or descriptions then one is engaged in an instrumentalist (and therefore anti-realist) interpretation:

In philosophy of science and in epistemology, instrumentalism is a methodological view that ideas are useful instruments, and that the worth of an idea is based on how effective it is in explaining and predicting phenomena. According to instrumentalists, a successful scientific theory reveals nothing known either true or false about nature's unobservable objects, properties or processes. Scientific theory is merely a tool whereby humans predict observations in a particular domain of nature by formulating laws, which state or summarize regularities, while theories themselves do not reveal supposedly hidden aspects of nature that somehow explain these laws. Instrumentalism is a perspective originally introduced by Pierre Duhem in 1906.[emphasis mine]

So, whereas the logical atomists presupposed that there was a straightforward correspondent theory of truth between propositional atoms and real things in the state of affairs in the world, an instrumentalist avoids that sort of commitment and might focus more instead on the coherence or the pragmatic grounding of truth.

As for the indispensability argument itself, nominalism rejects such a thesis in the same way that Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences meets incredulity and charges of psychological bias that invokes the counter-accuastion of the unreasonable ineffectiveness of mathematics. Hartry Field puts together one argument against indispensability, but there are broader attacks on the Platonic undercurrents of realism in mathematics such as those leveled by Brouwer and his intuitionism.

My believe is that philosophers of mathematics tend to asssert Platonic realism is powerfully intuitive, whereas those of us who are skeptical about such claims of facsimiles of a single causally-linked time-space tend to view such a position as a category mistake confusing the mental for the physical. I like to refer people to Korzybski's pithy "The map is not the territory." Kant muddied the waters by claiming the physical as we understand it is a phenomenalist construct of the mind which is drawn from empirical doctrine, such that the perception is built from the senses. The contemporary linguist Ray Jackendoff argues that the causal chain is noises trigger language faculties which associate concepts that draw upon perceptions of objects, so things have gotten considerably more complicated since Kant drawing from the cognitive sciences such as psycholinguistics and neuroscience.

Last PhilPapers survey I read, IIRC, is about 80% of philosophers commit to being realist, and 20% not. The question of where to invoke the property 'real' is still quite a lively debate, and Duhem, Quine, and Putnam, as well as Dennett, Davidson, Churchland, and others are still being sorted out. Dennett in his book Intentional Stance and his later paper Patterns attempts to take a middle position between extreme realist and eliminative materialist notions of belief which seems to me to casting the property real itself as a belief. Amusingly at the end of the essay, he refuses to categorize himself leaving the reader with rhetorical questions instead.

  • So... if I sift through all this, would you say that instrumentalists may possibly have leveraged that kind of attack on Quine-Putnam's indispensability argument by claiming that physics in particular, makes no ontological commitment?
    – Frank
    Apr 22 at 23:10
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    "My sense is that mathematicians tend to believe a word is a proof of thing" - my impression is that most mathematicians see mathematics as essentially abstract and not as "proof of thing". I have often seen mathematicians (and I am one by education & degree myself) bewildered by the in our view rather naive identification of mathematical results and physical truths that we find more in people who are non-mathematicians in the first place, even though maybe mathematically highly competent such as many physicists. Apr 23 at 12:25
  • @ChristianHennig Morning! I'm eager for first-person reports. I read, ie, Linnebo and take his testimony as true prima facie on the dominance of Platonic thinking. Do you take a real number to be real in the sense it has an objective existence independent of people? Or like Brouwer, do you see it as a construction of the mind with no independent existence?
    – J D
    Apr 23 at 13:33
  • @JD I'm actually not much interested in "real existence". I think it's a red herring. I for sure see it as a construction of the mind, so I am with Brouwer at least half the way, however I don't think that this necessarily implies that "it doesn't really exist". I'm agnostic on that one, and I ultimately think there is no way to know this so why bother? Apr 23 at 13:36
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    @JD I'm glad Hilbert spaces abstract some aspect(s?) of your experience. One of these days, I'll be curious to hear more about that.
    – Frank
    Apr 23 at 23:50

For Quine, and as linked in your article, ontological commitment is as follows: "to be is to be the value of a bound variable"

Since formalizations of physics typically involve quantification over mathematical objects, the Quinean is commited to all and only yhese entities which are quantified over. For a physics without numbers, see Fields.

So yes , a different notion of ontological commitent might not fare as well for Quibes argument. Lowe has a lucid discussion on various notions of ontological commitment if you are interested.

As for your example, the question is whether the new theories ultimately quanitfy over mathematical entities. If they do, the quniean will be unconvinced that they do not carry ontological commitment.

  • What does it mean to "quantify over mathematical objects"?
    – Frank
    Apr 22 at 22:30
  • The proposed line of attack though, is that there may well be no ontological commitment, of any kind, in the case of physics, but only descriptive models.
    – Frank
    Apr 22 at 22:36
  • 1) to quantify over them in some logic or model thereof 2) does your descriptive model quantify over mathematical entities? if so, the quniean beleives you have made an onotlogical commitment. Else, youre going the way of Field- nomiinzalization.
    – emesupap
    Apr 22 at 22:44
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    @JD Great. So now, instead of saying "I see a tree in my garden", I can say "I see something quantified over the set of plant objects" and sound smart. What has been gained? Besides being a pedantic way of saying "there is", is that "quantification" actually used to carry out some formal logical reasoning?
    – Frank
    Apr 23 at 15:48
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    @Papuseme No problem, I understand what Quine is getting at. I'm only complaining that he uses seemingly pedantic language.
    – Frank
    Apr 23 at 21:30

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