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1. Dodd (2007) and Friedell (2020) argue for this view; Juvshik (2018)argues against it.

2. The view that properties are tropes goes back at least to William ofOckham, and you might reasonably think that Locke, Berkeley, and Humeendorse this view. More recently, the view has been endorsed morerecently by Stout (1921), Williams (1953a,b), Campbell (1990), andMoltmann (2013b).

3. One might think that something is missing from this new version ofthe One Over Many, for one might think that part of the problem is toaccount for how two different things couldshare a nature.But people like Devitt would presumably say that if you can accountfor how objecta isF and also for howb isF, then youhave accounted for howa andb share a nature.

4. It is worth pausing to note why this principle says only thatwe’re committed by the singular terms insimplesentences and the existential quantifiers inexistentialsentences. The reason is that some singular terms and existentialquantifiers are not ontologically committing. For instance, if I say“If Santa Claus lives at the North Pole, then we should be ableto find him”, I have not committed to the existence of SantaClaus; and if I say “If there is a God, then there is a toothfairy”, I have not committed to the existence of a God or atooth fairy. This problem is avoided by concentrating on

1. (a) singular terms within simple sentences, or atomic sentences,i.e., sentences of form “a isF”,“a isR-related tob”,“a,b, andc areS-related”,etc.; and
2. (b) existential quantifiers in what we are calling existentialsentences, i.e., sentences in which the existential quantifier is themain logical operator, i.e., sentences of the form“There is an object such that…”.

Now, it would be wrong to say that these are the only sentences thatare ontologically committing—e.g., we’re committed by thesingular term and existential quantifier in the sentence “Marsis a planet and there exist some teeth”, and this sentencedoesn’t fit into category (a) or (b)—but for presentpurposes, we can ignore such sentences and concentrate on sentences incategories (a) and (b).

5. Maddy did not take mathematical objects to exist outside ofspacetime, so her view was not a version of platonism, as that viewhas been defined here. But she is included in this list because onsome alternative definitions, she might count as a platonist, and sheis often thought of as something like a platonist. In any event, wecan say that her view hasplatonistic leanings, as is clearfrom the fact that it entails the existence of mathematical objectsthat are non-mental and, in some sense or other, non-physical (inparticular, there is more to Maddian sets than the physical stuff thatmakes up their members).

6.Premise [5] is analytic and entirely trivial. And if we wanted to, wecould make[6] analytic by stipulating that “abstract object” is to meannon-physical, non-mental object. (You might think that,intuitively, there are other kinds of objects—e.g., socialobjects; but it can be argued that such objects reduce to physicalobjects, mental objects, and abstract objects.)

7. In addition to fictionalism, there is a second (much more radical)way to claim that our mathematical theories are not true: one couldendorse anon-cognitivist view of mathematics, claiming thatsentences like “3 is prime” don’t really sayanything at all and, hence, aren’t the sorts of things that havetruth values. One such view isgame formalism, which holdsthat mathematics is a game of symbol manipulation and that, e.g.,“3 is prime” is one of the “legal results” ofthe game of arithmetic. This view was defended by Heine (1872) andThomae (1898) and attacked vigorously by Frege (see Frege 1893/1903:§§88–131). One might also interpretWittgenstein’s (1956) philosophy of mathematics asnon-cognitivist, although this is controversial.

8. One might doubt that this paraphrase is really nominalistic; for itwould seem to commit people to things like sentences and mathematicaltheories, and one might argue that these things are best thought of asabstract objects.

9. Someone like Davidson (1967) might claim that Boris believes theEnglish sentence token (i.e., the one in the belief report) in virtueof the fact that it says-the-same as Russian sentence tokens that hebelieves. But the problem with this is that Boris has lots of beliefsthat don’t correspond to any Russian sentence tokens. Forinstance, he presumably believes that 17.427 is greater than 13.961;but it’s unlikely that there has ever been a Russian sentencetoken that says this.

10. It needs to be kept in mind here that in formulating this argumentfor the existence of propositions, platonists do not commit to theview thatall “that”-clauses are singular termsthat denote propositions. Consider, e.g., the sentence “Ralphfears that the killer is coming”. One might very well doubt thatthis sentence says that Ralph fears a proposition; after all, sincepropositions are abstract objects, they are causally inert, and sothey are not very scary—for instance, they can’t shootpeople. One might take this as evidence that the“that”-clause in the above sentence (that is, in“Ralph fears that the killer is coming”) does not denote(or purport to denote) a proposition, and indeed, one might take it asevidence that this “that”-clause isn’t a singularterm at all. There are a number of ways that platonists could try torespond to this: they might try to argue that despite appearances, theabove “that”-clause does denote a proposition; or theymight admit that it doesn’t denote a proposition (they mighteven admit that it’s not a singular term at all) and then arguethat this doesn’t undermine the claim that the“that”-clauses in belief ascriptions denote propositions.We won’t pursue the question of how platonists ought to respondto challenges of this kind.

11. Frege’s view entailed that for every predicate, there is aconcept and an extension of that concept, and as Russell pointed outin a letter to Frege, this leads to contradiction. ContemporaryFregeans, such as Boolos (1987), Wright (1983), and Anderson and Zalta(2004), have found ways to avoid the contradiction.

12. It might seem that if stories are best thought of as abstractobjects, then fictionalism about things like mathematical objects andpropositions is not a genuinely nominalistic view, becausefictionalism seems to commit to the truth of sentences like“‘3 is prime’ is true in the story ofmathematics”. But fictionalists can avoid this worry by taking afictionalistic attitude about stories, or fictions. It might seem thatthis would lead them into an unacceptable regress, but it doesnot.

13. This interpretation of Gödel is a bit controversial. Evidencefor it comes not just from his (1947), but also from his (1951). Formore on the interpretation of Gödel, see C. Parsons (1995) andvan Atten and Kennedy (2003).

14. For theories of how we could attain knowledge of consistency (andother logical knowledge) without this sort of information-transferringcontact, see, e.g., Berry (forthcoming) and Balaguer (2021: chapter6).

15. Leng (2022) argues that instead of starting with the assumption thatFBP is true, platonists could start by assuming Hume’sprinciple—i.e., by assuming that the number ofFs = thenumber ofGs if and only if theFs and theGs areequinumerous; she argues that, given this assumption, platonists couldconstruct a plausible naturalistic epistemology.