How can one believe that mathematical assertions are objectively true while denying the existence of mathematical objects?

Question

I had started reading a book called "A Historical Introduction to The Philosophy of Mathematics" where it began by outlying some common beliefs within the philosophy of Mathematics. One such case of belief is characterized by what it calls an "object nominalist" who is skeptical of the existence of mathematical objects while believing in the objectivity of non-vacuously true mathematical statements. My question is how can one believe a statement about an object that they believe to not exist be non-vacuously true? Doesn't the lack of existence of an object make any descriptions about it or its properties vacuous or meaningless?

Can you add more direct relevant quotation from your book, and perhaps the particular philosophers holding this kind of view? Is it the same school of thought about [mathematical nominalism](https://en.wikipedia.org/wiki/Nominalism#Mathematical_nominalism) from the WP source? — Double Knot, Dec 24 '21 at 05:35
The same way they can believe "Gandalf is a wizard" to be meaningful and non-vacuously true without believing that Gandalf exists. "True" and "exists" are ambiguous words that are often relativized to context, and one does not have to relativize them in the same way when talking mathematics, any more than when talking Tolkien, see SEP [Fictionalism](https://plato.stanford.edu/entries/fictionalism-mathematics/), [Nonexistent Objects](https://plato.stanford.edu/entries/nonexistent-objects/) and [Free Logic](https://plato.stanford.edu/entries/logic-free/). — Conifold, Dec 24 '21 at 06:34
@Conifold Are there also truth-value realists who believe that certain kinds of mathematical claims (say, the idea that certain propositions follow from certain axioms and inference rules) would be necessarily true in all possible worlds, while claims like "Gandalf is a wizard" would not, and yet don't feel the need to justify this in terms of an ontology of abstract objects? I would think some in the analytic tradition define objective reality more fundamentally in terms of the collection of all true propositions than in any ontological sense, — Hypnosifl, Dec 24 '21 at 19:34
(cont.) and there are ambiguities in [Quine's criterion for ontological commitment](https://plato.stanford.edu/entries/ontological-commitment/) which starts from some set of true propositions and tries to use them to define what "exists", so some philosophers might just say it's meaningless to ask which terms in the set of all true propositions refer to existing-things and which don't. Chalmers' piece in the book *Metametaphysics* refers to a position called "ontological anti-realism" which seems like it might be taking this approach, though I'm not sure I'm interpreting it right. — Hypnosifl, Dec 24 '21 at 19:34
@Hypnosifl My understanding is that truth value realists do not reduce truth to derivability from axioms by inference rules. But yes, they [include nominalists](https://plato.stanford.edu/entries/platonism-mathematics/#TruValRea) who deny necessity of justifying objective truth values by existence of abstract objects. Kant also famously grounded objective truths of mathematics in the transcendental subject's powers of "pure intuition" rather than any ontology, and Brouwer style intuitionists follow him in that. — Conifold, Dec 24 '21 at 22:40
@Conifold - I think Carnap might be an example of an ontological anti-realist (Chalmers mentions him as one) who is a truth value realist, and I've read that he justified his own interpretation of 'logicism' about arithmetic (differing from the logicism of Russell and others) by pointing to the fact that you could add a non-computable inference rule, the [ω-rule](https://encyclopediaofmath.org/wiki/Carnap_rule), to the standard inference rules of first-order logic, and in that case it can be shown that all the propositions of true arithmetic can be 'inferred' from the Peano axioms alone. — Hypnosifl, Dec 24 '21 at 23:02
@Hypnosifl I do not think either Carnap or Quine can be called truth value realists. Carnap relativizes everything to "linguistic frameworks". He does hold a robust analytic/synthetic distinction, but even if analyticity is cross-linguistic it would cover very little. Quine's ontologies (in the plural), even logic, are subject to change whenever pragmatic factors make it beneficial. His ontological criterion is for commitments of a *theory*, and ontology is tied to our "best theories" (of the day). He is a curious example of a "platonist" (self-described) who is not a truth value realist. — Conifold, Dec 24 '21 at 23:27
@Conifold *Carnap relativizes everything to "linguistic frameworks".* But doesn't he think there is a definite truth about what conclusions follow from such a linguistic framework, a truth which is independent of judgments made by actual human beings making use of the framework? Unless we discover some new physics, we humans can never actually use the ω-rule (which requires checking an infinite no. of cases) to judge whether a given WFF is derivable from the Peano axioms, if that WFF is not derivable in a computable way from the axioms and the usual inference rules of first-order logic. — Hypnosifl, Dec 24 '21 at 23:45
@Hypnosifl He does, but that is merely deductive determinism (the only author I can think of denying even *that*, that theorems are determined once axioms and rules are laid down, is late Wittgenstein), not truth value realism. Truth value realists believe in mathematical truth *non-relative* to a choice of framework, they envision some extra-linguistic anchor, be it ontology or intellectual faculties, that grounds mathematics. So Gödel, for example, insists on CH being true or false *despite* its proven undecidability in ZFC, and calls for discovering "new powerful axioms" to settle it. — Conifold, Dec 25 '21 at 01:56
@Conifold Thanks, I hadn't realized that truth-value realism was understood as necessarily going beyond deductive determinism. Can you recommend any books or papers/articles that lay out the distinctions between deductive determinism, truth-value realism, and mathematical platonism in philosophy of math? — Hypnosifl, Dec 25 '21 at 02:23
@Hypnosifl [SEP article](https://plato.stanford.edu/entries/platonism-mathematics/#TruValRea) contrasts truth value realism to platonism. Deductive determinism is not much discussed since it is only relevant to Wittgenstein, in that context see [Koshkin, Wittgenstein, Peirce, and paradoxes of mathematical proof](http://philsci-archive.pitt.edu/16924/). — Conifold, Dec 25 '21 at 03:05
@Conifold What I still don't understand, and what I think you discussed with is truth-value realists who take mathematical statements to be true in all worlds, but deny the existence of mathematical objects. The actual quote from the book said "Rationalism or Platonism is either the claim that mathematical objects like sets, numbers, geometric shapes exist or that mathematical claims can be non-vacuously true. We call the former object realism, and the latter claim sentence realism(or propositional-realism or true-value realism)". — Fumerian Gaming, Dec 25 '21 at 18:44
@Conifold It then goes on to say "Sentence realists who deny object realism are often motivated by the supposed spookiness of abstract mathematical objects .... So some philosophers believe that while some mathematical claims are true they do not refer to abstract objects; these are sentence realists who are not object realists; we can call them to object nominalists". What I'm very confused about is how one can believe that mathematical statements are objectively true, from what I understand in reality and not simply in some system, if they don't believe mathematical objects exists. — Fumerian Gaming, Dec 25 '21 at 18:48
@FumerianGaming Kant and intuitionists, for example, believe that there is an idealized subject with creative powers for mathematics that living mathematicians are approximations of. Mathematical statements are true in virtue of that subject's powers, and abstract objects are just a manner of speaking about something else, useful fictions. — Conifold, Dec 26 '21 at 21:20
@Conifold I'm still kind of confused. What do you mean by "Kant and intuitionists, for example, believe that there is an idealized subject with creative powers for mathematics that living mathematicians are approximations of."? Specificically by "idealized subject" and "creative powers". — Fumerian Gaming, Dec 27 '21 at 03:47
See [Brouwer's "Creating Subject"](https://plato.stanford.edu/entries/intuitionism/#CreSub). — Conifold, Dec 27 '21 at 08:26
What you're briefly describing in the OP can be the modern math [structuralism](https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)): *Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have (not, in other words, to their ontology)...Benacerraf contended that there does not exist a method for accessing abstract objects...* — Double Knot, Jan 23 '22 at 22:29
So it's like Whitehead's process philosophy where structures/processes are real ontic existence while sets/objects are of no ontic existence. It's also can be likened to higher-order functions in lambda calculus functional programming or type theory where everything is just function. Since you still at least commit to something existent, so the whole math statement still has truth value in a correspondent fashion. However, there're other metaphysic theory such as *Trope Nominalism* in my above WP ref which commit to no ontic existence even of statement's truth value as a type of deflationism. — Double Knot, Jan 23 '22 at 22:43
Your problem is, how are differing groups, & you, defining 'exist'? Consider whether money is 'real', or 'made up'. Is identity real if we are 'just' atoms - what kind of real is free will, as follow up. We use abstractions, like overlays projected onto the world, that help organise our experiences in salience-landscapes, which are intersubjective - not, objective. My breakdown of the issues here: 'The Unreasonable Ineffectiveness of Mathematics in most sciences' https://philosophy.stackexchange.com/questions/92058/the-unreasonable-ineffectiveness-of-mathematics-in-most-sciences/92064#92064 — CriglCragl, Aug 23 '22 at 16:24
It boils down, in me universe, to the question *what is math?* That said, it came as a pleasant surprise, math does have room enough for, as far as I can tell, limited personalization i.e. there can exist a *Kristian Berryian mathematics* as distinct from *Marco Ocramian mathematics*. — Agent Smith, May 20 '23 at 07:27

score 0 · Answer 1 · answered Dec 24 '21 at 11:48

One such case of belief is characterized by what it calls an "object nominalist" who is skeptical of the existence of mathematical objects while believing in the objectivity of non-vacuously true mathematical statements.

We can apparently believe that (A → B) ∧ A ⊢ B is true even though A, B and A → B are ideas in our mind and not objects we could observe in the world.

Of course, we can, and I do, take ideas in general, and so concepts in particular, to be the analogues in the mind of objects in the world. We can only observe objects in the world and we can only experience our own ideas. And we better not mix them up.

My question is how can one believe a statement about an object that they believe to not exist be non-vacuously true? Doesn't the lack of existence of an object make any descriptions about it or its properties vacuous or meaningless?

People are usually quite certain that their own ideas exist in some way, if not as objects in the physical world, and presumably they take at least some of them to be sufficiently clear and distinct that they can assert something about them, including that they be true. It would be particularly cumbersome if we were to decide that we could not take our own ideas to be true. It seems indeed a necessary feature of human cognition. And there is no reason that this should not apply to mathematical concepts.

We don't seem to experience mathematical objects as we do our ideas in our mind nor observe them as we do rabbits in the world, so we don't have any good reason to believe that they exist at all. So mathematical objects have the same status as fictional beings, such as superman and Pegasus. We can pretend that they exist and nobody could prove that they do not, but we could not prove that they exist even if we believed that they did. Thus, the debate about the existence of mathematical objects seems pointless until we find a way of proving that they exist if they do.

eaglese ha. good fun to read, thanks! – Jan 23 '22 at 12:50 — , Jan 23 '22 at 12:50

score 0 · Answer 2 · answered Jun 23 '22 at 08:42

Nominalists believe that statements about numbers are true for all collections of such. So 2+3 = 5 gets translated into any two things plus any three things yields five things.

Philosophers who are realists about numbers would also agree with this. But whilst the nominalist does not agree with the existence of numbers per se, the realist believes that they are. The Platonist believes, for example, that they live in a realm only intelligible to the intellect, one might say, through an inner sense rather than an outer sense.

score 0 · Answer 3 · answered May 20 '23 at 06:55

The root of the problem is that the question of the existence or non-existence of numbers is not as black and white as you take it to be. Numbers, and mathematics more generally, clearly 'exist' as ideas in the head, and clearly 'exist' as patterns in the Universe, and there are logical relationships between the patterns. If I have n eggs and break one, I have n-1 unbroken eggs left. If I measure the force between two charges I find that it decreases as I move the charges further apart in proportion to the square of the distance. All that is viewed by many as self-evidently true. The endless argument is about whether and how mathematics 'exists' in some other inescapably vague sense of the word, and there is no reason why the unresolved nature of that argument should prevent you from counting eggs.

In your example, it depends how you count your eggs. Mappings are not ineluctable. Makes me think of How The Laws of Physics Lie — CriglCragl, May 20 '23 at 07:53

How can one believe that mathematical assertions are objectively true while denying the existence of mathematical objects?

3 Answers3