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?
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.
Leibniz sheds light on the issue. In his maturity, Leibniz viewed only geometric entities used by the ancient Greeks as being possible. He viewed numbers and quantities that were beginning to play an increasingly important role in 17th century mathematics, such as surds (quadratic and other irrationals), imaginary roots, infinitesimals, and even negative numbers, as "fictions" that are non-contradictory and in this sense are only "accidentally impossible". This meant that they are not representable in what Leibniz referred to as the phenomenal realm. Accidental impossibility contrasts with, e.g., 2+2=5 which is contradictory and thus absolutely impossible.
Leibniz developed rules for infinitesimal analysis that give rise to a robust collection of procedures that are reliable in the sense of not leading to contradictions in the practice of analysis. As is well known, consistency (or truth) of the widely used packages such as PA or ZFC cannot be established without passing to an even stronger system. However, their practical reliability is established through extensive use encountering no contradictions. By contrast, issues of "existence" of mathematical entities are not testable in any comparable sense. Thus we can have reliable truth of mathematical procedures and assertions without committing ourselves to any specific ontology of numbers, in line with Leibnizian insights.
This answer is inspired by Misha Katz and provides more examples:
One can make the comparison with physics:
we know after a lot of testing (esp. quantum mechanics) that the laws of physics provide highly reliable mathematical descriptions of physical reality. However, whether at its core, reality is a continuum, the rationals, or even a finite set, is not know (or knowable, I conjecture). So we believe that our descriptions work, without knowing exactly what they are about. In particular, real analysis is used all the time in physics, but physicists generally do not claim that the real numbers somehow exist in reality.
One can make the comparison with computer programs:
one can write verified programs, but this is very hard and slow. Alternatively, there are extensive testing procedures that guarantee that programs run as expected in all but the most exotic cases. Unix and derivatives (macos) are examples of such robust programs. So we believe that our programs work and ascribe them certain (deterministic) properties, without knowing for sure the programs really have these properties.
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.
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.
Often enough, object-talk comes out to (A) substances-with-attributes-talk, whereas objectivity-talk comes out to (B) contrast-with-subjectivity-talk. So despite surface appearances, one can affirm (B) without affirming (A), or (A) and (B) together (perhaps not so much (A) by itself, although intuitionism perhaps could be glossed along such lines, if deviously).
This kind of distinction/taxonomizing applies with respect to mathematical structuralism, for example. The structures are not substances, not even made of some kind of "abstract matter," but they are systems of properties and relations in a kind of ontomorphic "free fall" through logical space. Now, if one adopted a Humean bundle theory of objects, one can have objects without pure substance underlying them, so one might have objects as sets of properties, and then the structures of structuralism come out as the objects of the theory, but if we wish to retain potentially helpful distinctions, then we will prefer to either drop object-talk in this context or avoid the Humean sense of the word "objects" as such.
So we might say: the substance-theoretic mathematical realist will be inclined to talk about ω and 2ℵ0, while the structure-theoretic mathematical realist will be inclined to talk about ℕ and ∞, although there are mutual interpretations and paraphrases and so on that can allow members of either school-of-thought to be more (or sometimes less) comfortable with exchanges of these symbolisms.
Further reading: Truth-value realism:
Truth-value realism is the view that every well-formed mathematical statement has a unique and objective truth-value that is independent of whether it can be known by us and whether it follows logically from our current mathematical theories. The view also holds that most mathematical statements that are deemed to be true are in fact true. So truth-value realism is clearly a metaphysical view. But unlike platonism it is not an ontological view. For although truth-value realism claims that mathematical statements have unique and objective truth-values, it is not committed to the distinctively platonist idea that these truth-values are to be explained in terms of an ontology of mathematical objects.