I don't think you understand what math realism is.

I don't think you understand what math realism is. — frank

Do you?

Well then, tell me. Say something. Commit.

You said your constructivism was compatible with realism, which would imply that the reality of numbers is a byproduct of social practices. Social practices are objective, and numbers are an aspect of them, so in that sense numbers are objective.

I handed you the problem with that theory, which is that a constructivist is forgetting about the existence of things like the extension of decimal pi.

You would notice that Quine struggled with the same issue and reluctantly agreed with platonism based on the indispensability argument. And if you think about it, Frege, Godel, Quine, and Putnam had time to sit around pondering it full time. You can't really half-ass your way to rejecting their arguments. :grin:

You said your constructivism was compatible with realism — frank

This?

This view preserves mathematical realism (mathematical statements have objective truth values) while avoiding the metaphysical commitments of Platonism (no need for causally inert, spatiotemporally transcendent entities). — Banno

Tell me what you think realism is - how you are here using it... Ontological realism (Platonism), Semantic realism, Quantificational or something else/combined? I've been pretty explicit that the 'reality' of numbers is little more than our ability to quantify over them.

...?

Basically what Quine said about ontological commitments.

ok.

The proffered alternative is that mathematical statements are true, and we can talk about mathematical objects existing, but this doesn't require positing some separate realm outside space and time where numbers "live." Instead, mathematical language works the way it does - we can truly say "there is a prime number between 7 and 11" - without needing to tell some grand metaphysical story about what makes this true. — Banno

Banno, the assumption that mathematical objects exist requires justification or else you're just talking through your hat. When anyone tries to justify their existence, Platonism is exposed in that attempt.

The truth of mathematical statements is connected to their role in our practices, proofs, and language games rather than correspondence to abstract objects in a Platonic heaven. — Banno

If your practise is to start with the premise that numerals refer to abstract objects, then the truth of this premise requires a platonic realm where these abstract objects exist. Otherwise any logic which follows is unsound, based in a false premise.

Here is the problem. For convenience sake, and common vernacular, we talk about numbers as if they are objects, and this in principle has no effect on mathematics, as mathematics is used. There is a clear separation between the talk about mathematics, people talking about numbers as objects, etc., and how the mathematicians are actually using the language of mathematics.

Describing mathematics in that way is just done to facilitate talk about mathematics. The talk about mathematics is in that way false, but it's a falsity of convenience, it facilitates our talk about mathematics. However, if the assumption that numbers are abstract objects makes its way into the axioms of mathematics (set theory), and this assumption is false, then we have a false premise within that logical system.

This view preserves mathematical realism (mathematical statements have objective truth values) while avoiding the metaphysical commitments of Platonism (no need for causally inert, spatiotemporally transcendent entities). — Banno

If it is the case, that within the axioms of mathematics, abstract objects are assumed, then "this view" which you present is a false description of mathematics. Clearly, set theory assumes within its axioms, abstract mathematical objects. Therefore the "objective truth" of mathematical set theory requires platonism.

You want to have it both ways (your cake and eat it, as frank says). You say that we can talk about numbers as abstract mathematical objects, though we know they really are not, and when we do mathematics the objective truth of mathematics is not dependent on this. That is fine in principle, if it is true. However, the truth about mathematics is that set theory assumes the existence of platonist objects, and the logical system is dependent on this assumption. This means that when we do mathematics using set theory, "abstract mathematical objects" is assumed, and the objective truth of mathematics is dependent on the "abstract mathematical objects".

So it is not just a matter of talking about numbers as mathematical objects, it is a matter of premising that numbers are platonic objects, and constructing a structure of mathematical logic with this premise as the foundation. That is set theory

Therefore, this talk about numbers as abstract objects, which we might recognize as false, yet still use, for simplicity sake, has been allowed to infiltrate and contaminate the system itself. We say that we recognize this assumption as not really a truth, but do we recognize the consequences? A vast logical structure, set theory, is based on what we recognize as a false assumption.

Platonism is not just "numbers exist", as Meta supposes. — Banno

Platonism is "numbers are objects". "Object" implies existing. When you propose that "X" stands for an object, or "2" stands for an object, the existence of that object must be justified. That's what Wittgenstein showed with the private language analogy. One can point to a chair, and say that is the object I'm talking about. But we can't point to a number this way. If I say that there is an object which is a number, this object must be independent from my mind, for its existence to be publicly justifiable, and that is platonism.

Otherwise the beetle in the box analogy applies. I have an object in my mind which I call "2", and you do too. We call them the same name, maybe even describe them in a very similar way, but your object is not the same as mine. therefore we do not have a proper "object" referred to with "2". The only way to justify 2 as an independent object is to assign platonic existence to it.

The response is not to reify the procedure that produces each digit; yet π is a quantified value within mathematics. It figures under quantifiers, enters inequalities, is bounded, approximated, compared, integrated over, etc. None of that is in dispute, and none of it commits us to Platonism. π is quantified intensionally, via its defining rules and inferential role — not extensionally, as a completed set of digits. — Banno

But you do not apply this principle infinite sequences. You do not say that each of these "is not a completed object. It is an instruction for producing digits". You insist on the very opposite, that these are completed objects That requires platonism to justify.