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.

What do you take Quine to have said about ontological commitment with regard to mathematical entities? It'd be helpful to understand how you think it differs from the view I expressed, which makes use of his "To be is to be the value of a bound variable."

Following Quine,

• There is exactly one whole number between one and three.
• Therefore, our theory quantifies over at least one whole number.
• Hence, we are ontologically committed to whole numbers.

Unfortunately, you do not agree, having said:

This supposition that you have, that there are numbers between numbers is very problematic. — Metaphysician Undercover

Quine's approach has a distinct advantage over your own, in that it allows us to do basic arithmetic.


Nothing in the above commits us to numbers existing independently, in the way of chairs or mountains. Nothing commits us to a hard platonic world of floating numbers. It is open for mathematical entities to be more akin to property, money or countries, a convenient way of talking about how things are.

Quine's approach does not commit us to Platonism in any robust or traditional sense.

What you post shows is your failure to follow the argument.

What do you take Quine to have said about ontological commitment with regard to mathematical entities? It'd be helpful to understand how you think it differs from the view I expressed, which makes use of his "To be is to be the value of a bound variable." — Banno

Would it be better to start a thread on ontological commitment? If so, I can do that.

Whatever it take for you to commit.

infinitely so. . . :-)

hatever it take for you to commit. — Banno

Dude. If you were committed to the topic, you would at least know what mathematical realism is. :razz:

You've claimed I don't, but haven't set out anything to support such a view. I have asked. What, for you , is realism? Technically, it's the commitment to statements being either true or false, with antirealism the view that some statements are neither true nor false. Meta, and perhaps you, suppose a slightly different realism in which truths are made true by a mind-independent domain of entities, whose existence and nature do not depend on our practices, languages, or activities.

But you are fishing again. What happened to indispensability?

What happened to indispensability? — Banno

You don't have the foundation to understand the argument. :confused:

:meh:

You know, Quine's dictum is a funny thing.

On the one hand, it seems to treat "there exists" as univocal, when discussions like this seem strongly to suggest different sorts of things exist in different ways.

But on the other hand, Quine's dictum does, in its own way, recognize that "is" is "substantive hungry". (Austin's phrase? It's the point that "Alfred is" strikes us as incomplete -- "Alfred is what?") Variables don't float around on their own in classical logic; even when not bound by a quantifier, they only show up governed by predicates.

("What about the domain of discourse? Surely that's just a collection of objects we have assigned names to." But Quine was also inclined to do away with names and use only predicates.)

I think we could follow Quine in saying that, so far as logic is concerned, "there exists" is univocal, while recognizing -- perhaps against his wishes -- that because bound variables are always governed by predicates, there is room for allowing that dogs exist the way dogs exist, numbers the way numbers exist, quarks the way quarks exist, and so on.

(I have complained on several occasions that our logic does not distinguish between predicates and sortals, and this looks like another one of those occasions. But we can similarly recognize that truth functions don't care about that distinction, even if sometimes we do.)

yes indeed. Existential qualification functions within a domain. So if it’s univocal then it’s univocal only within that domain...

So we might think that it moves the “question of existence” back a step, back to asking what it is to be part of the domain. And the domain is a construct; this or that counts as an item within the domain.

Yes, I think that's right.

In a sense, what the formalism of FOL identifies is that being a member of a domain, or not, and satisfying a predicate, or not, are the same operations for all domains and for all predicates.

In that sense, it is a just a further step along the path Aristotle discovered when he noted the structural similarity of classes of arguments, setting aside the specific contents of the premises and conclusions.