I believe this is also how we should see some mathematical truths, e.g. 2+2=4 is true. — Sam26

The rules of chess do not describe the truths of reality in the same way that "water freezes at 32 degrees F" does. Instead, they constitute the very framework within which true and false (correct and incorrect) can be assessed. — Sam26

But 2+2=4 is not arbitrary in the way that "bishops move diagonally" is.

Or rather, 2+2=4 follows the rules of adding in the same way that a diagonal bishop move follows the rules of chess. But the rules of adding are not mere convention, they capture some sort of truth that has not been stipulated into being, like the rules of chess were.

But 2+2=4 is not arbitrary in the way that "bishops move diagonally" is. — hypericin

Yes, you are correct; it's not the same. However, any system, whether epistemological (JTB) or formal mathematical systems, will have hinges (hereafter referred to as basic beliefs) that are true, but not in the JTB sense. All I'm saying is that both are basic statements of belief, and they function in similar ways. In both systems, these basic beliefs are bedrock to the system and function in a way that's not provable within the system. Some mathematical statements are accepted as true for the system to function.

Or rather, 2+2=4 follows the rules of adding in the same way that a diagonal bishop move follows the rules of chess. But the rules of adding are not mere convention, they capture some sort of truth that has not been stipulated into being, like the rules of chess were. — hypericin

Some basic beliefs are arbitrary and some are not, but both are basic and needed for the system to function. The difference in the use of truth is that one use is epistemological, and one is not. This is where part of the confusion lies, at least in OC. Another part of the confusion is the idea of hinge proposition, which is why I think they should be called basic beliefs.

I believe that mathematical systems are the product of minds and that anything created by that mind/s involving mathematics will have mathematical systems intrinsic to it. In other words, mathematics will be discoverable within that creation, by other minds, which is the case when we discover math as an intrinsic part of the universe. I'm an Idealist and believe that at the bottom of reality is consciousness (other minds). So, I believe, mathematical knowledge is intrinsic to this consciousness or mind. So, the answer to the age-old question, "Is mathematics discoverable or created by minds?" - it's ultimately a product of a mind/s, but it can be discovered as part of a creation too. So, again, if any mind uses mathematics to create something, then mathematics will be discoverable within that creation.

One can believe this as an idealist without believing in some religious doctrine.

Hinges aren't true in the epistemological sense, i.e., justified and true. However, one can use the concept of true in other ways, just as the concept know can be used in other ways. For example, someone might ask when learning the game of chess, "Is it true that bishops move diagonally?" You reply "Yes." This isn't an epistemological use of the concept — Sam26

So for example, when Moore raises his hand and says ‘I know this is a hand, and therefore it is true that it is a hand’, he is confusing an epistemological with a grammatical use of the concepts of know and true, because he considers his demonstration as a form of proof. Would you agree? But then what would be an example of a grammatical use of the word true in Moore’s case? Something like: ‘it is true that Moore is invoking a particular language game by raising his hand and saying he knows it is a hand?

This is slightly different but related. The rules of chess do not describe the truths of reality in the same way that "water freezes at 32 degrees F" does. Instead, they constitute the very framework within which true and false (correct and incorrect) can be assessed — Sam26

Wittgenstein seems to suggest that the intelligibility of ‘water boils at 100 C.’ depends on such a bedrock of hinge propositions ( a ‘whole way of seeing nature’).

291. We know that the earth is round. We have definitively ascertained that it is round. We shall stick to this opinion, unless our whole way of seeing nature changes. "How do you know that?" - I believe it.
292. Further experiments cannot give the lie to our earlier ones, at most they may change our whole way of looking at things.
293. Similarly with the sentence "water boils at 100 C. (On Certainty).

So for example, when Moore raises his hand and says ‘I know this is a hand, and therefore it is true that it is a hand’, he is confusing an epistemological with a grammatical use of the concepts of know and true, because he considers his demonstration as a form of proof. Would you agree? But then what would be an example of a grammatical use of the word true in Moore’s case? Something like: ‘it is true that Moore is invoking a particular language game by raising his hand and saying he knows it is a hand? — Joshs

The way I explain Moore's confusion, which is Witt's point, Moore's use of know is more akin to an expression of a conviction. In other words, a subjective feeling of truth expressed by emphasis or gesticulation. A feeling of certainty, not to be confused with objective certainty or knowledge.

Moore does consider "I know this is a hand," to be empirical proof, a self-evident truth. In the Wittgensteinian sense "This is a hand" would be a grammatical truth by virtue of the language game and context. However, Moore is saying something different, he thinks he has good reasons to suppose that he knows "This is a hand." Wittgenstein disagrees.

I edited this in later, so you may not have seen it.

This is slightly different but related. The rules of chess do not describe the truths of reality in the same way that "water freezes at 32 degrees F" does. Instead, they constitute the very framework within which true and false (correct and incorrect) can be assessed — Sam26

Wittgenstein seems to suggest that the intelligibility of ‘water boils at 100 C.’ depends on such a bedrock of hinge propositions ( a ‘whole way of seeing nature’).

291. We know that the earth is round. We have definitively ascertained that it is round. We shall stick to this opinion, unless our whole way of seeing nature changes. "How do you know that?" - I believe it.
292. Further experiments cannot give the lie to our earlier ones, at most they may change our whole way of looking at things.
293. Similarly with the sentence "water boils at 100 C. (On Certainty).

This should probably be in my thread on OC.

Moore using sign language before a deaf audience could emphasize the point.

I'm not sure it's needed though. Denial of an extra-self world seems like a philosophical (maybe psychological) problem alone, a Cartesian curse. Should we expect a purely deductive dis/proof?

↪Joshs This should probably be in my thread on OC. — Sam26

Done.

Math is a product of the human mind, and a very useful for modeling reality for human purposes. — Tzeentch

Well maybe God thought of numbers first so they exist in God's mind?

Here's a thing-

On the x-axis mark the line 0 to 1. You have one unit made of infinitely many points. These points are dimensionless; they have no width. But if you assert you are at 1 unit on the x-axis how did you get there? By lining up an infinity of dimensionless points? If this is the case you assert 0 + 0 + 0...for infinity add up to extension or width. So it would seem that an infinity of zero widths add up to a unit of width. Go figure.

Prima facie, it may sound counterintuitive to state that ‘there are infinitely many prime numbers’ is false. But if numbers do not exist, that's the proper truth-value for that statement (assuming a standard semantics). In response to this concern, Field 1989 introduces a fictional operator, in terms of which verbal agreement can be reached with the platonist. In the case at hand, one would state: ‘According to arithmetic, there are infinitely many prime numbers’, which is clearly true. Given the use of a fictional operator, the resulting view is often called mathematical fictionalism.

Can't one just say there are potentially infinitely many prime numbers? Also one can say that-
1/p1 + 1/p2 + 1/p3 +...diverges to infinity where these p are the primes.

But what does it mean to say there are infinitely many? What does 'are' mean? Does it mean they exist or potentially exist?

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

Saying mathematics exists independently of our minds and saying numbers exist independently are two different things. Mathematical truth, as it really is, may be something we have never imagined.

@Michael "3. If there exist any abstract mathematical objects, then human beings could not attain knowledge of them. Therefore,"

Why would consciousness be limited to physical spacetime? Mysticism asserts that consciousness transcends the physical.

Why would consciousness be limited to physical spacetime? — EnPassant

I don't believe it is, and the hard problem of consciousness suggests it is not, but naturalism assumes that it is. There was a lot of discussion earlier in this thread about that conflict.

I heard an argument related to this recently.

Bertrand Russell says something like: "mathematics is the field where we believe we know things most certainly, and yet no one knows what mathematics is about." By contrast, earlier mathematicians often did think their subject had a clear subject matter. Where they simply mistaken? Naive?

Here is the argument, the difference is one of equivocation. Barry Mazur has this really nice article called: "When is one thing equal to another?" https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://people.math.osu.edu/cogdell.1/6112-Mazur-www.pdf&ved=2ahUKEwiPlpKX5_WKAxXmhIkEHcOREwcQFnoECBgQAQ&usg=AOvVaw0j1f7DfoQP7OKuvRZ37rIU

If you read older mathematics though it might seem like they could be talking about different subjects, because there is often a strong distinction between magnitude and multitude, and both are primarily derivative of/abstracted from things. That is, there can be a multitude of things, e.g. 6 cats, or magnitudes related to things, e.g. a wood board that is twice as long as another. Mathematics is the consideration of the properties of magnitude and multitude in the absence of any other properties. For instance, a ratio would be understood as specifically a relationship of magnitude, never as a number.

Because of this, metaphysics and the philosophy of perception/epistemology end up bearing a closer relationship to mathematics.

Anyhow, on first glance, if one accepts this and a "study of magnitude and multitude," it seems like it may make various flavors of realism more plausible (immanent realism or platonism).

Mathematics is the consideration of the properties of magnitude and multitude in the absence of any other properties — Count Timothy von Icarus

Hmmm . . . never thought of it that way.

Mazur's article on category theory introduces one to modern mathematics. Not necessarily mathematics as practiced by a great many professionals. As time passes levels of abstraction increase and the subject seems more and more like philosophy and less and less like calculus, for instance.