MORE ON MATHEMATICAL PLATONISM (For me. Pls ignore.) “Famously, Tarski (1936) pr

MORE ON MATHEMATICAL PLATONISM

(For me. Pls ignore.)

“Famously, Tarski (1936) proved that no classical formal language could contain its own truth predicate, due to Liar’s paradox. As such, if we want to include a truth predicate, we are committed to a hierarchy of languages. Moreover, if consisting only of formal languages, this hierarchy does not collapse: at no level will a language Lm provide a truth predicate for a language Ln, where n ≥ m.”

CD: Yes, but I can see that this is starting to go south already, confusing sets with semantics…

“If one is not committed to strict formalism, there are far less

problems with Tarskian truth. In particular, the hierarchy of

languages can be collapsed. There are two ways of doing this. One

can either move from formal to informal languages – where Tarski’s

undefinability result does not hold in the strict sense – at some

point in the hierarchy, or one can hold some level in the hierarchy

to be of the language-to-world type. Philosophically these two

strategies are largely equivalent, since we seem to have no way of

describing the world outside language. This makes the job a lot

easier for the non-formalist. Rather than try to explain a

problematic relation between mathematical languages and mathematical reality, we can concentrate on characterizing the

connection between our formal and pre-formal mathematical

languages.”

“What proof is to formal mathematics, truth is to pre-formal. We

deal with mathematical proofs syntactically, but at the same time

we as human beings think about them semantically.

CD: Yes.

“We cannot deny pre-formal thinking, and its need for semantical truth. However, this alone is not enough to show a substantial difference between truth and proof. Even though the existence of pre-formal mathematics cannot be reasonably contested, there is always the possibility that when it comes to truth, it is essentially superfluous; whatever we can achieve with truth, we could also achieve with proof alone.”

CD: First, there is a very great difference between truth and proof if mathematics is platonistic and set based. But if it is marginally indifferent and non-platonic then there is no difference. So that’s my concern. But the question I have is, what externalities are produced? It’s a moral question. I know that’s hard to grasp. But a biologist who plays with viruses and a mathematician that teaches platonism both export risks onto others.

“The second problem that the lack of reference causes for

formalism is one that does not require semantical arguments, or

indeed any sophisticated philosophical devices.”

CD: I do not see that as a problem. Nor do I see the need for, or desire for, formalism.

“It could be plausibly claimed that human thinking as we know it could not exist without some mathematical knowledge.

CD: yes, this is correct. But the reason is not stated here.

“But if mathematics has absolutely no reference, what reason do we have for picking one theory over another? It must be remembered here that this reference does not have to mean anything resembling a Platonic universe of mathematical ideas. Simply put, if we believe that 2 + 2 = 4 rather than 2 + 2 = 3, we must believe in some kind of reference. (It must be noted that I do not mean to use “some” as a hedge word here. My point throughout this work is that the relevant dichotomy is reference against no reference, rather than no reference against Platonist reference.)”

CD: Yes, but if you wrote the argument as human actions in operational language you would not have this problem – which is purely linguistic. And obscurely so.