Magnus Anderson wrote: ↑April 30th, 2022, 7:04 am
I don't think that the so-called "correspondence theory of truth" is a theory. It's a definition. I don't think that definitions are theories. A definition is merely a statement about what meaning is assigned by some people to some word. What I'm saying is that the meaning assigned to the word "truth" by pretty much everyone -- literally everyone, whether they are aware of it or not -- is best captured by the statement "a belief that corresponds to the portion of reality that it refers to". That might be wrong, of course, but I wouldn't say it's a theory.
This is certainly a possible view.
The reason why I refer to it as the "correspondence theory of truth" is because it seems to be somehow the standard way of naming it:
https://en.m.wikipedia.org/wiki/Correspondence_theory_of_truth
Wikipedia on "Correspondence theory of truth" wrote:
In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world.
Magnus Anderson wrote: ↑April 30th, 2022, 7:04 am
I would actually go so far as to say that any given mathematical statement is true
precisely because it matches the portion of reality it refers to.
It matches a portion of the abstract universe constructed by its theory ("which interprets it's theory"). The term "reality" may be confusing in this context. A logic sentence does not seek to match a portion of physical reality.
Magnus Anderson wrote: ↑April 30th, 2022, 7:04 am
Let's take a very simple mathematical statement as an example e.g. "2 + 2 = 4". What is this statement actually saying? That's the question that seems to paralyze most people. What that statement is saying is that the symbol on the left side of the equation (which is "2 + 2") has the exact same meaning as the symbol on the right side of the equation (which is "4".) To answer that question, one actually has to look at what meanings were assigned by those who made that statement and whether or not they are in truth equal. It's a linguistic statement, i.e. a statement about language, fundamentally no different from statements such as "The English word "train" has the same exact meaning as the Spanish word "tren"".
Yes, agreed that it is about language, but disagreed that it would have meaning.
In the case of "2+2=4", we can prove from Peano Arithmetic Theory, by using the axiomatic recurrence relation a+S(b)=S(a+b), that:
S(1) + S(1)
= S(S(1)+1)
= S(S(S(1)))
= S(S(S(S(0))))
= the fourth successor to zero
When we look up the fourth successor of zero, we find the symbol "4".
Since "2+2=4" is therefore provable from arithmetic theory, and in application of the theorem of soundness, this logic sentence is true in all interpretations, i.e. models or universes, of Peano Arithmetic Theory, the intended standard of which are the natural numbers.
In that sense, the truth of this logic sentence is not really a consequence of its meaning but a mechanical (synctactic) consequence of the rules that construct arithmetic theory.
That is why a machine can trivially verify the provability of this logic sentence. It is still all about (formal) language, because verification rests entirely on meaningless symbol manipulation.
Wikipedia on "mathematical formalism" wrote:
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
Therefore, according to mathematical formalism, the logic sentence "2+2=4" is true but is deemed to mean nothing at all.