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Every proof in mathematics is an implication of the form A -> B where B is the proven statement and A is the premise (which can consist itself of many conjuncts like axioms, inference rules, theorems from axioms etc.) That means no mathemtical proof is unconditional.

But that means we'll never know that 'this and that is the case', all that we know is that 'if this and that is assumed' then 'this and that is the case'. As an example: we'll never verify unconditionally that V2 is irrational, only that within Q, certain rules and axioms, V2 is irrational and we also cannot verify unconditionally that at least 'within Q, certain rules and axioms, V2 is irrational' because we'd need some premises again to prove it.

It means that our proofs and justifications of statements (in math) can never get us to certainty to have grasped a truth. There is an eternal split between proof and truth that is so fundamental that it's even unprovable, it just reveals itself by occuring over and over again, without exception so far. Isn't that much more distressing for mathematics than Gödel's at least negative proofs of the Incompleteness Theorems?

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    You are absolutely right: we can never get us to certainty. Not in mathematics and logic, not in empirical and social science, not in everyday life. May 2 '20 at 16:55
  • But this fact has nothing to do with G's Theorem; G's Theorem is a mathematical theorem that has been proved exactly as every other mathematical theorem. So it is a mathematical truth: why bother about "certainty" ? May 2 '20 at 16:56
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    Not all theorems are conditionals, and there are proofs without premises that discharge all premises introduced in the proof, see e.g. How do I prove something without premises in a Fitch system? Yes, you have to work within some system/practice of proving to even make sense of what "proving" is, but within it you can prove unconditional theorems, like 2 is not the square of a rational. As for "unconditional truth", it is incoherent. One can not have "conditions" to make something meaningful and then turn around and remove them for truth.
    – Conifold
    May 2 '20 at 20:50
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    The issue you raise is basically that the rules of the "game" can't justify themselves. That's not inherently surprising. Godel's incompleteness theorem by conrast says that there is a particular nasty phenomenon which is guaranteed to occur within the game, as long as the rules are reasonable. Don't think of GIT as saying "There are some things we can't prove," since that's really too vague, but rather as a limitation on the possible behaviors of a broad class of systems in general even after we grant their assumptions. May 3 '20 at 1:41
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If the worry is, "We can never prove the axioms true by deduction," then the worry is misplaced---or inescapable, but without the consequences you fear. Of course an axiom can't be proven by deduction (if it really is an axiom, that is). What it comes down to is: you either see it, or you don't. If you have an intuitive grasp of basic set notation, you can "see" the succession from the empty set {} = 0 to {0} = 1, {{0}} = 2, and so on. (Or, you can see that it is either {} = 0, {0} = 1, {0, {0}} = 2, or the simpler nesting given above, or some other such nesting---the alternative is that you don't use the notation in the first place, either because you don't see the point of it (maybe you're an intuitionist!), or you see the point of some other notation (you're some other kind of formalist?), or whatever.) Is this intuitive in the end? Yes. But if you were hoping to avoid intuition no matter what, your hopes were misplaced.

Or think of mathematical questions as occurring in a sort of abstract game. Depending on our axioms, it will be easier or harder to deduce answers to these questions. But intuitively, we don't want to "rig the game" to deduce answers to complex questions by a process of deduction that is relatively simpler than the question is complex. At the limit, we try to avoid the adoption of axioms that directly solve complex problems. For example, we'd rather not just assume that the Continuum Hypothesis is true. We have the mathematical question, "What is the particular output of the powerset operation in the base case of 2 to the power of aleph-zero?" And this is a complex enough question that we'd like to find another axiom (besides the ones we already have in ZFC) that will allow a complex (but relevant) deduction of a specific value. As the doctrine of forcing showcases, we can't get the right kind of deduction in ZFC; and so far we haven't gotten almost anywhere, definitively, using large cardinal axioms; but then, to emphasize the point again, we don't want to "win the game" by rigging it.

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Note : one should be careful not to fall into this trap, namely, confusing

(1) It is not possible to know sth and to be wrong,

that is

necessarily ( If I know that P, then I am right in asserting P ) ,

and

(2) He who knows sth cannot possibly be wrong,

that is

if ( I know that P) then ( I'm necessarily correct in asserting P)

or,

if (I know that P) , then (I cannot possibly be mistaken in asserting P).

Proposition (1) is correct but proposition (2) is not.

Maybe emphasizing certainty in epistemologcal questions is sometimes linked to this modal confusion .

https://www.sfu.ca/~swartz/modal_fallacy.htm

also https://plato.stanford.edu/entries/certainty/


  • One can use Hobbes' distinction between historical knowledge( knowledge of facts) and scientific knowledge ( inferential knowledge).

Note : " history" is understood here as " knowledge of data, of what is given" ( not proved). See : Hobbes , Leviathan.

  • Proposiitons expressing facts are categorically true ( they relate you to reality) , but they cannot be absolutely certain. Maybe only the cartesian cogito , or better the cartesian " I am" is an absolutely certain categorical proposition.

  • Propositions expressing logical implications are not true of any particular state of affairs ( since they hold in all possible situation, all possible " world") , but they are absolutely certain : It is absolutely certain that in any possible situation if Euclid axioms/ postulates/ definitions hold, then the sum of the angles of a triangle is 180 degrees. What is absolutely certain here is the conditional ( taken as aa whole).

As one says in english, "you can't have it both ways": certainty is inversely proportional to truth ( i. e. relation to the world as it is factually).

  • You may also have a look at this link to a text by Russell ( " Epistemology" entry of Encyclopedia Britannica) : "All knowledge is more or less uncertain and more or less vague. These are, in a sense, opposing characters: vague knowledge has more likelihood of truth than precise knowledge, but is less useful."

https://www.marxists.org/reference/subject/philosophy/works/en/russell1.htm

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