Does truth exist without proof?

Question

When we prove something (e.g. in maths) we show that a particular statement is true. But if we couldn't prove that statement that doesn't mean that the statement would be false right? So is proof a process that let us know that a statement is true? For example when someone states "If P then Q" is true we could ask him "How do you know it is true?" or "Prove it". Also, usually we prove a statement by using other statements that have been proved. Do we this because it would take much time to prove all the other statements from scratch? Could we prove that particular statement without using any other proven statement?

Answer 1

Just to add the very trivial point that at the very least the Intuitionist Mathematician must also accept Definition in their sources of legitimate truths. These are taken to be in some sense “analytic” truths, declaring them not so much to be matters of substance than semantics, but our choices of definiens greatly influences the resulting mathematical structures that emerge from our proof processes!

Answer 2

For the realist point of view, see Truth-value realism.

Truth-value realism is the view that every well-formed mathematical statement has a unique and objective truth-value that is independent of whether it can be known by us and whether it follows logically from our current mathematical theories.

For a constructivist point of view, see See Enrico Martino, Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics, (Springer, 2018), Chapter 11 Temporal and Atemporal Truth in Intuitionistic Mathematics, page 97-on:

Martin-Löf (1991) distinguishes between actual and potential truth of a proposition. These notions would be explained intuitionistically by the notions of actual and potential existence of a proof. A proof of a proposition $A$ exists actually if, as a matter of fact, $A$ has been proved; it exists potentially if $A$ can be proved. Here possibility is not understood in the traditional intuitionistic sense as knowledge of a method to prove $A$, but as “knowledge-independent and tenseless” possibility. Accordingly, a proposition that has been proved becomes actually true, but it was potentially true even before having been proved, and it would be true even if, in fact, it had never been proved. In this way, according to Martin-Löf, the intuitionist can overcome the well-known objection that saying that a proposition becomes true just when it is proved is counterintuitive and in conflict with the standard use of the truth predicate: potential truth is not open to that objection.

According to this point of view, prior to the proof by Lindemann (1882), the transcendental nature of π was potentially true and become actually true with the 1882 proof.

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usually we prove a statement by using other statements that have been proved. Do we this because it would take much time to prove all the other statements from scratch? Could we prove that particular statement without using any other proven statement?

This is the gist of the Axiomatic method, that we are using since Aristotle and Euclid.

A proof requires a "logical machinery" (inference rules) to deduce new statement (theorems) from existing ones (previous proved theorems and axioms).

We have to start somewhere...

Answer 3

You may define a statement of a theory as "true" if it holds in every model, and as "provable" if there is a logical deduction from the axioms of the theory.

Then Goedel's first incompleteness theorem tells us, that any theory, which is as least as powerful as number theory, contains true statements which are not provable.

Hence "true" does not imply "provable" while the converse holds.

Answer 4

The mathematician Kurt Gödel proved that there are more arbitrary mathematical truths, which can only be stumbled upon, than there are mathematical truths which can be proved through rigorous logic.

Anybody claiming that truths do not exist until proven must explain how truths can nevertheless be stumbled upon without rigorous proof.

Cue much handwaving over "No, but what I mean by 'truth' is ..." -- there follows a statement of some belief which itself cannot be proven true.