# Does truth exist without proof?

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?

• Suppose A is a logical statement. then A or not A is true. We just managed to prove the statement "A or not A" without any assumptions. Then again, this is assuming that a statement is exclusively true or false. – Graviton Jul 27 '20 at 12:14
• It depends... According to a well-known point of vies (see realism) the sun exists also when we do not look at it. The same approach regarding "mathematical facts": they are "out there" irrespective of the fact that we know them or not. See Platonism. – Mauro ALLEGRANZA Jul 27 '20 at 12:17
• According to a different point of view (see Intuitionism) it makes no sense to assert that a mathematical statement is true if we have no proof of it. – Mauro ALLEGRANZA Jul 27 '20 at 12:18
• Please note that propositions like "if P then Q" are not true or false, but valid or invalid. If the set of logic rules that were agreed upon are applied properly when infering it (validity criterion), it just means "under those rules, when holding P to be true you have to also hold Q to be true". It's propositions like P or Q that can be true or false. – armand Jul 27 '20 at 23:31
• @armand Every statement in maths must have a truth value. The implication "if P then Q" has a truth value. Look here. Truth table – ado sar Jul 27 '20 at 23:59

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!

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.

• Impeccably put. Are there pragmatic implications to this asymmetry. – gonzo Jul 28 '20 at 0:04
• – Jo Wehler Jul 28 '20 at 6:29
• You say: "a statement S of theory T is true if it holds in every [after editing] model of T". For a first-order theory, like e.g. Peano arithmetic PA, the Completeness Theorem holds: every statement of T that is true in every model of T is provable from the axioms of T. So what is missing ? – Mauro ALLEGRANZA Jul 28 '20 at 12:09
• @Mauro In PA there are statements which are true but not provable in PA (Paris-Harrington theorem). See karlin.mff.cuni.cz/~krajicek/ph.pdf, cs.umd.edu/users/gasarch/TOPICS/largeramsey/bovINTRO.pdf – Jo Wehler Jul 28 '20 at 14:54
• They are true in N and not "true in every model" – Mauro ALLEGRANZA Jul 28 '20 at 15:02

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.

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...

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.

• Outside the bounds of propositional logic in the realm of experience of those 'things' which are termed 'actual' or 'actually existing', Spinoza maintains that the 'truth' or certainty of knowledge of reality is 'self-evidently' true and could not be true otherwise. But more to the point of your question, if any statement is not in some measure understandably true and requires a proof, then as you suggest, that proof would require a proof, and so on, in an infinite regression. – user37981 Jul 27 '20 at 17:29
• About G's Th: not exactly... The theorem holds for formal systems of a certain type (roughly speaking: containing some portion of arithmetic). "Truths" the are unprovable in the system are not "stumbled upon", but rigorously proved in a different system. G's Th itself "has been proved by the mathematician Kurt Gödel." – Mauro ALLEGRANZA Jul 28 '20 at 10:13