Let me use an example from my own field. The example is also well-known among philosophers, so it has the advantage of not seeming obscure.
Gödel proved that the set of mathematical proofs is a proper subset of the set of mathematical truths. In other words, Gödel proved that mathematical truth cannot be identified with axiomatic provability.
This forces a kind of mathematical Platonism, which does establish objective/ absolute truth. To quote Gödel:
"Finally it should be noted that the heuristic principle of my construction of undecidable number-theoretic propositions in the formal system of mathematics is the highly transfinite concept of 'objective mathematical truth,' as opposed to that of 'demonstrability,' with which it was generally confused before my own and Tarksi's work. Again, the use of this transfinite concept eventually leads to finitarily provable results, for example, the general theorems about the existence of undecidable propositions in consistent formal systems."
This is an example of the assumption of absolute truth -- what Gödel described as his heuristic principle -- leading to a profound discovery. It is very likely that the incompleteness results would not have been possible without this heuristic.
Do you believe that formal languages are comparable to natural languages in this respect?
Formal languages are based on axiomatic truths, and they restrict their functionality to describing the relationships of symbols.
They're not based on empirical observations, and they're not dependent on biophysical sensation or on non-axiomatic presumptions.
I don't see any equivalence in this comparison.
(Nice try, though...)
Thanks again for participating. I appreciate our conversation. May we continue?
My experience is in psych and philosophy; yours is mathematics.
What are the consequences of insisting that anyone who disagrees with my belief is absolutely mistaken?
And what would be the consequences of giving up insisting that our inferential beliefs are absolutely true?
Why do people isn't that their beliefs must be true?
Do you have any interest in our motives, @axiogenesis? Do you examine yours?
[wondering]
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I am interested in motives. I think the search for truth (including moral-aesthetic truth) should be a primary motive.
Can we expect anything beyond internal consistency from philosophy? If not, then the philosophical project seems to reduce to determining when two philosophies are isomorphic. Some philosophers might argue that we cannot expect more than internal consistency in any field, but I think I've given a counterexample to that.
Of course, I think truth is more than a value we assign to a proposition. Again, Gödel's theorems actually prove this. This ties in with your observation on axiomatic systems, so there does seem to be a kind of parallel with natural languages because both axiomatics and natural languages are insufficient to fully capture truth. But this view only makes sense on the assumption that we are approximating some truth.
Rescher developed a coherence theory of truth that is interesting, but I haven't looked at it in ten years.
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