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What is it that mathematicians, and more likely perhaps philosophers, give as an explicit justification that any method of formal logic, which is actually used by mathematicians, or even by automatic theorem provers, to prove anything, be it some theorem or some argument, would be effectively the best method to make valid deductions.

As far as I have been able to ascertain, there is no such justification.

Apparently, all logical proofs, formal and informal, seem to rely ultimately on the intuition of at least some human being as to what formulas are logical truths, and more likely on the consensus of the specialists since Aristotle as to what formulas are logical truths, consensus which itself seems to rely ultimately on the intuition each specialist may have as to what formulas are logical truths, such as for example p and q implies p, the Modus Tollens, Aristotle's syllogisms etc.

While closely related to my previous question on the justification of systems of "logical calculus", the focus here is not on the usefulness of the methods used but on the foundation of formal logic as articulated or even theorised from the perspective of each of the various methods used.

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From an intuitionist point of view there is no argument that any of these are the best way of doing anything. Brouwer proceeds from Kant's notion that there are limits to what human beings can understand. What disagrees with our most basic intuitions can be accepted, but it can never be realistically understood.

That doesn't mean any of this is efficient, or best, but that it is basic and we are stuck with it. This is one of the reasons the original hard-core Intuitionism just outright discards the strong sense of negation -- the idea that reaching a contradiction can be used to establish existence in particular.

To presume that we know all the things that accord with our intuitions are never going to lead to an unresolvable contradiction is to assert that we can understand everything, which seems unlikely, especially since we seem to have irresolvable paradoxes built into intuitions of free reference and universality (Russell's) or continuity and infinity (Tarski's). So even though negation itself is an intuition, it needs to be pared back in light of honesty.

(So mathematics becomes the work of deciding what the largest set of nicely consistent intuitions can be, and what those imply, making it a branch of psychology rather than something mysterious.)

This suggests the answer to the "unreasonable" effectiveness of mathematics as a model of the world via the anthropic principle. If it were only reasonably effective, we would not have survived this well.

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  • @jobemark Intuitively, contradictions don't by themselves don't speak of the existence or non-existence of anything in the ontological sense. A contradiction, however, does tell us that at least some of our assumptions are false. In that sense, a contradiction does establish that the thing that we have assumed cannot exist as such, and therefore doesn't exist in the ontological sense. And if we further assume that logic is a natural capability of the brain, we have the explanation of the effectiveness of mathematics and of the human brain.
    – Speakpigeon
    Jan 22, 2019 at 10:52
  • @Speakpigeon Classical mathematics contains proofs of existence via contradiction: en.wikipedia.org/wiki/Brouwer_fixed-point_theorem So your first comment is not true. If you do not modify the interpretation of contradiction there are contradictions that imply things exist. But it is not safe to use or reference such objects unless you think the machinery of mathematics is somehow flawless and we can really trust the Law of the Excluded Middle to apply to everything. Brouwer rejects this usage, leading to a more modest 'constructivist' mathematics.
    – user9166
    Jan 22, 2019 at 14:56
  • By tolerating a certain, clearly defined level of inconsistency, he avoids most of the major paradoxes of modern mathematics and preserves the naive interpretations of things like Set Theory instead of injecting a formalism. As a side effect, he produces a reasonable definition of mathematics without resorting to Platonism or over-valuing formalism.
    – user9166
    Jan 22, 2019 at 14:59
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Fundamental logic comes from observation. Nature behaves like that.

For example, when people draw circles to illustrate that if circle A is included in circle B and circle B is included in circle C then A is included in C, and then use this as an example for any object or set of objects A,B and C.... Then that rule emerges from all observations we've ever made including observing drawn circles on a piece of paper.

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  • Personally, I doubt we need to look at circles to know this logical truth. So, although what you suggest seems possible, I don't think that this is scientifically proven true. So, can you accept that for all you know, a logical truth, such as indeed that if it is true that A implies B and B implies C, then it is true that A implies C, may well be known entirely independently of the life experience a person can acquire through sensory perception?
    – Speakpigeon
    Jan 22, 2019 at 10:24
  • unless you are dealing with circular logic, everything has to have foundations outside of its area. Biology depends on chemistry, chemistry depends on physics, physics depends on math, math on logic, then logic hasnt much anywhere else to go but nature & language. Btw the circle thing isnt "scientifically proven true" because math and logic arent sciences. They are the foundations of science.
    – Manu de Hanoi
    Jan 22, 2019 at 10:49
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    Yes, I meant my suggestion as per the naturalistic paradigm you're describing. We all receive at conception a massive input of data in the shape of our DNA, and our DNA is basically the blueprint for what kind of logic is compatible with nature because it has been determined by natural selection over 525 million years. That has to have more authority than a paltry pair of millennia of mathematics and formal logic, and that explains the logical effectiveness of the human brain. We get the general truth of the "circles" because we already have the logic to interpret what we see.
    – Speakpigeon
    Jan 22, 2019 at 11:06
  • hanoi If you think that we learn logical truths through experience in life, then science should be able to prove it conclusively.
    – Speakpigeon
    Jan 22, 2019 at 11:09
  • @Speakpigeon science can't prove logic, but there is certainly a lot of science about kids developping logic along their development. That said, we get the logic of the circles cause we can see it on paper, nothing to do with formal logic.
    – Manu de Hanoi
    Jan 22, 2019 at 11:18

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