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In my studies of Mathematics and (mostly) Theoretical Computer Science I've encountered what is known as Munchhausen's Trilemma which purports to demonstrate nothing can be actually be proved because all proofs are one of the following:

  • The circular argument, in which the proof of some proposition is supported only by that proposition
  • The regressive argument, in which each proof requires a further proof, ad infinitum
  • The axiomatic argument, which rests on accepted precepts which are merely asserted rather than defended

My response: What's wrong with circular reasoning in the first place? I simply disagree fundamentally with the notion that we can't have "closed systems" of knowledge. Why is there this idea that we need some kickstarter axioms to prevent going around in circles? It's not that I don't appreciate the need for axioms from a convenience perspective. It's that I don't think that there's really no merit to going around in circles in the first place!

"I am God therefore I am God" is valid. Things don't need to make sense. After all sense is not an inherent property of a statement. It requires interpretation to judge that. Interpretation from another language - in the form of symbolic manipulation. It's all inherently meaningless - so why bother with making sense when you can just be valid.

This is my argument for why circular reasoning is a blessing. Instead of hunting down possibly ad infinitum the "source", the most "fundamental" axiom, the "point", we can prevent the existence of the Hydra altogether. It's either circular reasoning, or the same thing in disguise but with more stress on top of it - the difference is that in the latter the circle never runs out. Why choose stress?

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    Münchhausen trilemma is a good philosophical argument against the "dogmatist" point of view asserting that humans are able to have certainties, i.e. assertion proved without assumptions. Science and mathematics do not need the dogmatist point of view... – Mauro ALLEGRANZA Jul 1 at 13:00
  • From a philosophical point of vies, skepticism is as old as philosophy itself; see e.g. Pyrrhonism. – Mauro ALLEGRANZA Jul 1 at 13:01
  • But in "practical" life, as well as in science and mathematics, we rely heavily on assumptions and similar; the rational approach is to have a critical attitude with respect to our assumptions and theories. – Mauro ALLEGRANZA Jul 1 at 13:03
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    Hi Novicegrammer, welcome to the site. I took the liberty of editing down your question to help it focus in on what seems to be your key issue. If these edits are not helpful, please feel free to revert them. However, they may help your question stay open and attract better answers. – Chris Sunami supports Monica Jul 1 at 19:17
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    Because by arguing from axioms one can trace how things logically depend on each other, when going around in circles the exercise is a waste of time. Chosen axioms do not need to be "fundamental", axioms of modern mathematics are creatures of convenience, nor do "fundamental" axioms need to exist at all. The idea of a "closed system" is known as coherentism. But to make sure that things in them do cohere circles need to be broken somewhere. – Conifold Jul 2 at 3:34
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Informally, circular reasoning is seen as analogous to trying to lift yourself up by your own bootstraps. I can lift you up, or you can lift me up, but we can't both lift each other up at the same time, because it leaves us no foundation of support.

More formally, the way the concept of proof is constructed in systems of structured logic makes it impossible to "prove" something in such a system via circular reasoning. (Formal logic is an artificial system created largely to give a well-defined notion of proof, so it is allowed to place whatever qualifications on that it wants.)

With all that said, you have a valid point that in practice, outside of the world of formal logic, we often take things as true or as valid knowledge because they cohere with other things we think are true, not because we can draw a line of inference from them back to axiomatic first principles. You are not the only person to have noted this, it's one version of what's called the "coherence theory". However, it is vulnerable to the criticism that it is capable of certifying contradictory statements (or incapable of judging between them). "I'm beautiful because I'm beautiful" certainly coheres with itself. But it's no proof that I'm not ugly.

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  • I agree with your first paragraph and the last sentiment - but even a proof itself requires something before it. Any logic and foundation I've ever seen so far required something prior - at least metasyntax. So I guess you could say - my biggest issue is that the Tri-Lemma itself can't be understood without either one of the buckets it refers to. I only see proofs as symbolic manipulation and nothing else. They don't mean anything to me more than any other statement. I'll take a look into "coherentism", thank you – Novicegrammer Jul 2 at 8:13
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The Cartesian Circle is an example of circular reasoning being problematic. One thing leans on another, and there's no structural integrity created.

For me the solution to the trilemma is in the total structure, beginning where we are, and forming strange loops, both 'jumping out' into meta-thinking (at risk of infinite regress), circular reasoning but with such 'jumpings out' so one system can check another for instance, and choosing axioms which we still tweak as we stand back, or in a circular process, seeing the consequences our axiom choices have - like how the rules of a game try to be as simple as possible to make 'a good game', but prevent dead ends or boring tussles or the sense any one player has no chance but must keep playing etc. Strange-loop systems begin where they are, and go beyond the trilemma through self-reference and feedback loops. Creating a fabric, that can be judged by the total integrity, and expanded or repurposed in parts as needed. This is especially significant in accounting for what human minds can do, but computers cannot, yet.

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