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LudovicoVan

Posts: 1,435
From: London
Registered: 2/8/08

Claim: The Theory of All Is Equivalent to Inductive Logic
Posted: Mar 14, 2008 8:37 PM

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Claim: The Theory of All Is Equivalent to Inductive Logic

Inductive reasoning has been attacked several times, but we seem to have some good news here.

To your consideration: again, please do dis-prove.

=== The problem:

Two excerpts from Donald C. Williams, The Evils of Inductive Skepticism, 1947, [http://web.maths.unsw.edu.au/~jim/williams.html]:
> it is safe to say, with Whitehead, that 'the theory of induction is the despair of philosophy - and yet all our activities are based upon it'. [A.N. Whitehead, Science and the Modern World (New York, 1925, p. 35]
> [...]
> Western culture [...] depends on induction altogether and in principle. Spes est una in inductione vera. ["Our only hope is in true induction", Francis Bacon, Novum Organum, aphorism 14 in the first book.]

And, from the Wikipedia, [http://en.wikipedia.org/wiki/Inductive_reasoning]:
> Inductive reasoning has been attacked several times. Historically, David Hume denied its logical admissibility. During the twentieth century, thinkers such as Karl Popper and David Miller have disputed the existence, necessity and validity of any inductive reasoning, including probabilistic (Bayesian) reasoning.

=== The claim:
------------------------------
The Theory of All[*] (TOA) (as for now, a number theory) is _equivalent_ to Inductive Logic (IL) (a logical theory). This makes TOA a _general theory_.

Rationale: Zero exists, and from One we get Two
- Zero is the (self-)referential Empty
-- One is the (tauto-)logical Identity
--- Two is the (auto-)contradictory Choice
------------------------------

=== Sub-conjecture 1:
Zero,One,Two are equivalent to NOT,EQ,IIF operators, and so on.

=== Sub-conjecture 2:
Zero is _odd_ and One is _even_, and so on.

This said:

Would you second the claim? If not, why?

What about conjecture 1? (I'm more or less guessing on the operators, really wandering about some "natural formal language"...)

What about conjecture 2? (Is even/odd just conventional? I don't remember where in this forum, but there was something about an asymmetry with even/odd when the domain of analysis includes zero...)

BTW, I suppose there may be relevant connections with: "Representability Provability and Truth", [http://mathforum.org/kb/thread.jspa?threadID=1712092], and "Re: Godel proved maths inconsistent not incompleteness theorem", [http://sci.tech-archive.net/Archive/sci.logic/2008-03/msg00647.html].

For obvious reasons, I'll let you answer Popper and Miller. Then even Hume should be satisfied. :)

Overall, please consider that, strictly speaking, as long as the claim holds, we can _legally_ move outside of mathematics, though with a _valid_ foundation, both to _reasoning_ and to _applications_.

Thank you very much,

Julio

[*]TOA: http://mathforum.org/kb/thread.jspa?threadID=1708310