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-   -   Is Mathematics a contingent element in the ad infinitum of legitimacy? (http://www.physicsforums.com/showthread.php?t=490808)

 John Jones Apr16-11 12:01 PM

Is Mathematics a contingent element in the ad infinitum of legitimacy?

Mathematics and ordinary language cannot show that a set of signs is not mathematical or semantic.
For example x+2=2x-2.
This set of signs is assumed to be a mathematical expression by working with it, mathematically, to assess it for truth or falsity. The conclusion, that the set of signs (equation) is false, misses the target: the set of signs x+2=2x-2 is not false, but rather is not a legitimate mathematical expression. There are no false mathematical expressions.

In working out the truth or otherwise of a mathematical or semantic expression we must make the assumption (A) that what looks mathematical or semantic is mathematical or semantic. As this assumption is not one we take up on behalf of either maths or ordinary language, it follows that neither math, nor ordinary language, nor indeed any system of signs, can legitimately show, according to its rules or axioms, that a set of signs is true or not.

Here is an example of working with assumption A in practice, from a definition of proposition from Wiki:

In logic and philosophy, the term proposition (from the word "proposal") refers to either (a) the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. The meaning of a proposition includes having the quality or property of being either true or false, and as such propositions are claimed to be truthbearers.

The author of the above quote takes assumption A, on board, without knowing it, it seems. In b) a set of marks are offered as bearers (truth-trackers) of truth or falsity. Yet these marks cannot show themselves to be legitimate syntax without either assuming that what looks legitimate is legitimate, or else defaulting to position a).

To reiterate, the legitimacy of a set of marks, signs or syntax (whether a syntax is mathematical, etc, or not) cannot be demonstrated from its truth or falsity. Mathematical truth and falsity do not mirror the actual legitimacy of a proposition or set of signs that look mathematical.

And that is why Mathematics is a contingent element in the ad infinitum of legitimacy. Other elements are other systems of signs, and, as systems of signs are independent, are contingent. The ad infinitum refers to the fact that proofs for truth or falsity do not ultimately show that a set of signs belongs to a system - is legitimate.

 zhermes Apr16-11 02:16 PM

Re: Is Mathematics a contingent element in the ad infinitum of legitimacy?

Quote:
 Quote by John Jones (Post 3249805) Mathematics and ordinary language cannot show that ... For example x+2=2x-2. ... The conclusion, that the set of signs (equation) is false, misses the target: the set of signs x+2=2x-2 is not false, but rather is not a legitimate mathematical expression. There are no false mathematical expressions.
You're confusing colloquial conveniences and implicit assumptions for rigorous logic---which is incredibly false. Wittgenstein said, "Philosophy is a battle against the bewitchment of our intelligence by means of our language"

Quote:
 Quote by John Jones (Post 3249805) it follows that neither math, nor ordinary language, nor indeed any system of signs, can legitimately show, according to its rules or axioms, that a set of signs is true or not.
You are absolutely right, but this is old news, see the: Incompleteness Theorem

Quote:
 Quote by John Jones (Post 3249805) And that is why Mathematics is a contingent element in the ad infinitum of legitimacy.
Can't say I really know what that means (not to mention: I'm not entirely convinced it means much, no offense---see Wittgenstein quote above), but I think it just amounts back to Godel's theorem.

 John Jones Apr16-11 08:54 PM

Re: Is Mathematics a contingent element in the ad infinitum of legitimacy?

Quote:
 Quote by zhermes (Post 3249966) You're confusing colloquial conveniences and implicit assumptions for rigorous logic---which is incredibly false. Wittgenstein said, "Philosophy is a battle against the bewitchment of our intelligence by means of our language" You are absolutely right, but this is old news, see the: Incompleteness Theorem Can't say I really know what that means (not to mention: I'm not entirely convinced it means much, no offense---see Wittgenstein quote above), but I think it just amounts back to Godel's theorem.
(The multiquote button does not work, again.)

I was aware of the Godel comparison.
Godel made an identity between two different systems, which is a reductionist procedure. He then confused identity with relationship, and so was able to claim a procedural "proof". Godel's "consistency" arose as a privileged member of this relationship.

 zhermes Apr17-11 12:24 AM

Re: Is Mathematics a contingent element in the ad infinitum of legitimacy?

Quote:
 Quote by John Jones (Post 3250525) (The multiquote button does not work, again.)
Its for multiple posts.

Quote:
 Quote by John Jones (Post 3250525) Godel made an identity between two different systems, which is a reductionist procedure. He then confused identity with relationship, and so was able to claim a procedural "proof". Godel's "consistency" arose as a privileged member of this relationship.
So, you're claiming something that is a select subset/specific-case (perhaps a, 'contingent element') in the full, well-established and excepted (as common sense), Godel theorem.... but you're now complaining about the 'proof' of that theorem.

There's also really no need to use such senselessly pedagogical and pompous language to describe simple (and under-developed) concepts. E.g. 'Godel made an identity between two different systems, which is a reductionist procedure' is actually an incredibly empty statement lacking important, key elements. It would be better to phrase it as, 'Godel equated ____ and ____, but that is an unfair simplification because ______'. Your version is missing the key pieces (the blanks), and instead uses big words to compensate.

Stop tooting your own horn. That's not the point of these forums. If you'd like to ask a question, or present a topic for discussion---go ahead.

P.S. You're senseless use of quotations is really annoying. There is no need to quote "proof" when you are already explicitly stating that it is an improper term for the situation (further-more, single quotes should be used in such a case, not double quotes). There is no context for "consistency", thus the quotes don't make sense.
Correct usage would be: your use of "this" is an unclear antecedent in your final sentence; please clarify.

 John Jones Apr18-11 06:36 AM

Re: Is Mathematics a contingent element in the ad infinitum of legitimacy?

Quote:
 Quote by zhermes (Post 3250820) Its for multiple posts. So, you're claiming something that is a select subset/specific-case (perhaps a, 'contingent element') in the full, well-established and excepted (as common sense), Godel theorem.... but you're now complaining about the 'proof' of that theorem. There's also really no need to use such senselessly pedagogical and pompous language to describe simple (and under-developed) concepts. E.g. 'Godel made an identity between two different systems, which is a reductionist procedure' is actually an incredibly empty statement lacking important, key elements. It would be better to phrase it as, 'Godel equated ____ and ____, but that is an unfair simplification because ______'. Your version is missing the key pieces (the blanks), and instead uses big words to compensate. Stop tooting your own horn. That's not the point of these forums. If you'd like to ask a question, or present a topic for discussion---go ahead. P.S. You're senseless use of quotations is really annoying. There is no need to quote "proof" when you are already explicitly stating that it is an improper term for the situation (further-more, single quotes should be used in such a case, not double quotes). There is no context for "consistency", thus the quotes don't make sense. Correct usage would be: your use of "this" is an unclear antecedent in your final sentence; please clarify.

I never brought up Godel. If you check back on your own posts you will see that it was you who "tooted" (name-dropped) Godel.

Godel made an identity between two different systems. If that phrasing is unfamiliar then my addition that it was a reductionist procedure should make it familiar: we find a similar identity relationship in the (reductionist) identity theory that brain and mind are the same. So, the statement was conceptually precise, but pared down to the bone, and not "incredibly empty".

I did not use only the language of sets, subsets, and "equated", for a very good reason. This language, which you demand is the language I ought to be using, does not express the nature of elements considered in incommensurable systems. Now, if you want to stick to Godellian language then I'm afraid there will be unnecessary difficulties in the Godel philosophical debate.

Your objection to my use of quotations, as in "proof", doesn't stand. It was the first time the word was used, and I used it in the way in which Godel employed it. It had to be in quotes. But I think you knew that already. What you objected to, please tell me if I am wrong, was the fact that you thought I was insulting Godel, who perhaps is something of a favourite of yours.

There is evidence for that (but please correct me if I am wrong), for you misunderstood, in favour of Godel, what I wrote here:

"...it follows that neither math, nor ordinary language, nor indeed any system of signs, can legitimately show, according to its rules or axioms, that a set of signs is true or not".

You applied a Godellian interpretation to that, which was not my intention. A set of signs cannot be said to be true or not, not because it belongs to a system(s) a la Godel, which it does not in any case, and not entirely because different systems can't have their elements set up in relationship, but because truth and falsity only pertain to elements in a system, and elements are not a trans-system species. I can say more, although given your outbursts and demands I would hate to think that any interest you might have in pursuing the topic would be self-thwarted.

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