Philosophy question

[quinn]

Well, which is it?

[usmhot]

There are proofs, in a number of different forms, that were develoiped by a number of different people - most notably Godel and Turing - that in any sufficiently rich formal system - mathematical, logical, computational, etc - there are statements that can be made that can neither be proven nor disproven.

So, the question you're posing has already been answered - the answer being the third option you give.

[invictus33]

The first two options can't be true, so that only leaves the third option. Saying all things can be proven is thrown out the window when questions of irreducible complexity show up. By proving that nothing can be proven defeats itself and can never be a true statement. So that just leaves you with option #3.

[drudman]

The answer is number two.

All claims are synthetic propositions. Note that all sentences that purport to be claims about "God" or whatever metaphysical entity are not real claims, they are non-senses.

No synthetic propositions can be proved with 100% certainty.

All analytic propositions are true. "No claims can be proven" is an analytic proposition, so it escapes the pitfall of contradiction that another poster believed it had.

[Darryl_P]

Can I not claim a yet-to-be-proven but provable statement about prime numbers for example? If that wouldn't count as a claim then a lot of mathematicians don't know what the word "claim" means.

[drudman]

Mathematical propositions are analytic.

[Piers]

Proof is only meaningful inside a model defined by a set of axioms. For claims within such a system option three applies.

For a claim that concerns a real world phenomenon then it is necessary to encompass the claim within such a model for the concept of proof to have meaning. However as the mapping between model and reality is subjective I expect any claim can be proved or disproved depending on the framework in which you construct the proof. So option one will apply.

[kpux]

Number Three. In any axiom system there are statements that can neither be proven or disproven, ala Godel.

Taking an example from set theory, the cardinality of the set of all rational numbers (numbers that can be expressed as fractions, eg. integer/integer) is infinite. So is the set of real numbers (the rationals plus numbers like pi, e, the square root of 2, etc...). However, they do not have the same cardinality. There are more real numbers than rationals. For simplicity, call the cardinality of the rationals x1 and the cardinality of the reals x2.

The Continuum Hypothesis says that there is no set with cardinality say, x3, such that x1 < x3 < x2. Unfortunately, this is neither true nor false. I think it was Godel who proved that assuming the Continuum Hypothesis is true in conjunction with the normal axioims of set theory creates a perfectly logically consistent system. Then some other guy in the 60's proved that assuming the negation of the Continuum Hypothesis did the same thing.

So, the Continuum Hypothesis is neither provable nor disprovable.

[quinn]

[ QUOTE ]

So, the Continuum Hypothesis is neither provable nor disprovable.

[/ QUOTE ]

Axiom 1: The Continuum Hypothesis is right

Conclusion: The Continuum Hypothesis is right, by Axiom 1

--

Since you can use false axioms, everything is provable.

[kpux]

That isn't one of the axioms of set theory.

[drudman]

Axiomatic systems are analytic, and therefore do not count as claims, a la my previous reply.

The answer is still 2.

[invictus33]

[ QUOTE ]
Axiomatic systems are analytic, and therefore do not count as claims, a la my previous reply.

The answer is still 2.

[/ QUOTE ]

Would you care to define your terms then. So far all you've done is say that you're right and tell people that their ideas are wrong. What are you basing your opinion off of?

[kpux]

what is a claim?

i am drunk

[drudman]

[ QUOTE ]
[ QUOTE ]
Axiomatic systems are analytic, and therefore do not count as claims, a la my previous reply.

The answer is still 2.

[/ QUOTE ]

Would you care to define your terms then. So far all you've done is say that you're right and tell people that their ideas are wrong. What are you basing your opinion off of?

[/ QUOTE ]

http://www.dictionary.com

I am not expressing opinions.

[drudman]

[ QUOTE ]

All claims are synthetic propositions.

[/ QUOTE ]

Cheers Kpux.

[kpux]

hmmmm...then I call BS on this whole thing, and the Continuum Hypothesis argument would appear to be non sequitor

it seems all "claims" are delegated to the realm of metaphysics, which is crap

[Darryl_P]

Where does that definition say that a claim can't be analytic?

[runner4life7]

You guys are just overthinking. Use your brain efficiently. Religion - claim cannot be proven.

Gravity - claim - proven.

I dont know what the discussion is about...

[drudman]

[ QUOTE ]
hmmmm...then I call BS on this whole thing, and the Continuum Hypothesis argument would appear to be non sequitor

it seems all "claims" are delegated to the realm of metaphysics, which is crap

[/ QUOTE ]

As I stated in my first reply, all claims are synthetic propositions, which deal exclusively with phenomena.

[drudman]

[ QUOTE ]
Where does that definition say that a claim can't be analytic?

[/ QUOTE ]

Analytic truths are absolutely true. In is not a claim to say that one is true.

e.g.

"2+2=4 is true" is not a claim. It is redundant. All statements of the form "<analytic truth> is true" are logically equivalent to simply "<analytic truth>". Surely we can all agree that one does not "claim", "2+2=4". One says it, and it is true.

For this reason, claims are exclusively synthetic propositions.

[drudman]

[ QUOTE ]
You guys are just overthinking. Use your brain efficiently. Religion - claim cannot be proven.

Gravity - claim - proven.

I dont know what the discussion is about...

[/ QUOTE ]

Both are wrong.

"God" is not claim. A person who says "God" does not know what phenomenal observations could be made to verify "God".

The law of gravitation is also a claim, but is not proven. It is very extremely highly probable.

[Darryl_P]

Yours is a very trivial, obvious example of an analytic truth. There are others which are so complex, like Fermat's Last Theorem for example, that the world's best mathematicians worked on it for 300+ years without being able to solve it. Fermat "claimed" it was true, but it could only be proven 300 years later.

Also Godel proved that there are some mathematical propositions which cannot be proved, thereby introducing some serious fuzziness into the question of what constitutes an analytic proposition.

The word "claim", according to the dictionary.com definition applies to statements stated as fact but for which there is no proof available. Whether this is because the proof is simply not possible or because humans have not yet been able to do it is not mentioned and appears inconsequential. Mathematicians have widely used the word assuming this interpretation of the meaning anyway, so either you are wrong or a hell of a lot of mathematicians are wrong on this point.

[drudman]

Fermat proposes that for a^n+b^n=c^n there are no non-zero whole number solutions when n > 2. This is either true or false. In 1994, Wiles makes the elucidations necessary to confirm it as true. What is necessary to make these elucidations is the manipulation and logical progression of analytic truths.

To go back to my previous example, "2+2=4" is either true or false, in virtue of itself. One does not "claim" that it is true or false either way. You have mistakenly inferred (or at least, so it seems) that mathematical propositions whose truth or falsehood are not immediately apparent somehow need to be "proved" true or false to actually "be" true or false.* This is not the case.

Godel showed that, indeed, there are mathematical propositions that cannot be proved within their own system, but are nevertheless, true. This is the key - they are true. Being "provable" through manipulation of symbols and other analytic truths within a particular system is not a necessary characteristc of an analytic truth.

On dictionary.com, I found the following relevant definition of "claim" (the noun):

5. A statement of something as a fact; an assertion of truth: makes no claim to be a cure.

As I said in a previous post, to say "I claim that 2+2=4" (or to substitute the definition, "I state that 2+2=4 is fact"/"I assert that 2+2=4 is true") is redundant, and is equivalent to merely saying, "2+2=4".

*Note that if I mistakenly attributed this false implication to what you said, I apologize for doing so.

EDIT: In closing, I think I will simply say that one must determine whether or not synthetic AND analytic propositions are "claims", or whether only synthetic propositions are. I believe that it makes more sense logically to use the latter determination. If one chooses the latter, #2 is true. If one chooses the former, #3 is true.

[malorum]

Regards the God claims. It all depends which set of axioms you use.
I use a foundationalist epistemology (I submit that most people use some variant thereof at least as a working hypothesis, to function within percieved reality)
One of my base axioms involves the innerancy and historical accuracy of the Bible. Within the axiomatic framework thus created it is fairly straightforward to prove a number of claims about God. Equally some claims are a little harder to prove.
But it also depends on how you define 'prove'. The concept of proof in the Christian dogmatic tradition is not entirely in line with the magisterial application of logic that modern rationalist may be used to.

[Darryl_P]

[ QUOTE ]
You have mistakenly inferred (or at least, so it seems) that mathematical propositions whose truth or falsehood are not immediately apparent somehow need to be "proved" true or false to actually "be" true or false.

[/ QUOTE ]

What I inferred was not this exactly but something close (and no apology needed btw). I am aware that an analytic statement's truth or falsehood is right there in the statement itself, but I do consider the key moment of interest the point at which we know which it is. In other words I will behave as though it has no truth or falsehood until I know it but that is partly out of convenience and partly because I have egocentric tendencies. This type of egocentrism, however, is present in such a high percentage of the populace that it has been built into the English language, almost at the level of an axiom (or so it seems).

That's interesting what you say about Godel...so there are statements in an axiomatic system like mathematics that we know with 100% certainty to be true yet cannot be proven!? My first impression is that that's hard to believe but I'm really curious now and will surely look into it.

Or do you mean they are either true or false for sure but we will never know which? This is how I understood it up until now.

[snowden719]

please tell me you are joking.

[AleoMagus]

First, you are basing your opinion on an analytic/synthetic distinction being a correct distinction at all. It is not as cut and dry as you seem to suggest.

As Quine attempted to demonstrate in 'Two Dogams of Empiricism', this may be false. There may not be an analytic/synthetic distinction. Many (if not most) philosophers now hold this view.

Getting away from that altogether, the answer is still #3. I say this because clearly there are claims which cannot be proven but I can also think of many claims which can.

Your line so far has been to call claims non-analytic, based on a dictionary definition

[ QUOTE ]
5. A statement of something as a fact; an assertion of truth: makes no claim to be a cure.


[/ QUOTE ]

But this does not seem to imply what you are suggesting. How does this make 2+2=4 any less of a 'claim'. 2+2=4 is a statement of something as a fact, and 2+2=4 is an assertion of truth. It almost strikes me as sidestepping the actual question anyways. I think the OP is clearly more interested in whether 'statements' or whatever else you want to call them are 1,2 or 3.

Finally, we get into murly waters when we even define what it means to 'prove' something at all. there are differing standards of 'proof' than just the classical deductive ideas many of us are used to. I can concoct all kinds of logics in which proof operates differently, and make many 'claims' even as you descibe them 'provable'.

Geez, if I really wanted to, I could just take russells paradox, form a true contradiction from it. (the set of all sets that are not members of themselves. This set is both a member of itself, and also not a member of itself)

ex falso quodlibet gives us:

A and not-A
A
A or B (where B is any 'claim' whatsoever)
Not-A
------
B (by dysjunctive syllogism)

hmmmmm.... if I allow true contradictons, it seems that #1 may be the right answer.

Though for practical purposes (and logic is far more about practicality than many understand) it is #3

Regards
Brad S