Valid Methods for Proving Statements: Assumption and Contradiction

[WhyIsItSo]

I have read that a valid method for prooving a statement is to assume the opposite and show a contradiction.

This tells me the assumption is an "either or". If this is not true, then that must be.

Is this always valid?

 

[HallsofIvy]

Well that depends strongly on what this and that are, doesn't it?

It is true that two contradictory statements cannot be true. If assuming "P is not true" leads to an impossible situation (proving a statement and its contradiction) then P must be true.

A special case of this is proving the "contrapositive". If a statement say "if P is true then Q is true", then its contrapositive is "If Q is not true then P is not true". Proving the latter is equivalent to proving the former.

 

[CRGreathouse]

I suppose constructivists sometimes have problems with this in some cases (I don't know, I'm not one) but contrapositives are pretty widely accepted.

 

[WhyIsItSo]

Ok, it all comes down to the particular situation. The logic holds, but may not be of any help if the situation is not reflected by the logical rule.

The contrapositive case you mention, "if P then Q" therefore "if notQ then notP", is valid but of no value if you need to know conditions where Q may be false. "If P then Q" does not exclude "Q when notP".

It comes down to your first answer. It just depends on the "this" and the "that".

It would seem my OP was rather foolish

 

[Hurkyl]

I suppose constructivists sometimes have problems with this

Right; in an intuitionist logic "P or not P" is not a theorem. But it's "almost" a theorem, since "not (P or not P)" is, in fact, false.

 

[lokofer]

But how can we be sure that we really have a "proof" ?..for example a good way to prove Riemann Hypothesis would be to obtain a Hamiltonian (for example) $$H= p^{2} + V(x)$$ so (i think) $$\zeta (1/2+iH)|\Phi>=0$$ and this V(x) satisfy (exact or as an approximation) an Integral equation of the form:

$$g(x)=\int_{0}^{\infty}dy K(x,y,f(y))$$ ?..

then from the point of view of a mathematician or a logicist (is that the word?) Do the potential V(x) exist?..of course perhaps we could calculate it Numerically, or by other method, but how do we know looking at the "equation" above that the V(x) exist?..

 

[Hurkyl]

But how can we be sure that we really have a "proof" ?

By checking. Proof verification is, in principle, a trivial thing. You simply look at each line in the proof, and check to see if there's a rule of inference that allows you to infer that line from the previous lines and the hypotheses. If every line passes this test, then you really have a proof. If one of the lines fails this test, then you do not have a proof.

 

[lokofer]

Then Hurkyl..how could you be completely "sure" according to your reasoning and Eliyahu Rips'..there're secret codes in Bible..which seems completely stupid... but if you take a look into his/their jobs ( i can't since i don't know much probability) perhaps you can't find any fault.

 

[shmoe]

You would either learn the necessary probability and statistics to understand their paper for yourself, or you can defer to expert opinions (there's a fair bit on this):

http://cs.anu.edu.au/~bdm/dilugim/torah.html

I've understood enough of the debunking papers on the page above that I'm satisfied Rips was a load of bs. If it were somehow important to what I was doing myself, I'd make more of an effort to read Rips paper first hand.

If you don't have the necessary background to check your own work, then it's really up to you to learn the necessary background. You also might wonder what makes you capable of new research in a field that you aren't capable of judging the correctness of work you come up with. Everything you write could be complete nonsense and you'd have no idea, this isn't a very productive approach. I'd compare it with me randomly stringing together some Spanish words hoping that I write a lovely poem.

 

[CRGreathouse]

I'd compare it with me randomly stringing together some Spanish words hoping that I write a lovely poem.

You're on.

 

[lokofer]

You would either learn the necessary probability and statistics to understand their paper for yourself, or you can defer to expert opinions (there's a fair bit on this):

http://cs.anu.edu.au/~bdm/dilugim/torah.html

I've understood enough of the debunking papers on the page above that I'm satisfied Rips was a load of bs. If it were somehow important to what I was doing myself, I'd make more of an effort to read Rips paper first hand.

If you don't have the necessary background to check your own work, then it's really up to you to learn the necessary background. You also might wonder what makes you capable of new research in a field that you aren't capable of judging the correctness of work you come up with. Everything you write could be complete nonsense and you'd have no idea, this isn't a very productive approach. I'd compare it with me randomly stringing together some Spanish words hoping that I write a lovely poem.

I'm not saying the paper is correct or not..simply that the conclusions are simply "Nonsense" and that no serious journal should have published it...if i gave you a 100 pages paper about "The Quadrature of Circle" or "God exist because i have discovered a Wave function in Quantum Gravity that.." I'm sure that Scientific community didn't put any atention to them (my papers), it reminds me of NObel prize winner "Crick" who believed that DNA came from outer space... :laughing: :laughing: i suppose we're talking about science not Sci-Fi.

 

[HallsofIvy]

But how can we be sure that we really have a "proof" ?..for example a good way to prove Riemann Hypothesis would be to obtain a Hamiltonian (for example) $$H= p^{2} + V(x)$$ so (i think) $$\zeta (1/2+iH)|\Phi>=0$$ and this V(x) satisfy (exact or as an approximation) an Integral equation of the form:

$$g(x)=\int_{0}^{\infty}dy K(x,y,f(y))$$ ?..

then from the point of view of a mathematician or a logicist (is that the word?) Do the potential V(x) exist?..of course perhaps we could calculate it Numerically, or by other method, but how do we know looking at the "equation" above that the V(x) exist?..

eljose, why have you changed your name? As I have pointed out before, mathematics is not physics. If you had a proof of mathematical statement using a "Hamiltonian" or any other "energy" function, it could be phrased simply in terms of functions and differential equations having nothing to do with physics- and would be simpler to read.

You keep saying "a good way to prove Riemann Hypothesis" and then say "perhaps we could" but you don't know how to do any of those things. It is not "a good way" until you can do those things.

 

[shmoe]

I'm not saying the paper is correct or not..simply that the conclusions are simply "Nonsense" and that no serious journal should have published it...if i gave you a 100 pages paper about "The Quadrature of Circle" or "God exist because i have discovered a Wave function in Quantum Gravity that.." I'm sure that Scientific community didn't put any atention to them (my papers), it reminds me of NObel prize winner "Crick" who believed that DNA came from outer space... :laughing: :laughing: i suppose we're talking about science not Sci-Fi.

Their conclusions were nothing like "God exists", it was supposed to be a study of the probability of these equidistant sequences appearing and making sense. The many problems in their analysis weren't noticed by the referees and it was published. Being published is NOT a guarantee that a paper is correct. There's a reason any proofs of one of the clay prize problems has to stand published for a year (or two?) to give a wider audience a chance to scrutinize them.

 

[lokofer]

- "Hallsoftivy" the question is that someone "kicked" me out from the forum for expressing certain opinion... from now on i'll keep my opinions for myself.

- The "Hamiltonian" approach comes from the identity (¿?):

$$Z(u)=\sum_{n} e^{iuE(n)} \sim \iint dxdpe^{iuH}$$ $$H=p^2 +V(x)$$ so if we "knew" (i have not the faintest idea of what the shape of Z(u) is although the "eigenvalues" would be the imaginary parts of the Non-trivial zeros of $$\zeta(s)$$ ) integration over "p" we could get a NOn-linear integral equation.. that ¿exists? but...i can't go further

 

[matt grime]

I have this amazing function. Call it Amaz(t), and it is a function from the set {0} to the set {0,1}, now, if I could work out what Amaz(0) was I could prove the Riemann Hypothesis... The function, for those who haven't guessed it is defined by Amaz(0)=0 if there is a non-trivial zero of the Riemann zeta function off the critical line, and 1 otherwise. Now if only someone were smart enough to be able to evaluate it?

I've also got an even better function. SuperAmaz, and it is defined as a function from the space of zeta functions to the underlying field. SuperAmaz(zeta) is the product of all non-trivial zeros not on the critical line, with the empty product taken to be 0 (if the product of all zeroes diverges then we set it to be 1). If I could just work out if SuperAmaz evaluates to something non-zero on the Riemann zeta function! I don't even need to know its exact value either...

(Yes, it's sarcastic, eljose, but I'm really bloody tired of these stupid games.)

 

[lokofer]

-There's no need to be so rude...or ironic about this perhaps you could use some function involving a sum over non-trivial zeros $$\sum_{\rho} x^{\rho}$$ or something similar...

- Perhaps these discussions are just a nonsense..if the moderator desires can "erase" or delete my discussions about this topic.