To say: " Today, I am feeling positive!", -would be a view on your emotion.
To make such a statement, you had to have experience, of a previous emotion, that was contradiction to this "positive" feeling.-else you would not have mentioned how you are feeling today. It is therefore probable that you can measure how you feel, as a result of an opposite feeling. The reason why I prevent this as evidence, is because you have experience of what I mean:)
I'm not sure what you mean strangequark. Would the trivial proof that there are infinitely many primes work?
Well that proof commonly involves proof by contradiction but I am fairly certain there is a constructive proof for the existence of an infite number of primes as well, so I don't think that would work. I want to know just how important the law of the excluded middle is.
quark, I dont think i quite understand the question, because it is little complex, the words "the law of the excluded middle" When one work with decimals, one notice that there is a zero with a comma....like this 0,9999999999999 That zero would never ever change to a 1,0000000, unless someone in the infinite round it off, and then only it become one! It mean that between the number, 3 and 4 there are also infinity, and so on and so on. The question is, do we have the right to change figures for us to understand(meaning, should we round it off), and therefore exclude the middle...the middle is also relative to many things, but can it be that by doing so, that we actually tamper with what is, and make it fit to our understanding, rather than wanting to understand infinity itself? Are we wasting time with figures, like a record that is stuck...at the same time we want answers, but are we prepared to explore our own thoughts, to set it free?
Something that bothers me is, if it is really possible to make a 0...you might say, that we make it each day. you take a pen, and write it...but what a zero actually is, is an impossible shape...lets look at it, from the universe prospective....your pen, touch the paper to draw the zero, at a specific point in time, then you bring it around, to end at that same point in time again!!!! hallo!!!! take away the paper, and track the movement of the point in time where the pen touched a point in space...that point in space, is moving away from the pen at a rate, of the rotation of the earth, and also around the sun, and also around the galaxy and also away from wherever the big-bang takes us. the actual points that are joined on paper does not look like a zero at all...Question, are we thinking on the right tracks...do we want answers for things we create on paper. Would we consider, that we actually came to a situation, where we could understand our own shortcomings...I think that zero can therefore be excluded, its relative....perhaps we should get a common foothold from where we start to think, would that be at zero? Maybe we shouldn't exclude it, else we would have no place from where to create our own thoughts
Can one make a constructive proof proving that there exists some statement that cannot be proven constructively, but that only a proof-by-contradiction can establish such a proof?
I am assuming it is likely that such a proof would involve some diagonalization method of pairing the qualities in question? Because of course one can make a proof-by-contradiction that is still accepted in that it is still "constructive".
How weird, a friend of mine asked me almost this identical question last week. I would hunt down a logician. I doubt most mathematicians can answer this question.
Yes, I understand this is more of a foundational issue that most of us here would not ordinarily deal with. I wanted to try anyways.
Well I suppose some people may think it is cheating to make a constructive proof like a diagonalization which is also a reductio. This was just a guess as to how to proceed, and I don't suppose there could be other methods as well.
How about this way of turning a reductio ad absurdum into a normal one?
Take your reductio ad absurdum proof and transform it thus. Negate every statement, and reverse all the implications between them. The last statement was false, so it becomes true, so the implications are valid from the first to the last (opposite order to the reductio ad absurdum proof). The last statement, being the negation of the first statement in the reductio ad absurdum proof is the statement we want to prove.
This merely uses the rule:
if (P => Q) then (not(Q) => not(P))
Also known as proof by contrapositive?
Can one make a constructive proof proving that there exists some statement that cannot be proven constructively, but that only a proof-by-contradiction can establish such a proof?
I am assuming it is likely that such a proof would involve some diagonalization method of pairing the qualities in question? Because of course one can make a proof-by-contradiction that is still accepted in that it is still "constructive".