so first off i will state that math is a human language... whenever an article is published or a paper written and it is said to be true it must be sited for where the info came from and often times the source has to be checked as well because, who says they are right anyway. when a scientist finds mathematical proof that something is correct and others look over the work and see there are no mistakes it becomes accepted as proof and no sources are required... why does math always prove something correct? anybody got some good answers?
You mean: why do people accept a theorem from a reference once it is generally agreed upon that the proof of that theorem there is correct?
well yeah... i mean if someone comes up with a theorem... others check for errors and if the math is correct... it is generally accepted... with most arguements you have to argue sources and language terminology... but if you prove something mathematically it is pretty much correct by default.
Math proofs are derived from a very very small set of fundamental axioms. Fundamental, like 1+1=2 and two straight line segments in Cartesian space intersect at, at most, one point. There's not a lot ofpoint in taking up any science unless you grant these first few axoims. From those first few, all others can be directly created. A new, published math proof consists of showing that your new formula can be derived directly from existing axioms. No math proof is ever accepted until many colleagues have pored over it and concluded that there is no flaw. In a nutshell, if one accepts the initial underlying axioms, and the rest of the math is done correctly, one has no choice but to accept the new formula.
So to summarize, I guess the answer to "why do we accept a theorem once it is proven?" would be: "by definition of a proof".
What? No. The symbols used in math could, in some sense, be considered a language, but they are not math itself. If I make up my own symbols but solve the same problem, I would not be doing "different math", I would just be writing it a different way.
yes but if i make up my own symbols for english and speak them when i see the symbols im still speaking english just not writing english...
well i guess... but what i was getting at is that if you are arguing a point you need to site your sources and prove their credibility but in math you just need to prove that YOU didnt make any mistakes.
I think you're under a slight misapprehension. In mathematics it is very important to cite (not 'site') any sources you use - you're never going to prove everything from first principles. Of course many things are sufficiently self evident, or well known, as to require no citation - you wouldn't bother citing Fermat if you invoke Fermat's Little Theorem, for example, nor would you even name Lagrange if you said "since ord(x) divides |G|" these days.
Yes, in math you are trying to prove that YOU didn't make any mistakes. If you did use work from someone else (which you have to cite, of course!) you already assume that they are correct (because the steps in them were already checked or assumed correct from cited works, etc., all the way back to first principles.) Why is there a bar through the name?
you know the funny thing is, the very principle axiomatic foundations of maths is based upon "self-evedent truths" i.e. we can't prove it is, we just accept that it is, i.e. there is no rationality or logic to rationality or logic itself; Is logic and rationality ultimately purposeless? even that is even more twisted, imagine if you will, that you are asked to make a purposeless machine, can you make it? (a) Well if you make the machine, even if it does nothing, it was made with a purpose (i.e. to be purposeless) but that then makes it null, becuase in creating purposeless requires a purpose in the first instance (b) you don't create any machine at all, but then you contradict the initial point, i.e. you can't make it, but at the same time you "can't; can't make it" eek! :D
Well now, I don't know if it follows that it's not rational. Some of these axioms we actually define to be so. eg. 1+1=2 (and only 2). While we can't "prove" it, we have defined this to be so. Same with others.
Yes, we build math from axioms. You can take that as an axiom, if you want. I also don't know whether statements like "there is no rationality or logic to rationality or logic itself" are really relevant in this thread. I could argue about making assumptions and using intuition as a basis for science, but I won't :)
why? because the definition: We assume something, and we start from that assumption(like 1+1=2), and than we make this: if 1+1=2 than 2+1=1+1+1 so, 1+1+1=2+1==3.(and i've defined the 3 number with the premiss 1+1=2). I've just made a definition that, assuming 1+1=2, than 1+1+1=3. Can you disproof my proof? No, because is just a definition. However, from this definition, one can proof that exists infinite natural numbers, by a logical method called induction. Now, if you imagine that induction is like defining that pi=perimeter/diameter than you can't deny my proof of infinite natural numbers, because that would tell you that: pi≠perimeter/diameter or 1+1≠2. And that is contradicting your definitions. So, It must be true that exists infinite natural numbers. Now, the whole math is based in this kind of logical concept. Even the most complex idea like derivative, integral, limit, is based on basic axioms like that one, so, it is true because the assumptions you used were all derived from axioms(definitions), if the theorem isn't true, than some definition is contradicting other or itself. However, that mean that your definitions aren't always good. Exists basically 2 crisis in mathematical understanding. One was the definition that if you have 2 lines, you can always express the larger one as fraction parts of the other(made by the greeks that lead to the Zeno's Paradox and an re-formulation of great deal of mathematical definitions about real numbers) and the other that lead to the mathematical crisis of 19 century with the Russell's paradox or Burali-Forti paradox. that made some reformulation of great deal of maths definitions with Cauchy among others. Why, because definitions can't be made to contradict then-selfs.
I would be careful in stating that axioms are self-evident truths. This is only viable when there exists fairly simple physical representations of the deductive system at hand. For example, the axioms of incidence are intuitively self-evident when we represent lines as, say, rays of light (and points as photon small regions of these rays). However, attributing the terms of Euclidean geometry to other physical objects, the relationships described by the axioms could still hold without them being intuitively obvious.