Pondering a Proof

[QED Stepfiles]

Re: Pondering a Proof

Well, you're right, as Kurt Godel has already shown through the famous incompleteness theorem that mathematics (and all formal systems) is fundamentally and will always be incomplete.

You can prove axioms, but in order to do it you have to fabricate further axioms. I think Euclidian geometry is a good example. You need to formulate several axioms in order to complete the system, but in doing so you can no longer prove the axioms (although they are likely true simply from observation) unless you plan to fabricate additional unprovable axioms to prove them.


As such, I don't think a complete formal system is important. If the portions of the system which cannot be proven remain obvious through simple perception, then mathematics can adequately explain the perceived world. I don't see the point of describing anything else >_>

Yes I pretty much think that too, but this is really for the sake of discussion. For the purposes of keeping sane it's not really something we should worry about every time we solve an equation, but it's still interesting enough to think about every once in a while.

[kmay]

Re: Pondering a Proof

i said how the math works... what don't u get?

[QED Stepfiles]

Re: Pondering a Proof

well 1*1=1 so divide over the 1 it would be 1=1/1 umm its pretty simple...

i said how the math works... what don't u get?

Define division. What does it mean to divide 1/1? Is the operation well defined? Is there an inverse of "1" in the field? Is the inverse unique? What does "*" mean? Is "*" a function (Z x Z --> Z), or what? What are the integers? Some texts define the integers as the intersection of all inductive subsets the reals, but what's your definition? According to your definition, is it viable to say things like "1/1"? And is "*" well defined? And what does "=" mean anyways? Does 1/1=1 mean that 1/1 is congruent to 1 (i.e. in the same equivalency class), or that we may represent the same mathematical object "1" as both "1" and "1/1," or what?

There are a lot of things that are not precise if you want to structure a proof like that. Be careful.

Sidenote: Actually, on second thought, did you really say anything new? You essentially reformulated the proof of "1*1=1" into a proof of "1/1=1" (of course, assuming that these operations are all well define). If "1*1=1" is not obvious (as we are assuming for the purpose of discussion), then "1/1=1" is definitely not more obvious. So... what's the point here? We seem to be no further than where we had started.

[kmay]

Re: Pondering a Proof

after further thought i do see what u mean. with my "theory" it is saying the this could also equal 1. 1/1*1/1 but that would equal 2. when u multiply fractions u have to add across. so i see where the debate comes in. now i really want to know how this makes sense. its this it like Einstein's theories. if you cannot prove them wrong then they must have something in the theory that is one the right track. im pretty sure its just and accept thing as u said before. >.>

[Cavernio]

Re: Pondering a Proof

"It is definitely possible to think logically, and purely logically, and it's something that mathematicians have been doing for a long time."

Prove it. I'm not sure you can. Of course, that would be because our brain has developed from our senses, and any proof you'd offer would involve a general acceptance as a language being purely logical, which I say would not be possible because we're not purely logical beings. So I suppose I've asked you to do the impossible in proving that, but you might be able to convince me anyways.

The rest of that post is moot IMO because you have not proven the above statement. If we take math as representing our world around us, 1*1=1 does not need to be proven logically.

"Rather, mathematicians work more predominantly in "First Order Logic," a mathematical language that uses completely unambiguous language to a satisfactory level of precision."

And these are the same mathematicians who've been using logic 'for along time' as you say? My knowledge (which is obviously incomplete, but not necessarily wrong), is that 'formal' math in fact started as geometry, which is clearly an offshoot of our visual sense. Also, why is our language of math displayed visually?

Yes, I understand that my arguments are far from proof, however, that does not mean they're not valid to consider.

[QED Stepfiles]

Re: Pondering a Proof

"It is definitely possible to think logically, and purely logically, and it's something that mathematicians have been doing for a long time."

Prove it. I'm not sure you can. Of course, that would be because our brain has developed from our senses, and any proof you'd offer would involve a general acceptance as a language being purely logical, which I say would not be possible because we're not purely logical beings. So I suppose I've asked you to do the impossible in proving that, but you might be able to convince me anyways.

The rest of that post is moot IMO because you have not proven the above statement. If we take math as representing our world around us, 1*1=1 does not need to be proven logically.

"Rather, mathematicians work more predominantly in "First Order Logic," a mathematical language that uses completely unambiguous language to a satisfactory level of precision."

And these are the same mathematicians who've been using logic 'for along time' as you say? My knowledge (which is obviously incomplete, but not necessarily wrong), is that 'formal' math in fact started as geometry, which is clearly an offshoot of our visual sense. Also, why is our language of math displayed visually?

Yes, I understand that my arguments are far from proof, however, that does not mean they're not valid to consider.

Actually... the whole formal math starting from geometry thing may be accurate to some degree, but today mathematics is definitely not built up on geometry. Geometry is built up off of logic. Actually... euclidean geometry is a bit dated... there are a hell of a lot of assumptions in euclidean geometry (for example, why do we have to use the standard metric -i.e. measure distance and length the standard way- for the space?). Geometry is a result of logic.

First order logic was not developed in Ancient Greece or anything... the whole concept of mathematical logic and consistency is a relatively recent phenomenon. I'm not really up to writing a huge long post about first order logic, but suffice to say that it's precise and unambiguous. And, yes, language in general is not logical, but it is possible to fix all the ambiguities in language, and make it absolutely precise (thus, the creation of F.O.L). Humans may not be logical beings inherently, but we do have the capacity to think logically and unambiguously if we really try. And really trying in that capacity is essential to understanding mathematics.

[Cavernio]

Re: Pondering a Proof

I've studied the basics of first order logic.

It seems to be on topic if I asked what makes logic valid in the first place...seems almost like I'm asking why 1*1=1.