mathematics

[RootDBA]

Wait until we get to Godel's theorum...then things will get really interesting

[justin1987]

well, hi there root, u sure are intelligent.
are u a student, if so, in which class?
but one thing for sure, u certainly understand maths, u certainly do. keep posting.

[Imagine]

because music is very mathmatical. Listening to what you say is artistic. It's abstract nearly..... and I like it.

It's also tangible. I like pretend real numbers.

[Imagine]

mathematics, but it's new to me and it's really fascinating.

I'm giving myself 10 min to look at it and I'll look at it again later..

I could get totally into this stuff.

[RootDBA]

OK ! I'll do a brief introduction to Godel's theorum (On Formally undecidable propositions of Principia Mathematica and related Systems- to give it its full title).

I see, Imagine, that you are a fan of A.N.Whitehead- who along with Bertrand Russell, wrote Principia Mathematica and which is referred to in Godel's theorum.

Basicly, Russell and Whitehead believed, as did David Hilbert, that any Mathematical theorum could be reduced to a sequence of logical statements. Godel showed this was not possible- that any system of axioms would lead to statements that were true yet unproveable and it did not matter how many additional axioms you add, you will always end up with statements that cannot be proved. If you have a system where all statements can be proved, then it is inconsistent (that is, it will hold statements whereby than can be proved both true and false within the same system).

I am generalising a little here as what Godel actually wrote about was omega-consistent systems, which are a little different, but not importantly so here (in short,if a system is w-consistent, then it is consistent, but not vice versa), and anyway, have been generalised by people like Church to hold true of a more general definition of consistency.

Apart from it's huge implications in Mathematics, it hold very important results in Computer Science- for it may be that a computer is given a routine to check, that can never be proved (that is you could have a 'do...while' statement where it can never satisfy the condition, not because it is false but because the 'while' condition might contain an unprovable statement.

In short, Mathematics is not the edifice of truth it has been believed. It was thought at one time (until Andrew Wiles provided a proof) that Fermat's last theorum might actually be true, but not provable, and there is still a lot of research into whether not only whether Hilbert's original 10 theses that stood at the beginning of the 20th, Century are provable, but whether newer problems in Mathematics might ever have a proveable solution (You might be able to solve it, but not within the system that the problem was formulated in). It is a fascinating area of study and one that can be grasped by the non specialist, although it might give the brain cells a good work out, for like many things in Mathematics, what you think to be intuitively true, often turn out to be not the case.

Get you hand on W.V.Quine's 'Elementary Logic' it is hardly an elementary text, but gives a good overview of Godel and Church (in later chapters) without requiring you to have any understanding of things as exotic as the theory of recersive functions. A very good book, if rather dry.

[Imagine]

cirriculum for my study this week thank you!!


I appreciate your knowledge of these subjects.

I read your words..going.."oh! that's what that is"..bit of revelation..thank you..

sometimes i only know about the concepts but not what they are called.

[justin1987]

well, hi root, that certainly was enlightning. from where did u learn all that? i hadnt know them, maybe becvause they werent in our syllabus. nevertheless, thanks .
by the way, do u know calculus? maybe even tensile calculus?