Re: Chord connection


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RJ P asked

Just for Fun.. What would you think if someone asked you to prove
that
2
+
2
= 4 ?


lol. I'm not sure. I don't know how to prove it but I suppose my
response
is that its provable and I would tell them were they could find the
proof.

I would definately say that its not always the case that 2 + 2 = 4 and
give
them a counter example. On a practical side I might say that it
doesn't
matter if its true or not because its still useful. I'd then probably
go
off
on some long explaination about grammars, groups, operations, and
representation theory and probably try to include some history of man
type
of crap ;) (i.e., 1 + 1 = 2 hence 2 + 2 = 1 + 1 + 1 + 1 = 4 = 1 + 1 +
1
+
1).

It would all depend on if I thought they really wanted to know or was
just
trying to waste my time.


I, ofcourse, don't really know the proof but I have read that Hardy
and
Whitehead have proved it and its very complicated. Actually, at this
point
in my life I don't care to much about it. I don't think being a great
mathematician is going to save my soul or make me life for ever(which
I
guess is what I truely want... well, that or knowing everything).

caution ... not a simple question

It's one of those things where the reaction to the question is more
important than the question itself.

To me an analysis of why it would NEED to be proved is the candy here.

It's a thing where the journey is much more interesting than what's at
the
destination.

Try Peano's axioms and the field axiomms. It's not complicated if you
don't
get too picky about existential quantifiers.


I agree. Life would be boring if it was easy. If you knew everything then
what would be the point? I think part of the goal is the struggle. The
problem is that we only have a finite amount of time and so one has to
choose what is more important. Theres also a lot of other problems that
interact together to make it more difficult than it needs me(such as
survival(pretty much equal money), pride, ego, etc...)).

I do think that its human nature to want to understand the world around
us... else I think one would have a hard time explaining our
accomplishments. I'm glad that someone had the need to prove 1 + 1 = 2
as
I
feel it is necessary. Fortunately I have realized that there is much
more
to life than mathematics and it is only part of the equation if I want to
really understand life. (and in my adventure I feel that everything
seems
to be connected in one way or another)

Jon


My point is that for some reason a whole buncha guys were interested in
abstract number theory. there was a feeling that the state of mathematics
was simply a lot of tricks, which sometimes did not work
My entire undergraduate education in Math was stuff that was cool in 1790,
but there was almost a Starburst rennaissance about that ime in Math.


Sure. Mathematical education is always pretty far behind the times... I
think its suppose to be that way because it takes time to form a coherent
theory and to weed out the crap(and there happens to be a lot of it in math
and science in general).

It was the swimmers against the tide of complacency that made Math into
the
foundations of modern analysis.


Sure. I would imagine that complacency is pretty much evil in any
intellectual pursuit. Its very easy to become complacent and luckily there
are people that don't get conned by it. Unfortunately this tends to be the
case only with science and math.

To understand even Applied math, it is universally believed that the
intellectual basics need too be known at least by the first year of
Graduate
level.


Well, I'm not sure about this. Applied mathematics is a very general set of
subjects. I think that if one doesn't have a nack for mathematics when they
start undergrad then chances are they won't pick it up. If you go in to
math for the wrong reasons then you won't do well(in general atleast).


It doesn't really bother me that almost anyone would feel that Provng 2 +
2
= 4 is omphalism or solipsy, But it's not


It is to some people. Usually its the little cracks that need fixing... not
because they are worse than the big holes but because they are easier and
they allow you to then focus all your energy on the big holes. (because
maybe while your working on the big hole the little cracks get bigger)

What DOES bother me is that the folks who teach High School Math in
general
don't have the intellectual curiousity to have though about much except
what
page we aare on in the text.


Sure. When I was in my undergrad about 80% of the "math" students wanted to
become teachers. Most were complete idiots when it came to math. Its not
math's fault but societies. The same thing could be said about physics and
engineering. When I went to school I was expecting that people where there
to learn but it wasn't the case... unless you consider drinking and having a
good time learning.

Yes 2 + 2 = 4 needs to be proven, but the main reason is not "Why not
just
accept it as a Fact, and not be late for Footall Practice ", if you get my
drift .


Sure. But it depends on your goals. Some people don't give a *** about
much of anything except where they are going to go party at. It totally
sucks but I suppose thats life. I've realized that its possible maybe they
are right and that maybe one should just be content with simple things(since
it might not matter what you do in the end).

The "Moving rocks " proof is valid, and hopefully would trigger the
principles of abstraction in the connection of grouping rocks and the
Assoociative Principle
My fear is that it generally doesn't and "I weep for the youth of America"
( said by By the snobbish waiter about Ferris Bueller"

This is gettng way off topic, but the SYMBOL for 1 + 1 IS "2" ..
It need not be proven.


Sure. if you want to define it that way. Thats usually my quick and dirty
answer. The reason why this isn't perfect is it ignores the way symbols are
combined. How can we be sure that since 1 + 1 = 2 and 1 + 1 + 1 + 1 = 4 then
2 + 2 = 4? i.e. we have to prove that what we call operation of addition
holds and works inductively on the symbols.


One could say that 1 + 1 + 1 + 1 = 4, 2 + 2 = 4, 1 + 3 = 4 but then how can
be be mathematically certain that this method of construction is self
consistent? well, I guess simply by defining the structure recursively like
is done in the follow:

I found a simple proof here(although I have to admit that I just glossed
over it)(pretty much what you said before)

http://mathforum.org/library/drmath/view/51551.html

This is not the same proof that I remember reading about from Hardy and
Whitehead which was much more complex. (I could be wrong though as its been
several years when I read about it)

I think its just natural to cover all bases for some people... atleast the
best we can. We wouldn't want to run into issues that happened with
euclidian geometry(i.e. assuming that it is the only kind then finding out
later its not).


If you're interested, the "Completion of the Number System" is a pretty
standard exercise for any Ph.D Candidates in abstract Math.


Are you talking about the extention from naturals to rationals to reals to
complex? I don't think this is similar to the 1 + 1 = 2? The extention is
a natural generalization or enlargement of a system while the 1 + 1 = 2 idea
deals with the fundamental existence of the inital system. Similar to how
induction works where you have to prove for the base case then prove it
holds inductively after that. Its kinda pointless to prove it works
inductively if it doesn't work for the base case.


I think maybe the good thing that comes out of being anal about things like
this is sometimes you end up finding things that you didn't expect and that
you can be comfortable in that you have covered all the basis.

Jon


.

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