Archimedes and Pythagoras

You can’t define a real number such as sqrt2 without Archimedes Postulate (cuts or nests). If we assume the truth of the Pythagorean Theorem (sqrt2 exists), then Archimedes Postulate is a consequence of the Pythagorean Theorem.

But the Pythagorean theorem assumes the existence of a continuum, another neat discussion if anyone is interested.

For the time being, let’s postulate the existence of a continuum. This means, for example, you can stretch a row of atoms to fit any triangle, (not provable- you can create any right angle triangle so you must have been able to stretch a row of atoms to any length. But you can’t definitively prove you have a right angled triangle).

So we have:

(Archimedes Theorem iff Pythagorean Theorem) iff Continuum Hypothesis.

Yes Plato, now I am trolling. That’s why I am posting in the DISCUSSION forum.

EDIT: Is movement continuous? But the existence of a continuum requires that you specify any point in space using only moving objects (everything moves). I think we have reached the end of the road.

Re: Archimedes and Pythagoras

it is unclear to me how the Pythagorean theorem assumes the existence of a continuum. in order to "rectify" the diagonal length of a right triangle with its sides, one only needs to solve a quadratic equation:

A^{2} + B^{2} = C^{2}

and for such one only needs algebraic numbers, which do not form a continuum (in fact they are countable).

it is unclear whether a continuum is a good abstract model of "the universe we live in". i think it is, but studying the very fine structure of the actual world has some practical obstacles: the limits of our technology (it takes a lot of energy to break apart an atom, for example). from far enough away, discrete and continuous systems can look very much alike....and there is no way of knowing at the moment on which level of resolution from the "bottom layer" of the universe we reside.

the "real numbers" are, in a sense, poorly named...one tends to think they have some sort of distinguished existential property, which is no more true of them then any logical structure. while i am not a "fictionalist" per se (one who believes that all mathematics is merely a game humans play for diversion), such a philosophical stance does have the attraction of avoiding some of the usual ontological dilemmas that are part and parcel of mathemaitcs (what IS a number, how do we know we are RIGHT about how they behave?).

the Archimedian property of a number system is "stronger" (more restictive) than the Pythagorean theorem: in the hypperreals, for example, the Pythagoren theorem is "still true" (the usual algebraic rules still hold), but the Archimedean property fails (there are hyperfinite numbers larger than any integer). they are not "equivalent".

the continuum hypothesis, does not, by the way, actually assert the existence of a continuum...rather it says something about the size of a particular continuum (R, the real numbers, or equivalently the interval (0,1)): that it is "the next largest infinity" after the "first infinity" of the set of natural numbers. in other words, it asserts something about "the boundary of uncountablility" which has proved extremely difficult to prove. what is known (at least as i understand it) is that the axioms of set theory alone (that is Zermelo-Fraenkel plus the Axiom of Choice) are not sufficiently strong to force the continuum hypothesis on us. set theory says relatively little about "which sets exist", we have three main ways of "making bigger sets from smaller sets":

1) form the power set

2) take the union of a collection of already established sets

3) form the cartesian product of a family of sets

of these, only (1) affects "the level of infiniteness": (2) corresponds to "addition of cardinals" (that is |AUB| is bounded by |A|+|B|, and we have equality if A and B are disjoint), (3) corresponds to "multiplications of cardinals" |AxB| = |A|*|B|, while (1) corresponds to "exponentiation" (|P(A)| = 2^{|A|}).

talking about functions lifts us from (3) to (1). collections of functions are usually *much* larger than the sets they "act upon". for example, the real numbers may be thought of as a line (determined by a starting point 0, and locating a "unit length" 1 somewhere). but the set of polynomial functions on R (which is a very restricted class of functions) takes up "infinite dimensions" (we can choose any real number for every possible (non-negative) power of x).

i don't think the "iffs" are quite as cut-and-dried as you present them. existence of SOME irrational numbers does not entail existence of ALL irrational numbers: for example the existence of √2 does not help us understand how we might obtain pi. futhermore, one doesn't need "algebraicness" to get "large sets", the set of all sequences of natural numbers is already quite large, without any algebraic structure superimposed on it.

Re: Archimedes and Pythagoras

Thanks for response Deveno. Much appreciated.

Flatten out a couple of circles and make them the sides of a right triangle. Sorry, I know that's a little off-the-cuff.

Tied one shoe, Now if I can only get the other shoe on and out of the house without thinking. Sorry to abandon the thread Deveno, I need a troll, er, stroll.

Re: Archimedes and Pythagoras

well, no, i think i get it: make a circle out of (non-stretchy) string, let's say, and then pull two points on opposite sides of a diameter apart, until the circle collapses to a line segment. viola! a line segment of length pi.

i interpret this to mean: the concept of "distance (and/or length)" is very DEEP: it leads us to consider types of numbers we might not have thought about, straight off the bat. indeed, the very concept of "measurement" is quite profound, and complicated. i had a room-mate in college, Bill Ross, and he was talking about a girl in his "introduction to real numbers" class, and how she instantly and intuitively recognized the problem of "infinite recursion in measurement" (he was quite impressed), namely: how do we make an accurate ruler? what do we measure the measuring-sticks with? as i understand it, this is still an unsolved problem...the "current solution" depends on an assumption of the regularity of the speed of light in a vacuum...the trouble being now: where is a perfect vacuum so that we can verify our assumption (atoms are "leaky"...they give off all KINDS of radiations, and we only want the photons)?

a former moderator here once observed that the real numbers represent an "idealization" of measurement (in one spatial dimension). a serious consideration of this leads one to investigate metric topologies, and for any such topology to conform to our "intuitive" notions of length, it has to be complete (all lengths should be measurable). the trouble here is another twisted tangle: in order to define the structures that the real numbers would be ONE example of, we need the real numbers FIRST (for the metric).

historically, the way around this kind of "where do we start?" problem was through COMPARISON ("what's the difference"?). this leads to the idea of "ordering" (A is bigger than B, because when we compare them, A-B (the "leftover") is SOMETHING (bigger than nothing, or 0 = B-B)). this ordering by "size" is what's at the heart of turning rationals into reals (we "approximate", and estimate a bound on the "error" or "epsilon"). the Babylonians were very good at this sort of thing, and it's a shame the Greeks get most of the credit (Euclid probably proved very little of what's in his Elements, but he got the copyright). these simple notions of size cut quite deeply, a good deal of advanced analysis is just "taking them and running with it".

Re: Archimedes and Pythagoras

Thanks Deveno.

I know sqrt2 is not the only irrational number (sigh).

I was trying to make the connection: Archimedes "Postulate" iff existence of an irrational number; because the notion of limt requires Archimedes "Postulate."

The existence of sqrt2 seemed to me the most axiomatically obvious (because of general familiarity with pythagorean theorem), as opposed to the existence of, say, pi.

Re: Archimedes and Pythagoras

Oh, by the way, referring to your previous post, there is no such thing as the speed of light in a vacuum. There is only the speed of light rleative to its source, in a vacuum. This got me permanently banned on one physics forum and the disappearance of another: advancedphysics.org.

We better not persue that here: discretion is the better part of valor.

Re: Archimedes and Pythagoras

it does not seem to matter to me if "the real numbers are really real". equations often represent problems we want to solve. this is just a representation, not the actual problem itself.

however, we can do things with representations we cannot do with actual problems: manipulate them. sometimes such manipulations then provide us with insight into the original problem (solving a differential equation can tells us "good system values" for desired results).

triangles in a sense only exist in our imaginations. the "real" (by this i mean tangible) forces we might represent by a triangle aren't constrained by our ability to measure them, only our representation is. nevertheless, mathematical analysis often yields astonishingly good predictions of the actual behavior. we take this as evidence "we're on the right track" (although the possibility remains we are just REALLY lucky).

in some sense, the search for "the equations of the universe" is doomed to failure, actual things have idiosyncrasies our models often fail to capture. should we stop trying?