Talith wrote:You seem to have a problem with a geometric definition which relies on taking a limit of a sequence as the definition of pi, and yet your definition relies on a much more difficult concept of integration which has limits tied up right in the definition (take your favourite Riemann or Lebesgue integral and you'll soon find limits). Good definitions of objects in mathematics should strike a balance between 'intuitively obvious' in some sense but also 'easy to work with' from a practical point of view. This is why we sometimes take the definition of the trigonometric functions to be their Taylor expansion instead of their original geometric interpretation as angles of triangles inside a circle, because it's often nicer to work with.
Though that is the case however, we don't teach this definition of the trig functions at an early age for the obvious reasons and we can still get a 'mathematically sound' definition of the trig functions without having to know differential calculus or the theory of infinite polynomials and series (there's just no need). In the same way, the geometric definition of pi is a perfectly good, 'mathematically sound' definition in any way you could want (be it from axioms or just 'a good enough model to work with') and there doesn't seem to be any advantage at an early stage in education to change to your integral definition. In fact, there doesn't seem to be any reason to use your definition even at a later stage in education or research.
I think you miss my point (don't worry, i'm used to this by now). i am NOT suggesting we force this "integral" definition down the throats of kids in elementary geometry classes. they don't have the tools to comprehend it (a lot of ideas need to be fleshed out before-hand). i am saying that the "elementary definiton" of circumference/diameter relies on notions that cannot be established with the tools they have on hand. which is ok, i am not trying to re-vamp the educational system, here.
what i AM saying, is that when (if?) you reach the point (amd some people never do) where you are finally in a position to prove that pi is a real number, the integral definiton is then a clear one to use (one could, i suppose, go back to the geometrical proofs, armed with the new notion of "limit" and prove those rigorously. that ok, too). and that the integral definiton does provide some motivation for using pi, instead of tau.
your analogy with taylor series is apt. if one has complex analysis at one's disposal, then it's far more satisfactory to DEFINE the complex exponential as a certain complex power series, and obtain the trigonometric definitons as corollaries. the motivation behind the use of "e" and the transcendantal nature of the trigonometric functions becomes clearer, and proof of their properties are analytic, rather than geometrical. but that's not going to help someone taking trig for the first time, and i see no reason to abandon SOHCAHTOA on that account.
of course an integral definiton of pi isn't "elementary", and of course, some simpler definiton has to be given if we want students to be familiar with pi (or tau) at all. for those people will never use pi/tau all that much, any definition that is reasonably correct will serve. for those who will one day learn more math, one hopes that they will be trained in the habit of not using unproven facts. this entire thread is about the "reasonableness" or pi versus tau, and i was merely giving one rationale for pi (ironic, too, because tau makes sense to me).
let's back up a bit. assume we know "somehow" (in some unspecified way) that circumference/diameter is a constant. well, if it's rational, we know it's a number. but what if it's not? how do you even begin to say what a real, irrational number is (well, ok, algebraic numbers are roots of polynomials. too bad pi isn't algebraic) without some notion of what "convergence" is? perhaps dedekind cuts might be the most basic, although showing they have all the field properties we desire (so that we can, in particular, embed the rationals in them) is a bit messy. it's not a "given" that "pi is a number" until we a) say what "numbers are" and b)show pi is one of them. and sure, we gloss over that point in teaching number systems to begin with, because the explanation and proof would just be too hard.
Deveno wrote:and we had a valid number system for such "magnitudes" and a clear idea what "limit" was? gosh, someone should have told those guys who came up with limits and continuity, it was unecessary, they'd been pwned by archimedes. my point is, pi is irrational. that means there's no finite rational approximation to it. euclid's so-called proof depends on taking the limit of an n-gon, as n-->infinity. so does archimedes' "method of exhaustion". oh, but wait! we could use induction, right? because the greeks knew all about those peano axioms, didn't they?
Are you seriously suggesting that no one knew how to properly use anything in math until people came along centuries later and gave it rigorous logical foundations?
no, i'm suggesting that no one knew how to properly evaluate limits and define real numbers until the right concepts were created. yes, newton sometimes did calculus wrong. people had all sorts of erroneous ideas about infinity, limits, series and "magnitudes". before a proper inductive definiton of the natural numbers was put forth, the use of proof by induction had no logical justification. the best you could say about it was: "it always seems to work". that doesn't mean that no one came up with genuinely good results. euclid's elements is a masterpiece of deductive reasoning, euler, gauss, too many others to name all made great contributions to our mathematical understanding, even if the foundations of their investigations were uncertain. even now, we still don't really "know what a number is" (well, we might, if we actually knew what a set was). what we do know, however (and this is a great stride, and fairly recent), is how numbers have to behave. we know what numbers aren't, and we leverage that knowledge. when set theory was finally given some sort of shape, and arithmetic axiomatized, and topological ideas given form, we looked back, and saw that most of what had come before could be salvaged. this gives the hope that "we're on the right track".
again, i'm not arguing that young school-children should be versed in the logical foundational underpinnings of math. in many ways, they are recapitulating the historical development of math, working on faith that, ultimately, it does "make sense".