James Harris wrote: > > Paul Sperry <email@example.com> wrote in message news:<plsperry-3C9F3E.firstname.lastname@example.org>... > > In article <email@example.com>, > > firstname.lastname@example.org (James Harris) wrote: > > [...] > > > So, let me re-emphasize my point, by asking you a simple question: > > > > > > Can you start with integers and end up in the field of rationals with > > > fractions...your regular old garden variety fraction like 1/2...using > > > only addition and multiplication? > > > > > > > Yes, I can. The construction is standard, routine and well known: > > On the subset S of the set Z x Z of ordered pairs of integers consisting > > of all pairs (a,b) with b != 0, define (a,b)==(a',b') iff > > a * b' = a' * b where "*" is integer multiplicaion. > > "standard", "well known" and CIRCULAR.
Well known, yes, but not by you it seems.
> > Folks, he's saying that in integers, just say that 2b = 3, claim that > b "exists" (which he can't prove, so he must base it on some
No, Paul is not bringing b into existence by fiat. He is defining it from the raw ingredients: integers, ordered pair, and equivalence class. Specifically, b is the equivalence class containing the pair of integers 3, 2 (in that order). Let's denote that equivalence class by [3,2]. There are also equivalence classes [2,1] and [3,1]. We can define an operation of multiplication (let's call it x) on the equivalence classes such that [2,1]x[3,2] = [3,1]. Now, note that we are talking about the things that Paul _constructed_ NOT something that he magic'd into existence. Note also that with a change of notation: write [u,v] as u/v and u/v as u if v = 1, we get 2/1 x 3/2 = 3/1 or 2 x 3/2 = 3 which is _still_ a statement about arithmetic with those equivalence classes. Let's make a further change of notation and write the operation of multiplication of those equivalence classes (that's x) using the symbol for the operation of multiplication of integers: *. We get 2 * 3/2 = 3. So, by an abuse of notation, the left-most 2 and the right-most 3 _look_ like integers, but they are not. Is that why you thought that the definition was circular? Finally, using the embedding a -> [a,1] of the integers into the rationals, we _can_ take the left-most 2 and the right-most 3 to be the integers they look like. This isomorphism makes official, so to speak, the re-writing of [u,1] as u. You may have this worry: 3/2 corresponds to the ordered pair <3,2> in the equivalence class [3,2] but what if another representative, say <6,4> had been chosen? Luckily one can prove that the choice of representative makes no difference.
At this point it will do no harm if you go back and read Paul's construction of the rationals.
> principle, which basically just says it exists), and presto > magico!--you have a fraction. > > Of course, in the real world we use fractions as an intermediate step. > > The silliness of a circular definition like what he just gave is that > you can use if for just about ANYTHING. > > And then argue that people who question it are crazy. > > Oh yeah, speaking of "crazy" stuff. Since I'm downplaying fractions, > how do I handle transcendentals? I skipped past that before to say > they lie at infinity. > > So what does that mean? > > First off, remember how I said that decimals are humanity's way of > using really big counting numbers. Well, you could say that human > beings got tired of things like 1414, and realized it was more > convenient to *write* 1.414, because you see, the front of it doesn't > move so much.
Your language is so strange sometimes! I picture a car's odometer reading 1414, as we drive along the least significant 4 quickly changes to 5. The most significant 1 only changes at one thousandth the speed of that 4. A child (bright, but lacking the vocabulary) points to the odometer and announces "The front of it doesn't move so much".
> > Well, what pi actually IS, is the count of the minimum number of > elements necessary to make half the circumference of a circle. > > And you're going, hey, I knew that.
Are they? Then they understand you better than I do. Are you thinking of small measuring rods (called "elements"?) laid out end-to-end along the half-circumference and ditto along half the diameter (dunno why we've got two halves)? Let's say that there are 987 along the half-circumference and 314 along the radius ignoring in each case fractions. So we *almost* say that pi is 987/314 but not really because we know we've only done the job approximately. So we use a smaller measuring rod to do it more accurately. Do you want to use a measuring rod that fits along the radius some-power-of-ten times? Lets say 1000 and lets say the half-circumference is 3143. So pi is approximately 3143/1000 but since fractions are fictitious we'll just say 3143. Didn't Simon Stevin travel this route, didn't the Pythagoreans before him? Don't reinvent the wheel James, or even the half-wheel. What I don't understand is "_minimum_ number of elements".
> > Well, then then think about this: > > We know that there are other numbers besides 1. Think of pi as the > count with 1, and yes it's that count because human beings LOVE
I've changed my mind about what you might mean. Are you thinking: let the measuring rod be flexible and let it be as long as half the diameter and deem its length to be 1. Now as we lay it along the half-circumference we find it goes three times and falls short by a bit. Cut the measuring rod into ten equal pieces and use one of those pieces to measure that bit. One, and still a bit short. Cut into ten again and get four. And so on. But "_minimum_ number of elements"? And in this version what is an element?
> counting numbers. You see, they're really good with them. > > And they get started early with their fingers and toes. > > And their brains are PROGRAMMED to feel good with counting numbers, so
Do you think that what Chomsky claimed of a "deep structure" of language has a counter-part in mathematics? Are counting numbers and their "grammar" the deep structure of mathematics as a whole? Is the actual performance of mathematicians a flawed version of the "surface structure"?
> they feel COMFORTABLE with counting numbers :-). > > How about something outside of your regular programming? How many NON > discrete elements does it take to make half the circumference of a > circle?
I don't know! What's a non discrete element? Is it an infinitesimal by any chance?