The Definition of Points
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From: mmeron on 1 Apr 2007 13:41 In article <579ihiF2b85q4U5(a)mid.individual.net>, Bob Kolker <nowhere(a)nowhere.com> writes: >Tony Orlow wrote: > >> >> >> Bob - wake up. How do we know relativity is correct? Because it follows >> from e=mc^2? > >Correct in what sense. Mathematically, relativity theory is simply an >excercise in Poincare groups. As a physics theory, we insist on >empirical corroberation of the conclusions that are interpreted to say >something about the world. > And, just as an aside, relativity most certainly **does not** follow from e = mc^2, quite the other way around. Mati Meron | "When you argue with a fool, meron(a)cars.uchicago.edu | chances are he is doing just the same"
From: Virgil on 1 Apr 2007 15:01 In article <460fcad2(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > No, you were told that to define "an uncountable sequence" as "a > > sequence which is uncountable" makes about as much sense as defining > > "a quadriangle" as a "a triangle that has four sides". That is /not/ > > the same as being told "there is no such thing as a polygon with four > > sides". > > I defined it as a sequence where there exist elements infinitely beyond > other elements. But the meaning of "sequence" in mathematical contexts prohibits such, at least as far as the relative postions of elements within the sequence of elements are concerned. Between any two elements of a mathematical sequence there never more that a finite number of members of that sequence. If TO wishes to define something else with the properties he desires, all he has to do is call it something else. > > > > > If you'd pull your head out of, err, the sand, it's quite possible > > that your ideas can be formalized; but no one is going to accept that > > there exists a triangle with four sides. Not for "political" or > > "religious" reasons; but because it simply makes no sense - it's > > either false or gibberish. > > > > Straw man argument. How is it a straw man to say that misuse of terms is misuse of terms when you are misusing terms? > > >>> What is your definition of "number"? aleph_0 is called a transfinite > >>> number, but definitions, not names, are the important thing. > >> A number is a symbolic representation of quantity which can be > >> manipulated to produce quantitative results in the form of symbols. > > > > Great! All that's left then is for you to define "quantity", > > "manipulated", and "quantitative results" without using the words > > number, quantity, manipulated, and quantitative results. > > > > Don't be a boor. Does TO wish to reserve that tight to himself alone? > "Quantitative results" are quantities indicated by the > resulting symbolic expression. "Manipulate" means "produce a new string > from an existing one according to rules". A quantity is a point on the > real line. A number is a string that indicates a point. Arithmetic is > the manipulation of strings. But, you know all that. All of "that" is non-mathematical in extremis, and mathematically irrelevant. > > >> I > >> might be wrong, but I'm sure you can apprise me of the official meaning > >> of "number", mathematically. ;) > >> > > > > There really isn't one. Honest. Sure, there's a definition of natural > > number, rational number, algebraic number, adic number, complex > > number, Stirling number, number field, and so on. But by itself, the > > word "number" is too vague to have a useful mathematical definition; > > just like the word "size". > > > > Cheers - Chas > > > > That was tongue-in-cheek. I know there's no definition of "number",and > mathematicians seem quite satisfied with that for themselves, but insist > that I produce a definition of a word they use every day without knowing > what they even mean themselves. Since "number" is a sort of generic term with no clear boundaries, when a mathematician speaks of numbers, he or she is usually speaking of some specific instance of that generality which is is explicitely stated or clear enough from context. When TO uses the word with several mutually antagonistic meanings within limited context, we object.
From: stephen on 1 Apr 2007 15:19 In sci.math cbrown(a)cbrownsystems.com wrote: > On Apr 1, 8:07 am, Tony Orlow <t...(a)lightlink.com> wrote: >> cbr...(a)cbrownsystems.com wrote: >> > On Mar 31, 7:27 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> >> A number is a symbolic representation of quantity which can be >> >> manipulated to produce quantitative results in the form of symbols. >> >> > Great! All that's left then is for you to define "quantity", >> > "manipulated", and "quantitative results" without using the words >> > number, quantity, manipulated, and quantitative results. >> >> Don't be a boor. "Quantitative results" are quantities indicated by the >> resulting symbolic expression. "Manipulate" means "produce a new string >> from an existing one according to rules". A quantity is a point on the >> real line. A number is a string that indicates a point. Arithmetic is >> the manipulation of strings. But, you know all that. >> > Actually, much of what you just said is still unclear ("arithmetic is > the manipulation of strings"). But: > "Aleph_0" is a string; yet it does not "indicate" a point on the real > line. So that should answer your question "is it [aleph_0] a number?": > no, it isn't. And by that definition, i and all the imaginary numbers are not numbers, and most of the complex numbers are not numbers. But I doubt Tony has any problems calling imaginary numbers "numbers". Stephen
From: Virgil on 1 Apr 2007 15:21 In article <460fcfe5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > On Mar 31, 8:38 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> cbr...(a)cbrownsystems.com wrote: > >>> On Mar 31, 5:33 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > > > >>>>> What does it mean for an ordering to be "discrete" or "linear"? What > >>>>> does it mean for something to "occur in" an ordering? > > > >>>> Linear means x<y ^ y<z ->x<z > >>> Funny; everyone else calls that "a transitive relation". > >> Yes, is that unrelated? > >> > > > > It's certainly /necessary/ for "<" to be a partial order; but it's not > > sufficient. Just like it's necessary for my car to have gasoline to > > run; but not sufficient. A partial order is transitive; but not every > > transitive relation is a partial order. > > > > See: > > > > http://en.wikipedia.org/wiki/Partial_order > > > > It's just three simple rules, man. Sheesh! > > > > So, add a<b v b<a v a=b, and make it a total order. If we're trying to > address the real line and its subsets, we can assume total order. Those who know what a total order is and what it requires may make such assumptions, if they are careful, but TO assumes one property implies all the others, which is not the case. > > > > Right. That's why it's called a /partial/ order, and not a /total/ > > order; there are elements which are incomparable - i.e., they cannot > > be compared in the ordering. > > > > Is the true for R or N? No. Can TO prove that? > > >> That's why > >> there's a parallel route, and so the diagram is "nonlinear". It could be > >> all on a line, but there would be several possible ordering given the > >> stated relationships. > >> > > > > And sometimes, depending on the ordering, there is no particularly > > useful way to extend that ordering. > > > > Consider the subsets of {a, b, c}, ordered by inclusion. I can say > > that every subset A <= {a, b, c} in this ordering; and I can say that > > {} <= A for every subset A; but some subsets can;t be compared in this > > ordering; for example, {a,b} and {b,c}. > > > > That's a perfectly reasonable state of affairs; not every partial > > order is somehow "required" to be a particular canonical total order. > > > > There is an obvious and useful way to provide total order on the power > set, and that is to assign the value 2^n to the element n (starting at > 0), and order according to the sum of those values corresponding to the > elements of the subsets. And how does this work for the power set of infinite sets, i which every infinite subset corresponds to a divergent series of powers of 2? > For a set of size n, your subsets are numbered > 0 through 2^n-1. Power set is the same as binary. Only for comparing finite sets. > >> What I said was that a discrete order will have pairs of elements which > >> have no elements between them, whereas a continuous order will not. But, > >> I'm sure I'm wrong. :) > >> > > > > Not so much wrong as inconsistent to the point of incomprehensibility. > > There are /no/ two real numbers x, y in the set [0,1) union (1, 2] > > with x < y such that there exists no element z in [0,1) union (1,2] > > with x < z < y. 1 is /not/ an element of [0,1) union (1,2]. > > > > Um, you're being inconsistent, or at least unclear. Alright, let's see. > I defined a discrete order as one where there exist pairs of elements > not separated by any intermediate elements. It is a good deal more than that. The set S is discretely ordered if and only if for /every/ member except a largest member there is a unique next larger and for every member except a smallest there is a unique next smaller member. Discreteness is global property of the set, in a sense the extreme opposite of denseness. > You offered [0,1)u(1,2] as a > counterexample. Can you name two elements in the union which do not have > an element between them? Try again. AS it was only a counter-example to continuity, it is till valid, That union os not continuous since it does not contain the LUB of [-1,0). > > > > So when you say "discrete partions of R", you either mean something / > > different/ from saying "the ordering on that partition is discrete", > > or else you don't have a good sense of what you really mean yourself > > when you say "discrete". > > Funny. But it also shows that you can't actually answer: what "time" > > does 2^pi "happen" in the function f: R->R defined by f(x) = 2^x? > > > > What is your point? It happens after 2^3 and before 2^4. Is that 2^pi AM or 2^pi PM? Standard or Daylight? And in what time zone?
From: Virgil on 1 Apr 2007 15:26
In article <460fd109(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460f22e6(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> The obvious question is why haven't /you/ studied them; instead of > >>> making vague and uninformed statements about them (regardless of what > >>> you choose to call these ordered sets). > >>> > >> The question is, "is there an acceptable term with which to refer to > >> such uncountable linearly ordered sets?" > > > > The set of real numbers, whether with or without infinitesimals is an > > uncountable linearly ordered set, but of course not discretely ordered. > > > > And I cannot believe that TO, who is usually quite inventive, if not > > always accurate, cannot create one. > > I did, the H-riffics. > > > > >>>>>> However, where every element of a set has a well defined > >>>>>> successor and predecessor, it's a sequence of some sort. > > > > Not necessarily. If a set is partitioned into two or more subsets each > > with such an order on it, but with no order between partitions, then the > > set itself is not even an ordered set even though every member has a > > well defined predecessor and successor. > > It's certainly easy enough to order the partitions, though not without > infinite descending sequences either within or between the partitions, > in an uncountable set, as far as I can see. There is a distinction between "ordered" and "orderable" that is essential here. If two or more partitions are merely orderable, then they are orderable in more than one way and the consequences of different orderings are different. So that in the absence of any particular ordering of partitions, the set itself is NOT ordered even though each partition may be internally ordered. So that TO cannot see far enough. |