The Definition of Points
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From: Brian Chandler on 1 Apr 2007 12:38 Mike Kelly wrote: > On 1 Apr, 16:48, Tony Orlow <t...(a)lightlink.com> wrote: > > Brian Chandler wrote: > > > Tony Orlow wrote: > > >> Brian Chandler wrote: > > >>> Tony Orlow wrote: > > >>>> I'll give *you* a start, Brian, and I hope you don't have a heart attack > > >>>> over it. It's called 1, and it's the 1st element in your N. The 2nd is > > >>>> 2, and the 3rd is 3. Do you see a pattern? The nth is n. The nth marks > > >>>> the end of the first n elements. Huh! > > > > >>>> So, the property I would most readily attribute to this element Q is > > >>>> that it is the size of the set, up to and including element Q. > > >>> Euuuughwh! > > >> Gesundheit! > > > > >> I seeee! Q is really Big'un, and this all jibes with my > > >>> previous calculation that the value of Big'un is 16. Easy to test: is > > >>> 16 the size of the set up to and including 16? Why, 1, 2, 3, 4, 5, 6, > > >>> 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 - so it is!! > > > > >> Well, that's an interesting analysis, but something tells me there may > > >> be another natural greater than 16.... > > > > > Indeed. So your "characterization" of Q isn't much use, because it > > > doesn't distinguish Q from 16. > > > > If the size of N is Q, then Q is the last element of N. It doesn't exist. > > Irrespective of what notion of "size" is being used? I think it's easy to see that Tony's notion of "size" is based on his all-powerful intuition, honed by looking at literally millions of finite sets. He knows what size is when he sees it. If you count some collection of elements, the size is the count when you finish. So obviously the "size" of the pofnats (is this what N is here?) doesn't exist. Which is perfectly true of course. I think he has at least realised that he's safest shoving all the confused and nonexistent bits of his "theory" off to beyond the left or the right, so only thos capable of "reaching" infinity will ever be able to discuss them with him. The real mystery is why he bothers sci.math with this. Brian Chandler http://imaginatorium.org
From: cbrown on 1 Apr 2007 12:59 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: > >> step...(a)nomail.com wrote: > >>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>> step...(a)nomail.com wrote: > >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> step...(a)nomail.com wrote: > >>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>> step...(a)nomail.com wrote: > >>>>>>> So in other words > >>>>>>>>>>>>> An actually infinite sequence is one where there exist two elements, one > >>>>>>>>>>>>> of which is an infinite number of elements beyond the other. > >>>>>>> is not your "correct" definition of an "actually infinite sequence", > >>>>>>> which was my point. You are so sloppy in your word usage that you > >>>>>>> constantly contradict yourself. > >>>>>>> If all you mean by "actually infinite" is "uncountable", then > >>>>>>> just say "uncountable". Of course an "uncountable sequence" > >>>>>>> is a contradiction, so you still have to define what you mean > >>>>>>> by a "sequence". > >>>>>> Please do expliculate what the contradiction is in an uncountable > >>>>>> sequence. What is true and false as a result of that concept? > >>>>> A infinite sequence containing elements from some set S is a function > >>>>> f: N->S. There are only countably infinite many elements in N, > >>>>> so there can be only countably infinite many elements in a sequence. > >>>>> If you want to have an uncountable sequence, you need to provide > >>>>> a definition of sequence that allows for such a thing, and until > >>>>> you do, your use of the word "sequence" is meaningless, as nobody > >>>>> will know what you are talking about. > >>>> Oh. What word shall I use? Supersequence? Is that related to a > >>>> subsequence or consequence? > >>> As long as you define your terms it does not matter to much what you > >>> call it. You could just call it an uncountably infinite sequence, but you > >>> need to define what that is if you want anyone to know what you are > >>> talking about. Why are you so reluctant to define your terms? > >> I did that, and was told no such thing exists. Gee, then, don't talk > >> about unicorns or alephs. > > > 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. > Right. And taken using the usual meanings of those terms, your definition makes as much sense as defining an uncountable sequence as a 4 sided triangle; because there are no natural numbers which are "infinitely beyond" any other natural number. > > > > 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. > A straw man argument is one in which I address an argument that you / didn't/ make. What is the straw man here? Defining an uncountable sequence as a sequence having a property that is impossible for a sequence to satisfy is to define a term for which there is no mathematical referent; just as defining a quadriangle as a triangle having a property that it is impossible for a triangle to satisfy is to define a term which has no mathematical referent. So saying "S is an uncountable sequence" is either false, or gibberish. And you /did/ claim that mathematicians were contending with your definition for political and/or religious reasons; yet that's not the case. It's because your definition makes no sense. > >>>>>>> Aleph_0 is not a finite number. Care to try again? > >>>>>> It's also not the size of the set. Wake up. > >>>>> It is the cardinality of a set. > >>>> Is that a number? > >>> 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. "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. > >> 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. You asked "Is [aleph_0] a number?". /Because/ there is no standard definition of "number", Stephen naturally asked what you meant, because the definition of the term depends on context. Your response was unclear; but it appears you meant "real number"; which aleph_0 is not. Cheers - Chas
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 |