The Virgin Birth of Points
in [Physics]
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From: Jan Burse on 25 Dec 2007 19:28 Lester Zick schrieb: > Which "real number line" do you have in mind? Obviously pi does not > exist on any real number line in common with other numbers; it only > exists on circular arcs as demonstrated by Archimedes approximation > for pi. > > The strange thing is I've written extensively on this aspect of modern > math and "the real number line" many times, most recently on the > thread "Quoth the Raven where lies Pi?". Even Bob Kolker agreed with > me on this aspect that there is no real number line in formal terms. > Yet you use the phrase "real number line" as if it had some meaning. Take a can. Paint a vertical line on it. Roll it over a white sheet of paper along a line. You will get a number of points where the lines intersect. The points have pi distance from each other provided the can had 1 in diameter. pi is tangible. Bye
From: Lester Zick on 25 Dec 2007 20:03 On Thu, 20 Dec 2007 14:34:12 -0800 (PST), David R Tribble <david(a)tribble.com> wrote: >My working definition of point is a geometric object >isomorphic with a real number. But this raises a couple of interesting issues. If points are indeed isomorphic with real numbers and real numbers lie on a straight "real number line" presumably points would too. So what about points in the rest of space? And second, how do you define curves? ~v~~
From: Lester Zick on 26 Dec 2007 13:49 On Wed, 26 Dec 2007 01:28:00 +0100, Jan Burse <janburse(a)fastmail.fm> wrote: >Lester Zick schrieb: >> Which "real number line" do you have in mind? Obviously pi does not >> exist on any real number line in common with other numbers; it only >> exists on circular arcs as demonstrated by Archimedes approximation >> for pi. >> >> The strange thing is I've written extensively on this aspect of modern >> math and "the real number line" many times, most recently on the >> thread "Quoth the Raven where lies Pi?". Even Bob Kolker agreed with >> me on this aspect that there is no real number line in formal terms. >> Yet you use the phrase "real number line" as if it had some meaning. > >Take a can. Paint a vertical line on it. > >Roll it over a white sheet of paper along a line. > >You will get a number of points where the lines intersect. You're ignoring slippage. >The points have pi distance from each other > >provided the can had 1 in diameter. > >pi is tangible. Of course pi is tangible. That's not the problem. Pi lies on circular arcs not straight lines. That's the problem. And circular arcs are every bit as "tangible" whatever that may mean, as straight lines. There are infinitely many more points on curves than straight lines. That's why curves define transcendentals rather than merely square roots of rationals defined on straight lines. This is also the reason curves have variable tangents defined at only the point of tangency whereas straight lines have a constant tangent defined at all points on the line. And it's also the reason all straight lines are congruent with one another and curves of different types are not. I suspect you and others misunderstand the nature of the problem. Curves have variable tangents with different tangents at each point on the curve. Straight lines have a constant tangent at all points. This means ratios between curves and straight lines and square roots of rationals defined on them are not constant so that there can be no exact definition between any curve segment and straight line segment. The difficulty is that between any two points the only thing definable exactly between them are square roots of rationals because tangents are constant and the difference is linear. Which is what makes them straight lines to begin with. Curves on the other hand aren't straight lines because tangents are variable, which is what makes them curve. I won't suggest the problem here is peculiar to modern mathematics; however I don't think modern mathematics has helped comprehend the nature of the problem with its emphasis on arithmetic to the exclusion of geometry. The fact is you can approximate the length of curves in relation to the square roots of rationals but you can't commensurate them exactly because there are infinitely many more points on curves than straight lines and modern math hasn't facilitated the study of either class of infinity when modern mathematikers naively assume the existence of a "real number line" inclusive of all possible ratios for rationals and irrationals, together with the implication curves and straight lines have the same infinite number of points just because they're infinite. In other words there is one infinity of points associated with each straight line tangent and a different infinity of points associated with all infinities of variably different straight line tangents. And that's why there can be no exact commensuration between curves and straight lines. At best there can only be an exact commensuration between tangents to curves and straight lines. ~v~~
From: Lester Zick on 27 Dec 2007 12:02 On Wed, 26 Dec 2007 18:08:13 -0800 (PST), David R Tribble <david(a)tribble.com> wrote: >David R Tribble wrote: >> "Not" is a unary logical operator. So, by definition, "not" >> cannot stand by itself as a well-formed logical statement. > >Lester Zick wrote: >> Your form of argument is defective. I say X is true since alternatives >> to X are false. You say X is false because X isn't Y and you assume Y >> to be true. > >I said that "not" is a logical operator, and you say that >my argument is defective. So you must be saying that >"not" is not a logical operator, but something else. By your own words right above you don't just say that "not is a logical operator". You say that "not is a unary logical operator". Consequently my comment is rather obviously directed at the idea "not" is not "unary" not that it is not logical. >Alternatives to "not" are other logical operators. >Or are you saying that's not the case? I'm saying "not" is not unary just as differences are not unary and are taken between things and can stand by itself as a well formed logical statement. >> You're interested in Y only because assumptions of truth can be false. > >If by "assumptions of truth" you mean "axioms", no, that's not >the case. Mathematical axioms are always true. Incorrect by your own words. You say elsewhere: "An axiom is an assumed truth. Period." "Assumed truth" and "always true" are different predicates. >If by "assumptions of truth" you mean "definitions", no, that's >not the case either. Definitions are not true or false. Well definitions are always definitions. Doesn't make them true or false. Contradictions between predicates in definitions make them false as in "squircles are square circles". ~v~~
From: Lester Zick on 28 Dec 2007 12:36
On Thu, 27 Dec 2007 11:03:35 -0800 (PST), David R Tribble <david(a)tribble.com> wrote: >David R Tribble wrote: >> "Not" is a unary logical operator. So, by definition, "not" >> cannot stand by itself as a well-formed logical statement. > >Lester Zick wrote: >> Your form of argument is defective. I say X is true since alternatives >> to X are false. You say X is false because X isn't Y and you assume Y >> to be true. > >David R Tribble wrote: >> I said that "not" is a logical operator, and you say that >> my argument is defective. So you must be saying that >> "not" is not a logical operator, but something else. > >Lester Zick wrote: >> By your own words right above you don't just say that "not is a >> logical operator". You say that "not is a unary logical operator". >> Consequently my comment is rather obviously directed at the idea "not" >> is not "unary" not that it is not logical. > >If "not" is not a unary operator, is it a binary operator? >Or perhaps it's a logical value, or maybe a free variable, >since logical statements are composed of operators, >logical values (true and false), and variables, and a >few other kinds of convenience punctuation. "Not" is a predicate designating "alternatives to" "different from" or "contradiction of". Beyond that I can't say because I can't make out exactly what your argument is. I might say "A is not A" which would be self conradictory. I might also say "not is not" which would also be self contradictory or the "alternative to alternatives" or "different from differences" which would likewise be self contradictory. But what this has to do with being "unary" or not "unary" is your argument and is beyond me. ~v~~ |