The Virgin Birth of Points
in [Physics]
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From: Lester Zick on 28 Dec 2007 12:41 On Thu, 27 Dec 2007 11:03:35 -0800 (PST), David R Tribble <david(a)tribble.com> wrote: >David R Tribble wrote: >> Alternatives to "not" are other logical operators. >> Or are you saying that's not the case? > >Lester Zick wrote: >> 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. > >The only thing that can stand by itself in a logical statement >is a simple value, such as "true". "Not" is not a value, it's an >operator, and therefore cannot "stand by itself". "Not" is a predicate and can be analyzed on that basis alone in terms of itself for the purpose of determining whether it is true or self contradictory in combination with other predicates or itself. >Differences are values, not operators. They are the result >of applying a binary difference operator to two arguments >(operands). The statement "d = a - b" has a value, d, >equated to the result of a binary operator "-" taking two >arguments, a and b. > >I can't see how "not = (whatever)" can be a well-formed sentence. >What is "not" equal to as a value? It is equal to false? You can always make any predicate a sentence just by prefixing it with "it is". ~v~~
From: Lester Zick on 28 Dec 2007 13:03 On Thu, 27 Dec 2007 11:03:35 -0800 (PST), David R Tribble <david(a)tribble.com> wrote: >Lester Zick wrote: >> You're interested in Y only because assumptions of truth can be false. > >David R Tribble wrote: >> If by "assumptions of truth" you mean "axioms", no, that's not >> the case. Mathematical axioms are always true. > >Lester Zick wrote: >> Incorrect by your own words. You say elsewhere: >> "An axiom is an assumed truth. Period." >> "Assumed truth" and "always true" are different predicates. > >Let's just expand that, then, to: > Axioms are statements of truth, and are always true, and are > therefore always assumed to be true (obviously). So you're seriously suggesting the predicates "assumed to be true" and "are always true" mean the same? In your example you've got it exactly backward. If an axiom is true we can certainly assume it to be true. But the reverse is not true that what you assume true, mathematically or otherwise, is always true. I can certainly think of any number of "assumptions of truth" which are not "always true" and even "not ever true". >If an axiom is assumed to be false, it is not an axiom. >In other words, a false statement cannot be an axiom. >Hence axioms cannot be false. You know, you really need a refresher course in Logic. You seem to consider "assumptions" the only predicate present. It doesn't run that way. "Axioms are either considered true or false" and then progress to whether "false statements" can be axioms. The proper sequence is to begin with "assumptions of truth" you label "axioms" and then analyze whether they're "true" or "false" and hence whether they can be considered axioms. The problem is you never progress to the second step. You just begin with "assumptions of truth" as axioms and let the matter rest because you have no way to determine the truth of any assumption you make. >You seem to be hung up on the "assumed" part. Wouldn't any mathematician worthy of his salt get hung up with any student who just "assumed" a theorem is true? The problem with your use of "axiomatic assumptions of truth" is you have no way to analyze whether they're true or not so you just assume they're true instead of false. If you're going to use the predicate "true" whether assumed or demonstrated, you have to be able to use the predicate "false" also. ~v~~
From: Lester Zick on 28 Dec 2007 14:25 On Thu, 27 Dec 2007 11:03:35 -0800 (PST), David R Tribble <david(a)tribble.com> wrote: >David R Tribble wrote: >> If by "assumptions of truth" you mean "definitions", no, that's >> not the case either. Definitions are not true or false. > >Lester Zick wrote: >> 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". > >Contradicting predicates makes a definition void or vacuous, >but it does not make it false. Definitions are never false or true. Except I don't understand the difference between vacuous, invalid, void, invalid, incorrect, etc. and simply false. The only difference between definitions and propositions is that the former just involve isolated predicate combinations while the latter entail predicate combinations from different sources. >"Squircles are square circles" is a vacuous definition; there >are no geometric objects that meet the definition of squircle. >Yet the definition itself is neither true nor false; it just is. However there are many definitions for which that would be true. You can only recognize the vacuous nature of the definitions for squircles on cursory inspection because everyone understands that squares are different from circles in some fundamental way. But if I simply said X is Y Z you would have no way to know that the definition was valid, invalid, or simply false without analyzing each of the predicates associated with Y and Z in combination to determine whether there were any basic contradiction involved. In other words you cannot just pronounce a definition valid, invalid, true, or false without specialized knowledge. >Consider "a Goldbach non-integer is an integer greater than >2 that is not the sum of two primes". We don't know if there >are any integers meeting this definition, but that does not >make the definition false, or true, or unknown, or whatever. >The definition itself is completely valid. Unfortunately you have no way to know that without analyzing each and every predicate in combination with each and every constituent predicate to decide whether there's a mutual contradiction involved. It may be that the predicates which characterize "sums" and "primes" considered in combination with one another preclude a Goldbach non integer. But we can't know that except through the evaluation of each in combination with each and every other predicate in the definitions. In other words there is no categorical distinction between definitions and propositions in mathematics or anywhere else which precludes the possibility of mutual contradiction between predicate combinations in the aggregate of predicate combinations in definitions or propositions because both are made up of predicates and predicate combinations. Consequently any distinction between the two is artificial and there is no reason to say one can be false and the other not to the extent there can be mutual contradictions involved in each. In fact the only relevant distinction I can think of is that false propositions do not necessarily invalidate any of their constituent definitions although there would have to be some mutual inconsistency among definitions. In other words one might say that a false definition would entail self contradiction and a false proposition mutual contradiction between otherwise non self contradictory definitions. But even this concession would not necessarily invalidate the possibility of self contradictory predicates in definitions. For that matter just consider the Peano axiomatic generation of the naturals. You maintain points are isomorphic with the reals. However the Peano axiomatic generation of the naturals doesn't say how it is successive naturals are supposed to fall on any straight line in going from point to point because there is no guarantee successive points would lie in any particular direction. In fact the only way such a thing could be possible is if you had the straight line to begin with and successively divided it to produce the naturals. Hence I consider the Peano axiomatic generation of the naturals invalid, incorrect, and "false" to the extent it entails a false assumption: that a succession of points has to lie in any specific direction. ~v~~
From: Lester Zick on 29 Dec 2007 14:01
On Wed, 26 Dec 2007 11:49:31 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >>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. I'd like to correct a misuse of terminology in my reply above which might engender misunderstandings. Above I consistently refer to points, infinities of points, relative sizes of those infinities, etc. In doing this I was adopting the conventional vernacular and perspective on points as constituents of lines just for simplicity and not to argue that points are in fact constitutents of lines. To be accurate what I should have said was "infinitesimals" instead of points. Points have zero dimensionality; lines have zero width; and surfaces zero thickness. Points delimit or define line segments; lines delimit or define surfaces; and surfaces delimit or define solids. And although points require no dimensionality to delimit or define lines, lines themselves require an infinitesimal dimensionality to be lines.There has to be some directionality associated with constituents of lines and that directionality has to have a reciprocal relationship between lines and their constituents. In other words derivatives of lines are infinitesimals and conversely the integral of infinitesimals constitute lines. The same is true of lines with respect to surfaces and surfaces with respect to solids. Lines have zero width with respect to the surfaces they delimit or define. But surfaces are constituted of infinitesimals integrated between lines which delimit those surfaces. And the same is true of solids which are constituted of infinitesimals integrated between surfaces which delimit or define those solids. I hope this clarifies what might otherwise be misunderstood. The relative magnitudes of infinities of infintesimals and not points is what I had in mind and what I should have been describing. Points are boundaries which define lines and are not constituents of them. ~v~~ |