Galileo's Paradox
in [Math]
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From: Tony Orlow on 9 Dec 2006 23:16 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>> >>>> What,in mathematics, has a solution which is neither a real measure, or >>>> the measure of truth of a statement, 0, 1, or somewhere in between? >>> "Find all pairs of distinct naturals (x,y) such that x^y = y^x". The >>> solution to which is the set {(2,4), (4,2)}, which doesn't appear to be >>> a "real measure", nor a "measure of the truth of a statement" (as far >>> as I can understand your meaning of the terms). >> Those are specifications for two points in a 2D array of naturals, the >> values within each pair denoting the distance of each point in each of >> the two directions, from the origin. > > That's one way of imagining it. Another interpretation would be that > each pair (x,y) is a specification for the point on the real line whose > distance from the origin is denoted by 2^x/3^y (this is a dense set). > > Yet another interpretation is that the pair (x,y) is a specification of > the set of all x by y chessboard positions containing 4 non-attacking > rooks. > > Are you saying that the "solution" of {(2,4), (4,2)} in these > interpretations would be somehow different from the solution in your > interpretation? Why is an interpretation required at all? > Huh? I'm just saying it boils down to quantity (including truth values), language, and operators. >> Are you saying x and y and 2 and 4 >> are not taken to be quantities? That's a rather strange position to take. >> > > Well, your phrase "real measure" is a little vague. I really don't > understand what you mean by it. > > Sure, I understand that 2 and 4 are quantities in the usual sense of > the word: "2 apples", "4 apples". And I understand that when I add 1/2 > cup of apple juice to 1/2 cup of apple juice, I get 1 cup of apple > juice; "1/2" is a "real measure". > > But the "solution" to the problem given is not 2 or 4 or 1/2; it is a > set of ordered pairs of naturals. > > Why is the set {(2,4), (4,2)} a "quantity" or a "real measure"? Can I > have "{(2,4), (4,2)} apples", or "{(2,4), (4,2)} cups of apple juice"? > It's a construction in the language using operators and real values. Remember, I included the language of expression, and the operators allowed in the language, in mathematics. >>> I assume that you offer more than the trivial observation that all >>> mathematical statements P, including the statement "2^4=4^2", are >>> examples of the (boolean) truth valued statement "it can be proved that >>> P". If so, I would claim that the mathematical question is "Find a >>> proof of P, or proof of not P". And the solution is not "0" or "1"; the >>> solution is either a proof of P, or a proof of not P. >> First of all, that trivial observation is plenty. All logic is subsumed >> under math as a calculation of truth values between 0 and 1. > > I think you are confusing boolean algebra with mathematical proof. > I think you are forgetting that any computer can perform deduction with simple boolean algebra. There are automatic proof generators based on that principle, no? >> Further, a >> proof is precisely this calculation of truth value for a given >> statement. To say "find a proof" is to say "define a sequence of >> operations on the given statements to produce another such that its >> value is 1". > > A proof is a series of calculations such as: from the strings "(p->q)" > and "p" use the rule modus ponens to produce the string "q". It ends > with a step which produces the string corresponding to the assertion of > the theorem. Yes, that can be performed using Boolean algebra. > > These are not calculations producing the value 1, except in the trivial > sense that the statement "It is the case that George W Bush is the > current president of the US" can also be called "a calculation > producing the value 1". ((p->q)->(p->q))=1. That's all you're saying. > > In which case, the "solution" to just about any question you like is "a > measure of truth of a statement, 0, 1, or somewhere in between"; so I > don't see how it's a particular property of /mathematical/ questions as > opposed to other types of questions. > Who said it was? It's the quantitative foundation of logic, whether applied to statements about quantities, languages, or hillbilly presidents. >> Truth is a form of quantity. > > In different contexts there are different axioms and different rules of > inference; so "truth" is certainly not a "quantity" that is fixed for > some statement such as "there exists x such that 2*x = 3". > That depends on what restrictions you have on x, and whether they are compatible with that statement. If x is a natural, then the statement has value 0, if real or rational, then 1. >>> "Find all finite groups G having a maximal subgroup S, and having a >>> subgroup T which is isomorphic to S but not maximal". This question was >>> asked in sci.math a few days ago. The groups in question are not >>> "numbers" at all; and a set of them possesses no particularly natural >>> total orderings. They can be partially ordered by "size" (number of >>> elements); but there are, in general, multiple distinct groups on a set >>> of any given size. >> Assign each element a bit, and every group corresponds to a binary >> string, which corresponds to a value. > > #1: There is 1 group of order 1. > #2: There is 1 group of order 2. > #3: There is 1 group of order 3. > > There are two groups of order 4, both are commutative. I will > arbitrarily number them as > > #4 : 0 + x = x; x + x = 0; 1 + 2 = 3; 1+ 3 = 2; 2 + 3 = 1 > #5 : 0 + x = x; 1 + 1 = 2; 1 + 2 = 3; 1+ 3 = 0. > > But how does this numbering help me figure out what the groups of order > 5 are? > > I'm not arguing that it is impossible to count these groups (i.e., > assign a unique natural to each one). I'm arguing that /before/ you can > count them, you must produce them. /Producing/ them is then the > "solution" to the problem, not counting them /after/ they have been > produced. > > I think you haven't been exposed to much mathematics where this is the > case; and I'm guessing that that's why you seem to assume that > mathematics is about "measure". Perhaps. > > For example, "find all squares of naturals", is a case where we /can/ > produce them by /counting/: "the first is 1*1 = 1, the second is 2*2 = > 4, ..., the nth one is n*n = n^2, ..." > > A more complicated example: we can produce the Fibonacci numbers (1, 1, > 2, 3, 5, 8, ...) by counting them via the function F(n) = (phi^n - (1 - > phi^n))/sqrt(5). > > But there is no similar "closed form function" that maps naturals onto > groups in this way. To find the groups with 5 elements up to > isomorphism, one can examine one by one each element of the set of 5^25 > (approximately 10^17) binary functions on 5 elements, which can > certainly be enumerated. > > But it's far faster (and of more mathematical interest) to use a proof > to show that there is exactly 1 such group (up to isomorphism), because > 5 is a prime number. > Okay, well, I am not particularly familiar with group theory, so it's hard to comment on that. >> These letters on your screen are >> numbers. >> > > And so every statement communicated through written language is a > number, and therefore everything is mathematics? This again becomes > true of just about any statement. And yet, I feel that there is a > difference between the branches of human endeavor "history" and > "mathematics". > Of course. > <snip> > >> That's fine. I am still of the opinion that math boils down to measure, >> the language of measure, and the operations allowed on that language. >> > > Your statements would be easier to understand if you explained what you > mean by "measure", "the language of measure", and the operations you > allow on that language. > > Cheers - Chas > Measure is difference. Language is a set of strings from an alphabet. Operations transform one string to another.
From: Tony Orlow on 9 Dec 2006 23:22 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Fine. Please state your rule. Let's take a look. >>>> I've been over this a lot, but hey, what's one more time. Practice makes >>>> perfect. >>>> >>>> We start with the notion of infinite-case induction, such that a >>>> equation proven true for all n greater than some finite k holds also for >>>> any positive infinite n. >>> What is a "positive infinite n"? >> A value greater than any finite value. >> >>>> Inequalities can also be proven true for >>>> infinite n, but only provided that the difference between expressions >>>> which forms the inequality does not have a limit of 0 as n grows without >>>> bound. If it does, the inequality holds only for finite n. Now, given >>>> this extension of classical inductive proof, we can easily prove such >>>> facts as, say, 2<n <-> 2 < n < 2n < n^2 < 2^n < n^n, and this ordering >>>> will be true for any infinite n. Thus we have a full spectrum of >>>> infinite expressions which can be ordered, provided we have some common >>>> infinite n with which to express them. Okay so far? >>> No. See above. >> Why do you have a problem with the mere suggestion of an infinite value? > > I don't have a problem with infinite values. However, you have to do > more than merely say "positive infinite n". Assuming your positive > infinite things aren't the same as something that we already know about > (in which case, you should just say so), you either need to give a > construction of these positive infinite things or you need to specify > their properties. You haven't done either. For example, how many of > these things are there? How do they relate to each other? How do they > interact with the natural numbers? Are the operations of addition and > multiplication defined for them? > Well, I have been through much of that regarding such specific language approaches as the T-riffic digital numbers, but that's not necessary for this purpose. It suffices to say that, if a statement is proved true for all n greater than some finite k, that that also includes any postulated infinite values of n, since they are greater than any finite k. I don't need to construct these numbers. Consider them axiomatically declared. Big'un is the number of reals per unit interval. As far as operations of addition and multiplication, they obviously apply, if I am saying that arithmetic formulas using those operators are valid for infinite values of n. I mean, is "a number greater than any finite number" so hard to understand? How many reals are in the unit interval? Some infinite number.
From: Tony Orlow on 9 Dec 2006 23:28 Virgil wrote: > In article <457b5606(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> There is nothing wrong with saying E and N have the same cardinality. >>>>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse >>>>>> Function Rule, which indeed can parametrically compare sets mapped onto >>>>>> the real line, using real valued functions, as a generalization to set >>>>>> density. It works for finite and infinite sets. So, what is wrong with >>>>>> trying to form a more cohesive theory of infinite set size, which >>>>>> distinguishes set sizes that cardinality cannot? There certainly seems >>>>>> to be intuitive impetus for such a theory. >>>>> Fine. Please state your rule. Let's take a look. >>>> I've been over this a lot, but hey, what's one more time. Practice makes >>>> perfect. >>>> >>>> We start with the notion of infinite-case induction, such that a >>>> equation proven true for all n greater than some finite k holds also for >>>> any positive infinite n. >>> What is a "positive infinite n"? >>> >> A value greater than any finite value. > > But if "n", is, as usual, reserved for indicating natural numbers, those > which are members of every inductive set, there is no such thing. > What, now mathematics has declared what the single letter n means? It's a variable. I could use x or p or any other string. In this case, n is any real or hyperreal value. >> Why do you have a problem with the mere suggestion of an infinite value? > > It is ar\ithmetical operations with infinite values absent any > definition of what those operations mean or pwhat properties they have, > to which we make legitimate objections. > Not really. If the expressions used can themselves be ordered using infinite-case induction, then we can say that one is greater than the other, even if we may not be able to add or multiply them. Of course, most such arithmetic expressions can be very easily added or multiplied with most others. Can you think of two expressions on n which cannot be added or multiplied? >> Surely you must have guessed enough what I meant to follow the >> paragraph? (sigh) > > Guessing is not a reliable way of finding things out. Of course, reading is a little more reliable, but when one gets stuck on such words as "positive" and "infinite", maybe one needs to do a little guessing.
From: stephen on 10 Dec 2006 00:01 Tony Orlow <tony(a)lightlink.com> wrote: > Tonico wrote: >> So the above, and very specially the despise and offensive tone many >> idiotic trolls/cranks use to refer to PROFESSIONAL mathematicians just >> because they don't abide by their whims is what makes me call you >> people what you are: trolls/cranks. > There is nothing wrong with expecting science to satisfy intuition. Other than the fact that science again and again has proven to not satisfy intuition. It was intuitive that heavier objects fall faster than lighter objects. It was intuitive that heat was a fluid. It was intuitive that an object in motion must have a force acting on it. It was intuitive that light was a wave in a physical medium. It was intuitive that objects obey Gallilean transformations. Stephen
From: David Marcus on 10 Dec 2006 00:55
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Why do you have a problem with the mere suggestion of an infinite value? > > > > I don't have a problem with infinite values. However, you have to do > > more than merely say "positive infinite n". Assuming your positive > > infinite things aren't the same as something that we already know about > > (in which case, you should just say so), you either need to give a > > construction of these positive infinite things or you need to specify > > their properties. You haven't done either. For example, how many of > > these things are there? How do they relate to each other? How do they > > interact with the natural numbers? Are the operations of addition and > > multiplication defined for them? > > Well, I have been through much of that regarding such specific language > approaches as the T-riffic digital numbers, but that's not necessary for > this purpose. It suffices to say that, if a statement is proved true for > all n greater than some finite k, that that also includes any postulated > infinite values of n, since they are greater than any finite k. I don't > need to construct these numbers. Consider them axiomatically declared. Then list the axioms for them. -- David Marcus |