Alex

Is Beauty Quantifiable ?

144 posts in this topic

Is there a means of easily calculating the square roots of numbers?

I use my calculator, and that works fine, however, given pen and paper is there a solution to triangles that involve incommensurables?That is, has the method of finding the exact numbers of square roots, to specified precision, ever been found?

Your knowing all about Greek mathematics would imply knowing that of course there are methods of finding square roots and that the Greeks figured out how to do it. Furthermore, if no such method had "ever been found" then a calculator couldn't do it either. Calculators, like larger computers, are programmed with specific numerical algorithms, which requires first having a method to do it. They are not magic.

"Solution to triangles" has no meaning in this context; "triangles" are not a problem to be solved. You have to pose a specific problem before asking if there is a "solution", let alone one that "involves incommensurables". In the proper context the phrase "solving triangles" usually means solving trigonometric problems about triangles. Incommensurables are not relevant to that; the methods work the same for all numbers.

Are you asking how schoolchidren are taught to do it when they don't have a calculator? I'm not sure if school children are still taught how to calculate square roots by hand, but I certainly was when I was young (calculators didn't exist yet, especially today's modern handheld calculators).

If calculators didn't exist when you learned to do it by hand, then were you taught the method analogous to long division? That is the method I learned in grade school and remember as finding very tedious even though I was interested to know that it could be done and how to do it if I had to. (Using a slide rule later for calculations that didn't require excessive precision was a big improvement on the practical side.)

The "long division" type of method in fact originated with the ancient Greeks, who also used our actual long division algorithm that we use today, and is the only systematic computational method for finding square roots that the Greeks are known to have had except for an older, more specialized method that was restricted to sqrt(2) only.

The method is based on the mathematical relation: (x+h)^2 = x^2 + 2*h*x + h^2. Given x, find h. Mathematically, we can rewrite: 2*h*x = (x+h)^2 - x^2 - h^2. We can estimate h as [(x+h)^2 - x^2] / 2*x by neglecting h^2. The idea is to keep approximating h to smaller and smaller increments, making the h^2 term successively less significant (approaching zero). The method computes a square root one digit at a time, iterating once for each desired digit.

That is the basis for the "long division" method used by the Greeks and by school children like me (and what you learned?), but it was not used in the way you describe here and in your subsequent post with a more detailed account. Like the method used in long division, the method for square roots required setting up a table and filling it out by trial and error, one digit at a time, following certain rules. No explanation for why it works was provided to us, but the formula you cite does provide the basis for it.

First, to clarify the notation: Omitted from the discussion so far is that given an approximation x to sqrt(A) we are looking for an improved value x + h and that we want the value of h so that

A = (x+h)^2 = x^2 + 2*h*x + h^2,

which is used, replacing (x+h)^2 with A, transposed to the form

2*h*x + h^2 = A - x^2

(Betsy, please implement latex on the Forum so these thing can be more readable!)

The original approximation must be close enough so that h^2 is significantly smaller than the other terms.

But rather than iterate with rational numbers, the Greeks (like me in the 4th or 5th grade) restricted the use of the formula to integers, using it to compute the next digit (decimal or, often for the Greeks, sexidecimal = base 60 instead of 10). -- more like the example you worked out in the later post To find x = sqrt(A) the Greeks -- and our old grade school method -- required starting by using trial and error to find the largest value of the highest order part of sqrt(A), a whole number x, such that x^2 < A as the first approximation. Then you find the next digit using x as a starting point to find the biggest h so that

A > 2*x*h + h^2

The new value is essentially h = (A/2x) truncated to an integer, then you check (indirectly) to see if it is the largest integer satisfying the inequality.

The new approximation is x+h truncated with exactly one more digit than you start with, so you don't actually calculate the full (A/2x), but only observe by trial what the single digit is (like in long division). This new value of x is then used the same way to get the next value with one additional digit in the next cycle, so the process systematically adds one digit at a time just like long division, but using a new pair in the string of digits given in A at each step instead of only the next digit in the dividend as in long division.

That is an iterative process, but it is coupled with trial and error and is restricted to a single new digit in each cycle. Though it is based on the same formula you cited, the Greeks didn't have our concept of rational (or real) number and tended to use methods involving whole numbers as additional digits to progressively improve the accuracy. And as a practical matter, using the full recursion would involve extra long division at each cycle, rather than truncating to a single digit without even dealing with the rest of the digits in A until you need them in a subsequent step.

If that explanation of the mathematical analysis of the method looks unfamiliar, it is because the actual procedure as taught in practice is not expressed that way; it uses a tabular display with rules for the trial and error criteria similar to the way long division is done by hand, with no formulas as a guide.

In contrast, the full iterative method you described in which the small value h^2 is omitted in a full recursion (is that literally what you were taught?) is in fact equivalent to the later Newton method, independently mentioned by Bob Kolker's post. The iteration for h that you described is the equivalent of x_(n+1) = h + x_n, which for A = sqrt(x) simplifies to a combination of an arithmetic and a geometric mean to compute the next iterant in the form Bob gave:

h	   = 1/2 * (A/x_n - x_n) so that
x_(n+1) = 1/2 * (A/x_n + x_n)

If x_n is an underestimate of sqrt(A) then A/x_n is an overestimate (and vice versa); these two estimates bracketing sqrt(A) are averaged to obtain a better approximation (but that is not the way Newton derived it, he used his calculus).

Some Greeks also knew of this form, too, but apparently there is no record of it being used repeatedly in true iteration past the first or second approximant.

When used fully as a true Newton iteration instead of truncating to compute only the next digit, the convergence of the method (as shown by modern numerical analysis) is quadratic. The truncation error for x^2 ~ A is asympotically quadratic in h = (x_n+1 - x_n), and for approximants x_n already relatively accurate, the error for x_n at each step is the square of the previous one, so the number of additional digits approximately doubles at each step in the iteration cycle.

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Such methods go all the way back to the Greeks, probably even before Eudoxus. The earliest method is reguli falsi, or false position. One has two approximations to the root one larger and one smaller. Take the average and apply again until the differences become smaller than a specified number.

What you describe -- halving the interval and reusing one of the two half-intervals to bracket the sign change at the root, and then successively repeating the process with ever smaller intervals -- is called the "bisection method", not "regula falsi".

"Regula falsi" is another early method that uses a secant line to approximate the curve, solving directly for where that line crosses the axes to estimate the root, then repeating the process. "Regula falsi" and the "secant method" both use a straight line secant approximation to the curve but differ in how the end points for the next secant line are chosen for the next subinterval: regula falsi always brackets the sign change (like bisection) and the secant method retains the last two approximants as the new range. The computational efficiency index of the secant method is higher than regula falsi by (1+sqrt(5))/2 = the so-called Golden ratio, so rationalist Greek mystics can infer that it is also more beautiful. :-)

While it is reasonable to wonder if someone back then may have used such simple methods for square roots, especially bisection, there is no historical record indicating that the ancient Greeks used bisection, or regula falsi, or any other systematic method other than the "long division" tabular approach. Have you seen otherwise? Pythagoreans supposedly knew of a specialized iterative method for calculating (only) sqrt(2) to unlimited precision, but the general "long division" style method for sqrt(A) discussed in the previous post is only inferred to be known as far back as Ptolemy, with no traces before that (including in Eudoxus). A lot of what was done in such early times, including the Greeks, is entirely unknown or not known in any detail because the historical record is typically only a few fragments left to us from later works as much as hundreds of years later or more, leaving a lot to conjecture and to inference based on what later writers reported or revised in copying over the centuries.

The purely geometric method is to construct ...

My border collie suggests a simple geometric method based on drawing the A and B scales of a slide rule..., but the Greeks didn't know about Napier and you need logs for the construction. (Border collies sometimes fall into stolen concepts as easily as falling off a log, but at least the method would work for roots so she's not barking up the wrong tree.)

A quicker method with quadratic convergence goes back (at least) to Isaac Newton, the Newton-Rapheson method. If one wishes to solve

x^2 = L let x\sub 1 be the first approximation. Then recursively

x\sub n+1 = 1/2(x\sub n + L/x\sub n). This is a special case of getting the root of a general polynomial in one real variable.

As described in the previous post, this is the same as the iteration formula cited by System Builder as the basis of the method taught in elementary school, so the Newton iteration for square roots was in fact known much earlier than Newton, but it wasn't used by the ancient Greeks as a systematic iteration. Furthermore it was derived from calculus for roots of more general functions than polynomials.

It can be used with asymptotic quadratic convergence for any function with a continuous second derivative because (as Bob probably knows) it is based on a Taylor series truncated to the linear term. If y(x) is the function whose root you want, such as y(x) = 0 for y(x) = x^2- A to find sqrt(A), then the truncated series to first order is

y_n+1 = y_n + y'_n * h

The value of h = x_n+1 - x_n that makes y_n+1 zero at x_n+1 leads to the formulation used today (due to Simpson in 1740)::

x_n+1 = x_n - y_n/y'_n

So through the use of calculus a tangent line is used to approximate the curve (with slope y') instead of a secant line in the regula falsi or secant methods.

(If there is a multiple root you need more terms in the series because the lower order derivatives are zero at a multiple root.)

For square roots, substitute y = x^2 - A, then the value of h that makes x^2 - A zero at x_n+1 results in the equivalent formulas discussed here previously for the square root by both Newton's method and the "long division" kind of iteration:

x_n+1 = x_n - (x_n^2 - A)/(2*x_n)
x_n+1 = (x_n + A/x_n))/2

More rigorous analysis using the Taylor remainder term for the truncated series demonstrates the asymptotic convergence as quadratic. But Newton's method is not guaranteed to converge, depending on the starting value, and can in practice can cause trouble with polynomials when roots are close together (including roots in the complex plane).

The quadratic convergence of Newton's method only becomes practical if the starting value is x_0 is "close enough" to the root. The use of bisection can be faster to get "close enough", then you switch to Newton for higher accuracy much faster to finish the job to the precision required.

There are endless variations based on modifying and combining these methods, use of higher order polynomial approximations, and entirely different approaches. Many of them also generalize to multiple dimensions. Bob Kolker must have used a lot of these in "fortranacus-TO" (if that is Greek to most Forum members, he will get it -- "TO" being the ancient Greek numerals for "77").

If you want to brush up on your math there are several series of instruction books that are fairly gentle. The have titles like "Algebra for <demeaning term>" or some such (no slight or insult intended!). The price new is generally under $20 and even less expensive if you buy used from Amazon or such like vendor online.

I don't think Algebra for dummies explains the numerical analysis of why and how well different methods work, but it might describe the rules for using the "long division" method for square roots. But it may not be in there at all if the method isn't taught as elementary mathematics anymore now that we all have magic calculators that don't require anyone, including those who design the calculators, to know the methods :-).

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My objection to your use of the word "subjective" was not because it was not properly defined. It was that the concept as defined was invalid because it package-dealed characteristics that do not always occur together in reality. The given definition was "Subjective means subject to the observers interpretation" and it was contrasted to "objective" as if anything that is personal and/or subject to an observer's interpretation was NOT objective. This isn't true.

Personal values can be chosen and pursued objectively. Emotional reactions can be based on true premises. It is important, especially when defending one's most important personal values to not allow them to be dismissed as "subjective," but to make the distinction between the personal and the subjective.

For more discussion of this point, see this and this and this or use THE FORUM's search engine to lookup posts by me containing the word "subjective."

The concept as defined was not a package deal. I did not combine two different meanings of the word. I did not imply that "anything that is personal... is not objective." I was actually careful to point out otherwise. I defined one concept and used it distinctly without muddying the boarder with a different concept. My point is that the word subjective does not need to be abandoned. It has a perfectly valid use as it is defined in the dictionary. As an example, "subjective" is used in philosophy of the mind/consciousness to distinguish between experiences that are necessarily contextual to a first person perspective and outside activities which are not. It's not necessary to replace this use of the word "subjective" with "personal". It is not packaging any contradictory concepts together. The word "subjective" still has valid meaning despite its misuse by subjectivists.

Your posts did in fact use invalid concepts in several ways, as was explained in detail here and previous to that, but left unacknowledged. The misuse of concepts cannot be corrected with word substitutions attempting to use traditional terminology while relegating the Objectivist concepts and principles to "connotations" the way you tried to explain it away.

Invalid concepts do not equate to package deals. Your entire post here is irrelevant to what you are responding to. Further, nothing I said relegated Objectivist concepts to connotations. I am not invalidating the theory of objective evaluation, I am highlighting the role of personal opinion in that theory and defending that from someone who seems to think that objective contradicts personal. You seem to read into my argument things that are not there. If so those are your connotations not Objectivisms.

Comparison of spacial measurement vs. esthetic evaluation is not a matter of an alleged distinction between the "intrinsic" that is "external" to the observer, versus something "internal" to the observer. The philosophical dichotomy you made is false. Both are objective measurements; each depends on both the facts of reality and man's means of cognition.

That comparison is valid. There is a valid distinction between external comparisons and internal comparisons. I never implied that either is independent of facts of reality or man's cognition, only that the two are different in a way that makes distinguishing them useful.

Comparisons of "height" are not "intrinsic" to reality; objects are what they are without regard to "taller" or any other form of relation, measurement or comparison through commensurable measurements, which are only possible through man's methods of comparison based on the facts of what things are.

Certainly the comparison doesn't exist until man is there but as you said, objects are what they are. A tree and a cat each have their heights independently of man's existence. This is the definition of intrinsic. Again the word intrinsic still has valid use if you're careful to distinguish it from the false concept which intrincisicts have applied.

Neither is esthetic evaluation possible without regard to what things in fact are independent of our opinion.

I never implied that it was. I actually gave specific examples to the contrary.

You seem to believe that the general philosophical values are objective, but not the narrower esthetic evaluation (such as "beauty"), arguing that only the former are "true for everyone" . The distinction of the general philosophical versus the esthetic, the role of both in art, and why both are objective are explicitly discussed in OPAR, which you continue to deny.

Yes. General values, and their representation in art are true (or not true) for everyone, and some esthetic evaluations are not. If someone says a painting is beautiful because the distorted misery is shown with great power this is wrong for everyone. But if someone says a given painting is not beautiful because its passive strokes and colors are boring they are right for themselves but not right for everyone.

This leads us to the phrase: "Beauty is in the eye of the beholder".

Like goodness, therefore, beauty is not "in the object" or "in the eye of the beholder." It is objective. It is in the object -- as judged by a rational beholder.

I agree completly with this statement. But I do not think Dr. Peikoff here is arguing against the common use of the phrase. If someone says "beauty is in the eye of the beholder" and implies that beauty has no reference to reality then of course they are wrong. But this is not what people generally imply when they say it. As the original post in this forum attests, this phrase highlights personal opinion in evaluation.

Notice here that Ayn Rand is careful to distinguish which interpretation she refutes:

Now since this is an objective definition of beauty, there of course can be universal standards of beauty—provided you define the terms of what objects you are going to classify as beautiful and what you take as the ideal harmonious relationship of the elements of that particular object. To say, “It’s in the eyes of the beholder”—that, of course, would be pure subjectivism, if taken literally. It isn’t [a matter of] what you, for unknown reasons, decide to regard as beautiful. It is true, of course, that if there were no valuers, then nothing could be valued as beautiful or ugly, because values are created by the observing consciousness—but they are created by a standard based on reality. So here the issue is: values, including beauty, have to be judged as objective, not subjective or intrinsic.

Notice that she says "if taken literally", because of course the phrase should not be taken literally. It was not so by the original poster. It is not so generally. I did not use it so. I am arguing that "beauty is in the eye of the beholder" is a valid statement in that it highlights the role of personal opinion and that personal opinion can be different from person to person but still equally valid.

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(Betsy, please implement latex on the Forum so these thing can be more readable!)

The Invision software doesn't have a facility for that as far as I know, but I'll check it out.

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My objection to your use of the word "subjective" was not because it was not properly defined. It was that the concept as defined was invalid because it package-dealed characteristics that do not always occur together in reality. The given definition was "Subjective means subject to the observers interpretation" and it was contrasted to "objective" as if anything that is personal and/or subject to an observer's interpretation was NOT objective. This isn't true.

Personal values can be chosen and pursued objectively. Emotional reactions can be based on true premises. It is important, especially when defending one's most important personal values to not allow them to be dismissed as "subjective," but to make the distinction between the personal and the subjective.

For more discussion of this point, see this and this and this or use THE FORUM's search engine to lookup posts by me containing the word "subjective."

The concept as defined was not a package deal. I did not combine two different meanings of the word. I did not imply that "anything that is personal... is not objective." I was actually careful to point out otherwise. I defined one concept and used it distinctly without muddying the boarder with a different concept. My point is that the word subjective does not need to be abandoned. It has a perfectly valid use as it is defined in the dictionary. As an example, "subjective" is used in philosophy of the mind/consciousness to distinguish between experiences that are necessarily contextual to a first person perspective and outside activities which are not. It's not necessary to replace this use of the word "subjective" with "personal". It is not packaging any contradictory concepts together. The word "subjective" still has valid meaning despite its misuse by subjectivists.

Your posts did in fact use invalid concepts in several ways, as was explained in detail here and previous to that, but left unacknowledged. The misuse of concepts cannot be corrected with word substitutions attempting to use traditional terminology while relegating the Objectivist concepts and principles to "connotations" the way you tried to explain it away.

Invalid concepts do not equate to package deals. Your entire post here is irrelevant to what you are responding to.

Your posts in this thread used several invalid and misused concepts, which includes 'package dealing'. The relevance is explained in previous details that were referred to and which you do not acknowledge or address.

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Further, nothing I said relegated Objectivist concepts to connotations. I am not invalidating the theory of objective evaluation, I am highlighting the role of personal opinion in that theory and defending that from someone who seems to think that objective contradicts personal. You seem to read into my argument things that are not there. If so those are your connotations not Objectivisms.

This is what you wrote regarding Objectivist "connotations":

Betsy Speicher commented that she had issue with my use of "subjective". While I don't think that dictionary definitions should be left at the door, I understand that in an Objectivist forum the word carries a unique connotation.
There are number of essential concepts and principles that you appear -- and not just from this statement alone -- to not understand and not realize the significance of, and whose meanings and importance cannot be relegated to "connotations" as some kind of variation or emphasis on traditional notions. These have been explained previously.

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Comparison of spacial measurement vs. esthetic evaluation is not a matter of an alleged distinction between the "intrinsic" that is "external" to the observer, versus something "internal" to the observer. The philosophical dichotomy you made is false. Both are objective measurements; each depends on both the facts of reality and man's means of cognition.

That comparison is valid. There is a valid distinction between external comparisons and internal comparisons. I never implied that either is independent of facts of reality or man's cognition, only that the two are different in a way that makes distinguishing them useful.

Comparisons of "height" are not "intrinsic" to reality; objects are what they are without regard to "taller" or any other form of relation, measurement or comparison through commensurable measurements, which are only possible through man's methods of comparison based on the facts of what things are.

Certainly the comparison doesn't exist until man is there but as you said, objects are what they are. A tree and a cat each have their heights independently of man's existence. This is the definition of intrinsic. Again the word intrinsic still has valid use if you're careful to distinguish it from the false concept which intrincisicts have applied.

You have again left out what my response was in reference to. Go back and deal with it directly, fully and in full context, not by rewriting as if you had said something else, and not by trying to salvage invalid "subjective" and "intrinsic" notions while ignoring the objective in the realm of epistemology and esthetics.

Trees have the attribute height, but metaphysically do not have the numerical or any other kind of measurement or comparison; they do not have "taller", which is what you wrote. Measurements, including esthetic evaluations, are objective, not intrinsic vs. subjective, and measurements and comparisons are epistemological, not metaphysical. In the context of Objectivist discussion you can't use "intrinsic" and "subjective" or "internal" and "external" without regard to this, and if you understand why then you can see why "intrinsic" vs. "subjective" or "internal" vs. "external' is not a valid way to make a distinction you are trying to with regard to the status of esthetic and geometric measurement.

Your previous assertion claiming the objective is "partially" subjective [Jan. 1, 2010] was also wrong. Here is a relevant sequence that you never addressed showing a confusion over intrinsic vs. subjective vs. objective. You cannot discuss this issue without full and proper regard to the distinctions in and meaning of the trichotomy.

An evaluation is not by its nature "partially subjective". The objective is not a combination of the subjective and the intrinsic, somehow being "partially" both. It is a rejection of both. An evaluation is objective when the object is evaluated in accordance with rational standards. The evaluation must also be rational. An irrational, subjective assessment of "the facts" is not objective. Beauty is also not "partially objective and partially subjective". Dependence on context does not make something "partially subjective" and does not limit it to being only "partially objective". The partially objective is not objective.
I would say that, because an esthetic context varies from person to person, esthetic responses are always personal and may be objective or subjective depending on whether or not the evaluation is based on the facts of what the work being evaluated actually is.

I realize that my use of the term subjective was confusing.

Confusion of your readers is not the issue. You are at the least misusing the term. Objective evaluations are not "partially subjective". Objectivism stresses that objectivity is an alternative to both subjectivism and intrinsicism. It is not a mixture of them.

And here is the (latest) relevant sequence regarding the objective-intrinsic-subjective trichotomy, addressing the fact that you denied its role in Objectivism:

Proper esthetic evaluation is objective, not intrinsic or subjective.

I am not making an argument against Objectivism's theory of value. I defined what I meant by subjective and gave plenty of examples which illustrate that aesthetic evaluation is not arbitrary. Betsy Speicher commented that she had issue with my use of "subjective". While I don't think that dictionary definitions should be left at the door, I understand that in an Objectivist forum the word carries a unique connotation. Read my original post and in the word "objective" read "not dependent on personal opinion" and the word "subjective" read "dependent on personal opinion".

You have left out and ignored what my statement was in response to. This was the sequence you omitted:

Beauty is not "in the eye of the beholder" any more than it is "in the object". It depends on both the facts and the standards of evaluation.

What do you think "Beauty is in the eye of the beholder" means? It is a distinction between a property like height which has external evaluation (one thing can be intrinsically taller than another) and beauty which has internal evaluation (one thing can not be intrinsically more beautiful than another). It means one person's opinion does not invalidate another. I pointed out in my post that someone's opinion of beauty can be objectively right or wrong but is not necessarily so.

That alleged distinction is not correct. Measurements, e.g., height, are objective, not intrinsic. See Ayn Rand's Introduction to Objective Epistemology. Proper esthetic evaluation is objective, not intrinsic or subjective.

Your claim that you are "not making an argument against Objectivism's theory of value" is not responsive to that.

This is not a matter of loose "connotations", leaving the "dictionary" "at the door" or mechanical word substitutions. These concepts and principles have specific, well developed meanings that you appear not to be aware of or understand.

Comparison of spacial measurement vs. esthetic evaluation is not a matter of an alleged distinction between the "intrinsic" that is "external" to the observer, versus something "internal" to the observer. The philosophical dichotomy you made is false. Both are objective measurements; each depends on both the facts of reality and man's means of cognition. Comparisons of "height" are not "intrinsic" to reality; objects are what they are without regard to "taller" or any other form of relation, measurement or comparison through commensurable measurements, which are only possible through man's methods of comparison based on the facts of what things are. Neither is esthetic evaluation possible without regard to what things in fact are independent of our opinion.

In response to my statements -- that your "alleged distinction is not correct. Measurements, e.g., height, are objective, not intrinsic. See Ayn Rand's Introduction to Objective Epistemology." -- you later wrote (misappropriating it to a different context) that "OPAR and ItOE do not support these claims". They absolutely do. The role of the objective as opposed to the false alternative of the intrinsic and the subjective are fundamental and essential, in both works, for all the branches of philosophy, and in particular are prominent both in concept formation based on measurement and in esthetics. You haven't looked, have you? You should go back and read or reread them and see for yourself rather than continuing to deny it.

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Neither is esthetic evaluation possible without regard to what things in fact are independent of our opinion.

I never implied that it was. I actually gave specific examples to the contrary.

You took that snippet out of context of its explanatory role and the error you had made.

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You seem to believe that the general philosophical values are objective, but not the narrower esthetic evaluation (such as "beauty"), arguing that only the former are "true for everyone" . The distinction of the general philosophical versus the esthetic, the role of both in art, and why both are objective are explicitly discussed in OPAR, which you continue to deny.

Yes. General values, and their representation in art are true (or not true) for everyone, and some esthetic evaluations are not. If someone says a painting is beautiful because the distorted misery is shown with great power this is wrong for everyone. But if someone says a given painting is not beautiful because its passive strokes and colors are boring they are right for themselves but not right for everyone.

That is not the distinction for the status of esthetic evaluations in Objectivism; both general philosophic values and narrower esthetic values are objective. This is explained in OPAR. There are standards in art regarding the artistic presentation, not just the subject matter portrayed. There are some realms of the optional depending on one's experience and valid optional values, but not for "man at his best" and not based on a distinction between general philosophical values versus esthetic portrayal.

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This leads us to the phrase: "Beauty is in the eye of the beholder".
Like goodness, therefore, beauty is not "in the object" or "in the eye of the beholder." It is objective. It is in the object -- as judged by a rational beholder.

I agree completly with this statement. But I do not think Dr. Peikoff here is arguing against the common use of the phrase. If someone says "beauty is in the eye of the beholder" and implies that beauty has no reference to reality then of course they are wrong. But this is not what people generally imply when they say it. As the original post in this forum attests, this phrase highlights personal opinion in evaluation.

Of course he is denying the common use. And despite your professed agreement with it, you your argument shows that you still don't understand his explanation. You are rewriting his explanation to make the words into what you have wanted all along. "Beauty in the eye of the beholder" is an issue of esthetics; it does not mean there is nothing in reality at all that the subjectivist claims is beautiful. It is the difference between standards in art and abdication. This is not about metaphysics.

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Notice here that Ayn Rand is careful to distinguish which interpretation she refutes:
Now since this is an objective definition of beauty, there of course can be universal standards of beauty—provided you define the terms of what objects you are going to classify as beautiful and what you take as the ideal harmonious relationship of the elements of that particular object. To say, “It’s in the eyes of the beholder”—that, of course, would be pure subjectivism, if taken literally. It isn’t [a matter of] what you, for unknown reasons, decide to regard as beautiful. It is true, of course, that if there were no valuers, then nothing could be valued as beautiful or ugly, because values are created by the observing consciousness—but they are created by a standard based on reality. So here the issue is: values, including beauty, have to be judged as objective, not subjective or intrinsic.

Notice that she says "if taken literally", because of course the phrase should not be taken literally. It was not so by the original poster. It is not so generally. I did not use it so. I am arguing that "beauty is in the eye of the beholder" is a valid statement in that it highlights the role of personal opinion and that personal opinion can be different from person to person but still equally valid.

If you don't mean it literally then why say it, and why repeat it with such insistence while emphasizing "personal opinion"?

Ayn Rand's statement [Ayn Rand Answers, p. 227] is fully consistent with but briefer than the explanation in OPAR. She refers to "universal standards" for a given category.

Note that throughout this discussion we are emphasizing the epistemological status of standards in esthetics as a branch of philosophy, not personal reactions or even personal objectivity based on limited knowledge or experience, and not the particulars of the content of the standards in her philosophy.

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(Betsy, please implement latex on the Forum so these thing can be more readable!)

The Invision software doesn't have a facility for that as far as I know, but I'll check it out.

Thanks. It looks like Prodos set it up on his tewlip forum but I don't know what he had to do to get it to work: http://tewlip.com/viewtopic.php?f=10&t=22 Is he using the same forum software?

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(Betsy, please implement latex on the Forum so these thing can be more readable!)

The Invision software doesn't have a facility for that as far as I know, but I'll check it out.

Thanks. It looks like Prodos set it up on his tewlip forum but I don't know what he had to do to get it to work: http://tewlip.com/viewtopic.php?f=10&t=22 Is he using the same forum software?

Prodos uses PHPbb software. I tried the [tex] ... [/tex] bbs codes but they don't work with our software.

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(Betsy, please implement latex on the Forum so these thing can be more readable!)

The Invision software doesn't have a facility for that as far as I know, but I'll check it out.

Thanks. It looks like Prodos set it up on his tewlip forum but I don't know what he had to do to get it to work: http://tewlip.com/viewtopic.php?f=10&t=22 Is he using the same forum software?

Prodos uses PHPbb software. I tried the [tex] ... [/tex] bbs codes but they don't work with our software

I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

Without it beauty will not be quantifiable and quantifying cannot be beautiful. :-)

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

Without it beauty will not be quantifiable and quantifying cannot be beautiful. :-)

:D

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

Without it beauty will not be quantifiable and quantifying cannot be beautiful. :-)

But, I bet you never saw number you didn't like.

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

Without it beauty will not be quantifiable and quantifying cannot be beautiful. :-)

But, I bet you never saw number you didn't like.

You never saw an uninteresting number either. Suppose that uninteresting numbers exist. Then by the Well Ordering Principle there must be a least uninteresting number. But that number is rather interesting being the least of the uninteresting numbers. The contradiction carries the proposition. Q.E.D.

Bob Kolker

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(Betsy, please implement latex on the Forum so these thing can be more readable!)

The Invision software doesn't have a facility for that as far as I know, but I'll check it out.

Thanks. It looks like Prodos set it up on his tewlip forum but I don't know what he had to do to get it to work: http://tewlip.com/viewtopic.php?f=10&t=22 Is he using the same forum software?

Prodos uses PHPbb software. I tried the [tex] ... [/tex] bbs codes but they don't work with our software.

I may have missed the discussion on this thread, however, doesn't MS Word have mathematics typography capability [while not having intelligent mathematics capability] ?

I know that this remark regarding software is not apropos of expression or demonstration of beauty by means of mathematics. Its just extra information. Intelligent physics, chemistry, and manufacturing [physics and chemistry] process software are on the market. Business economics calculation models based upon manufacturing process software [involving physics and chemistry] are now appearing in industry. For example, in a process design if one changes an ingredient or add a certain energy level and quantity of photons to water and you get a quantity of steam at a certain temperature.

Returning to the matter of giving number or characteristics to beauty.

Mathematics demonstrations of universal principles always yield results that are at least as universal as the starting expression and no more. To achieve particular results and particular numbers, substitutions of numerical values for symbols in equations will result in particular number expressions. Even if some particular numbers may only be expressed in specific terms, e.g., base-pi numbers.

The maximized particular demonstrations of mathematical principles always result in number. The demonstration of the [supposed] concept of 'infinity' may only be accomplished in particular finite terms of specific numbers. The concept of infinity can never reach an infinite universal. Once that limit has been reached, the end of the process is a particular finite number - and it is no longer a universal.

Pythagoras discovered that the principle that purportedly leads to the huge result or number of infinity always is either an expression of a universal concept or continuous functioning of the principle. Once a number is increased by +1 the result is always finite. Infinity can never be reached. It is always a claim of a continued functioning of a principle that can never result in a particular or finite number. That's a contradiction. The concept that Pythagoras discovered that is superior to the [supposed] concept of infinity is that od the 'continuation' of a functioning principle. The result of a demonstrated continuing of a principle is always finite.

Note that the pro-infinity advocates always claim the existence of a continued functioning of a principle that is supposed to lead to infinity. That is a Stolen Concept. The 'must-have-ists' of infinity require continuity of principle to lead them to infinity, when the concept of continuity, which is a corollary to the principle of universality, is the end concept. That also is a corollary to the concept of the continuity of existence in the universe, to which the pro-infinity advocates desire and end. Forget the anti-concept of infinity, and instead, reawaken the concept of the continuity of the functioning principle, e.g., of finite existence, or of a triangle.

The universal geometrical demonstration system developed by Pythagoras results in drawing, and the mathematical principles that were modeled could be modeled either in 2D or 3D. The limit of the Pythagorean geometrical system was that it could only accommodate mathematical principles that could be modeled or drawn. Pythagoras did permit, as Euclid later stated, that a line may be extended by a selected or chance amount. The principles that could be accommodated in the system could be rendered in the real world and the finite drawn and modeled results could be made to be perceptible. All roots and square roots could be conceived in the Pythagorean geometrical demonstration system, and the results that could be rendered were always finite.

Note that infinity, being always a finite demonstration with finite results, can never be modeled or made perceptible.

Enter architecture. Demonstrations of mathematical principles once arranged in terms of a perceptible design must always be expressed in terms of particular numbers. If you want a multi-column facade on your temple you have to pick the precise finite number of columns. You cannot have any type of infinite expression of numbers, e.g., eight, seven, and six, columns at the same time, place and respect. Nor is an arbitrary number of columns possible.

Note that the Aristotelean concept of non-contradiction, and either-or, that are necessary in the Pythagorean Hekatompedon system have a nice fit with the Hekatompedon way of demonstration of geometric principles. The echoes of the Hekatompedon are stated in Aristotles comments.

Particular expressions of mathematical principles yield finite numbers. Particular expressions of geometric principles yield finite model or drawing demonstrations. Add the requirements of a building, e.g., the worship of the idea of a woman of ideal characteristics, or practical matters of construction economies, with a design concept, complete the construction, and the result may be measured.

The beauty of a universal principle is theoretical.

The beauty of a measured particular principle is demonstrated or expressed in terms of a modeled design concept, and among the characteristics that may be modeled are specific numbers or measurements.

There is nothing infinite about the Parthenon. It is as concretely perceptible as all get out. There are many principles that went into its making, and the result is a beautifully perceptible arrangement of elements. One may say that the result is imposingly finite and perceptible.

May I say that mathematical principles may be beautiful from the standpoint of being complete, inventive, unique, new, ordered, demonstrable, provable, succinct, or even, expressible, to name just some of the possible characteristics. That's one classification of mathematical beauty that doesn't get enough press.

The other classification type of mathematical beauty involves the perceptible actualization of principles, that is, demonstration. The proof of the pudding. The beauty may be found in terms of the interesting principles that are being demonstrated.

Machine drawings may yield fascinating images, and computer drawn real-time dynamic images of planets and satellites orbiting a star have a different result. The principles there also involve the dynamical inter-relationships of mass, gravitational accelerations, and velocities. That's way beyond the making of a drawing of a continuous circle figure by means of compass and stylus.

Truly interesting dynamic traces are evident to the senses. When I saw the sunlight reflecting USA satellite pass over head one dark night, I saw an even-ness of speed that was most fascinating. The line traced by the motion of the aluminized point was produced with incredible precision. It was there, then there, and just when I thought that I had captured it again, it was now over there - all with great even-ness. Finite demonstrations of mathematical principles may be made to be actual and perceptible. The motion was incredulously smooth and even.

We know that the Pythagorean geometrical system has generated the means of creating architectural perceptible characteristics and measurements. AG wares are well-know to have been made, measured, and shaped according to those principles. Systems of perceptible proportions are given evidence in the design of the Parthenon. The design there is all about how much of everything of importance there is in the design.

What we may infer from the design of the Parthenon, and that is due to its ever-present precision of designing and making, is the underlying geometry that is among the causes of its design and making. Once those selected characteristics are there, and have been carved in marble, they may be measured. Centuries later, the building has been measured, and its secrets have been re-discovered. Thanks to the use of mathematics to govern or control the shapes of the design, and to their accurate stating in terms of stone. we may reverse-engineer the design of the building. That work was probably the first ever study of reverse engineering. A goodly amount about what we know about the geometry system of the Hekatompedon of Pythagoras has been revealed to us by measurements taken by archeologists and historians of architecture from the actual building. And by the innocent eye.

Inventor

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

Once again, I may have missed the sense of what is being discussed.

Mathematical typographical symbols are offered on MS Word.

Open a document.

Open tab, "Insert".

To the right on the main menu of the insert tab are two sub-menus:

Click on "Equation" or "Symbol" to see a display of math symbols.

[...]

Above, I tried to copy one of them from MS Word to the Forum. That didn't work.

Could the MS Word expression be saved to a file format that may be brought into The Forum?

This one worked:

±

=======================================

Now, ± , is a good one for Platonic infinity advocates who require that a concept of number, or a series of numbers, be extended in two directions, e.g., to "100" and to "-100", simultaneously. A straight line is required for the demo.

However, they don't require that the line be limited by the second points beyond the origin point in either direction. Out there in the " + " direction there is only infinity, and out there in the " - " direction there is only infinity. Note that there are no defining second points which would enable the basic concept of a straight line.

A straight line to an infinity advocate would be stated:

<-infinity><origin point><+infinity>.

It also appears that the Binswanger concept of a single point plus vector definition of a straight line also requires at least one "infinity", and probably two "infinities".

That would be anti-Pythagorean, because Pythagoras identified that continuously functioning universal principles are and must be demonstrated, or are expressed, in terms of finite elements or number values.

Inventor

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I think he had to install it as an add-on package or feature, so no tex codes could be expected to work without that. But that doesn't mean it is necessarily available for the Forum software. Is there someone you could ask?

Yes. There is an Invision Forum for Invision Forum admins. (Natch!) I'll ask there. Even if we can't do it now, if I ask, they may add it to a later upgrade.

Once again, I may have missed the sense of what is being discussed.

Mathematical typographical symbols are offered on MS Word.

Open a document.

Open tab, "Insert".

To the right on the main menu of the insert tab are two sub-menus:

Click on "Equation" or "Symbol" to see a display of math symbols.

[...]

Above, I tried to copy one of them from MS Word to the Forum. That didn't work.

Could the MS Word expression be saved to a file format that may be brought into The Forum?

This one worked:

±

You did completely miss the sense of what is being discussed. Some special characters can be embedded in a Forum post and the code will be properly recognized. This has nothing to do with file formats or with equation editing and formatting as is done in latex. The Forum software does not read usoft word document "files". You can only display image files or link to external web files. This has nothing to do with formatting within a Forum post.

=======================================

Now, ± , is a good one for Platonic infinity advocates who require that a concept of number, or a series of numbers, be extended in two directions, e.g., to "100" and to "-100", simultaneously. A straight line is required for the demo.

What "demo" are you talking about? ±n means that the value n may be either positive or negative. It has nothing to do with "infinity advocates". Infinite lines and sequences are open-ended and are conceptually extended as a potential. An infinite extent designated by ± means only that the extension is in both directions. Anyone can do this to "100" or any other magnitude, which has nothing to do with Platonism. Can't you conceive of a sequence of integers simultaneously extending to both -100 and +100?

However, they don't require that the line be limited by the second points beyond the origin point in either direction.

Neither does anyone else for either a fixed line segment or an infinite line with open-ended extent.

Out there in the " + " direction there is only infinity, and out there in the " - " direction there is only infinity. Note that there are no defining second points which would enable the basic concept of a straight line.

A straight line to an infinity advocate would be stated:

<-infinity><origin point><+infinity>.

What "infinity advocate" or anyone else do you know who specifies a straight line by a single point or who thinks that + or - means there is nothing else but infinity in either direction? A line through a point has a specific direction. "Infinity" has nothing to do with it and does not itself specify a direction for anyone.

The abstract "point at infinity", as used for example in stereoscopic projection, is a single "point" on a sphere mapping to all infinite extents in the plane which are used (for example in conformal mapping) as the abstract point at infinity, which is not an actual point because there is no such thing, only a limiting process. It has nothing to do with specifying the direction of a straight line and the mathematical concept is not Platonism.

It also appears that the Binswanger concept of a single point plus vector definition of a straight line also requires at least one "infinity", and probably two "infinities".

There is no such "Binswanger concept". Everyone knows that a straight line may be specified by two points or by a point and a direction. This has nothing to do with requiring one or "probably two" "infinities". You do not appear to understand any of this mathematics.

That would be anti-Pythagorean, because Pythagoras identified that continuously functioning universal principles are and must be demonstrated, or are expressed, in terms of finite elements or number values.

There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

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There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

The Reductio ad Absurdum for the Pythagorians was the discovery of numbers which are not the ratio of integers (the so-called irrational numbers). The first number proven to be irrational was the length of the diagonal of a square. The unitary case is the square root of two.

PS: For those who claim one cannot prove a negative, I love bringing up the case of the square root of two. One can prove that two is not the square of the ratio of two integers. That is as negative as can be and it can be proven.

Bob Kolker

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There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

The Reductio ad Absurdum for the Pythagorians was the discovery of numbers which are not the ratio of integers (the so-called irrational numbers). The first number proven to be irrational was the length of the diagonal of a square. The unitary case is the square root of two.

Interesting. That concurs with what I know. What he also did in addition to what you say is to identify a concept of number, that is a series of numbers that are generated according to a specified principle, is to create a concept of number that has as its base the irrational hypotenuse of the integer that was its cause. E.g.:

the diagonal of the 1:99^.5 rectangle = 100^.5, =10

the diagonal of the 1:100^.5 rectangle = 101^.5,

the diagonal of the 1:101^.5 rectangle = 102^.5 ... and so on.

The rectangles are placed on top of one another and all rectangles have a common origin point, and the height of all the rectangles is unity. The rectangles were said by the AGGs to be constructed between parallels, that is between parallel straight lines. That is part of what was known to Pythagoras and the earlier Ancient Egyptian Geometers as the Hekatompedon geometric system.

The diagonal lines of all rectangles laid upon integers, e.g., 1,2,3,4,5,6,7,8,9,10, . . . . can be radii for circles. The AGG teachers used to ask of their students, "Have you done your 'circles'?"

Those same diagonal lines may then be the bases of rectangles or squares, depending upon what you are speaking of. These figures are a variant of the "Whirling Rectangles" of the AGGs. The newly drawn figures that have either this or that specific properties are called rectilineal figures.

Aristotle later added his 2cents and said that since the Whirling Squares [or other specific concepts] functioned according to continuous universal principles [or motions] the general classification of concepts could be termed, "Recilinear Motion". Rectilinear [or Rectilineal] Motion is the concept that separates things that have classifiable properties from things that have continuously existing substance. That is, the Identity Axiom was separated from the Existence Axiom.

PS: For those who claim one cannot prove a negative, I love bringing up the case of the square root of two. One can prove that two is not the square of the ratio of two integers. That is as negative as can be and it can be proven. Bob Kolker

Another interesting coincidence or not, is the relationship of the components of the basic square of the Golden Mean Rectangle and the generation of the series' of squares and rectangles, of either commensurable or incommensurable properties.

Draw a square between parallels with the origin of the square placed upon the lower parallel line. Find the midpoint of the base of the square, and produce a straight line to the upper right corner point of the square. From the origin construct a radius line with a length equal to the length of the diagonal line. Find that length on the lower parallel line from the origin. Add that length to the length that is equal to 1/2 the side of the square. Extend a medial line vertically from that new length point to the second parallel line, and construct a new rectangle.

The length of the new rectangle in a ratio to unity is the Golden Mean Ratio. The diagram that was drawn between parallel lines was called the Golden Section.

Something interesting was found. The Pythagoreans tried to build a new concept of number that would have the special points [the first one being the mid-point] at places that were subdivisions of the base of the square, e.g., at 1/2, 1/3, 1/4, 1/5, . . . . . 1/10 positions. They had a new number system, however it had little practical use. The Golden Mean Rectangle and 1:2^.5 rectangle were the only interesting results. As the student would be constructing new special points closer and closer to the origin point the principle of selection or of sufficiency was invoke and work stopped.

As the base of the square was subdivided into smaller and smaller units, e.g., 1/2, 1/3, 1/4, 1/5, . . . . . 1/10000, the diagonal was busy causing rectangles of Golden Mean : unity height ratio and smaller, all the while the diagonal length approached the length of the diagonal of a square.

If the special points are propagated to form sub multiples on the base line of the square in the opposite direction from the midpoint first mentioned, the length of the diagonal line produced gets progressively shorter and approaches the length of the base of the square, or unity, in value.

That path of investigation did come up with interesting side-lights, however, the result was mostly one of the discovery and demonstration of properties of lines drawn between parallels. Except for the root : square root and Golden Mean ratio incommensurables, there was no discovery of a new concept of number.

Zeno had his observation regarding the impossibility of approaching the origin point. Aristotle later commented that the principle of infinity, there the goal of a process of a continued subdivision rather that of multiplication or addition, was a process of unattainably. Infinity could never be actualized or demonstrated in terms of a finite number. Infinity would be a matter of a futile quest for the actualization of a universal, first a, then u, first a, then u, then a, then u, then a, then u, and so on, by continuous application of the principle being used. There can be no end to the process of seeking infinity, either by multiple or by sub-multiple. The process focuses the student's mind upon the contradiction and not on the continuous functioning of the universal.

[The task of the AG student was to find whether universals were prior to particulars, or vice versa. --- In particular and eternally. ]

The universal then was also able to function as a causal principle in proofs and syllogisms. Via Pythagoras the Golden Mean principle was one of the several paths that lead to the systematization of the deductive syllogism.

I realize that this condensation makes intelligibility difficult, and that an illustrated expansion of the ideas would be helpful.

Inventor

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That path of investigation did come up with interesting side-lights, however, the result was mostly one of the discovery and demonstration of properties of lines drawn between parallels. Except for the root : square root and Golden Mean ratio incommensurables, there was no discovery of a new concept of number.

Not quite. Eudoxus was able to define proportions even for non-rational quantities. His method came within a whisker of producing what we moderns call the Real Numbers. The method of Eudoxus was reproduced and extended in a rigorous fashion by Dedekind (the so-called Dedekind Cuts) to give us our beloved real numbers. August Cauchy, the French mathematician introduced an alternative method of defining the reals, by means of converging sequences of rational numbers. But wait! That's not all! There is even a method more directly based on Eudoxus for constructing the reals, without first constructing the rationals from the integers. See arXivmath/04054v1 24 May 2004.

-The Eudoxus Real Numbers- by R.D. Arthan

Here is the abstract:

This note describes a representation of the real numbers due to Schamuel. This representation lets us construct the real numbers from first principles. Like the well-know construction of the real numbers using Dedekind's Cuts, this idea is inspried by the ancient Greek theory of proportion due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers. ........

For the original Eudoxus method, see Book VI of Euclid's Elements.

Bob Kolker

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There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

The Reductio ad Absurdum for the Pythagorians was the discovery of numbers which are not the ratio of integers (the so-called irrational numbers). The first number proven to be irrational was the length of the diagonal of a square. The unitary case is the square root of two.

PS: For those who claim one cannot prove a negative, I love bringing up the case of the square root of two. One can prove that two is not the square of the ratio of two integers. That is as negative as can be and it can be proven.

Bob Kolker

Nice try, but you need to study logic a little closer. The fallacy is not simply that one can't prove a negative, the fallacy is "he who asserts the positive has the burden of proof." Your statement, although grammatically negative, is not logically negative. If you assert, "two is not the square of the ratio of two integers" then it is up to you to prove it. In other words, until you prove it with evidence, I do not have to disprove it with evidence that I don't have. To summarize, I do not have to prove "two IS the square of the ratio of two integers" (the logical negative of your statement) to reject your statement before you prove it.

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There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

The Reductio ad Absurdum for the Pythagorians was the discovery of numbers which are not the ratio of integers (the so-called irrational numbers). The first number proven to be irrational was the length of the diagonal of a square. The unitary case is the square root of two.

PS: For those who claim one cannot prove a negative, I love bringing up the case of the square root of two. One can prove that two is not the square of the ratio of two integers. That is as negative as can be and it can be proven.

Bob Kolker

Nice try, but you need to study logic a little closer. The fallacy is not simply that one can't prove a negative, the fallacy is "he who asserts the positive has the burden of proof." Your statement, although grammatically negative, is not logically negative. If you assert, "two is not the square of the ratio of two integers" then it is up to you to prove it. In other words, until you prove it with evidence, I do not have to disprove it with evidence that I don't have. To summarize, I do not have to prove "two IS the square of the ratio of two integers" (the logical negative of your statement) to reject your statement before you prove it.

Ooops. Make that "square root" in the two cases above.

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That would be anti-Pythagorean, because Pythagoras identified that continuously functioning universal principles are and must be demonstrated, or are expressed, in terms of finite elements or number values.

There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

The Reductio ad Absurdum for the Pythagorians was the discovery of numbers which are not the ratio of integers (the so-called irrational numbers). The first number proven to be irrational was the length of the diagonal of a square. The unitary case is the square root of two.

Inventor has consistently confused the elementary mathematical successes of the Pythagoreans with their mystical philosophy, refusing to acknowledge the latter, which they most certainly did not "demonstrate". They reified numbers to metaphysical status. Their discovery of incommensurable numbers apparently drove them nuts. The most fundamental reductio ad absurdum was the implication for their metaphysics, not for numbers!

PS: For those who claim one cannot prove a negative, I love bringing up the case of the square root of two. One can prove that two is not the square of the ratio of two integers. That is as negative as can be and it can be proven.

The epistemological principle that one "cannot prove a negative" pertains to the irrationality (epistemological, not mathematical!) of demanding that "proof" must be provided to reject the arbitrary. The burden of proof is on he who asserts the positive. It does not mean that the "negative" isn't a valid concept, that it cannot be identified when it arises, or that the law of the excluded middle cannot be used in deductions such as simple reductio ad absurdum. There is nothing arbitrary about classifying numbers as "rational" vs. "irrational" (meaning expressible as a ratio of whole numbers or not). The Pythagoreans started with identifiable magnitudes then found by logical means that they weren't all commensurable.

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That path of investigation did come up with interesting side-lights, however, the result was mostly one of the discovery and demonstration of properties of lines drawn between parallels. Except for the root : square root and Golden Mean ratio incommensurables, there was no discovery of a new concept of number.

Not quite...

"Not quite" is the least of it; it is isn't even close. The "square root and Golden Mean ratio incommensurables" were not a "concept of number" at all. They Pythogreans only knew that some numbers were not commensurable with others.

Eudoxus was able to define proportions even for non-rational quantities.

To clarify this, the Pythagoreans' theory of proportions was arithmetical in character and was restricted to numbers commensurable with the same unit; Eudoxus' use of geometrical expression of magnitudes for a full theory of proportion came much later.

His method came within a whisker of producing what we moderns call the Real Numbers. The method of Eudoxus was reproduced and extended in a rigorous fashion by Dedekind (the so-called Dedekind Cuts) to give us our beloved real numbers.

Caution: the following requires an understanding of both the mathematics and Objectivist epistemology, so if anyone is left...

Dedekind cuts provide a systematic way for sorting out how to consistently deal with real numbers by a precise method that can also be understood visually, but it does not provide any conceptual clarification for what a real number is, i.e., understanding the concept in terms of "the problem of universals". To try to take the Dedekind cuts theory literally for such an interpretation would drive you crazy -- cast in terms of sets of points or segments of a line it would result in saying that a real number is everything except for what it is -- in terms of the set of everything that it isn't as a way to provide for its existence.

August Cauchy, the French mathematician introduced an alternative method of defining the reals, by means of converging sequences of rational numbers.

This approach to pinning down the mathematics of real numbers is much closer to how the concept actually arises as a specialization of the concept of number than the geometrically based theory of Dedekind cuts, and is therefore less rationalistically artificial for conceptual understanding in such terms, but (for those interested) the mathematical formulation still does not provide a full conceptual understanding even though it has "omitted measurements" written all over it, making the epistemology obvious for anyone who understands the IOE answer to the "problem of universals". But as the theory is normally presented mathematically it does not explain the number concept at all. To try to take it that way literally results in (again) confusing sets with concepts, requiring that you think that a "real number" is a "set" -- actually an equivalence class consisting of a set of sets -- with no mental integration of the referents into a concept referring to the elements in a sequence.

The mathematical approach is sensible and with a little thought can be made conceptually sensible in addition to -- like Dedekind cuts -- its technical success, but it was not due to Cauchy. Cauchy contributed much to a rigorous analysis of limit, greatly improving the mathematical analysis of the abstract continuum, but if I remember correctly, his theories of analysis were not refined sufficiently for him to realize that convergence to a limit is not equivalent to the (weaker) idea of what we now call Cauchy convergence -- in which elements of the sequence become progressively closer to each other. The first implies the second, but the latter does not demonstrate that the limit exists ("completeness") and he treated it as if it did, begging the question of existence -- in the mathematical technical sense in addition to the conceptual sense that requires another, higher level of abstraction. The full mathematical theory of real numbers making the distinction between convergence and Cauchy convergence, and using that distinction explicitly in the theory, came later with the two related methods of Cantor and Weierstrass in the last part of the 19th century, with Cantor's being most like the way it is done today (despite his crazy mysticism in other realms). The method uses sequences that converge in the sense of Cauchy, but Cauchy himself did not use it to formulate the theory of real numbers.

But wait! That's not all! There is even a method more directly based on Eudoxus for constructing the reals, without first constructing the rationals from the integers. See arXivmath/04054v1 24 May 2004.

-The Eudoxus Real Numbers- by R.D. Arthan

Is that on the web?

For the original Eudoxus method, see Book VI of Euclid's Elements.

Eudoxus' copyright seems to have expired recently so you can now find the Heath translation online. But the Eudoxus theory of proportions is in Euclid Book V, not VI. (Other theories of his are in other books of Euclid.) Search the web and you will even find animated cartoons carrying out the geometrical constructions.

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But wait! That's not all! There is even a method more directly based on Eudoxus for constructing the reals, without first constructing the rationals from the integers. See arXivmath/04054v1 24 May 2004.

-The Eudoxus Real Numbers- by R.D. Arthan

Is that on the web?

Log onto arxiv.org and use the search function with the document number as the argument. Also you can search on Arthan. It is a nifty, nifty paper.

BTW, thank you for the correction. Yes Book V, not Book VI.

And it is here: http://aleph0.clarku.edu/~djoyce/java/elem...ookV/bookV.html

complete with little Java Applets.

Bob Kolker

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