Alex

Is Beauty Quantifiable ?

144 posts in this topic

For the record, the second sentence,

"...therefore, principle [1] and principle [2] logically follow....",

should read,

"...therefore, principle [1] and principle [3] logically follow...".

[...]

One more problem that scientists have equally with writers of

philosophy or esthetics is logic. For example, if the writer says that

because principle [1] and principle [2] are parts of relationship in a

first premise, and because principle [2] and principle [3] are parts

of a similar relationship in the second premise, that, therefore,

principle [1] and principle [2] logically follow in the appropriate

relationship. That conclusion, if the three principles have no

relationship to the checkable facts of reality, is patently false.

Just because the form of, a=b, b=c, therefore, a=c, is followed,

that does not mean that, in the absence of demonstrated factual

identities for a,b,c, the physical identities are true. That error is

common where demonstrated and proved definitions have not

been identified and set forth.

[...]

Inventor

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Multiple clauses.

While I'm at it, I want to add one more type of writing error that is

commonly encountered in writings on esthetics. That is, that

circularity results when far too many sentence sub-clause modifiers

are used to explain a point. What happens, especially when several

undefined or high level abstract concepts are used, is that the

arguement is pointed to several paths that are ultimately difficult

to reconcile. Gray-area meanings and svelt grammar may cloak the

digressions. Multiple circularity is a result. Some authors seem to

claim that if they say enough, and digress enough, and use enough

erudite terminology, that their circular discussion reinforces itself.

Those authors seem to provide that because the concepts they use

overlap that the general drift of what they say is correct. When a

critical eye is used, however, it may often be found that an entire

paragraph may be reduced to a single sentence. A closer inspection

may find that the writer is saying the same thing over and over again.

One solution for this problem of multiple clauses is to write shorter

sentences. Limit the number of modifying clauses. The circular

writer will, instead, have to keep the number of his words to a

minimum.

Inventor

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BTW, one of the most difficult tasks of writing about esthetics is

Non-Contradiction. The writer cannot say that a thing is black and

non-white at the same place. Just stay with one type of statement,

for example, that a glossy sphere is black, and order everything

in the writing accordingly. That simplifies a work to one-half the

verbiage, and in some cases, when you really had nothing to say,

to nothing at all.

One more problem that scientists have equally with writers of

philosophy or esthetics is logic. For example, if the writer says that

because principle [1] and principle [2] are parts of relationship in a

first premise, and because principle [2] and principle [3] are parts

of a similar relationship in the second premise, that, therefore,

principle [1] and principle [2] logically follow in the appropriate

relationship. That conclusion, if the three principles have no

relationship to the checkable facts of reality, is patently false.

Just because the form of, a=b, b=c, therefore, a=c, is followed,

that does not mean that, in the absence of demonstrated factual

identities for a,b,c, the physical identities are true. That error is

common where demonstrated and proved definitions have not

been identified and set forth.

This supposed explanation of how the alleged "problem that scientists have equally with writers of philosophy or esthetics is logic" is itself incomprehensible.

For the record, the second sentence,

"...therefore, principle [1] and principle [2] logically follow....", should read,

"...therefore, principle [1] and principle [3] logically follow...".

The entire passage remains unintelligible. The problem is not the simple principle of transitivity; it is your writing. Maybe more is wrong, too, but we can't tell without knowing what thoughts are intended to be expressed.

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[...]

The entire passage remains unintelligible. The problem is not the simple principle of transitivity; it is your writing. Maybe more is wrong, too, but we can't tell without knowing what thoughts are intended to be expressed.

Unintelligible? OK.

I wouldn't deny an actual criticism reqarding my writing, however, without a specific example I cannot evaluate and change what is there. The claim that my writing may include intransitive relations is too excessive to deal with. I have never heard of intransitive relations until you mentioned it. Btw, shouldn't the word in proper grammar be "relationship" rather than "relation"?

Next time just ask. I would be the first to say that my writing tends more often towards the inconsistently coherent and is only rarely excellent.

Your last sentence, quoted here, is no marvel of grammatic excellence, either.

Thanks for the reminder. My writing does need work, and I think that it could be 100 percent better if I would refine the writing offline prior to posting. Often I have excellent theoretical discoveries, and I don't verbally develop the explanations in order that they will be logically functional.

I have read too much Euclid or Aristotle, and that sometimes provides forms, unique technical terms, or structures that others may not understand or appreciate.

Context is everything. For example, I once wrote a scientific commentary reply, and I said that the science work innvolved needed to be proved using deductive proofs. The scientist writer said that I was not clear, after all, proofs are not used much any more. In another example, I asked a psychologist if logic had anything to do with psychology. The answer was, "No." Some answer about one being comfortable with what one says or with the reactions of others was offered. My point here is that unless one is a party to the context of the offered statements one may have to not understand what is said. That's the way the cookie crumbles.

If you misunderstood what I said, that may be due to the error in the Barabara Sylogism that I presented. In the subsequent post I provided the correct text. Correct logic is an essential cause of intlligibility.

You didn't pick up on the error of logic in the text I wrote. The error is in the same passage that you said was unintelligible. Have you taken a course in deductive logic? If not, I strongly recommend the one offered by Leonard Peikoff. It isn't your task to find errors, and, you have every right to expect correct theory, logic, grammar, and revisions, where appropriate.

Inventor

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[...]

The entire passage remains unintelligible. The problem is not the simple principle of transitivity; it is your writing. Maybe more is wrong, too, but we can't tell without knowing what thoughts are intended to be expressed.

Unintelligible? OK.

I wouldn't deny an actual criticism reqarding my writing, however, without a specific example I cannot evaluate and change what is there.

The 'example' was quoted in full.

The claim that my writing may include intransitive relations is too excessive to deal with. I have never heard of intransitive relations until you mentioned it. Btw, shouldn't the word in proper grammar be "relationship" rather than "relation"?

I referred to your use of the "principle of transitivity", not "intransitive relations". The problem, as already stated twice, was the rest of the 'rambling' in two long paragraphs which made it impossible to see what you were trying to do with a simple logical principle. You seem to ignore or forget what has been written and 'respond' with rambling discourses that no one can follow, let alone identify the relevance of. For example, who cares about gratuitous analysis of the grammar of statements that were never made?

Next time just ask. I would be the first to say that my writing tends more often towards the inconsistently coherent and is only rarely excellent.

It wasn't a question. I don't think anyone here is asking you questions expecting knowledge from you, only hoping that you will express clearly what you choose to say so that we can decipher it.

Your last sentence, quoted here, is no marvel of grammatic excellence, either.

Sentences express coherent thoughts; their purpose is not to exhibit "marvels of grammatic[al] excellence". The one you referred to -- "Maybe more is wrong, too, but we can't tell without knowing what thoughts are intended to be expressed" -- is clear enough, and also grammatical.

Thanks for the reminder. My writing does need work, and I think that it could be 100 percent better if I would refine the writing offline prior to posting. Often I have excellent theoretical discoveries, and I don't verbally develop the explanations in order that they will be logically functional.

"Cognition precedes communication". No signs of excellent theoretical discoveries here. If there are any buried someplace, you need to cogently describe what you think you are saying before attempting "explanations" or expecting anyone to evaluate your claims.

Whatever you mean by "logically functional", all that is required here is coherent description so we know what you are talking about, then the explanation can be explored as the next step.

Yes, you could try "refining the writing off line before posting", but this is all informal writing, and all that is required is basic clarity of thought and expression without worrying about "refinements". First there would have to be something specific to refine. Just try putting what you write aside for a while and coming back to read it later before making changes and posting so you have a better chance of seeing it yourself as it was written, and not as whatever you had in your head at the time you wrote it, which is otherwise invisible to everyone else. The cycle of delay/re-read/re-write is one means to help achieve objectivity in writing -- iton teh Foru is useful because it is often hard to see your newly written words objectively, i.e., as they are on the screen apart from what is in your head when you write them.

I have read too much Euclid or Aristotle, and that sometimes provides forms, unique technical terms, or structures that others may not understand or appreciate.

I see no similarity with Euclid or Aristotle. Appreciation of "forms" etc. is not the issue -- just simple clarity of thought objectively expressed.

Context is everything. For example, I once wrote a scientific commentary reply, and I said that the science work innvolved needed to be proved using deductive proofs. The scientist writer said that I was not clear, after all, proofs are not used much any more. In another example, I asked a psychologist if logic had anything to do with psychology. The answer was, "No." Some answer about one being comfortable with what one says or with the reactions of others was offered. My point here is that unless one is a party to the context of the offered statements one may have to not understand what is said. That's the way the cookie crumbles.

Whatever these are supposed to be examples of, the "context" here on the Forum is the posts on the Forum you claim to be responding to, not extraneous discourses and whatever else you have in your mind from someplace else. When you want to introduce external "contexts", it is up to you to identify and explain them so we know what you are talking about.

If you misunderstood what I said, that may be due to the error in the Barabara Sylogism that I presented. In the subsequent post I provided the correct text. Correct logic is an essential cause of intlligibility. You didn't pick up on the error of logic in the text I wrote. The error is in the same passage that you said was unintelligible.

As I wrote: "The problem is not the simple principle of transitivity; it is your writing. Maybe more is wrong, too, but we can't tell without knowing what thoughts are intended to be expressed." The full two paragraphs are unintelligible. If the words are gibberish then the 'logical forms' dimly referred to are irrelevant and no one cares or can figure out if there was an error in that because we don't know what you are trying to say anyway. (The "Barabara Sylogism", presumably the "Barbara Syllogism", is not the issue here, but is very similar to what is more commonly called "transitivity".)

Have you taken a course in deductive logic? If not, I strongly recommend the one offered by Leonard Peikoff. It isn't your task to find errors, and, you have every right to expect correct theory, logic, grammar, and revisions, where appropriate.

I am not seeking to "find errors", nor do I expect impeccable grammar; the first step is very basic: simply understanding what you intend to express.

Meanwhile, you are in no position to be recommending that other people take courses in logic, suggesting that if only we were not so ignorant of logic you would be understood. Cogent recommendations of courses, books etc. may be of interest, but not for that purpose. Leonard Peikoff's logic course in not a prop supporting your writing -- which itself might, however, benefit from his course on "objective communication".

And yes, to answer your question, I did attend Leonard Peikoff's course on logic in addition to reading and other course work in logic and mathematical logic.

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Beauty . . . . .

is an evaluation of existential properties [of an object] by an individual within the context of his own sense of life and hierarchy of values.

That is the best new definition of the concept, beauty, that I have seen yet. That may be a first for Objectivism.

I added a grammatical item that I thought made a better isolated sentence, and that specifically points to the referent that concerns the individual of context.

That brings to mind my questions. Are such new definitions, that are not included within Ayn Rand's Objectivism, to be considered to be part of Objectivism?

May one say for the purposes of clarification that the definition is of objective beauty?

Inventor

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Beauty . . . . .
is an evaluation of existential properties [of an object] by an individual within the context of his own sense of life and hierarchy of values.

That is the best new definition of the concept, beauty, that I have seen yet. That may be a first for Objectivism.

Actually, it is a description of some essential characteristics of esthetic judgment, not a definition of beauty. A definition would be something like "Beauty is a physical characteristic of an object that evokes a positive emotional response in an individual within the context of his own sense of life and hierarchy of values."

That brings to mind my questions. Are such new definitions, that are not included within Ayn Rand's Objectivism, to be considered to be part of Objectivism?

Although it depends on Ayn Rand's concept of sense of life, it wouldn't be part of Objectivism because Ayn Rand did not identify nor endorse it. Instead, it would be more accurate to describe it as Objectivism-based or Objectivism-compatible.

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Inventor, your preaching to the quire when you say that beauty is objective. But I don't agree with how you interpret 'objective'.

Beauty is recognized and identified as an esthetic and esthetic-emotional response to the charateristics and qualities of the existing or created object, or scene.

This is not true. Beauty is not a response. It is an evaluation of characteristics. You don't say I feel beauty or I am beauty. You say that thing is beautiful..

I find that there is something right in what you say. Yes, beauty isn't just a response, although that may be a result that merits contemplation or adds to happiness. And, yes, beauty is an evaluation of characteristics. However, Objectivism, and Objectivist-based psychology, if not the prior observations, provides that evaluations of the data provided by the sense-perception biological system may be evauated biologically causing emotions, and/or evaluated intellectually, which may yield ideas, emotions, and esthetic emotions. In music, for example we may anticipate and hope for certain happenings, and then be rewarded when the composer demonstrates the principle of the piece in a wonderful finale.

The Consciousness axiom is here appopos. To be conscious is to be conscious of something. Thats a broad statement that includes identifications of beauty and beautiful objects and characteristics of objects, however, it doesn't get down to the nitty gritty of what's really beautiful, and why. That's for esthetics.

What's beautiful about some item may be a matter of unexpressed descriptions regarding the characteristics of the object. Some objects, e.g., the Parthenon and some symphonies are so complex that there may be many elements that may be of interest in several different respects to more than one person. When they can verbalize what they agree of, and provide both principle and example, then the discussion really gets interesting.

I would hate to have to say why I enjoy, and find beautiful, a favorite symphony, to a professional musician. My hemming and hawing would make me seem the idiot. On the other hand, I can think of a modern eclectic composer, who writes the most vile music, and my small talk about music far exceeds his damaged demonstrations.

What are the specifics of beauty?

What characteristics of the object, meaning what causes , forms [Aris., not Platonic], and principles, that are integrated and demonstrated within the functioning work, meaning examples, yield what results of evaluations, emotions, esthetic emotions, pleasures, and happiness.

What this here now is fun and beautiful. And why?

LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIIIIIIIIIIIIIIIIIIIII

Inventor

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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".

"Happiness and heroism vrs depression and mediocrity" are not objects subject to the attribute "beauty" and provide no commensurate distinction with assessment of use of color and texture. You appear to have confused philosophical evaluation with a narrower esthetic evaluation. Again, see OPAR.

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

OPAR and ItOE do not support these claims. Happiness, heroism, depression, and mediocrity are not objects but are, as themes in art, subject to beauty. Height is a quality that exists independent of man's existence, or intrinsically.

If you would read OPAR on the subject you would see why "beauty is in the eye of the beholder" is not correct. You don't have to read it if you don't want to, but don't contradict it here, apparently without even realizing it, and then in a snide non sequitur like the above statement demand that no one who rejects your position should be "giving advice" on what to read. Look up the subject in the index, or don't, but don't tell anyone not to refer you to OPAR on specified topics as the answer to your post.

I gave arguments as to why the phrase "beauty is in the eye of the beholder", if properly understood, is true .This does not contradict anything in OPAR. I'm not telling "anyone" not to refer me to OPAR. I am disagreeing with your interpretation. It was absolutely not "non sequiter" and only snide in its deflection of your original attitude.

Instead of chasing each other around with long quoted posts, why don't we start back at the beginning? Do you agree with the statement, to which my objection was addressed?

personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder

I do not agree with this statement and I gave reasons why. If you do agree with the above statement than we can start there and actually discuss it.

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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.

----------------------------------------------------

"Happiness and heroism vrs depression and mediocrity" are not objects subject to the attribute "beauty" and provide no commensurate distinction with assessment of use of color and texture. You appear to have confused philosophical evaluation with a narrower esthetic evaluation. Again, see OPAR.

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

OPAR and ItOE do not support these claims. Happiness, heroism, depression, and mediocrity are not objects but are, as themes in art, subject to beauty. Height is a quality that exists independent of man's existence, or intrinsically.

You have again omitted what you wrote that I responded to, this time also editing statements of mine out of sequence giving a false impression of what you are responding to and making it look like one of my statements was in response to another of my own statements, without identifying who wrote it and where. You extracted part of what you quoted from the discussion referred to in the segment above, which shows it in its proper context. Here is the rest of the actual sequence showing what you omitted:

There is a realm of personal options in art, but beauty is not deuces wild in accordance with arbitrary standards depending only on personal choice. That is not what "context" means in objective evaluation.

I never implied that beauty is "deuces wild" or "arbitrary". I actually pointed out that it is not. "Some values are objectively good or bad (happiness and heroism vrs depression and mediocrity) but some are subjectively good or bad (swift brush strokes and red hues vrs careful and precise brush strokes and blue hues)...in each of these choices, one might be objectively better for an individual." I am highlighting the difference between what is true for one individual vrs true for everyone. Something can be beautiful for one person, and not beautiful for another without objective contradiction. This is what is meant by "Beauty is in the eye of the beholder."

You relied on each individual's own "context" to claim lack of objectivity in evaluation of art. That is not what dependence on "context" means in pursuing objectivity. Nor are properly optional assessments of inessential features "subjectively good or bad".

"Happiness and heroism vrs depression and mediocrity" are not objects subject to the attribute "beauty" and provide no commensurate distinction with assessment of use of color and texture. You appear to have confused philosophical evaluation with a narrower esthetic evaluation. Again, see OPAR.

In particular you continue to confuse general philosophical values (such as "happiness") with the esthetics of how they are portrayed in art. 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. You haven't looked, have you?

----------------------------------------------------

If you would read OPAR on the subject you would see why "beauty is in the eye of the beholder" is not correct. You don't have to read it if you don't want to, but don't contradict it here, apparently without even realizing it, and then in a snide non sequitur like the above statement demand that no one who rejects your position should be "giving advice" on what to read. Look up the subject in the index, or don't, but don't tell anyone not to refer you to OPAR on specified topics as the answer to your post.

I gave arguments as to why the phrase "beauty is in the eye of the beholder", if properly understood, is true .This does not contradict anything in OPAR.

Your statement most certainly does contradict OPAR. If you would look it up you would find that OPAR literally and explicitly says the exact opposite of your claim and explains why. You haven't looked, have you?

I'm not telling "anyone" not to refer me to OPAR. I am disagreeing with your interpretation. It was absolutely not "non sequiter" and only snide in its deflection of your original attitude.

Again, here is the statement you made that you now omit:

If you believe that beauty is not in the eye of the beholder than you should not be giving advice about which references need rereading.

It does not follow that disagreement with your statements, which you also confuse with "attitude", implies that I should not advise you to read or reread the discussion in OPAR that contradicts what you seem to think is compatible with Objectivism.

----------------------------------------------------

Instead of chasing each other around with long quoted posts, why don't we start back at the beginning? Do you agree with the statement, to which my objection was addressed?
personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder

I do not agree with this statement and I gave reasons why. If you do agree with the above statement than we can start there and actually discuss it.

The only "chasing around" if have had to do is track down the statements you edited or omitted while you improperly claim to respond to what I wrote.

I don't know what the original phrase "personal opinion drops as a criterion" means. I have been objecting to your statements of what you claim is correct yourself and which you seem to think is Objectivism. I agree with Leonard Peikoff's discussion of the objective nature of a proper esthetics and why he wrote:

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.

If you want to start over then I again suggest that you start by thoroughly reviewing OPAR and IOE, particularly with regard to the topics I mentioned previously: "the nature of values, the objective vs. the subjective and intrinsic, the role of context, and the standards of evaluation in art in accordance with both esthetic principles and philosophical principles regarding what the art is representing".

Then state more clearly the meaning of what it is you object to that you saw in Inventor's original post (regardless of what he may have personally meant behind the ambiguities), and then contrast it with what you think is right in terms of the well-developed concepts and principles of Objectivism -- not by word substitutions in invalid concepts that we are expected to translate as if the Objectivist conceptsand principles were not of any importance beyond "connotations".

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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.

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."

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Here's a very short analogy.

Suppose you go to a nutritionist for diet advice. He could give you a lot of objective information on that basis, given the objective nature of calorie counts, the amount of protein and fats and carbohydrates in various foods, etc. He might advise you to eat more fish and less red meat, more green vegetables and less starchy root vegetables.

But he'd be totally out of line to tell you that you should like, say, tilapia better than salmon -- and to assert that there's something wrong with you if you don't. The late Robert A. Heinlein once made a similar argument: which is "better" -- a chocolate malt or a strawberry malt? (I like coffee ice cream better than chocolate, but I'd be a fool to argue against the chocolate lovers!) I think it's the same with our experience of the arts -- there's a range of normal objective taste. Indeed, there is a range of normality in a lot of things. Nobody would assert that a man with an !Q of 101 is a genius and a man with an IQ of 99 is a moron. Nobody would consider heights of six feet two or five feet ten abnormal on an objective basis.

Anyway, if we can have divergent tastes in food, why not in beauty? Even though we can agree that there's an objective difference between good food and rotten food, between beauty and ugliness. I'm not sure what "quantifying" it would entail: Bo Derek was touted as a 10, but there were are other women just as beautiful who didn't/don't look just like her -- consider the women of other races. Can we rate every sunset precisely on a scale of one to ten? Every painting? Every piece of music? Every poem? Every novel? It's enough for me to know when I'm experiencing beauty.

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I want to know what makes you think that Pythagoras was influenced by Thales, and those other claims about Pythagoras.

Pythagoras

Thales, ~624 BC to ~546 BC, was later called the, 'father of philosophy', due to his axiomatic approach to ideas and to their systematization. He was also called, 'the father of geometry', due to his cataloguing and ordering of all the known theorems of geometry. Thales was a wealthy ship owner, and he owned ninety six ships that plied the Eastern Mediterranean, and who knows where else. He was known as "He" to the geometers of the ancient world. He collected navigational geometry techniques from his ship captains, and he catalogued them. He also traveled to Egypt where he studied geometry with the Priests.

Thales work is less known in writings than it is in the form of Pythagorean geometrical theory. Thales said, and I paraphrase, "For all who may want to continue this work of geometry they should study with the priests of Egypt, for they know all the theorems that there are to know." The Egyptians had, by that time, collected virtually all the theorems that the Babylonians had collected.

Pythagoras was a young man at the time of Thales death, and he chose to study Thales work. Pythagoras traveled to Egypt, and he spent fifteen years at Memphis and Thebes studying with the priests. The study of what we know to be fractions, was to the Ancients the study of 'ratios'. Pythagoras collected a large amount of geometry work, and he systematized it. Pythagoras set up a school in the Greek area of lower Italy, and he became famous for teaching his geometrical system. He kept close wraps on his ideas that were his stock in trade, and students were forbidden to discuss the ideas.

The Pythagorean system was given evidence in the 3D geometry of the Parthenon, and virtually every proportional measurement of the building was based in and generated from the Pythagorean system called the Hekatompedon. The building is known AKA today as the Hekatompedon. The minor temple within the building is dedicated to the geometrical system of Pythagoras.

The Pythagorean system was carried down by the later geometers at Plato's school, for example, by Eudoxus and Aristotle. Eudoxus expanded upon the system. Aristotle critiqued the system, not in its entirety, to remove the non-scientific parts of the system and to justify it in the terms more well suited to writing and that were again in concord with the extended system of proofs first developed by Thales. The geometrical system was, after Aristotle, combined into a hierarchy of proofs that was commonly in use. All science, epistemology, and validation was geometry at that time. Aristotle revolutionized geometry by causing the concepts to be ordered into a revised system of knowledge based upon genus and differentia definitions, and upon Eudoxus refinement and development of the sub-science of ratios, that was called by the AGs, "Eudoxus' famous theory of equiproportionality."

Aristotle removed the primacy of coincidence and demonstration from the Pythagorean geometrical system, and his critiques are not too obvious unless you have been schooled in the then current events of the Pythagorean system. Then, also, those parts of Aristotle's writings make sense. You can't study geometry with an understanding of Pythagoras' theories.

Aristotle's theory of knowledge revised all knowledge, and from that time on, geometry became just one of the sciences, and science was then the more general concept. Aristotle also separated the sciences of concept formation, validation of concepts, and proof from geometry, and those studies became part of the more general science of epistemology. For the same reason, and with the development of mathematics after the later Archimedes, geometry became a part of mathematics. Aristotle changed the world of ideas, and Pythagoras, had earlier built some of the foundations for those ideas.

Until Eudoxus and Aristotle, who broke away from the mysticism of the school of Plato to contemporaneously run their own schools, all geometry was discussed in terms of the Pythagorean system. After Aristotle and before Archimedes Euclid produced the work that we know to be, "The Elements," or 'principles'. Euclid catalogued the distillate of the known concepts of geometry, and he organized them in terms of Aristotle's theory of knowledge, definitions, and logic. Euclid's "Elements, Book V", contains a good selection of the principles of geometry that are attributed to Pythagoras. Pythagoras system was really a means of holding geometrical relationships in mind, and for the purposes of further development, for example, that was done by Eudoxus. Euclid does express homage to the Pythagorean system, however, serious science had to be developed. That's about the last of the Hekatompedon system, however, the separate principles of geometry, and hence, mathematical concepts, discovered or developed by Pythagoras were retained by Euclid. Eudoxus' theories that were devised in the context of the Pythagorean Hekatompedon system were also presented, re-proved, and catalogued in Euclid's "Elements". With Archimedes, a student of both Aristotle and Euclid, the principles of the then ancient science of ratios, and also Euclid's systematization of the methods of inquiry and proof, became algebra.

Today, architects know the Hekatompedon system of Pythagoras as the system of "dynamic symmetry". To historians of geometry those words have the same meanings that they did then. I attended a lecture by Mies van der Rohe on the topic of how the geometry of proportions is the cause for the organization and ordering of architectural shapes so to create a beautiful result. Pythagoras' Hekatompedon and root rectangle - triangle system was there, too.

Where are the works of Pythagoras written?

In, (a^2 + b^2)^.5 = c^2 , the Parthenon, a:b::b:c, deductive logic, and in New York City.

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Where are the works of Pythagoras written?

In, (a^2 + b^2)^.5 = c^2 , the Parthenon, a:b::b:c, deductive logic, and in New York City.

A minor correction either you write a^2 + b^2 = c^2

OR

(a^2 + b^2)^(1/2) = sqrt (a^2 + b^2) = c

Pythagoras theorem was known is several places in the world long before Pythagoras was born. The Egyptians had special cases: The used the 3:4:5 knotted rope as a T-square to cut square corners. The Chinese had this relation also as did the Babylonians. What made the Greek handling different was that eventually the "Pythagorian" relation was proven from a small number of axioms. Deductive mathematics is a Greek original and Thales was one the the earliest to prove theorems. You might say that Thales invented the theorem. This great idea was spread by the Greeks after the time of Alexander to other parts of the world. The deductively proved theorem is in the tool kit of humanity.

Bob Kolker

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I want to know what makes you think that Pythagoras was influenced by Thales, and those other claims about Pythagoras.

------------------

In, (a^2 + b^2)^.5 = c^2 ,----

Shouldn't that be "=c"?

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Pythagoras theorem was known is several places in the world long before Pythagoras was born. The Egyptians had special cases: The used the 3:4:5 knotted rope as a T-square to cut square corners. [...]

Thanks for the corrected formula regarding "a, b, c".

In addition to the collected geometry theorems Thales also collected ratios. That is, the specific ratios that were known to the Egyptian priests and to his sea captain navigators. Solutions to triangles were part of the task, and the general solution to triangle problems were always sought by the AGGs. Many of the ratios were organised into what we know to be trigonometry tables, and at THales' time the tables were grossly incomplete.

Anaximander was a student of Thales, and also a teacher to Pythagoras. The study of ratios in the sense of trigonometry was a subscience of geometry and the quest for general practical methods of finding trigonometric ratios persisted in the ancient world. Pythagoras developed of the study of ratios, and he organized numerical ratios into the commensurables and incommensurables, e.g., 10=100^.5 and c=3^.5 . Pythagoras found a graphical method for finding ratios, and that demonstrated the general theory of triangles that was part of his more general graphical educational theory. The difficulty was that for numbers of great size the line lengths that were the solutions to the problems were difficult to measure with any precision given the tools available. Graphical means to find trigonomteric ratios were nearly totally impractical. Arithmetical calculations provided better approximations, and the tables for most practical uses were completed.

Isn't it true that the principle of the numerical solution to triangles is known conceptually, and that only the commensurables may be easily calculated? Isn't it true that the incommensurables could not be calculated without approximation or by exhaustion?

My question are these:

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?

Several discoveries of numeric and graphical coincidences contributed to the discovery of the Golden Mean ratio, and these discoveries are still being made and developed. Once the slope, or run : rise, of the unique ratio has been found is not the ratio of the Golden Mean also easily found for all cases?

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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?

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).

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.

Refer to the Wikipedia entries on "square root" and "methods of computing square roots" for additional info.

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That is, has the method of finding the exact numbers of square roots, to specified precision, ever been found?

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. The purely geometric method is to construct a square whose side is equal to the number who square root you want. The square root is one half the diagonal of that square. That follows directly from the Pythagorean Theorem. (Euclid Prop 47 Book I).

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.

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.

Bob Kolker

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The purely geometric method is to construct a square whose side is equal to the number who square root you want. The square root is one half the diagonal of that square. That follows directly from the Pythagorean Theorem. (Euclid Prop 47 Book I).

I'm afraid I don't quite follow this. If S is the length of a side of the square, then the diagonal is S*sqrt(2). Half the diagonal would be S/sqrt(2). That is not the same as sqrt(S). Can you clarify?

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The purely geometric method is to construct a square whose side is equal to the number who square root you want. The square root is one half the diagonal of that square. That follows directly from the Pythagorean Theorem. (Euclid Prop 47 Book I).

I'm afraid I don't quite follow this. If S is the length of a side of the square, then the diagonal is S*sqrt(2). Half the diagonal would be S/sqrt(2). That is not the same as sqrt(S). Can you clarify?

You can't follow it, because it is WRONG. Mea culpa, Mea culpa. The way you get the square root geometrically is to construct the geometric mean. Given a, b the geometric mean x is defined by a:x = x: b. Suppose b is the number whose square root you want to find and a = 1. then 1:x = x:b or x^2 = b.

How does one find the geometric mean. The operative theorem is that if a right triangle is inscribed in a circle and the hypotenuse is a diameter then drop a perpendicular from the point on the circle where the right angle is. This perpendicular divides the diameter into two portions a and b and the length of the perpendicular segment is the geometric mean of a and b. This is proved by the similarity of triangles. Now if you are given b construct a line segment whose length is 1 + b and make that the diameter of a circle. The where the unit segment joins the b length segment raise a perpendicular to intersect the circle just constructed. The length of that perpendicular is the square root of b since its length x satisfies 1:x = x:b.

Once again, sorry for the error. I hope you can follow this correct.

Bob Kolker

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Ah, yes, I remember now the C^2 = A*B relation for a perpendicular on a diameter of a circle, where C is the length of the perpendicular, and A and B are the two parts of the diameter on either side of C. I had never thought of that as a method for finding a square root, but it certainly makes sense.

In case anyone is still curious about calculating square roots by hand, here is some additional detail. (My previous description was a bit vague.) Suppose we want to calculate the square root of S, and we have an approximation, x. We can define the "remainder" as the difference between S and x^2: R = S - x^2. Next, we want to add an adjustment, h, to x, and calculate how much the remainder is reduced as a result of adding h:

Old R = S - x^2.

New R = S - (x+h)^2 = (S - x^2) - h*(2x + h).

The "new R" formula allows us to work from the old R instead of having to calculate (x+h)^2 anew at each step and subtract it from S directly.

Since the goal is to reduce the remainder successively closer to zero, we can decide on the value of h from R/2x. I.e., set "new R" equal to zero, and neglect the "h" in "2x+h," leaving 0 = old R - h*2x. This is an approximation for h, but the calculation for R is exact. We don't lose any accuracy in successive values of R.

As an example, here is how the square root of 29 might look using pencil and paper, starting with 5 as an initial estimate of the square root (". . ." is for column alignment):

. . . . . . . . . 29.000000 . . . . . . Initially, x=5

. . . .. . . . . -25

. . . . . . . . ====

10 . . . . . . . 4.0 . . . . . . . . . . . h1 = .3 (.4 would be too much)

10.3 . . . . . -3.09

. . . . . . . . . =====

10.6 . . . . . . 0.91 . . . . . . . . . . h2 = .08

10.68 . . .. . -0.8544

. . . . . . . . . =======

10.76 . . . . . 0.0556 . . . . . . . . h3 = .005

10.765 . .. . -0.053825

. . . . . . . . .=========

10.770 . . . . 0.001775 . . . . . . h4 = .0001

10.7701 .. . -0.00107701

. . . . . . . . . ==========

. . . . . . . . . . 0.00069799

The value of x calculated so far is the sum of the h terms: x = 5.3851.

The remainder (difference between S and x^2) is 0.00069799.

Check: x^2 = 28.99930201, and 29 - x^2 = 0.00069799.

The pairs of numbers at the left edge are 2x and 2x+h, respectively, for each successive increment of h. The value of 2*(x+h) can be calculated simply as (2x+h) + h. For example, 10.6 is 2*(5.3), 10.68 is 10.6 + h2, and 10.76 is 10.68 + h2 again.

Each successive reduction in the remainder is just h*(2x+h), as in 0.08 * 10.68 = .8544.

If desired, we could continue the process to obtain even more significant digits for x, and a still smaller remainder.

Speaking of calculation by successive refinement, another fascinating math challenge is to calculate the value of pi. For that, I like Archimedes' method of inscribing a regular hexagon inside a circle, then successively doubling the number of sides to get a polygon perimeter that becomes ever closer to the circumference of the circle. It doesn't take very many steps to get a huge number of sides, since the number of sides doubles with each step. We can use the Pythagorean Theorem to compute the length of a side for each successive polygon, based on the length of a side in the previous polygon.

Additional information on Archimedes' discoveries and methods can be found in the following Wikipedia topics:

"Archimedes Palimpsest."

"Measurement of a Circle" (e.g., computing pi using inscribed and circumscribed 96-sided regular polygons).

"On the Sphere and Cylinder" (see especially Arcimedes' formula for the volume of a sphere, using methods long predating modern integral calculus and modern theory of limits).

Since this thread began with the topic of beauty, I would like to point out that discoveries and validations in mathematics and other sciences often are so superbly elegant as to be rightly regarded as works of beauty by those who are familiar with them. The more I learn about the work of Archimedes, the more impressed I am with the brilliant cognitive beauty of so much of it.

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[...] Since this thread began with the topic of beauty, I would like to point out that discoveries and validations in mathematics and other sciences often are so superbly elegant as to be rightly regarded as works of beauty by those who are familiar with them. The more I learn about the work of Archimedes, the more impressed I am with the brilliant cognitive beauty of so much of it.
This thread is called, "Is Beauty Quantifiable ?, Golden Ratio as Applied to Esthetics".

We now find that there are two provinces for beauty. One is the discovery and identification of mathematical principles. The other is the validation and demonstration of the principles.

The formula for a circle requires the functioning of the concept of the radius, and the parts of the circle are everywhere on it equidistant from the center according to the radius length. That intellectual identification and demonstration results in the imagination, or in mind. In physical reality the circle may be drawn using the unalterable length of the radius using a compass with stylus. The circle may be circumscribed about a "selected or chance number" of placements of radii (straight lines of the same lengths). The principle of the universality of a concept was also discovered by one of the AG geometers, and more likely, philosophers.

Concepts, e.g., mathematical concepts, may be discovered, identified, and validated in mind, and they may also be demonstrated and proved in physical reality by drawing or modeling. Concepts known to the AGs that were identified by the use of drawing were conceptualized in mind. On the other hand, concepts that were discovered in mind were often demonstrated by the use of drawings. Many concepts were too complex or that required too much accuracy to be drawn by the AGGs were identified in word concepts.

The many possible ways that the mathematical concept of the Golden Mean Ratio and its many properties and possible relationships of the objects made according to the Ratio have been drawn in designs, and also the unique properties of the principle have been discovered in natural objects. And also in mathematically demonstrated objects.

E.g., 'The Application of Areas' principle. [The 'Application of Areas' was also used as a general method for the proof of other mathematical [geometrical to the AGs] concepts.] Algebra symbolically identifies a given mathematical concept, and, via the principle of the 'identity', may graphically or physically demonstrate properties and relationships.

The Application of Areas was one of the long sought concepts of the ancient world. We know it to be: a^2 + 2ab + b^2 = c^2. The AGGs knew it as a graphic relationship of the multiples of straight line lengths of a, and b.

Draw 'between parallels', meaning orthogonally to the AGGs:

A square of side c, and call it square [c].

A second square of side a within square [c] a starting at the lower left corner of square [c].

A third square of side b within square [c] a starting at the upper right corner of square [c] so that the opposite corners of squares [a] and touch.

The area of square [c] will be found to equal the sum of,

The area of square [a],

The area of square ,

The area of rectangle with sides a,b,

The area of rectangle with sides b,a.

The identity of the sum is dependent upon the invariant graphical relationship of the squares as drawn.

Problems identified by the AGGs that were difficult to solve were:

The Application of Areas

The Duplication of the Square

The Squaring of the Circle

The Duplication of the Cube

The Whirling Squares (spiral of the Golden Mean Ratio)

These concepts were the prior art for Archimedes' discovery of the concept that a volume that had no regular shape could be identified by a number, and that number could be identical to the number of the volume of a selected cube.

His 'numbers" were the values of objects that had several other properties that were ignored. That discovery may be one of the first examples of "Measurement Omission", and Archimedes really meant, 'properties selection' with a corollary of 'properties omission'.

More importantly, he devised the principles that in physical reality one could identify the relationship of physical measurements a and b, that is, a=b.

The mass / weight of one volume of a specific material could equal the mass / weight of another different shape of the same material density and volume.

The AGGs sought the identities of the properties of many regular plane figures and solids, e.g., the volumes of cones to spheres, or the area of a square that equals the surface area of a sphere.

Archimedes founded the science of mathematical physics.

He did so based upon the work of the geometers beginning with the discovery by an Ancient Egyptian scholar of geometry that the number value of one group of specific fractions was equal to the number value of another different group of specific fractions. The discovery had its corollary that the number value of some groups of fractions did not necessarily equal the number value of other groups. The principle of the identity was discovered, and the corollary was that of, 'Either Or'.

When the 3D form of the 'Application of Areas', that is, what we may term, the 'Application of Volumes', was set forth it was done so by Pythagoras within the complicated wireframe / solid structure of placed geometrical relationships of the Hekatompedon. That was a system of parallel lines that was developed in a matrix of X, Y, and Z dual directions. Pythagoras fitted most of the known principles of geometry into the educational [grid] system, and all the geometrical concepts of the time that were conceptualized and demonstrated within circles or between parallel lines, were placed within a solid geometrical [wireframe] system of lines, planes, cubes, and spheres, to name the basics. Eventually the system reached its limit of capability, and elements more complex than spheres, e.g., spirals, could not be fitted within the system.

Concepts held in mind by means of economy, became prevalent, and as we may see in the geometry figures of the time, the literal drawing or diagramming of the work became less prevalent. Except that invention, engineering, science, and design became more prevalent.

The science of mathematics and engineering, and the architecture and design of manmade objects became a matter of reality.

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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.

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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.

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