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It's not clear whom you are addressing. Regardless, I made no assumption it was "high fine art", nor did I compare it to the "awesome heroic sculpture." But I wonder how amusing your methods, whatever they are, would be in the professional domains of others, e.g. advanced math or science. If I'm one of your targets, I welcome your professional opinion on art, but not haughtiness.

I should have done this earlier, but it is not too late. I looked at the definition of mathematics in my Webster's New World Dictionary (copyright 1970) and here it is. mathematics - the group of sciences (including arithmetic, geometry, algebra, calculus, etc.) dealing with quantities, magnitudes, and forms, and their relationships, attributes, etc., by the use of numbers and symbols I like it. The same definition is here: http://www.yourdictionary.com/mathematics Regarding "magnitude" as covering "measurement", it is clear that mathematics is far wider than "the science of measurement."

There is clearly much overlap between math and logic, but I don't regard all of logic, e.g. term logic and informal logical fallacies, to be simply a subset or type of math.

I've already answered that. But you have yet to make a case that group theory has nothing to do with quantity or numbers. Absent that, we are just "spinning the tires" and I'm ready to "exit the car". Incidentally, in post #14 you used "quantity" or "quantities" nine times, so it would take a long and steep uphill drive to convince me group theory has nothing to do with quantity.

I don't know the background of the illustration on the cover of The Romantic Manifesto, or Rand's role in its selection. Unlike Dragonfly (post #12) and Jim Shay (posts #14 and #23), I don't find it ridiculous at all. Instead it strikes me as expressing the human desire to fly like a bird. I've heard that this is one of the most common dreams people have. I remember as a kid I often flew in my dreams. I'd stretch my arms out to the side and soar. In reality man can fly -- not fully like a bird -- but a person flying on a hang glider or regular glider is very much like a bird soaring. Of course, man's ability to fly has been with us only about a century. Before then it was merely a dream. I find the Icarus allusion plausible, but not very convincing. There is no sun on the cover or melting wax wings. Indeed, having either would destroy the said dream. It would be akin to a mole on the painting of a beautiful woman. Like Rand wrote somewhere in the book (I believe), a mole would make the painting a monstrosity.

Baal said 'Group theory is the kind of mathematical system used to express and describe symmetries. Groups, qua groups are not bound to systems of quantities." Maybe Baal could elaborate, my group theory knowledge is limited to one course some 25 years ago and it's a huge field of study. "Not bound to systems of quantities", or in other words "does not necessarily involve quantities", is quite different from "has nothing to do with quantity." It's like "Some S is not P" versus "No S is P."

Fine. However, you quoted Korzybski saying group theory has "nothing to do with quantity".

Really? What does Group Theory have to do with quantity or measurement, for example? Correct me if I'm wrong, but I assumed by "group theory" you meant algebraic groups. Examples of such groups are (1) the set of all integers and addition and (2) the set of all rationals excluding 0 and multiplication. Numbers are quantitative.

The theory of groups, projective geometry, and the theory of numbers have nothing to do with quantity? Pretty bizarre in my view.

Ba'al, It is refreshing to see somebody on an Objectivist forum not confounding measurement with other kinds of quantification. I think this is too broad, since logic would satisfy the definiens. I think a good definition of math is "the science of quantity and quantifiable structures."

and pp. 23-24 (my bold): Yes, there is sleight-of-hand in that passage. The distinguishing characteristics of these items of furniture are not measurements, but their specific purposes as follow: table - for putting other objects on and flat horizontal top chair - for a person to sit on bed - for a person to sleep in cabinet - for storing other objects in and has shelves door - for entering or exiting a building or room window - for seeing in or out of a building or room picture - for showing an image drape - for covering a window Note that each has some specific purpose, but may have any specific purpose that would make it furniture.

Is this your intuition of the essence of Rand's mind again? I read Rand literally, just as she advised reading other philosophers. Of course, philosophers sometimes use words metaphorically, cavalierly, even sloppily. That is what Rand did with "measurement." You have bent over backward trying to make "measurement" essential to concept formation. It does not work. If measurement were essential to concept formation, then forming a concept before learning to count would be impossible. Yet children do form concepts before they learn about numbers -- which they first acquire by learning to count and is a prerequisite to even a minimal understanding of measurement. It depends on what you are measuring in those structures in order to arrive at a concept. Michael This reply makes as much sense as talking about the DNA of a rock. A non-metric structure is not measurable by definition.

1. Hmm. That's exactly what I've long thought about the analytic-synthetic distinction (dichotomy). It's trivial and uninteresting. 2. One can't draw any conclusions about "things" from such a statement? Well, strictly speaking, no, it's not specific enough, just as x - y = 9 is not specific enough to conclude what specific numbers x and y represent. (Of course, it's obvious x is greater than y.) However, "a thing acts according to its nature" is a summary of a vast number of more specific statements, e.g. fish by nature bite on worms on fishhooks. Clearly one can draw a conclusion from it. If you want to catch some fish, use worms as bait and you will likely catch some fish.

Not so. Consider children. They learn the meaning of all or most words ostensively.

I responded to your post here. DB:"Uh huh. And then how do you resolve disputes over the "true" meanings of words "ostensively"? To use the example from Popper's footnotes to that chapter, I say "puppy" and point to a young dog; you say "puppy" and point to an arrogant young man. Which one is the "true" definition? By what method do you decide?" I didn't reply then due to lack of relevance. I might even answer them as you would. What's is Popper's answer?

Really? Then why did he write the following? "This seems to suggest that any preoccupation with meaning tends to lead to that result which is so typical of Aristotelianism: scholasticism and mysticism." I didn't claim it was. Indeed, I said he completely missed the boat because it wasn't.

Huh? Rand did not adopt the the Aristotelian method -- Middles Ages scholasticism -- referred to by Popper in "Two Kinds Of Definition". About the only thing that essay shows is that Popper was badly confused about definitions as I explained here

Bissell's guess may be correct, but it remains speculative that Ayn Rand took geometry and trig without specific evidence or at least knowing what a high school education in Russia was like about 1920. If she did take them, it strikes me as odd that she studied algebra late in life. Also, the following webpage lists Ayn Rand's college courses. None in math. http://www.nyu.edu/projects/sciabarra/essays/randt2.htm So it's pretty clear she was relatively uneducated in math, and I don't regard use of "ignorant" as improper. Here is a dictionary definition. ignorant - 1. having little knowledge, education, or experience - 2. lacking knowledge (in a particular area or matter) Some may take the term to be offensive. But I would not be offended if somebody said I was ignorant in chemistry. I took chemistry in high school and one college course in it, but have forgotten most of it since.

Merlin, Does non-metric topology use any units? Michael Yes, "unit" meaning "member of a set", but not meaning "a basis of measurement". I comment about Rand's ambiguous use of "unit" in the 2nd link in msg #198.

Rand's defining mathematics as "the science of measurement" is too narrow. One of the rules for a proper definition -- e.g. see David Kelley's logic book -- is that it not be too broad or narrow. Indeed, "the science of measurement" doesn't even encompass arithmetic in my view. "Science of quantity" is far better, but still too narrow. "Ignoramus" usually means "ignorant and stupid." So I would only call Rand ignorant in math. Face the facts. Late in life she studied algebra -- a middle school subject for better students. I'd guess she didn't even take high school math (geometry, trig, etc.), let alone basic college level math (linear algebra, calculus, etc.) and advanced college math (advanced calculus, differential equations, probability and statistics, real analysis, abstract algebra, topology, etc.). Yes, there is advanced math that doesn't involve measurement, e.g. non-metric topology. As to why Rand used measurement as fundamental to concept formation, I can only speculate. Perhaps it was "marketing". Perhaps she accepted a corrupted definition of measurement -- see here: http://rebirthofreason.com/Articles/Jetton...asurement.shtml In any case it's clear to me that much of concept formation does not require measurement omission. There is some measurement omission, but it is overwhelmed by qualitative omission. I cover the topic much more extensively here: http://www.objectivistcenter.org/events/ad.../JettonOaM3.PDF

Merlin, Contra? I thought I said precisely that. Really? My response was to the the bold part in the following, which is contra Rand's saying "the essential characteristics implies all the characteristics of the units." Not only is this claim of Rand's overstated, it conflicts with what she says three pages later. "Metaphysically, a fundamental characteristic is that distinctive characteristic which makes the greatest number of others possible; epistemologically, it is the one that explains the greatest number of others" (ITOE 45) "Implies all" and "explains the greatest number" are inconsistent.

How do you know? Do you believe you have intuited the essence of Rand's ideas? Combining these two claims, the essential characteristics implies all the characteristics of the units. Ergo, in the case of man and contra MSK, 'rational' or 'animal' implies a thumb. I believe "a definition implies all the characteristics of the units" is far overstated. A correct formula seems to be that the definition refers to the units, and the reference is to all the units' characteristics, not just the ones condensed into the definition. Essential characteristics can't imply contingent characteristics, e.g. the definition of 'boat' doesn't imply it has sails.

I don't own The Open Society and Its Enemies , so I borrowed a copy from the library to read Popper's unessential rambling about definitions in Chapter 11. Of course, there are two kinds of definitions -- essentialist being one kind -- but Popper completely "misses the boat" about the second kind. The second kind is ostensive definition. If one can point to what one means by a particular word, then no definition in terms of other words is needed. Popper's "undefined terms" he thinks are so prevalent in science are simply words that are assumed to be already commonly understood, and thus need no further clarification in terms of other words (an essentialist definition). Popper characterizes an essentialist definition as "left to right" and opposes it to a nominalist definition in science as reading "right to left". This is a distinction without a difference. An essentialist definition is presented "left to right", but it is understood "right to left" by someone seeking understanding of the term defined. In philosophy defining one's terms is more necessary because key terms used often have very wide application and the different meanings people have for them can vary widely.

Then Popper's position, too, which he calls as "modified essentialism", must be part of the problem. Please tell us your solution to what you deem the relevant question to be.

1. You're welcome. 2. I used "so-called" because Mill's name is most attached to them, even though he did not originate at least three of them. 3. We (you, I and a few others) could, but the tag "Mill's methods" is quite entrenched (and shorter).