The Mind-Body Problem

Cal wrote (to Ed),

If you reread my post #84, you'll see that I even mentioned an infinite-dimensional complex vector space. I asked that question while you stated that you and Bill know the reason of the usefulness of mathematics (and I supposedly not). This is a mere assertion, so I asked you about a concrete example, so that you could demonstrate your knowledge. I suppose that your opinion is that such an infinite-dimensional space can't be useful, as there are no infinite dimensions in reality.

Absolutely not. I have never said that. Nor is a knowledge of higher mathematics necessary in order to grasp this issue. I would never presume to "demonstrate" my knowledge of higher mathematics (of which I am largely ignorant) as a basis for the view that abstractions must have a tie to the real world. I wasn't sure what your point was regarding a super-dimensional sphere, so something obviously got lost in translation. I have no problem with the concept of super (or infinite) dimensions, if the concept has a useful relevance. Nor does Rand, whom you branded condescendingly as a "quack," after telling Ellen Stuttle how much you appreciated her for not writing in a bombastic and condescending manner. Rand's view on this (and related) issues is evident in the following dialogue:

Prof. C: An imaginary number is supposed to be that number which when multiplied by itself is equal to minus. There is, in fact, no real number which when multiplied by itself gives you minus one.

AR: What is its purpose?

Prof. C: It turns out that it has a great usefulness as a device mathematically for solving problems of a real kind--for instance, problems involving electrical circuits. But I personally do not see the validity of this concept. There is nothing in reality to which it corresponds. Nothing is measured except by real numbers.

AR: But here there is a certain contradiction in your theoretical presentation. If you say that these imaginary numbers do serve a certain function in measurement, then--

Prof C: Excuse me, not in measurements of anything, but in computation--in solving an equation.

AR: The main question is: do they really serve that purpose?

Prof. C: In practice, yes.

AR: If they serve that purpose, then they have a valid meaning--only then they are not concepts of entities, they are concepts of method. If they have a use which you can apply to actual reality, but they do not correspond to any actual numbers, it is clearly a concept pertaining to method. It is an epistemological device to establish certain relationships. But then it has validity. All concepts of this kind are concepts of method and have to be clearly differentiated as such.

Whenever in doubt, incidentally, about the standing of any concept, you can do what I have done in this discussion right now. I asked you, "What, in reality, does that concept refer to?" If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept. But if you tell me, yes, this particular concept, although it doesn't correspond to anything real, does achieve certain ends in computations, then clearly you can classify it: it is a concept of method, and it acquires meaning only in the context of a certain process of computation.

Therefore, when in doubt about the classification or nature of a concept, always refer ultimately to reality. What in reality gives rise to that concept? Does it correspond to anything real? Does it achieve anything real? Or is it just somebody's arbitrary theory. (IOE, pp. 304-306)

You continue:

However, this is just a description of the Hilbert space of quantum mechanics, the most successful scientific theory ever. So this is a clear example of some highly abstract construction that has turned out to be enormously successful, also in practical applications (for example, you wouldn't be able type on your computer if QM hadn't made it possible).

Cal, do you see any contradiction between what you're saying and what Rand is saying? I don't, at least not in this context.

- Bill

Post 101

Tuesday, February 7, 2006 - 4:41pm

CORRECTION: In the previous post, I quoted Prof. C as follows:

Prof. C: An imaginary number is supposed to be that number which when multiplied by itself is equal to minus. There is, in fact, no real number which when multiplied by itself gives you minus one.

The first sentence should read: "An imaginary number is supposed to be that number which when multiplied by itself is equal to minus ONE." This was probably clear form the context, but I just wanted to make it explicit so there's no misunderstanding, as the previous post was too old to edit.

- Bill

Post 102

Wednesday, February 8, 2006 - 8:29am

Bill:

Cal wrote (to Ed),

If you reread my post #84, you'll see that I even mentioned an infinite-dimensional complex vector space. I asked that question while you stated that you and Bill know the reason of the usefulness of mathematics (and I supposedly not). This is a mere assertion, so I asked you about a concrete example, so that you could demonstrate your knowledge. I suppose that your opinion is that such an infinite-dimensional space can't be useful, as there are no infinite dimensions in reality.

Absolutely not. I have never said that.

Read the first line of your own post. Are you Ed?

Cal, do you see any contradiction between what you're saying and what Rand is saying? I don't, at least not in this context.

Oh yes, there is a big difference. Rand says:

Therefore, when in doubt about the classification or nature of a concept, always refer ultimately to reality. What in reality gives rise to that concept? Does it correspond to anything real? Does it achieve anything real? Or is it just somebody's arbitrary theory.

She apparently doesn't realize that many "arbitrary theories" achieve "something real" only much later than the time they were conceived as a purely abstract construction without any reference to the physical world. What today is a completely "useless" abstract theory may turn out later to have some very practical application in physics. The theory of Hilbert spaces already existed before physicists discovered that it could be used for the formalism of QM. Tensor calculus existed long before Einstein discovered that it could be applied to his formulation of general relativity - he had to learn the theory from Levi-Civita and Grossmann. It is a serious error to think that abstract scientific or mathematical theories are just some useless flights of fantasy, and that only applied mathematics and applied science are worthy of consideration. Those applied sciences are the offshoot of those "useless" abstractions "in Plato's heaven", which may have existed years, decades or even centuries before they found an application in real life.

Post 103

Wednesday, February 8, 2006 - 9:42am

Cal,

I think you are confounding two different usages of "concept". There is a common usage that means "idea" or "theory" and there is Rand's specific usage as "a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted."

Post 104

Wednesday, February 8, 2006 - 11:55am

Rick, it isn't clear to me what you want to say. I even didn't use the word concept myself. It is Rand who talks about the concept of imaginary numbers. (BTW Bill, an imaginary number is not a number that multiplied by itself equals minus one, for example 3i is an imaginary number, but multiplying it by itself results in -9).

Rand:

If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept.

and:

Therefore, when in doubt about the classification or nature of a concept, always refer ultimately to reality. What in reality gives rise to that concept? Does it correspond to anything real? Does it achieve anything real? Or is it just somebody's arbitrary theory

In the first sentence she talks about building a theory on the concept, in the second sentence she even uses both terms for the same thing ("is that concept somebody's arbitrary theory"). Anyhow, apart from semantic quibbling about the terms "theory" and "concept", it's clear that Rand dismisses theory and the concept that the theory uses if the concept doesn't correspond to anything real or doesn't achieve anything real. Now "achieving something real" is a very vague notion. I suppose she means "if you can use it in practical applications (like calculations on electrical circuits)". If that is not the case and if the concept does not correspond to anything real, it is in her view an invalid concept.

Now Rand is contradicting herself, as she stated elsewhere (ITOE): It is crucially important to grasp the fact that a concept is an "open-end" classification which includes the yet-to be-discovered characteristics of a given group of existents. Similarly Peikoff (ITOE): The fact that certain characteristics are, at a given time, unknown to man, does not indicate that these characteristics are excluded from the entity - or from the concept. How can she then ever declare that a concept is invalid while it doesn't correspond to anything real and doesn't achieve anything real? She would have to be omniscient to know whether a certain mathematical concept will be highly useful in 100 years while it now doesn't have any practical application. I find her view extremely narrow and short-sighted: if a concept and/or a theory at this moment have no practical applications and don't refer to the physical universe, they are dismissed as an "arbitrary theory". Apparently she doesn't realize how much progress in science and technology is ultimately the result of such "arbitrary theories" (her term, not mine, as mathematical and scientific theories are far from arbitrary).

Post 105

Wednesday, February 8, 2006 - 9:02pm

I wrote,

Cal, do you see any contradiction between what you're saying and what Rand is saying? I don't, at least not in this context.

He replied:

Oh yes, there is a big difference. Rand says: "Therefore, when in doubt about the classification or nature of a concept, always refer ultimately to reality. What in reality gives rise to that concept? Does it correspond to anything real? Does it achieve anything real? Or is it just somebody's arbitrary theory." She apparently doesn't realize that many "arbitrary theories" achieve "something real" only much later than the time they were conceived as a purely abstract construction without any reference to the physical world. What today is a completely "useless" abstract theory may turn out later to have some very practical application in physics. The theory of Hilbert spaces already existed before physicists discovered that it could be used for the formalism of QM. Tensor calculus existed long before Einstein discovered that it could be applied to his formulation of general relativity - he had to learn the theory from Levi-Civita and Grossmann. It is a serious error to think that abstract scientific or mathematical theories are just some useless flights of fantasy, and that only applied mathematics and applied science are worthy of consideration. Those applied sciences are the offshoot of those "useless" abstractions "in Plato's heaven", which may have existed years, decades or even centuries before they found an application in real life.

I really think that something is being missed here. Rand is not saying that a coherent mathematical theorem or construction must have a practical application in the real world, otherwise it's invalid. The point she is making is that the theory, idea or concept must have some basis in reality, which tensor calculus obviously does, even if when it was discovered, it didn't have a direct, practical application.

Look, you've got to be a little sympathetic to what Rand is saying and try to understand the point of her answer. Let's take, for example, an arbitrary concept like God, which has no basis in reality. Would you say that that's a valid concept? No, of course, you wouldn't. The same is true for other kinds of supernatural ideas, such as the soul's surviving death and migrating to heaven or to hell, the transubstantiation of bread and wine into the body and blood of Christ, and so on. Even if the concept of imaginary numbers didn't have a practical application in the real world, it would still be a legitimate concept, unlike the religious concepts just mentioned, because it would serve the purpose of a mathematical method--a method of calculation.

Incidentally, in a subsequent reply to Rick (Post 104), you wrote:

(BTW Bill, an imaginary number is not a number that multiplied by itself equals minus one, for example 3i is an imaginary number, but multiplying it by itself results in -9).

I never said it was. Go back and read the post from which you got that statement, and you'll see that I was quoting Prof. C. It was he who made that statement, not I. In the same post, you also stated:

[Rand] would have to be omniscient to know whether a certain mathematical concept will be highly useful in 100 years while it now doesn't have any practical application. I find her view extremely narrow and short-sighted:...

That's because it's not her view.

...if a concept and/or a theory at this moment have no practical applications and don't refer to the physical universe, they are dismissed as an "arbitrary theory".

Not true!

Apparently she doesn't realize how much progress in science and technology is ultimately the result of such "arbitrary theories" (her term, not mine, as mathematical and scientific theories are far from arbitrary).

Well, well. Look what we have here. First, you assume that her use of "arbitrary theory" applies to mathematical theories that have no practical application, when in fact that is not her position at all. Then you turn around and deny that mathematical theories with no practical application are arbitrary, while continuing to accuse her of holding that they are, when it was you who attributed that position to her in the first place. I don't know if there's a term for this kind of fallacy, but if there isn't, there should be.

In any case, this discussion has gotten completely off track, as the original points that were raised applied to the existential basis of the foundation of numbers, and the Platonic epistemology that underlies the idea that numbers have no existential referents. Cal, I'm having trouble understanding how a scientific guy like yourself can believe that numbers, instead of having a basis in concrete reality, refer to a Platonic archetype in another dimension. Perhaps you don't actually hold that belief, although you seemed to express sympathy with it in a previous post. But if you don't, then what, in your view, does a number like 5 refer to? As I've indicated, it cannot refer only to itself, so what does it refer to? Doesn't it have to refer to five somethings--to five existents of some kind? The referent of 5 is not 5; it is | | | | |.

- Bill

(Edited by William Dwyer
on 2/08, 9:38pm)

Post 106

Wednesday, February 8, 2006 - 11:27pm

===================
Cal, I'm having trouble understanding how a scientific guy like yourself can believe that numbers, instead of having a basis in concrete reality, refer to a Platonic archetype in another dimension. Perhaps you don't actually hold that belief, although you seemed to express sympathy with it in a previous post. But if you don't, then what, in your view, does a number like 5 refer to? As I've indicated, it cannot refer only to itself, so what does it refer to? Doesn't it have to refer to five somethings--to five existents of some kind? The referent of 5 is not 5; it is | | | | |.
===================

My sentiments exactly.

Ed

Post 107

Friday, February 10, 2006 - 6:08am

Bill:

Incidentally, in a subsequent reply to Rick (Post 104), you wrote:
(BTW Bill, an imaginary number is not a number that multiplied by itself equals minus one, for example 3i is an imaginary number, but multiplying it by itself results in -9).
I never said it was. Go back and read the post from which you got that statement, and you'll see that I was quoting Prof. C. It was he who made that statement, not I.

You wrote:

The first sentence should read: "An imaginary number is supposed to be that number which when multiplied by itself is equal to minus ONE." This was probably clear form the context, but I just wanted to make it explicit so there's no misunderstanding, as the previous post was too old to edit.

No, you didn't make that statement (and I never said you did), but you took the trouble to make a correction in it - "so there's no misunderstanding" - without telling us that you think that the new version is still incorrect. That can only mean either that you were in agreement with that statement or that you just weren't aware of the error. In the latter case you should be glad that I pointed it out to you, instead of reacting so defensively.

Well, well. Look what we have here. First, you assume that her use of "arbitrary theory" applies to mathematical theories that have no practical application, when in fact that is not her position at all.

Really? Conveniently you omit the relevant statement by Rand, so I'll repeat it here:

If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept. [emphasis added]

This after Prof. C had said:

It turns out that it has a great usefulness as a device mathematically for solving problems of a real kind--for instance, problems involving electrical circuits.[emphasis added]

So there's no misunderstanding about what is meant by "reality" in this dialogue: it is the physical world. And it's obvious that Rand thinks a mathematical concept is invalid if it can't be applied to solve problems in the physical world. If I understood you correctly, you'll agree that this is a ridiculous notion. But the fact remains that it is exactly what Rand said here, no matter how you try to rewrite history by telling us that Rand didn't mean it.

Cal, I'm having trouble understanding how a scientific guy like yourself can believe that numbers, instead of having a basis in concrete reality, refer to a Platonic archetype in another dimension.

I don't believe in "Platonic archetypes in another dimension", I only used "Platonic" as a metaphor for an abstract world.

But if you don't, then what, in your view, does a number like 5 refer to? As I've indicated, it cannot refer only to itself, so what does it refer to? Doesn't it have to refer to five somethings--to five existents of some kind? The referent of 5 is not 5; it is | | | | |.

You still don't get it: in mathematical number theory numbers do not "refer to somethings", they are an abstract construction that may have once originated from the kindergarten arithmetic you mention, but no they longer depend on such primitive notions for their definition (and sentiments have nothing to do with it).

Post 108

Friday, February 10, 2006 - 1:06pm

=====================
I don't believe in "Platonic archetypes in another dimension", I only used "Platonic" as a metaphor for an abstract world.
=====================

An abstract world.

Ed
[not being the light here, only the mirror]

Post 109

Friday, February 10, 2006 - 2:18pm

I hesitate to jump in here, but I just got my account validated and I'm eager to break it in.

Looking at the passage from Ayn Rand that Bill quoted in message 100, I see that Rand herself has too restrictive a view of proper concepts. This is evident in the words she puts into "Prof. C"s mouth (Prof. C is plainly a stand in, no actual professor of mathematics would make those mistakes).

First, there is an equivocation between real numbers and the "real numbers". Unfortunately, there is a set of numbers called "Real"; that shouldn't be taken to imply that other numbers (such as i or the rest of the complex numbers, or even more exotic constructions) are any less "bona fide".

Historically, the numbers have been plagued by this.
natural numbers.
negative numbers. It was once truly meant pejoratively.
rational and irrational numbers.
real and imaginary numbers.

To (almost) any modern mathematician, these are just names.

Carrying on, the "purpose" of i, if a number can be said to have a purpose, is certainly not to solve circuit diagrams, although they can do that. It was to be the solution to this equation: X*X + 1 = 0. None of the "real" numbers will fit there, so some olden mathematician invented (or discovered, depending on your metaphysics) i.

Similarly, such rarified mathematical concepts as infinite dimensional complex vector spaces are understood not in terms of their presence in the physical world, but as being logical extensions of more accessible concepts, such as a 3-dimensional real number vector space (your usual euclidian vectors which do have common physical referents).

When Rand asks "What, in reality, does that concept refer to?" a proper mathematician would answer "Nothing in your mundane world; I'm making beauty here."

Post 110

Friday, February 10, 2006 - 2:54pm

Which is the flaw, since the 'beauty' is in a platonic world - a floating abstraction.

Post 111

Saturday, February 11, 2006 - 4:33am

Cal,

I wanted to say exactly what I did say.

In response to Rand's comments on concepts your reply talked about theories. Why would you do that unless there were some equivalence in your mind? Of course you could have been just trying to muddy the waters.

Post 112

Saturday, February 11, 2006 - 5:42am

Rick, in post 104 I showed that Rand herself used those terms interchangeably ("is it somebody's arbitrary theory"), so if anybody is muddying the waters it is Rand. Anyway, I don't see a problem here. Rand also talks about the theory built on a certain concept, and that is of course essential here, as the concept (and its validity) in this case is inextricably linked with the theory, in fact the theory defines the concept. Her standard definition of a concept as a mental integration of two or more units, that you quoted in an earlier post, doesn't work here ("hey, this thing here and that thing there have something in common, let's call them imaginary numbers"), and she realized that, when she wrote: they are not concepts of entities, they are concepts of method and All concepts of this kind are concepts of method and have to be clearly differentiated as such.(emphasis added)

Post 113

Saturday, February 11, 2006 - 3:38pm

Cal,

You misread what Rand wrote. The antecedent of the pronoun in "Or is it just somebody's arbitrary theory?" is "What in reality gives rise to that concept?", not the concept itself. It is the same antecedent as in the two intervening sentences.

What you call Rand's "standard theory" of concepts is actually her definition and applies to all concepts. The units are not necessarily physical objects — sometimes they are relations among physical objects or relations among other concepts — but they are still concepts defined like all other concepts.

Rand is very precise in her word usage. A concept and a theory are two very different things.

A theory is based on a concept, a theory does not define a concept. A theory is described using a concept.

Post 114

Sunday, February 12, 2006 - 1:27am

I quoted Cal, "(BTW Bill, an imaginary number is not a number that multiplied by itself equals minus one, for example 3i is an imaginary number, but multiplying it by itself results in -9)." And replied, "I never said it was. Go back and read the post from which you got that statement, and you'll see that I was quoting Prof. C. It was he who made that statement, not I."

He replied:

You wrote, "The first sentence should read: 'An imaginary number is supposed to be that number which when multiplied by itself is equal to minus ONE.' This was probably clear from the context, but I just wanted to make it explicit so there's no misunderstanding, as the previous post was too old to edit." No, you didn't make that statement (and I never said you did), but you took the trouble to make a correction in it - 'so there's no misunderstanding' - without telling us that you think that the new version is still incorrect."

I was correcting the quotation -- the attribution -- not the accuracy of the Professor's statement itself, since I misstated what Prof. C had said. You're right, I wasn't aware that he had misstated the theory. And I do appreciate your pointing it out. I'm not a mathematician, and didn't give the accuracy of his statement a second thought, since it didn't bear on the main point of the dialogue. I was concerned only with making sure that I had quoted him accurately. Sorry if I sounded defensive, but I didn't appreciate the suggestion that I agreed with the professor's erroneous definition, since I hadn't even considered the accuracy its content.

I wrote, "First, you assume that her use of 'arbitrary theory' applies to mathematical theories that have no practical application, when in fact that is not her position at all. Then you turn around and deny that mathematical theories with no practical application are arbitrary, while continuing to accuse her of holding that they are, when it was you who attributed that position to her in the first place. I don't know if there's a term for this kind of fallacy, but if there isn't, there should be."

You replied,

Really? Conveniently you omit the relevant statement by Rand, so I'll repeat it here:

If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept. [emphasis added]
Cal, I think that by "reality" in this context, she is referring to the fact that a process of abstract measurement can have a potential application in the real world, even if it doesn't presently have one. I think the context bears this out, if you go back and reread the entire dialogue. (See below)
This after Prof. C had said:

It turns out that it has a great usefulness as a device mathematically for solving problems of a real kind--for instance, problems involving electrical circuits.[emphasis added]

So there's no misunderstanding about what is meant by "reality" in this dialogue: it is the physical world. And it's obvious that Rand thinks a mathematical concept is invalid if it can't be applied to solve problems in the physical world. If I understood you correctly, you'll agree that this is a ridiculous notion. But the fact remains that it is exactly what Rand said here, no matter how you try to rewrite history by telling us that Rand didn't mean it.

Nobody's trying to rewrite history here. Let's recap the dialogue:

Prof. C: It turns out that it has a great usefulness as a device mathematically for solving problems of a real kind--for instance, problems involving electrical circuits. But I personally do not see the validity of this concept. There is nothing in reality to which it corresponds. Nothing is measured except by real numbers.

AR: But here there is a certain contradiction in your theoretical presentation. If you say that these imaginary numbers do serve a certain function in measurement, then--

Prof C: Excuse me, not in measurements of anything, but in computation--in solving an equation.

AR: The main question is: do they really serve that purpose?

Prof. C: In practice, yes.

AR: If they serve that purpose, then they have a valid meaning--only then they are not concepts of entities, they are concepts of method. If they have a use which you can apply to actual reality, but they do not correspond to any actual numbers, it is clearly a concept pertaining to method. It is an epistemological device to establish certain relationships. But then it has validity. All concepts of this kind are concepts of method and have to be clearly differentiated as such.
Let's stop here and consider what has just been said. Observe that Prof C says, "...not in measurements of anything, but in computation--in solving an equation." To which Rand replies, "The main question is: do they really serve that purpose?" Prof. C answers, "In practice, yes. To which Rand replies, "If they serve that purpose, then they have a valid meaning...
Whenever in doubt, incidentally, about the standing of any concept, you can do what I have done in this discussion right now. I asked you, "What, in reality, does that concept refer to?" If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept. But if you tell me, yes, this particular concept, although it doesn't correspond to anything real, does achieve certain ends in computations, then clearly you can classify it: it is a concept of method, and it acquires meaning only in the context of a certain process of computation.

I believe that by "do anything in reality," Rand would include the potential practical application of a process of computation. Remember, this is an extemporaneous dialogue, so we have to cut her a little slack here. When Prof. C says, "Not in measurements of anything, but in computation," Rand is clearly in agreement, which suggests that she doesn't regard the actual measurement of existents to be a necessary condition for the validity of the concept. She considers abstract computation to be sufficient. She clearly recognizes that mathematics is a science of method, which may not have an immediate practical application, but could have in the future.

I asked Cal, "'[W]hat, in your view, does a number like 5 refer to? As I've indicated, it cannot refer only to itself, so what does it refer to? Doesn't it have to refer to five somethings--to five existents of some kind? The referent of 5 is not 5; it is | | | | |." He replied,

You still don't get it: in mathematical number theory numbers do not "refer to somethings", they are an abstract construction that may have once originated from the kindergarten arithmetic you mention, but no they longer depend on such primitive notions for their definition (and sentiments have nothing to do with it)."

The point is that they must refer to some quantitative unit, but may refer to any, which is what makes them abstract rather than concrete. The concept 5 refers to what is common to any group of 5 units. The concept is unintelligible without this reference. You can't make sense out of it unless you understand it in terms of a certain number of discrete units. They can be any kind of units, but they must be some kind. The five vertical lines are stand-ins for any five units, whether they be apples, oranges, cats, dogs, whatever. If you don't agree with this, then please tell me what 5 refers to, and what you understand it to mean.

- Bill

Post 115

Sunday, February 12, 2006 - 4:59pm

Rick:

You misread what Rand wrote. The antecedent of the pronoun in "Or is it just somebody's arbitrary theory?" is "What in reality gives rise to that concept?", not the concept itself. It is the same antecedent as in the two intervening sentences.

I don't think so. She wrote: What in reality gives rise to that concept? Does it correspond to anything real? Does it achieve anything real? If your interpretation were correct, she would in fact say: "does that thing in reality that gives rise to that concept correspond to anything real?", "does that thing in reality that gives rise to that concept achieve anything real?", which is complete nonsense (do you know a thing in reality that does not correspond to anything real?). It's obvious that the last two sentences don't refer to the question "what in reality gives rise to that concept", but that she in all the three sentences looks for a link between the concept and reality and that therefore "it" in the last two sentences refers to "the concept". And of course she asks whether the concept does achieve anything real, because that is one of her criteria to decide whether the concept is valid.

What you call Rand's "standard theory" of concepts is actually her definition and applies to all concepts.

Definitely not. The definition you refer to is "a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted." This definition would be nonsensical for concepts of method, like mathematical definitions and operators. It's rather ironical that it would in fact suppose that all mathematical notions discovered and still to be discovered are somewhere residing in a Platonic heaven, where we stumble upon them, discover similarities between them and mentally integrate them, omitting "their particular measurements". Well, in a sense we do discover mathematical notions in a kind of Platonic realm, namely in the sense that everyone who explores that realm will get the same results. But here concepts are defined by the theory. Take the example of the number i. The concept of i is not a mental integration of several units with the same characteristics with their particular measurement omitted - as Craig already remarked, it is defined as a solution to the equation x * x + 1 = 0. Rand was no doubt clever enough to understand that her standard definition of a concept would not apply in this case, so there's no reason to be more Catholic than the Pope.

Post 116

Sunday, February 12, 2006 - 5:00pm

Bill:

I believe that by "do anything in reality," Rand would include the potential practical application of a process of computation. Remember, this is an extemporaneous dialogue, so we have to cut her a little slack here. When Prof. C says, "Not in measurements of anything, but in computation," Rand is clearly in agreement, which suggests that she doesn't regard the actual measurement of existents to be a necessary condition for the validity of the concept.

Oh, but she does: If they have a use which you can apply to actual reality, but they do not correspond to any actual numbers, it is clearly a concept pertaining to method.[emphasis added]

She considers abstract computation to be sufficient. She clearly recognizes that mathematics is a science of method, which may not have an immediate practical application, but could have in the future.

Perhaps you'd like to think so, but the evidence points the other way. The whole dialogue would become meaningless if she accepted the notion of possible applications in the future, because there's no way to decide which notions and theories will ultimately have applications in reality. The whole discussion about what imaginary numbers are and can do would be superfluous, as she would have to accept a priori all mathematical theories and concepts in that case as valid.

About the number 5: I'll try to think of a way of making it more clear, but that will be later.

Post 117

Monday, February 13, 2006 - 2:14pm