Ayn Rand

Introduction to Objectivist Epistemology

Expanded Second Edition, Edited by Harry Binswanger and Leonard Peikoff, Meridian Books, 1990, pp 184-222

 

Measurement, Unit, and Mathematics, p184

 Abstraction from Abstractions, p204

 

 

 

 

Measurement, Unit, and Mathematics

 

 

Measurement

 

Prof. F: I would like to raise a question about the measur­ability of attributes. Length is obviously an attribute and it is measurable. And I am sure that everyone agrees that hardness and temperature are measurable. But now let's take the example of triangularity. And let's ask if there are such things as degrees of triangularity. It seems to me that a given entity either is a triangle or it is not a triangle.

 

And there's a related question that you might want to treat at the same time. Sometimes you speak as if every individual, every concrete, is a unit. Did you mean to say that?

 

AR: No. Every concrete is a unit when regarded as a separate member of a group of two or more similar con­cretes. A unit is a concrete, an existent, regarded in a certain manner, regarded in a certain relationship. Every concrete is a unit when it is so regarded. But that doesn't mean that every concrete can serve as a standard of mea­surement. Because "unit" here has two different meanings. A unit selected as a standard of measurement e.g., the inch has to be a given quantity of a given attribute, not of an entity.

 

But now what do you mean by "degrees of triangularity"?­ because there is no such thing.

 

Prof. F: It follows then that for this attribute, triangular­ity, there is no unit in terms of which it can be measured.

 

AR: No, it does not. Triangularity is one form of two­ dimensional shape, and shape can be measured. Triangular­ity-isn't a special attribute; the attribute is shape. In the case of triangles it is a triangular shape; in the case of squares it is something else. And all of them have to be measured in terms of linear measurement. I even referred to that as an example.

 

Prof. E: It's interesting that you asked that, because it's the identical question that I once asked, and I remember what Miss Rand's answer was at the time, which made it perfectly clear. Now please correct me, Miss Rand, if my formulation is not one you would endorse.

 

At the time, she distinguished between a simple and a complex attribute. She said that there are things, such as triangularity, which are attributes, in the sense that they can't exist independently, but which nevertheless have more than one measurable aspect. And that to measure a complex attribute is not to take a unit of that attribute in itself as the standard of measurement, but to measure the various distin­guishable aspects of that attribute. So that for triangles, you'd have to measure the number of sides, the length of the sides, the angular relations between them-which sum of aspects constitutes the triangularity. So to measure a triangle is no more than to measure the distinguishable aspects of the attribute.

 

In effect, there's such a thing as a complex attribute, which is still an attribute metaphysically, but is measurable by a different procedure than a simple attribute like length. Would you agree?

 

AR: In a sense. Is that the question about whether you take "little triangles" to measure every triangle?

 

Prof. E: That's right.

 

Prof F: So there are complex attributes and simple attributes.

 

AR: Well, put it in a somewhat more relevant way. All attributes, in order to be measured, have to be reduced to the kind of unit which we can perceive and by means of which we can establish a [quantitative] relationship. So if we perceive two triangles, by means of measuring them qua triangles we will never get anywhere. It is not possible to measure shapes that way. What would we have to do? Reduce them to linear measurement.

 

But, a point I want to make very clearly: let us not make metaphysical distinctions on the basis of our methods of cognition. In other words, to say that you have to measure shape, for instance, in terms of, ultimately, reducing it to linear measurement is not to say that various shapes possess different attributes metaphysically. That's only creating con­fusion. And to tell you the truth, I do not quite understand the relevance or the consequences of distinctions of this kind-such as simple attributes versus complex ones. The attributes are what they are; our methods of measuring them may be simple or complex.

 

Prof. F: All right. What we called a "complex attribute" is merely due to a complex method of measurement, right? Would that be correct, to leave it just as an epistemological distinction?

 

AR: I think that, for precision, we'd better say "complex method," not "complex attribute." Because "attribute" re­lates to the existent, "method" relates to our form of measurement.

 

Prof. F: So can we then conclude this: not every concrete is a unit?

 

AR: Units serving as standards of measurement? Or concretes regarded as units? Which do you mean? Not every concrete can be taken as a standard of measurement qua concrete. A triangle cannot be taken as a standard of mea­surement for triangles. But a triangle can be regarded as a unit when we form the concept "triangle."

If we observe various shapes and find a difference be­tween triangles and squares, how do we separate the two categories? By regarding all triangles as units of one group and observing that they have a characteristic in common-a certain kind of shape-that distinguishes them from an­other group which are squares. In that sense we do regard triangles as units. And in that sense every existent - not only every concrete, but also every attribute, every action, every relationship-is regarded as a unit when it is unified into a concept.

 

But if I understand you correctly, your question was more pertaining to methods of measurement. And in that sense, you are correct when you say not every concrete can serve as a unit of measurement. And I have indicated that it isn't a concrete entity that one has to use [as the standard], but an attribute. We can't measure by means of concrete enti­ties; we measure only by relating attributes of certain enti­ties to a selected standard of measurement which is the concrete unit selected as this standard-like an inch or a meter or a mile. That's a concrete unit of measurement, which represents an attribute, not an entity. Is that your question?

 

Prof. F: Essentially, yes. I just wonder why the term "unit" should be used at all, except in those cases where the thing can be used as a unit of measurement. I'm suggesting that it might be best to simply use the term "concrete" and not the term "unit.”

 

AR: But the [implicit] concept "unit" is essential to concept ­formation. The essence of the first two pages of the book is to point out that only when we learn, in infancy, to regard concretes as units-only then can we begin to form con­cepts. So if there is any verbal confusion, I would say it

would be better not to use the word "unit" for "standard of measurement." There's only one difficulty that would oc­cur there: that which we select as a standard of measure­ment has to be a unit.

 

Prof. F: Yes. It may be a verbal problem here. You see, what I'm trying to say is this. A triangle cannot be regarded as a unit of triangularity, but ultimately it must be analyzed in terms of extension, length, and so forth.

 

AR: Right.

 

Prof. F: And some of the words that you use led me to think that you might want to say that a given triangle, because of the fact that it is a unit, is therefore a potential standard of measurement. And this seemed to me obviously wrong, and that's why I raised it.

 

AR: But in cases like this, I think we have to rely on the context to establish the meaning. Because it isn't arbitrary or purely linguistic that I use such a term as "unit of mea­surement." The mental relationship involved is the same as in regarding individual existents---concretes-as units when we form a concept. The relationship, the aspect from which we regard it, is the same. But that doesn't necessarily imply that every concrete existent can be a unit [of measurement]. If we say that only units of attributes can serve as units of measurement, need there be any confusion? I don't believe so really.

 

A "unit of measurement" means one concrete, belonging under a concept, which is taken as the standard compared to which you then measure all the other concretes belonging under that concept. If you take an inch as a unit of measure­ment of length, an inch is a specified unit of length when regarded against other lengths. But once you have selected the standard of measurement, thereafter you determine, you actually denote, the length of other objects in relation to that object whose length you have chosen to be one inch long.

 

Prof. H: So in both senses "unit" is used as one of a group.

 

AR: Yes, but this tine it is chosen as a standard by which you measure.

 

Prof. F: You just used the phrase "unit chosen as a standard." I want to call your attention to the sentence, on page 7, where I expected you to use that phrase, but where you used the opposite: "Measurement is the identification of a relationship-a quantitative relationship established by means of a standard that serves as a unit." Why didn't you say there "unit that serves as a standard"?

 

AR: That's purely verbal. I did not intend any different meaning. I think it was because my emphasis here was on the fact that the standard chosen will be a unit of the measurement which has to be performed.

 

Prof. F: Your statement that all entities are measurable in terms of their attributes raises this question in my mind: do you think all measurements in the last analysis come down to things like measurements of length, velocity, and so on, in a reductionstic sort of way? For instance, you stated that colors can be measured in terms of wavelengths of light. And if we do that with all attributes, it seems to me that ultimately we will come down to a few fundamental measure­ments-measurements, of gravity perhaps, electromagnetic measurements­ –

 

AR: You mean reductionism in the materialistic sense?

 

Prof. F: No, I am not going that far at the moment. I am merely suggesting that all material measurements may come down to a small list of measurements, ultimately.

 

AR: But isn't that a scientific question more than a philo­sophical one? Whether everything in nature is ultimately reducible to subatomic particles and [whether measurement is reducible] to counting them-isn't that a question for science to determine? Philosophy cannot answer it one way or another. I would never attempt to say-because it would be completely arbitrary-whether all measurements are ulti­mately relatable to only one set. That would be a scientific issue in regard to the nature of the universe. I would claim that all existents-since they are part of one universe-are measurable. But I have never said that all standards of measurement are ultimately translatable one into another or reducible one to another. That would be a claim for which nobody has any evidence one way or the other.

 

[In response to a question about Norman Campbell's di­chotomy of "intensive qualities" and "extensive qualities"-the two requiring, he claimed, different systems of measurement:]

 

AR: I object in principle to the idea of making metaphysi­cal distinctions of any kind on the basis of our ignorance. The fact that we can measure certain attributes but cannot be as precise in regard to others does not justify the idea of saying that entities possess two different categories of attri­butes, some of which are "extensive," others "intensive."

 

Campbell's standard here is not the nature of the entities but our capacity to measure-which means our state of knowledge, which is greater in one case than in another. That is what I mean by taking our ignorance as a metaphysi­cal standard.

 

What would one say properly about the situation? Merely that some attributes are easily related to one another quan­titatively, and others, are harder to relate, and we have not yet learned how to measure them. I don't see how that has any metaphysical significance. By "metaphysical" I mean: in the nature of these entities. Are we to say these entities have attributes essentially different from one another-with the standard of difference being our ability to measure them? I don't think that is the proper methodology for establishing the nature of entities.

 

 

 

Exact Measurement and Continuity

 

Prof. D: Your definition of "measurement" involves a realist conception of the relationship of measurement to the world. And measurement involves not only some quantity but also a standard of measurement, a quantitative standard of measurement, involving a unit that can be repeated over and over.

 

AR: That's right. But don't leave out the rest of it. I said "some quantity, but may exist in any quantity." I specifi­cally never said what kind of quantity, expressed in what mathematical figures, or achieved by what method.

 

Prof. D: But what if we suppose that any numerical system is going to be discrete in nature and rest upon discrete items?

 

AR: It has to.

 

Prof. D: Whereas suppose someone maintained that real­ity is continuous. For instance, Bergson wanted to maintain that. If Bergson is right, then there is this gap between any numerical system and reality: reality is continuous, numeri­cal systems are discontinuous, discrete. Then it would be the case that in a strict sense there wasn't any exact measure­ment possible, because your discrete units would never be able to handle this continuum. You would always have those little infinitesimals left over.

 

AR: Bergson wasn't the first to argue that. Zeno did the same thing. Would you answer for Bergson if you can, since he can't be called upon: how did we get to the moon? Without measurement?

 

Prof. D: Oh, I am not denying the practicality of mea­surement.

AR: How can it be practical if it doesn't relate to or correspond to the metaphysical nature of reality? How can we achieve fantastic things in regard to the material world and yet suppose for one minute that what we are doing is arbitrary and has no absolute, unquestionable relationship to the facts of reality? Because the Bergsonian position amounts to denying the validity and the existence of mea­surement. Now, are we going to argue on that point?

 

Prof. D: Not on the practical value of measurement.

 

AR: What is the distinction between the practical and the theoretical? That's a distinction which I do not recognize. "Practical" means acting in this world, in reality. If what we do works, how is that possible if it doesn't correspond to reality?

 

Prof. D: Well, suppose the correspondence were just a gross one. For instance, I can approximate, and my approxi­mations are not terribly gross ones. I use a ruler, and 1 say that this room is fifteen feet long and ten feet wide. And if I order lumber to build another room of this size, I will get approximately enough lumber. So someone could maintain that the measurements are practical enough, even though metaphysically there isn't a correspondence.

 

AR: But I would not know, and I wish you would explain to me, what is meant by "metaphysically" in this kind of context. There are two different things involved here. If what we do is only approximate-which I do not grant, but let's assume it is-that reflects merely on our capacity to perform exact measurements. How do we assign our inca­pacity to the metaphysical nature of reality? And more than that, if you say that we cannot, with a simple ruler at home, measure invisible submicroscopic lengths, will that invali­date the measurements which we are able to perform?

 

And more than that, my main point is this. I would like somebody to explain to me, and I am not being just rhetori­cal, what is meant by "a continuous reality." And in that context, what does Bergson think, if that is not a contradic­tion in terms, about the process of measurement? - about the discrete vs. the continuous? I have made it clear ­and this is really only common sense - that when you per­form a process of measurement you take a ruler and you decide this is the standard you are going to use. Now, does it mean that, if you then proceed to measure a mile by means of that ruler, there is some kind of "discontinuity" in the mere fact that you have to move that ruler over and over and take your measurement in installments? It merely means that you cannot measure the whole mile at once. Now expand the same principle to interplanetary distances and it still applies.

 

The fact that we isolate a unit is precisely the point I had in mind when I wrote-let me quote this because it is relevant "A unit is an existent regarded as a separate member of a group of two or more similar members. Two stones are two units; so are two square feet of ground if regarded as distinct parts of a continuous stretch of ground." The mere fact that we cannot encompass the whole of the universe at a glance does not mean that, when we attempt to measure it or to establish relationships one step at a time, we somehow destroy the "continuity of existence." You would not say that, if we measure this table in so many motions of moving the ruler, we have broken it up, whereas in reality it is a continuous table.

 

Prof. D: No, but one could say that if reality, space, and so on, are continuous, then no discrete unit of measurement would ever be able to take care completely of measurement. Because no matter how small you made your unit, there could still be a little bit left over or short. And if then you make even a smaller unit of measurement to take care of that little bit left over, still there would be a little left over or short. You could carry out the decimals as far as you wanted to and there'd still be a discrepancy.

 

AR: From the viewpoint of whom? Let me now go mysti­cal, like Bergson, and suppose we were a different size, the size of little beings inhabiting an atom (you know the meta­phor that the atom is like a solar system). If we were that size, we could with the naked eye perform minute measure­ments which we cannot do with our instruments now.

 

Prof. D: It wouldn't be exact though, if space is continuous.

 

AR: Now, we have to define terms. What is the standard of exactness here? What is the standard of exactness that Bergson was discussing? That is the crux of the issue. We use the term "exact," and now Bergson challenges it. What did he mean by "exact"? And what did he mean by "continuous"?

 

Prof. D: He would mean by "exact" I think this: it would involve what existed in reality, and an exact measurement would be one which corresponded, without any more or less, to what existed in reality.

 

AR: Now, since it is an exact measurement, it presup­poses a consciousness that is doing this. Whose consciousness?

 

Prof. D: Well, say mine.

 

AR: All right. And if you are able to measure it, and you are able to grasp relationships by means of measurement which you didn't invent, that is exactness. And if you are able to grasp that maybe some milli-milli-parts of a millime­ter are not correct and you are not able to bring it to a greater precision, who grasped that? You did. Therefore, your concept is correct, does correspond to reality, and it is reality that you have been consulting in discovering that perhaps you can't measure submicroscopic quantities.

 

Prof. F: So the very concept of "exactness" is a contex­tual concept. Suppose I say to you that I will meet you in this room exactly one year from now. If when you arrive I take out a stopwatch and say you are a tenth of a second late, this is dropping the context.

 

AR: Exactly. Everything that we discuss, everything, is done from the human viewpoint and has to be, because there is no such thing as "reality in itself." That is one of the concepts of Kant's that we have to be very careful of. If we were omniscient like God, we would still have to perceive reality by our God-like means of perception, and we would have to speak of exactitude from that viewpoint. But "things in themselves"-as separated from consciousness and yet discussed in terms of a consciousness-is an invalid equivo­cation. That would be my widest metaphysical answer to any construct a la Kant and Bergson.

 

Prof. D: So you answer this question by saying that con­textually, for our purposes, the measurement will do. In saying that the sides of this right triangle are each 1 foot long and the hypotenuse is 1.414 feet long, that will do.

 

AR: What is inexact about it?

 

Prof. D: Well, geometrically it is inexact but it will do for building a platform.

 

AR: No, that is not what I am saying. I am saying that when we speak of measurement, we begin with a perceptu­ally given unit, and that unit is absolute and exact [within the context of our means of perception]. Then conceptually we may refine our methods and we may measure such things as milliseconds and a part of a subatomic particle, which we can't do perceptually. But the standard of these measure­ments, the base from which conceptual complications may later be derived, is that which we perceive directly on the perceptual level; that is what measurement means, that is its base. Therefore, when I say that for measurement there has to be a unit of measurement, I mean that even when you take a submicroscopic, conceptual type of measurement, that type ultimately has to be reduced back to our stan­dard of measurement, which is the perceptually given, and nothing more or less.

 

With scientific development you might discover that, mi­croscopically, the edge of this piece of paper is ragged and has tiny mountain peaks and valleys. That is not relevant to your [macroscopic] process of measurement, because you had to use the perceptual method as a start in order to get to your microscopic instruments of measurement.

 

Prof. I: On the exactitude of measurement, is this chain of reasoning correct? We measure this book first, say, in inches. And we find that it is six inches long, plus a little bit. Then we subdivide the unit "inch" into sixteen equal subdi­visions, measure it again, and find that it is six and three­ sixteenths inches long. Then we fix up some fancy apparatus by which we can measure it by means of light waves, and we get accuracy to twenty decimal places. Then we ask, well, what is the relationship to the unit "inch" really? Well, it is what it is, but if we want to say that it is really six plus the square root of two inches long, we are saying this indepen­dent of any possible measurement that we can actually per­form. As such, we are attempting to abstract consciousness away from a concept of consciousness, and therefore it is invalid. In short, isn't it meaningless to ask, "What is the relationship to the inch really, out of context of a given instance of measuring?"

 

AR: Yes, in the sense of going beyond the point where more minute measurement is possible. Because then you would say that under any circumstances there will be sub­subquantities which you can't measure by the same ruler. In that sense it would be an improper switch of the term "measurement." When you speak of measurement, you al­ways have to define contextually your method of measure­ment. So that if you say it is so much measured by a ruler, or it is something else measured by some fancy apparatus, you have complied with the requirement of absolute corre­spondence to reality. You have said it measures so much by such and such means.  But to talk about what it would measure without any consciousness there to measure it, that would be improper.

 

Prof. E: Every measurement is made within certain speci­fiable limits of accuracy. There is no such thing as infinity in precision, because you are using some measuring instrument which is calibrated with certain smallest subdivisions. So therefore there is always a plus or minus, within the limits of accuracy of the instrument. And that's inherent in the fact that everything that exists has identity. Now if that's so, you can measure up to any specifiable degree of precision by an appropriately calibrated measuring rod.

 

If exactness in measurement is defined in such a way that you have to get the last decimal of an infinite series, by that definition no measurement can be exact. The concept of "exact measurement" as such becomes unknowable and mean­ingless, and therefore what would it mean to say a measure­ment is inexact? Exactness has to be specified in a human context, involving certain limits of accuracy. Is that valid?

 

AR: Yes, in a general way. But more than that, isn't there a very simple solution to the problem of accuracy? Which is this: let us say that you cannot go into infinity, but in the finite you can always be absolutely precise simply by saying, for instance: "Its length is no less than one millimeter and no more than two millimeters."

 

Prof. E: And that's perfectly exact.

 

AR: It's exact. If an issue of precision is involved, you can make it precise even in non-microscopic terms, even in terms of a plain ruler. You can define your length-that is, establish your measurement-with absolute precision.

 

 

 

Numbers

 

Prof. J: What measurements are omitted in forming con­cepts of particular numbers, for example, the concept of "seven"?

 

AR: In a certain sense the measurements omitted from the concept of numbers are the easiest to perceive. What you omit are the measurements of any existents which you count. The concept 'number" pertains to a relationship of existents viewed as units - that is, existents which have cer­tain similarities and which you classify as members of one group. So when you form the concept of a number, you form an abstraction which you implicitly declare to be appli­cable to any existents which you care to consider as units. It can be actual existents, or it can be parts of an existent, as an inch is a part of a certain length. You can measure things by regarding certain attributes as broken up into units-of length, for instance, or of weight. Or you can count enti­ties. You can count ten oranges, ten bananas, ten automo­biles, or ten men; the abstraction "ten" remains the same, denoting a certain number of entities viewed as members of a certain group according to certain similarities.

 

Therefore, what is it that you retain? The relationship. What do you omit? All the measurements of whichever units you are denoting or counting by means of the concept of any given number.

Here the omission of measurements is perceived almost at its clearest. And I even give the example in the book-it's an expression I have heard, I did not originate it-that an animal can perceive two oranges and two potatoes but can­not conceive of the concept "two." And right there you can see what the mechanics are: the abstraction retains the numerical relationship, but omits the measurements of the particulars, of the kind of entities which you are counting.

 

Prof. B: Does that mean that the referents of numerical concepts are not the entities as such, but entities regarded a certain way? In reality, each entity is one-that's metaphysi­cal. For you to have two, three, or four requires an act of consciousness to view them in a certain way?

 

AR: That's almost correct, except that you can't say that in reality there's only one. As entities, each one is only one, but when you view them as seven, let's say seven men, in reality there are seven men. This is the important thing, otherwise it becomes subjective. In reality there are seven men. Why do you identify them as seven men, and you don't include in that four men, two potatoes, and one street­car? In order to count them, you have to classify them as having something in common. It's from that aspect that you can count them.

 

Prof. B: The Greeks used to say that two is the first number, that one had a kind of special status. Entities in reality, apart from consciousness, are individual entities, and a group doesn't have any higher metaphysical status.

 

AR: None whatever.

 

Prof. B: But it's as if the numbers higher than one are tools of integration and not direct designators of-I can't say it without making a mistake!

 

AR: That means you are on the wrong track. The number "two" is crucially important epistemologically, because to form concepts you need two or more existents between which you observe similarities. It's in that sense that the number two is very important, epistemologically. Metaphys­ically, it is all equal-there is no metaphysical hierarchy between one and a million.

 

Prof. B: The reason it comes up is that any object that you choose in reality can be viewed objectively as two of one kind of units or four of another kind of units or what­ever, depending on how you divide it up or how you view it. For instance, we can view this book as one entity, as two halves, as one hundred pages, etc.

 

AR: That's right.

 

Prof. B: But isn't there some special metaphysical status to the fact that it is one entity? To say that this is one is somehow a more metaphysical statement, and it is that distinction that I'm trying to pin down.

 

AR: More metaphysical than what? Than saying that it is one hundred pages, or that it is two halves of one book?

 

Prof. B: But to say that this is one book and to say I have ... no, it remains metaphysically the same.

 

AR: It remains the same. But you know where you might sense a distinction? It's that the term "one" is the concept "entity." And the concept "entity" is the base of your entire development. It has that great epistemological significance.

 

***

 

Prof. E: Is there a distinction in meaning or referent between the concept "unit" and the concept "one," in the sense, for instance, that you grasp that this is one ashtray, or one book? To regard it as a unit is to regard it, as you say, as a member of a class of similar things. Is that same perspective involved in grasping that it is one?

 

AR: Before you have a concept of numbers?

 

Prof. E: Yes.

 

AR: You will perceive that it is one, as an animal would, but you couldn't grasp the concept "one" without a concept of more than one-without a concept of numbers.

 

Prof. E: Perception gives you directly a certain kind of quantitative information.

 

AR: Yes.

 

Prof. E: Even prior to either implicit or explicit concepts.

 

AR: That's right.

 

Prof. E: And is it true that that quantitative information is presupposed, before you form even the implicit concept "unit"? In other words, a young child would have to per­ceive that this is one, even though it has no implicit concept of that, before it could even form the implicit concept "unit."

 

AR: Of course, and here is where we have to be Aristote­lian: everything that exists is one. "Entity" means "one." But we couldn't have the distinction between what we mean by "one" vs. what we mean by "entity" if we didn't have the concept of numbers more than one which, after all, are only multiplied ones or divided ones.

 

Prof. E: So you get quantitative information by percep­tion; then, via the process of grasping similarities and differ­ences, you form the implicit concept "unit." You then rise to the general conceptual level, at which point you are able to form conceptually for the first time the concept of various numbers, including "one."

 

AR: Right.

 

Prof. E: Am I correct in saying that "one" and "many," as concepts, are metaphysical, while "unit," as a concept, is epistemological?

 

AR: That's right.

 

Prof. E: Is it correct to say that "quantity" is a metaphysi­cal concept and "measurement" an epistemological one, in the sense that if human beings and consciousness were erased, there would still be quantities, but there would no longer be such a thing as measurement. Measurement involves a hu­man act of establishing relationships.

 

AR: Of establishing quantity, that's right.

 

Prof. E: And is that why you formulate the nature of concept-formation in terms of omitting measurements rather than omitting quantities?

 

AR: Right.

 

Prof. E: Because you omit the relationships that you could establish?

 

AR: Yes. But the quantities continue to exist whether you measure them or not.

 

Prof. H: This is related to the issue of forming number concepts. On page 9, you say that the stage at which a child learns to count is when he is learning his first words.

 

AR: It comes a little later, as a matter of observation. It is almost simultaneous but not quite. Before a child can be taught to count, he has to have the beginnings or the rudi­ments of the vocabulary.

 

Prof. H: You meant counting explicitly, as in counting how many people are in this room, not just in the sense of perceiving the quantity.

 

AR: No, literally to count, as a conscious activity. He perceives the quantities, but he has to first form some con­cepts identifying objects, and then he can begin to count the objects explicitly.

 

Prof. H: You say that it occurs shortly afterward. From what I have observed, it seems to occur quite a bit later, so that it seems to be a much higher-level process.

 

AR: It isn't so much higher-level, but the fact is that you cannot begin to count objects until you have learned to distinguish them, and you cannot distinguish them firmly until you have learned some words-i.e., formed some con­cepts. Therefore, it is part of the same general develop­ment. But a child does have to acquire some conceptual vocabulary, meaning: learn to identify some concepts in reality, before he can begin to count.

 

Prof. H: I was taking it too literally.

 

AR: No, if I said "when he is learning his first words," I meant in the same general period of development.

 

 

Mathematics

 

 

Prof. B: You have said [in a section here omitted] two things about the mathematical field. One was that once the base has been established, one can proceed without direct reference to perceptual reality. The second point was that the mathematical field was more precise than the concep­tual. Would both of those facts be due to the particular nature of mathematics-that it is a science of method?

 

AR: In part. Also it is a science that defines the entities it deals with very simply. For instance, all you have as the basis of your operation is the arithmetical series. You don't need any further definitions as a base. From then on you work with that base. Whereas in other conceptual knowl­edge you deal with such a complexity of phenomena that your definitions can change as your knowledge expands, and your definitions may be very imprecise indeed. That's one of the differences.

 

Prof. B: In other words, you have all of the material before you from the beginning in mathematics. There's no new information which you are going to integrate into your concepts. Rather you are going to build up abstractions from abstractions, such as "function," "limit," and so on.

 

AR: That's right. This is not to imply that non-mathematical concepts necessarily have to be in some way less exact than mathematical concepts. No. The ideal to aim at is to bring your concepts into exactly that kind of precision. At least those concepts you know-you cannot have omniscience, and you cannot guarantee that you will not expand your knowledge (as I explain in Chapter 5) and change a con­cept's defining characteristic.

 

But the proper epistemological ideal is to have your con­ceptual knowledge, as far as it extends, in as precise a form as mathematics. Or as mathematics used to be, prior to Rus­sell. When I say "mathematics," I really don't mean the modern status of the science, but proper mathematics, ratio­nal mathematics.

 

Prof. B: In other words, it is not in the nature of the two fields that one must be more precise than the other. You are talking journalistically.

 

AR: I am talking not journalistically, but empirically, of the difficulty of the job involved. But this is one of the very vague suggestions of why I think that mathematics has some­thing to do with the essential pattern of concept-formation ­that it serves as an ideal.

 

But I don't want to sound Platonic here. It is simply that the kind of perfection which mathematics used to have (and applied mathematics still seems to have) is the pattern for concept-formation and concept-use. That is the way our conceptual equipment should be. But it's much harder; more is involved.

 

Prof. B: When you say it is the pattern, are you saying it is just an illustration or m some sense it serves that role? That's not mathematics' function-you wouldn't define math­ematics in those terms if you ever worked this out.

 

AR: Oh no. It's simply that mathematics, being a science that deals predominantly with concepts, and clearly defined concepts whose definitions do not change, gives you the pattern of precision that you have to bring into your concep­tual equipment; which latter, dealing with a much more complex field of knowledge, is much more prone to error or ignorance or change, change on the basis of newly discov­ered and relevant knowledge. So that mathematics as a science which deals with firmly defined entities can serve as a model.

 

Prof. C: Don't the definitions of mathematical concepts change with the growth of our knowledge?

 

AR: No. Philosophically, you may have much better defi­nitions of mathematical terms than they have today. But that's merely due to the fact that there's been no real philosophy of mathematics to speak of. But the actual defi­nitions do not change.

 

Prof. C: I don't see why you say that. For example, in the case of "number," first there were natural numbers (1, 2, 3, 4, etc.), then fractions were included under the concept "number," because they were similar. Then they invented imaginary numbers and other numbers which have even a more dubious status, like trans-finite numbers, etc. But in any case, the definition of the concepts would change in this wider context.

 

AR: Well, not of the concept "number," for instance.

 

Prof. C: Why?

 

AR: Because look right in your presentation: first you have natural numbers. And then you have fractions. Well, that's not the same concept, it's a subdivision. It's a new elaboration of what you know about the science of number­ing. But the addition of fractions, for instance, hasn't al­tered in any way your understanding of the basic number series. "One," "two," "three," etc. remain the same. But then you might have new combinations or new relationships numerically which you identify as fractions, and then powers or roots. But your knowledge of "number" hasn't changed. [Just as forming subdivisions of "man"-such as "farmer" or "brother"-does not change the concept of "man" or its definition.] What you can do with numbers, or what type of measurements you can discover, that's a development in your use of numbers, but not a change in the definition of "number" itself, in the way that the definition of "man" can change.

 

Prof. B: Take "seven." You don't learn more information about "seven" which leads you to change the definition of "seven."

 

Prof. E: You don't discover new phenomena previously unencountered which require you to distinguish "number" in a new way, the way you do in the case of the child's expanding definitions of "man."

 

 

 

 

Abstraction from Abstractions

 

First-Level Concepts

 

Prof. F: I have a fundamental question about the hierar­chy of concepts. On page 22 you say, "The meaning of `furniture' cannot be grasped unless one has first grasped the meaning of its constituent concepts; these are its link to reality." Now, what about the meaning of "table": can we say that the meaning of "table" cannot be grasped unless one has first grasped the meaning of "dining table," "con­ference table," "writing table," and so forth? Are these its constituent concepts? Or is the concept "table" a kind of privileged concept that comes at a kind of absolute bottom in the hierarchy of concepts and has a direct relationship to reality?

 

Or would you say that where a concept comes is deter­mined by the context of one's own learning? For instance, might a person form the concept of "furniture" without having formed the concept of "table" before? Might he form the concept of "living being" before he has formed the concept of "animal"?

 

AR: In a sense, yes. There is a big problem here, how­ever, whether this applies all the way through the concep­tual chain-which I would claim cannot be the case. But, on the level we are discussing, there is a certain element of the optional. Because when you first form your concepts, you might conceivably first form in a very loose way the concepts "living entity" versus "inanimate object," and later subdivide into "man," "animals," "plants," etc. (and "ta­bles," "rocks," "houses," on the other hand). In a loose way, that can be done, but only up to a certain level. Because, suppose you started with the concept "living being." You would then find that that is too generalized a category, and you would have to say, in effect, "By living beings I mean men, animals, and plants."

 

Therefore, understanding what your original semi-concept "living being" meant would depend on what you mean by the constituents, such as "man," "animal," and "plant."

 

What then is the ultimate determinant here? What I call the "first level" of concepts are existential concretes-that to which you can point as if it were an ostensive definition and say: "I mean this." Now, you can point to a table. You cannot point to furniture. You have to say, "By furniture I mean . . ." and you would have to include all kinds of objects.

 

Prof. F: Why wouldn't one have an equal difficulty when one came, let's say, to the concept of "bird"? Why wouldn't one have to say, "By bird, I mean eagles, penguins, and hummingbirds"?

 

AR: Because, in fact, one doesn't. And that is the differ­ence between subcategories of concepts and first-level con­cepts. Because, you see, you could not arrive at the differences between eagles, hummingbirds, etc., unless you had first separated birds from other animals.

 

Even if chronologically you may learn those concepts in different orders, ultimately when you organize your con­cepts to determine which are basic-level concepts and which are derivatives (in both directions, wider integration or nar­rower subdivision), the test will be: which objects you perceive directly in reality and can point to, and which you have to differentiate by means of other concepts.

 

Prof. F: Then you are suggesting that metaphysically there are certain lowest species or infima species: certain concepts that are directly tied to concretes. Whereas, on top of them, we continually build higher-order concepts, which refer, in turn, to the lower.

 

AR: Yes, if you mean, by "metaphysical," existential objects-entities which exist qua entities.

 

Prof. E: I'd like to ask a follow-up question. This is the kind of question I get all the time, which I do not fully know how to answer. I will give the example: "table" is first-level, and then you can go up to "furniture" or down to "living­room table," etc.

 

AR: That's right.

 

Prof. E: Then I get this kind of question: Is it theoreti­cally possible for someone to start by first conceptualizing living-room tables (he wouldn't, of course, be able to call it "living-room table" since he wouldn't yet have the concept "table") and then "desk," etc. and have separate concepts for all of what we call subcategories of "table," and then one day, in effect, grasp in an act of higher integration that they have something uniting them all, and reach the concept "table"?

 

AR: Theoretically, maybe; existentially, no. By which I mean that in order to do that, if that is how a child starts, he would have to live in a furniture store. He would have to have observed an enormous number of certain kinds of tables so that he isolates them first and then arrives at the overall category, which is "table."

 

Here, the process is directed by what is available to the child's observation when he begins to form the concept.

 

Prof. E: Would the state of his ability to discriminate also be relevant to defining what is a first-level concept? In other words, he couldn't perhaps discriminate subtler distinctions before he had the gross category.

 

AR: Exactly. And he has to have, and this is very essen­tial, a sufficient number of examples of a given category differentiated from other dissimilar entities before he can form a concept.

 

Prof. E: What do you say about this objection? People say you can't point to table, all you can point to is living­ room table, or dining-room table, etc., and, therefore, how do you distinguish "table" from "furniture" in this respect?

 

AR: The answer is in the Conceptual Common Denomi­nator. If you point to table and you say, "I mean this," what do you differentiate it from? From chairs, cabinets, beds, etc. You do not mean only a dining-room table but not an end table. What is involved here, in the act of pointing, as in everything about concepts, is: from what are you differ­entiating it?

 

Prof. B: Isn't the issue then what similarities and differ­ences you are able to be aware of? And wouldn't that be a function of two things: the actual properties of the objects plus the context that you are in?

 

AR: That's right.

 

Prof. B: Take the earlier question of whether you could form the concept of "furniture" before the concept "table." In order to do that, you would have to perceive the similari­ties uniting all items of furniture before you perceived the difference between a table and, say, a bed. And the ques­tion is: how could that ever come up?

 

AR: The difficulty here is that the infant or child would have to have a much wider range of perception than is normal to a beginning consciousness. He would have to consider objects outside of the room, objects moving in the street, and then conclude: by "furniture" I mean the objects in this room. Even subverbally, if this is what he observes, he has already made an enormously wide range of observa­tions, which is not likely as a beginning. In logic, there would be objections to that, because how would he differen­tiate furniture from, let's say, moving vehicles in the street? How did he get to that wide a range without first observing the immediate differences and similarities around him?

 

Prof. B: If he looked at a bed and a dresser, let's say, he would have to see them as different before he saw them as similar.

 

AR: That's right. Also, remember that we use "table" as an example because that is the object most likely to be one of the first perceived by a child in our civilization. But now suppose a child has to grasp the concept "coconut." In our civilization that would be a much later development. He would probably first grasp "food," then maybe "apple" and "pear," until some day he discovers an unusual food-a coconut. But now take a child in a primitive society, in a jungle. He never heard of tables, and he might be bewil­dered when he first sees a table in the home of the local missionary. But "coconut" might be one of the first con­cepts he forms because coconuts are all around him.

 

The overall rule for what is first-level is: those existential concretes which are first available to your consciousness. But they have to be concretes. A first-level concept cannot be one which, in order to indicate what you mean by it, requires other concepts, as is the case with "furniture." "Furniture" is not a term designating concretes directly. It is a term designating different kinds of concretes which all have to be conceptualized, as against another very broad category, such as moving vehicles, let us say.

 

In other words, if, after you have acquired a conceptual vocabulary, a given concept cannot be understood by you or communicated by you without reference to other concepts, then it is a higher-level concept, even if maybe somehow you grasped it first (and I question the issue of whether you could grasp it first). But the hierarchy that you will establish eventually when you are in the realm of a developed lan­guage, the hierarchy of which concept depends on the other, will not be determined by the accidental order in which you learned them, because that can have a great deal of the optional element and depends on what is available in your immediate surroundings.

 

It is after you are in the realm of language, when you can organize your concepts and say what you mean by "table," what you mean by "furniture" - it is at this level, logically and not chronologically, that you can determine which are concepts of the first order and which are derivatives.

 

Prof. C: I have a follow-up on this same issue. You state,

 

"Observe that the concept `furniture' is an abstraction one step further removed from perceptual reality . . ." What I would like to get at is: what is that "one step"? It seems to be the fact that to identify the units subsumed by "furni­ture," you need to grasp the objects' function, which is something that one does not perceive directly. The function is a more abstract characteristic.

AR: May I point out something here? I said, in this sentence, an abstraction one step further removed from perceptual reality. Now, remember, abstractions also are real. Abstraction itself is only our epistemological process, but that which it refers to exists in reality; but it would not be available to us by direct perceptual means. And, there­fore, the term "perceptual reality" is very important here. I don't mean that higher abstractions are a step removed from reality. I mean they cannot be perceived by perceptual means; in order to grasp them, we need concepts.

 

A child can observe perceptually the function of an item of furniture. But it cannot be the first thing that he observes, because, before he can observe a function, he has to isolate the objects of which this function is a characteristic. In order to observe that a table is something on which you put objects, a bed is something on which you lie down, he first has to conceptualize those objects; then he conceptual­izes what he can do with those objects-what their func­tion is.

Prof. C: But how does one focus on the function of a thing? Just looking at a chair or a bed, for instance, doesn't tell you what its function is. So how does a child recognize the similarity in this case when there isn't some perceivable characteristic like color that tells him that those things go together?

 

AR: What do you mean here by the function? The use which you can make of it, in the case of furniture. Well, you can observe that directly, after you have conceptualized or isolated those objects. How does a child discover what a table is for? Suppose he tries to lie down on the table and finds he is uncomfortable, but he lies down on the bed and  he is comfortable. He observes that he is put on the bed at night, but dishes are put on the table for dinner. That is observed perceptually. But to conceptualize it, he would first have to isolate and grasp such a thing as table versus bed.

 

The function itself is observable directly, but to conceptu­alize it he first has to conceptualize the objects as objects. Because here the function is an action-concept. It is either what the thing can do or what you can do with it. And an action-concept cannot precede an entity-concept. He first has to conceptualize the objects-the entities-then, the kinds of action they can perform or he can perform with them.

 

Prof. C: So you don't agree with my distinction that there are characteristics like color and shape that are directly given in perception, as opposed to a set of other characteris­tics for which one must have a vaster amount of knowledge or data?

 

AR: I thought I agreed in a certain sense, if I understood you correctly. If your question pertains to the order in which it is possible to form concepts, then I would agree with you that certain concepts, such as concepts of function or action, even though perceptually observable, cannot be conceptualized without a prior conceptualization of the act­ing or functioning entities. So, if you are asking, "Can concepts of function be formed ahead of the necessary ante­cedent concepts?" I would say "no." In that sense I agree with you. When you say there is something extra that is required-the something extra is the conceptualization of the entities involved, which is required before you can take the next step of conceptualizing the functions.

 

But we are referring here to the order of concept-forma­tion. The fact that functions can be observed perceptually is not the essential issue here. A child can see a moving object directly, but he cannot form the concept "motion" until he has formed the concept "object." Therefore, it is the order of possible conceptualization that is different here. This is what makes concept-formation hierarchical. This is what forms the dependence of certain concepts on certain others in human conceptual development.

 

But as to which part of this conceptual hierarchy involves direct, perceptual observation and which is purely conceptual­ that is a somewhat different question. On the lower levels of conceptualization-lower in the sense of first to be concept­ualized-all that first material is available to direct percep­tion but cannot be conceptualized indiscriminately, although certain optional elements exist on the lower level.

 

The higher concepts, the abstractions from abstractions, come when you have to integrate perceptual concretes with concepts of consciousness or concepts of human action. For instance, such a concept as "marriage" cannot be grasped perceptually. Even if you observed all the actions of a couple, that wouldn't give you the concept "marriage," because here certain relationships, actions, and processes of consciousness are involved.

 

So the distinction regarding the hierarchy in concept ­formation is not what can or cannot be perceived perceptu­ally, even though it is true that on the higher levels the referents cannot be perceived by exclusively perceptual means. The issue here is: which concepts can be formed first and which depend on other concepts that had to come before you reached that level. And the distinction between an object and its function or action is one of those. You cannot grasp the function before you have conceptualized the object.

 

Prof. D: On the question about "furniture" versus "ta­ble" versus "coffee table," I'm not clear as to the answer you gave to the question of whether or not it was contextual.

 

AR: My answer is that although there is an element of the optional for certain first-level concepts, the logical determi­nation of which concept is primary, or first, and which is derivative depends on whether that second concept required the conceptualization of the first, before it could be concep­tualized.

 

The hierarchy here refers to your concept-forming process -in other words, it's a hierarchy of epistemology, not of metaphysics. Furniture exists on the same level as tables and chairs. But the question is, which concept depends on which? And the answer here is: that concept is second-level which, to be grasped, requires prior conceptualization of its constituents.

 

So even if we suppose that some child grasped "furniture" before he grasped "tables" and "chairs" (which is highly unlikely), in organizing his concepts later, he would have to place "furniture" second, and "tables" and "chairs" first. But he wouldn't include "dining tables" and "coffee ta­bles," etc., as first-level, because those are subdivisions; he would have to form the concept "table" before he could subdivide it into particular classes of tables.

 

Prof. E: Then is it true that while there is a certain area of option, chronologically speaking, as to which concept is formed first and which is formed by derivation, either by subdivision or as a wider integration, it is nevertheless true that once the conceptual apparatus has been developed and you establish a logical hierarchy, that hierarchy is invariant for human beings, being dictated by the nature of the con­cepts, with no option as to which concept is higher-level and which is lower-level?

 

AR: Correct.

 

Prof. E: Now to distinguish your view here from Aristot­le's. Aristotle would also say that you could arrange con­cepts on a hierarchical level, in effect from "table" on up or "man" on up. But he would say that what qualifies as a first-level concept is exclusively dictated by metaphysical considerations, and that subtypes of "man" or subtypes of "table" have, in effect, a lesser metaphysical status.

 

AR: Correct. We do not say that.

 

Prof. E: Whereas, would it be correct to say that, for Objectivism, once we have the logical hierarchy, the desig­nation of concepts as first-level within the logical hierarchy is dictated by a combination of metaphysical and epistemo­logical considerations, and are in that sense objectively first­ level, if I can use that terminology, as against intrinsically, in the Aristotelian system?

 

AR: Exactly right.

 

Prof. B: I don't understand the difference between the chronological and the logical order. In establishing the logi­cal order, you consider whether the second concept requires the first one in order to be formed. But if it is possible to form the concept of "furniture" before "table"-I don't think that is possible, but if it is-then why, in your logical hierarchy, would "furniture" be derivative?

 

AR: Yes, but if chronologically you form "furniture" first, you would have only a very approximate, very woozy lump­ing together of certain objects as against something else, say moving objects. Therefore, your concept at that stage would be enormously imprecise, and one could say almost tenta­tive. Before it can become a fully clear concept in your mind, so that you know what you mean by "furniture" fully, you would have to know which objects you call furniture as against vehicles, architectural features, etc.

 

Therefore, it is the precision of your concept, which you need in order to firmly differentiate it from everything else in your context of knowledge, that determines the hierarchi­cal status of your concepts. So, you can form a vague approximation, but that is not yet a concept.

 

Prof. D: Well, someone might say you can't form the concept "flower" clearly unless you know the specific kinds of flowers. I must say "flower" is a kind of woozy concept in my mind because I don't know peonies from roses, etc.

 

AR: No, that isn't wooziness. The issue is: can you tell a flower from an animal or a man? You see, the clarity depends on whether you are able to differentiate and draw clear lines of demarcation, within the context of your knowl­edge, between one concept and another. If you can, that is conceptually valid. If not, then it is approximate or at a preconceptual stage.

 

May I ask one general question? Is it of great importance what happens on the first levels of concept-formation? That is not important. The order in which we can or cannot form concepts and which we can do first is really more psycholog­ical than cognitive.

 

Prof. E: I think it is a question of preserving the objectiv­ity of the logical hierarchy that is the crucial question.

 

AR: But the logical hierarchy depends on which catego­ries you regard as depending on which. Do you know where it would have importance? Only in regard to issues such as the "stolen concept," where somebody will claim that some concept exists while denying the concepts on which it de­pends. But aside from that, I don't think it is significant which concepts we form in what order, except that when we have reached the full conceptual level, when we can form sentences and can differentiate concepts consciously, when the process has become self-conscious, then it is important to organize the relationships.

 

Prof. F: I am very disturbed by this, because it seems to me that you either are or are not asserting that there are some concepts whose formation does not require lower-level concepts. I think this is a major position. You either are or are not saying that there are some concepts whose forma­tion requires no lower-level concepts. Which is it?

 

AR: All the things which you can perceive directly [and conceptualize] without presupposing in that concept some other conceptual material, those are the first-level concepts. And if you want to form "animal" first and species later or vice versa, that is optional. But you couldn't have the con­cepts "love," "truth," "justice" as first-level concepts. You don't perceive them perceptually, directly.

 

Prof. F: So Objectivism holds that there are first-level concepts?

 

AR: Epistemologically, not metaphysically.

 

Regardless of what a given man did chronologically, once he has his full conceptual development, a very important test of whether a concept is first-level would be whether, within the context of his own knowledge, he would be able to hold or explain or communicate a certain concept without referring to preceding concepts. For instance, if a man formed the concept "furniture" directly from perception and then found that in communication he had to say, "Well by 'furni­ture' I mean tables, chairs, and other objects," he's classi­fied it as second-level.

 

Prof. B: But you can give a definition of "furniture" without referring to types of furniture. Take the man who formed "furniture" directly from the perceptual level. If you ask him what he means by "furniture" he can answer: "movable man-made objects within a human habitation. . ."

 

AR: Oh, that he couldn't do. That I can say with assur­ance. Because he couldn't arrive at that kind of definition while bypassing the identification of the objects he means. He could conceivably memorize that definition if he's heard it, but he couldn't form it himself. Because you'd have to ask him, "Well, which objects do you mean?"

 

Prof. B: But then he'd just point to items of furniture without having to use the classification­ --

 

AR: While seeing no distinction between tables, beds, chairs, etc.? Seeing only their distinction from architectural features or small objects, but no distinction between them?

 

Prof. B: He'd have to point to more than one. He couldn't point to just this table.

 

AR: No, if he means "furniture" he'd have to point to several different items of furniture. And then the question arises, psychologically, is it possible to form that kind of differentiation while never conceptualizing the particular things? The answer would be "no," because he cannot point to those objects if he hasn't conceptualized them.

 

Prof. B: Can you say why that is? It has to do with similarities and differences, doesn't it?

 

AR: It has also to do with what is more immediately and easily available to his consciousness, when he's starting to conceptualize, as against that which is much harder to iden­tify and separate, and requires a wider context. To separate furniture from architectural features is a much more com­plex issue of observations and requires a certain subtlety, which is why I say it is not likely-if a man has that much subtlety, he would certainly not fail to observe the differ­ences between tables and chairs.

 

 

 

Lower-level Concepts as Units in Relation to Higher-level Concepts

 

Prof. C: You say that one forms wider concepts by taking lower-level concepts as units. I was somewhat perplexed; I would have liked to see a different phrase used, namely that the wider concepts are formed from the knowledge of the lower concepts. Because some people, in mathematics for example, take a certain level of abstraction and quit refer­ring to reality thereafter and deal with nothing but the concepts.

 

AR: That would be psychology, or psychopathology, and I couldn't go into that. That some people would take lan­guage improperly-there's no protection against that.

 

Prof. C: But t