Peikoff - Concept-Formation - Concept-formation As A Mathematical Process

Question 1

AR solution to the problem lies in her discovery that there is an essential connection between:

Answer 1

CONCEPT-FORMATION and MATHEMATICS.

==> Since mathematics is the science of measurement, let us start by considering the nature and the purpose of MEASUREMENT.

Question 2

Measurement is …

Answer 2

THE IDENTIFICATION OF A RELATIONSHIP.

==> A quantitative relationship established by means of a standard that serves as a UNIT.

Question 3

The process of measurement involves 2 concretes:

Answer 3

1. The existent being measured.

2. The existent that is the standard of measurement.

Question 4

In every case, the primary standard is …

Answer 4

Some easily perceivable concrete that functions as a unit.

==> One measures length in units, say, of feet; weight in pounds; velocity in feet per second.

Question 5

The unit must be appropriate to the attribute being measured:

Answer 5

One cannot measure length in pounds or weight in seconds.

==> An appropriate unit is an instance of the attribute being measured.

==> A foot for example is itself a length; it is a specified amount of length.

==> Thus it can serve as a unit to measure length. Directly or indirectly, the same principle applies to every type of measurement.

Question 6

In the process of measurement, we …

Answer 6

Identify the relationship of ANY instance of a certain attribute to a SPECIFIC instance of it selected as the unit.

==> The former may range across the entire spectrum of magnitude, from largest to smallest; the latter, the (primary) unit, must be WITHIN THE RANGE OF HUMAN PERCEPTION.

Question 7

The epistemological purpose of measurement is best approached through an example:

Answer 7

Consider the fact that the distance between the earth and the moon is 240.000 miles.

==> No creature can perceive so vast a distance; to an animal, accordingly, it is unknowable and unfathomable.

==> Yet man has no difficulty in GRASPING (and now even transversing) it.

==> What makes this cognitive feat possible is the HUMAN method of establishing relationships to concretes we CAN DIRECTLY PERCEIVE.

Question 8

We cannot perceive 240.000 miles, but …

Answer 8

The distance is expressed in miles, and a mile is reducible to a certain number of feet, and a foot is: THIS (I am pointing to a ruler).

(It works in the other direction also)

Question 9

In both directions, AR holds, and in regard to countless attributes, the …

Answer 9

PURPOSE of measurement is to expand the RANGE of man’s consciousness, of his knowledge, BEYOND THE PERCEPTUAL LEVEL.

==> Beyond the direct power of his senses and the immediate concretes of any given moment.

==> Process of measurement is a process of integrating an unlimited scale of knowledge to man’s limited perceptual experience.

==> A process of making the universe knowable by bringing it WITHIN THE RANGE of man’s consciousness, by ESTABLISHING ITS RELATIONSHIP TO MAN.

Question 10

Measurement is an ANTHROPOCENTRIC process, because …

Answer 10

MAN IS AT ITS CENTER.

==> HIS scale of perception-the concretes HE can directly grasp-is the base and the standard, to which everything else is related.

Question 11

This brings us to AR’s momentous discovery:

Answer 11

THE CONNECTION BETWEEN MEASUREMENT AND CONCEPTUALIZATION.

==> The 2 processes have the same essential purpose and follow the same essential method.

Question 12

In both cases (measurement and conceptualization), …

Answer 12

Man IDENTIFIES RELATIONSHIPS AMONG CONCRETES.

==> He takes PERCEIVED CONCRETES AS THE BASE, to which he relates everything else, including innumerable existents outside his ability to perceive.

==> The result is to bring the whole universe within the range of HUMAN knowledge.

Question 13

And now a further, crucial observation:

Answer 13

In both cases, man relates concretes by the same method-by QUANTITATIVE MEANS.

==> BOTH concept-formation and measurement involve the mind’s discovery of a MATHEMATICAL relationship among concretes.

Question 14

AR’s seminal observation is that the similar concretes integrated by a concept differ from one another only …

Answer 14

QUANTITATIVELY.

ONLY IN THE MEASUREMENTS OF THEIR CHARACTERISTICS.

Question 15

When we form a concept, therefore, …

Answer 15

OUR MENTAL PROCESS CONSISTS IN RETAINING THE CHARACTERISTICS, BUT OMITTING THEIR MEASUREMENTS.

Question 16

Example of “length” (pencil, stick etc).

Or, more precisely, if the process were identified in words, it would consist of the following:

Answer 16

Length must exist in SOME quantity, but may exist in ANY quantity.

==> I shall identify as “length” that attribute of any existent possessing it which can be quantitatively related to a unit of length, WITHOUT SPECIFYING THE QUANTITY.

Question 17

This is the process-performed by the mind wordlessly-which enables the child not only to integrate the first instances of “length” that he observes, but also to identify future instances, such as the length of a pin, a room, a street.

All such instances are …

Answer 17

COMMENSURABLE.

==> THEY CAN BE RELATED QUANTITATIVELY TO THE SAME UNIT.

==> THEY DIFFER ONLY IN THEIR SPECIFIC MEASUREMENTS.

Question 18

To omit measurements, AR stresses, does NOT MEAN …

Answer 18

TO DENY THEIR EXISTENCE.

==> IT MEANS THAT MEASUREMENTS EXIST, BUT ARE NOT SPECIFIED.

Question 19

That measurements MUST EXIST is an ESSENTIAL part of the process.

The principle is:

Answer 19

THE MEASUREMENTS MUST EXIST IN SOME QUANTITY, BUT MAY EXIST IN ANY QUANTITY.

Question 20

Example of “table”.

The concept “table” integrates all tables, past, present, and future, regardless of these variations among them.

How can it do so?

Answer 20

When we form the concept, we RETAIN THE ABOVE CHARACTERISTICS.

==> There must be a surface of some shape, the legs must have some position in relation to the top, there must be some height, weight, and so on.

==> But the varying characteristics are NOT SPECIFIED.

==> From THIS perspective tables are INTERCHANGEABLE.

==> One is ABLE TO FORM A MENTAL UNIT THAT SUBSUMES ALL OF THEM.

Question 21

Neither a child nor an adult knows all the characteristics of tables.

For example, a child forming “table” may not yet have discovered the attribute of weight.

Speaking literally, such a child …

Answer 21

Cannot omit measurements of weight.

His mind, however, is governed by A WORDLESS POLICY APPLICABLE TO ALL FUTURE KNOWLEDGE.

==> This essence of the conceptual process, amounts to the following:

“I know certain attributes of tables. Whatever other attributes I discover, the same process will apply: I will retain the attribute and omit its measurements.”

==> In this sense, in the form of an EPISTEMOLOGICAL STANDING ORDER, the concept may be said to retain ALL the characteristics of its referents and to omit ALL the measurements (these last within an appropriate range).

**This principle applies even in regard to characteristics UNKNOWN at a given stage of development.

Question 22

The grasp of SIMILARITY as we have seen, is essential to conceptualization.

But what is similarity?

Answer 22

In ordinary usage, objects are described as similar if they are partly the same, partly not.

“Similarity” denotes “partial identity, partial difference.”

Question 23

In the context of concept-formation, the differences among similars concretes are apparent.

The puzzle has been: what is the SAME?

AR’s profound new answer is that the relationship among similars is …

Answer 23

MATHEMATICAL.

When two things are similar, what is the same is their characteristic(s).

==> What differs is the magnitude or measurement of these.

==> SIMILARITY, in this context, is the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree.

Question 24

A man’s grasp of similarity is actually his mind’s grasp of …

Answer 24

A mathematical fact:

The fact that certain concretes are COMMENSURABLE.

==> That they are reducible to the SAME UNITS OF MEASUREMENT.

==> A man can relate such concretes to one another, bracketing them mentally into the same group, because his mind can RELATE EACH ONE QUANTITATIVELY TO THE SAME STANDARD.

==> The ONLY difference is the measurement of this relationship in the several instances.

==> Given this perspective his mind, in order to proceed to form a new unit, need merely REFRAIN FROM SPECIFYING THE MEASUREMENTS.

Question 25

Such is the essence of ABSTRACTION, according to Objectivism:

Answer 25

Men abstract attributes or characteristics from THEIR MEASUREMENTS. ==> The result is an outlook on existents that permits A NEW SCALE OF INTEGRATION.

Question 26

The process of measurement-omission is performed for us by the ...

Answer 26

Nature of our mental faculty, whether one identifies it or not. ==> To FORM a concept, one does not have to know that a FORM of measurement is involved. ==> One does NOT have to measure existents or even know HOW to measure them. ==> On the conscious level, one need merely observe SIMILARITIES.

Question 27

Measurement as a CONSCIOUS process presupposes ...

Answer 27

A substantial CONCEPTUAL DEVELOPMENT. ==> It presupposes that one has already conceptualized separate attributes, knows how to count, and has defined suitable units and a method of relating objects to them in numerical terms. ==> The measurement involved in forming concepts, however, which may be described as “implicit” measurement, does NOT REQUIRE SUCH KNOWLEDGE.

Question 28

When we (first) conceptualize, we focus on an attribute ...

Answer 28

PERCEPTUALLY, NOT CONCEPTUALLY.

Question 29

Nor do we need a knowledge of numbers:

Answer 29

For concept-formation, we need to discover COMMENSURABILITY, not specific quantitative data. ==> The ESSENCE OF THE PROCESS IS THE OMISSION OF SUCH DATA.

Question 30

To discover commensurability, we need to ...

Answer 30

Observe variations in degree or amount, such as long/much longer/shorter/much shorter, hotter/colder, lighter/darker, rougher/smoother and so on. ==> Such variations are observed well before we know how to measure them explicitly or precisely.

Question 31

For example, we can SEE that some objects ...

Answer 31

Extend further or much further than others from a given point, before we know numbers or the concepts “length” or “foot”. ==> In the act of apprehending such a CONTINUUM of more-or-less, we are grasping the place within it of any particular length. ==> We are thus grasping-in implicit, approximate form-that ANY PERCEIVED INSTANCE can serve as the standard. ==> In other words, in the process of concept-formation, ANY PERCEIVED UNIT of the future concept can serve as the unit of measurement.

Question 32

Such is the means by which we are able to grasp, without the need of numbers or any other ANTECEDENT concepts, that ...

Answer 32

ALL the relevant concretes are REDUCIBLE TO A COMMON UNIT.

Question 33

To learn how to express in numerical terms the implicit measurements involved in concept-formation is a later development, which is sometimes simple and sometimes not:

Answer 33

It was relatively simple, for example, once men had acquired a conceptual vocabulary, for them to demarcate “foot” or some equivalent as a unit of length and learn how to deploy a ruler. But an advanced science was required to discover a unit by which to measure color (the wavelengths of light), or to discover a method of measuring the area of complex curvilinear figures (such a method is provided by integral calculus).

Question 34

A form of measurement, in sum, makes concept-formation possible-and concepts in turn ...

Answer 34

Make NUMERICAL measurement possible. ==> This INTERDEPENDENCE reflects a fundamental fact about human cognition: ==> The perspective essential to both processes-the QUANTITATIVE REDUCTION TO A UNIT-is the SAME.

Question 35

So far, we have been considering measurement primarily in regard to the INTEGRATION of concretes. Measurement also plays a role in the FIRST step of concept-formation:

Answer 35

The DIFFERENTIATION of a group from other things.

Question 36

Such differentiation cannot be performed arbitrarily. For example, one can form a concept by distinguishing tables from chairs, but not by distinguishing tables from red objects:

Answer 36

THERE IS NO BASIS on which to bring these 2 sets of concretes together before the mind, and no way to identify a relationship between them. ==> The reason is that the relationships required for concept-formation are established QUANTITATIVELY, by means of (implicit) measurements. ==> And there is no unit of measurement common to table-shaped objects and red objects. ***THE ATTRIBUTES OF SHAPE AND COLOR ARE INCOMMENSURABLE.

Question 37

AR proceeds to develop the concept of the Conceptual Common Denominator (CCD):

Answer 37

CCD = The characteristic(s) reducible to a unit of measurement, by means of which man DIFFERENTIATES two or more existents from other existents possessing it.

Question 38

For example, one can differentiate tables from chairs or beds, because all these groups possess a ...

Answer 38

COMMENSURABLE characteristic ==> Shape. This CCD, in turn, determines what feature must be chosen as the DISTINGUISHING characteristic of the concept “table”: ==> Tables are distinguished by a specific KIND of shape, which represents a specific category or set of geometric measurements within the characteristic of shape-as against beds, eg, whose shapes are encompassed by a different set of measurements. ***Once the appropriate category has been specified, one completes the process of forming “table” by omitting the measurements of the individual table shapes within that category.

Question 39

The above is merely a passing mention of a complex topic, but it indicates from a new aspect the mathematical basis of concept-formation. MEASUREMENT is essential to both parts of the process:

Answer 39

We can differentiate groups only by reference to a commensurable characteristic(s). We can integrate into a unit only concretes whose differences are differences in measurement. ==> NO ASPECT OF THE PROCESS IS CAPRICIOUS. ==> In BOTH its parts, concept-formation depends on our mind’s recognition of OBJECTIVE, MATHEMATICAL RELATIONSHIPS.

Question 40

AR’s formal definition of “concept” condenses into a sentence every key idea discussed above:

Answer 40

A concept is a mental integration of 2 or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.

Question 41

In her treatise, AR covers all the main kinds of concepts, including concepts of motion, relationships, and materials. In each case, she explains how the principle of measurement-omission applies. Instead of pursuing this illustrative material, I want to turn to another question. Since the mind omits measurements whether a man knows it or not, one may ask, what is the PRACTICAL purpose of the Objectivist theory of concepts?

Answer 41

In part, the answer is that philosophers have to know the mathematical aspects of concept-formation in order to define the rules to guide the CONSCIOUS aspects of a thought process, the ones that ARE within men’s deliberate, volitional control.

Question 42

In deeper part, however, the answer is that ...

Answer 42

The theory of measurement-omission is essential to the validation of conceptual knowledge and, therefore, to the validation of REASON itself. ==> In the long run, a scientific civilization cannot survive without such validation.

Question 43

So long as men remain ignorant of their basic mental process, they have no answer ...

Answer 43

To the charge, leveled by mysticism and skepticism alike, that their mental content is SOME FORM of revelation or invention detached from reality. ==> This kind of viewpoint can go into remission for a while, thanks to the remnants of a better past. ==> Ultimately, however, if it is not burned out of men’s souls completely by an EXPLICIT philosophic theory, it becomes the most virulent of cancers. ==> It metastasizes to every branch of philosophy and every department of a culture, as is now evident throughout the world. ==> Then, THE BEST AMONG MEN BECOME PARALYZED BY DOUBT; while the others turn into the mindless hordes that march in any irrationalist era looking FOR SOMEONE TO RULE THEM.

Question 44

A proper theory of concepts is not sufficient to save the world. But ...

Answer 44

IT IS NECESSARY. ==> The fact that concepts are valid tools of cognition whether we know it or not will not save us-NOT UNLESS WE DO KNOW IT.

Question 45

What the objectivist theory of concepts accomplishes practically is the ...

Answer 45

Defense of man’s mind on the level of fundamentals, along with the philosophic disarmament of its worst enemies. ==> The key to this historic achievement lies in AR’s demonstration that concepts ARE based on and DO REFER to the facts of reality.

Question 46

Now (AR writes) we can answer the question: To what precisely do we refer when we designate three persons as “men”?

Answer 46

We refer to the fact that they are living beings who possess the SAME characteristic distinguishing them from all other living species: A RATIONAL FACULTY. Though the SPECIFIC measurements of their distinguishing characteristic qua men, as well as of all their other characteristics qua living beings, are different. ==> As living beings of a certain kind, they possess innumerable characteristics in common: the same shape, the same range of size, the same facial features, the same vital organs, the same fingerprints etc and ALL THESE DIFFER ONLY IN THEIR MEASUREMENTS.

Question 47

A concept is not a product of arbitrary choice, whether personal or social:

Answer 47

IT HAS A BASIS IN REALITY. ==> The basis is not a supernatural entity transcending concretes or a secret ingredient lurking within them. “Manness” is MEN, the real men who exist, past, present, and future. ==> It is men viewed from A CERTAIN PERSPECTIVE.

Question 48

A concept denotes ...

Answer 48

Facts-as processed by a HUMAN METHOD. ==> Nor does the method introduce any COGNITIVE DISTORTION. ==> The concept DOES NOT OMIT or ALTER any characteristic of its referents. ==> It includes EVERY FACT ABOUT THEM, including the fact that they are COMMENSURABLE. ==> It merely REFRAINS FROM SPECIFYING the varying relations they sustain to a unit(s).

Question 49

The answer to the “problem of universals” lies in AR’s discovery of ...

Answer 49

The relationship between UNIVERSALS and MATHEMATICS. ==> Specifically, the answer lies in the brilliant comparison she draws between CONCEPT-FORMATION and ALGEBRA. ==> This is more than a mere comparison, as she shows, since the underlying method in both fields is the SAME.

Question 50

The basic principle of concept-formation (which states that the omitted measurements must exist in SOME quantity, but may exist in ANY quantity) is the ...

Answer 50

Equivalent of the basic principle of algebra, which states that algebraic symbols must be given SOME numerical value, but may be given ANY value. ==> In this sense and respect: PERCEPTUAL AWARENESS = ARITHMETIC. CONCEPTUAL AWARENESS = THE ALGEBRA OF COGNITION. ==> LET THOSE WHO TRY TO INVALIDATE CONCEPTS BY DECLARING THAT THEY CANNOT FIND “MANNESS” IN MEN, TRY TO INVALIDATE ALGEBRA BY DECLARING THAT THEY CANNOT FIND “a-ness” IN 5 OR IN 5,000,000.

Question 51

For centuries, rationalist philosophers have venerated mathematics as the model of cognition. What they have admired about the discipline is ...

Answer 51

ITS DEDUCTIVE METHOD. ***Objectivism, too, regards mathematics as an epistemological model, but for a different reason.

Question 52

The mathematician is the ...

Answer 52

Exemplar of conceptual INTEGRATION. He does professionally and in numerical terms what the rest of us do implicitly and have done since childhood, to the extent that we exercise our distinctive HUMAN capacity.

Question 53

Mathematics IS ...

Answer 53

The SUBSTANCE OF THOUGHT written large, as the West has been told from Pythagoras to Bertrand Russell. ==> It does provide a unique window into HUMAN NATURE. ==> What the window reveals, however, is not the barren constructs of rationalistic tradition, but man’s method of extrapolating from observed data to the total of the universe.

Question 54

What the window of mathematics reveals is NOT the mechanics of DEDUCTION, BUT ...

Answer 54

OF INDUCTION. SUCH IS AYN RAND’S UNPRECEDENTED AND PREGNANT IDENTIFICATION IN THE FIELD OF EPISTEMOLOGY.