Measure and Quantity and Rand's notion of mathematics

[BaalChatzaf]

There are two related notions that occur in a mathematical context: measure and quantity.

Let us examine the notion of quantity. There are roughly three kinds of quantity

a. Ordinal. An ordinal quantity indicates where in a sequence something is. First, Second, etc.

b. Cardinal. A cardinal quantity answers the question -- How Many.

c. Measure. A measure quantity answers the question -- How Much.

Ordinals and Cardinals are represented by positive and non-negative integers (respectively) in the finite case. So we have the first item, the second item etc. for ordinals. Or zero items, one item, two items etc. for cardinals.

Measure is different. First of all Measure is generally compact and continuous and not discrete. Measure is handled in physics by the use of -real numbers-. The set of real numbers is linearly ordered, is compact and dense (no holes, no gaps, no jumps).

That are other mathematical objects that are not quantities but have quantities associated with them. For example vectors. Vectors have a magnitude (represented by a non-zero real number) and a direction (conventionally a real number between 0 and 2*pi in the radian system for measuring angles.

When Rand spoke of measurement omission she implied the underlying quantities were measures and involved linearly ordered, dense and compact systems, the simplest of which is the real number systems. If compactness is not absolutely required (i.e. we do not require sequences to converge), then the rational numbers will do. They form a dense but not compact system. When Rand held that mathematics is the science of measurement she was effectively restricting mathematics to the system of real numbers or rational numbers. Mathematics is much broader than that. For example there are mathematical systems that deal with symmetries of all sorts. Group theory is the kind of mathematical system used to express and describe symmetries. Groups, qua groups are not bound to systems of quantities.

One could say mathematics is the discipline or art of abstract structures and relations. For the more technically informed, mathematical systems can be expressed as categories of one kind or another. [Category her is not the general term meaning class; in a math context, category is a particular kind of mathematical system, and I do not intend to discuss categories here]

Ba'al Chatzaf

[tjohnson]

One could say mathematics is the discipline or art of abstract structures and relations.

Interesting post. Korzybski would say mathematics is the discipline or art of abstract structures and relations capable of exact treatment at a given date. All knowledge consists of abstract structures and relations but mathematics differs in the sense that they are exact and so, ideal.

Traditionally, too, since Aristotle, and, in the opinion of the majority, even

today, mathematics is considered as uniquely connected with quantity and

measurement. Such a view is only partial, because there are many most important

and fundamental branches of mathematics which have nothing to do with quantity

or measurement—as, for instance, the theory of groups, analysis situs, projective

geometry, the theory of numbers, the algebra of ‘logic’, .

Sometimes mathematics is spoken about as the science of relations, but

obviously such a definition is too broad. If the only content of knowledge is

structural, then relations, obvious, or to be discovered, are the foundation of all

knowledge and of all language, as stated in the division of words given above. Such

a definition as suggested would make mathematics co-extensive with all language,

and this, obviously, is not the case.

[merjet]

Ba'al,

It is refreshing to see somebody on an Objectivist forum not confounding measurement with other kinds of quantification.

One could say mathematics is the discipline or art of abstract structures and relations.

I think this is too broad, since logic would satisfy the definiens. I think a good definition of math is "the science of quantity and quantifiable structures."

[merjet]

Traditionally, too, since Aristotle, and, in the opinion of the majority, even

today, mathematics is considered as uniquely connected with quantity and

measurement. Such a view is only partial, because there are many most important

and fundamental branches of mathematics which have nothing to do with quantity

or measurement—as, for instance, the theory of groups, analysis situs, projective

geometry, the theory of numbers, the algebra of ‘logic’, .

Sometimes mathematics is spoken about as the science of relations, but

obviously such a definition is too broad. If the only content of knowledge is

structural, then relations, obvious, or to be discovered, are the foundation of all

knowledge and of all language, as stated in the division of words given above. Such

a definition as suggested would make mathematics co-extensive with all language,

and this, obviously, is not the case.

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

[tjohnson]

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

Really? What does Group Theory have to do with quantity or measurement, for example?

[merjet]

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

Really? What does Group Theory have to do with quantity or measurement, for example?

Correct me if I'm wrong, but I assumed by "group theory" you meant algebraic groups. Examples of such groups are (1) the set of all integers and addition and (2) the set of all rationals excluding 0 and multiplication. Numbers are quantitative.

[tjohnson]

Correct me if I'm wrong, but I assumed by "group theory" you meant algebraic groups. Examples of such groups are (1) the set of all integers and addition and (2) the set of all rationals excluding 0 and multiplication. Numbers are quantitative.

From Wiki;

"Group theory is the mathematical study of symmetry, as embodied in the structures known as groups"

Whether or not some number systems are groups is another issue.

[merjet]

From Wiki;

"Group theory is the mathematical study of symmetry, as embodied in the structures known as groups"

Whether or not some number systems are groups is another issue.

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

[tjohnson]

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

Baal said 'Group theory is the kind of mathematical system used to express and describe symmetries. Groups, qua groups are not bound to systems of quantities." Maybe Baal could elaborate, my group theory knowledge is limited to one course some 25 years ago and it's a huge field of study.

[merjet]

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

Baal said 'Group theory is the kind of mathematical system used to express and describe symmetries. Groups, qua groups are not bound to systems of quantities." Maybe Baal could elaborate, my group theory knowledge is limited to one course some 25 years ago and it's a huge field of study.

"Not bound to systems of quantities", or in other words "does not necessarily involve quantities", is quite different from "has nothing to do with quantity." It's like "Some S is not P" versus "No S is P."

[tjohnson]

"Not bound to systems of quantities", or in other words "does not necessarily involve quantities", is quite different from "has nothing to do with quantity." It's like "Some S is not P" versus "No S is P."

I don't know what else to say, Group Theory is about the study of symmetries. Just because a number system is an example of a group doesn't change this.

A group G is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements A, B, C, ... with binary operation between A and B denoted AB form a group if

1. Closure: If A and B are two elements in G, then the product AB is also in G.

2. Associativity: The defined multiplication is associative, i.e., for all A,B,C in G, (AB)C==A(BC).

3. Identity: There is an identity element I (a.k.a. 1, E, or e) such that IA==AI==A for every element A in G.

4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element. Therefore, for each element A of G, the set contains an element B==A^(-1) such that AA^(-1)==A^(-1)A==I.

Now what has this to do with quantity or measurement?

[BaalChatzaf]

From Wiki;

"Group theory is the mathematical study of symmetry, as embodied in the structures known as groups"

Whether or not some number systems are groups is another issue.

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

There exist groups in which the elements are not associated with quantities or are quantities. Then there are groups in which the elements and or structures are very much involved with quantities. For example the group of linear transformations on a vector space. If the vector spaces are real, the transformations are uniquely represented by non-singular matrices whose elements are real numbers.

The definition of a group is not inherently bound to any system of quantities. For example the Klein 4-Group which is the set of strings over the alphabet {x, y} (these are two distinct symbols, that is all) with the defining relations

x*x = e (the identity)

x*y = y*x.

This group happens to be isomorphic to the set of symmetries on a (general) rectangle. It has four distinct elements:

e (the identity), x, y and x*y

Ba'al Chatzaf

[BaalChatzaf]

Ba'al,

It is refreshing to see somebody on an Objectivist forum not confounding measurement with other kinds of quantification.

One could say mathematics is the discipline or art of abstract structures and relations.

I think this is too broad, since logic would satisfy the definiens. I think a good definition of math is "the science of quantity and quantifiable structures."

Logic is a mathematical system.

Special case (for example): Boolean algebras (the algebra of propositions) is a distributive complemented lattice.

Ba'al Chatzaf

[tjohnson]

Below is a discussion about the relationship between group theory and physics. We can see that quantity and measurement are involved in the work of the physicist as he establishes some relation and the mathematician is using numbers to establish the "same" relation. Group theory goes beyond this, however, and into invariance of relations.

The role of groups in physical theory is best described by quoting Professor

Rainich. (Remarks in brackets are mine.) ‘A physicist, we may take it, is a person

who measures according to certain rules. Let us denote by a the number he obtained

in a given situation by applying the rule number one, by b the number obtained in

the same situation by measuring according to rule number two and so on (a may be

e.g. the volume, b the pressure, c the temperature of gas in a given container). The

physicist finds further that the results of measurements of the same kind undertaken

in different situations satisfy certain relations, we may write, for instance:

 r(a,b)=c

.

A mathematician is busy deducing from some given propositions other

propositions; this usually leads to numbers which we may call A, B, C, . . . . These

numbers also satisfy certain relations, say

R(A,B)=C.

Then comes, as Professor Weyl says, a messenger, a go-between who may be a

mathematician or a physicist, or both, and says: “If you establish a correspondence

between the physical quantities and the mathematical quantities, if you assign A to

a, B to b, etc., the same relations hold for the physical quantities as for the corresponding mathematical quantities so

that R≡r.” [similarity of structure.]

‘It may happen, and in fact it happens often, that the same mathematical theory

can be applied to the same physical facts in more than one way; for instance, instead

of assigning to the physical quantities a, b, . . . the mathematical quantities A, B, . . .

we might have assigned to them A', B', . . . with the same results, that is, the

relations for physical quantities are the same as for the mathematical quantities

corresponding to them now (think of space considered from the experimental point

of view—and of coordinate geometry; different ways of establishing a

correspondence result from different choices of coordinate axes). If this happens it

means that the mathematical theory possesses a peculiar property, namely, that if A'

is substituted for A, B' for B and so on, no relation of the type

R(A,B)=C

which was correct before the substitution is destroyed; in other words, there are substitutions or

transformations for which all relations are invariant. All such transformations

constitute what we call a group; the existence and the properties of such a group

present a very important characteristic of the mathematical theory. Moreover it is

clear that if two different mathematical theories can be applied—in the sense

described above—to the same physical theory, the groups of these two theories will

be essentially the same, so that the groups reflect some of the most fundamental

properties of physical systems.’2

[BaalChatzaf]

Below is a discussion about the relationship between group theory and physics. We can see that quantity and measurement are involved in the work of the physicist as he establishes some relation and the mathematician is using numbers to establish the "same" relation. Group theory goes beyond this, however, and into invariance of relations.

The role of groups in physical theory is best described by quoting Professor

Rainich. (Remarks in brackets are mine.) ‘A physicist, we may take it, is a person

who measures according to certain rules. Let us denote by a the number he obtained

in a given situation by applying the rule number one, by b the number obtained in

the same situation by measuring according to rule number two and so on (a may be

e.g. the volume, b the pressure, c the temperature of gas in a given container). The

physicist finds further that the results of measurements of the same kind undertaken

in different situations satisfy certain relations, we may write, for instance:

 r(a,b)=c

.

A mathematician is busy deducing from some given propositions other

propositions; this usually leads to numbers which we may call A, B, C, . . . . These

numbers also satisfy certain relations, say

R(A,B)=C.

Then comes, as Professor Weyl says, a messenger, a go-between who may be a

mathematician or a physicist, or both, and says: “If you establish a correspondence

between the physical quantities and the mathematical quantities, if you assign A to

a, B to b, etc., the same relations hold for the physical quantities as for the corresponding mathematical quantities so

that R≡r.” [similarity of structure.]

‘It may happen, and in fact it happens often, that the same mathematical theory

can be applied to the same physical facts in more than one way; for instance, instead

of assigning to the physical quantities a, b, . . . the mathematical quantities A, B, . . .

we might have assigned to them A', B', . . . with the same results, that is, the

relations for physical quantities are the same as for the mathematical quantities

corresponding to them now (think of space considered from the experimental point

of view—and of coordinate geometry; different ways of establishing a

correspondence result from different choices of coordinate axes). If this happens it

means that the mathematical theory possesses a peculiar property, namely, that if A'

is substituted for A, B' for B and so on, no relation of the type

R(A,B)=C

which was correct before the substitution is destroyed; in other words, there are substitutions or

transformations for which all relations are invariant. All such transformations

constitute what we call a group; the existence and the properties of such a group

present a very important characteristic of the mathematical theory. Moreover it is

clear that if two different mathematical theories can be applied—in the sense

described above—to the same physical theory, the groups of these two theories will

be essentially the same, so that the groups reflect some of the most fundamental

properties of physical systems.’2

This interesting observation would not apply to the two major theories of gravitation, Newtonian and Einsteinian (General Theory of Relativity). The underlying mathematical structure of the theories are extremely distinct and different from each other. Newtonian gravitation is based on a scalar potential function. Einsteinian gravitation is based on a tensor potential. The equations that follow are different in number and kind. The underlying geometries are also quite different and distinct. The genius of Einstein's theory is that it -explains- why the Newtonian theory is so close to right in a sufficiently "flat" manifold. Close to right, but not right. It is no accident that Newton's Law of Universal Gravitation has worked as well as it has (even though it is not right).

A lovely account of this can be found in -Introduction to General Relativity:Spacetime and Geometry- by Sea Carroll, pp 286-293. Interested readers will need some math background but it is not overly daunting.

Ba'al Chatzaf

[merjet]

Now what has this to do with quantity or measurement?

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

[merjet]

Logic is a mathematical system.

Special case (for example): Boolean algebras (the algebra of propositions) is a distributive complemented lattice.

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

[tjohnson]

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

The mathematician is not concerned with quantities or measurement, it is the physicist, etc., whoever is APPLYING the mathematics.

[merjet]

I should have done this earlier, but it is not too late. I looked at the definition of mathematics in my Webster's New World Dictionary (copyright 1970) and here it is.

mathematics - the group of sciences (including arithmetic, geometry, algebra, calculus, etc.) dealing with quantities, magnitudes, and forms, and their relationships, attributes, etc., by the use of numbers and symbols

I like it. The same definition is here: http://www.yourdictionary.com/mathematics

Regarding "magnitude" as covering "measurement", it is clear that mathematics is far wider than "the science of measurement."

[Michael Stuart Kelly]

Merlin,

Isn't magnitude a kind of measurement?

Michael

[tjohnson]

Regarding "magnitude" as covering "measurement", it is clear that mathematics is far wider than "the science of measurement."

There are many ways one could define 'mathematics'. Korzbski defines it as "Mathematics consists of limited linguistic schemes of multiordinal relations capable

of exact treatment at a given date." Maybe we should discuss the merits of each definition.

[merjet]

Merlin,

Isn't magnitude a kind of measurement?

Michael

Off-hand, I think a better way of saying it is that one measures magnitudes.

A measurement is a ratio of one magnitude to another of the same type. For example, the length of a football field is 100 times the length of one yard.

Edited October 19, 2007 by Merlin Jetton

[merjet]

There are many ways one could define 'mathematics'. Korzbski defines it as "Mathematics consists of limited linguistic schemes of multiordinal relations capable of exact treatment at a given date." Maybe we should discuss the merits of each definition.

One of the rules of a good definition is that it not be vague or obscure. I note that you misspelled his name, omitting the "y". If you were to correct it, would that be doing math?

[tjohnson]

There are many ways one could define 'mathematics'. Korzbski defines it as "Mathematics consists of limited linguistic schemes of multiordinal relations capable of exact treatment at a given date." Maybe we should discuss the merits of each definition.

One of the rules of a good definition is that it not be vague or obscure. I note that you misspelled his name, omitting the "y". If you were to correct it, would that be doing math?

Well I'm glad that you took the time to notice that spelling error I am not aware of any "rules of a good definition", nor do I think definitions are static affairs. Mathematics has changed incredibly in the last 2000 years and so maybe the definition must change as well. I believe mathematics grew out of quantity and measurement but evolved into, as per Korzybski, the study of exact relations, more or less. In applied mathematics we don't deal with exact relations in our measurements and quantities, they can only be considered approximate.

[merjet]

I am not aware of any "rules of a good definition"

They are given in many introductory logic texts, e.g. David Kelley's The Art of Reasoning.

There are 5 rules here: http://en.citizendium.org/wiki/CZ:Definitions

Kelley's book has these 5 plus one more: A definition should include a genus and a differentia.