2 = +2 = 2/1 = 2.0 = 2.0 + j x 0.0?

"Typically 2 the integer is used for counting, whereas 2 the real number is used for measuring.

But in higher mathematics there's a technical sense in which integers aren't real numbers -- we say instead that they can be identified with real numbers." -- Timothy Gowers

1.0 Introduction

Numbers, in some sense, are only defined in mathematics up to isomorphisms. This post runs quickly through some math to explain what this means.

I begin by assuming knowledge of the natural numbers, {0, 1, 2, ...}, as characterized by the Peano axioms. I also assume an understanding of what it means for two natural numbers to be equal, for one to be greater than another, and for two to be added or multiplied together. Other operations could be built on top of this structure, as needed.

2.0 Integers as Equivalence Classes of Ordered Pairs of Natural Numbers

Consider ordered pairs (a, b) of natural numbers. The point of this may be cryptic for a while. Part of the point is to check that no manipulations of these ordered pairs or definitions rely on anything that is not defined in the natural numbers.

I define equality between two ordered pairs:

(a, b) = (c, d)

if and only if

a + d = b + c

At this point, one should check that, by this definition, an ordered pair is equal to itself (reflexive) and that if an ordered pair is equal to another ordered pair, that ordered pair is also equal to the first (symmetry). One also wants to show that this definition is transitive.

if (a, b) = (c, d) and (c, d) = (e, f) then (a, b) = (e, f)

A relation with these three relations is called an equivalence relation. An equivalence relation breaks a set into non-intersecting equivalence classes. All elements of an equivalence class are equal to one another, and no element in the set outside an equivalence class is equal to any element in that class.

Here are three equivalence classes, by this definition: {(1, 0), (2, 1), (3, 2), ...}, { (0, 0), (1, 1), (2, 2), ...}, and {(0, 1), (1, 2), (2, 3), ...}. Is it becoming clear what is going on here?

Next, I want to define a total order:

(a, b) ≥ (c, d)

if and only if:

b + c ≥ a + d

I claim that ≥ is well defined. That is, if

• (a, b) = (a', b')
• (c, d) = (c', d')
• (a, b) ≥ (c, d)

then:

(a', b') ≥ (c', d')

I also want to show that ≥ is reflexive, transitive, and antisymmetric. This relation is antisymmetric if and only if for all ordered pairs of natural numbers,

if (a, b) ≥ (c, d) and (c, d) ≥ (a, b) then (a, b) = (c, d)

Furthermore, the relation is complete. For all pairs of ordered pairs, (a, b) ≥ (c, d) or (c, d) ≥ (a, b). With these properties, ≥ orders equivalence classes, just as well as ordered pairs of natural numbers. And one could use the above to define a relation >.

The next step is to define addition:

(a, b) + (c, d) = (a + c, b + d)

This definition also applies to equivalence classes. The sum of elements from two equivalence classes is in the same equivalence class, whichever elements you start with. Notice that:

(a, b) + (0, 0) = (0, 0) + (a, b)

So one might as well define:

+0 = {(0, 0), (1, 1), (2, 2), ...}

The negative numbers are those equivalence classes less than zero. That is, if (a, b) is in an equivalence class where a > b, then that equivalence class is a negative number. The positive numbers are equivalence classes greater than zero. More on this below.

Addition has some other properties of interest. But I am going to move on to define multiplication:

(a, b) * (c, d) = (ad + bc, ac + bd)

Multiplication is well-defined for equivalence classes. Multiplication also has an identity element:

(a, b)*(0, 1) = (0, 1)*(a, b) = (a, b)

So define +1 as {(0, 1), (1, 2), (2, 3), ...}.

Now, associate every natural number n with an equivalence class:

f( n ) = +n = {{(0, n), (1, 1 + n), (2, 2 + n), ...}

The function f is one-to-one and onto for the set of non-negative integers. It preserves equality:

If n = m, then f( n ) = f( m )

The first equality is a relation in the set of the natural numbers. The second equality was defined above. The function f also preserves order, addition, and multiplication:

• If n ≥ m, then f( n ) ≥ f( m )
• n + m = f( n ) + f( m )
• nm = f( n ) * f( m )

The function f is an isomorphism.

Suppose proofs can be given for propositions stated above. Then I have (unoriginally) constructed the integers out of the natural numbers. Negative numbers are not some mystical entities. In this construction, zero is the additive identity. Every integer has an additive inverse, that is, one can negate every integer. Addition is associative and commutative. The integer +1 is the multiplicative identity, and the additive and multiplicative identities are distinct. Multiplication is commutative and associative. Multiplication distributes over addition. Finally, if (a, b) * (c, d) = +0, then (a, b) = (c, d) = +0.

Any set with two binary operations with these properties is an integral domain. I think the set of polynomials with coefficients that are rational numbers, with the usual definitions of polynomial addition and multiplication, is also an integral domain.

So the natural number 2 has been shown to be isomorphic to the positive integer +2. I think this proof of the first equality in the post title is fairly typical of math. It is not particularly difficult, but requires what is called "mathematical maturity". One has to keep track of what can be inferred from definitions, and not let your intuition leap ahead. Even so, your intution will guide you. When writing the above, I kept thinking of (a, b) as "b - a". I would like to say this is the kind of mathematics that Bertrand Russell amused himself with when the Brits put him in prison during World War I. But this would all come after the proof of 1 + 1 = 2 in Principia Mathematica.

3.0 The Field of Quotients of an Integral Domain

Now consider ordered pairs (a, b) of the elements of an integral domain, where b is not +0, the additive identity for the integral domain. Here, I define equality by:

(a, b) = (c, d) if and only if a*d = b*c

I have put aside the structure of the elements of the integral domain, whether they are equivalence classes of ordered pairs of natural numbers or polynomials with rational coefficients or whatever. Addition is defined by:

(a, b) + (c, d) = (a*d + b*c, b*d)

One can show that this definition is well-defined for equivalence classes. The additive identity is the set of ordered pairs {(0, a), a a non-zero element of the integral domain}. Multiplication is defined as:

(a, b)*(c, d) = (a*c, b*d)

The multiplicative identity is the set of ordered pairs equal to (1, 1). Every non-zero ordered pair (a, b) has the multiplicative inverse (b, a).

Define an isomorphism to a subset of the quotient field defined above:

g( a ) = a/1 = {(c, d), where (c, d) = (a, 1)}

Obviously, I have skipped over a lot of steps. But the above is an outline of how to construct the rational numbers and to prove that +2 = 2/1.

4.0 The Reals as Equivalence Classes of Cauchy-Convergent Sequences of Rationals

The set of real numbers is the set of limit points of all convergent sequences of rational numbers. I guess below is Cantor's approach to constructing the real numbers. Dedekind had another approach, and Rudin (1973) presents this construction based on "cuts".

Consider a sequence of rational numbers (a₀, a₁, a₂, ...) where for all ε greater than zero, there exists a natural number N such that for all n greater than N,

| a_{n + 1} - a_n | < ε

In some sense, the terms in the sequence get closer together. Such a sequence is known as Cauchy-convergent.

Two sequences (a₀, a₁, a₂, ...) and (b₀, b₁, b₂, ...) are equal if and only if for all ε greater than zero, there exists a natural number N such that for all n greater than N,

| a_n - b_n | < ε

This definition will yield equivalence classes. Addition of sequences is defined termwise. Should multiplication also be defined termwise? What about order?

Anyway, for any rational number r, the equivalence class of the sequence (r, r, r, ...) is isomorphic in the reals to that rational number.

5.0 Complex Numbers as a Field Extension

I have already stepped through a construction of the field of complex numbers.

6.0 Observations and Higher Mathematics

So that is what numbers are, in some sense. In each of these constructions, I have attempted to preserve some properties of the more primitive domain. I do not know how much freedom one would want to permit me for the properties of, say, equality addition, or multiplication.

Does "+", the symbol for addition have the same meaning when adding natural numbers, integers, rationals, real numbers, and complex numbers?

"This discussion prompts us to ask: do the arithmetical operations have the same meaning in each of these calculi? For instance, in the domain of integers, is subtraction the same operation as in the domain of natural numbers? Furthermore, what do we mean here by 'the same'? If this implies that the operations must satisfy the same conditions, then the questions must be answered negatively. For in the domain of natural numbers the expression a - b is admissible only if a > b, while in the domain of integers this restriction is removed; obviously this is an important distinction. Consequently, there is not, strictly speaking, one subtraction but as many different operations with this name as there are domains of numbers. We should not be deceived regarding this situation by the fact that we use the same signs +, -, :, etc., at the various levels. If we put the statements of these concepts side by side, it becomes clear how far the analogy between them goes and where it stops." -- Friedrich Waismann (1951: 61).

I am not at all sure, however, that Waismann would answer the question in the post title in the negative:

"In the construction above, the integers were first constructed and then rational numbers. Is there an innate necessity for this sequential order? Couldn't we first introduce the rational numbers without signs and then the distinction between positive and negative numbers? Certainly! We would not thereby obtain another system of rational numbers; rather the system so constructed would prove to be isomorphic to the one considered above, since every relation of one system could be mapped on a similarly constructed relation in the other, and conversely." -- Friedrich Waismann (1951: 65).

Waismann famously drew on Wittgenstein in developing his views, though Wittgenstein might have come to disavow them.

Some advanced mathematics raises further questions about how symbols can have meaning. Gödel's first incompleteness theorem seperates the notion of provable from a set of axioms from the question of truth. According to his second incompleteness theorem, consistency, if true, cannot be proven within a sufficiently interesting system of axioms and derivation rules. The Löwenheim-Skolem theorem shows that axioms are not enough to fix the meaning of mathematical objects. I was explaining to a colleaque that a power set is always "bigger" than the original set. And I raised the question of whether the set of all subsets of the natural numbers can be put in a one-to-one correspondence with the reals. (I can understand what the quesion is, at least, given a naive acceptance of the construction of the reals.) I told my colleague that it was not clear what the answer was or even what it would mean for the question to have an answer. Apparently, Gödel thought that mathematicians might one day come to agree on an answer.

I think how mathematical terms are used in these systems affects the meaning of mathematical terminology in ordinary life. Some strict separation of these uses cannot be drawn, as far as I am aware. I conclude by noting that some classics (for example, Putnam 1981 and Kripke 1982) of Anglo-American analytical philosophy argue difficulties in understanding how mathematics means extends to (much of?) the remainder of language.

References

• John B. Fraleigh. 2003. A First Course in Abstract Algebra, 7th edition.
• Kurt Gödel. 1947. What is Cantor's continuum problem? American Mathematical Monthly 54: 515-525. Reprinted in Kurt Gödel: Collected Works, Volume II.
• Saul A. Kripke. 1982. Wittgenstein on Rules and Private Language Harvard University Press
• Hilary Putnam. 1981. Reason, Truth and History Cambridge University Press.
• Walter Rudin. 1976. Principles of Mathematical Analysis, 3rd edition
• Friedrich Waisman. 1951. Inroduction to Mathematical Thinking: The Formulation of Concepts in Modern Mathematics.

2 + 2 = 5

"One must be able to say at all times - instead of points, straight lines, and planes -- tables, chairs, and beer mugs." -- David Hilbert (as quoted by Constance Reid, Hilbert, Springer-Verlag, 1970: p. 57)

Consider the Fibonacci sequence: 1, 2, 3, 5, 8, 13, ... The first two terms in this sequence are 1 and 2. After this, each term is the arithmetic sum of the previous two terms. Let A be the set of elements in this series. Let s be the function mapping an element in A onto another element of A, where s(n) is the term in the sequence following n. I want to read s as "successor". (I do not claim originality for the following use of the Fibonacci series.)

Notice that the set A and the successor function satisfy the following properties:

1. 1 is in the set A.
2. For any n in the set A, its successor s(n) is in the set A
3. There does not exist an elment n of A such that its successor s(n) = 1.
4. For all n and m in A, if s(n) = s(m), then n = m.
5. Suppose a set B contains 1. And further suppose that for any n in B, the successor s(n) is also in B. Then the set A is a subset of B.

The above five properties are the Peano Axioms. The last property is known as the principle of induction.

Some, such as the formalist David Hilbert, would say that mathematical objects are defined by axioms. And the above axioms define the set of natural numbers. Taking the succesor of a number is defined as adding one. And you can define taking the succesor of the successor of a number as adding 2. So, by these definitions, one has:

1 + 1 = s(1) = 2

2 + 2 = s(s(2)) = 5

(By the way, Hilbert was what the kids these days call "woke":

"David Hilbert sought to bring [Emmy Noether] into the mathematics department at the University of Göttingen in 1915, but other faculty objected. 'What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?' one professor complained. Hilbert was indignant. 'I do not see that the sex of the candidate is an argument against her admission,' he retorted. 'We are a university, not a bath house.' (See here.))

Bertrand Russell objected to formalism. He thought that whatever mathematicians meant by numbers, these meaning should allow us to assert that most people have one nose, two eyes, and five fingers on each hand. I find his Introduction to Mathematical Philosophy the most approachable statement of his logicist position. Russell found a way to define numbers based on set theory. He required the universe to contain a countable infinity of things, in some sense. His theory of types was a weakness of his approach. Frank Ramsey comes into the story here, with a theory I do not understand. Anyways, various ways can be found to define number in Zermelo-Fraenkel set theory, which I gather avoid the problems with the theory of types.

But this story is incredible. Did people not know how to count or what they were doing until the twentieth century? Maybe a logical foundation for mathematics is not needed. Rather, one might try to provide an anthropological description of what people do when they are counting, adding, multiplying, manipulating infinite series, and so on. Does this capture the "must" in following a rule, though? When a judge consults a law book, he does not treat it as a work in anthropolgy. And who is in doubt on how to proceed when they come to a sum they have never calculated before? Here I am gesturing towards the work of Ludwig Wittgenstein.

I am near my intellectual limits, but I know I have hardly exhausted positions that have been taken over the last century on the philosophy of mathematics. By the way, if I take Jean Piaget seriously, the most advanced mathematics some tried to teach me is crucial to understanding how children think and important for anybody that wants to attempt to design curricula for mathematics.

This post should have a bibliography.

Theses For Debate In Reading Marx

I present four claims about Marx's Capital. I strive for topics more general than, for example, squabbles about the transformation problem. I suggest that some of these claims present a useful focus for reading Marx's book, even if part of your focus is arguing why the claim is wrong. If this were more than a blog post, I would need to cite various Marxists and scholars that inspired me.

Thesis I: Capital is organized around a model of a pure, two-class capitalist economy.

I think the above claim is helpful in making sense of the opening chapters of Volume 1 and of Volume 2. In Volume 2, I am thinking of the analysis of the analysis of various circuits, as well as the models of simple and expanded reproduction.

This claim separates out the historical material and the analysis more sharply than some commentators on Marx accept. I guess it is consistent with some of Marx's use of Blue Books filed by factory inspectors in Britain. Historical material that goes beyond a model of pure capitalism includes the analyses of primitive accumulation in pre-capitalist formations and of the development of machinery and manufacture. I think of the replacement of the putting-out system, handicraft, and domestic industry by factories.

Thesis II: Capital continues the tradition of classical political economy; it does not represent a sharp break with this tradition.

One can argue Marx saw William Petty, Francois Quesnay, Adam Smith, and David Ricardo, for example, as having applied a scientific method of abstraction to identify essences that lie behind the surface phenomena of market prices. Of course, Marx had many criticisms of his predecessors. He thought Smith had not sufficiently distinguished labor that was and was not productive of surplus value. Even Ricardo did not distinguish (abstract, social) labor from labor power. Marx argued his distinction between constant and variable capital was more fundamental, in some sense, that the classical political economy distinction between fixed and circulating capital. And the classical did not talk about surplus value in general, instead of manifestations in the form of profits, interest, and rent.

This claim of continuity can also be argued to be consistent with Marx's contrast of vulgar and scientific political economy. Not everybody in the time of the classics, including Adam Smith, were thoroughgoing in the application of their scientific method.

But some of what Marx has to say about illusions generated by competition is in tension with this claim of continuity. He was interested in what social conditions made possible the development of political economy. The classical political economists championed the rising bourgeois before the social question became sufficiently biting. And what about the sarcasm and irony in Capital.

Thesis III: The system of labor values is a reality behind the appearance of freedom in market transactions.

In some sense, labor values provide a sub-basement underlying a building more obvious to our sight.

A counter thesis would be based on a Wittgenstein-like reading of Capital. Nothing is hidden, but markets, like languages, are befuddling. Marx is presenting arrangements in a therapeutic treatment to dissolve confusions. This also gets into some readings of Sraffa's work.

Thesis IV: One can accept the analysis in Capital as a way of understanding the world, independently of a any position on the desirability of changing it, either through a revolution or otherwise.

Classification of Finite Simple Groups: A Proved Theorem?

Figure 1: Lattice Diagram for Group of Symmetries of the Square

"I shall now mention something I obviously do not understand." - Ian Hacking (2014, p. 18)

1.0 Introduction

This has nothing to do with economics. It is my attempt to get my mind around a place where I can get a glimmer of some exciting mathematics being done in my lifetime.

Mathematicians have stated a theorem for classifying finite simple groups. Whether they have proven this theorem is an intriguing question in the philosophy of mathematics.

A finite simple group is a group with a finite number of elements and no proper normal subgroup. This definition contains several technical terms. In this post, I try to explain these terms and the setting of the theorem for classifying simple groups. This preamble raises several questions:

• What is a group? A proper subgroup? A normal subgroup?
• How can a finite, non-simple group be factored into a composition of simple groups?

I try to clarify the answers to these questions by means of a lengthy example. You can probably find this better expressed elsewhere. In working this out, I relied heavily on Fraleigh's textbook, which is the only book in the references that I have read, albeit mostly in the second edition.

2.0 The Group of Symmetries of the Square

A group is a generalization, in some sense, of a multiplication table. Formally, it is a set with a binary operation, in which the binary operation satisfies three axioms. A finite group is a group in which the set contains a finite number of elements.

To illustrate, I consider the set of symmetries of the square (Figure 2). These eight elements of the set are like the numbers along the top and left side of a multiplication table. Each element is an operation that can be performed on a square, leaving the square superimposed on itself. Each operation is described in the right column of Figure 2. The third column provides a picture of the operation. The four vertices of the square are numbered so that one can see the result of the operation. The second column specifies each operation as a permutation of the numbered vertices. The first row in each permutation lists the vertices, while the second row shows which of the original vertices ends up in the place of each vertex. The first column introduces a notation for naming each operation. The remainder of this post is expressed in this notation.

Figure 2: Elements of a Group

The group operation, *, is function composition. Let a and b be elements of the set {ρ₀, ρ₁, ρ₂, ρ₀, μ₀, μ₁, σ₀, σ₁}. The product a*b is defined to be the single operation that is equivalent to first performing the operation a on the square and then performing the operation b on the result. (Many textbooks define functional composition from right-to-left, instead.) Table 1 is the multiplication table for this group, under these definitions. For example, rotating a square 90 degrees clockwise twice is equivalent to rotating the square clockwise through 180 degrees. Thus:

ρ₁ * ρ₁ = ρ₂

Table 1: The Group D₄

*	ρ0	ρ1	ρ2	ρ3	μ0	μ1	σ0	σ1
ρ0	ρ0	ρ1	ρ2	ρ3	μ0	μ1	σ0	σ1
ρ1	ρ1	ρ2	ρ3	ρ0	σ0	σ1	μ1	μ0
ρ2	ρ2	ρ3	ρ0	ρ1	μ1	μ0	σ1	σ0
ρ3	ρ3	ρ0	ρ1	ρ2	σ1	σ0	μ0	μ1
μ0	μ0	σ1	μ1	σ0	ρ0	ρ2	ρ3	ρ1
μ1	μ1	σ0	μ0	σ1	ρ2	ρ0	ρ1	ρ3
σ0	σ0	μ0	σ1	μ1	ρ1	ρ3	ρ0	ρ2
σ1	σ1	μ1	σ0	μ0	ρ3	ρ1	ρ2	ρ0

A group is defined by the following three axioms:

• The binary operation in the group is associative. That is, for all a, b, and c in the group:

(a * b) * c = a * (b * c)

• The group contains an identity element. There exists an element e in the group such that for all a in the group:

e * a = a * e = a

• Every element of the group has an inverse. For all a in the group, there exists an element a^-1 in the group such that:

a * a^-1 = a^-1 * a = e

Associativity is tedious to check for D₄. Associativity implies that one can drop parenthesis below. ρ₀ is the identity element. Every row and column in the multiplication table for D₄ contains ρ₀; thus, every element has an inverse.

An Abelian group is one in which the binary operation is commutative. The group of symmetries of the square is not Abelian. For an Abelian group, the multiplication table is symmetric across the principal diagonal; it does not matter to the result in which order one performs the operation for two arguments. The following two equations illustrates that D₄ is not Abelian:

μ₀*ρ₁ = σ₁

ρ₁*μ₀ = σ₀

In words, flipping a square around its horizontal axis of symmetry and then rotating it ninety degrees clockwise is not equivalent to rotating it ninety degrees clockwise and then then reflecting it across that axis. The result of the first composition of operations is equivalent to reflecting the square across the diagonal axis of symmetry running from the south west to the north east. The second composition of operations is equivalent to flipping the square across the other diagonal.

One can also set up equations in a group, for example:

ρ₁*ρ₂*x = μ₀

Then x must be σ₀. Solving a Rubik's cube is analogous to solving such an equation.

3.0 Proper and Improper Subgroups

Some rows and columns in Table 1 can stand alone as a group. The entries in these restricted row and columns all appear as headings in the rows and columns. These entries form a subgroup of the original group. One-fourth of the table in the upper left of Table 1 provides an example. {ρ₀, ρ₁, ρ₂, ρ₃} is a subgroup of D₄ (Table 2).

Table 2: A Subgroup of D₄ with Four Elements

*	ρ0	ρ1	ρ2	ρ3
ρ0	ρ0	ρ1	ρ2	ρ3
ρ1	ρ1	ρ2	ρ3	ρ0
ρ2	ρ2	ρ3	ρ0	ρ1
ρ3	ρ3	ρ0	ρ1	ρ2

The group D₄ has ten subgroups, as shown in the Lattice Diagram in Figure 1 above. Subgroups have been defined such that, for any group G, the group G is a subgroup of itself. Another trivial case, the one-element group consisting of the identity element, also provides a subgroup of G. These two subgroups are known as improper subgroups. All other subgroups are proper subgroups.

One can make a couple of observations about subgroups. The binary operation in the group is the same as the binary operation in the subgroup. The property of associativity carries over from the group to the subgroup. Since a subgroup is a group, it must contain an identity element. And that identity element must also be the identity element for the group containing the subgroup. Thus, every subgroup of D₄ contains ρ₀. Likewise, for every element of a subgroup, the subgroup must also contain its inverse. Finally, the number of elements in a subgroup must evenly divide the number of elements in the group.

I have shown above how the eight elements of D₄ can be defined in terms of permutations. As a matter of fact, the set of permutations of (1, 2, ..., n) form a group under the operation of function composition. This permutation group is designated as S_n, and it contains n! elements. Thus, S₄ contains 24 (= 4x3x2x1) elements. Not only can one find all the subgroups of D₄, one can extend the group such that D₄ is a subgroup of that extended group.

4.0 Isomorphic Groups

In a group, the order of rows and columns in the multiplication table are of no matter. Likewise, the names of the elements are irrelevant to the structure of the group. Two groups are isomorphic if the multiplication table for one group can be mapped into the multiplication table for another group by reordering and renaming the elements of, say, the first group. As an example, consider the groups {ρ₀, ρ₂, μ₀, μ₁} and {ρ₀, ρ₂, σ₀, σ₁}. They each have the same number of elements, which is necessary for an isomorphism. Table 3 defines the group operation for the first group. Suppose that, in Table 3, μ₀ is renamed σ₀, and μ₁ is renamed σ₁ throughout. The resulting table will match the operation for the second group. Thus, the two groups are isomorphic.

Table 3: The Group {ρ₀, ρ₂, μ₀, μ₁}

*	ρ0	ρ2	μ0	μ1
ρ0	ρ0	ρ2	μ0	μ1
ρ2	ρ2	ρ0	μ1	μ0
μ0	μ0	μ1	ρ0	ρ2
μ1	μ1	μ0	ρ2	ρ0

The groups in Tables 2 and 3 are NOT isomorphic. They each contain four elements. Each element, however, in the group in Table 3 is its own inverse. This is an algebraic property, preserved no matter how the elements of the group are renamed. And the group in Table 2 does not have this property. As a matter of fact, only two groups containing four elements exist, up to an isomorphism. In other words, any group with four elements is isomorphic to either the group in Table 2 or to the group in Table 3.

Furthermore, only one group, up to isomorphism, contains two elements. Its operation is defined by Table 4. All the subgroups of D₄ containing two elements are isomorphic to this group and, ipso facto, to each other. The text colors of the subgroups in the lattice diagram (Figure 1) express these isomorphisms.

Table 4: The Unique Group (Up To Isomorphism) With Two Elements

*	0	1
0	0	1
1	1	0

5.0 Normal Subgroups, Factor Groups, and Homomorphisms

Certain additional patterns are apparent in Table 1. I have already pointed out that the first four rows and columns constitute the subgroup with the operation shown in Table 2. Notice that none of the entries in the last four columns for the first four rows are in this subgroup. Likewise, none of the entries in the first four columns for the last four rows are in this subgroup. On the other hand, the entries in the remaining rows and columns in the lower right are all in this subgroup. Can you see that these observations reveal the pattern expressed in Table 4? Mathematicians express this by saying that the factor group D₄/{ρ₀, ρ₁, ρ₂, ρ₃} is isomorphic to the group with two elements.

A subgroup is normal if it can be used to divide up the rows and columns in the multiplication table for the group like this. For another example, consider the subgroup {ρ₀, ρ₂}. Table 5 shows a reordering of the rows and columns in Table 1 to facilitate the calculation of the factor group for this subgroup. Consider dividing this grid up into 16 blocks of two rows and two columns each. Each block will contain two elements of the group D₄, and which element is paired with each element does not vary among these blocks.

Table 5: The Group D₄ Reordered

*	ρ0	ρ2	ρ1	ρ3	μ0	μ1	σ0	σ1
ρ0	ρ0	ρ2	ρ1	ρ3	μ0	μ1	σ0	σ1
ρ2	ρ2	ρ0	ρ3	ρ1	μ1	μ0	σ1	σ0
ρ1	ρ1	ρ3	ρ2	ρ0	σ0	σ1	μ1	μ0
ρ3	ρ3	ρ1	ρ0	ρ2	σ1	σ0	μ0	μ1
μ0	μ0	μ1	σ1	σ0	ρ0	ρ2	ρ3	ρ1
μ1	μ1	μ0	σ0	σ1	ρ2	ρ0	ρ1	ρ3
σ0	σ0	σ1	μ0	μ1	ρ1	ρ3	ρ0	ρ2
σ1	σ1	σ0	μ1	μ0	ρ3	ρ1	ρ2	ρ0

These observations can be formalized by the function defined in Table 6. For an element a of D₄, let f(a) denote the map defined in Table 6. To find the value of this function, locate a in the first column. Whether this value is 0, 1, 2, or 3 is determined by the corresponding entry in the second column. For all a and b in D₄:

f(a * b) = f(a) o f(b)

A map from one group to another with this property is a homomorphism. An isomorphism is a homomorphism, but a homomorphism is a more general concept. Homomorphisms do not need to leave the number of elements in the group invariant.

Table 6: A Homomorphism from D₄ to {0, 1, 2, 3}

Elements of D4	Image
ρ0, ρ2	0
ρ1, ρ3	1
μ0, μ1	2
σ0, σ1	3

The factor group D₄/{ρ₀, ρ₂} is easily calculated. Replace each element of D₄ in Table 5 by its image under the homomorphism in Table 6. Collapse each pair of rows and columns. One ends up with Table 7, where I have renamed the group operation, as above. The factor group D₄/{ρ₀, ρ₂} is isomorphic to the group with four elements with the operation shown in Table 3 above. The number of elements in a factor group is the quotient of the number of elements in the original group and the number of elements in the subgroup used to form the factor group.

Table 7: The Factor Group D₄/{ρ₀, ρ₂}

o	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

The two improper subgroups for any group are normal and yield trivial factor groups. The factor group D₄/D₄ is isomorphic to the one-element group whose only member is the identity element. The factor group D₄/{ρ₀} is isomorphic to D₄. The factor groups for improper subgroups provide no information about the structure of a group.

6.0 A Subgroup that is Not Normal

Not all subgroups are normal. The subgroup {ρ₀, μ₀}, for example, is not a normal subgroup of D₄. Table 8 proposes a map from the elements of the group to the first four natural numbers. And Table 9 illustrates another reordering of the rows and columns in Table 1, with the entries replaced by the natural numbers to which they map. If one confines oneself to the first two columns, each pair of rows could be collapsed into one, with the label from the row taken from the map. But this process breaks down for the next two and the last two columns.

Table 8: A Map from D₄ to {0, 1, 2, 3} that is Not a Homomorphism

Elements of D4	Image
ρ0, μ0	0
ρ1, σ0	1
ρ2, μ1	2
ρ3, σ1	3

Table 9: Another Reodering of The Group D₄

*	ρ0	μ0	ρ1	σ0	ρ2	μ1	ρ3	σ1
ρ0	0	0	1	1	2	2	3	3
μ0	0	0	3	3	2	2	1	1
ρ1	1	1	2	2	3	3	0	0
σ0	1	1	0	0	3	3	2	2
ρ2	2	2	3	3	0	0	1	1
μ1	2	2	1	1	0	0	3	3
ρ3	3	3	0	0	1	1	2	2
σ1	3	3	2	2	1	1	0	0

Suppose a subgroup contains n elements. To determine if the subgroup is normal, it is sufficient to examine the first n rows and the first n columns in the reordered table. This capability follows from a theorem about what are known as left and right cosets for a subgroup.

The permuation group S₄ provides another example of a subgroup that is not normal. By my calculations, D₄ is NOT a normal subgroup of S₄.

7.0 The Composition Series of a Group

At this point, I have completed my explanation of the lattice diagram at the top of this post, including circles, text colors, and boxes. I draw from these results to illustrate how a non-simple group, namely D₄, can be expressed as a composition of factor groups.

Table 10 lists twelve series of subgroups of the group of symmetries of the square. Each series has the following properties:

• The leftmost group in the series is the one-element group containing the identity element.
• The rightmost group is D₄.
• Each group in the series (except D₄) is a proper normal subgroup of the group immediately to the right of it in the series.

A series with these properties is known as a subnormal series of the group D₄. If every group in the series is also a normal subgroup of D₄, the series is a normal series of the group D₄. By the last property in the bulleted list, one can calculate a factor group for each pair of immediately successive groups in the series.

Table 10: Twelve Normal and Subnormal Series for D₄

Number Factor Groups	Series	Normal Series
1	{ρ0} < D4	Yes
2	{ρ0} < {ρ0, ρ1, ρ2, ρ3} < D4	Yes
2	{ρ0} < {ρ0, ρ2, μ0, μ1} < D4	Yes
	{ρ0} < {ρ0, ρ2, σ0, σ1} < D4	Yes
	{ρ0} < {ρ0, ρ2} < D4	Yes
3	{ρ0} < {ρ0, ρ2} < {ρ0, ρ1, ρ2, ρ3} < D4	Yes
	{ρ0} < {ρ0, ρ2} < {ρ0, ρ2, μ0, μ1} < D4	Yes
	{ρ0} < {ρ0, ρ2} < {ρ0, ρ2, σ0, σ1} < D4	Yes
	{ρ0} < {ρ0, μ0} < {ρ0, ρ2, μ0, μ1} < D4	No
	{ρ0} < {ρ0, μ1} < {ρ0, ρ2, μ0, μ1} < D4	No
	{ρ0} < {ρ0, σ0} < {ρ0, ρ2, σ0, σ1} < D4	No
	{ρ0} < {ρ0, σ1} < {ρ0, ρ2, σ0, σ1} < D4	No

The definition of an isomorphism for a subnormal series builds on the definition of isomorphism for groups. Consider the factor groups arising in each series from successive pairs of subgroups in each series. Two series are isomorphic if they contain the same of number of factor groups, in this sense, and these factor groups are isomorphic. The order in which the factor groups arise can vary among isomorphic subnormal series.

I have collected isomorphic series together, in Table 10, by means of horizontal lines in the first column. The series with one factor group is not isomorphic to any other series. The first series shown with two factor groups is not isomorphic to the other three series with two factor groups. And those three series are isomorphic to one another. All of the series with three factor groups are isomorphic to one another.

The series with three factor groups have another property. All factor groups in these series with three factor groups are simple groups. That is, they contain no proper normal subgroups. A subnormal series of a group in which all factor groups formed by the series are simple is known as a composition series. By the Jordan-Hölder Theorem, all compositions series for a group are isomorphic series. This theorem justifies one in speaking of THE composition series for a group. Finding the factor groups in a the composition series for a group is somewhat analogous to factoring a natural number. Note that D₄ contains eight elements and each of the three factor groups in the composition series contain two elements. Furthermore,

8 = 2³

For a natural number, the prime factors can be combined to yield the original number. Here the analogy apparently breaks down. The factor groups in a composition series for a group constrain the structure of the group, but two non-isomorphic groups can have the same composition series. But still, mathematicians have solved various problems in group theory for finite non-simple groups by use of the classification of finite simple groups.

Composition series apparently have an application in solving polynomial equations. The composition series for the permutation group S₅ contains a factor group that is non-Abelian. This is connected with the insolvability of the quintic. There are formulas for zeros for cubic and fourth order polynomial, analogous to the quadratic formula. But there is no such formulas for poynomials of the fifth degree and higher.

8.0 Classification of Finite Simple Groups

At this point, I have explained how finite simple groups arise as factor groups for the composition series of any finite group. I hope that this gives some hint of why the following theorem is of interest.

Theorem: Each finite simple group is one of the following, up to an isomorphism:

• A group of prime order.
• An alternating group.
• A Lie group.
• One of 26 sporadic groups not otherwise classified.

I am aware that this this theorem uses technical terms I still have not explained, including one that I simply do not understand myself.

The sporadic groups are finite simple groups that do not fall into the other categories, although, I gather, some sporadic groups are related to one another.The sporadic group with the largest number of elements is called the Monster group. It has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements.

9.0 History of the Theorem

In 1972, Daniel Gorenstein proposed that mathematicians could complete a classification of all simple groups. By the early 1980s, mathematicians had stated the theorem and those specialists who had pursued Gorenstein's program believed they had proven it. The proof, however, was scattered among (tens of?) thousands of pages in hundreds(?) of papers in many mathematics journals. No one person had probably ever understood the proof or read it in its entirety.

The proof, however, was discovered even then to be incomplete. Steve Smith and Michael Aschbacher worked on closing this gap, relating to quasithin groups. They succeeded by 2004.

Meanwhile, a number of mathematicians have been trying to simplify the proof and to restate it in one location. The ambition of these mathematicians is to produce a "second generation" proof of only a couple thousand pages.

Has a theorem been proven if only one or two mathematicians have read the proof in its entirety? How about if nobody has, which would have been the case in the 1980s if the proof had indeed been valid? Certainly, the proof of the classification theorem is not surveyable, in Wittgenstein's sense. Do mathematical results need to be established by a social process? If so, how can such social processes be characterized?

Appendix: Terms Defined or Illustrated Above

Abelian group, Associativity, Composition Series, Factor Group, Finite Group, Group, Homomorphism, Identity Element, Improper Subgroup, Inverse, Isomorphic Groups, Isomorphic Subnormal Series, Lattice Diagram, Normal Series, Normal Subgroup, Permutation Group, Proper Subgroup, Subgroup, Subnormal Series.

References

• Michael Aschbacher (2004). The Status of the Classification of the Finite Simple Groups, Notices of the AMS, V. 51, No. 7 (Aug.): pp. 736-740.
• Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon (2011). The Classification of Finite Simple Groups: Groups of Characteristic 2 Type, American Mathematical Society.
• Nicolas Bourbaki (1943). Elements of Mathematics: Algebra I: Chapters 1-3.
• J. H. Conway and S. P. Norton (1979). Monstrous Moonshine, Bulletin of the London Mathematical Society, V. 11, no. 3: pp. 308-339.
• John B. Fraleigh (2002). A First Course in Abstract Algebra, 7th Edition, Pearson.
• Daniel Gorenstein, Richard Lyons, and Ronald Solomon (1994). The Classification of the Finite Simple Groups, American Mathematical Society.
• Ian Hacking (2014). Why is there Philosophy of Mathematics at all?, Cambridge University Press.
• Daniel Kunkle and Gene Cooperman (2007). Twenty-Six Moves Suffice for Rubik's Cube, ISSAC'07, 29 Jul. - 1 Aug., Waterloo, Canada.
• Tomas Rokicki (2008). Twenty Five Moves Suffice for Rubik's Cube.

Obscure Postmodern Language

I try here to outline certain postmodern¹ doctrines that, in a full development, might result in one using obscure terminology. None of this is to say that every postmodern writer using polysyllabic terminology is expressing complicated ideas in the most effective way. Nor do I want to argue that it is impossible to ever write clearly² about (some subset) of these ideas.

People have a tendency towards reification³, towards talking as if certain abstract ideas are concrete realities. For example, they might tend to confuse relationships between people with relationships between things⁴. And people tend to think dualistically, or at least to categorize things into pre-existing categories. And with dividing things into two categories, one may tend to elevate one over the other, or to define the inferior in terms of the negation of the properties of the superior⁵. One might think that these confusions become embedded in our language⁶. It is not as if we have access to a language appropriate for a "view from nowhere", where nature is carved at its joints⁷.

Furthermore, current classifications and fundamental ideas embodied in current language have a history; our current language does not reflect how people always thought. In looking at past patterns of language and governance, one should try not to read our current way of thinking into the past⁸.

One might also think current classifications have a functional relationship to class structure, hegemonic⁹ ethnicities, patriarchal relationships, or whatever¹⁰.

I have deliberately been abstract here. But, I suppose, I might mention some examples. In economics, I think one is confused if one looks at capitalism as catallaxy, that is, purely in terms of market relationships, in which all parties are free. Furthermore, many things have been said to be socially constructed. I think here of money¹¹, race¹², gender¹³, and sex¹⁴.

In fully trying to explicate these ideas, one can be expected to struggle with bewitchments brought about by language. One might look for multivocalities in past texts. How have current suppositions been read into them? How might they be read from a subaltern position? How might language be expanded so as not to deny normalcy to currently marginalized groups? So reasons exist why academics thinking along postmodern trends might express themselves obscurely.

The above is not to say that these ideas cannot be criticized¹⁵.

Update (21 December 2015):

• Am I agreeing or disaggreeing with what Robert Paul Wolff says here?
• Noah Smith has a knee-jerk reaction to postmodernism.
• The blogger with the pseudonym "Lord Keynes" has often complained about left-leaning postmoderns.

Footnotes

1. For purposes of this post, I do not distinguish between deconstruction, post structuralism, various trends in the social studies of science, etc.
2. Richard Rorty is an example of a postmodern philosopher known for clear - but not necessarily easy - writing.
3. The popularity of the term "reification", in postmodern discourse, comes from Georg Lukás.
4. This is how Marx defined commodity fetishism.
5. I am thinking of how Simone de Beauvoir, early in The Second Sex, describes women being defined as the Other.
6. Here I point to Ludwig Wittgenstein's later work, unpublished in his lifetime.
7. I guess this relates to Jacques Derrida's claim, "There is no outside the text."
8. Michel Foucault, in particular, offers provocative studies of changing European thought in the classical age, between the Renaissance and the nineteenth century.
9. The popularity of the term "hegemony", in postmodern discourse, comes from Antonio Gramsci.
10. As Marx said, "The ruling ideas are the ideas of the ruling classes."
11. This is an example of how something can both be socially constructed and real. Obviously, money has quite real effects in modern societies.
12. Think of the use of the words "Black" and "Colored" in South Africa and in the USA. In the former, they are not synonyms, while among older Americans of a certain sort, they are.
13. I gather Judith Butler originated the concept of gender as performative.
14. Judith Butler also questions whether sex is necessarily a biological division. People might be classified based on chromosomes, hormones, genitalia, and secondary sex characteristics. More than two categories exist in many of these classifications, and they do not always line up. Philip Mirowski observes somewhere that, for the International Olympic Committee (and the International Association of Athletics Federations), these classifications are a quite practical issue. After all, they are structured to find exceptional humans.
15. For explicit references below, I only give critiques. I am sympathetic to the idea that the popularity of postmodernism among academics was connected to an inability to successfully improve material conditions for many.

References

• Samir Amin (1998). Spectres of Capitalism: A Critique of Current Intellectual Fashions, Monthly Review Press.
• Terry Eagleton (1996). The Illusions of Postmodernism, Blackwell.

Bertrand Russell, Crank

On the Post Topic

Some great thinkers compare their work to the works of Nicolaus Copernicus or of Galileo:

"The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds must be abandoned as beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion." -- Bertrand Russell, Our Knowledge of the External World as a Field For Scientific Method in Philosophy (1914).

"...an imagination better stocked with logical tools would have found a key to unlock the mystery. It is in this way that the study of logic becomes the central study in philosophy: it gives the method of research in philosophy, just as mathematics gives the method in physics. And as physics, which, from Plato to the Renaissance, was as unprogressive, dim, and superstitious as philosophy, became a science through Galileo's fresh observation of facts and subsequent mathematical manipulation, so philosophy, in our own day, is becoming scientific through the simultaneous acquisition of new facts and logical methods.

In spite, however, of the new possibility of progress in philosophy, the first effect, as in the case of physics, is to diminish very greatly the extent of what is thought to be known. Before Galileo, people believed themselves possessed of immense knowledge on all the most interesting questions in physics. He established certain facts as to the way in which bodies fall, not very interesting on their own account, but of quite immeasurable interest as examples of real knowledge and of a new method whose future fruitfulness he himself divined. But his few facts sufficed to destroy the whole vast system of supposed knowledge handed down from Aristotle, as even the palest morning sun suffices to extinguish the stars. So in philosophy: though some have believed one system, and others another, almost all have been of opinion that a great deal was known; but all this supposed knowledge in the traditional systems must be swept away, and a new beginning must be made, which we shall esteem fortunate indeed if it can attain results comparable to Galileo's law of falling bodies." -- Bertrand Russell, ibid.

The "new logic" Russell refers to is set out in, for example, Russell and Whitehead's Principia Mathematica. So Russell is comparing himself to Galileo.

An Approach to a Book Review

I'm glad I read this book, although I think it is basically mistaken. Not surprisingly, given their interactions at Cambridge before World War II, Russell's exposition reminds me of Ludwig Wittgenstein's Tractatus Logico-Philosophicus. Although clearly written, Russell's book has a quite different literary style than Wittgenstein's gnostic utterances and hierarchical structure. Both argue that everyday observations about, say, tables and chairs, should be decomposed into logical conjunctions, negations, and disjunctions of atomic facts, which cannot be further broken down. Russell and Wittgenstein differ on the nature of these atomic facts. For Wittgenstein, the referents for entities in atomic facts are quite mysterious; the specification of what these entities are is not a matter of logic, but of its application. Russell is quite clear that these entities include unintegrated sensations, something like "red patch here now."

Russell outlines how one might combine statements about such entities to construct entities that we see, hear, taste, smell, or feel. He goes on to analyze claims about other minds. The analysis of time leads to comments on Zeno's paradoxes and the mathematical theory of continuity. He also explains the idea of infinity, explaining the then recent theory of Cantor. He tries to present a popular overview of these topics. He acknowledges that some of his exposition is more mathematics than philosophy. But, as you can see above, he thinks previous philosophers and many of his contemporaries stumbled into error because they did not possess these logical and mathematical tools. For later developments along the lines, I gather one can look at such works of logical positivism as Rudolf Carnap's The Logical Structure of the World. I have never read Carnap, but I have read A. J. Ayer's Language, Truth, and Logic.

I recently stumbled somewhere across an argument that Noam Chomsky's approach to linguistics supercedes Russell's application of logic to philosophy. Russell and Chomsky agree that sentences of very different structures can have a close surface appearance, and that the same structure can be exhibited in sentences of different surface appearances. In deciding whether or not propositions are true, or even make sense, one should supposedly concentrate on the meaning captured by this deeper structure. But in trying to analyze the meaning of such propositions as, "The king of France is bald", Russell takes an a priori approach. The adequacy of grammar, however, to characterize sentences in a language is an empirical question. And semantics should be based on the parse trees derived from grammatical analysis of the surface appearances of language, not a logical analysis of the surface appearance. This approach, as I understand it, is analogous to how compilers operate. They apply a semantic analysis to a computer program only after first completing a parsing phase. And Chomsky's approach, I gather, has been influential in Artificial Intelligence.

One can argue that just as Wittgenstein, in Philosophical Investigations, showed his earlier approach in the Tractatus was mistaken, so he also showed Chomsky's approach in linguistics to be mistaken. A fortiori, AI is not possible either. Exposition of the parallelism between Russell and Chomsky's analysis of language makes these claims a bit more clear to me. (I guess Sraffa was not too impressed by Chomsky, either.) I suppose one might look at Norman Malcolm's Wittgenstein: Nothing is Hidden, for a fuller argument against Chomsky along these lines. (I did not get much out of Malcolm when I read him years ago.)

Illusions Generated By Markets Like Those Created By Language On Holiday

I have been reading a book, edited by Gavin Kitching and Nigel Pleasants, comparing and contrasting Ludwig Wittgenstein and Karl Marx. This is the later Wittgenstein of the Philosophical Investigations, not of the Tractatus. The authors of the papers from the conference generating this work do not seem too concerned with arguments about the differences between the young Marx and the mature Marx, albeit many quote a passage from the German Ideology about language. (I think this post is more disorganized than many others here.)

Anyways, I want to first consider a reading of Capital, consonant with the approach of Friedrich Engels and the Second International, but at variance with an analogy to Wittgenstein's later philosophy. One might think of the labor theory of value as a scientific approach revealing hidden forces and structures that are at a deeper level than observed empirical reality. Think about how, for example, physicists have an atomic theory that explains why tables are hard and water is wet. Even though a table may be seem solid, we know, if we accept science, that it is mostly empty space. Somewhere Bertrand Russell writes something like, "Naive realism leads to physics, and physics shows naive realism is wrong. Hence naive realism is false". Similarly, you may think purchases and sales on markets under capitalism are made between equals, freely contracting. But the science of Marxism reveals an underlying reality in which the source of profits is the exploitation of the workers.

Wittgenstein, in rejecting his early approach to language, rejects the idea of a decontextualized analysis of the sentences of our languages into an ultimate underlying uniform atomic structure which explains their meaning. Rather, in his later philosophy, he gathers togethers descriptions of the use of language, to dispel and dissolve the illusions characteristic of traditional philosophy. He is hostile to ideal of an ultimate essence for meaning, and points out the multifarious uses to which language is put. Some of his famous aphorisms include, "Nothing is hidden" and his explanation of the point of his philosophical investigations as "To show the fly the way out of the fly bottle". Some of his descriptions are not from actually existing societies, but from imagined primitive societies. Some of these imagined societies are described near the beginning of the Philosophical Investigations, much as in the first chapter of Piero Sraffa's Production of Commodities by Means of Commodities.

Can Marx be read in an analogous manner, as attempting to dispel illusions, while claiming that no hidden essence or foundation underlies capitalist economies? Such a reading, I think, will emphasize Marx's remarks on commodity fetishism and "real illusions" that come with non-reflective participation in a market economy. It also makes sense of Marx's literary style. Both Marx and Wittgenstein are attempting to encourage a fundamental change so that our form of life will not generate these illusions.

Perhaps such a reading is in tension with the view of Marx's account of exploitation as descriptive, not normative. What about Wittgenstein's saying that philosophy "leaves everything as it is"? How can one read Wittgenstein and Marx as pursuing complementary projects when Marx writes, "Philosophers have hitherto only interpreted the world in various ways; the point is to change it"? Various essays in this book address these issues. I guess what concerns me more is Marx's Hegelian style, quite different from Wittgenstein. (I rely on English translations.)

This book also alerted me to some issues in Wittgenstein interpretation. When Wittgenstein writes of a form of life, is he writing of human life in general (in contrast, say, to the form of life of a lion)? Or would different human cultures and societies have different forms of life? Does Wittgenstein encourage a political quietism since he does not provide an external standpoint outside of language to criticize rules? (I think the last objection draws lines more firm than is compatible with Wittgenstein's comments on family resemblances.)

I also have two new books to look up, Gellner (1959) and Winch (1963). Gellner sounds like an unscholarly polemic that yet was influential in turning philosophy away from the linguistic philosophy of the later Wittgenstein, J. L. Austen, and Gilbert Ryle. Winch seems to argue those studying society must use the terms that members of a culture use, and with the same understanding. So perhaps this is a Wittgensteinian argument that social science is not possible, or at least must lower its aims. But I have not read it yet.

References

• Ernest Gellner (1959). Words and Things: A Critical Account of Linguistic Philosophy and a Study in Ideology London: Gollancz.
• Gavin Kitching and Nigel Pleasants (editors) (2002). Marx and Wittgenstein: Knowledge, Morality and Politics, London: Routledge
• Peter Winch (1963). The Idea of a Social Science, London: Routledge and Kegan Paul.

What Is Mathematics - And Sraffa

An Unsurveyable Rule For Generating A Real Number In Binary Format

Noah Smith offers a definition: "Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules." ("Unlearning Economics" has also recently written on mathematics in economics). To me, the manipulation of meaningless symbols is a powerful form of reasoning. Taking this definition as is, I think two questions can be raised here:

• What is the interest that mathematicians find in these rules and these symbols in the historical circumstances current at the time?
• What does it mean to follow a rule?

Ludwig Wittgenstein is the philosopher most known, I think, for raising the question of what it means to follow a rule. Any summary of his views will be controversial, but I suppose one can fairly say that he adopted an anthropological point of view, at least for some purposes. Describing how to follow a rule by another rule raises the prospect of an infinite regression. Rather, one might show how people do actually follow a rule, how these uses and practices work pragmatically in some form of life. I find it difficult to see how such description conveys the logical must, so to speak, of many rules. But Wittgenstein was alive to this difficulty. He notes that a judge does not seem to treat a statute book as a manual of anthropology.

Furthermore, Wittgenstein spent quite some time in elaborating how these ideas relate to the philosophy of mathematics. His views on the foundations of mathematics seems to have been constructivist and included questioning whether mathematics needs a foundation. Wittgenstein has frequently been labeled an anti-foundationalist. From this viewpoint, one might question whether existence proofs that do not specify how to construct the relevant object can be reformulated. And one even ends up doubting the meaningfulness of defining the real numbers as, say, any set isomorphic to a set of certain equivalence classes of Cauchy-convergent sequences of rational numbers. The use of the notion of infinity remains, I guess, as a standard topic in the philosophy of mathematics.

It seems one of my favorite economists, Piero Sraffa, was an important stimulus in Wittgenstein's development of these views. Sraffa has been said to have led Wittgenstein to see the importance of an anthropological point of view. Sraffa's masterpiece, The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, is written in a unique style, not less in the presentation of the mathematics underlying the economics in the book. Sraffa frequently provides outlines of algorithms for constructive existence proofs, maybe most famously for the Standard Commodity. So Sraffa and Wittgenstein might be said to have shared a certain attitude to the philosophy of mathematics, although I do not expect to ever see oral discussions on this topic to be well documented. Sraffa's book can also be said to address only a limited range of topics in economics. An earlier statement of his seems to suggest that he thought room should exist in economics for non-formal treatment of some topics:

"The causes of the preference shown by any group of buyers for a particular firm are of the most diverse nature, and may range from long custom, personal acquaintance, confidence in the quality of the product, proximity, knowledge of particular requirements and the possibility of obtaining credit, to the reputation of a trademark, or sign, or a name with high traditions, or to such special features of modelling or design in the product as - without constituting it a distinct commodity intended for the satisfaction of particular needs - have for their principal purpose that of distinguishing it from the products of other firms. What these and the many other possible reasons for preference have in common is that they are expressed in a willingness (which may frequently be dictated by necessity) on the part of the group of buyers who constitute a firm's clientele to pay, if necessary, something extra in order to obtain the goods from a particular firm rather than from any other." -- Piero Sraffa (1926). "The Laws of Returns Under Competitive Conditions", Economic Journal (Dec.): pp. 544-545.

Whatever you think of the speculations in this post, I think some conclusions are nearly inarguable. Advocates and opponents of the use of mathematics in economics do not neatly divide between mainstream and non-mainstream economists. In particular, one important non-mainstream economist, Piero Sraffa, demonstrated one approach to mathematical economics, while still being aware of the limits to formalism in economics. Furthermore, any comprehensive scholarly study of the philosophy of mathematics will necessarily look at his work as long as Wittgenstein's later views are considered germane to such scholarship.

Parallel Thoughts By Wittgenstein And Sraffa

Apparently Wittgenstein wrote the following in 1937:

"The origin and the primitive form of the language game is a reaction; only from this can more complicated forms develop.

Language - I want to say - is a refinement, 'in the beginning was the deed'." -- Ludwig Wittgenstein, Culture and Value (Translated by Peter Winch) (1980)

And Sraffa, I guess, wrote the following in the early 1930s:

"If the rules of language can be constructed only by observation, there can never be any nonsense said. This identifies the cause and the meaning of a word.

The language of birds, as well as the language of metaphysicians can be interpreted consistently in this way.

It is only a matter of finding the occasion on which they say a thing, just as one finds the occasion on which they sneeze.

And if nonsense is 'a mere noise' it certainly must happen, as sneeze, when there is cause: how can this be distinguished from its meaning?

We should give up the generalities and take particular cases, from which we started. Take conditional propositions: whan are they nonsense, and when are they not?" -- Piero Sraffa as quoted by Heinz D. Kurz, "'If some people looked like elephants and others like cats, or fish...' On the difficulties of understanding each other: the case of Wittgenstein and Sraffa", The European Journal of the History of Economic Thought, V. 16, n. 2 (2009): pp. 361-374

Wittgenstein to Sraffa

The March 2008 issue of Harper's Magazine reports that a new edition of Wittgenstein's letters in Cambridge is being published by Blackwell. They include the following 31 January 1934 letter:

Dear Sraffa,

The following are some remarks I've put down on the topic of our last conversation. I hope they won't be too disconnected and that you'll read them to the end.

You said, "The Austrians can do most of things the Germans did." I say, How do you know? What circumstances are you taking into account if you say they can? "This man, Austria, can remove the wedding ring from his finger." True, it's not too heavy and doesn't stick to his finger. But he may be ashamed of doing it, hiw wife may not allow it, etc.

You say, "Learn from what happened in Italy." But what should I learn? I don't know exactly how things happened in Italy. So the only lesson I can draw is that things one doesn't expect sometimes happen.

I ask, How will this whose face I can't imagine in a rage looks when he gets into a rage? And can he get into a rage? What shall I say when I see him in a rage? Not only, "Ah, so he can get into a rage after all," but also, "So this is the way he can be in a rage; so this is how it connects up with his former appearance."

You say to me, "If a man is in a rage, the muscles a, b, c of his face contract. This man has the muscles a, b, c, so why shouldn't they contract? If you, Wittgenstein, wish to know what he will look like in a rage, just imagine him with those muscles contracted. What will Austria look like when it turns Nazi? There will be no Socialist Part, there won't be Jewish judges, etc., etc., etc. That's what it'll look like."

I reply, This gives me no picture of a face; apart from the fact that I don't know enough about the workings of things to know whether all these changes that you point out will happen together. For I understand what it means to say that the muscles a, b, c will contract, but what will become of the many muscles, etc., between them? Can't the contraction of the one in this particular face prevent the contraction of the others? Do you know how in this particular case things interact?

You may say, "Surely the only way to tell the future physiognomy is to know more and more exactly the contractions, etc., of all, not only the main, muscles."

I say, I don't think this is the only way; there is another one, although the two ways meet. I may ask a physiologist what the face will be like, but also a painter. The two will give different answers - the painter by drawing the angry face - although if they both are correct they will agree. Of course, I know that painters have to study anatomy. I want to know the painter's answer, and I also want to know what the physiologist can tell me to check the painter's answer.

I am interested to know what phrases the Austrians will use when they'll have turned Naze. Supposing their patriotism is only talk, then I'm just interested in their future talk.

I wish to say one more thing. I think that your fault in a discussion is this: YOU ARE NOT HELPFUL! I am like a man inviting you to tea in my room, but my room is hardly furnished; one has to sit on boxes, and the teacups stand on the floor, and the cups have no handles, etc., etc. I hustle about fetching anything I can think of to make it possible that we should have tea together. You stand there with a sulky face, say that you can't sit down on a box and can't hold a cup without a handle, and generally make things difficult. At least that's how it seems to me.

Yours,

Ludwig Wittgenstein

Send A Letter

King’s College
Cambridge
14 March 1938

Dear Wittgenstein,

Before trying to discuss, probably in a confused way, I want to give a clear answer to your question. If as you say it is of “vital importance” for you to be able to leave Austria and return to England, there is no doubt – you must not go to Vienna. Whether you are a lecturer at Cambridge or not, now you would not be let out: the frontier of Austria is closed to the exit of Austrians. No doubt these restrictions will have been somewhat relaxed in a month’s time. But there will be no certainty for a long time that you will be allowed to go out, and I think a considerable chance of your not being allowed out for some time. You are aware no doubt that are now a German citizen. Your Austrian passport will certainly be withdrawn and then you will have to apply for a German passport, which may be granted if and when the Gestapo is satisfied that you deserve it.

As to the possibility of war, I do not know: it may happen at any moment, or we may have one or two more years of “peace”. I really have no idea. But I should not gamble on the likelihood of six months’ peace.

If however you decided in spite of all to go back to Vienna, I think: a) it would certainly increase your chance of being allowed out of Austria if you were a lecturer in Cambridge; b) there would be no difficulty in your entering England, once you are let out of Austria (of Germany, I should say); c) before leaving Ireland or England you should have your passport changed with a German one, at a German consulate: I suppose they will begin to do so in a very short time; and you are more likely to get the exchange effected here than in Vienna; and, if you go with a German passport, you are more likely (though not at all certain) to be let out again.

You must be careful, I think, about various things: 1) if you go to Austria, you must have made up your mind not to say that you are of Jewish descent, or they are sure to refuse you a passport; 2) you must not say that you have money in England, for when you are there they could compel you to hand it over to the Reichsbank; 3) if you are approached, in Dublin or Cambridge, by the German Consulate, for registration, or change of passport, be careful how you answer, for a rash word might prevent you ever going back to Vienna; 4) take care how you write home, stick to purely personal affairs, for letters are certainly censored.

If you have made up your mind, you should apply at once for Irish citizenship – perhaps your period of residence in England will be counted for that purpose: do it before your Austrian passport is taken away from you, it is probably easier as an Austrian than as a German.

In the present circumstances I should not have qualms about British nationality if that is the only one which you can acquire without waiting for another ten years’ residence: also you have friends in England who could help you to get it: certainly a Cambridge job would enable you to get it quickly.

I shall be in Cambridge till Friday: afterwards letters will be forwarded to me in Italy, so take care what you say, that you may be writing for the Italian censor.

My telephone is 3675: you will find me available before noon and in the evening after 10.

Yours
Piero Sraffa

Excuse this confused letter.

Ludwig Changes His Mind

I had this as a draft post before the discussion on this over at Max's place.

"Wittgenstein and P. Sraffa, a lecturer in economics at Cambridge, argued together a great deal over the ideas of the Tractatus. One day (they were riding, I think, on a train) when Wittgenstein was insisting that a proposition and that which it describes must have the same 'logical form', the same 'logical multiplicity', Sraffa made a gesture, familiar to Neapolitans as meaning something like disgust or contempt, of brushing the underneath of his chin with an outward sweep of the finger-tips of one hand. And he asked: 'What is the logical form of that?' Sraffa's example produced in Wittgenstein the feeling that there was an absurdity in the insistence that a proposition and what it describes must have the same 'form'. This broke the hold on him of the conception that a proposition must literally be a 'picture' of the reality it describes." --Norman Malcolm (1966). Ludwig Wittgenstein: A Memoir. Oxford University Press: 69

(I'm quoting second-hand.)

By the way, Pierangelo Garegnani has been permitting scholars to quote Sraffa's unpublished notes for a number of years. For example, Luigi Pasinetti has examined them and reported on them. Why shouldn't Sraffa have chosen a literary executor who is as slow to publish as he was? I am looking forward to their publication, though.

• Luigi L. Pasinetti (2001). "Continuity and Change in Sraffa's Thought: An Archival Excursus", in Piero Sraffa's Political Economy: A Centenary Estimate (edited by Terenzio Cozzi and Roberto Marchionatti), Routledge

Wittgenstein and Marxism

Ralph Dumain has compiled a bibliography on Wittgenstein, Marxism, and Sociology. It contains, for example, the Moran article in the New Left Review that I had previously noted, but neither the Eagleton nor the Robinson article in my list. But it does contain lots more to read.

Wittgenstein and Soviet Communism

On being recommended a couple of sources, I have been reading about Wittgenstein's political opinions, especially as in regard to Russian communism. Apparently, he was inspired by Tolstoy and thought a classless society sounded like a fine idea.

While I was looking up old articles in the New Left Review, I took a gander at the "Special Dossier" on Sraffa in the November-December 1978 issue. Sraffa and Wittgenstein seem to have this in common: the documentary evidence on their views on a whole host of interesting topics is slim.

References

• Eagleton, Terry (1982). "Wittgenstein's Friends", New Left Review, N. 135 (Sep.-Oct.): 64-90
• Ferrata, Giansiro (1978). "An Argument with Gramsci in 1924", New Left Review, N. 112 (Nov.-Dec.): 67-71
• Garegnani, Pierangelo (1978). "Sraffa's Revival of Marxist Economic Theory: An Interview with Pierangelo Garegnani", New Left Review, N. 112 (Nov.-Dec.): 71-75
• Moran, John (1972). "Wittgenstein and Russia", New Left Review, N. 135 (Sep.-Oct.): 64-90
• Napoleoni, Claudi (1978). "Sraffa's 'Tabula Rasa'", New Left Review, N. 112 (Nov.-Dec.): 71-77
• Napolitano, Giorgio (1978). "Our Debt to Sraffa", New Left Review, N. 112 (Nov.-Dec.): 65-67
• Ranchetti, Fabio (1978). "Keynes, Sraffa and Capitalist Crisis", New Left Review, N. 112 (Nov.-Dec.): 78-80
• Robinson, Christopher C. (2006). "Why Wittgenstein is Not Conservative: Conventions and Critique", Theory and Event, V. 9, N. 3
• Roncaglia, Alessandro (1978). "The 'Rediscovery' of Ricardo", New Left Review, N. 112 (Nov.-Dec.): 80-82
• Sraffa, Piero (1978). "An Unpublished Letter from Piero Sraffa to Angelo Tasca", New Left Review, N. 112 (Nov.-Dec.): 82-83

Libertarianism Versus "Libertarianism"

One might meet, in certain precincts of the Internet, soi-disant libertarians. As far as I am concerned, "libertarians" of this stripe are victims of commodity fetishim, and they have stolen the label. Traditionally, a libertarian is an anarchist, that is a kind of socialist. For example, Maureen Stapleton plays a libertarian in Warren Beatty's movie Reds. Anarchists, generally, do not have Ludwig Von Mises in their pantheon of heroes.

"...readers should take ... particular warning that I am absolutely not against freedom. On the contrary, I am for it. Libertarians ... think they are for freedom but they don't know what freedom is. In reality, their doctrine is so contrary to freedom that it ought to be entitled 'anti-libertarianism'. The thief comes in innocent disguise, but the beautiful garment is stolen. (The Right are good at that sort of thing.) So, if you want to make your copy of this book read more accurately, you should delete 'libertarian' and 'libertarianism' throughout, substituting 'anti-libertarian' and 'anti-libertarianism' as you go. For 'anti-libertarianism', etc., you should substitute 'anti-anti-libertarianism'. Unfortunately, this would make the book cumbersome to read, so I haven't followed the advice myself except in my choice of title, where my subject is named according to its true nature." -- Alan Haworth, Anti-Libertarianism: Markets, Philosophy, and Myth, Routledge, 1994: 5

Haworth does have more substantial points. Warning: this is political philosophy for those who think "If a lion could speak, we would not understand him" is a thesis worth discussing and who are comfortable with thought experiments which might lead one to be willing to say that a rock feels pain. Nevertheless Haworth is quite readable. (As an example of unreadable philosophy strongly following the later Wittgenstein, I cite John Wisdom's Other Minds.) I realize that those interested in political philosophy and "Libertarianism" should also read Robert Nozick.