A Turing Machine for a Binary Counter

Table 1: Tape in Successive Start States

Input/Output Tape	Decimal
tb0	0
tb1	1
tb10	2
tb11	3
tb100	4
tb101	5

1.0 Introduction

This post describes another program for a Turing Machine. This Turing machine implements a binary counter (Table 1). I do not think I am being original here. (Maybe this was in the textbook on computability and automata that I have been reading.)

2.0 Alphabet

Table 2: The Alphabet For The Input Tape

Symbol	Number Of Occurrences	Comments
t	1	Start-of-tape Symbol
b	Potentially Infinite	Blank
0	Potentially Infinite	Binary Digit Zero
1	Potentially Infinite	Binary Digit One

3.0 Specification of Valid Input Tapes

At start, the (input) tape should contain, in this order:

• t, the start-of-tape symbol.
• b, a blank.
• A sequence of binary digits, with a length of at least one.

The above specification allows for any number of unnecessary leading zeros in the binary number on the tape. The head shall be at the blank following the start-of-tape symbol.

4.0 Specification of State

The machine starts in the Start state. Error is the only halting state. Table 3 describes some conditions, for a non-erroneous input tape, that states are designed to satisfy, on entry and exit. For the states GoToEnd, FindZero, CreateTrailingOne, Increment, and ResetHead, the Turing machine may experience many transitions that leaves the machine in that state after the state has been entered. When the state PauseCounter has been entered, the next increment of a binary number appears on the tape.

Table 3: States

State	Selected Conditions
	On Entry	On Exit
Start	The head is immediately to the left of the binary number on the tape. (The binary number on the tape at this point is referred to as "the original binary number" below.)	Same as the entry condition for GoToEnd.
GoToEnd	The head is under the first digit of the binary number on the tape.	Same as the entry condition for FindZero.
FindZero	The head is under the last digit of the binary number on the tape	If all digits in the original binary number are 1 and that number has not been updated with a leading zero, the head is under the first digit of the binary number on the tape. If the original binary number contained at least one digit 0, the head is under the location of the last instance of 0 in the original binary number, and that digit has been changed to a 1. Otherwise, the head is under the first digit in the binary number now on the tape, and that digit is now a 1 (having once been a leading zero).
CreateLeadingZero	All the digits in the original binary number are 1. The head is under the first digit of the binary number on the tape.	Same as the entry condition for CreateTrailingOne
CreateTrailingOne	All the digits in the original binary number are 1. The first digit in the original binary number has been replaced by 0. The head is under that first digit.	The original binary number has been shifted one digit to the left, and a leading zero has been prepended to it. The head is under the last digit of the binary number now on the tape.
StepForward	If all digits in the original binary number are 1, that number has been shifted one digit to the left, that number has been updated with a leading 0 which is now a 1, and the head is under that digit. Otherwise, the last instance of 0 in the original number has been updated to a 1, and the head is now under that digit tape.	Same as the entry condition for Increment.
Increment	If all digits in the original binary number are 1, that number has been shifted one digit to the left, that number has been updated with a leading 0 which is now a 1, and the head is under the next location on the tape. Otherwise, the last instance of 0 in the original number has been updated to a 1, and the head is now under the next location on the tape.	Same as the entry condition for ResetHead. All the 1's to the right of the 0 updated to a 1 have themselves been updated to a 0.
ResetHead	The head is under the last digit of the binary number on the tape, and that number is the successor of the original binary number.	Same as the entry condition for PauseCounter.
PauseCounter	The head is immediately to the left of the binary number on the tape, and that number is the successor of the original binary number.

I think one could express the conditions in the above lengthy table as logical predicates. And one could develop a formal proof that the state transition rules in the appendix ensure that these conditions are met on entry and exit of the non-halting tape, at least for non-erroneous input tapes. I do not quite see how invariants would be used here. (When trying to think rigorously about source code, I attempt to identify invariants for loops.)

5.0 Length of Tape and the Number of States

Suppose the state PauseCounter was a halting state. Then this Turing machine would be a linear bounded automaton. In the Chomsky hierarchy, automata that accept context-sensitive languages need not be more general than linear bound automata.

The program for this Turing machine consists of 10 states. The number of characters on the tape grows at the rate O(log₂ n), where n is the number of cycles through the start state. I gather the above instructions could be easily modified to not use any start-of-tape symbol. Anyways, 20 people seems more than sufficient for the group activity I have defined, for this particular Turing machine.

Appendix A: State Transition Tables

This appendix provides detail specification of state transition rules for each of the non-halting states. I provide these rules by tables, with each table showing a pair of states.

Table A-1: Start and GoToEnd

Start				GoToEnd
t	t	Error		t	t	Error
b	Forwards	GoToEnd		b	Backwards	FindZero
0	0	Error		0	Forwards	GoToEnd
1	1	Error		1	Forwards	GoToEnd

Table A-2: FindZero and CreateLeadingZero

FindZero				CreateLeadingZero
t	t	Error		t	t	Error
b	Forwards	CreateLeadingZero		b	b	Error
0	1	StepForward		0	0	Error
1	Backwards	FindZero		1	0	CreateTrailingOne

Table A-3: CreateTrailingOne and StepForward

CreateTrailingOne				StepForward
t	t	Error		t	t	Error
b	1	FindZero		b	b	Error
0	Forwards	CreateTrailingOne		0	0	Error
1	Forwards	CreateTrailingOne		1	Forwards	Increment

Table A-4: Increment and ResetHead

Increment				ResetHead
t	t	Error		t	t	Error
b	Backwards	ResetHead		b	b	PauseCounter
0	Forwards	Increment		0	Backwards	ResetHead
1	0	Increment		1	Backwards	ResetHead

Table A-5: PauseCounter

PauseCounter
t	t	Error
b	b	Start
0	0	Error
1	1	Error

Noam Chomsky And Norman Mailer Share A Jail Cell For A Night

No joke. This happened as a result of an October 1967 march on the Pentagon to protest the Vietnam war. I find I had misremembered this passage. I recalled Mailer as being much less modest, as not acknowledging that technical linguistics used mathematical methods that might be beyond him at that stage of his life, no matter how much time he put into it. (I haven't actually read all of the technical works by Chomsky in the references below.) I have always liked Mailer's reporting and essays better than his novels, an opinion that I probably share with many and that he did not appreciate.

"Definitive word came through. The lawyers were gone, the Commissioners were gone: nobody out until morning. So Mailer picked his bunk. It was next to Noam Chomsky, a slim-featured man with an ascetic expression, and an air of gentle but absolute moral integrity. Friends at Wellfleet had wanted him to meet Chomsky at a summer before - he had been told that Chomsky, although barely thirty, was considered a genius at MIT for his new contributions to linguistics - but Mailer had arrived at the party too late. Now, as he bunked down next to Chomsky, Mailer looked for some way to open a discussion on linguistics - he had an amateur's interest in the subject, no, rather he had a mad inventor's interest, with several wild theories in his pocket which he had never been able to exercise since he could not understand what he read in linguistics books. So he cleared his throat now once or twice, turned over in bed, looked for a preparatory question, and recognized that he and Chomsky might share a cell for months, and be the best and most civilized of cellmates, before the mood would be proper to strike the first note of inquiry into what was obviously the tightly packed conceptual coils of Chomsky's intellections. Instead they chatted mildly of the day, of the arrests (Chomsky had also been arrested with Dellinger), and of when they would get out. Chomsky - by all odds a dedicated teacher - seemed uneasy at the thought of missing class on Monday.

On that long unwinding passage from the contractions of the day into the deliberations of the dream, Mailer passed through a revery over much traveled and by now level ground where he thought once more of the war in Vietnam, the charges against it, the defenses for it, and his own final condemnation which had landed him here on this filthy blanket and lumpy bed, this smoke-filled barracks air, where he listened half-asleep to the echoes of Teague's loud confident Leninist voice, he, Mailer, ex-revolutionary, now last of the small entrepreneurs, Left Conservative, that lonely flag - there was no one in America who had a position even remotely like his own, who else could indeed could offer such a solution as he possessed to such a war, such a damnable war. Let us leave him as he passes into sleep. The argument in his brain can be submitted to the reader in the following pages with somewhat more order than Mailer possessed on his long voyage out into the unfamiliar dimensions of prison rest..." -- Norman Mailer (1968).

References

• Noam Chomsky (1959). On certain formal properties of grammars, Information and Control, V. 2: pp. 137-167.
• Noam Chomsky (1965). Aspects of the Theory of Syntax, MIT Press.
• Noam Chomsky (1969). American Power and the New Mandarins, Pantheon Books.
• Noam Chomsky and M. P. Schützenberger (1963). The algebraic theory of context-free languages, in Computer Programming and Formal Systems, North Holland.
• Norman Mailer (1968) The Armies of the Night: History as a Novel, the Novel as History, New American Library.

A STEAM Experience For A Flash Mob

1.0 Introduction

STEAM stands for Science, Technology, Engineering, Arts, and Mathematics. This post describes a possible plan for a crowd of many people to participate in. Roles for players consist of:

• A Recorder.
• State Actors.
• Holders of letters in a line.

I once read Terry Eagleton suggesting that part of the definition of art is that it be "somewhat pointless."

2.0 Equipment

Equipment to be provided consists of:

• A six-sided die.
• Two balls. They could be soccer balls, beach balls, volley balls, or so on. One ball is called the Head, and the other ball is called the State Pointer.
• Six sets of equipment, labelled 1 through 6. A set of equipment consists of:
• A set of cards, where each card is a "letter" from an alphabet. Letters can be, for example, "Blank", "(", ")", ";", "End", "0", and "1". Many letters must have many cards with that letter.
• A set of state placards. Each state placard contains:
• An arbitrary label. These labels are arbitrary, but not repeated. They could be in high elvish, for all it matters, as long as participants can pronounce each label.
• Either the word "Halt" or a set of rules. The placards for the halting states may also contain a short phrase. Each rule in a set of rules is designated by a letter from the alphabet.
• Guidelines for setting up. These guidelines include:
• Optional guidelines for the geographical distribution of states.
• A specification of which State Actor initially holds the State Pointer.
• Guidelines for forming an initial line of letters from the alphabet. These guidelines must include a specification of which holder of a letter initially also holds the Head.

3.0 Playing the Game

3.1 Preliminaries

The Recorder throws the die and chooses the corresponding set of equipment. One might create only one set of equipment, and this step would be omitted.

The Recorder distributes the state placards. A audience member comes up for each placard. He collects it, and becomes a State Actor. The State Actors all gather, with some distance between them, in a designated region. (One might break down the region into sub-regions, for subsets of the states, if one wants.)

The Recorder gives the State Pointer to the State Actor holding the placard for the initial state.

The Recorder reads out the guidelines for the initial line of letters. Audience members come up and form a line, accordingly. As an example, the guidelines might say:

The first player sits in the line and holds the "End" letter. The second player stands behind the first player. He holds a "Blank" and the Head. A number of players sit in the line behind the second player. They should each hold "0" or "1", as they choose. A person should sit after these players, and she holds a ";". Another number of players sit in in a row behind her. They also should each hold a "0" or "1".

The Recorder writes down the sequence of letters in the initial set up. This step is optional.

Play can now commence. Play consists of a sequence of clock cycles.

3.2 A Clock Cycle

The player holding the Head commences a clock cycle. This player calls out the letter he is holding.

The state actor holding the State Pointer now plays. He looks at his rules and finds the rule corresponding to the letter that has been called out. Each rule has two parts. The first part is either a letter from the alphabet or the word "Forward" or the word "Backward". The second part is the name of a state. That state could be the label on the state placard that this State Actor is holding. Or it could be another state.

If the State Actor calls out a letter, an audience member comes up. He selects that letter from leftover letters in the initial set of equipment. He replaces the player holding the Head in the line. And that player hands the new player the Head.

If the State Actor calls out Forward, the player holding the Head hands it to the player holding a letter in front of him and sits down. The player now holding the Head stands up. There would be no such player if the player holding the Head at the start of the cycle is standing at the front of the line. In this case, an audience member picks up a "Blank" from the leftover set of equipment. That player accepts the Head and stands at the front of the line.

If the State Actor calls out Backward, the player holding the Head hands it to the player holding a letter behind him and sits down. As you might expect, that player now holding the Head stands up. This step might also result in a new player coming up from the audience and joining the line. And this new player would join the line at the back.

The State Actor holding the State Pointer now calls out the state listed on the second part of the rule he is executing. If that state is not the state listed on his state placard, he hands the State Pointer to the appropriate State Actor.

The Recorder writes down the new state that the State Pointer has now transitioned to. (This step is optional.)

3.3 Ending the Game

The game ends either when the players become convinced it could go on forever, or it ends when a State Actor holding a placard for a halting state receives the State Pointer. If the game ends in a halting state, the State Actor reads the corresponding phrase from the state placard. That phrase might be something like:

You have been a Turing machine computing the sum of two non-negative integers, written in binary.

Or it could be:

You had at least one unmatched parentheses in your initial line.

If you want, the Recorder could have more audience members come up to recreate the initial line. You can then review, if you like, the computation. For example, you might check that the sum of the two numbers separated by a comma in the initial line up is equal to the number now represented by the final line up on the stage.

4.0 Much To Do

Obviously, much would need to be done to flesh this out. In particular, equipment sets need to be constructed. Some choices to think about:

• Would one want to include an equipment set in which the simulated Turing machine does not terminate for some initial line of letters? Or would one want to, at least for the first performance, only have rules that are guaranteed termination for all (valid?) inputs?
• Might one want to emulate automata for languages lower down on the Chomsky hierarchy? For example, one might create a stack to be pushed and popped before the start of the line. Here I envision that a subset of the states specify subroutines. And the State Actors defining these subroutines might be grouped separately from the other actors.
• Would one want to share alphabets among more than one equipment set? Maybe all six sets should have the same alphabet.
• How would one describe the initial line up for a Turing machine that is to decide or semi-decide whether a given string is in a given language? The specification of a grammar for generating a string can be quite confusing to beginners.
• I am thinking that one would not want to create rules for a universal Turing machine. Even some of the suggestions above might be too long to play.

An interesting variation would be to simulate a non-deterministic Turing machine. For some clock cycles, the line would be duplicated. And one would introduce another Head and State Pointer.

5.0 Instruction and Theatrics

This activity could serve pedagogical purposes. Suppose the players are different cohorts of students. Could the older students be directed to write the rules for some other computation at the next meeting? Could a set of recursive functions be built up over many meetings? Maybe one would end up with a group engaging in real-time debugging in a joint activity.

One could set up an accompanying talk or lecture. Many topics could be broached: The Church-Turing thesis and universality, uncomputable functions and the halting problem, computational complexity and the question of whether P equals NP, linguistics and the Chomsky hierarchy, etc. Or one might talk about the British secret service and reading the Nazi's mail. I guess there is both a Broadway play and a movie to go along with this activity.

One could introduce some sophistication in showmanship, depending on where this concept is instantiated. I like the idea of the alphabet players wearing different colored shirts, with each color corresponding to a character. Zero could be red, and one could be green. Blanks would be a neutral color, such as white. The State Actors could be in a dim area, with a spotlight serving as the State Pointer. The State Actors or the letter holders could be members of an orchestra, with some tune being played for every state transition or invoked rule. At termination, the entire derivation written down by the Recorder could be run-through. I imagine it would be difficult to design a set of rules that results in an interesting tune. At any rate, I guess the interests of an observing mathematician, the participants, and a theatergoer would be in tension.

I hope if somebody was to try this project, they would give me appropriate acknowledgement.

Reference

• Lou Fisher (1975). "Nobody Named Gallix", The Magazine of Fantasy and Science Fiction (Jan.): pp. 98-109.
• Andrew Hodges (1983). Alan Turing: The Enigma, Princeton University Press.
• HarryR. Lewis and Christos H. Papadimitriou (1998). Elements of the Theory of Computation, 2nd edition. Prentice Hall.

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.)

What Is A "Special Interest"?

I do not want to compare and contrast analytically precise definitions that answer the question in the post title. (Socrates, as reported by Plato, always asked for a definition after being given examples.) Instead, I give two lists, where I trust the reader to see family resemblances among the items on each list:

• Ethnic groups like African-Americans; women; the poor; organized labor; and lesbians, gays, bisexuals, and transgenders.
• Corporations, especially those operating in specific industries (e.g., big oil); Corporate Executive Officers; and owners of small businesses.

I suggest that the policies and culture of a country would be quite different, when the dominant understanding of the phrase, "special interests" was consistent with one or another list.

I think somewhere or other Noam Chomsky has asserted that the second understanding reflects the true meaning or the term, or at least a meaning consistent with what the Founding Fathers of the United States wrote. This quote does not have the look back to classical liberals:

"...these questions have been asked for a long time in polls, a little differently worded so you get some different numbers, but for a long time about half the population was saying, when asked a bunch of open questions - like, Who do you think the government is run for? would say something like that: the few, the special interests, not the people. Now it's 82%, which is unprecedented. It means that 82% of the population don't even think we have a political system, not a small number.

What do they mean by special interests? Here you've got to start looking a little more closely. Chances are, judging by other polls and other sources of information, that if people are asked, Who are the special interests? they will probably say, welfare mothers, government bureaucrats, elitists professionals, liberals who run the media, unions. These things would be listed. How many would say, Fortune 500, I don't know. Probably not too many. We have a fantastic propaganda system in this country. There's been nothing like it in history. It's the whole public relations industry and the entertainment industry. The media, which everybody talks about, including me, are a small part of it. I talk about mostly that sector of the media that goes to a small part of the population, the educated sector. But if you look at the whole system, it's just vast. And it is dedicated to certain principles. It wants to destroy democracy. That's its main goal. That means destroy every form of organization and association that might lead to democracy. So you have to demonize unions. And you have to isolate people and atomize them and separate them and make them hate and fear one another and create illusions about where power is. A major goal of this whole doctrinal system for fifty years has been to create the mood of what is now called anti-politics." -- Noam Chomsky, Class Warfare: Interviews with David Barsamian Common Courage Press (1966): p. 138.

But there is another literature, a post modern literature, that also looks at how people come to associate examples with words. People generally do not think logically, following the rules of predicate calculus. One trying to understand culture should realize this. One might talk about the The politics of the signifier. How does one or another definition, or set of examples, become hegemonic? (For what it is worth, I think Slavoj Zizek is a very intelligent, very well-read, self-aware clown.)

Knowledge/Power

Figure 1: Paul Krugman And Bill O'Reilly Talk To Tim Russert

"My problem was ... to pose the question, 'How is it that at certain moments and in certain orders of knowledge, there are these sudden take-offs, these hastenings of evolution, these transformations which fail to correspond to the calm, continuist image that is normally accredited?' But the important thing here is not that such changes can be rapid and extensive or, rather, it is that this extent and rapidity are only the sign of something else - a modification in the rules of formation of statements which are accepted as scientifically true. Thus, it is not a change of content (refutation of old errors, recovery of old truths), nor is it a change of theoretical form (renewal of a paradigm, modification of systematic ensembles). It is a question of what governs statements, and the way in which they govern each other so as to constitute a set of propositions that are scientifically acceptable and, hence, capable of being verified or falsified by scientific procedures. In short, there is a problem of the regime, the politics of the scientific statement. At this level, it's not so much a matter of knowing what external power imposes itself on science as of what effects of power circulate among scientific statements, what constitutes, as it were, their internal regime of power, and how and why at certain moments that regime undergoes a global modification.

It was these different regimes that I tried to identify in The Order of Things, all the while making it clear that I wasn't trying for the moment to explain them, and that it would be necessary to try to do this in a subsequent work. But what was lacking here was the problem of the 'discursive regime', of the effects of power peculiar to the play of statements. I confused this too much with systematicity, theoretical form, or something like a paradigm. This same central problem of power, which at that time I had not yet properly isolated, emerges in two very different aspects at the point of junction of Madness and Civilization and The Order of Things." -- Michel Foucault, "Truth and Power", reprinted in The Chomsky-Foucault Debate: On Human Nature, The New Press (2006), pp. 144-145.

I do not know that I understand Michel Foucault, and I have not read much that he wrote towards the end of his life. I had thought Foucault's discursive formations were to be grouped with Thomas Kuhn's paradigms and Imre Lakatos's scientific research program. To me, economics is like medicine, psychiatry, and penology. It fits in well with the disciplines that Foucault analyses. Superficially, the epistemic status of these disciplines is more questionable than a hard science. And they have been used to help nation states categorize, partition, and rule their subjects since, say, the eighteenth century. But I want to drop talk of science for now. I look at a concrete example to help me understand what Foucault might mean when he talks about government, power, the political economy of the sign, a discursive regime, and politics. Doubtless, I will miss many, many nuances here.

You can see many commentators and supposed experts in the media, although, for many, I am none too clear in what area they are expert. (I have in mind such people as Rush Limbaugh, Bill O'Reilly, and even Wolf Blitzer and Thomas Friedman.) Many write best-selling books. A book store will classify them, when they come out, in a section labeled "current events". I suppose libraries will put them somewhere in social sciences. People with the sort of media presence I have in mind can be said to benefit from a sort of power for their statements. From an analytical point of view, you might know a drunk at the end of your local bar who is more worth listening to. Yet these commentators react to one another, take each other seriously, and end up having effects on laws that are passed. At least one kind of power circulates among their statements, a power that is not easily available to those taking their own way at your local.

Foucault also writes about power being productive, not solely a matter of prohibitions. How does the above clip illustrate this theme? Those who have power circulating their statements can sometimes dismiss others as living in a "fantasy world". But I think the power we see in right wing commentators in America extends to individuals in communities across the country. You can find many who think they keep informed by watching TV news. And they will have conversations with one another, maybe conversations that you could not participate in without being seen as rude, dismissive, and condescending. Some of these people might even participate in governing your community by participating in, say, the school board, the city council, or state government. Within such groups, you might find an intellectual who has read, for example, Hayek's Road to Serfdom on Glenn Beck's recommendation. So this power I am vaguely pointing at helps form local communities, as well as national discourse.

I consider Krugman to be at a different level of seriousness than the other two people in the above clip. Still these questions arise for him. What power gets his statements listened to and to circulate widely? It would be a mistake to classify his statements solely as part of the academic discipline of economics. For example, his newspaper columns about the Iraq war do not have much to do with economics. And a regime in which he occupies the acceptable left wing of the public face of economics seems quite limiting to me. When Krugman debates Keen through their blogs, it seems clear who is doing the other more of a favor to acknowledge the existence of the other's work, whatever you make think of the outcome of that debate.

I trust that one can see that in merely acknowledging the existence of political power that allows Krugman's statements to circulate, I am not thereby criticizing their content or what Krugman does with this non-personal power. In fact, I think Krugman has quite often acknowledged the power of his platforms and talked about how that influences his topics. As far as I know, he does not read Foucault. (Has not Brad DeLong written a bit on Foucault?) I am not sure what Krugman has said about his willingness to participate in the sort of hurlyburly babble seen in the above clip, other than that he sometimes has a book to promote. I suppose the bit where he leans back and rolls his eyes at the ceiling is comment enough on his particular antagonist there. I think Krugman would even be receptive to claims about the lack of agency of the author. He is rarely as forthright as in the above clip about calling a lie, "a lie". And he knows that his conventions do not allow him to comment on nonsense spouted by his fellow columnists, except very elliptically.

By the way, the video clip above is not directly from a major network. Apparently, it was put on YouTube with annotations added by Jim Gilliam. And, of course, I do not claim the power of those you might see babbling on your television.

Positions On The Philosophy Of Math

What does a mathematical proof show? In what sense, if any, do the objects which mathematicians reason about exist? A program of study on the philosophy of mathematics might consider some of the following views:

• Bourbaki - Structuralism
• Brower - Intuitionism
• Frege - Logicism (?)
• Gödel - Platonism
• Hilbert - Formalism
• Lakatos - Proofs and refutations in the tradition of Popper
• J. S. Mill - Math as empirical generalization
• Poincare - Intuitionism (?)
• Russell - Logicism
• Wittgenstein - Constructivism (?)

And one might look at more recent academic interpretations and commentary, such as Putnam or Kripke's interpretation of Wittgenstein.

Philosophers of math often discuss various interesting bits of mathematics. These include the construction of real numbers as equivalence classes of Cauchy-convergent sequences of rationals, of rational numbers as equivalence classes of ordered pairs of integers, and of integers as functions mapping (subsets of) the natural numbers to the natural numbers. Each construction comes with definitions of <, +, and *. The notion of an isomorphism is important in these constructions.

Those who think mathematics is in need of a foundation have often looked for one in terms of logic and set theory. Different axiom systems have been offered for sets. Russell's theory of types contrasts with the Zermelo-Fraenkel (ZF) system. In these set theories, there is an infinity of orders of infinity. I've always like the proof that the power set of a set, that is, the set of all subsets of a set, cannot be put in a one-to-one relationship with the original set. Thinking about applying that theorem recursively to the set of the natural numbers soon exhausts my imagination. Someday I would like to understand Gödel's proofs that if ZF is consistent, then ZP with the axiom of choice (ZFC) is consistent. And if ZFC is consistent, then Cantor's continuum hypothesis is consistent with ZFC. Paul Cohen went further. He proved, in 1963, the continuum hypothesis is independent of the axioms of ZFC. I guess this relates to model theory. I gather the Löwenheim-Skolem theorem is a surprising result.

Philosophers of mathematics often discuss certain important results from comutability theory and the theory of automata. Among these are Gödel's imcompleteness theorem. (Barkley Rosser, Sr., generalized Gödel’s work, from ω-consistency to consistency.) I gather the unsolvability of Diophantine equations, in general, follows from Gödel's theorem. The existence of uncomputable functions is of interest. Every computer programmer should be aware of the halting problem. I find interesting the Church-Turing thesis, the Chomsky hierarchy, and the question of whether the set of problems that can be solved in polynomial time by a deterministic Turing machine is equivalent to the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.

Quoting Joan Robinson

I might as well note that Mark Thoma puts up a long quotation from Joan Robinson.

Gabriel Mihalache, with his confusion of methodological individualism and political individualism, makes the comments, basically, all about him.

Gavin Kennedy cannot seem to handle that a leftist like Joan Robinson is quite aware of the contrast between popular portrayals of Adam Smith and what Smith actually said. He assigns Joan Robinson's words to Mark Thoma.

This should further explode Gavin Kennedy's head:

"I didn't do any research at all on Smith. I just read him. There's no research. Just read it. He's pre-capitalist, a figure of the Enlightment. What we would call capitalism he depised. People read snippets of Adam Smith, the few phrases they teach in school. Everybody reads the first paragraph of The Wealth of Nations where he talks about how wonderful the division of labor is. But not many people get to the point hundreds of pages later, where he says that division of labor will destroy human beings and turn them into creatures as stupid and ignorant as it is possible for a human being to be. And therefore in any civilized society the government is going to have to take some measures to prevent division of labor from proceedings to its limits...

[A mixture of stuff that I agree is a good interpretation of Smith and stuff I don't think is all that good an interpretation]

The version of him that's given today is just ridiculous. But I didn't have to do any research to find this out. All you have to do is read. If you're literate you'll find it out. I did do a little research in the way it's treated, and that's interesting. For example, the University of Chicago, the great bastion of free market economics, etc., etc., published a bicentennial edition of the hero, a scholarly edition with all the footnotes and the introduction by a Nobel Prize winner, George Stigler, a huge index, a real scholarly edition. That's the one I used. It's the best edition. The scholarly framework was very interesting, including Stigler's introduction. It's likely he never opened The Wealth of Nations. Just about everything he said about the book was completely false. I went through a bunch of examples in writing about it, in Year 501 and elsewhere..." -- Noam Chomsky (1996). Class Warfare: Interviews with David Barsamian, Common Courage Press.