TURING'S THEOREM REVISITED
Anyone who tries to use Adam Reed's intellectual ammunition ("An Obituary for Machines of Loving Grace," October) will find it exploding in his face. Mr. Reed's argument contained a fatal flaw. To put it simply, his human cheated. (Reed's article was not a presentation of Turing's famous theorem, which is correct.)
Reed attempted to show that there is no program which can answer a certain question about programs in every case in which a human can answer it. His "proof" is by reductio ad absurdum. The existence of such a program (Program A) is assumed. Then A is modified in a very special way to produce Program B. Finally Reed shows that Program A cannot answer the question about Program B, but a human being supposedly can. From this he concludes that Program A cannot exist.
When Program A is called upon to answer a question about Program B, A is given just the number of Program B. (All programs can be and are assigned numbers in the formal proofs in computability theory.) If Reed or any other human were given only B's number and were not told the assumptions made about Program A and how B was constructed from A, the human couldn't answer the question either. Conversely, there exists a program which could prove all that Reed proved, namely that any program, B, satisfying the conditions in Reed's argument cannot produce a definite answer.
Reed runs into more problems when he tries to apply his argument to the issue of determinism. Turing machines are deterministic in the sense explained by Reed—the state and input at any given step uniquely determine the next state. However, one can define a non-deterministic Turing machine by allowing the current state and input to determine a set of possible next states, rather than one specific next state. Hence, at any step, the machine may make any one of several possible moves. We say the Turing machine computes the right answer if there is some sequence of correct choices the machine could make at each step in order to get the answer. It is a theorem in computability theory that non-deterministic Turing machines are no more powerful than deterministic Turing machines. That is, any problem that can be solved by a non-deterministic Turing machine can also be solved by some deterministic one. (This theorem shouldn't surprise anyone who thinks about the definitions for a while.)
I will not draw conclusions about the philosophical issue of determinism from what I have said. Anyone who does so must understand that when the terms above ("deterministic," "non-deterministic," even "compute" and "answer") are used by computability theorists, they are technical terms with mathematically precise definitions. When the philosopher or layman uses the same terms he usually has other (or no) definitions in mind, and great care must be taken when trying to transfer results from one field to another.
Department of Mathematics
Calif. State University
San Diego, CA
MR. REED REPLIES: Baase's letter contains two objections to my outline of Turing-type proofs. Her first objection assumes incorrectly that "assumptions about A and how B was constructed from A" cannot be included in the description of B referred to in my outline (and presented to the computer in the form of the number she mentions). In fact, however, any statement which can be made in English can also be encoded as a number, fed into a computer, and used by its programs accordingly. One may stipulate that in the context of my outline all program descriptions include the assumptions about A and a description of the process, if any, for transforming A into the given program. The contradiction remains, unchanged, when the descriptions (or numbers) are thus augmented. The assertions in Baase's third paragraph are thus shown to be false.
Baase's second fallacy is even more elementary. All deterministic entities may be simulated by deterministic Turing machines, but it does not follow that nondeterministic Turing machines can simulate all non-deterministic entities. Thus while the general limitations of Turing machines must apply to nondeterministic Turing machines, they need not limit the cognitive range of other nondeterministic entities, such as human beings. I agree with Baase's statement that great care must be taken when trying to transfer results from one field to another. I would add the suggestion that the same standard of care be applied when criticizing such work.
BRIDGE-BUILDING WITH CONSERVATIVES
I am writing this in protest against a disturbing trend towards a sort of ideological bridge-building with conservatives, particularly those associated with YAF. It is evidenced most clearly in the acceptance of conservatives into the libertarian camp (e.g., David Brudnoy of YAF as a contributing editor to REASON), and the failure to differentiate clearly between the two groups.
A good example of this is the uncritical acceptance of Congressman Steven Symms (R-ldaho) by the libertarian media. Is Symms a conservative or a libertarian? In your REASON Profile (April 1973), you note his stands against the minimum wage and the postal monopoly, and his attack on big government in general. Since then he has introduced legislation to legalize gold and prohibit sending foreign aid to North Vietnam. Great stuff—but not good enough. All of these positions are common to both camps.
Unlike the "free enterprise" conservative, the libertarian applies the concept of laissez-faire to all non-coercive social relations. This is the acid test of the libertarian. The conservative is more interested in maintaining a certain king of moral and social order, and supports "free enterprise" because it is a traditional part of the national culture.
It is time to ask those "friends" of ours some very pointed questions: Do they oppose censorship of "obscene" material? Do they support the full legalization of drugs—including heroin—or do they gabble about "decriminalization" for users? Do they support amnesty for draft resistors? Do they support the Equal Rights Amendment? Do they oppose an interventionist foreign policy?
This issue is not mere ideological nitpicking, but a profoundly practical matter. Conservatives have a certain image in the public eye: a group of socially uptight and pious nationalists, church-going bigots, cantankerous Middle Americans, etc. We can't afford to be contaminated with that kind of image. Libertarians must take the utmost pains to distinguish themselves from conservatives at every opportunity, so that we may project a clearly different image: that of freethinkers and cultural innovators, rather than relics of the past.
So far, we libertarians have had little problem being identified as liberals. Instead, we have been lumped with conservatives as representatives of the Right. We must keep up an indignant protest until this error is corrected. To this end, we should adopt the following rule: The only conservatives accepted in our ranks shall be ex-conservatives; the rest shall be treated as lepers.
Bruce A. Ramsey
Editor, THE SHOCKWAVE
Libertarian Party of Washington State
MORE ON CONSERVATIVES
I am writing to protest the appearance of conservative writers in the pages of REASON, namely: Gary North and David Brudnoy. I protest by reason of their incompatibility with libertarianism.
(1) In the October REASON, Mr. North writes on the characteristics of bureaucracy. He essentially tells us what we have known all along: bureaucracy is self-serving, made possible only through coercive financing, and is inefficient.
This warmed-over insight is simply not worth the effort of exposition. Furthermore, his advocacy of "peeing on the court-house steps" is facetious, and therefore worthless. If he is really so pessimistic of eventual reform, he ought rationally to concede defeat rather than spin useless fantasies.
Considering further: Mr. North's article in the 31 August 1973 NATIONAL REVIEW ("Pornography, Community, Law") in which he defends the censorship of "obscene" material, makes it doubtful that he is even sincere in his denunciation of bureaucracy. Not only is Mr. North willing to propagate a bureaucracy dedicated to the suppression of pornography (in the name of "the 'collective conscience' of the community"), he also denies the protection of the First Amendment to anything other than political speech and publication. Mr. North is not so much interested in "live and let live" as he is dedicated to "the censorship of…cultural nihilism."
(2) Mr. Brudnoy's column in the November REASON hashes over another well- understood argument: the economic disaster associated with minimum wage laws. The reiteration of established libertarian arguments is also shown in the April NEW GUARD, wherein he writes on "Decriminalizing Crimes Without Victims: The Time is Now." An admirable topic, but laced with the usual conservative confusions and contradictions: (a) although he speaks of the injustice of victimless "crimes", he still bows to the necessity of some kind of censorship; (b) he misconstrues the libertarian position to be one of "let anybody do anything"; and (c) he declares that "orthodox" libertarianism accepts freedom as an end in itself. One cannot fault Mr. Brudnoy's intentions in writing his articles, but his understanding of victimless "crimes" and of libertarian philosophy is retarded, compared to the libertarian mainstream. Unless he has radically transformed in the meantime, there is no evidence to believe he can enlighten libertarians.
(3) In opening REASON to conservative writers, the editorial staff of REASON is obliged to defend this decision. Libertarians deliberately distinguish themselves from conservatives and liberals because both the Right and the Left are inconsistent in their advocacy of human liberty. Libertarianism is unique in its consistent commitment to human rights, and eschews compromise of this commitment. Conservatives, as reflected by North and Brudnoy, still advocate coercion on some level. They offer libertarianism nothing, and profit by the mantle of prestige and popularity which the libertarian movement has gained over the last five years. If this precedent is set, why should REASON refuse to print Galbraith, Kleindienst, or Mark Rudd, if they should coincidentally agree with libertarians on some specific issue?
I did not become a libertarian with the intention of blurring my uniqueness from conservatives and liberals. I did not subscribe to REASON with the expectation of reading the sentiments of those whom I measure among the oppressors of society.
There are many libertarians vying for a chance to speak. How is it they are being pre-empted by representatives of the establishment?
THE EDITORS REPLY: REASON does not do background checks on authors, nor do we maintain an "enemies list." (Frankly, we were as offended by Mr. North's defense of censorship in NATIONAL REVIEW as was reader Dunn.) We strive to publish articles that rationally defend and advance the principles of human liberty, whatever their source. We have no intention of requiring a test of a person's ideological purity, so long as what he has to say in the pages of REASON is reasonable. Thus, REASON's pages are open to "conservatives" (Gary North), "liberals" (John Holt), "pacifists" (Robert LeFevre), "limited statists" (John Hospers), "anarcho-capitalists" (Murray Rothbard), et al., on this basis.
Moreover, the use of labels to classify the acceptability of a person's thinking is far from precise; and a person considered to be a "conservative" might well take rational profreedom approaches on many issues.
Mr. Ramsey poses some excellent questions for our "friends." To the extent our "friends" identify with a libertarian position on such issues, we are pleased; in other cases, our "friends"—whether Rand, Hospers, Symms, etc.—are not REASON's friends as to issues on which they take a statist position.
As for contributing editor David Brudnoy, readers Dunn and Ramsey might be interested in corresponding with Brudnoy regarding his growing attraction to libertarian principles and his thoughts on communicating them to conservatives. We believe such communication is extremely valuable at a time when many intelligent conservatives are re-thinking their values in light of the Nixon debacle.
ROTHBARD AND HAYEK COMPARED
I would like to respond briefly to Martin Glasser's comments (Letters, December) on my review (September) of Dr. Rothbard's book FOR A NEW LIBERTY. While Mr. Glasser's comments about Mises are fully valid, I cannot say the same for his comments concerning the epistemological views of Hayek and Rothbard. Even though they might agree on a vast number of conclusions, the epistemological views of Rothbard and Hayek are light-years removed from one another. My specific comment in my review referred to Hayek's own classifications, particularly in THE CONSTITUTION OF LIBERTY, where he heatedly attacks rationalism. In my view, Hayek's concept of reason, which is central to many of his works, is totally false; in the broadest philosophical sense, Hayek is a Kantian in his epistemological views, and has been in every book from THE SENSORY ORDER: AN INQUIRY INTO THE FOUNDATIONS OF THEORETICAL PSYCHOLOGY to his most recent book, LAW, LEGISLATION AND LIBERTY, Vol. 1. In this sense, in a fundamental sense, Hayek is not an innovator in epistemology; in all of his works, his philosophical categories are taken straight from textbooks devoted to the history of philosophy. Although there are some details new with Hayek, he is, largely, arguing from within categories defined and defended by other, earlier, writers. Hayek would say that the notion of an "objective reality" in the Aristotelian or Objectivist sense is meaningless, holds that sense-perceptions are acts of classification, and argues for a notion of what he calls "the primacy of the abstract." Perceptions, to Hayek, are derived from abstractions—a basically Kantian approach to epistemology. His position on "rationalism" is more obscure: he attacked rationalism for many years in favor of a brand of traditionalist-empiricism of the Humean, Burkean, variety. But in his book STUDIES IN PHILOSOPHY, POLITICS AND ECONOMICS, he argues that perhaps he was wrong, that perhaps there are various kinds of rationalism, but in any case the focus of his attacks remain the same: the use of explicit reason to define and solve problems. I think that all of this rests on serious errors, but this is not the place for me to attempt to correct them. I shall publish a lengthy and fully detailed consideration of the Hayekian system, with all of its virtues and flaws, after Hayek has finished publishing the three volumes of his projected magnum opus, LAW, LEGISLATION AND LIBERTY.
Glasser's oversimplified representation of the Hayekian position is simply out-of-touch with the broader fundamentals and the more intricate philosophical views on which Hayek's views are based. I refer him particularly to Hayek's THE SENSORY ORDER, the volume of STUDIES mentioned, and the essay "The Primacy of the Abstract," in Koestler and Smythies, eds., BEYOND REDUCTIONISM.
Now for a note on the views of Mises and Hayek. Mises' work in epistemology is not as detailed as that of Hayek, but he has placed himself squarely in the Kantian tradition. Even economic laws, or, more properly, their analytical bases, are not facts of objective reality to Mises: they are "categories of the understanding," clearly an extension of Kantian thinking to the realm of economics. The position of Rothbard is crystal clear, if not fully detailed: he is the exact opposite of Mises and Hayek in this sense; he is an Aristotelian who defends the axioms of economics as being both empirically derived and necessarily true, thus rejecting many of the dichotomies which Mises and Hayek implicitly rely on, and the whole Kantian-Positivist frame of reference as a basis for valid debate. Rejecting these dichotomies, Rothbard in fact places himself outside of the Misesian-Hayekian categories, despite his agreement with them on certain more substantive points of methodology (the focus on deduction from axioms, for example) and in economics proper. Rothbard has never himself emphasized this radical departure from Misesian and Hayekian lines of thought, but it exists nonetheless, and it is a very basic difference between them. As far as agreeing with Hayek is concerned, it is simply not true of Rothbard that he does. When THE CONSTITUTION OF LIBERTY was first finished a decade and a half ago, Rothbard wrote a long and unpublished critique of it which many libertarians, including myself, have been privileged to read. In it he took Hayek mercilessly to task for his epistemological views, and for their consequences in the Hayekian political philosophy. On most substantive issues of methodology and substantive political philosophy, Rothbard and Hayek are at opposite poles.
R.A. Childs, Jr.
Editor, BOOKS FOR LIBERTARIANS
Los Angeles, CA
This article originally appeared in print under the headline "Letters".