Wittgenstein's Remarks on the Foundations of AI

Stuart G. Shanker

• Publisher: Routledge
• Year: 2002
• eBook ISBN:
  9781134859917
• Topic: Filosofia
• Subtopic: Intelligenza artificiale (IA) e semantica

Wittgenstein's Remarks on the Foundations of AI is a valuable contribution to the study of Wittgenstein's theories and his controversial attack on artifical intelligence, which successfully crosses a number of disciplines, including philosophy, psychology, logic, artificial intelligence and cognitive science, to provide a stimulating and searching analysis.

1
WITTGENSTEIN’S RESPONSE TO TURING’S THESIS

The real problem is not whether machines think but whether men do.

(B.F. Skinner, Contingencies of Reinforcement)

§1 Turing’s machines ‘are humans who calculate’

The title of this chapter suggests two contentious claims: first, that Wittgenstein was aware of the developments in recursion theory which took place during the 1930s; and second, that he disputed the version of Church’s Thesis (hereafter CT) that Turing presented in ‘On Computable Numbers’ (Turing 1936). It will be best to concede at the outset that both themes represent something of a critical liberty or, perhaps, a corollary. For the subject of this chapter is really Wittgenstein’s attack on the mechanist terms in which Turing interpreted his computability results. But one of the central points that Turing made in his 1947 ‘Lecture to the London Mathematical Society’ was that the Mechanist Thesis is not just licensed by, but is in fact entailed by his 1936 development of CT. Wittgenstein’s argument thus demands careful scrutiny of both the relation of Turing’s argument to CT, and the cognitivist implications that the founders of AI read into CT as a result of Turing’s influence.

Before we consider these matters, however, we must first satisfy ourselves that Wittgenstein was, in fact, intent on repudiating Turing’s mechanist interpretation of his computability thesis. It has long been a source of frustration to Wittgenstein scholars that no overt mention of this issue is to be found in Lectures on the Foundations of Mathematics: Cambridge 1939, despite Turing’s prominent presence at these lectures. Indeed, until recently, it might have been thought that the title of this chapter makes the still further unwarranted assumption that Wittgenstein was even aware of ‘On Computable Numbers’. But any such doubts were laid to rest by the discovery of an off-print of the essay in Wittgenstein’s Nachlass, and, even more important, the following mystifying reference to Turing Machines which occurs in Remarks on the Philosophy of Psychology:

Turing’s ‘Machines’. These machines are humans who calculate. And one might express what he says also in the form of games. And the interesting games would be such as brought one via certain rules to nonsensical instructions. I am thinking of games like the ‘racing game’. One has received the order ‘Go on in the same way’ when this makes no sense, say because one has got into a circle. For any order makes sense only in certain positions. (Watson)

(RPP I:§1096)

The latter half of this passage is clear enough: Wittgenstein is saying that Turing’s Halting Problem is no more epistemologically significant than any other paradox in the philosophy of mathematics.¹ The confusing part is the opening sentence. On first reading, this sounds hopelessly obscure: a clear demonstration of Wittgenstein’s failure to grasp the significance of Turing’s Thesis vis-à-vis recursion theory. Yet another way of describing the goal of this opening chapter, therefore, will be to see what sense can be made of this curious remark.

To see what Wittgenstein was driving at here, we have to work our way through a prolonged discussion of the nature of calculation in Remarks on the Foundations of Mathematics (see §2). But before we look at this material, it will be salutary to fill in some of the background to Wittgenstein’s thought. In a widely quoted passage from The Blue Book, Wittgenstein told his students:

The problem here arises which could be expressed by the question: ‘Is it possible for a machine to think?’ (whether the action of this machine can be described and predicted by the laws of physics or, possibly, only by laws of a different kind applying to the behaviour of organisms). And the trouble which is expressed in this question is not really that we don’t yet know a machine which could do the job. The question is not analogous to that which someone might have asked a hundred years ago: ‘Can a machine liquefy a gas?’ The trouble is rather that the sentence, ‘A machine thinks (perceives, wishes)’ seems somehow nonsensical. It is as though we had asked ‘Has the number 3 a colour?’

(BB: 47)

Whether or not Turing had read The Blue and Brown Books, he would almost certainly have been aware of Wittgenstein’s opposition to the Mechanist Thesis. Indeed, it seems plausible to suppose that the Turing Test represents Turing’s opposition to Wittgenstein’s critique, using a Wittgenstein-like argument.

Wittgenstein’s last point—a veiled allusion to Frege—is that, unlike the question of whether or not a gas can be mechanically liquefied, the question of whether a machine thinks is problematic in much the same way as the question of whether or not 3 is coloured; i.e. both are troubling because they transgress rules of logical grammar, and not because we do not as yet possess the means to answer them. But Turing might have regarded this argument as simply a request for the criteria whereby one would respond to each question. That is, in Turing’s eyes, the proper response to make to Wittgenstein’s argument might have been that, if there is—at present—no method to ascertain whether or not 3 has a colour, then the latter question might well be regarded as a category-violation. But the point of the Turing Test is to argue that there is no a priori reason why we should not apply the same criteria to the question of whether or not such-and-such a machine can think as to the question of whether or not such-and-such a subject can think, given that both inferences are based on observable behaviour. In other words, whether or not the answer is affirmative, we can at least see that the question of whether machines can (or ever will be able to) think is empirical.

Perhaps the most striking aspect of the above passage from The Blue Book is simply the date at which it was written. This was in 1933, nearly ten years before Turing began to think seriously about the Mechanist Thesis. Close to the same time, Wittgenstein wrote in Philosophical Grammar.

If one thinks of thought as something specifically human and organic, one is inclined to ask ‘could there be a prosthetic apparatus for thinking, an inorganic substitute for thought?’ But if thinking consists only in writing or speaking, why shouldn’t a machine do it? ‘Yes, but the machine doesn’t know anything.’ Certainly it is senseless to talk of a prosthetic substitute for seeing and hearing. We do talk of artificial feet, but not of artificial pains in the foot.

‘But could a machine think?’—Could it be in pain?—Here the important thing is what one means by something being in pain [or by thinking].

(PG: 105)

Here is yet further evidence—already familiar from the writings of Curry and Post—that the Mechanist Thesis was in the air at least a decade before Turing began serious work on it. From Wittgenstein’s point of view, ‘On Computable Numbers’ represents a misguided attempt to integrate independent issues in mathematical logic and the philosophy of mind. And it was precisely Turing’s bridging argument which concerned Wittgenstein.² Turing discusses the nature of Turing Machines (‘computing machines’) at two different places in ‘On Computable Numbers’: §§1—2 and §9. In the first instance he defines the terms of his argument, and in the second he takes up his promise to defend these definitions. Thus, in his justification, Turing naturally shifts from mathematics to epistemology, and, in so doing, he introduces a theme which constitutes the focus of Wittgenstein’s remarks.

In essence, Wittgenstein objects that the mathematical and philosophical strands in ‘On Computable Numbers’ are not just independent of one another but, indeed, that the epistemological argument misrepresents the mathematical content. The remark that Turing’s machines ‘are humans who calculate’ is only concerned with the prose presented at §9; but its corollary is that we must go back and look at the mathematical content of Turing’s achievement in such a way as to avoid Turing’s philosophical confusion.

Turing introduces Turing Machines in §§1—2 with the following argument:

We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q₁ q₂, …, q_R which will be called ‘m-configurations’. The machine is supplied with a ‘tape’ …running through it, and divided into sections…each capable of bearing a ‘symbol’. At any moment there is just one square, say the r-th, bearing the symbol G(r) which is ‘in the machine’. We may call this square the ‘scanned symbol’. The ‘scanned symbol’ is the only one of which the machine is, so to speak, ‘directly aware’.

(Turing 1936:117)

This reiterated warning of the anomaly—’so to speak’ and inverted commas around ‘directly aware’—highlights the premise which Turing felt constrained to justify.³ When read as a strictly mathematical affair, §§1—2 operate as an abstract outline for how to construct a machine that can be used to execute calculations, with no reason to present the argument in terms of the various cognitive terms with their attendant qualifications.

Turing’s Thesis reads as follows: suppose it were possible to transform a recursive function into binary terms. It would then be possible to construct a machine that could be used to compute analogues of those functions if it used some system which could encode those ‘0s’ and ‘1s’. Both the function (the table of instructions) and the argument (the tape input) must first be encoded in binary terms, and then converted into some mechanical analogue of a binary system. Turing speaks of the machine scanning a symbol, but that is entirely irrelevant to the argument; for how the binary input is actually configured and how the program/tape interact is not at issue.

Turing was not presenting here the mechanical blueprints for a primitive computer but, rather, a computational design, which only five years later he sought to implement using electrical signals to represent the binary code. And he did this with his version of CT, which demonstrates that:

All effective number-theoretic functions (viz. algorithms) can be encoded in binary terms, and these binary-encoded functions are Turing Machine computable.

The thesis which Turing defends at §9 is not this, however, but rather the epistemological premise that ‘We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q₁, q₂, …, q_R which will be called “m-configurations”’ He begins by analysing the conditions which govern human computing. The crucial part of his argument is that:

The behaviour of the computer at any moment is determined by the symbols which he is observing, and his ‘state of mind’ at that moment…. Let us imagine the operations performed by the computer to be split up into ‘simple operations’ which are so elementary that it is not easy to imagine them further divided. Every such operation consists of some change of the physical system consisting of the computer and his tape. We know the state of the system if we know the sequence of symbols on the tape, which of these are observed by the computer…and the state of mind of the computer…. The simple operations must therefore include:

• Changes of the symbol on one of the observed squares.
• Changes of one of the squares observed to another square within L squares of one of the previously observed squares….

The operation actually performed is determined…by the state of mind of the computer and the observed symbols. In particular, they determine the state of mind of the computer after the operation is carried out.

We may now construct a machine to do the work of this computer. To each state of mind of the computer corresponds an ‘m-configuration’ of the machine….

(Ibid.: 136–7)

This passage raises a highly intriguing question à propos Wittgenstein’s objection that Turing’s machines ‘are really humans who calculate’. For this looks like the exact opposite: viz. that Turing has actually defined human calculation mechanically, so as to license the application of quasi-cognitive terms to describe the operation of computing machines. So why, then, did Wittgenstein not make the converse point that ‘Turing’s humans are really machines that calculate’? The answer to this question, as we shall see in the following section, lies in Wittgenstein’s discussion of the nature of calculation in Book V of Remarks on the Foundations of Mathematics.

WITTGENSTEIN’S RESPONSE TO TURING’S THESIS

§2 ‘Does a calculating machine calculate?’

In a passage that can be read as a direct response to Turing’s argument at §9, Wittgenstein asks:

Does a calculating machine calculate?

Imagine that a calculating machine had come into existence by accident; now someone accidentally presses its knobs (or an animal walks over it) and it calculates the product 25×20.

I want to say: it is essential to mathematics that its signs are also employed in mufti.

It is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into mathematics.

Just as it is not logical inference either, for me to make a change from one formation to another (say from one arrangement of chairs to another) if these arrangements have not a linguistic function apart from this transformation.

(RFM V: §2)

Wittgenstein’s point here is that it is essential to what we call ‘calculation’, or ‘inference’,⁴ that we employ a proposition like ‘25×20 = 500’ in our everyday interactions, and that we don’t just treat this as an interesting pattern (which, as Wittgenstein says elsewhere, might serve as the central motif on a piece of wallpaper). That is, we say things like ‘If S owns 25 shares of XYZ which is currently trading at $20, then S must have $500 invested in XYZ’. And if asked to justify this, we do not insist on taking receipt of the certificates and counting them out; rather, we simply respond that 25×20 = 500.

Interestingly, Wittgenstein seems to be denying here the one part of Turing’s argument that no one has ever questioned: the idea that recursive functions are mechanically calculable. The point that Wittgenstein is driving at is that the concept of calculation cannot be separated from its essential normativity. This becomes clear two passages later, when he asks us to

Imagine that calculating machines occurred in nature, but that people could not pierce their cases. And now suppose that these people use these appliances, say as we use calculation, though of that they know nothing. Thus e.g. they make predictions with the aid of calculating machines, but for them manipulating these queer objects is experimenting.

These people lack concepts which we have; but what takes their place?

Think of the mechanism whose movement we saw as a geometrical (kinematic) proof: clearly it would not normally be said of someone turning the wheel that he was proving something. Isn’t it the same with someone who makes and changes arrangements of signs as [an experiment]; even when what he produces could be seen as a proof?

(RFM V: §4)

That is, we do not say that S ‘...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• PREFACE
• ACKNOWLEDGEMENTS
• ABBREVIATIONS
• 1: WITTGENSTEIN’S RESPONSE TO TURING’S THESIS
• 2: THE BEHAVIOURIST ORIGINS OF AI
• 3: THE RESURGENCE OF PSYCHOLOGISM
• 4: MODELS OF DISCOVERY
• 5: THE NATURE OF CONCEPTS
• NOTES
• BIBLIOGRAPHY