The Hollow Universe (extracts)

Charles de Koninck

Laval, 1964


These lectures are not intended as a brief course in the philosophy of science. They are concerned only with views of thought and nature suggested by certain advanced interpreters of the scientific outlook. The final opinion of these men is by no means accepted by all scientists, and indeed is not even the common view amongst them. Yet it is the one which, in the popular mind, is thought the most truly scientific. We have all heard of computer-men who would lead us to believe that they are on the way to constructing machines as truly capable of mathematical thinking as any man has ever been, with the implication that we shall have to change our mind about the nature of man as characterized by reason. As another sample of the advanced views, we have Bertrand Russel's pronouncement that physics proves man to be a mere collection of occurrences, that 'Mr. Smith' is in fact a collective name for a bundle of events. Again, there is the assertion of certain outstanding biologists that 'what life is' has become a meaningless question; that if the behaviour of organisms is to be explained at all, this will have to be done in terms of physics and chemistry.

We do not question the competence of those who hold such views: all of them have made signal contributions in the field of science and its philosophy. But it may be opportune to determine the import of their opinions by comparing them with doctrines which they are believed to replace. Every conception which seems the consequence of genuine progress in science, no matter how nihilistic and disturbing, must be honestly faced, and this is all that I mean to attempt. It should be clear, then, that my intention is not to attack the new positions, except where they are intended as total and final, Rather, my aim is to show that unless we examine them in the light of earlier views, we shall fail to appreciate what has actually been achieved; i also hope to make plain that if we generalize this new scientific outlook as it is commonly understood, if we accept it as the one true way of thinking about nature, the only wonder left to us will be wonder at the hollowness of the universe, both of nature and of thought. Even Einstein's conception of the material world as the manifestation of a wondrous intelligence would be out of date. A friend tells me that his little daughter, having seen the title of these lectures, begged him to ask me: 'Is the hollow in the universe or in our heads?'

The World of Symbolic Construction, or Two is One Twice Over

You may have recently noticed in the newspapers that groups of psychologists, neurophysiologists and linguists gathered, late in November, at Britain's National Physical Laboratory for an international conference on The Mechanization of Thought Processes. The scientific correspondent of the Manchester Guardian reported on this occasion

that it is rash to rule out the likelihood that one day it may be possible to design machines which can for themselves (given suitable experience of practical problems) form the kind of judgment which characterizes the higher flights of creative intellectual activity. It will be a long time, of course, before a machine will do for modern physics what Einstein did for it at the turn of the century. But it is important that at this stage there appears to be no obvious reason why this should not turn out to be possible.

I am not shocked at this bold anticipation. The challenge of the machines is a fine thing for us all if it drives us to investigate more clearly what thought is, and, if need be, to disengage the 'higher flights of creative intellectual activity' from the machinery which attends them in the human brain. Meanwhile, in the sense in which logic and mathematics are understood nowadays, the new electronic hardware certainly proves that machines can be splendid logicians and mathematicians.

The foundation of our ultra-modem logic and mathematics was already discerned, and very clearly, by a Greek philosopher, Democritus, some 2,400 years ago. The position which Aristotle attributes to this predecessor of his is again so disarmingly simple that I have some misgivings in offering it as the basis of the vast edifice which in the last few centuries has been reared upon it. But that was the way with Greek philosophers: in pondering the simplest things and searching in them for the basis of whatever needs to be explained, they showed themselves to be possessors of true wisdom. Edgar Allan Poe makes an interesting remark somewhat to this effect in The Purloined Letter:

The principle of the vis inertiae . . . seems to be identical in physics and metaphysics. It is not more true in the former, that a large body is with more difficulty set in motion than a smaller one, and that its subsequent momentum is commensurate with this difficulty, than it is, in the latter, that intellects of the vaster capacity, while more forcible, more constant, and more eventful in their movements than those of inferior grade, are yet the less readily moved, and more embarrassed and full of hesitation in the first few steps of their progress.

I waive what Poe calls 'metaphysics' and come to Democritus's opinion as reported by Aristotle: 'One thing cannot be made out of two nor two out of one.' Democritus meant something which may appear quite trivial, viz., that the number two is exactly the same as one plus one, so that to say 1 + 1 = 2 is just another way of saying 1 + 1 = 1 + 1. Two, then, is nothing new over and above one and one. Though Democritus is the first to put it so clearly, the same idea was already held by Thales, who believed that numbers were actually just heaps, or, as one would say today, mere classes or bundles.

The statement of Democritus may seem trivial, but I hasten to urge that it is not. Aristotle was fully aware of its grave implications and would have recognized at once how it illustrates the most basic principle of what our age intends by 'logic' and 'mathematics', as distinguished from some earlier meanings of these names. It contains, and not merely in embryo, the modern conception of number as a mere collection or aggregate, as a logical fiction and symbolic construction. Nearly half a century ago Bertrand Russell, in his Introduction to Mathematical Philosophy, explaining to the English public Frege's definition of number, observed quite pointedly 'that number is a way of bringing together certain collections . . . '. However, the implication of this conception of number was eventually more fully realized by that early pupil of his, to whom we have already referred, Ludwig Wittgenstein. In My Mental Development Lord Russell says that he himself 'had thought of mathematics with reverence, and suffered when Wittgenstein led me to regard it as nothing but tautologies'. Russell's disappointment must not obscure the fact that Democritus's way of treating number provides the most exact of the sciences with an indispensable tool, one that had been available of course from the earliest days of arithmetic as the art of calculation, but the vast potentialities of which had passed unnoticed. Plato called the technique 'logistike'; Aristotle's term was 'logismos'. But I must beg you to notice that both distinguished this technique of calculation from the science which it served. Science was the effect of reasoning by way of syllogistic demonstration, as exhibited later and more completely in the Elements of Euclid. Demonstration, in this sense, was not the same as calculation—although in mathematics one could not demonstrate without computing. But the computation itself was no proof. We shall return to this point later. Meantime, let it suffice that mathematics nowadays is almost wholly identified with and confined to the art of calculation. This is what is implied by the conception of number as 'a way of bringing together certain collections'. Numbers are defined by the operations that can be performed upon them. As Hermann Weyl put it: 'their being exhausts itself in the functional role which they play and their relations of more or less.' As a result, 0, 1, — 1, 1/2, √2, etc. are in this sense just as much numbers as 2 or 3.

That mathematics is now largely equated with its technique, or rather that all mathematical entities are defined by the technique for manipulating them, is an observation based upon what mathematicians now attend to, especially those who reflect upon what they are doing. This assimilation of mathematics with the mushrooming art of calculation was explicity asserted by the late John von Neumann: The new calculus, discovered by Newton and Leibniz, or rather all of analysis, which sprang from it, must be held 'the first achievement of modem mathematics, and it is difficult to overestimate its importance. I think it defines nore unequivocally than anything else the inception of modern mathematics, and the system of logical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.'


Mental Construction and the Test of Experience


Let us get on with our hollow universe. We have seen what is meant by the hollowness of the world of mathematics. We have shown what this emptiness means in connection with number, which is sufficiently clear in the elementary example of 2 conceived as exactly the same as 1 + 1.


Various mathematical tools had been tried for digging down to the basis of the physical world, and the last quarter-century has shown that group theory is a most suitable tool. To show just how mathematics first gets a grip on basal entities whose nature and activities are essentially unknowable, I can only refer to that lecture of Eddington's from which we have just quoted. For the present we shall accept his authority.

The application of group theory in quantum physics actually gives more weight to the observation by Einstein, made a few years earlier, which we cited in our first lecture:

It is my conviction that pure mathematical construction enables us to discover the concepts and laws connecting them which give us the key to the understanding of the phenomena of Nature. Experience can of course guide us in our choice of serviceable mathematical concepts; it cannot possibly be the source from which they are derived; experience of course remains the sole criterion of the serviceability of mathematical construction for physics, but the truly creative principle resides in mathematics.

We must be careful, however, not to confuse this mathematics, and its use, with the mathematics of traditional philosophy and its application to nature, as in Aristotle's ingenious theory of the rainbow. As already suggested, 'mathematics' in Aristotle's day meant a very different type of knowledge, whose demonstrations were derived from definitions of what the subjects were apart from the way in which they were given, Even the 'operational demonstration' (the expression was used by Aquinas), in which we construct an equilateral triangle to show that it exists (that is, as an abstract subject entailing definite properties, of which we assert neither that it exists in reality, as Plato thought, nor that it does not), is not concerned with the triangle as a construct: construction and construct in an operational demonstration are only means of getting at what an equilateral triangle is. For the old philosophers, mathematical abstraction was of a very special kind, differing from that of all the other sciences, and one little understood by Aristotelian scholars even to this day. In our time, however, the construct is the subject qua construct, and it is only this operational aspect of mathematical entities that is applied to the investigation of nature.

Now, sheer mathematical constructs, endowed with that hollowness which renders them so efficient as tools, must make over into their own likeness the natural world to which they are applied. The science which uses these symbolic devices, mathematical physics, thus finds itself able to reveal the world of nature only to the extent in which it can be caught in what has been termed the selective screen of symbolic construction. As Eddington remarked: 'We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the foot-print. And lo! it is our own.' Nor must we forget Einstein's warning, that pure mathematical construction 'can give us no knowledge whatsoever of the world of experience; all knowledge about reality begins with experience and terminates in it. Conclusions obtained by rational processes are, so far as Reality is concerned, entirely empty.' If our rational constructs and processes are not entirely empty, then, it is thanks to experience alone. This is what we mean by the test of experience.

There is a further sense, one even more generally agreed upon, in which the world of mathematical physics is hollow, or, as Eddington called it, 'a shadowy world of symbols'. This aspect of the matter is implied in the basic definitions of the science, and it was Einstein who did most to reveal it. I am referring to the methodological principle that lies at the very foundation of Relativity Theory, namely, that the mathematical physicist must define his terms by describing the way in which he reaches them, that is, by a process of mensuration which results in a measure-number; a symbol whose meaning is defined by reference to a certain operation and the incidentals that attend it. For instance, length could be defined as 'what is extended in one dimension'; and such a definition is intended to establish 'what length is,' no matter what length it may be. Whatever may be intended by such definitions, as distinguished from mere interpretations of names or symbols, they are of universals only (it does not matter for the present what universals are, whether they are, or in what sense they would be if they were). The point is that the mathematical physicist, if held to such a definition, even though not obliged to reject it, can make no headway. What he means by a length is 'when we take a reasonably fair copy of a certain platinum-iridium bar kept in Paris at a controlled temperature of O0 centigrade, etc., and apply it, once or more, successively or by division, to know a given distance between A and B, the result of the operation will be expressed by xm'. But in this mode of defining, the standard of length can of course have no length, when there is no prior standard. Length, then, only appears when the measurement is actually made. It is not an abstract operation upon something abstract: it is performed here and now, or then and there; and the time-factor may be essential; nor, if the metre be his standard, may the physicist forget the reference to Paris. He cannot afford abstraction of universal from particular. Now, to this whole process, means and result, to this complex operational unity, he cannot give a true name, but he can embrace all of it by the particular value of m, let this be 1/4 or 1000. Such length will turn up on the graduated scales of weighing machines and clocks, by which weight and time are defined in their turn, and even on the scale of the thermometer which entered into the definition of length. As I have noted elsewhere, in this type of definition, the crucial term is when. If the physicist said 'length is ...' instead of 'length is when . . . ' he would revert to a mode of definition that tells, or pretends to tell, 'what' a thing is absolutely. However, having defined length in the only way profitable to him, the physicist may assert that 'this is length—much as Russell handled the number 2: 'In fact, the class of all couples will be the number 2, according to our definition.' But the physicist who so takes length can only mean that this understanding of it is the only one with which he will concern himself. Allow him this sort of definition and he will astonish you with what he can do with it. Let us notice, however, that if this type of definition were the only valid one at all levels of science, the definition of man would have to be something like this: 'when I bump into something and it produces a series of sounds like "Where do you think you're going?", this is man'— which would be a possible enough interpretation of the name.


It may be noted as well that scientists are most often bewildering because they use words where there is no need for them, when they should stick to their symbols; and where words in fact turn out to be as meaningless as some of them say they are. Allow me to illustrate such a case. Bertrand Russell's famous Mr. Smith is, he assures us, really a mere bundle of occurrences, a collection of events; and his name is hence no more than a collective one for such a swarm. This, according to Russell, does away with 'substance', reducing it to a mere linguistic convenience. On his understanding of 'substance', this opinion is unassailable. The meaning he reads into substance is a very strange one, one that I am sure never occurred to Aristotle, and which in fact makes no sense. Substance, for Russell, is something that supports accidents in the way in which a floor supports a table: if you could scratch off or peel away the accidents, he thinks, you would be left with something reassuringly solid, like a stone for an unscientific imagination, chock full of stone through and through, with no hollow left. Swiss cheese would therefore be rather insubstantial; and a sponge, or a flurry of snow, far less substantial than Sir Arthur Eddington's familiar table. Now, it may well be that some people associate such an irrelevant image with the notion of substance, but nowhere shall we find support for it in Aristotle.

Here is how the Philosopher reached his conception of substance. He first observed our way of speaking of things. When speaking, we always have something in mind, yet it may be no more than the sounds or shape of the words, which may nevertheless be used with an air of erudition. (This could be true of philosophers ut in pluribus.) But he goes on to show that we do say things like 'Socrates is pale', or that 'he is five feet eight inches tall', or that 'he weighs two hundred pounds, knows Greek', and so on. All of this we may say of Socrates, but we never say him of anything else. We do not think of saying that 'five feet eight inches tall is Socrates'. (except in Latin syntactically, or in poetry!) And, though in a proposition of identity, we can say Socrates of Socrates, we can never say him of his wife, Xanthippe.

Now, has this way of speaking, and indeed of thinking, some foundation in reality? Is Socrates Socrates? Does he have colour, size, weight, and grammar? Is he the husband of Xanthippe? Is he a corrupter of youth (for showing them that their thoughts about virtue were somewhat confused)? Can this way of speaking be true? When I say that he is seated, and he is in fact sitting down, is it not true as I say it? What is there about Socrates that would make it false to say that he is his wife ? Or to say that he is grammar? or that the grammar he knows is Socrates? Not even Russell, surely, would allow this manner of speaking—at least not in the cases just mentioned.

As we see and say that Socrates is pale, our mind forms a certain relation expressed by this way of speaking. This immanent product of our mind, a work that remains within and could not possibly exist outside the mind, is called a second intention or understanding: a something with which our own mind invests the subject, as known to us and in our mind (whatever 'mind' may stand for). Now it is invitingly easy to define realities by means of these second intentions, but the fatal shortcoming of this sort of definition is that it tells only how the reality is to be thought or spoken of, not. how it is. To define substance in this way, by its mode of predication, is to define substance logically (in the ancient sense of this term). Substance, logically, is a mere intention of reason, something on a plane with linguistic conveniences. Yet if it is true to say that Socrates knows his grammar, and false to say that the grammar Socrates knows is Socrates, there must be some grounds for this: our way of speaking is dictated by something in this pale, literate Socrates himself, A lengthy dialectical pursuit will bring us to what is called a natural or real definition of substance: 'that which is in itself and not in another.' And whatever we may profess about this general definition of substance, as found in man, horse, or tree, our normal language supposes it, and we certainly treat our neighbour as if he were such a thing. Of some objects, like stones or snowflakes, we may not feel so sure—since the physicists have upset us by showing that, in a sense, these are bundles of non-stones or non-snowflakes—but even these we consider to be substantial at least in the way that a crowd is, though we cannot identify the kind of unity they have. One thing is plain despite the difficulty of certain examples: we never feel that substance needs the reinforcement of solidity in order to be substantial, that it is like Parmenides' geometrical sphere, or that it demands a resistant material in order to exist. To be a substance, I no more require what Russell calls by that name than the earth requires an elephant to rest upon.

That I am a swarm of sparsely scattered electric charges, rushing headlong in nearly empty space, the combined bulk of the swarm less than a billionth of myself, does not in the least distress me as a substance. Neither they, in their mad rush, nor I, have any doubts as to whom they belong. Though I keep on losing and acquiring them in billions, I sit here quite unperturbed by the frightful turmoil, nor do I feel any more divided up by their multiplicity than by having two eyes, two legs, and a heart, all outside of my feet. I am a loose enough assemblage of limbs, but does that prevent me from being myself? Indeed, how could I be myself without these odds and ends? But if I were truly described by these multiplicities as such, if I deserved nothing more than Russell's collective name, then, speaking literally, I should always use the royal 'we': 'We' the bundle of events.

In plain English, what are we asserting? That the physicist need not know what a man is any more than the shipping agent weighing trunks need be aware of what is in them. Far less, in fact. To the physicist, Mr. Smith is part and parcel of the shadowy world of symbols; and how else can a physicist treat of Mr. Smith ? Indeed, it would mean a much closer approach to what Mr. Smith is to state that he can be turned into a good soap—which is demonstrable fact. So all we ask is that the mathematical physicist realize what his science supposes: that if he wishes to deal with men he cannot permit himself the use of words. It is precisely the improper use of words which makes Russell's statement concerning Mr. Smith so delightfully comic.


The prevailing 'scientific outlook', above all in the Commonwealth, is now more than ever dominated by Hume. His critique does not affect science as a tool; indeed, mathematics is now largely recognized for what he thought it was—a tool, and a quite reliable one. But his treatment of induction and causality is now being used to snuff out that first type of wonder : wonder about what a cause is, what is cause of what, what movement is, what place and time are, and so on. His apparently cold analysis has met with considerable popular success; and its effect is to drive from the human mind that primordial curiosity, the parent of all other motives of inquiry, which Aristotle describes in his account of the beginnings of science and wisdom.


Surely it is disheartening to reflect that we live in an age when it can be necessary, not merely to explain Einstein's speculative goal, but even to defend it against another type of mind which would have it that his time might have been better spent in the practice of plumbing. But the spirit of intellectual nihilism is gaining ground. It is frightening to think of the extent to which people are now being encouraged to banish from the minds of their children great questions as devoid of all meaning; to dispel the wonder which is a young mind's birthright; to confine their spirit to petty problems that can be answered once and for all to the satisfaction of reasoners incapable of raising a question to begin with. We now have a philosophy to show that there are no problems but those which it has shown to be no problem; and to decree that there is no philosophy other than one that is a denial of philosophy. Under the twinkle of a fading star, Hollow Men rejoice at a hollow world of their own making.

The Lifeless World of Biology

THE title of this lecture may seem a contradiction, and perhaps also a piece of impertinence. Yet the judgement which it makes is both true and moderate. Indeed, it should become clear in the course of our discussion that, to the adjective lifeless, must be added the adjectives inorganic and functionless. Modern biology, if some of its distinguished representatives are to be believed, dare not call itself true science unless it avoids and ignores all that naturally comes to the minds of ordinary people when they think of familiar animals and plants. Nor have I been provoked to this general comment by one or two radical works like Mr. N. W. Pirie's The Meaninglessness of the Terms Life and Living (1937), or his more recent The Origin of Life (1953). Long before I was aware of his opinion, I had pointed out in an Introduction a l'etude de l'ame (1947), that neither the courses in biology followed by myself more than a quarter of a century ago. nor anything I have read since, offered any reason why the terms ' life ' and ' inanimate ' should be used at all except as 'linguistic conveniences'. The reason is that the biology I am talking about is resolved to be sternly empirical, while it can find nowhere any definite, empirically defined property able to separate, once and for all, the animate from the inanimate. Irritability, self-repair, nourishment, growth, and reproduction, as described in typical modern treatises, can be no more than provisional hypotheses, if they amount even to this. It may of course be granted that, as a matter of method, we can and should attempt to explain so-called living phenomena in terms of what we call the inanimate, as far as possible; and that, when we cannot, we should at least keep an open mind on the question. However, even this apparently broad view will lead to difficulties. 'Inanimate', after all, is a negative term: I mean that, linguistically, it is a negation of 'animate', so that it looks none too easy to get rid of the living when, without at least the idea of it, the 'non-living' cannot be named.


Taking for granted our ordinary-acceptance of ' living ' and ' non-living ', these writers, from the start, resolve to explain them in terms of the kind of life we know least about, that is, in terms of the so-called lowest animate forms. Once this method is adopted to the exclusion of any other, there is no escaping Professor Beck's conclusion:

As perceptual objects, plants are plants whether we call them living or not: 'life' is a conceptual object. In other words, Pirie is correct: 'life' is beyond rigorous definition—but he, I, we will speak of life because we all know what it means in the large area of nonambiguity. The errors to be avoided are compulsive rigidity and failure to be happy in the company of uncertainty. When asked what viruses are and what they do, we can answer. When asked, what is life, we must reply with no more or no less than an enigmatic smile.


[Now] we can be reasonably sure about the distinction to be made between a live Socrates and a dead one; but we cannot be anything like so sure whether this particular organism is an animal or a plant; nor whether this object, at this moment, is even a plant or something not alive at all. Now, our objection was that the man who hopes to arrive at some definition of life, enabling him to set life apart from non-life, should never begin with the study of what is alive very obscurely, if alive at all. Why not begin with horses? He can see them without a microscope. Or why not start with the kind of thing that asks what horses are? which eventually constructs microscopes and finds itself faced with the obscure forms of life?

The full chapter of the Lifeless World of Biology can be found (together with other articles) at the Institute for the Study of Nature.


Reckoning with the Computers

If the word 'thought' be taken in the meaning conferred upon it by the late A. M. Turing, there is no doubt that our computing-machines think; just as, according to a certain meaning of 'select', our potato-sorting machines do select tubers of various sizes. There is no reason to find fault with writers like Turing over such use of words. When computer-men agree to call the operations performed by a calculating-machine 'thinking' and to speak of them as 'logical', they are exercising the technician's right to impose new meanings on old names, or perhaps to revive worn-out meanings recorded only in etymological dictionaries. De nominibus non curat sapiens. So we may easily receive Turing's opinion:

The original question, 'Can machines think?' I believe to be too meaningless to deserve discussion. Nevertheless I believe that at the end of the century the use of words and general educated opinionw will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.

But we are perhaps closer to the end of the century than Turing knew. As the word 'thought' is now used frequently enough by philosopher or scientist, it is difficult to see why its meaning cannot be verified of things other than man. Carrots, apparently, 'know how' to grow, and even wheelbarrows how to be pushed around. The human mind may be more complicated than UNIVAC, and UNIVAC more complicated than a wheelbarrow, but the position of the computer-men is that all three are machines in the same sense, with operations equally mechanical. After making this point, Turing might have been expected to let the matter rest. He does not. Certain difficulties may be raised against the identification of 'thinking' with both the mental act of man and the mechanical movement of the machine, and these difficulties he is determined to face. The first opinion opposed to his own he calls the 'theological objection', and he formulates it as follows: "Thinking is a function of man's immortal soul. God has given an immortal soul to every man and woman, but not to any other animal or to machines. Hence no animal or machine can think". To this argument he appends a footnote: "Possibly this view is heretical. St. Thomas Aquinas (Summa Theologica, quoted by Bertrand Russell, A History of Western Philosophy, Simon and Schuster, New York, 1945, p. 458) states that God cannot make a man to have" no soul. But this may not be a real restriction on His powers, but only a result of the fact that men's souls are immortal, and therefore indestructible.'

Turing observes that he himself is

not very impressed with theological arguments whatever they may be used to support. Such arguments have often been found unsatisfactory in the past. In the time of Galileo it was argued that the texts, 'And the sun stood still . . . and hasted not to go down about a whole day' (Joshua, x, 13) and 'He laid the foundations of the earth, that it should not move at any time ' (Psalm cv, 5) were adequate refutations of the Copernican theory. With our present knowledge such an argument appears futile. When that knowledge was not available it made quite a different impression.

The argument which he professes to cite is of course a hopeless one. But the first question is why he should concern himself with theological objections at all. Even when all appeal to authority is ruled out, as it ought to be in philosophy as such, his theory, one would think, might still be discussed in terms of what we already know and understand. However, it is precisely here that the difficulty lies nowadays. The new conceptions of scientific rigour are such that no acceptable account of what we already know and understand seems possible. Lord Russell, Ludwig Wittgenstein, and Kurt Goedel constitute empirical proof of this retreat, or advance—for it may be looked upon as either. An advance it is at least in this respect that it reveals, once for all, what happens when certain types of question are entirely put aside. These I would describe as the kind which, historically, have led to endless controversy; I mean philosophical questions, and all of them, at least in the Platonic or Aristotelian sense of philosophy. This initial exclusion of all the classic problems was plainly formulated as a principle by the late Richard von Mises when he said: 'It is impossible to accept as the basis of mathematics merely statements that seem self-evident, if only because there is no agreement as to which statements actually belong to this class'. This of course means the immediate rejection of mathematical science as understood by the ancients, who considered it a branch of philosophy and the very model of what they meant by a disciplina. And, though it may be objected that the mathematics which von Mises is thinking of has almost nothing to do with that of Plato or Aristotle, this objection is actually irrelevant. The distinction between science and calculus, like every similar distinction, is ultimately based on some self-evidence or other, about which, according to the attendants of the new mechanical brains, there may always be disagreement. Well, what line of inquiry is left to us? Where do we go from here? This question too has been taken care of, not by the computer-men, but by thinkers equally up-to-date. It seems that, to find out where to go, or even where to start, would be to deprive oneself of the spirit of adventurous inquiry. People who want to be sure of something—or even, more modestly, of nothing-have no place in our time. There exists a type of psychoanalysis to show that they are victims of a morbid craving to return to the vegetative night of the embryo. As for debate about the soul and its destiny, there is reason to fear that, if the men behind the calculators take part, the result can only be a comedy of errors. For instance, if by soul is understood what the latest critics of ancient philosophy mean by this word (a meaning easily derived from the context of their criticism), any Aristotelian, whether Averroist, or Thomist, or whatever his dissension, will realize that he owns no such thing, personally or collectively, and will shudder at the prospect of ending his days with an 'immortal' one. The modern interpretation of this term, for instance, is as remote from Aristotle's psyche (originally 'breath', 'wind', 'breeze') in Book III of the De Anima, as would be 'exhaust', used of the exhaust of a modern engine. So we cannot make out, any better than Lord Russell can, why the dissolution of that tedious bundle of events, Mr. Smith, should be trailed by some kind of perpetual exhaust. No living thing stands in need of such a soul.

So, to pursue the rigour demanded by the bundle-of-events-and-computer-philosophy is to be led to this utter impasse. No questions can now be asked or answered; no statement made which is not a tautology; no mental act performed which cannot be matched by a machine. Since we are at a full stop anyhow, we may as well spend our time trying to understand what we have done, and why the result should be such a dead end. The attention paid by Turing to an argument supposedly taken from theology invites us to inquire what this science, which is not so easily silenced as philosophy, might be able to teach about the reasons for our plight. In the same ancient Literature to which he makes reference, there are several passages which throw a curious illumination upon that bundle of dust and that mechanical calculator which man has made of man. 'It is an old tale now', proclaims the prophet Jeremiah, 'how thou didst break in pieces the yoke of my dominion, didst sever all the bonds between us, crying out, / will not serve!' (ii.20). And Ezekiel adds the warning: ' .. . Such a fire I will kindle in the heart of thee as shall be thy undoing, leave thee a heap of dust for all to gaze at' (xxviii.18). Now St. Thomas, upon whom Turing draws for his specimen of theological reasoning, explains that 'the activity [of him to whom these words were addressed], averted from what is one and first, was bent upon the many of inferior things, and it was their primacy that he coveted'. This mysterious personality directed all his thought and action towards the scattered and sheerly multitudinous, his aim being to reduce everything to that alone. A lesser creature, like man, can pursue such a goal only in thought and, when he does so professedly, is called a sophist. Because the sophist, in his elaborate mental traffickings, uncontrolled by any standard of truth, has the infinite store of what is merely incidental, of ens per accidens (as well as an infinity of logics) to draw upon. He may thus make the worse reason appear the better, that which is least seem greatest, whatever there is most of seem synonymous with that which most truly is and, in the end, lead his hearers to conclude that nothing is but what is not.

What about poor Mr. Smith, then? 'Dust thou art and unto dust thou shalt return', the ancient writings tell him, in a phrase which acknowledges the 'thou' that he is, as well as the dust that he is made of. But the ancient writings care no more about a scientific attitude than they do about Copernican astronomy. The modern choice is to be scientific and, scientifically, Mr. Smith is a bundle of occurrences and that is all that can be known of him. The choice is simple enough, in all conscience: him is cast aside in favour of a 'mere bundle' of fleeting 'occurrences' which happen to nobody, as being far more a sheer many opposed to the one than the particles of dust helping to make up the per se whole of a human being. To meet this new scientific standard of reality, to announce the primacy now desired, how can one do better than to present the rational animal as a mere bundle rather than as a substance? If, even of those particles of dust, Mr. Smith holds in himself a number greater than the number of men on earth, as a bundle of events, he is multitudinous beyond imagination. So, with substance forever banished as 'a hopelessly muddle-headed notion', Mr. Smith, 'a collective name for a number of occurrences', no matter how you look at him, is Legion. Every single one of us is a crowd of something or other. 'I' always stands for a bundle, like the pronoun 'we' in 'We, the Cricket Club'. And everything else is Legion too. If a house could speak for itself, for example, without falling into the snare of mere linguistic convenience, it would say something like 'We, the boards, bricks, mortar, nails, etc.'; while each board in turn would cry 'We, the woodfibres, cells, etc.'; each brick in turn 'We, the molecules of calcium, silica, etc . . .', and so on, with not a thing knowing where to stop. Besides, whether there is a 'thing', and a 'where to stop', are now distressingly meaningless questions (unless stated in language exhibiting how meaningless, and then quite unimportant). It is some consolation to note that even the old non serviam, if mere linguistic phenomena be ignored, is now quite emptied of its ego. If 'ego' is only a collective sign for a bundle of non-egos, there just isn't anybody there to utter a defiance.

Heaps of dust, disconnected thoughts, bundles of occurrences with no string to tie them, all these are bad enough; but there is worse to come. We have not yet measured the full scope of the perverted desire to accord primacy to the things which have most of all the nature of the many. This appetite is not sated until it has triumphantly embraced the order of a multitude which is negative—its final feat, as it were, and one which can procure man's deliverance from anything at all that he chooses to be delivered from. In his discussion of the classic arguments 'professing to prove the existence of God', Russell offers the frightening spectacle of a thinker become the contented captive of a logical fiction, namely, of the kind of infinity which can have no meaning whatsoever outside the art of calculation. Here are his own words:

All of these [arguments], except the one from teleology in lifeless things, depend upon the supposed impossibility of a series having no first term. Every mathematician knows that there is no such impossibility; the series of negative integers ending with minus one is an instance to the contrary.

So, in the order of pure science, of speculative activity, man is asked to find ultimate satisfaction in seizing an infinity which is only a negation.

In the order of operation, as we have seen already, the new science asks man completely to renounce thinking as a power peculiar to him, and to persuade himself that he stands on the same level with his own tools, that is, of the complex tools called machines. We have come a long way from non serviam, at any rate, since the only thing tools actually can do is to serve, and in the strict sense of the word, as a hammer serves to drive nails; for a tool, like any instrument, is of its nature movens motum. Hence, Aristotle held that we ourselves, in one way or another, are the agent and final cause of artificial things. This opinion is no longer acceptable to science, of course, not only because it distinguishes between art and nature, but more especially because it distinguishes between instrument and principal agent. For the computers, if we can believe their interpreters, present us with an utterly new kind of tool: an instrument divorced from any principal agent, like a sign that does not signify, or a relation adrift without terms. The point is that the tool-maker, the agent, in whose interest the machine was built, has faded out of sight, he is 'refined out of existence'. And he has been deprived of all grounds for existing by the possibility of one computer giving birth to another, a notion which Russell might accept, since he has proved something to the effect that we may have tools of tools and nothing but tools, to no end, without end, like a series without a first term. Non serviam seems a sinister kind of boomerang.

Whatever the ultimate value of those venerable scriptures to which Turing referred, it is clear that they utter a strophe which he ought to find deeply significant, a strophe for which we have been attempting to provide an antistrophe. We have thought it our duty to expose the implications, both of non serviam, and of the accompanying desire that primacy be granted to the sheer many. And we have seen that the new 'light-bearers' do not express their rejection of ultimate order in the romantic fashion of Marx quoting Prometheus; nor by simply making over some of their own servitude to the machines by using these for 'the more repulsive drudgery' of science; but by actually identifying knowledge and science with the mechanical process which may attend them, by insisting that what goes on in the computer is the same as what goes on in man's more abstract thinking. If man could accept this identification, he would never again need to face any question with power to disturb him, so that it is not surprising that those who make this reduction do so with an air of triumph. As for the primacy of the many, its most entitative expression becomes the mere bundles that are Mr. Smith, Russell, and so on. Logically, of course, the dispersion must be carried on and extended to all things, and ultimately to the universe itself—that supreme bundle out-bundling all.

Lord Russell warns us that we may some day blow up our pieces of the cosmic bundle. The curious thing is that he is appalled at the prospect. If the grand scattering be surveyed from the scientific point of view, what reason can there be for emotion? For Russell, the time is neither in nor out of joint, so that it is difficult to see what significance he can attach to an eventual dislocation. Under his tutelage we ought to have learned, surely, that for all living creatures without exception, destruction implies only that it shall be as if they had never been, and that to attach any significance to the annihilation of such things is to betray a foolish devotion to that 'organismic conception of nature' which furnishes the world with substances, animate and inanimate, rational and irrational. Further, if Mr. Smith, like the good scientist he is, can accept his own impending dispersal as he would the scattering of a set of ninepins, one is at pains to see why his equanimity should be disturbed where the whole bundle of humanity is concerned. Such inconsistency looks suspiciously like nature thwarting theory. And even though Mr. Smith does not accept the destruction of all mankind as calmly as he does the effect of a well-aimed ball on the ninepins, what can it matter? The final catastrophe would take place in strict fidelity to law, and with merciful suddenness. Those who expected the event would be neither surprised nor mistaken, as Russell once explained:

Suppose you are walking in a thunderstorm, and you say to yourself, 'I am not at all likely to be struck by lightning.' The next moment you are struck, but you experience no surprise, because you are dead. If one day the sun explodes, as Sir James Jeans seems to expect, we shall all perish instantly, and therefore not be surprised, but unless we expect the catastrophe we shall have been mistaken.

Suppose we did precipitate that transformation of all matter into pure radiation, that 'stupendous broadcast' which was Arthur Eddington's version of a possible end of the world, why call it a catastrophe? Upheavals in the universe are an everyday occurrence. Life thrives on them. On our little planet, is anyone troubled by the fact that the sun is in continuous explosion? And any loss incurred may not be irremediable since, according to some people, when thermodynamic equilibrium is reached, the universal process of degradation will sweep into reverse, so that eventually the whole farce will be acted out again. And an almighty farce it has been and would be a second time, if it is a world where the distinctions between living and non-living, between rational and irrational, are to be rejected as evidence only of man's basic vanity, and even of his unfeeling cruelty towards what Turing calls 'the rest of creation'; where man is accused of 'brutality' because he cares more for the living than for the lifeless, and for man than for beast; a farce to which no words will ever do justice, if machines are brought to prove that rationality, the 'specific difference' of the human being, finds its proper home at last in its opposite.

Philosophical writings of Charles de Koninck