PART V—
SCIENTIFIC KNOWLEDGE—ITS SCOPE AND LIMITS

Preface to Part V:
Scientific Knowledge—Its Scope and Limits

In this part I revert to the knowing subject and the nature of his or her knowledge, beginning in chapter 18 with some well-known epistemological challenges (such as the Gettier counterexamples to knowledge as justified true belief). The goal here is to arrive at a robust definition of (scientific) knowledge as a characteristic of the knower . It turns out, if I may so put it, to be an ability rather than a commodity. The definition of knowledge given in this chapter is already to be found in The Philosophy of Science: A Systematic Account , but here it is modified and strengthened.

The other chapters in this part look at some generally assumed characteristics of scientific knowledge and question their adequacy. Chapter 19 examines the assumption, underlying a great deal of thought about the "exact" sciences, that science has privileged knowledge of quantitative properties of things. The relation between the quantitative and the qualitative is however one of the least well understood in the domain and I give what is perhaps a new view of it.

Chapter 20, as remarked in the preface to part III, is more speculative. Relativity theory has long been understood to imply that known characters of objects—particularly quantitative ones!—change at relativistic distances and speeds, but again what that really means has not always been clearly thought through. In particular what does it mean to make an adjustment here in the properties of an object there ? Observers there would not have the sense of moving at a very great speed relative to us, any more than we here now have the sense of moving at very great speed relative to them. So what does "relative motion" amount

to, in relation to our local understanding of motion? The issue here is one that recurs in the last chapter of this part, and is taken up briefly in an article that could not be included in this book because it is an entry in an encyclopedia, the Encyclopedia of Physics : "Our perceptions and naive thoughts," I say there, "are adapted to the scale of our bodies, our days, and our lives; relativistic and quantum phenomena have no direct bearing on them, and being equipped to envisage such phenomena would have been of no evolutionary advantage." The term "phenomena" is used in a loose sense here—strictly speaking, relativity and quantum theory offer us nothing phenomenological.

In chapter 21 I take up the problem of the self-enclosed character of scientific knowledge, another variant on the theme of hypothetical realism, and give reasons why circularity in knowledge is not necessarily vicious.

The final chapter is of a different order from the others—it is devoted to the work of a French philosopher of science and of literature, who saw as clearly as anyone has what is involved in the restriction of the imaginable to the local and macroscopic. "Imaginable" is used in a strong sense, in keeping with the meaning of the cluster of terms in French built on this root; we tend to use "imagination" to include the having of bold ideas of any kind (for example, one of the formulations in Science and the Theory of Value was "science is imagination controlled by experiment"), but, like the term "idea" itself, it has lost in English its close association with the visual image. Bachelard stresses throughout his work on the philosophy of science that it is reason, rather than imagination in the strong sense, that gives access to the theoretical structures of science, but in doing so he makes room for a different function of the imagination, namely, a poetic function, whose correlative status to the activities of science is not stressed by writers in English because it falls for most of us altogether outside the domain of inquiry. He is one of those very rare practitioners of the human sciences whose contributions to the understanding of the humanities on the one hand and of the hard sciences on the other are in a working equilibrium, and he deserves to be better known to the profession than he is.

18—
Is There (Scientific) Knowledge? Who Knows?

The title of this paper is complex—it packs in several layers of suggestion by the use of typographical devices. The first suggestion is that if "scientific" is an optional adjective for knowledge then if there is any knowledge—which is the question at issue—then some of it needn't be scientific. Or, to put it the other way round, I am asking the question about ordinary as well as scientific knowledge. The second suggestion, however, is that if the adjective "scientific" is an optional part of the question at issue, then the two kinds of knowledge, scientific and nonscientific, stand or fall together: if there is knowledge then there can be scientific knowledge; if there can't be scientific knowledge there can't be any knowledge. In other words, to be scientific is a permanent possibility of knowledge, if there is any. The third suggestion, taking "who knows?" in its colloquial sense, is that there is some doubt as to whether there is knowledge or not, and that this doubt isn't particularly easy to dispel, so that one might be inclined to throw up one's hands over the question. But the fourth suggestion, taking "who knows?" in a plain and straightforward sense, is that the answer to the question whether there is knowledge has something to do with the particular individuals who have it.

Epistemology is the central discipline of philosophy, and every philosopher has to come to terms with it. The stakes are high. Recently the very idea of knowledge as a reflection of the way the world actually is has come under attack from neopragmatists like Richard Rorty, who wish to discourage us from thinking that we are getting closer to the truth through the efforts of scientific inquiry, to encourage us in a kind

of benign floating in the stream of culture, from which we cannot escape. Actually, this program is rather appealing, especially in the later stages of a decaying civilization, which ours may well be; at least it does nobody any direct harm, which can hardly be said for bombs and industrial effluents. But it gives up too easily, and in my view for inadequate reasons, an old hope that knowledge cannot only be reliably acquired but also put to safe use for the benefit of people who really need better control of their world.

One of the things that has made the old standard of confirmed scientific knowledge vulnerable to the criticisms of the neopragmatists has been its conceit, its assumption (or the assumption of the people who thought they had it) that it would conquer everything, produce the answer to all problems, be total or absolute. It was no doubt the risk of this God-like pretension that the ancient Jews had in mind when they told the story of the Tree of Knowledge. But the fact that human knowledge is limited hardly seems a good reason for trying to discredit it altogether. What we need is a genuinely modest but at the same time sturdy conception of knowledge that will avoid the extremes of megalomania and despondency but serve us adequately in the middle region between nothing and infinity, between birth and death—it is after all our knowledge that is in question, not some abstract entity or substance independent of us.

Suppose we start with this idea of what is in question , and ask, what sort of thing have people thought they were talking about when they used the word "knowledge" or its ancestors? Here I want to take the ancestors seriously. It is not always helpful to ask where a word of ours came from, since that may have very little to do with what it means now, but there are some things about the derivation of the word "knowledge" that help to get it in perspective at least. First of all the "kn-" at the beginning: this puts the word in a whole family deriving from a Greek root, "gno-"; I think of them as the g-n (or c-n or k-n) words. One possibly unexpected member of this group is "noble"; the g-n shows up in its negative, "ignoble," the two words meaning in effect who's known, and who's beneath notice. Another is "king," the king being in ancient times wise as well as strong; another is "cunning," a kind of low knowledge, but still part of the family.

Greek gnosis means a kind of knowledge of the sort that we might call acquaintance, and indeed "acquaintance" is part of the family too (call it a q-n word), having come via Latin cognitio , somewhat distorted by a passage through old French. (We have this Latin root in a less distorted form in "cognition.") However there was another Greek word for knowledge, episteme , which translated into Latin as scientia , meaning in both cases not knowledge by (casual) acquaintance but careful

or serious knowledge. It is tempting to try to see a connection here between Greek episteme and Latin scientia on the one hand, and Greek temno and Latin scindo respectively on the other, both the latter meaning "to cut," thus to divide into categories, to distinguish—distinguishing among the things with which we are acquainted being an important step on the way to more adequate knowledge. In this way the "sci-" of "science" and the "sci-" of "scissors" would be related and a point easily made about the sharpness and exactitude of scientific knowledge. But this connection is uncertain. In any case Latin scio , "to know" in the sense of scientia , gradually gave way to another verb, sapio , originally meaning "to taste" and thus eventually in its own way "to distinguish." Remnants of that root are scarce in English, though Homo sapiens is familiar enough.

At all events we have a double history here, whose two parts are however related to one another: becoming acquainted with things in the world on the one hand, and finding out about them more carefully and exactly on the other. "To know" and "knowledge" in English carry both burdens, helped out in the latter case by the intensifier "scientific," scientific knowledge being an especially knowledgeable kind of knowledge.

But it is worth looking once again at the word "knowledge" itself, since it has a component that does not derive from Latin or Greek. The suffix "-ledge" seems to come from Old English "-lac," which survives in only one other modern English word. The suffix denoted a kind of action or proceeding, and it often had playful connotations, as in games. The basic idea seems to be of a state or condition into which one enters which enables (or entitles) one to engage in a certain sort of activity or practice, sometimes serious, sometimes not. The other survival is in "wedlock."

Now again I do not wish to burden you with surplus etymological baggage, but I have a feeling that the Old English were on to something when they assimilated knowledge to a class of activities including at the time dancing, fighting, robbing, pledging (the original meaning of "wedlock"), etc. Knowing is an activity, it involves skill and can be done well or badly, it can be celebratory or destructive, it can commit to consequences. (There is a line in Eliot's "Murder in the Cathedral" in which Becket says, "After such knowledge, what forgiveness?") I mention all this so that it will be vivid as we plunge into some of the grey matter of philosophy.

What have philosophers in fact said about knowledge? There is a whole history here and I will not enter into it but begin fairly recently. Some time ago it was a standard move in analytic philosophy (that is, philosophy that engages in the analysis of concepts, which all philoso-

phy ought to do at least some of the time) to offer as an analytic equivalent of the concept of knowledge the concept of "justified true belief." Roughly speaking the argument was that belief is an attitude we have to propositions when we think they are true, and if we think we know something we must at least think it is true; we may of course be doubtful, realizing that many of the things we think true may not be, but at all events we aren't going to offer as a candidate for knowledge something we think isn't true. If something we think true turns out actually to be true then we'll move it up from the status of mere belief to the status of the special kind of belief we call knowledge, and we won't do this otherwise. But how can we be sure that it actually is true? We have to have some basis for this conclusion, be able to offer a justification. Hence knowledge as justified true belief.

In 1963, in a very brief article in Analysis ,^[1] Edmund Gettier blew this view out of the water with a couple of telling counterexamples. I might believe something, it might be true, and I might be justified in believing it was true, but it might turn out that I didn't know it after all. Gettier's cases hinged on contingencies and ambiguities but they were telling nonetheless. The kind of strategy he uses can be illustrated as follows: Suppose I claim to know that my car is parked opposite my house. I believe this to be the case, it is the case, and I have a justification for believing it to be the case, namely, that I parked it there this morning. However, unknown to me my wife used the car to run an errand at lunchtime, and she returned the car to a slightly different spot, still opposite the house. So the three conditions are still met; yet nobody would claim that I know the original proposition.

Since 1963 a good many people have had a crack at this problem. In 1984 Richard Kirkham published an article in Mind^[2] in which he claimed that no "analysis of knowledge can be found which is (a ) generous enough to include as items of knowledge all, or most, of those beliefs we commonly regard as knowledge, and (b ) rigorous enough to exclude from the class of knowledge any beliefs held in real or hypothetical cases which we would agree on reflection are situations where the epistemic agent does not know the belief in question." If Kirkham is right—and I think he is—then either we need a new analysis of knowledge or we'll have to conclude we don't have any. Kirkham takes the latter position, but he says it shouldn't bother us as long as "we remember that a belief or proposition does not become less valuable merely because we can no longer apply the 'hurrah' word 'knowledge' to it. Only the discovery that it had less justification than we thought it had can cause it to lose epistemic value."

Now something very odd is going on here, something quite characteristic of some recent moves in philosophy, which can throw light

on the comfortably skeptical neopragmatist phenomenon to which I referred earlier. The problem is that nothing can be absolutely nailed down, so firmly that it can't be budged by anyone—or rather that's not the problem, since I don't see how we could possibly expect, knowing what we know (and I use the words deliberately), that anything ever could; the problem is that because things can't be nailed down absolutely, people tend to throw up their hands and assume that nothing is even approximately in place. This is sometimes called a crisis in the foundations, and the position to which pragmatism opposes itself called foundationalism (I drop the "neo-" here because it is clumsy and because the old pragmatism made the same claim for the same reasons).

For myself I'm not too much concerned about foundations; since Copernicus we've had to get used to the idea of being freely suspended in physical space, and I think there is a lesson for the intellectual domain in that. Pragmatism in fact seems to me to be an essentially foundationalist move; the situation is like that of theism and atheism—as Sartre once said of an atheist friend, he was "a God-obsessed crank who saw His absence everywhere, and could not open his mouth without uttering His name, in short a gentleman who had religious convictions."^[3] Pragmatists keep saying that we should give up the old silly ways of talking about the differences between knowledge and conjecture, between facts and interpretations, between rationality and irrationality, and so on, thus emphasizing the very concepts they reject. But if we can still make sense of them there seems no reason why we should follow this advice. One might turn the tables and say that historically foundationalism was an essentially pragmatist move: there was a problem about certainty, and trying to make knowledge fixed and absolute seemed like a good solution to it, especially when the means were at hand (thanks to the belief in God) to do so convincingly. Spinoza has as I recall an argument to this effect in his essay on the improvement of the understanding.

At all events we are dealing with something we all think we have some of, with respect to which however our confidence has been shaken because of devious and cunning counterexamples devised by tricky philosophers. I seem to be making light of their work; in fact I respect it highly, but want to get it in the kind of perspective that the lively and even playful attitude to knowledge we encountered in Old English would facilitate. The kinds of objection to the possibility of knowledge that I have already outlined are supplemented by objections to general truths in science because of the skepticism about induction that we owe originally to Hume; this has led careful philosophers of science as soundly empiricist as Hempel, for example, to admit that there are no scientific explanations, only explanation sketches (because in a strict

explanation the explanans would have to be true and contain a general law, something that can never be known to be true).

I think Hume was right about induction: we don't, in the end, know why the things that go together in Nature ultimately do so. (For complex things we sometimes know it in terms of the properties of their parts; in the end, at some level, we can only accept the fact that the parts behave as they do, we cannot explain it further.) But that doesn't mean that we have to give up the word "knowledge" as correctly characterizing something we have and can do. Let me dwell for a while on the philosophical situation in which we find ourselves. For some perverse reason it keeps reminding me of an unfortunate Scottish lady my family knew when I was a small boy. Her name was Mrs. Catterall. One of Mrs. Catterall's misfortunes was to discover in a local store, and to buy, something that was labeled "unbreakable china." This was a misfortune, in fact, only relatively to a second misfortune: that of being married to Mr. Catterall. Mrs. Catterall brought home her purchase and said excitedly to Mr. Catterall, "James! Only think! Unbreakable china!" "Unbr-r-reakable!" said Mr. Catterall, seizing some of the china and dashing it with all his force against the tiled floor of the kitchen, where it unthinkably broke.

The point of this story, of course, is that, while "unbreakable china" is as audacious a designation as "certain knowledge," Mrs. Catterall wasn't altogether silly for buying it, only for telling Mr. Catterall what it claimed to be. What "unbreakable china" meant was that it wouldn't break under normal or even rigorous conditions of use, conditions under which ordinary cheap china might break, not that it couldn't be broken by a large and irate Scotsman if he put his mind to it. Of course there's some hyperbole in the "unbreakable" and if we made china we'd be disinclined to call ours that, just as we might be disinclined to call our knowledge "certain." But in the places and under the conditions in which we have to use it—building machines, curing diseases, etc.—our knowledge seems to function pretty well and we don't want to be told we don't have it. And we might feel, as G. E. Moore came to feel, that the principles on which the radical criticism of knowledge rests are in fact themselves far less certain than the principles on which the knowledge itself rests.

How can we characterize a kind of knowledge that might withstand radical criticism and justify our reliance on it under normal conditions of use? One move is to concede everything the radical critics want and then go patiently back to where we left off. Descartes was the modern initiator of the radical move, which took the form for him of doubting everything except the fact that he doubted. We might be asleep, he said, we might be mad, God might be deceiving us. We might be brains

in vats, some more recent thinkers have suggested. Well, suppose we grant that possibility: in our sleepy or crazy or deceived or wired state we can still ask the question whether there is a difference between what we know and what we do not know, and ask how we know the difference. And, until we wake up or get cured or undeceived or unplugged, philosophy can proceed as before.

Another possible move is to reject any claim for our knowledge that takes it further than what is before our eyes, unless we have carefully argued grounds for extrapolation. This was the original strategy of what came to be called positivism: refuse to assert anything you don't positively know to be the case. It is very hard to do this, since the most elementary judgments about states of affairs take us beyond the immediate, and call for resources of logic and language we certainly didn't learn from observation. Still we can profit from the idea behind the strategy. Suppose we resolved not to extend our claims to knowledge, in the first instance, beyond the kinds of thing and event we are directly acquainted with, beyond the places and times of our familiar life, and then built very cautiously out from the initial claims towards wider ones, holding ourselves ready to modify them at any time in the light of new evidence? That would seem modest and safe enough. What would be wrong with it?

In some people's eyes, I suppose, its very modesty would be what is wrong with it. We want knowledge of the whole universe, and are tempted to claim it on the slimmest of evidence. Science seems to do this systematically. Newton adopted as "rules of reasoning in philosophy" (by which he meant what we call science) that "to the same natural effects we must, as far as possible, assign the same causes," and that "the qualities of bodies . . . which are to be found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever." And these have been essential assumptions for the development of science. But in fact, in spite of their sounding imperialistic, they themselves embody just the kind of modesty we are looking for: "as far as possible," says Newton, "within the reach of our experiments"^[4] —if it isn't possible, when our experiments reach further, we'll be happy to change our minds. (Newton has taken a lot of criticism from antiscientists because his view of science is taken to have locked us into grandiose and inhuman claims; such claims have been made by others on the basis of what he worked out but they need not be imputed to him.)

It did indeed look for a long time as if scientific knowledge would prove to be unlimited. The turn away from this hope has been partly due to a misunderstanding. It was thought that the knowledge we already had would extend to the limits of the universe; when it became

clear—thanks to Einstein and Planck and Heisenberg and others—that it didn't, this was taken to be a blow to science. But actually the discovery that it didn't was itself a scientific discovery. The general point to be made here—and to be learned from the positivist program—is that we can't jump to the limits and work back, we have to start from the middle and work out. But that was what science, properly understood, always did. The paradigm case of scientific knowledge (I use the term "paradigm" in its old sense, not in Kuhn's sense) is for me something that lies at the very beginning of the development of modern science: it is Galileo's demonstration of the relations between distance and time for bodies moving in a gravitational field (not that he called it that). He says he will find an equation that actually describes what happens, and he does. The equation matches the behavior of the moving body; the behavior of the moving body matches the equation. What happens in Galileo's laboratory in Padua; the equation he writes down he writes down there.

The lesson I want to draw from this case is this: that in the first instance all knowledge is local . It involves a matching of a description and a state of affairs. The relation between the two is open to radical challenge—there might be something wrong with our eyes, we might not be able to count straight, we might be brains in vats. Or again we might not—the claim that any of these things is the case is at least as implausible as the claim that all is normal. So we put that challenge courteously aside and get on with our work. From what is established locally we extrapolate at our own risk, and provisionally. How badly do we need to do so? Well, that depends on the application we wish to make of the knowledge in question. Perhaps we would like to apply it globally—but what is the motivation here? This is where we have to learn restraint. There is a parallel situation with meaning, which gets people into all sorts of psychological difficulties. I learn meaning locally, in episodes of effort or enjoyment or human contact; forthwith I am tempted to require that the universe as a whole should have the kind of meaning I have learned locally, that my life should have it, that life in general should, or history, or human striving. When I discover that they don't, I may come to think that the local episodes didn't have meaning either. But this would be a sad mistake—it is a sad mistake, made sadder by the fact that so many people make it.

A concept I have found useful in dealing with these questions is the concept of what I call the "flat region." The floor of my room is flat; localities generally are, or their declivities can be measured in relation to a flat surface. I learn the geometry of things, the earth-measurement, in the flat region. As I now know—thanks to other people mainly, I have to admit, and this is a point to which we will have to return—what

I learn here won't work if I try to extrapolate it for more than a few miles; I'll have to correct for the curvature of the earth. But that's only if I want to talk about some distant place while staying physically here. If I actually go off around the earth in search of its curvature, I discover a curious thing: wherever I stop, it's flat again. Of course if I can get off into space and look back at the earth I'll see it as curved, but for the purposes of my metaphor that's cheating, although we could extend the metaphor to accommodate it—space is curved too, and four-dimensional, but however far I go looking for that curvature, my spacecraft will remain "flat" in three dimensions.

The metaphor of the flat region applies to other domains as well. The flat region is where we are, locally, in the middle of things; if we push to the edges, to microscopic or cosmic dimensions, speeds near that of light, etc., the things we've learned locally won't apply. Why should we ever have thought they would? Up to a point, when we'd had no experience at all of anything nonlocal, the expectation was understandable, but that was a long time ago and by now there's no excuse for it. Yet people keep exclaiming over the fact that at the quantum level things don't look and behave like macroscopic objects. Are they waves? Are they particles? Why can't we measure their position and momentum at the same time? But waves and particles, positions and momenta, are things we learned about in the flat region; off at the limits we can by now expect things to be different. Even in logic and mathematics something like this occurs; locally, with ordinary proofs and other inferences, things work perfectly well, but when we push to limits of consistency or completeness we run into self-referential or self-descriptive problems.

The conclusions that are drawn from these altogether expectable failures of flat-region concepts to work after we've pushed inquiry over the edge remind us again of pragmatist pessimism about knowledge. Because of Heisenberg people wanted to throw out physical causality, because of Gödel they wanted to throw out logic. It is true that the advocates of these drastic revisions generally had an axe to grind, about freedom or the inadequacy of language, although it also usually turned out that they could have got the results they wanted without recourse to spurious technicalities. But they hung tremendous weight on what seem to me fairly banal conclusions, to the effect that middle-size people like us, who become acquainted with the world in a middle-sized context, don't learn in the course of coming to terms with their middle-sized world all the refinements they are going to need when they set off towards the very large, the very small, the very distant, the very complicated, and so on.

Let me return to a point I made just now, about the earth's being flat

wherever on its surface I happen to be. The general observation to be made here is that I regularly take my flat region with me . That is because I can't myself go very fast or become very small or very big; it's always the other fellow who is moving or distant, never myself. I speak of relativistic motion here, though we could with a bit of perversity sustain the view for local motion—in the morning the University rolls in my direction, at night my house does. Some medieval thinkers, notably Nicholas of Cusa, had exactly the same idea, which is the essential point of relativity theory, centuries before Einstein. We are fixed in relation to the world, always here, always now; where we are is always in the middle of the perceived universe for us, however far from home we may be.

What does all this have to do with knowledge? I said earlier that in the first instance all knowledge was local. The point of the flat region analogy is to insist that the failure of a form of knowledge to carry undistorted to the edge of the universe is no argument against its adequacy in the local context. The fact that definitions of mental illness, for example, break down in borderline cases, doesn't mean that we're in any doubt as to the insanity of the patient in four-point restraints. So we've laid down two lines of defense for our sturdy local concept of knowledge—one against radical, brain-in-the-vat type criticisms, and another against inadequacy-in-limit-cases type criticisms. We can now go back once more to square one, this time with some hope of being able to get to the straightaway without being tripped up, and ask once again the old question: what is knowledge, assuming it to be possible?

Our Old English friends wanted to make knowledge an enabling or entitling, an ability to engage in some practice. Let's say, taking a hint from the justified-true-belief school, that the ability in question is telling the truth . Having knowledge means that I can tell the truth (if I want to; nothing prevents me from lying, or from just keeping my mouth shut). So knowledge is an ability, in the first instance an ability to assert true propositions . But "telling the truth" has a double meaning, not just telling people true things but also telling what is true from what is false. This is the "justification" clause of justified true belief, and it will eventually shift the discussion to the concept of truth. Meanwhile, however, the Gettier objections really are, as I said earlier, telling in their own way; how should we deal with them?

Knowledge is an ability to assert true propositions and defend their claim to truth. "Defend" here means, among other true things, against the Gettiers of this world—that is, we have to be able to come back again and again to the defense when challenged, until we drop if necessary, build in safeguards against accidental fulfillment of the epistemic conditions, and so on. Also we may have to specify the kinds of knowl-

edge claim we are prepared to defend in this way (excluding perhaps as not worth the trouble anecdotal assertions about unspecified members of groups who have coins or tickets in their pockets, but including certainly claims about the regular behavior of significant classes of object in the flat region). What this means in effect is that we must be prepared to justify our justification, up to as many levels as may be required; if the justification breaks down at level n we will have to accept as a consequence that the knowledge all the way down to level zero is wiped out, but, at least for small n , we won't let that happen until we've tried level n + 1. The series of levels of justification constitute a system of the adequacy of the mind to things, to use an old formula, and the proposition whose true assertion entitles us to claim knowledge of what it asserts has to belong to the system if the claim is to be valid.

This view gives incidentally an answer to some relativists who claim that we are culturally biased in what we accept as scientific method; a favorite counterexample to our so-called "Western" concept of knowledge is a method, used among the Azande for determining the truth in vexed cases, known as the chicken oracle. The Azande know a poison that is marginally fatal to chickens; to consult the oracle a standard amount of this poison is administered to a chicken; if it kills the chicken the answer is yes or no as the case may be (I forget which), if the chicken survives the answer goes the other way. And why, ask some self-critical Western social scientists, shouldn't the Azande believe their oracle just as we believe our scientific oracles? The answer is that you can't reliably ask the oracle to pronounce on its own reliability (that's not the sort of question you can put to it, since it is used mainly to ferret out witches), whereas scientific method belongs to a complex of argument and inference in which it is possible to raise the question of its reliability, and of the reliability of our estimate of its reliability, and so on up for as many levels as you like (not too many, generally, since the higher-order questions have been debated by philosophers of science for whole classes of cases).

Does this whole edifice lie open to radical skepticism? Of course it does. By now, does this disturb us? It does not. The usefulness of radical skepticism lies in the fact that it forces us to stare it down. The disagreement between us is not, as far as that goes, as deep as it looks. It amounts (to go back to the language of the article by Kirkham cited above) to a difference of judgment as to when it is necessary, if ever, to say "hurrah!" Skeptics have a view of a philosophically perfect kind of knowledge; they think we can't have it, though if we did it would be worth saying "hurrah!" about it; meanwhile nothing else will do, everything falls short, so we should stop claiming to have any knowledge. I agree that we can't have that kind of knowledge, but I think

that only a thoroughgoing Utopian would ever even dream of having it; meanwhile it seems to me quite reasonable to claim as knowledge, until further notice, whatever, having earned its place in the system of justification, enables us to play our part in the truth-telling game. Of course we'll have to be sensitive to different possible moves in the game, to judge prudently how far out we may venture on excursions away from the flat region; in home territory, however, we are entitled to a certain confidence.

From here there are several directions in which we can go. Recall that ordinary knowledge and scientific knowledge were said at the beginning to stand and fall together; the kind of care science compels us to bring to the formulation of our knowledge can be exercised with respect to any subject-matter whatever, though there are many cases in which it would hardly be worth the trouble. One thing worth noticing, though, is that the natural sciences on the one hand and the social sciences on the other result from special care exercised on two different kinds of everyday knowledge: one of states of the world that are independent of our interest in them and one of states of the world that are to some degree created by our interest in them. To this distinction correspond two different conceptions of truth. Truths in one category are accepted as such because they are forced upon us by observation: they obey Tarski's semantic criterion. Truths in the other are forced upon us because they are required in order to preserve the fabric of intelligibility in discourse—they can't not be true on pain of the incoherence of our whole scheme. That this desk is hard or this room illuminated belong to the first category; that it is Monday or that John is a friend of mine (or even that he is John) belong to the latter.

The person known as "John" is not John in the way that the desk is hard. The desk is called "hard" by a linguistic convention, and hence by something our interest created, that is true, but it is independently what we call "hard" (and what the French call "dur," etc.), whereas John isn't independently anything called "John" in the same way; there is no property of Johnness he could have or fail to have and still remain himself (though after acquaintance with a social object we may begin to attribute properties of this kind, saying, for example, "John isn't himself today," and so on). However, if we doubt that he is after all properly called "John Smith," we pose a radical challenge to the stability of the social structure, just as if we doubt that this day is properly called "Monday, January 13," we challenge the whole worldwide system of names and dates. There is nothing about this day, as I look around in it, to label it Monday, January 13. The fact that I'm beginning a course of lectures tonight is confirmation that it is, since the first lecture is announced for this date, but the alignment of earth and sun

that makes this day rather than night, winter rather than summer, is supremely indifferent to my lecturing schedule. We have to do things to make days into what they are for us, but we don't have to do anything to make the table hard, once it is (that somebody made it means that its existence as a table is a social fact; its hardness however isn't a social fact but a natural one).

This distinction is useful when it comes (as it often does) to charges of the cultural relativism of knowledge. It is in fact the failure to keep clearly in mind the distinction between the natural and the social sciences—a distinction that for several perverse reasons nearly everyone has been at pains to suppress—that has led to a lot of the confusion about the possibility of knowledge. If we talk about the truths of society or history or any branch of what Sartre so usefully called the "practicoinert," then of course these change from culture to culture, from generation to generation, although even in those cases there are some striking continuities in the mainstream from its beginnings in Greece and Judea until our own day. But if we talk about the truths of science or nature then however different cultural formulations may be they prove in the end to converge, to be intertranslatable.

This casual assertion on my part goes against a whole recent tradition that casts doubt on the convergence of scientific discovery, on the grounds that there have been revolutions, that Einstein has displaced Newton, etc. However, let me repeat that I still have my feet firmly planted in the flat region—indeed I'm confident I'll never leave it—and the kinds of truth I'm talking about are not megalomaniac claims about how the whole universe is but are the stuff of the sturdy local knowledge on which I rely when I go to the dentist or use my word processor. This is not a regression to some sort of naïve realism—indeed, it is consistent with a quite radical theory of perception—but it does involve the claim that it's pointless to refuse the title of knowledge of truth to what has always counted as such in our long adaptation as curious and discursive organisms to the local conditions in which we evolved. This view implies no rejection of or even disrespect to scientists whose business it is to push inquiry far from the flat region, towards big bangs or quarks, but it does—to repeat what has already been said—insist that difficulties encountered only there need cast no doubt on the reliability of local knowledge.

This is the obvious point at which to deal with an objection that has no doubt occurred to many readers. My radical opposition of the natural to the social sciences, in terms of their objects as (to put it succinctly) mind-independent as opposed to mind-dependent, may seem to collapse because physicists have been saying for a long time that at the quantum level observation partially determines what is observed, etc.

But again this is a problem of a region of extreme curvature, in which all that is available to us in the way of claims to knowledge involves complex theoretical reconstructions (undertaken, I may remind you, by macroscopic scientists in macroscopic offices and laboratories). The paradoxes that have led some physicists to posit one form or another of an "anthropic principle," according to which a condition for the development of the physical world was the eventual appearance of human beings capable of observing it, seem to me extreme cases of the disproportionality I have already referred to, hanging global consequences on small discrepancies. The discrepancies mean, to be sure, that our present knowledge isn't yet, and may never be, absolutely and completely and universally valid. But we gave that up a while ago in favor of knowledge as relatively and partially and locally valid. This is not, however, to be interpreted in a minimal sense—on the contrary, these deficiencies are acknowledged only out of a principled concession to modesty. They are marginal, not central.

There is room here for a version of the old legal maxim, "hard cases make bad law"—limit cases cannot be allowed to overturn principles established centrally. Of course this does not mean that no other central principles are conceivable—that would be to play into the hands of critics like Feyerabend who want to make all received views into forms of fascist oppression. But until such conceptual replacement actually occurs I want to conclude that the flat region remains Euclidean, Newtonian, causal, etc. and that we know this as well as we know anything. We know it, I repeat once more, of the flat region , and it is our knowing it there that makes departures from flat-region principles intelligible, when we move inquiry in the direction of the limits. Classical physics makes modern physics possible. But this is getting repetitive and that suggests that it is time to go on to a conclusion.

Most scientific inquiry in the history of the race has been conducted in the flat region. In fact the analogy with the surface of the earth is inexact in its proportions, since we don't really have to go very far before plane Euclidean geometry becomes a bad basis for geographic surveys, whereas we have to go very far indeed before relativistic or quantum considerations impose themselves more than marginally. "Ignoring second- and higher-order terms" is a standard and perfectly safe rubric for most actual computations. Nothing in biology, and in practice hardly anything even in chemistry or physics, requires us to resort to anything other than normal science (here Kuhn's term is unobjectionable, as long as we realize that most scientists live practically as if the revolution had never occurred). Nothing a scientist can personally do goes beyond the limits of normal science, only what instruments sometimes do (and usually the most expensive instruments—there is an ex-

ponential relation between distance from the flat region and the cost of research). Even space travel, so far, has been entirely within the flat region; no navigational computations for any spacecraft to date have required the insertion of relativistic terms.

Also we all live in the flat region, and it is our knowledge we are talking about. This brings me back to the last element of my title. I want to claim that there is no such thing as knowledge in general, only someone's knowledge, and that each knower—I will take myself as the paradigm case—has not only two kinds of knowledge of the world, but knowledge of two kinds of world. I distinguish between my world, which will die with me, and the world, which I suppose to have been there before I was born and which I expect to remain after my death. I learn my world and carry it with me; it is my locality; it is, in my metaphorical sense of the term, flat, although I could probably distort it to some degree, if I wanted to, by ingesting mind-altering chemicals. Thanks to the fact that there are other people in my world (the sense in which they are "in" it needs to be elucidated by the theory of perception to which I referred just now, but that will have to be on another occasion) and thanks to the evolutionary accumulation of the knowledge they pass on to me (each bearing some part of it, or directing me to elements of the practico-inert on the basis of which I can reconstruct it), I come to learn a good deal about the lineaments of the world, at least locally, where it too is flat.

That the world is locally flat has in fact to be a truism, because "flat" for me just means conforming, again thanks to evolutionary adaptation, to local conditions. Whether I ever want to push my knowledge off in search of the curvature of the mind-independent world depends on my inclination; most people don't. But I won't be able to do that unless I have come exactly to terms with the structure of my world, which is to a first approximation the local structure of the world. (What we know of the world can only be structural—there is no reason to think it shares the vividness of the material contents of our worlds.) And that means making some of my knowledge scientific, that is, exercising care in its formulation and attending to its empirical adequacy and its logical consistency.

Nobody can do that for me, although I can profit readily enough from what they have done for themselves, especially if I am lucky enough to have access to them and it—which is exactly what universities exist to make possible. So the question "Is there scientific knowledge?" really has to be posed differently; it should be "Is any of the knowledge I have scientific?"—that is, have I cared enough about exactitude and consistency to be willing to do the work necessary to make it so? For if there is to be scientific knowledge, if it is to survive and have the

useful effects it is capable of producing, individuals will have to continue to choose to do that work, to attend carefully to what they know, to organize and perfect it. We ought not to discourage them from doing so by belittling the possibility of knowledge. Indeed we ought—but with this I would need to start another lecture—to get them to pay such careful attention to knowledge in domains not generally thought scientific: religion, politics. At all events we should exercise such care ourselves, making our work in these domains at least commensurate, in the level of seriousness and responsibility we bring to bear on it, with the work of the natural sciences. For scientific knowledge, if it is not the answer to everything (and it isn't), does at least set a standard.

19—
The Law of Quantity and Quality, or What Numbers Can and Can't Describe

Origins

Before there was writing, any culture carried by language had to be transmitted orally. People memorized poems that incorporated the knowledge that was to be passed on to future generations. A poem is something made (poiein is "to make"), something made with words and remembered, not just words uttered for an occasion and forgotten. Now, we are accustomed to think, things have changed: there are texts and chronicles, and the art of memorization has gone almost entirely out of use. We don't need it for the storage or transmission of knowledge, and the old chore of learning poems by heart in school has been almost entirely dispensed with. Feats of memory, outside some technical contexts (in the theater or in medicine, for example) have become curiosities, useful to intellectuals who are unexpectedly imprisoned and need something to keep them sane, but otherwise merely freakish or decorative.

It is worth noting, though, that in fact there are still at least two poems that everyone who has the most rudimentary education learns and remembers. Learning them indeed is a condition for participation in the literacy that makes the old feats of memory unnecessary. One of them is the alphabet, and the other is the series of names for the integers.^[1] They don't look like poems, but on reflection they obviously are poems: words that belong together, to be remembered and recited in a given but not intuitively obvious order. The order is important, and must be learned exactly; later on it will seem intuitively obvious, but

that will be only because it was thoroughly learned before the concept of the obvious (or not) had been acquired.

The elements of these poems have iconic representations, in our case respectively Roman and Arabic—a significant detail, this, and relevant to the separation of the quantitative from other predicates in our scheme of concepts. The Greeks and Romans used letters for numerals; in Greek they were accented, but in both cases it was clearly enough understood that the combinatorial rules were different as between literal and numerical uses, whether ordinal or cardinal. We however learn different poems and not merely different rules, so that they seem from the beginning to belong to different domains, mixing the elements of which creates awkwardness, though it is easier for us in the ordinal than the cardinal case. We may identify, and if desirable order, paragraphs, buses, telephones, postal codes, registered automobiles, etc., alphabetically or numerically or by a combination or alternation of the two, and be comfortable with this, but the alphanumeric notations sometimes used in computer programming (such as the hexadecimal, which inserts A through F between the usual 9 and 10, 1A through 1F between the usual 19 and 20, 9A through FF between 99 and 100, and so on) still seem intuitively strange to most people.

Of course it is not only alphanumeric notations that perplex—so do purely numeric ones to bases less than ten. That is because the number poem is a poem to base ten; the sequence 1, 10, 11, 100, 101, 110, 111, 1000 in the binary system would have to be read "one, two, three, four, five, six, seven, eight," not "one, ten, eleven, one hundred," etc., in order to refer correctly in ordinary language to the numbers in question, and this strains the intelligibility of the written characters. Something of the same sort happens with Roman numerals—most of us have to make a more or less conscious translation of MDCXLVII into 1647 as we read it off, much as we do with familiar words in an unfamiliar script, Cyrillic for example (try reading "CCCP" as "SSSR").

So far these considerations are purely discursive—they do not bear on the properties these two systems of representation may serve to articulate, but only on the existence of the systems themselves, and the ordering and legibility of their elements, the letters and numerals. But it is evidently not just a curiosity that these systems should exist, and it is worth reflecting on what brought them into being. Letters were the issue of a long evolution of modes of representing what could be conveyed in speech, pictorially and then pictographically and then hieroglyphically. (No doubt at the same time speech itself developed to express distinctions that had shown up graphically.) At some point the connection between the system of representation and the content of what was said gave way to a connection between the system of repre-

sentation and the sound of what was said. This reinforced a separation between discourse and the world that had begun far earlier with the abandonment of any necessary connection between sound and sense, a move from motivated sound elements to merely differential ones.

With the numerals the story was somewhat different. They seem to have been invented (if etymology is to be trusted at all) in connection with a special social activity, the acquisition and distribution of goods (Latin numerus is connected with Greek nemo , to deal out, dispense, thence to hold, possess, etc.; one of the derivatives of this verb is nomos , meaning among other things a law that assigns lots and places to people and things, from which in turn philosophers of science have derived "nomological," thus indirectly reinforcing the connection between mathematics and the laws of nature). This activity necessarily involved on the one hand gathering and counting, on the other dividing, apportioning and so on, and one can imagine the closeness of the attention paid to the sizes and quantities of things in these processes. The concepts of more and less are attached to powerfully affective modes of relating to the world, involving property and justice, security and self-esteem. It has been noticed by educators among others that people with apparently undeveloped mathematical talents may be quantitatively knowledgeable or even sophisticated when their interests in fair shares or sums of money are engaged.

Two Kinds of Predicate

There is an interesting difference in the uses of these two systems, hinted at above in the remark about ordinality and cardinality. Either system can be exploited for the purposes of ordering , on the basis of the conventional structure of its poem: we know that K comes before L just as we know that 11 comes before 12. But the development of the alphabetic system goes in the direction of arbitrary associations of letters and sequences of letters with sounds, and thence of the arbitrary association of sounds with conceptual contents, that is in the direction of language and its "double articulation." The development of the numeric system, on the other hand, goes in the direction of systematic combinations of numbers and thence of their systematic interrelations among themselves; insofar as they are associated with conceptual content this remains external. Words mean by referring to things in the world; numbers do not—they mean only themselves, though they can be attached to and modify the referents of associated linguistic elements. If I say "ten grey elephants," the terms "grey" and "elephant" refer to each of the entities in question or to their properties, but the

term "ten" doesn't refer to any of them or even to all of them as the entities they are ; it refers only to the cardinality of the collection to which they happen to belong. If I had said "ten grey owls," it would make sense to ask, of "grey," whether it was the same grey as in the case of the elephants or a different grey; but it wouldn't make sense to ask if it was the same ten, or a different one.

Is "ten" an adjectival property of "ten grey elephants" in the same sense that "grey" is? That, in a nutshell, is the problem of the qualitative and the quantitative. It certainly looks as if there is a radical difference here: they couldn't be elephants if they weren't grey but they could certainly be elephants if they weren't ten. Well, they couldn't be ten elephants, but that sounds tautological. Wait a minute, though—why not say similarly that the only thing ruled out by their not being grey is their being grey elephants (they might still be pink elephants)? Even so we are tempted to feel that the greyness (or pinkness as the case may be) inheres in the elephants in a way that the quality of being ten does not; going from ten to eleven is a contingent and external move, requiring nothing more exotic than the arrival of another elephant, whereas going from grey to pink seems like an essential and internal move, requiring a general metamorphosis on the part of all ten elephants.

"The quality of being ten"—this expression sounded natural enough when I used it a few lines back. It wasn't a quality of the elephants exactly, but rather of the collection they happened to constitute, which however might quite as well have been constituted by penguins, nebulae, or abstract entities. Call it a set: students of elementary abstract set theory have to get accustomed to the irrelevance of the obvious properties of the members of sets as individuals, to dealing with sets whose only members are, say, {Napoleon, and the square root of minus one}, or {the empty set, and the Lincoln Memorial}, and to recognizing that the cardinality of these sets, which we call two , is the same (and the same as the cardinality of the set that contains both of them —and of the set that contains {the empty set, and the set that contains both of them}).

The quality of cardinality is something that only sets have: what it permits is an unambiguous classification of sets according as they have more or fewer members than, or the same number of members as, other sets. Perhaps I should have said, the qualities of cardinality, since two is different from three and both are different from 10¹⁰ . At the lower end of the scale of cardinals (it doesn't have an upper end) these qualities are perceptible and have common names: pair or couple, triad or threesome, etc. Other names for numbers (dozen, score) are generally survivals from alternative poems rather than directly descriptive predi-

cates: applying them correctly normally requires counting out. In special (pathological?) cases the perception of cardinality can apparently go much higher: the neurologist Oliver Sacks recounts (an expression that in itself reflects the overlapping of the qualitative and the quantitative in ordinary language) an episode in the lives of a pair of idiot savant twins in which someone drops a box of matches and they both say at once, "111!" When asked how they could count so quickly, they say they didn't count, they saw .

They seemed surprised at my surprise—as if I were somehow blind: and John's gesture conveyed an extraordinary sense of immediate, felt reality. Is it possible, I said to myself, that they can somehow "see" the properties, not in a conceptual, abstract way, but as qualities , felt, sensuous, in some immediate, concrete way?^[2]

The choice of words here reinforces the suggestion that the rest of us too think of small numerical attributes as qualitative, and that they become properly quantitative only when the numbers are too large to be attributed without counting.

If we now revert from speaking of sets as such to speaking of their members, we say that there is a quantity of them—but they don't thereby acquire any new qualities . Things get complicated, though, when this habit of switching attention back and forth from sets to members of sets follows the development of the number system from the integers or natural numbers, in connection with which the idea of cardinality was first defined, to rational, real, or even complex numbers. By introducing the concept of unit , which makes some standard embodiment of a quality such as length or weight (the standard meter, the standard kilogram) the sole member of a set of cardinality one , and specifying a rule of matching (laying end to end, piling up in the scale of a balance) that will generate sets of higher cardinality whose members will be units (fractions of units being relegated to fractional scales, where the new units are fractions of the old: a tenth, a hundredth, etc.), cardinality comes to be attached by courtesy to other objects embodying the quality in different degrees. Instead of "longer" and "shorter" we now have "11 meters" and "10.3 meters," which define whole classes of longers and shorters among indefinitely many such possible classes. Our interest in "11 meters" as a defining property of some object was initially, no doubt, a desire to know what it was longer or shorter than, or the same size as, but "11" came to attach to it as a predicate along with "blue," "soft," "glutinous," and whatever other qualities our postulated 11-meter object may be supposed to have. And before we knew it, our language was stocked with ratios, averages,

angles, temperatures, coefficients, dates, times, indices, prices, and other numerically-expressed predicates as familiar and useful, in our commerce with things in the world, as any other qualities by which they might be distinguished from one another.

The specifications of degree among objects sharing a given quality, which quantitative predicates make possible, have been available in some technical contexts for a long time, but their general invasion of daily language is relatively recent. That the temperature should be "in the sixties" has of course been a possible determination only since the invention of the Fahrenheit scale and the general availability of thermometers, i.e. since the early eighteenth century. But a temperature "in the sixties" has nothing to do with the number 60 or the cardinality it represents, it has to do with spring and light coats, while "in the twenties" means bitter cold and "in the nineties" intolerable heat. Note that the expressions "in her twenties," and "in his nineties," coexist with these unambiguously, as indeed do "in the twenties," "in the sixties," and so on, as applied to the years in a given century, but that these in their turn mean young and beautiful or old and wizened, flappers and flower children, rather than anything quantitative. It is interesting to find that although these latter expressions have been available for much longer than is the case with the weather, birthdays and calendars having been marked by cardinals for centuries, they were not in fact used until about the same time; whether it be temperatures, ages, or years, the first occurrences of the expressions "twenties," "thirties," etc., up to "nineties," are all given by the Oxford English Dictionary as falling between 1865 and 1885.

It was at about this time, in 1878 to be exact, that Frederick Engels, in Herr Eugen Duhring's Revolution in Science (commonly known as the "Anti-Duhring"), gave popular form to the principle, introduced by Hegel and utilized by Marx, of the passage of quantity into quality. Hegel speaks of "nodal lines" in nature, along which incremental quantitative changes are accompanied, at the nodes, by qualitative shifts. Such a shift is "a sudden revulsion of quantity into quality," and Hegel offers as an example "the qualitatively different states of aggregation water exhibits under increase or diminution of temperature."^[3] Engels too cites this as "one of the best-known examples—that of the change of the state of water, which under normal atmospheric pressure changes at 0°C from the liquid into the solid state, and at 100°C from the liquid into the gaseous state, so that at both these turning-points the merely quantitative change of temperature brings about a qualitative change in the condition of the water."^[4]

This "brings about," however, is highly misleading. It gives the impression that temperature is a property of water that is causally re-

lated to its state: change the (quantitative) temperature, and the (qualitative) state will change. The fact is that at the boiling and freezing points the temperature can't be changed until the state has changed. What happens is this (I will take the case of boiling, which applies mutatis mutandis to freezing also): steadily supplying enough heat energy to water will raise its temperature to 100°C; at this point supplying further energy will not change the temperature but will dissociate the molecules from one another so that they become steam at 100°C; when all the water has been changed to steam then, assuming a closed system, the supply of still further energy will raise the temperature of the steam above 100°C. But if the process begins at room temperature it will take about seven times as long to change all the water into steam as it took to raise the water to the boiling point.

So there are two things wrong with the Hegel-Engels account: first, it isn't changing the temperature that changes the state, and second, the change is not sudden. As I have pointed out elsewhere,^[5] when water boils because it is heated from the bottom, the change of a small amount of it into steam makes dramatic bubbles, and this is not a bad analogy for repressed change, which was one of the popular senses in which the dialectical principle of quantity and quality came to be understood: history will accumulate exploitation and repression incrementally, until crisis and revolution suddenly ensue. And this may indeed happen—only the quantity/quality distinction has nothing to do with it. Water froze and boiled long before temperatures were thought of, and when we talk about "the boiling point" and attach a number to it (note by the way that it is impossible to measure the boiling point at standard atmospheric pressure in degrees Celsius, since 100°C is defined as the boiling point of water at standard atmospheric pressure), the number by itself does not refer to anything that is true of the water, but (as before) only to the cardinality of a collection of units.

This point can be driven home in various ways, One of the remarkable and useful features of the exact sciences is that quantities can be measured and the measurements plugged into computations. The qualities whose degrees are attended to in the process of measurement (or predicted by the outcome of the computation) are sometimes thought to enter into the computations. Thus in the most elementary case of a freely falling body initially at rest we have the equation:

which means "the distance fallen is equal to half the acceleration of gravity multiplied by the square of the time elapsed." But a moment's thought will show that this can't possibly be what is meant: times can't be squared; only numbers can. Nothing can be multiplied by an acceler-

ation. The expression is only a shorthand way of saying that measurements of the distance, the acceleration, and the time, using compatible units, will yield numbers that stand in the required arithmetical relation. In the algebraic expression given above s isn't a distance at all, it's a variable that can take numerical values, and so for the other elements.

The coincidence of Engels's popularization of dialectical doctrine on the one hand, and the emergence of numerical expressions as descriptive in ordinary language on the other, suggests that the latter paved the way for the general confusion represented by the former. We can use numbers to describe things, but unless the thing described is a set or collection with a given cardinality, they won't be functioning as numbers, just as predicates to be defined in the ordinary way and eliminable by substitution. Their use will he a metaphorical use. Yet in the last hundred years or so people have thought of themselves as getting hold of a special numerical or even mathematical feature of things when they use numbers in this way, a quantitative feature at any rate. And when the numbers change concomitantly with some notable qualitative change we have all the appearances of a passage from quantity to quality.

Notable and Just Noticeable Differences

The idea of concomitant change ("concomitant variation," to use Mill's phrase) is basic to the scientific enterprise: we want to know, if we make some change in the world, what else will also change, so that we can achieve or avoid it. Changes can be large or small, dramatic or marginal. Group sizes change by the addition or subtraction of members, other properties by augmentation or diminution, intensification or dilution, etc., or by outright metamorphosis, one property being replaced by another. Cumulative marginal changes, each of which is hardly noticed, may eventually result in states so altered that they require altogether different descriptions. But this phenomenon is context-dependent and works on both sides of the qualitative-quantitative boundary. If a large surface, a wall for example, has always been red, but suddenly overnight is painted yellow, the change is startlingly obvious, but if its red color is modified very slowly, through an imperceptible shift in the direction of orange and progressively through lighter and lighter shades, until finally the last trace of red has vanished and the wall is pure yellow, the fact that it has changed at all may dawn only slowly, and then only on an observant witness with a good memory (imagine the change stretched out over centuries, so that in any one

witness's life it was just an orange wall). Psychologists speak of "jnd's" or "just noticeable differences" as a measure of the refinement of perception (similar to "resolving power" in optics), a threshold below which changes cannot be perceived, so that several subliminal moves may be possible before anything is noticed—and indeed if they are made at suitable intervals nothing may ever be noticed.

Something very similar happens on the quantitative side if the sets in question are sufficiently large. If one person is in a room and another enters, the change is obvious enough, and similarly if a third joins a couple, but if forty people are watching a parade, let us say, the arrival of the forty-first may go entirely unremarked. Still if people keep coming, one by one, sooner or later we have a huge crowd, a demonstration, a triumph—and when exactly did this happen? There is an ancient paradox called The Heap: a grain of wheat is set down, then another grain, and so on; eventually there is a heap, but which grain was it that turned a scattering of grain into a heap? This paradox was presumably intended to remain paradoxical—no empirical research was done, as far as I know, to find out when impartial observers would start to use the term "heap" without prompting. (My guess is that four grains, in a tight tetrahedral array, would qualify as a very small heap, whereas if the procedure were to scatter randomly over a given area, say a square yard, there would be a range of many thousands of grains over which the status of the accumulation as a heap could be disputed.) The point the paradox makes is that categoreal boundaries, for example, between "scattering" and "heap," are fuzzy, but that surely comes as no surprise and hardly makes a very convincing foundation for philosophical doctrine, whether metaphysical or revolutionary.

The dialectical law of the passage of quantity into quality, like its companions, the law of the interpenetration of opposites and the law of the negation of the negation, is thus seen to be an entertaining but nonessential red herring. There are cases in which cumulative imperceptible changes in x lead to the emergence of y , and there are cases in which they just lead to more x —and either x or y can be indifferently qualitative or quantitative predicates; everything depends on the particular case, and can only be learned by looking. Adding atom after atom to a lump of uranium 235 eventually produces an atomic explosion and an assortment of vaporized fission products; adding atom after atom to a lump of gold just produces a bigger lump of gold. Water when refrigerated changes into ice; iron when refrigerated gets colder but doesn't change into another form. No general law can be established that would be of any reliable predictive value; as in any empirical situation, the correlations cannot be generalized in advance but must be learned for each case or class of cases. That solids will eventually melt on heating,

and liquids vaporize, can be expected within limits, but even there other forms of dissociation may take place, and nothing whatever is gained by claiming these phenomena as examples of the dialectic in nature.

The contingency of the relation between quantitative and qualitative change, its dependence on the state of the system, can be illustrated by the following thought-experiment, in which A is a pedestrian walking slowly towards the edge of a cliff C:

Cumulative quantitative displacements of A in the direction of the arrow will lead to a dramatic qualitative change in his situation at point C (call it the "falling point"), but nobody would seriously think of attributing this to the quantitative change as such, only to its taking place near the edge of the cliff.

These considerations do not abolish the differences between qualitative and quantitative but they do suggest fresh ways of thinking about them. In particular it is not clear that they need be accepted as dividing the field when it comes to determinations of the state of the world in various respects. Both derive from members of a family of Latin adverbs beginning with "qu-," all of which have interrogative uses, whose form was presumably determined by the verb quaero , to seek, ask, inquire. So qualis ? from which "qualitative" derives, means in effect, "I ask: what sort?" while quantus ? similarly means, "I ask: how much?" We may think of this "qu-" prefix as a kind of question mark, and translate qualis and quantus respectively as "(?)sort" and "(?)degree." However there are lots of other possible questions, and Latin provides for them: (?)manner will give quam or quomodo ; (?)time, quando ; (?)elapsed time, quamdiu ; (?)reason, quia or quare ; (?)distance, quoad; (?)place, quo ; (?)number, quot ; (?)frequency, quoties ; (?)number in series, quotus , and so on. Why should there not therefore be quamitative, quanditative, quaritative, quotative and quotitative inquiries, as well as qualitative and quantitative? And yet these last two are the only survivors to have made it into our ordinary language, and this means, if we are to take Austin seriously, that only one difference or opposition out of this whole crew was important enough to be preserved. The question is, what opposition was it?

I shall suggest that it was not the sort of opposition that divides the

world into a part that is qualitative and a part that is quantitative, or that allows the transition of one sort of predicate into the other according to any law, no matter how dialectical. The world is as it is and its states are amenable to description on condition of our having a suitable language at our disposal; every element of every state invites the question of what sort of thing it is, what sort of thing is going on. Let this be the general question, the descendant of qualis ? to which the answer may be in diverse modes: spatial, temporal, causal, numerical and so on. If the last among these rather than the others singles itself out for special attention, why might this be?

Separation of the Mathematical Apparatus

It should be noticed at once that something is slipping here—if numerical properties had been the issue surely quot rather than quantus should have been the root of our own expression. This slippage indicates, I think, where our own confusion lies. The questions "what sort?" and "how much?" are both required if the entity or event under investigation is to be estimated correctly in relation to other things; both are differential questions, and the answers to them provide the coordinates that locate the object in an array of types and magnitudes: the first distinguishes it from other objects of different sorts, the second compares it with other objects of the same sort. The latter purpose, however, can be served in diverse ways—within a given category there can be more than one dimension of variety. So a series of possible orders may be envisaged, in which the members of the category might be arranged, and for each order an ordinal sign may be assigned to each member. For this purpose we are not unlikely to call upon one of the poems with which we began. And the discovery that if we choose the number poem we may also be able to make use of cardinality, and even perform computations that will accurately predict some features of the ordering in question, will come as a surprise and a revelation.

It is just this formal and computational aspect of the matter that brings in the quantitative as it has generally come to be understood. One of the earliest discoveries along these lines was made by the Pythagoreans, who correlated the ratios of lengths of stretched strings with musical intervals. They thought this discovery sacred, and indeed it is hard to imagine the awe and astonishment it must have produced. I suspect (indeed I remember) that something like it can happen in childhood when elementary mathematical truths suddenly dawn, but that is an expected step, an entry into a known domain, not as for them

the opening up of something novel and incredible. Pythagorean doctrine concluded that the world was at bottom numerical , which involved a category mistake but nevertheless set the tone for a long tradition. The beginning of modern science was marked by Galileo's resolve to make the "definition of accelerated motion [i.e., its mathematical expression] exhibit the essential features of observed accelerated motions,"^[6] a scrupulous formulation that seems unnecessary to us, because obvious, but that required new clarity on his part. The comparable claim in his case was that "the book of nature is written in the language of mathematics," which does not involve a category mistake but does assume a parallel between an intelligible domain (the book and its mathematics) and a sensible one (nature); here also Galileo was scrupulous and clear, though his remark has frequently been interpreted as meaning that "nature is mathematical," which brings back the mistake.

These episodes represent steps in a process of realization that reached its full formulation with the Turing machine: the realization that all relations between exactly specifiable properties of all the things in the world can be modeled to as close an approximation as desired in logico-mathematical language. This development is recounted with great perspicuity in Husserl's The Crisis of European Sciences and Transcendental Phenomenology , in which he speaks of "Galileo's mathematization of nature," and in a brilliant image describes a tendency to "measure the life-world—the world constantly given to us as actual in our concrete world-life—for a well-fitting . . . garb of symbols of the symbolic mathematical theories."^[7] The success of this program of measurement however leads to "the surreptitious substitution of the mathematically substructed world of idealities for the only real world, the one that is actually given through perception, that is ever experienced and experienceable—our everyday life-world."^[8]

The properties of things in the life-world are what we would normally and generally call "qualities," and the only qualities that permit of direct mathematical expression are precisely the properties of sets or collections already discussed; all the others have to be translated into sets or collections, through the specification of units and combinatorial procedures. This process has been called "substruction" by Paul Lazarsfeld, independently, I take it, of Husserl's use of the term (cf. the quotation above); it "consists essentially in discovering or constructing a small number of dimensions, or variables, that underlie a set of qualitative types."^[9] The actual carrying out of the process will involve distinctions between ranked and scalar variables, discontinuous and continuous scales, ratio and interval scales, etc.;^[10] fitting the life-world with its mathematical garb is a busy and demanding industry.

Only sets or collections, properly speaking, can be said to have quantitative properties, and these in the end will all turn out to be numerical—Husserl speaks of the "arithmetization of geometry," of the transformation of geometrical intuitions into "pure numerical configurations."^[11] (This claim is no doubt oversimplified—there may be topological features, such as inclusion or intersection, that have nonnumerical expressions, though these would not normally be called quantitative.) What are thought of as quantitative properties of other entities, such as length, temperature, density, etc., are so many qualitative properties with respect to which however an entity may change its state over time, or otherwise similar entities may differ from one another. Such differences are themselves qualitative, though they may be given numerical expression. It is important to realize that, for example, the difference in height between someone five feet tall and someone six feet tall is not a numerical difference, even though the difference between five and six is a numerical difference. At every given instant every entity is in the state it is in, with the qualities it has. These may include vectors of change or becoming. Whether such vectors essentially involve quantities—that is, whether becoming involves at every infinitesimal moment a change in the size of a collection—is a question as old as Zeno, which however need not be answered in order to characterize a momentary state.

Of course collections may change their cardinality with time, and we can over suitably large time intervals make other changes into changes in the cardinality of collections by choosing to represent them numerically. In counting and measuring we have two ways of generating numerical predicates out of determinate qualitative situations. The numbers so generated can be inserted into more or less complicated mathematical expressions and made the objects of computation; the numerical outcome of the computation may then by a reverse process be applied to a new qualitative feature of the original situation, or to the same feature of a transformed situation. The rules according to which all this is done (the generation of the numbers, the computations, and the application of the results) have to be learned empirically, as Galileo realized; in this way a number of mathematical relations and formulae are selected from the potentially infinite store of such things and given physical meaning by courtesy. But the mathematical work is entirely carried out within mathematics; measurement shifts attention from quality to quantity, crossing the boundary between the sensible and the symbolic. This shift corresponds to what Braithwaite, in his Scientific Explanation , called the "separation of the mathematical apparatus."^[12]

Qualitative and Quantitative Revisited

Qualitative and quantitative do not divide up a territory; they both cover it, overlapping almost totally. But one is basic and the other optional. Everything in our world is qualitative; but virtually everything is capable—given suitable ingenuity on our part—of generating quantitative determinations. Whether we want to expend our ingenuity in this way is up to us. The United States Bureau of the Census, whose main business might seem to be quantitative, has nevertheless an interest in questions of "the quality of life," and has devoted a good deal of attention to efforts that have been made to translate expressions of satisfaction or dissatisfaction into numerical measures. The standard trick is to develop an ordinal ranking and then assign cardinal values to the positions within it for the purpose of drawing graphs, performing statistical computations, etc. The SIWB scale, for example (the initials stand for Social Indicators of Well-Being), assigns the integers i through 7 to "terrible," "unhappy," "mostly dissatisfied," "mixed," "mostly satisfied," "mostly pleased," and "delighted."^[13]

One possible use of the results of inquiries on such bases (or improved ones—the Census people seem realistically aware of the shortcomings in the state of their art) might be to produce correlations between these measures and quantities that permit of objective assessment, such as income, energy consumption, cubic feet of living space, number and horsepower of automobiles, etc. These might throw light on some aspects of our common systems of value. But it is worth noting that the starting-point here is not an experimental procedure but an appeal to the judgment of an individual. The individual does not need the quantitative apparatus, only in the first instance an awareness that better or worse conditions are possible, and a subjective conviction of distress or euphoria as the case may be. This is what I mean by saying that the quantitative is optional: our lives would be in some important respects just what they are if we did not know the date or the time or the temperature, or perhaps even our ages or bank balances or IQ's or cholesterol counts. In some significant respects they might be better. I do not mean this as a regressive criticism of measurement or computation, without which we would be at the mercy of old forces from which they have helped to deliver us, but rather as a comment on the use of the metaphorical language of number.

The French used to make fun of tourists who insistently wanted to know the population of this city, the height of that building, by calling them hommes chiffres , "number people." It is worth asking what use is to be made of numerical information. Sometimes numbers are reas-

suring or threatening, as when they mean that I can expect to live a long time, or that I run such and such a risk of having a certain sort of accident. Sometimes they give me a sense of solidarity with a community, sometimes a sense of inferiority or superiority. Sometimes there is an effect of scale, as when the numbers of people killed at Hiroshima or in the Holocaust boggle the imagination—genuine cases, perhaps, of a psychological transformation of quantity into quality (and with nothing metaphorical about the numbers either). But in every case, even these, I or other individuals must prosper or suffer singly. The quality of pain or terror or despair involved in a quite private injury or death or betrayal may match anything any individual can feel or have felt in a mass event.

The value of genuinely collective measures—aggregates, averages, and the like—remains unquestioned, but the question as to role of numerical determinations in the descriptive vocabulary remains open. Part of my argument has been that when these come about as a result of measurements they are to be understood not as quantities but as disguised qualities. Their use as such has drawbacks as well as advantages. There is a short story of Hemingway's, "A Day's Wait," that may serve as a closing illustration. An American child who has lived in France falls ill, and overhears the doctor telling his father that he has a temperature of 102°, upon which he withdraws into himself, stares at the foot of the bed, and won't let people near him. Only at the end of the day does it dawn on his father that he takes this 102 to be in degrees Celsius, a scale on which he has been led to believe a temperature of 44° to be surely fatal, and that he has been quietly preparing for death. The story ends on a happy ira shaky note. But in a world where plunges in the stock market index have been known to provoke plunges from high windows there may be room for the renewed cultivation of quality unmediated by quantity, leaving the quantities to do their undeniably useful work in their proper domain.

20—
On Being in the Same Place at the Same Time

"Nobody has ever noticed a place except at a time," says Minkowski, "or a time except at a place."^[1] One might add, "and nobody has ever noticed a place except here , or a time except now ." With this addition, what was meant as an innocent argument for the interdependence of space and time becomes a serious obstacle to all cosmologies in the traditional sense. This paper is an attempt to draw out some of the philosophical consequences of the fact that the observer must always be located here and now. It is a commentary on some aspects of the theory of relativity which seem still, after half a century, to be misunderstood.

The quotation from Minkowski is taken from his paper on space and time in which he introduces the postulate of the absolute world : "the substance at any world-point may always, with the appropriate determination of space and time, be looked upon as at rest." His decision to use the term "absolute" to describe the four-dimensional world of space-time seems curious, since the theory which led to this view of the world was a theory which promised freedom from absolutes and their replacement by relativistic determinations, but it illustrates just that ambivalence in the theory of relativity with which this paper will be concerned. The loss of absolute rest and motion in absolute space and time was a serious shock to physics, comparable to the loss (in more recent developments) of certain aspects of conservation and symmetry; in both cases the immediate reaction was to look for some way of restoring, in a slightly modified form, what had become psychologically indispensable. In the relativistic case the new version appeared to be even better than the old; the effect of Minkowski's world-postulate is,

as Cassirer points out, that "the world of physics changes from a process in a three-dimensional world into a being in this four-dimensional world."^[2] The world-postulate has its first and most obvious application at the place and time where the observer happens to be, but it was of course assumed that it might be applied equally well anywhere else in the universe. The observer may be considered at rest, but then for purposes of argument any other point may just as well be considered at rest. In fact, however, considering other points as at rest is only a game—for serious scientific purposes the observer must be considered at rest. There is no such thing as a moving observer.

This conclusion was foreseen by as early a thinker as Nicholas of Cusa. "As it will always seem to the observer," he says, "whether he be on the earth, or on the sun or on another star, that he is the quasi -motionless center and that all the other things are in motion, he will certainly determine the poles of this motion in relation to himself. Thus the fabric of the world will quasi have its center everywhere and its circumference nowhere."^[3] The last sentence is as succinct a statement of the theory of relativity as could easily be found. For Cusa the quiescence of the observer poses no problem, but that is because he too believes in an absolute, namely, God, in whom all opposites are reconciled—motion and rest, center and circumference, maximum and minimum. Apparent contradictions are tolerable when there is a divine guarantee of their ultimate resolution. But such mystical resources are no longer available to us, and the denial of the possibility of a moving observer—the claim that such a being is a contradiction in terms—is intended here as something more than an exemplary paradox.

The assertion appears paradoxical, in fact, only because we are all conditioned to Newtonian modes of thought. In a homogeneous three-dimensional universe all vantage points will be equivalent, and motion from one to another is possible without any distortion of phenomena. To put the same thing in another way, observers are interchangeable. And in the Newtonian system God is over all, the generalized observer whose omnipresence is a guarantee of the universality of the laws of motion. Every Newtonian observer could take God's point of view (i.e., any point of view removed from his or her own) and from it regard the world, observer included, sub specie aeternitatis . The reference to Spinoza is deliberate; Newtonian mechanics was a physical counterpart of Spinoza's ethics, and each rested on the possibility of seeing the world in God. Unfortunately a belief in this possibility has persisted; although contemporary scientists would hardly describe it in quite that way, many of them feel that the task of science is to give an account of the world which shall be independent of any particular perspective, But this is quite impossible.

The reason why the theory of relativity was widely thought to provide another absolute account was that it did in fact offer a formulation of the laws of nature invariant between observers, whatever their state of motion with respect to one another. (It is to be remembered that it is always the other observer who is moving.) Laws of nature had always been thought of as rules obeyed by the universe as a whole, and an invariant formulation of them was taken to be a new and more compendious way of saying what the universe, as a whole, was like. But with the new theory came a new insight into the nature of scientific law. A law (and this is by now so familiar that it seems hardly worth repeating) is simply a generalized relationship between observations, each made at a particular time and at a particular place; and the invariance of a formulation of such a law means simply that it can be applied to sets of observations taken in different times and places with equal success. But these observations can never be mixed, and if we wish to insert data from an observation at A ́ into calculations based on observations made at A , they will first have to be transformed according to some set of transformation equations appropriate to the shift from A ́ to A . There is no law which is capable of application to the universe as a whole.

Such a law, if it existed, would in any case be far too powerful for any practical purposes. The function of laws is to provide explanations, and there is only one world which calls for explanation, namely, my own world. It would be presumptuous to suppose that that constitutes more than an insignificant fragment of the world as a whole. In my world I am always at rest. Other bodies move about, and I get information about their movements from observations made, as always, here and now; the larger their velocity with respect to me, the odder the transformations they undergo—increases in weight, the speeding up of time, the contraction of lengths, etc. I should find such changes extremely inconvenient, and it is fortunate that I am not called upon to experience them. It is not that I do not move fast enough, but that I do not move at all (for the relativistic effects of small velocities are only quantitatively different from those of large velocities, and are equally inadmissible). I may get reports from other observers who are in motion relative to me, but I do not accept them until they have been transformed according to the equations mentioned above. Oddly enough, these reports never make any reference to the inconvenient consequences of motion from which I congratulate myself on being preserved; these appear only if I make observations from my own point of view on the physical system of the other observer regarded now not as an observer, but as an object of observation. Occasionally, it is true, other observers attribute to me anomalous states of motion, etc., but

these attributions are contradicted by my experience and are soon corrected by applying the appropriate transformation equation.

These considerations bring up in a novel form the whole question of the relationship of theory to observation. In theory, theoretical observers may be in motion; one well-known cosmological theory refers in fact to fundamental observers moving outwards from a point of mutual origin in such a way that none of them is at rest. Such theories are, however, purely hypothetical, and they have nothing to do with the real world except insofar as their consequences are projected upon the real world. To quote Minkowski again, "only the four dimensional world in space and time is given by phenomena, but . . . the projection in space and in time may still be undertaken with a certain degree of freedom."^[4] To say that the four dimensional world is given by phenomena is, however, to use the term "given" in a special sense, since a complex process of reasoning separates the conclusion that the world is four dimensional from the observational evidence for it. According to more familiar usage, what is given by phenomena is what has to be explained, and this is done by taking a projection of a theory which is precisely not given by phenomena, but which is freely constructed by the scientific imagination. For an observer at (x, y, z, t ) a theory is confirmed if its projection at (x, y, z, t ) agrees with observations made there, i.e. if it satisfies the boundary conditions at (x, y, z., t ). It is the possibility of different projections of the same theory, according to the different space-time situations of different observers, which Minkowski asserts in the passage quoted.

But the real world can never be the world of theory; only parts of the real world may coincide more or less exactly with parts of the world of theory when the latter are submitted to boundary conditions. And this imposition of boundary conditions has to be done afresh every time an observer makes an observation. Retrospectively, the fit of theory to the real world is remarkably good, on account of the fact that (at least in principle) those elements of theory which do not fit are discarded. But every future application of theory, even of a theory which has proved itself without exception in the past, has to be validated at the time when it is made. Such validated bits and pieces of theory remain, nevertheless, the best way of grasping the real world as it presents itself to me in bits and pieces; for my experience, while it validates theory cognitively, validates reality existentially. "The world can not exist," says Sartre,

without a univocal orientation in relation to me. Idealism has rightly insisted on the fact that relation makes the world. But since idealism took

its position on the ground of Newtonian science, it conceived this relation as a relation of reciprocity. Thus it attained only abstract concepts of pure exteriority, of action and reaction, etc., and due to this very fact it missed the world and succeeded only in making explicit the limiting concept of absolute objectivity. This concept in short amounted to that of a "desert world " or of "a world without men"; that is too a contradiction, since it is through human reality that there is a world. Thus the concept of objectivity, which aimed at replacing the in-itself of dogmatic truth by a pure relation of reciprocal agreement between representations, is self-destructive if pushed to the limit.^[5]

This would still be true philosophically even if our world were really Newtonian, but in that case a self-consistent theory of physical objectivity would be possible, and a useful reinforcement of the philosophical point lacking.

The appeal to Sartre is again deliberate. It is not, I think, too fanciful to say that, just as Spinoza was said to be a moral counterpart of Newton, Sartre is a moral counterpart of Einstein. Both Spinoza and Newton devised absolute deductive systems; Sartre, like Einstein, recognizes the necessity of reducing all questions to the level of the individual observer. The data of science, no less than those of ethics, require phenomenological analysis, since human beings in their capacity as knowers depend on their bodies for entry into the physical world just as basically as, in their capacity as agents, they depend on them for entry into the moral world. As a matter of fact, most intuitive objections to the thesis of the immovable observer rest on phenomenological grounds; what makes it implausible is not the theoretical possibility of motion but our frequent consciousness of it. But this "motion of the observer" always takes place with respect to a more or less confined framework, an environment which is itself taken to be at rest and which is always of modest and human dimensions. This is part of our psychological orientation to the world which we inhabit, and only goes to show that we need to feel anchored and located in a setting which is, by comparison with ourselves, stable and enduring. The change of attitude characteristic of the shift from Newtonian to relativistic science reflects a change in the answer to the question whether the comforting characteristics of this familiar and local world can be extrapolated beyond it. The belief that they can turns out historically to be tantamount to a belief in God.

In the light of contemporary science the conclusion seems inescapable that human beings, condemned to carry their own perspective on the world always with them—to be each (not all!) in the same place at the same time—are denied the vicarious view of a domesticated universe once provided by God. The trouble is that the scientists always

lag behind the philosophers in their understanding of the relation of human beings to God. While Newton clung to his conception of the "Lord over all, who on account on his dominion is wont to be called Lord God pantokrator , or Universal Ruler,"^[6] Spinoza had already arrived at his Deus sive natura ; and when Sartre had come to recognize that a universal consciousness of universal being was a contradiction in terms, Einstein still held explicitly a Spinozistic view of "a superior mind that reveals itself in the world of experience."^[7] It would of course be foolish to take this disparity of outlook too seriously, especially since some philosophers as well as scientists share Einstein's pantheistic conviction of the intelligibility of the world in an objective sense, i.e., independently of the perspective from which we view it. But if scientific theory is only the means of rendering intelligible the world as it appears to me from my irreducibly singular point of view (and any stronger claim seems to entail quite unjustifiable assumptions) then nothing is gained by putting into theory the possibility of my own motion except a spurious and slightly megalomaniac feeling of all-inclusive understanding. Nothing is lost by it either as long as the motion is slow compared with the velocity of light; the ordinary language of local movement does not have to be given up. The foregoing argument is addressed to the relativistic case. The immobility of the observer can be carried through for local motion too, but for relativistic motion it must be.

21—
On a Circularity in Our Knowledge of the Physically Real

In this essay I wish to raise a comparatively innocent-looking problem and explore the consequences of taking it seriously. The problem, briefly stated, is this: is there an essential circularity in our knowledge of the physical world? If so, does it matter? That is, does it have a systematically self-defeating effect on our attempts to understand that world? It will be seen as we proceed that a similar question can be raised for all claims to knowledge, but for the time being I restrict my attention to the epistemology of science. By way of an approach to the problem, consider an example that embodies it. Suppose we take some book about the physical world, for example, Henry Margenau's The Nature of Physical Reality .^[1] It is an object in the physical world and has all the properties of such an object—location, cohesion, relative impenetrability, mass, motion, and the rest. Also, it consists of an ingenious and compact ordering of plane surfaces, about 1.7 × 10⁵ cm² of them, that allows the display of an arrangement of some 9.5 × 10⁵ marks, of roughly 75 basic types, by means of a technique of impregnating the surface at the appropriate points with a preparation that changes its reflective power. These marks constitute a code that can be decoded by certain other physical objects, namely, human beings, which share the same basic properties—location, cohesion, and the rest—but have in addition special facilities for receiving and analyzing visual signals and processing and storing information. The book that I hold in my hand is a member of a class of books (called The Nature of Physical Reality ) which is in turn a member of a class of such classes of books (called just books). It is a characteristic of the members of the class of

books called The Nature of Physical Reality that they are virtually indistinguishable from one another, except for accidental marks of individuation—stains, the yellowing of pages, tears, marginal notations, dedications. By contrast, it is a characteristic of the members of the class of human beings that they are essentially distinguished from one another, i.e., that each is a unique individual. This contrast may not be as fundamental as it appears, however, depending as it does largely on differences in the mode of production in the two cases: if books were still copied out by hand they would have much greater individuality; if humans underwent some standardizing or screening process that straightened out differences in genetics or education they would have much less. A more fundamental difference between books and people than their degree of individuation (and one that largely explains that difference) is the fact that books, once produced, are inert and do not change, either in themselves or with respect to one another, except by the operation of adventitious forces, whereas human beings not only start out with marked genetic differences but also continue throughout their lives to change in themselves and with respect to one another as well as being acted on by adventitious forces. Also, they are far more sensitive to the adventitious forces, being affected by changes in their environment and by incoming stimuli to which books remain completely indifferent. It is just because the sequences of internal changes and especially of adventitious forces are never the same for two individuals (although they may be closely similar in the case of identical twins), and because the changes are for the most part irreversible and the effects of the forces cumulative, that progressive physical individuation of human beings goes on continuously. I pursue perhaps to the point of absurdity this insistence on the common physical status of books and human beings because the interaction between them to which I now wish to draw attention is purely physical in nature, consisting as it does of the scanning of the pages of the book, under suitable conditions of illumination, by the eyes of the human being, and the transmission of the coded sequence of arrays of light and dark surface areas to the brain, where it is processed by means of rapid chains of electromagnetic and chemical events in and between some of the 0.5 × 10¹⁰ neurons normally found there. It is the configuration of the links and potential barriers between these neurons that determines the individuality of the person whose brain they constitute, as apart from the individuality of that person's body (which is more like the individuality of a book). The person is a mental structure. This structure is made progressively definite by what we call the person's "experience," which can be thought of as a long syntagmatic sequence of sensory inputs, some of them coming into the brain from the rest of the body more or less

directly (as in proprioception, or in looking at one's own hands, hearing one's own voice, touching one's face, etc.), but the greater proportion coming from external objects in the form of light, sound, convection or radiation of heat, pressure, and the like. In the case of humans who have grown up in a social context, an extremely important, indeed the dominant, part of this syntagma consists of spoken or written or printed words , that is, arrays of marks or sequences of sounds that occur in repeating patterns and set off, singly or in combination, essentially similar neural reactions on each occurrence. Depending on the state of the mental structure at the time, the other types of input that arrive simultaneously, and so forth, these reactions may in turn trigger others, and the cumulative influence of these inputs and reactions determines eventually the structure of the actions that the person carries out. Let me refer once again to the present case: my own verbal input syntagma has included two readings of Margenau's The Nature of Physical Reality , one almost twenty years ago, the other quite recent. Partly as a result of these experiences, and partly under the stimulus of other inputs—reading philosophy and talking about it, feeling gratitude in respect of certain events in my life and wishing to cooperate in a particular enterprise, receiving letters and telephone messages from editors, etc.—it came about at a quite definite time and in a quite definite location that the physical body with which I am associated sat down, took up a physical pen, and began to make marks on a plane surface by running an inked ball over it that left a trace, detectable because of the differential reflecting powers of the surface and the ink. "In this essay," I wrote, "I wish to raise a comparatively innocent-looking problem and explore the consequences of taking it seriously."

I have now constructed a circle in this essay, and it is easy to see how it mirrors a circle in the processes of our knowledge. For all knowledge that can be recognized and defended as such eventually finds its way into written (or at any rate spoken) form, and the physical character of the mode of its expression will locate it in the world alongside the things of which it constitutes knowledge. The Nature of Physical Reality is not, it is true, a physics book, so that the allusion to "things in the world," of our knowledge of which it constitutes the expression, may be needlessly confusing. I would be prepared in another context to defend the view that there is only one world, namely, my own, and that everything that I can know finds its place there—and to argue further that this does not involve solipsism—but for the time being wish to concentrate on the case of physical knowledge: books about the physical properties of physical objects are themselves physical objects, and may indeed be composed of some of the physical objects—fundamental particles, atoms, molecules, etc.—about which they speak. This

is clearly self-referential, but its essential circularity may not be so evident. That, however, is because we have to include ourselves in order to complete the circle. We believe that our eyes and brains, as well as the ink and paper, are made of physically real particles, but clearly we would not be able to know about physically real particles if it were not for the use of our eyes and brains, of the ink and paper.

Now it is a mistake to think of all circularity as being paradoxical or even undesirable. In fact, there is a long tradition in philosophy which suggests that circularity in argument is inevitable. An early hint of it is given in the Euthyphro , where every attempt at clarity seems to result in bringing the argument back to its starting point;^[2] at the other end of the historical scale Wittgenstein, in the preface to the Philosophical Investigations , speaks of traveling "over a wide field of thought crisscross in every direction. . . . The same or almost the same points are always being approached afresh from different directions."^[3] The fact is that the world in which philosophy operates is a closed world; every attempt on the part of sentient beings to understand the world in which they find themselves is bound to be circular if it is carried far enough. This inevitability arises out of the fact that, whatever the starting point of the inquiry which is to lead to understanding, sooner or later the starting point itself will become an object of the inquiry. Circularity can, it is true, always be refused by resort to a priori assumptions; the trouble with these, however, is that the resulting linear argument may not intersect with another linear argument constructed by somebody else on different assumptions. If my assumptions contradict yours, our conversation is ended before it is begun. (There are people who are not troubled by this kind of standoff, but this I suspect is either because they are too lazy to be interested in alternatives or because they are too dogmatic to entertain them.)

The question is, what is the character of this world which is thus closed in upon itself? Is there room for physical reality in it? In the first instance it is a world marked by discourse, but since there are parts of it that have not been adequately articulated discourse cannot be the whole story. The best way of characterizing it may be to say that it is a thought world—not a world of thoughts , but a world every element of which is thought by somebody: in my case, by me, in yours, by you. The arguments for closure have still not been better put than by Berkeley in the Three Dialogues between Hylas and Philonous .^[4] All sensation, as well as all conception, all language, and so on, belong on the side of mind, as is clearly evidenced by the fact that in the absence of all mental activity no trace of the world remains. This is not to say that no world remains; it is only to remark on the thorough assimilation of whatever world may independently exist into mental form for pur-

poses of human consumption. In the thought world there is clearly no room for material, if by material we mean something that is not thought.

But there is no need to identify the physical with the material; physis means, after all, nothing more than the nature of things naturally generated. And yet I think that most people who use the word mean by the "physical" something that is at any rate independent of their minds, and the independence of the material is no more problematic than, in Berkeley, the independence of the mind of God. Even for Hegel we are only the Absolute knowing itself, and its knowing itself does not in any way entail our knowing it. The problem of physical reality remains as acute as ever for the individual in his or her private circle, and the philosophical circle can be described in idealist or in materialist terms indifferently. I started this essay with a book as an object in the world, but I could have equally well have started it with the concept of the book—the point is that in both cases the exercise was an exercise in thought, a conclusion not to be escaped because it happened to be an exercise in discursive thought and even in written form.

The choice of the medium, then—ideas or matter—leaves the question of reality untouched. The problem of the real, as I understand it (and this is not, as we shall see, exactly the way that Margenau himself understands it), is the problem of clarifying the status of what there is, insofar as this is independent of my experience. My experience itself is of course also real, but this is not problematic, nor has it been since Descartes—that is, since the establishment of the distinction between the actuality of experience and its significance. To say that the problematic part of the real is the part of it that is independent of my experience does not mean that none of it could be in my experience, only that it would be as it is whether it were in my experience or not. But by far the larger part of it—everything that happened before I was born, everything that will happen after I die, everything that now happens elsewhere, or behind my back, or beneath the surface of things—is and remains outside my experience. Reality is infinitely richer, ontologically speaking, than my world (even though my world can contain a conception of the whole of reality).

I wish now to deal with two kinds of attempts that have been made to solve the problem of the relationship between the content of my experience on the one hand and reality on the other. One strategy has been to try to construct the elements of reality out of the elements of experience, or at any rate out of classes of elements, typical examples of which are found in experience. The most elegant expositions of this strategy are to be found in Russell and Carnap. Russell poses the problem as "the construction of physical objects out of sense-data," and he arrives at the notion of things as classes of their aspects. We have seen

the table from a few points of view, under a few conditions of lighting, etc.—imagine how it would look from all possible points of view under all possible conditions, and we have an aggregate that will exhaust without remainder all possibilities of perceptual knowledge of the table.^[5] Carnap's program is more ambitious: from the perceptions of a single observer he constructs, in ascending order, autopsychological, physical, heteropsychological, and cultural objects.^[6] (He uses for the purpose a fictional character A, the first half of whose life is spent in experiencing the world without analysis, the second half in analyzing the data thus gathered, without further experience.) A third attempt along somewhat similar lines—although all three have radical differences from one another—is that of Whitehead, who by his method of extensive abstraction sought to isolate real space-time points as the termini of converging series of experienced space-time regions.^[7]

The second strategy is that of Margenau himself. He approaches the question from a diametrically opposite point: assuming, in effect, that if the real shows up in our experience at all it will bear the marks of its independence (its coherence, its connectedness, and so on), he limits the ascription of reality to those parts of experience that have been successfully systematized by means of scientific constructs. "To us," he says in The Nature of Physical Reality , "reality is not the cause but a specifiable part of experience."^[8] Now both these strategies—Russell's and Carnap's on the one hand, Margenau's on the other—seem to me unsatisfactory, in the former case because of ontological extravagance, in the latter case because of ontological poverty. On the one hand, Russell thinks nothing of introducing infinite sets into the real, and he does so with an abandon that would have made Ockham shudder. Margenau, on the other hand, excludes from the real undeniable facts of experience, if these have not been "normally standardized into scientific knowledge." "As yet," he says, "they have not been united into an organized pattern comparable with the structure of physical reality, and it would be pardonable for the scientist to suggest that the name reality be at present denied to them."^[9] If we subscribe to the Cartesian principle referred to above, this puts him in the odd position of saying that reality is only a part of a part of itself. Of course, the motivation for introducing a limited and specific meaning for the expression "physical reality" is quite clear in Margenau's work, and if we view it as an epistemological rather than as an ontological limitation his strategy becomes perfectly acceptable. And yet from the ontological point of view it does seem overcautious.

My own inclination is to try a completely different approach, by asking whether there could be anything with respect to which we could know that it lay outside our experience as well as being independent of

it. If it were just a question of our confronting the physical world as individuals, there would be no possible way in which we could know that anything lay outside our individual experience, and anyone who wished stubbornly to maintain that view throughout the following argument could perfectly well do so. There is, however, a class of events of which we can come to have knowledge that would strike all but the confirmed solipsist as satisfying my criteria: I mean the contents of other people's experience. We come to know these by being told about them, and we have to admit that, for the most part at any rate, our existence makes no difference to them and their immediacy is out of our reach. Other people report to us what happened in the past, and what happened elsewhere, and what happened when we were not looking, and so we extend the basis of our knowledge of the real. I cannot here go into the details of the process according to which scientific knowledge is generated out of this collective resource; suffice it to say that the result is not a monolithic body of scientific theory or even of knowledge of the everyday world, but a series of free-standing conceptual schemes each associated with a single knower, yet overlapping in such a way that groups of people can in various times and places be taken to embody such theories and such knowledge.

Every conceptual scheme is closed upon itself just in the way in which experience was earlier said to be—in fact, every such scheme is precisely the consequence of a certain experience—and so is the aggregate of conceptual schemes, incoherent as this is bound to be even if limited to the domain of a particular science. Yet, when the scheme of an individual knower has been critically rethought, when out of intuitive systems scientific constructs have been formed, when some part of the scheme has been converted into a system , the overlapping with other people's schemes proves to be more precise than before, so that among the practitioners of a given science it becomes reasonable to speak of an isomorphism between individual conceptual systems. And it is this that leads us back, in the end, to the idea of an external and independent physical reality. "It is forced upon us by the constant pattern of the isomorphic systems. But this in itself does not necessitate the postulation of an external: perhaps the isomorphic systems just hang together that way. All of them, however, except the postulated natural reality, are in mind; they constitute a universe of thought. The reason why it seems to me sensible to postulate a universe of being also is that in principle the universe of thought could be entirely capricious, and it is not. The simplest explanation of this constancy of form between the isomorphic systems is the existence of another system isomorphic with all of them—rather such that they are all imperfect isomorphs of it—underlying them and proceeding independently of them."^[10]

The standard objection to this formulation is that it makes physical

reality hypothetical, and that this hypothetical status then extends to every element of it, even those we take to be reflected in our experience. This leads to the paradoxical consequence that the real components of our experience are made up of hypothetical parts. But this is just a set of confusions. The real elements of our experience are not made up of anything, unless of parts really experienced at the same time (there is nothing paradoxical about an experienced person's having experienced arms and legs). What is hypothetical is the object as a whole under its real, as opposed to its experienced, modality, and there is nothing odd about a hypothetical object's having hypothetical parts. What science enables us to see is that there are more hypothetical objects than experienced ones: electrons lie in the hypothetical real world, and not in the real world of experience—but the formulation of that sentence is meant to underline the proposition that we are not talking about two worlds, only about one, most of which, however, remains outside our experiential range.

The upshot of this analysis is that the cycle of our knowledge, from perceptions to hypotheses and back to perceptions, while it is itself inscribed in the real, does not enclose the reality it claims to know. Reality lies outside the circle, rather in the way that tracks lie outside the train. (Imagine a train running on a single circular track, and passengers able only to look horizontally out of the side windows—they could tell by the repetition of the landscape that they were on a closed track even if they could never actually see it.) The idealist temptation is to identify reality with the circle, but this seems unnecessarily limiting as an ontological principle, besides being immodestly self-centered. Assuming a reality infinitely vaster than our knowledge of it, we can still make at least one assertion about it with complete assurance, albeit a negative assertion: namely, that the real has not been such as to require other experiences than those we have in fact had. In closing I offer two formulations of a somewhat more positive kind, each of which sums up the result of the argument, and the choice between which can be left to philosophical temperament. The first is skeptical, and yet conditionally affirmative: we do not know the real at all, but what we assert about it would be knowledge if there were any way of getting out of the circle . The second, which I prefer, has obvious affinities to Margenau's position, but avoids some unfortunately relativistic aspects of that position: the real is the objective correlate of those aspects of conceptual structure that most people have in common or in such form that it is translatable into a common structure . This assumes that the development of science is on the right track, not necessarily that it has arrived. The possibility remains, of course, that we might all be mistaken together.

I conclude, then, that there is an essential circularity in our knowl-

edge of the physical world—that every line of demonstration sooner or later turns back upon itself. But I conclude also that this is not a self-defeating circularity, and that two things about it even give grounds for philosophical satisfaction. One is that, as always happens in philosophy, the state of the knowers , if not of their knowledge, changes as they go around the circle, so that they understand each proposition anew in the light of the propositions that have intervened since their former understanding of it. The other is that if the relative sequence of the propositions remains unchanged—if they are always there, as it were, when the argument comes round to them—that in itself may be evidence of an underlying stability in the real, of which, after all, the propositions and the argument and for that matter we ourselves are an integral part.

22—
Truth and Presence:
Poetic Imagination and Mathematical Physics in Gaston Bachelard

In the stacks of the Sterling Library at Yale University, thirty years ago, I happened as a graduate student in philosophy to be reading Gaston Bachelard's L'activité rationaliste de la physique contemporaine while my closest friend at the time, a graduate student in French, was reading his L'eau et les rêves . This coincidence was gratifying, although it did not seem remarkable; neither of us found the other's interest alien. I refer to it not from romantic nostalgia but because it now occurs to me that this personal conjunction of science and the humanities antedated by five years C. P. Snow's The Two Cultures and the Scientific Revolution ,^[1] an essay which suggested that it ought to have seemed remarkable, since according to Snow a great gulf was, if not fixed, at least being busily dug, between the domains to which these works belonged. Of course Snow believed rather complacently that he himself embodied a rare and difficult combination of the two, but he seems not to have realized how thoroughly his problem had been anticipated, or how satisfactorily it had been solved, by a professor at the Sorbonne who had begun his career as a provincial French postman.

As far as that goes my own double interest, in science and in poetry, antedated by many years my encounter with Bachelard. Bachelard somewhere acknowledges a debt to his father in the matter of building fires; I owe a debt to mine both for his habit of reciting Milton and for his curiosity about the sciences, especially astronomy. He possessed some of the works of those great popular writers, both distinguished scientists, Sir James Jeans and Sir Arthur Eddington, and I read them while I was still in school; in the latter's The Nature of the Physical

World is a passage that Bachelard may have known and would certainly have liked. "One day," says Eddington, "I happened to be occupied with the subject of 'Generation of Waves by Wind.' I took down the standard treatise on hydrodynamics, and under that heading I read," (and there follows a paragraph of mathematical symbols):

And so on for two pages. At the end it is made clear that a wind of less than half a mile an hour will leave the surface unruffled. At a mile an hour the surface is covered with minute corrugations due to capillary waves which decay immediately the disturbing cause ceases. At two miles an hour the gravity waves appear. As the author modestly concludes: "Our theoretical investigations give considerable insight into the incipient stages of wave-formation."

On another occasion the same subject of "Generation of Waves by Wind" was in my mind; but this time another book was more appropriate, and I read:

There are waters blown by changing winds to laughter
and lit by the rich skies, all day. and after
      Frost, with a gesture, stays the waves that dance
And wandering loveliness. He leaves a white
      Unbroken glory, a gathered radiance,
A width, a shining peace, under the night.

The magic words bring back the scene. Again we feel Nature drawing close to us, uniting with us, till we are filled with the gladness of the waves dancing in the sunshine, with the awe of the moonlight on the frozen lake. These were not moments when we fell below ourselves. We do not look back on them and say: "It was disgraceful for a man with six sober senses and a scientific understanding to let himself be deluded in that way. I will take Lamb's Hydrodynamics with me next time." It is good that there should be such moments for us. Life would be stunted and narrow if we could feel no significance in the world around us beyond that which can be weighed and measured with the tools of the physicist or described by the metrical symbols of the mathematician.^[2]

Eddington suggests here that the business of life will draw one's attention now to the scientific side of things, now to the poetic; there is no thought that the two functions will be exercised by different people, or belong in the life of the same person to separate periods, say, youth and maturity.

Critics are fond of chopping great thinkers into two, the early and the late, and this is nearly always misleading, as the most obvious examples show (Marx, Wittgenstein, and Sartre come immediately to mind). Some people have tried to do this with Bachelard, as if he turned from science to poetry, but even the sequence of published works is more complicated than that. If it is necessary to identify periods there

are at least four, the first two overlapping: (1) an initial preoccupation with scientific thought, from Essai sur la connaissance approchée (1928) to La philosophie du non (1940); (2) the working through of the elements and the corresponding forms of the imagination, from La psychanalyse du feu (1938) to La terre et les rêveries du repos (1948); (3) a reconsideration of the thought processes of science in the light of a new rationalist epistemology, which includes Le rationalisme appliqué (1949), L'activité rationaliste de la physique contemporaine (1951), and Le matérialisme rationnel (1952), three works that Roch Smith has called a "trilogy," a view supported by Bachelard himself ("je considère que [ces] trois livres . . . ont une unité de vue";^[3] and (4) the new poetics of the three last works, La poétique de l'espace (1957), La poétique de la rêverie , and La flamme d'une chandelle (both 1961). So at the beginning of this paper I state my confidence in two kinds of unity: that of Bachelard's career, and that of the possible embodiment of both science and poetry in a single individual that that career exemplified.

In the stacks of the Sterling Library, however, the rest of the Bachelardian corpus was still in my future. I was reading L'activité rationaliste for a quite specific reason, namely, to advance an inquiry into the ontological status of fundamental entities in physics. Electrons, protons, and the rest are never observed directly, so they remain theoretical constructs; what we observe are the consequences of interactions in which we suppose them to have participated—bubble chamber tracks, clicks from Geiger counters—and these consequences are always macroscopic and more or less familiar. This is still a topical problem, though not in the form of bewilderment about waves and particles that Eddington dramatized with his "wavicle," which was a wave, as I remember, on Mondays, Wednesdays, and Fridays, and a particle on Tuesdays, Thursdays, and Saturdays. Now, with the benefit of hindsight, I would rather be inclined to say: why did we ever suppose that the habitual images experience equips us with in the local "flat region" of macroscopic observation would be adequate to remote reaches of physical reality—the microscopic, the cosmological, the relativistic? Getting physical theory right means being ready to leave the comforts of the flat region, to depart from the simple image.

Now two things about Bachelard seem to me particularly memorable and important: on the one hand the tenacity of his rootedness in what I am calling the "flat region," the familiar, the everyday, the down-to-earth, but on the other hand the audacity of his speculative departures from this solid base, his persistence in following his arguments where they led, whether into the gloom of psychoanalytic depths or the vertigo of relativistic speed and distance. The polarity of his work between

science and poetry is, as I have already noted, notorious; I find no less remarkable the polarity between the postman and the philosopher. On the whole it seems to me that it would be a good thing for more philosophers to have been postmen. The metier may not be accidental: apart from the letter-scales Bachelard refers to as having given him his idea of weight, there is a hermetic side to the postman's activity—he is the point of contact with the world beyond, he brings sealed messages from distant origins, there is no knowing what marvels or portents they may not contain; at the same time nothing can surprise him, he is the very image of persistence and reliability, of local intimacy and homely order. And when the postman himself leaves for the outside world—for Dijon, for Paris—he takes with him this imperturbable sense of the familiar, and his concern continues to be with the firm materiality of the world, now from the scientific point of view.

It is, however, the point of view of a new science, a "nouvel esprit scientifique," one of whose effects is gradually to undermine that materiality. The old science, beginning with Galileo, say, made its object the mathematical representation of observable relations; Newton added the modern concept of force, but that had its own familiar representation in muscular effort. Microscopes and telescopes, etc., merely extended the flat region; they did not lead outside it. It was towards the end of the nineteenth century that the existence of entities hitherto unsuspected, with entirely new properties, began to force itself on scientific attention. The electron was discovered when Bachelard was eleven, and he was a young man during the heady days at the beginning of the century when relativity and quantum theory were undergoing their dramatic development from marginal conjectures to fundamental disciplines of physics.

The initial reaction to the opening of these new domains was sometimes overdone, and Bachelard did not escape the temptation to which so many of his contemporaries succumbed of making a mystery out of the absence of an imaginable substantiality at the quantum level. In Le nouvel esprit scientifique he says,

Instead of attaching properties and forces directly to the electron we shall attach quantum numbers to it, and on the basis of the distribution of these numbers we shall deduce the distribution of the places of the electrons in the atom and the molecule. The sudden dissolution of realism should be clearly understood. Here number becomes an attribute, a predicate of substance . . . . Thus chemistry, which was for a long time the "substantialist" science par excellence , finds the knowledge of its own matter progressively dissolving. If we judge the object according to the proofs of its objectivity, we must say that the object is mathematizing itself and that it manifests a singular convergence of experimental and mathemati-

cal proof. The metaphysical gulf between mind and the external world, so unbridgeable for a metaphysics of immediate intuition, seems less wide for a discursive metaphysics that attempts to follow scientific progress. We can even conceive of a veritable displacement of the real, a purging of realism, a metaphysical sublimation of matter. Reality first transforms itself into a mathematical realism, and then mathematical realism comes to dissolve itself in a sort of realism of quantum probabilities. The philosopher who follows the discipline of the quanta—the schola quantorum —allows himself to think the whole of the real in its mathematical organization, or better, he accustoms himself to measure the real metaphysically in terms of the possible, in a direction strictly the inverse of realist thought. Let us then express this double supremacy of numbers over things and of the probable over numbers by a polemical formula: chemical substance is only the shadow of a number (l'ombre d'un nombre ).^[4]

This is terribly confused. It is simply misleading to suggest that there are numbers in the objective world and that they somehow replace a materiality that has dissolved away. If the world ever was material, it has not ceased to be so just because we can't picture its materiality. Before, we could have a pictorial representation as well as a mathematical one; now we can manage only the mathematics, but it is no more constitutive of the world than in the former case. The epistemological basis of science is still in ordinary macroscopic objects; our immediate world is still Euclidean and Newtonian; but we have learned that the rough-and-ready world-picture of the flat region, with its colours and sounds, its solids and spaces, is inadequate for the representation of basic physical truths.

What gets in the way of a relaxed and uncomplicated acceptance of this limitation seems to be a need on our part to have an image of matter. It is difficult to attribute reality, materiality, or substance to the world there physically is without attributing to it the imaginative contents that have hitherto accompanied these ideas. There is no way of getting rid of these imaginative contents but their existence poses a problem for scientific understanding. The fact that La formation de l'esprit scientifique and La psychanalyse du feu were published in the same year is not accidental: in the former Bachelard is concerned not only with the proper formation of the scientific mind but also with the fact that it is de formed by its habitual expectations, while in the latter he looks at a particular case, the habitual association of substantiality and fire. "In this book when we talk of our personal experiences we are demonstrating human errors," he says in the Introduction to La psychanalyse du feu , and he continues.

Our work is offered, then, as an example of that special psychoanalysis that we believe would form a useful basis for all objective studies. It is an illustration of the general theses put forward in our recent book, La formation de l'esprit scientifique . The pedagogy of scientific instruction would be improved if we could demonstrate clearly how the fascination exerted by the object distorts inductions . It would not be difficult to write about water, air, earth, salt, wine and blood in the same way that we have dealt with fire in this brief outline. . . . If we succeeded in inspiring any imitators, we should urge them to study, from the same point of view as a psychoanalysis of objective knowledge, the notions of totality, of system, of element, evolution and development. . . . In all these examples one would find beneath the theories, more or less readily accepted by scientists and philosophers, convictions that are often ingenuous. These unquestioned convictions are so many extraneous flashes that bedevil the proper illumination that the mind must build up in any project of discursive reason. Everyone should seek to destroy within himself these blindly accepted convictions. Everyone must learn to escape from the rigidity of the mental habits formed by contact with familiar experiences. Everyone must destroy even more carefully than his phobias, his "philias," his complacent acceptance of first intuitions.^[5]

It is clear from this passage, among other things, that Bachelard's project at this time was a full-fledged deconstructionism avant la lettre .

There are now two directions in which the Bachelardian work must obviously go—towards the dissolution of the scientific image, and towards the exploration of what this turn uncovers, namely the richness of the material image in its own right, and not just as an obstacle to scientific understanding. What led to the other works on the elements was just the realization, which dawned after (but no doubt as a result of) the writing of La psychanalyse du feu , that the domain of the imagination has its own constructive materiality ("quand j'ai écrit le Feu je ne me rendais pas compte du rôle de l'imagination matérielle").^[6] The former direction is taken in La philosophie du non , and leads from the image to the concept, not now as a mathematized abstraction but as a postulated object more real than anything merely imaginable. Just as in surrealism (in which Bachelard at this time was deeply interested, to such a degree that Breton called him "the philosopher of surrealism"), the domain of the everyday is transcended, by an appeal to the unconscious, towards the poetically marvelous, so in Bachelard's "surrationalism" the familiar image is transcended, by an appeal to critical reason, towards the physically fundamental.

In one way or another, what is cut away from the image has to be found in the rectified concept. We could therefore say that the atom is exactly the sum of the criticisms to which its first image has been submitted.

Coherent knowledge is a product not of architectonic reason but of polemical reason. By its dialectics and its criticisms, surrationalism in a certain way determines a surobject . The surobject is the result of a critical objectification, of an objectivity that preserves of the object only what it has criticized. As it appears in contemporary microphysics the atom is the very paradigm [type ] of the surobject. In its relations with images, the surobject is exactly the nonimage. Intuitions are very useful: they are good for destroying. In destroying its first images, scientific thought discovers its organic laws. The schema for the atom proposed by Bohr a quarter of a century ago has in this sense behaved like a good image: nothing remains of it.^[7]

(I translate "surobjet" as "surobject" rather than as "superobject" to maintain consistency with "surrealism"—and hence "surrationalism"—even though it is a rebarbative term. The use of this prefix in recent thought presents some interesting contrasts: "Ueberich" in German becomes "surmoi" in French but "superego" in English, which seems right—but if "surréalisme" had by the same token become "superrealism" I cannot help feeling that the understanding of the movement would have been very different, perhaps indeed improved.)

But if for science nothing remains of the image, the images that nevertheless remain lose nothing of their poetic value. Since this is the aspect of Bachelard's thought that has become the most familiar, I can afford to dispense with a catalogue of what those images are and concentrate on some problematic aspects, with the remark however that if he had done nothing but identify the species of the material imagination, that would have been enough to establish him as one of the century's seminal figures in the domain of poetics. It is perhaps not without significance that this work had its origins in a therapeutic situation, the psychoanalysis of fire described in an earlier citation.

Fire is the least material of the elements, and its elemental status is the most obviously unscientific. If we ask what fire is, the scientific response is quite straightforward: it is the hot and therefore visible gaseous product of an exothermic chemical reaction, usually one of oxidation; and this is as far as it could possibly be from the poetic response, in which it is warmth, passion, domesticity, life. The two poles do not interfere. What this means is that it is relatively easy to perform the required psychoanalysis; we are not really aux prises with materiality (indeed as remarked above the material imagination is not in play at the time of La psychanalyse du feu ). However as Bachelard works through the elements things get stickier, as it were, and by the time of La terre et les rêveries de la volonté there is a kind of collision of matter and imagination that seems to compromise the distinction between science and poetry. "Reverie that looks for substance under

ephemeral aspects," confronted with the three lighter elements (fire, water, and air or sky),

was in no way blocked by reality. We really confronted a problem of imagination ; it was a matter precisely of dreaming a profound substance for the fire, so lively and so brightly colored; it was a matter of immobilizing, faced with running water, the substance of this fluidity; finally it was necessary, before the counsels of lightness given us by breezes and flight, to imagine in ourselves the very substance of this lightness, the very substance of aerial liberty. In short materials no doubt real, but mobile and inconstant, required to be imagined in depth, in an intimacy of substance and force. But with the substance of the earth, matter brings with it so many positive experiences, the form is so evident, so striking, so real, that it is hard to see how to give body to reveries touching the intimacy of matter. As Baudelaire says, "The more positive and solid matter is in appearance, the more subtle and laborious is the task of the imagination."^[8]

The resolution of this conflict is to be found in the admission that the substantiality of earth is just as imaginary as the substantiality of any of the other elements—that is, material and imagination belong together on the side of poetry, neither has anything to do with science. To the question whether images of density, hardness, massiveness, substantiality, etc., tell us anything at all about how the physical world really is, the brutal answer is no. They tell us about our world, with its vertigo and its viscosity, but not about the world science has to deal with. This doctrine is hard to accept because we want science to be about ordinary objects, not "surobjects" inaccessible to us, or accessible only through the operations of reason, and because as Bachelard says the impression of contact with the real material of things is so strong. But science is under the rule of reason and it does compel us to conclude that the physical world is beyond the reach of the material imagination; and Bachelard believes that this conclusion has to be accepted according to what he calls

the cogito of mutual obligation, [which,] in its simplest form, should be expressed as follows: I think you are going to think what I have just been thinking, if I inform you of the episode of reason which has just obliged me to think beyond what I previously thought.^[9]

What we have to "think beyond" is, once again, the image. It is not just images of materiality that are suspect; in contemporary physics nothing is given to the imagination, not even something "hidden"—what there is seems less discovered than invented. In the works of the trilogy "surrationalism" gives way to "applied rationalism," a

more modest way of handling the same problem, and the atoms of an earlier citation from La philosophie du non have been generalized into particles, but the message, though expressed differently, is by now familiar:

Particles are situated at the boundary between invention and discovery, just where we think applied rationalism is active. They are precisely "objects" of applied rationalism. When we studied matter in an attempt to resume it in its four elements, in its four kinds of atom, phenomenology offered seductive images: fire has a spark, water a drop, earth has a grain, air can be felt in the movement of dust. Here, nothing. No natural "corpuscularisation." Nothing, absolutely nothing in common knowledge that could set us on the track of the isolation of a particle. And all the images are deceptive [et toutes les images sont trompeuses ].^[10]

By now the point seems sufficiently established. Yet there is something unsatisfactory about it even from the scientific point of view. It is as if, in looking for the truth about the world, which is now to be expressed in formal rather than materially imagistic terms, we had somehow forgotten that it was there . The parts of the world—its particles—are yielded only by the application of reason and only when I am attending to them with a certain concentration of thought and from a particular point of view. But all the while the rest of the world is there, as it were, peripherally; I can't, precisely, be attending to it , and yet its being there is a condition of my having anything to attend to in the first place.

In a remarkable paper delivered to a philosophical congress in Lyon in 1939 Bachelard speaks of "the idea of the Universe [which] presents itself as the antithesis of the idea of the object," and introduces the lapidary formula: "The Universe is the infinite of my inattention." The truth about objects has to be complemented by the presence of the world, immediately and globally; our sense of this presence is a matter of intuition rather than of knowledge, it comes not from the accumulation of facts but from a kind of phenomenological totalization.

Experience of the Universe, if we admit that this concept has a sense, prepares no multiplication of thought; as far as I am concerned the idea of the Universe immediately and definitively dialectizes my objective thought. It breaks my thought. The I think the world ends for me with the conclusion: therefore I am not .

In other words, the I think the world puts me outside the world . Meditate on the other hand on the axiom of the philosopher of the universe: everything is in everything. Listen to him sing, like a poet, his Einfühlung among the forms and the light, the breaths and the perfumes. Look at him in his paradoxical attitude: it is in opening his arms that he embraces the world! But—strange conclusion—this Universe that totalizes all qual-

ities keeps none of them as a specific quality. Or at least if it does keep one, one soon sees that it is only as the valorization of a reverie.^[11]

This is where the image comes back into its own. The quality of the Universe is in effect the quality of the moment of my apprehension of it, not now with scientific concentration but with poetic openness; it is the product of the nonspecific awareness that Bachelard calls reverie, waking but not active, alert but not intentional. The image, specifically the literary image, offers us this kind of relation to the world, or rather offers us a new content for it. Literature is significant, and its significance derives in part from its lending new significance to the world. In Bachelard this process goes through three stages, in which the image is first directly signifying, then metaphorical, and finally a creator of its own "unreality." The first is found in L'air et les songes :

How can we forget the signifying action of the poetic image? The sign here is not a reminder, a memory, the indelible mark of a distant past. To deserve the title of literary image it has to have the merit of originality. A literary image is a sense in the state of being born; the word—the old word—comes to receive from it a new signification. But this is not yet enough: the literary image must enrich itself with a new oneirism . To signify something other, and to make for other dreams, such is the double function of the literary image.^[12]

"To make for other dreams": it is not that we needed the image to have dreams in the first place, to live the reverie that yields the Universe in the mode of presence rather than (scientific) truth, but it offers us a renewal of that presence under a different sign. However, the relation between signs that this originality of the literary image generates is nothing other than metaphor, and some years later, in this passage from La terre et les rêveries du repos , Bachelard suggests that poetry gives access through its metaphoric shifts to something like a true dream, a truth of its own:

In all its objects, Nature dreams. From this point, if we faithfully follow the alchemical meditation of a chosen substance, a substance always gathered in Nature, we arrive at this conviction of the image which is poetically salutary, which proves to us that poetry is not a game, but rather a force of nature. It elucidates the dream of things. Thus we understand that it is the true metaphor , the doubly true metaphor: true in its experience and true in its oneiric thrust.^[13]

The imagination here, however, is still, as Bacon might have said, "hung with weights," held down in this as in the other earth book

(cited above) by the evident reality of the material, convinced by its experience rather than freely adventuring. It is only in the period of the last poetics that the imagination is given a power of its own, liberated not only from the burden of experience but from metaphor itself. Thus, in La poétique de l'espace , Bachelard says,

Academic psychology hardly deals with the subject of the poetic image, which is often mistaken for simple metaphor. Generally, in fact, the word image , in the works of psychologists, is surrounded with confusion: we see images, we reproduce images, we retain images in our memory. The image is everything except a direct product of the imagination. . . .

I propose, on the contrary, to consider the imagination as a major power of human nature. To be sure, there is nothing to be gained by saying that the imagination is the faculty of producing images. But this tautology has at least the virtue of putting an end to comparisons of images with memories.

By the swiftness of its actions, the imagination separates us from the past as well as from reality; it faces the future. To the function of reality , wise in experience of the past, should be added a function of unreality , which is equally positive, as I tried to show in certain of my earlier works.^[14]

Such a "function of unreality" is clearly incompatible with scientific truth, whose concern must in the end be with the real even if on the way to its formulations it passes through the philosophie du non . But it is not incompatible with presence, especially if we construe the prae of praesens as temporally before; the future is axiomatically unreal, but it is the task of the imagination to face it, not in the mode of knowledge and the determination of parts but in the mode of creativity and transcendence towards the whole. So Bachelard quotes with approval these words of Jean Lescure: "Knowing must be accompanied by an equal capacity to forget knowing. Non-knowing is not a form of ignorance but a difficult transcendence of knowledge. This is the price that must be paid for an oeuvre to be, at all times, a sort of pure beginning, which makes its creation an exercise in freedom."^[15]

The poetic presence to the world that is always a pure beginning transcends scientific knowledge but does not thereby belittle or annul it. I revert now to the duality from which I began, between science and poetry, in the light of Bachelard's itinerary. We left the truth about the real, some pages back, in the care of a strictly unimaginable but mathematically compelling "applied rationalism," in order to pursue the power of the image towards an immediate presence to being. This presence is characterized in La poétique de l'espace as a possession of the subject by the image, as a reverberation that constitutes a "veritable

awakening of poetic creation . . . in the soul of the reader."^[16] These two extremes—on the one hand mathematics with no image at all, on the other an image that fills the whole space of subjectivity—seem to stand in complete opposition to one another, to have nothing in common. For Bachelard, however (as for Eddington), they are clearly not opposites but complementaries. It may be helpful in closing to consider their complementarity through the mediation of language.

Language is a common resource of science and of poetry, but the roles it respectively plays in them illustrate at once their separation and their continuity. Language—the language of logic and of mathematics—is the only medium we have for representing the truth about objective physical reality, inaccessible as it is to the imagination. On the other hand language is incapable of representing the immediacy of presence, which is yielded only by the imagination, although in poetry it can as it were prepare the imagination for presence. Language, in Heidegger's terms, is "the house of Being," by which we are to understand that if we make (poiein ) a place for being, by means of poetry, Being may come to dwell in it. Presence to Being however is not linguistic, it is not the same as presence to poetry—the latter is merely propaedeutic to it. Bachelard seems to have had an independent understanding of this in his doctrine of the reverberation of the poetic image, the image that "has touched the depths before it stirs the surface."^[17]

These two functions—the discursive ground of science that is constituted by language and the unspoken intentionality of poetry that is prepared by it—are both eminently human functions. The subject does not vacillate between them but occupies their intersection, an intersection that is not a point but a place , the place where our life, with all its scientific complexity and poetic intensity, takes place. What Bachelard reminds us, in his person no less than in his writings, is that the complexity and the intensity are departures from, and equally rooted in, the familiar materiality of the simple image; that, given a willingness to do the necessary work, whether rational or imaginative, scientific truth and poetic presence are both accessible, to postmen as to philosophers.