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The Scope and Limits of Human Knowledge

D.M. Armstrong

Where do we have knowledge and where do we fall short of knowledge? If we take this question in a broad and general way - and, of course, as a mere philosopher I cannot take it in any other way - then I believe that a fairly definite answer can be given, one moreover from which important consequences follow for the conduct of our intellectual life.

What are the proper foundations of our knowledge? Descartes, as I am sure everybody here knows, thought that all he was entitled to start from was his own existence as a thinking being. He then treated us to the unedifying spectacle of an attempt to reconstruct the body of what he ordinarily took himself to know from this absurdly limited base. We are only now beginning to recover from the damage done to epistemology that his enterprise fostered.

The foundations for knowledge that I uphold are much more extensive and at the same time much more mundane. These foundations were argued for by the Cambridge University philosopher, George Edward Moore, in the earlier part of the recently ended century. Moore spoke of the propositions that he wished to uphold - primarily against other philosophers! - as the deliverances of common sense . Common sense, someone might immediately say, can be - has sometimes turned out to be - common error. But Moore was thinking truisms, obvious truisms of which we can be certain. One mark of such truisms is that, in ordinary circumstances, it is almost embarrassing to mention them. Here are some Moorean truths (as I like to call them) that apply to us in this building this evening. We are in Sydney University, which is in the city of Sydney, in the state of New South Wales, one of the states of Australia. We are human beings and we all have bodies, and heads, and each is likely to have a full quota of four limbs. (Moore used to hold up his hand and say "Here is a hand".) We were once children, but are now adults, some of whom have children themselves that they have begotten. Eventually, we will die. As you will immediately recognize, I could go on indefinitely boring you with truths of this sort, truths of which we are perfectly certain. We know them to be true.

But, you might say, perhaps with memories of a first-year course in philosophy, is it not possible that a deceitful demon is manipulating our mind in such a way that these things I allege to be truths are not truths at all? In the technologically updated version of this nightmare, is it not possible that we are brains in a vat being fed a totally false scenario by some evil scientist? Moore had what I think is a very strong argument to give against this and any other considerations that might be brought against the body of Moorean truths. It may help if we focus on one particularly fundamental case: the existence of motion.

It is one of the most obvious facts about the world that things move, that there is motion. Just to take our own case, it is surely absurd to say that there is no crawling, walking, running, driving in vehicles, flying in aeroplanes. In this case, though, over and above arguing from the bare possibility that we are completely deluded in this matter, it happens that a systematic and argued case against the reality of motion has been made. You are no doubt aware of Zeno of Elea's wonderful paradoxes that seek to show that motion cannot exist. (Some of you may have seen or read Tom Stoppard's play Jumpers, where the hero, a philosopher named George Edward Moore - but, oddly, not identical to our great George Edward Moore - rehearses Zeno's argument about the Flying Arrow and concludes "so we see that the arrow cannot move, and St. Sebastian died of fright".)

What was the (real) Moore's reply, as applied to this particular case? It was to point out that if we take the premisses of Zeno's arguments, and compare them with the conclusion "motion does not exist" then we are very much more sure of the falsity of the conclusion than we are of the truth of the premises. Zeno's arguments are really very tricky - he was, obviously, an excellent philosopher - but the rational response to them is not to give up on motion but rather to conclude that there must be something wrong with the argument in each case even if we cannot spot what is wrong with it. That, I think with Moore, is the correct strategy for attacks on Moorean truisms. They are more certainly truths than any evidence that is brought against them. I doubt if the bare possibility of being mistaken about these truisms is really evidence at all against them. But even if we grant that it is some sort of evidence it cannot compare with the certainty that attaches to the truisms. That is why I, following Moore, think that we are justified in cleaving to the truisms, the truism that there is motion and innumerable other truisms. And they are, in my view, the proper foundations for our knowledge.

You will notice that Moore's argument seems not, alas, to be itself a piece of Moorean knowledge. It is a philosophical argument, and there is, I'm sorry to say, no knowledge in philosophy, much less Moorean knowledge. But it does seem to me to be an excellent argument, even if a philosophical one.

But we are not finished with the case of motion yet. Moore has more to teach us. Moore, in his greatness, was always ready to insist on what we might call the shallowness of truistic or Moorean knowledge. The way he put it himself was that while, for instance, it is a truism that there is motion, nevertheless that knowledge could co-exist with ignorance of what was the true analysis of motion. I will put his point by saying that we can know very well that motion exists, yet at the same time not know just what the true nature of motion is. Motion is an utterly familiar phenomenon, we know it when we see it, or feel it, but our understanding of it, I think, is very far from complete. Perhaps philosophers can be of some help here - I can think of two opposed and each very interesting philosophical theories of motion - but I myself expect that is to fundamental discovery in physics from which we will get the greatest enlightenment about motion.

The point just made about motion applies very widely within the Moorean corpus, perhaps even to every scrap of it. Moorean knowledge is vague and imprecise in many dimensions and it takes us very little way into the true nature of things. But that is where we start. And indeed, at one time in the history of the human race there was no other knowledge except knowledge of this sort.

But at this point I have to admit that I have been exaggerating a bit. Is it really true that every Moorean truism is true and known to be true? Let us consider a couple of cases. It is a truism that the sun rises and sets. But is that not false? The sun does not describe an arc over our head during the day. Rather, the earth is turning and so the sun appears to move. In the words of the Latin inscription under the statue of Copernicus in his home town of Torun in Poland: the earth moves, the sun stands still.

The case must be conceded. But look how little needs to be conceded. What we see really happens. We do see that sun and the portion of the earth on which we live, are changing their relations to each other. That is certainly knowledge. But we are quite naturally led by this correct perception to acquire a bad theory about which partner in the relation is moving. The moral I draw from this is that Moorean truisms may sometimes come encumbered with false theories that later discovery may reveal to be false. But even so there is a core of knowledge present.

There is a worse case than this, though, one well known to philosophers. It is traditional from the time it was introduced by the astronomer and cosmologist Sir Arthur Eddington to introduce it by discussion of a table. Eddington pointed out that if we want to take the discoveries of physics seriously, as I think that we should, then it has to be agreed that a table contains far, far, more empty space than it contains matter. A striking bit of evidence is this. A beam of electrons can be shot through it and the beam will go perfectly straight. Collisions of the beam with any of the matter of the table are extraordinarily rare affairs. (I'm assuming here, against instrumentalists and operationalists about the entities postulated by physics, that there really are such things as electrons.) Yet is it not a Moorean truism that your ordinary table is an object that completely fills up a certain area of space without any holes? That is the way it seems to be, but this is revealed to be mere appearance. W.V. Quine sums up the position elegantly: "Even the tightest object, short of an elementary particle, has a scattered substructure when the physical facts are in."

I think that this is a case of a false Moorean truism. But notice first that even here many important truisms survive. The table is an object that excludes other ordinary objects in our environment from penetrating it. Your body and the table exclude each other from certain continuous volumes. There is no mistake there. In that sense, the table does fill up a certain area. It fills the area up for the purposes of ordinary life. But even more importantly than this, consider the scientific evidence that leads the physicist to say that there are electrons - one of the various sorts of particle that atoms are made up of - and to say that beams of these particles can be fired through the table with little risk of hitting anything. This theory is, of course, evidenced. And ultimately this evidence must come from experiments and observations made in laboratories and elsewhere and afterwards communicated to the community of physical scientists and others. If something of sort had not been done, we would never have any reason to think that tables were so strangely empty.

But now consider the vast mass of Moorean truisms that have to be true and known to be true in order for us to think we have good reason to give up on the literal space-filling nature of tables. If physicists did not know that they were working in certain laboratories, using certain instruments, getting certain readings, co-operating with other workers, and so on and so on indefinitely, then they could not begin to think that they had evidence that would convict tables of emptiness. This is really Moore's point all over again, but presented in a more sophisticated way that allows for the occasional truism to be discarded as false. The Moorean truisms that we ordinarily accept as true, and are certain that we know, constitute, by and large, allowing for the occasional rotten apple, a body of propositions that we are more certain of than anything else. And even the bad apples can only be identified as bad by putting our trust in the rest of the corpus.

Here, I wish to assert, is the true and proper foundations of our knowledge. And, as I have already said, until quite recent times, until the last few thousand years, that was all the knowledge that anybody had. But now things have changed. Using the Moorean truisms as a base, a base that is always necessary, we have in all sorts of disciplines set out on paths that have led us to a continuously expanding body of knowledge. The exponential growth of our knowledge (not necessarily ours personally, of course) shows no present sign of slowing.

We need to be a little careful here. There is plenty that is controversial and unsettled in the sciences. That is one of the things that make them so fascinating. In these controversial matters there can be no more than degrees of rational belief. To go to the assured knowledge you have to go to the boring bits, boring within the sciences in the way that the Moorean truisms are boring in ordinary life. Here is something that has come to be known within the last two centuries. Water is made up of molecules which in turn are made up of two atoms of the element hydrogen and one atom of the element oxygen. And behind this there is a good deal of knowledge about what atoms, elements, hydrogen, and oxygen are. I have taken the example because it is one that we are all familiar with. It is hardly at the cutting edge of scientific progress. But it is precisely at the cutting edge that knowledge is likely to be unavailable. But anybody who has a good grasp of any established scientific discipline will be in possession of a huge amount of non-Moorean knowledge.

There is an old, and I think good, distinction between the rational and the empirical sciences. The rational sciences are mathematics and logic. (Here I will just talk about mathematics.) They are dubbed "rational" because they are developed a priori as we philosophers say. That is to say, here is no appeal or at worst a minimal appeal, to experience. Characteristically, you start from premisses, very often quite simple, and advance from these to conclusions without any appeal to observation. What you get in these sciences is a proof. Given the premises, the conclusions must follow. Proof has only been with us relatively recently, Euclid's geometry being the first mathematical work that attempted to prove conclusions rigorously - and came very close to achieving that ideal. The Indian mathematical genius Ramanujan was brought to Cambridge by the mathematician G.H. Hardy. He worked intuitively, and Hardy had to teach him proof theory. Proofs are the nearest thing we have to a totally reliable knowledge. In the last decade, the first attempted proof of that famous problem, Fermat's last theorem, that the English mathematician Andrew Wiles produced turned out to be flawed when examined by the experts. But he went back to work with a collaborator and eventually got it right. The experts checked it and it was a genuine proof. At that point you can put down your glasses, as the horse racing commentators say. We now know that Fermat's conjecture is true. And mathematics and mathematical proof now advances continually on hundreds of fronts. With each proof we can add all these advances to the store of human knowledge.

Philosophical mysteries remain. What is the metaphysical status of numbers and other entities in which mathematics deals? Is there a mathematical realm set apart from the realm of nature, as some great thinkers, including relatively recently the great mathematician Kurt Gödel, have believed? Or is some more down-to-earth, down-to-space-time, account of mathematical reality correct? Again, there is a mystery lying between mathematics and the natural sciences, what Eugene Wigner called very neatly "the unreasonable effectiveness of mathematics in the natural sciences". Why does mathematics work so well in unlocking the secrets of the physical world? I don't think that anybody knows the answer to these questions, although, of course, people have theories.

We do not have proofs in the natural sciences in the way we have them in mathematics. A proof is a sort of knock-down argument that you cannot get away from. But in the natural sciences we do have observations (which can be checked) and experiments (which can be repeated). Experiment is, indeed, a particularly sophisticated sort of observation that has turned out to have the greatest power in either confirming or falsifying our theories about the world. Confirming hypotheses is less conclusive than falsifying them, as Karl Popper has taught us. But, against Popper, certain sorts of confirmation, especially confirmation from multiple and independent sources, can yield us non-Moorean knowledge about the physical world.

And as with mathematics, the natural sciences have yielded us collectively, even if not individually, a huge and ever-increasing body of knowledge. (The human race could lose this knowledge. We could fall back into barbarism, if you will allow me the politically incorrect term, if things go very badly for us. But for now we are living in a knowledge explosion.) You have to be careful when recent theorizing seems to look good, but has not been exposed to a sufficiently long period of testing. For instance, the big bang hypothesis is looking pretty good these days as a story about the origins of our space-time. But it is not clear to me that it yet deserves to be treated as something we definitely know to be the case. But providing you stay a bit behind the frontiers of advancing science, claims to knowledge look pretty good.

At this point, I'm in a position to put forward a hypothesis about the scope and limits of human knowledge. My suggestion is that what I have gone through: the Moorean truisms (with a little trimming to exclude false truisms), the proofs that mathematicians (and logicians) come up with, and, finally, the securer results of the natural sciences, give us the totality of our knowledge. Philosophy, religion, the pronouncements of mystics, other systems of belief, may be fine things to have and engage with. But, I suggest, none of them contain anything beyond belief. Or, to be on the safe side, if they do contain any knowledge, it is not reliable knowledge in the sense that it is not socially identifiable in the way that knowledge in the rational and the empirical sciences is identifiable. For instance, just maybe you know that God exists, or just maybe I know that God does not exist (we can't both be knowers in this matter of course), but, situated as we are, there is no way to settle the question between us. Neither of us, I think, can rationally claim to have knowledge, even if one of us does have it.

At this point it is important to say something about belief that falls short of knowledge. It is really inevitable that there should be a lot of this around, particularly if I am right in claiming that the scope of our knowledge is as relatively limited as I have asserted. If you consider much of what people believe, say, about their family and the rest of the people they know, their views on social and political matters, their views on religion, it is easy to see, at least in the case of other people as opposed to oneself, that temperament, upbringing, social environment and so on, determine one to believe all sorts of things that, even if they happen to be true, run far beyond what one can count as knowledge. We are all aware also, even in our own case, that, except at the edges and here and there, we cannot change this situation very much. Our beliefs stick with us.

But not only are we stuck with many beliefs that fall short of knowledge, but this situation is not in every way a bad thing. This is clearest in the sciences. We all now, in the present intellectual climate, understand the need, if understanding is to advance, for starting with hypotheses that can then be tested more or less rigorously. But these hypotheses, if they are to receive proper consideration - their day in court, so to speak - are going to have to be cherished, to be articulated and strengthened as far as is possible, sheltered to a degree by taking on additional auxiliary hypotheses - the so-called protective belt - and so on. It is near impossible psychologically to do this without becoming a partisan of the hypothesis, that is to say, coming to believe it.

It is true that in the fields of morality, politics and religion we are all aware of the intolerances generated, and the dreadful consequences of these intolerances, when beliefs clash with contrary beliefs. If the argument presented has been correct, then these are for the most part cases where the beliefs involved are not known to be true, even if by chance they happen to be true. This reflection may even serve to moderate dogmatism a little. I think it has done so in my own case, as I shall indicate shortly. And supposing that dogmatism can be moderated - a rather unlikely assumption in a great many cases, I hasten to admit - then the existence of opposing beliefs, and opposing systems of belief, may have something of the same sort of value that opposing bodies of hypothesis have in the sciences. There should be controversy about doubtful matters. "Oh! Let us never, never doubt, what nobody is sure about!" as Hilaire Belloc, a thinker not without his own dogmatisms, remarked in one of his comic poems.

You will perceive that, on the basis of my epistemological argument, I am beginning to moralize a bit. Let me continue to do so. Here is the position that I wish to defend in epistemic morality. Supposing that some matter is controversial, and further is controversial among those who have genuinely studied the matter, who know the literature and/or other sources of evidence, and are aware of the different positions that may be taken up. Suppose, that is, that the matter is controversial among those who have some right - some epistemic right - to speak on the matter. Then, I suggest, persons who come down on one side of the controversy ought not to claim to know that their point of view is correct. I have already allowed that perhaps some persons in the controversy may know the truth of the matter, and if we are involved in such a controversy one cannot be condemned for trusting that we are the ones who do know the truth. But I doubt that in such a situation we can know that we know, and in any case I do not think that we should claim knowledge, even to ourselves.

An obvious place to apply the maxim I have just been arguing for is in matters of religion. To get a fairly precise question, we can ask whether there is, or is not, a transcendent God who is the creator and sustainer of the natural world. I myself do not believe in God and I have always thought that, although the question cannot be settled definitively in this life, there is not much case for the existence of such a deity. That remains my position. But what should I think if I consider the question in the light of the epistemic principle I have argued for in this lecture? I have to recognize that that there are a large number of persons who are in as good an epistemic position as I am, but who do not agree with me. Some of them are philosophers. They have thought about the matter, have gone over the familiar arguments pro and con, and have studied the literature. They are certainly as intelligent as I am, in some cases, I am able to recognize, more intelligent. There are, furthermore, plenty of natural scientists, of logicians and mathematicians, all of them at least as rational creatures as I am, who find themselves in disagreement with me. I must conclude that I do not know the answer to the question, or at least that I am in no position to claim knowledge that there is no God. As a result I feel compelled to lace my atheism with a degree of agnosticism. I ought not, I think, regard the matter as settled.

(I can't forbear telling you about the joke made by the Cambridge philosopher C. D. Broad, active in the earlier part of the 20th century. He was inclined to think that there was some, though by no means conclusive, reason to think that there was life after death. He concluded a judicious discussion of the matter by saying "We must just wait and see. Or, alternatively, not see." You will note, by the way, that the joke is on the disbeliever in life after death. Supposing him to be right, he doesn't get to discover that he is right.)

One case that does worry me is the Creationist doctrine. It denies what I take to be fact: the immeasurably long evolution of ourselves and the rest of life from some primeval mix of organic compounds. (To make things easier, let us abstract from, or bracket, the hypothesis, which I do in fact support, that the mechanism of this process was nothing but natural selection plus chance.) Consider perhaps the most distinguished creationist of them all, the able English 19th century geologist Philip Gosse. He wanted to believe the view put forward by Archbishop Ussher, based on biblical genealogy, that the world was created in 4004 B.C. At the same time he was very worried by all the geological evidence, including the fossil evidence that suggested that the earth was far, far, older. He wrote a book designed to reconcile Christian doctrine and science. The world was created in 4004 B.C. But God, to test our faith, salted the mine. He created the earth just as if it had been in existence for uncounted eons, fossils and all. (Gosse didn't get many takers for his theory.)

What should one say about Gosse's hypothesis, say at the time it was put forward? He seems to meet my criteria for granting the matter to be a disputable one. So would it have been rational to refrain from claiming to know that his view must be wrong? This is a hard case for me. Can I weasel out by saying that principles like mine will always involve hard cases at the boundaries, and that hard cases make bad law? I don't know.