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PROBLEMS

As you may remember, I started and finished my course last time! I told you all about scientific method: that it doesn’t exist as a teachable technique, which is the way that it was originally

conceived, and that all that we can say is that scientists start with problems and propose theories as solutions to them, that they criticise these solutions and try to eliminate their errors, and that what we call ‘science’ are those solutions that have been criticised in a reasonably severe way and have not yet been found too badly wanting. And that is all. It would already be false to say that science consists of the solutions that have been severely criticised and found in every respect satisfactory. There is very little in the science of any period that is found in every respect satisfactory. There are always lots of difficulties, lots of unsolved problems and

unanswered criticisms, at any moment and in every branch of science. And so we can really represent scientific method with my little schema:

P1 ->  TT ->  EE -> P2

Now this is my first and last word in this course—though I shall have to repeat it in almost every lecture—and all we have left to do is to discuss certain aspects of it.

For example, I say that we start in science with problems. So you may ask, ‘Where do we get these problems?’ My answer is simple. Some of the problems are already there, and you will be introduced to them when you are first introduced into a science. These may be well-known problems that are on everybody’s lips. Or they may be problems that only your teachers are aware of and are trying to solve. But there are also other problems, namely, the ones that you

discover for yourself.

You may, for example, find that there are difficulties in a certain presentation, something that you don’t quite understand. Others may not even notice these difficulties. And you may at first think that the difficulty is really in you, and that the material is just too hard for you to understand. But as you look into it more closely, you may find that these difficulties are due to problems in the material itself. There may, for example, be gaps in an explanation. Or there may be actual inconsistencies—perhaps in what a theory says, or perhaps between what a theory says and what you observe. These are problems that you discover for yourself. But even so, they may not be entirely original. If you look into it a little more closely, you may find that the discovery of your problem had been anticipated by somebody else long ago.

And it is quite generally so. If you are a young scientist keen upon making a contribution, then you will find, as a rule, that most of what you can think of has already been thought of by others before you.

Still, to find a problem is already a very important step—and it is not like finding a beetle, or a diamond, or a ten-pound note. It’s not anything like that. Problems are typically found through other problems: one problem leading to another, and that problem, yet again, to another still. So while it isn’t always easy to find a problem, there will always be plenty of problems to find. This it due to the fact—and it is a matter of historical fact—that most of the solutions to our problems, and especially good solutions, raise more problems in science than they actually solve. The more important a problem, the more interesting will be its solution. And the more interesting its solution, the more new problems it will raise. In fact, a good solution to one problem will generally raise about twenty new problems in its place. So you need not be afraid

that there will be no problems for you to find.

Nevertheless, it is still a problem to see these problems, and, also, to find a problem for which your powers are more or less suited. Open problems—the problems that everybody knows about but nobody has been able to solve—may simply be too difficult for a beginner to tackle. And I am sure that there have been some very capable people—people who might otherwise have made great discoveries—who became discouraged and gave up on science, all because they tried to tackle a problem that they were not yet ready to solve. So the problem of finding a problem is really your first problem. And if you ask me where or how you can find one, I would say that you can generally find them by studying the prevailing intellectual situation of the moment, plus your own work, and the work of others provided, of course, that you have the

interest to do so.

So this would be your starting point. But again, I must tell you at once that what I have just said is in complete disagreement with what most other people would say about these things. Almost

every book that deals with the question and that contains a sentence beginning with the words ‘Science starts with....’ continues the sentence with the word ‘observation’—not with ‘problems’, but with ‘observation’. So while I personally believe what I have just told you, I must warn you not to accept it, because almost everybody else would say, ‘No, science does not start with a problem. Science starts with observation’—or, perhaps, ‘with observation and experiment’.

Now this view—that science begins with observation—is one of the primary things that I want to criticise. But let me postpone that criticism for just a moment. For some of you must now be

thinking, ‘All right, I see how you get your problems. But how do you get your solutions?’ My answer is that there is no recipe for that at all. There are, in fact, so many different ways of getting solutions that it is impossible to enumerate them all. I won’t even try. One person does it by drinking coffee. Another by drinking whisky. Others drink nothing at all. Some people just wake up in the morning and have the solution. None of these are methods that I would particularly recommend. I am, on the contrary, a kind of nihilist about it. I would say that there are ever so many ways of getting solutions, and that they are all ever so personal. But one

can certainly give some general advice.

My advice is to devote your mind to your problem—to think about it, to read about it, and to see what other people have thought about it. Read the history of your subject and see how other people have tried to solve your problem. And then study the solutions they have offered! I think that all of this is better than black coffee or whisky.

But I wouldn’t say that it is a method that will lead to greater success. None of these methods, in my opinion, can be regarded as more promising than any of the others. The real answer, which is hardly a method at all, is to have as many ideas as you possibly can, and to be as critical about your ideas as you possibly can.

But I think that this is a very unsatisfactory answer. For how does one have ideas? I really don’t know. Sometimes another person’s ideas may stimulate ideas in you. So you may get ideas by talking to other people, or by reading what they have written. But the point is not so much to get new ideas, as it is to get good ideas. And I don’t think that anybody has ever had a particularly good method for getting good ideas. This is one of those things for which nobody has given an answer, except, perhaps, the following. If you already have ideas, then there may be a good one amongst them. To find this good idea, you have to criticise all your ideas. And by criticising all your ideas, you may in the end select one that is better than the others.

So we are again thrown back to having ideas and to criticising them. And, in fact, I know of no better advice for how to get a good idea than what I have just told you. My advice is to have ideas and to criticise them. If you are lucky, a good idea may be amongst them. If you are not lucky, then a good idea will not be amongst them. And that is all. The disappointment that you may now be feeling is the primary reason why I started this course by saying ‘There is no such thing as a scientific method’. There is no method for having good ideas, and there is no real advice that anyone can give someone as to how he can get good ideas. Since this is obviously the very heart of the whole matter—having good ideas—it is a strong argument for saying that there is no such thing as a scientific method.

But this, again, is in complete disagreement with the accepted view. You will read in most books that you must start with observation, and that by making careful observations you will then be able to see what is common to these observations, and to generalise from them.

So you don’t need to have any problems or any ideas at all! You just need to make observations and to generalise from your observations. By generalising from your observations, you will arrive at a scientific law. And having in this way arrived at a scientific law, you will have more or less done your business. This is the view that seems to be generally accepted—if I may generalise from my observations of the textbooks on the matter—and this very awkward question of having new ideas or good ideas doesn’t even arise. You are supposed to just look at your observations, and if you have made enough observations and if you have looked long enough at your observations, then a scientific law will somehow jump out of them.

Now the only thing that I can agree with here is that the formulation of universal laws is about the most important step in the whole procedure. For if the universal law is true, then we will have made a discovery and will have really managed to get somewhere. So what I have called ‘solutions’ may, in a very general way, really consist in finding some universal law. This I am ready to concede. But what I am not ready to concede is that we find these universal laws by looking at observations. I do not think that we start with observations, or that we find a universal

law by generalising from them. I don’t want to say that we cannot find a universal law by looking at observations. If we can find one by drinking black coffee or whisky, then we can find one by

looking at our observations. But this just means that anything may lead to the proposal of a universal law. And while finding a solution is the most important step in the procedure, I would say and here again I am in disagreement with the usual position that this is not the end of our work, but only the beginning, because now comes the criticism.

So although I don’t agree that you start with observation, or that this story has any truth in it at all, I really don’t mind. Let us assume that it were true. Then I would say, ‘All right, go and get

your universal law, and then we will begin’. So although I don’t concede this story, I could concede it. I could say, ‘All right, you have told us how we begin in science. We make observations. Good. Go and make your observations. I don’t care how you do it. Just do it. Then, you say, we make generalisations. Good. Go and generalise from your observations. Again, I don’t care how you do it. Just do it. The real point is that now, after you have made your

generalisations and have proposed a scientific law to us, we still have to ask, ‘Is this proposed law really true?’ and ‘What can we do to find out if it is true?’ And the answer to the second question is, ‘Criticise it! Look at it closely. Try to pick holes in it. If from every point of view we can’t find any, then perhaps it may be true’.

So even though I don’t believe that science begins with observations and then proceeds by generalizing from observations, I really wouldn’t mind if it were so. I would stress, however, that

the work of science begins only afterwards.

But this is not the generally accepted view. The generally accepted view goes roughly like this: if you have really proceeded in the proper way—starting with observations, and nothing else, and then reading the universal law off of these observations if you have really done your job thoroughly, then you need not worry any more, because your universal law will be true. Of course, it may not always be possible to do your job thoroughly, or properly. A chemist friend of mine in New Zealand used to tell the following story. He had studied his chemistry in Germany, and his old German Professor had told him about science. ‘Observation’, he had said, ‘is not enough. You must make sequences of observations. But sequences of observations are not enough. You must make sequences of sequences of observations. But even sequences of sequences of observations are not enough. You must make sequences of sequences of sequences of observations, and then you will get your result’.

This is a view of science that can explain away every failure at once by saying, ‘You haven’t yet done the work properly. Go back and start again. Make your observations. Repeat your observations. Repeat them again. And if you repeat them often enough—and if you have done it all properly—then your result will be true’. This, in brief, is the accepted view. So let me repeat, I do not believe that there is one word of truth in it. I think it is a complete myth. But it

really doesn’t matter even if it is true. For the real work of science begins only afterwards.

But now, some of you must be thinking, ‘Isn’t this a bit too strong? For how can you possibly have a problem if you have never observed anything?’ My answer is that a new born child may well have a problem if he is not fed, or if he is cold!and not because he has made some observations and is now trying to generalise from them, but because he has some inborn expectations that have not yet been satisfied. The child is probably not even conscious of its problem—I have not said that we are always conscious of our problems, and there is little doubt that we are often not conscious of our problems. Even in science, where we are all adults, and can talk, and can thus articulate our problems, we are still very often unconscious of them. This is why we write articles that try to clarify our problems that try to say what it is that we have really been worrying about for the last five years.

This attempt to clarify our problems is one of the things that scientists most frequently do, and it is quite an important thing to do. For we had a problem all along without knowing what it really

was. We felt uneasy about something, but were not quite able to see what our uneasiness was all about. And then someone points it out, and suddenly it is clear. So it is perfectly possible to have all sorts of problems without being conscious of them. You may wake up in the middle of the night with the dull feeling that something is wrong. It may be that you are cold. But it may take quite some time before you realize that you are cold, let alone observe that your blanket has fallen down. Certain observations may play a role in all of that. But you don’t start with the observation that your blanket has fallen down, or at least not usually. I would be more inclined to say that you start with vague feelings of uneasiness, that you then discover that you are cold, and that only later do you proceed to the observation that your blanket has fallen down. But I

would not want to rest my reputation as a methodologist on this particular problem of whether or not you start with the observation that your blanket has fallen down.

What I wanted to show you was this. The usual belief is that we do start with observations. And the argument that supports it appears at first to be unanswerable. For how can I have a problem before I have made any observations? The problem must somehow or other pertain to my observations. So if I haven’t made any observations, then I cannot have any problems. For there is nothing in my mind about which I could possibly have a problem about. But all of this

depends upon the theory that my mind is necessarily empty before I make observations, so that there can’t be anything in my mind if I have never made an observation. Since there cannot be anything in my mind, there cannot be a problem in my mind. Nor can there be a hypothesis. So obviously we must start with observations.

This is not only the usual theory, it is also a very old theory. It goes back to the early Middle Ages, and the Latin formulation given to it is ‘Nihil est in meme quod non prius in sensu’—’Nothing is in the mind that has not been previously in the senses’, in the eyes, the ears, the nose, and so on. This theory, which also has a Latin name, is called the tabula rasa theory. ‘Tabula rasa’ means empty blackboard. And according to this theory, the mind is like anempty blackboard: it remains empty so long as nothing is inscribed upon it, and inscriptions can be made upon it only through sense observation.

So this is a theory of venerable age, and it is still the theory that most people hold either because they have thought of it themselves, or because they have somehow or other imbibed it

from what others have thought. It seems to be a very obvious theory. And I usually present it in a slightly more picturesque way as The Bucket Theory of the Mind. This is the theory that our mind is a kind of bucket, and that the bucket is empty until something enters it through the various holes—the eyes, ears, mouth, nose—that we call the senses. These senses are our openings to the world through which all sorts of things enter and somehow get collected down in the bucket. And the accumulation of what settles down in the bucket is our knowledge.

Here a diagram of the head as a bucket.

This, in part, is meant ironically—but also, in part, very seriously. For I think that almost everybody believes in the bucket theory of the mind, and that our educational system is almost entirely based upon it. This can be seen, for example, in examinations. In an essay examination the idea is to pour out of your bucket whatever has been poured in. And in a short answer examination a sort of ‘dipstick’ is put into your bucket to find out what the level is. In

either case, the point of the examination is to see how much you have in your bucket!

But if the people who wrote examinations were just a little more thoughtful, they would put you in a library with all the books available to you—because one of the main things that you should

learn at a university is how to use a library. An intellectual is not a person who never goes into a library. He is not someone who has everything stored in his head and who pours everything out upon demand. On the contrary, an intellectual is a person who knows how to acquire knowledge when necessary, a person who is used to cquiring knowledge and then using it.

Now I think that the fundamental proposition of the bucket theory is true, but that its significance has been vastly exaggerated. The basic idea is that our ability to learn about the world is somehow dependent upon our senses, so that if we had no senses we would not be able to learn anything at all. I admit that this is true, but I would go even further and say that if we had no senses then we would not even be alive. And I think that this is much more important than the fact that we would not be able to learn. What we call a living organism is something that has both sensitivity and reflexivity. In order to say of something that it is alive, it must somehow be sensitive to certain changes in its environment, and it must betray this sensitivity through its behaviour, because we judge that a thing is alive by its reactions.

So if we had no senses, we wouldn’t be alive. And if we weren’t alive, then we would obviously have no knowledge. So there is this trivial sense in which we would have no knowledge without our senses. And I am prepared to admit, in this very trivial sense, that our senses are necessary for our knowledge, since they are obviously necessary for our life.

But this is not the sense in which the bucket theory says that we get knowledge through our senses. The bucket theory says that we must first make observations before we can have any hypotheses, or theories, or any thought at all. For a hypothesis must be about something, and it is impossible for us to know this something that the hypothesis is about without first having knowledge. This is obviously another instance of the question ‘What comes first, the chicken or the egg?’ And here, the bucket theory says that what comes first is observation. It says that before we can have hypotheses, we must first have observations—for otherwise we cannot know anything about which we can possibly make a hypothesis.

So what does the newly born child do first? According to the bucket theory, he opens his eyes and observes. And somehow or other he in time obtains out of his observations a slightly more

organised sort of knowledge. And then, within this more organised knowledge, there arise such things as problems, and then, perhaps, hypotheses about how to solve these problems. But all of these things, problems and hypotheses, are of a somewhat theoretical nature, and they must certainly come after his observation. And this, according to the bucket theory, holds for science as well as for the newborn child. How does science proceed? Science begins with observation. You can read that in hundreds and hundreds of textbooks. Science begins with observation, and from observation it then arrives at a later stage, and after a long and drawn-out process, at generalisations, or laws, or theories.

This is the popular psychology that practically everybody today still accepts. It is the theory that still seems to rule education, and I will have a lot to say about it later. I think that there is hardly a

crumb of truth in it. But to get over it we must first begin to see knowledge and learning not as the accumulation and generalisation of observations, but as a modification of inborn expectations andinborn reactions that are under the influence not only of sensations but also of the organs by which we react. I would say of a newborn child that its whole organism is so constructed that it expects to be fed and to be warmed—that the organism of the newborn child

expects to be fed and warmed. And I think that every organism is born with very many expectations. I would say that learning and experience in the ordinary sense of the words, in the sense in which we learn from experience, is a modification of these inborn expectations and reactions under the influence of our sensations. In this sense, we have inborn knowledge, and this inborn knowledge develops, under the influence of experience, through trial and error.

Here, the most important thing to remember is that trial and error is an active process, that the typical trial and error movement is a movement by which we actually make our experiences. This, in fact, is one of the main mistakes of the bucket theory: it forgets that we can act, and instead conceives of us as entirely passive. The fact of the matter is that trial and error is an active give and take between an organism and its environment by which the organism explores its environment and makes its experiences. If the newborn child begins to explore his environment, it is typically because he has a problem, because he wants to be fed or to be warmed. In exploring his environment, he may acquire experience and knowledge—with very many trials and very many errors, and with very many disappointed expectations.

And this is another thing that the bucket theory forgets. For the bucket theory, every observation is a success because every observation is knowledge gained. But from my point of view, things are quite different. If we learn from trial and error, then error plays a major role in our learning. Our expectations lead us constantly to make movements that very often result in disappointment. And when they do, they lead us to modify our expectations. This is what we mean when we say that something has been learned through experience. If we say of a man that he is experienced, we mean that he has been disappointed. We can, in fact, say that the path to knowledge is paved with disappointment.

So this is roughly the picture of learning that I want to put before ou. And I think that it is a far more convincing picture than the bucket theory. If we ask what really comes first, it is not an

observation or even a hypothesis, but an entire organism in which there are built-in expectations and built-in ways of reacting. These expectations and ways of reacting are themselves a kind of built-in way of knowing. And our sensations and observations really have

meaning only relative to them. Jet planes, for example, do not, as a rule, even disturb the birds. A human being might look up when he hears a plane, but a cat would not. This is because the plane lies outside of a cat’s framework of expectations. It just doesn’t ring a bell. And in order to observe something, it has to ring a bell, which means that you observe only those things that your organism is prepared to observe by your inborn reactions and earlier experiences.

Still, it really wouldn’t matter even if the bucket theory were true. All of that is like whether you drink whisky or coffee. From my point of view, and from the point of view of scientific method, it

doesn’t really matter at all how you get the solutions to your problems.

The important thing is to remember that the real work begins only after you have produced a solution. The important thing is to not forget the real work, to not forget to criticise.

So where are we? In science we start from problems and we offer solutions to these problems. How do we get these solutions? We just have some idea!we cannot really explain how a man gets a new idea, though one can say a few things about it!and then, after a solution to the problem has been offered, we begin the hard work of testing it. The solution—we can call it a theory or a hypothesis, it doesn’t matter—the hypothesis is tested by criticising it. We criticise the hypothesis and try to find out how to refute it. Then we try to refute it, and as a rule we succeed. We succeed in, say, 10,000 cases for every one in which we don’t succeed. And when we don’t succeed, we try to test it even more severely. And if we still don’t succeed, and for as long as we don’t, the hypothesis is tentatively accepted as a part of science. And perhaps one day we do succeed in refuting it. And then it goes out. That is roughly the

procedure in science.

What I want to do next, always remembering my ‘spiral’ method that I told you about, is to tell you a bit more about the usual view of science, about this bucket theory. But this time I will describe it as inductivism. First I will tell you about inductivism, and then I will try to criticise it from various points of view.

I want to emphasize from the outset that my own views do not depend upon my criticism of inductivism. On the contrary, they stand on some very simple arguments that are quite independent of the whole issue of inductivism. So I could omit this entire discussion of inductivism—my explanation of the views of inductivists and my criticism of them—from the point of view of explaining my own views on the matter. It is not essential, and I need never mention it if I didn’t want to. But I feel that I have to tell you about it for two reasons. First, because inductivism is the generally accepted view. And second, and much more importantly,

because you probably all accept it unconsciously. I should say that amongst a hundred students who come here, there is not one who is not already a confirmed inductivist. Practically everybody believes in it. It is a very commonsensical view of scientific method, though one should add that the commonsensical approach to abstract problems is not always terribly reliable, and that this is a somewhat abstract problem.

The fundamental idea of inductivism is already expressed in the bucket theory of the mind. It goes something like this. How do we learn? We learn in the main by observation. And from the

observation of singular instances, and from the repetition of observations of singular instances, we then arrive at generalisations. This is the view that I call inductivism. And I want to explain at once why it has a sort of historical significance.

Observations can always and only be observations of something singular—something that happened at one place and at one time. If you say that we know by observation that the sun rises every day, or that we know by observation that if I open my fingers the chalk will fall to the ground, then you are actually taking a bit of a leap—for observations are obviously only of singular instances. You can observe the sun rising one day or another, but you cannot observe that the sun always rises. Observation is of something that happened at a particular time and at a particular place. Everything else is no longer observation but is, according to the view that I am explaining, already inferred from observation. You see a little boy throwing things out of his pram, and according to inductivists, every time he throws something out he makes a certain

observation, and from these many observations that he has thrown something out and it fell to the ground he then generalises that whenever he throws something out of his pram it will fall to the ground.

Now observation is essential for an inductivist, and I believe that it is essential too. And I also agree that observation can only be of singular instances, and that this is one of the most important things to be clear about. Not everybody realises this. In a loose way, we can say that we have observed that the sun rises every day, or that things fall to the ground. But this is a loose way of speaking. We cannot observe generalities, we can observe only the singular

instance.

But the inductivist says that the general case is somehow derived from these singular instances. Inductivism says that we proceed from the singular instances to generalisations. And this is where the inductivist and I disagree.

According to inductivists, induction is the inverse of deduction, like subtraction is the inverse of addition. Now in deduction we typically use a generalisation and a singular instance to obtain

knowledge about another singular instance. Here is the classical rubber stamp example:

All men are mortals. - General premise

Socrates is a man. - Singular premise

Socrates is a mortal. - Singular conclusion

This is a valid deductive inference. We call it a syllogism, which means that it has two premises and a conclusion, and we know that instead of ‘Socrates’ I could have said any other proper name, and I would have been able to validly draw the same corresponding conclusion.

Now induction is conceived in the following way. We start with the syllogism’s conclusion ‘Socrates is a mortal’. We then notice that it is the same with Aristotle and Plato!that they are both men and mortals!and from many such cases we then obtain the generalization that ‘All men are mortals’. So from the deductive syllogism ‘All men are mortals, and Socrates is a man, therefore Socrates is a mortal’ we obtain the inductive syllogism, namely ‘Socrates is a man, and Socrates is a mortal, therefore all men are mortals’.

But of course, the inductive syllogism must have a large collection of singular instances, and not simply one. And you can put it this way if you like: the deductive syllogism has one singular premise, and the inductive syllogism has many. So in that sense, the inductive syllogism is not really a syllogism, since it has more than two premises. But you can also look at it as many inductive syllogisms all having the same type of premises and the same conclusion. And it doesn’t really matter. In either case, you will see that the premises of the inductive syllogism are in a sense symmetrical. There is no difference in form between the premise ‘Socrates is a mortal’ and the premise ‘Socrates is a man’. So if we can derive ‘All man are mortals’ from these premises, then we should be able to derive ‘All mortals are men’!because the premises don’t show any difference in structure and are, in that respect, symmetrical. Since ‘All mortals are men’ is certainly false, these premises give us no reason to believe that ‘All men are

mortals’ is true.

If from the same premises and by the same kind of arguments you can obtain an obviously false conclusion, then it is only an accident that you should ever obtain a true conclusion from them.

This is a very simple argument and one that I believe smashes this particular version of induction. But I want to stress at once that it certainly does not smash the idea of induction. It says only that things are not quite as simple as that.

There is a famous passage in Bacon where he speaks about simple inductions where no counter-instances can be found. And most inductivists would say, ‘Oh no, we didn’t really mean it like that. We meant, of course, that the conclusion holds only if we don’t know of any counter-instance. There are many counter-instances in the case of ‘All mortals are men’—a dog is a mortal and not a man, and a bird is a mortal and not a man—so we have here counterinstances that prevent us from drawing an inductive conclusion’. So we can repair induction a bit and say that it holds only when no counter-instance can be found.

But many of the typical examples used to support induction suffer from counter-instances. Inductivists say that nobody doubts for a minute that the sun will rise tomorrow. And this is supposed to be a powerful argument. But the example that the sun will rise has really been out of date for about two or three thousand years.

A contemporary of Aristotle, an astronomer and a great navigator called Pytheas—Pytheas of Massalia—travelled far to the north. He observed the midnight sun and the frozen sea and he reported that in some regions there are months and months when the sun doesn’t rise at all. He thereby falsified the law that the sun rises and sets every day. So what happened? His reports were rejected as travellers’ tales, and Pytheas became the proverbial liar. His report that the sea froze? Another absolute lie, of course. Pytheas, in fact, had the reputation of a liar almost until the Middle Ages. People would say ‘He lies like Pytheas’, and continue to use the example of the rising sun as an example of induction. It is still used this way even today.

Now if I say this to an inductivist he will of course say, ‘We mean only between the polar circles, that is to be understood’.

But it was not meant like this. The whole point of induction is to say something that goes beyond experience. The main point about the sun always rising is that it goes beyond observation. And when somebody went to Norway and observed that the sun didn’t rise for

months and months he was said to be a liar, on the grounds of what we know through induction. Now he turns out to be right and inductivists should acknowledge that the induction was invalid.

But they still use the example of the rising sun, though they now, of course, mean it differently.

Of course!

But we can also criticise this view, and criticise it fairly effectively. We can say that the conclusion of an inductive argument rests not on its singular premises, but on the absence of

any counter-example. So you can forget about the inductive argument entirely and pass, instead, on whether or not there are any counter-examples. And if you take this seriously, you will come to exactly the same view that I previously sketched to you as my own. For we now see that everything depends on the presence or absence of counter-examples, so that we have no induction, but only hypotheses that we try to criticise, or refute, by finding counter-examples. And if we take this view seriously, it then leads even further: to the view, namely, that our present inability to find counter-examples does not mean that we shall always be unable to

find counter-examples.

Our present inability to find counter-instances can establish the thing only for a time, so that there is only a provisional way of asserting anything.

We assert it only until we have found a counter-instance, and with the knowledge that we can never know in advance whether or not we shall find one.

But let us now proceed a little bit further with the theory of induction. I have just pointed out to you that inductive inferences are not generally valid, which means that an inductive inference of

exactly the same form may lead to a false conclusion. This, of course, has been recognized. And inductivists have said, quite early on, that an inductive argument cannot be expected to lead to certainty, that it can be expected to yield only probability, and that an inductive conclusion is not a certain conclusion, but only a probable conclusion. So in this way the idea of probability was introduced at a very early stage into the theory of induction.

I shall not assume that any particular theory of probability or any particular theory of inductive argument is part of inductivism. This would be unfair since there are some inductivists who say that this is all very difficult and not yet quite solved. But one of the things I shall try to show you is that either one says that probability is not applicable at all, or, if one applies it, one has the result that in a sufficiently big world all generalisations will have the probability zero. This shows that the probability theory does not help us at all. For there does not exist a theory of probability or a theory of probable inference that explains how a generalisation by the accumulation of instances gains a probability equal even to one half. But less than one half is less than probable. And if a statement has a probability of less than one half, then we do not say that it is

probable, but that it is improbable.

Inductivists, as a rule, start by thinking that it is all very easy, and only in their old age do they find that it is really very difficult. Almost all inductivists, when they are young, believe that they will easily be able to show that the probability of a generalisation will, with the accumulation of instances, get closer and closer to one.But when they die, this is still left undone, and they haven’t even shown that we can ever reach a probability even of one half, which, of course, they would have to do in order to show that a generalisation is even more probable than not. Most of them, if they live long enough, reach the point where they say, ‘It is all very difficult. My previous theories about which I was so confident must now be modified, and I don’t know exactly how to modify them.’ But they seldom go further than that.

Bacon, who in modern times reinvented induction, seems to have reached this stage. He left his work on the subject unfinished, and published, without really explaining, his theory of induction. And things haven’t changed much since. The last great exponent of induction, a friend of mine named Rudolf Carnap, also reached this stage. After publishing a very thick book called The Logical Foundations of Probability and Induction, he finally said that it is all very difficult and that we need a lot more work and that some of our theories have to be modified.

But inductivists never, as a rule, get to the point where they say that it is all a myth and that there isn’t such a thing as induction. They usually die before they reach that point. That is the usual

procedure!

But the probability of a generalisation is supposed to be the quotient obtained from dividing the number of positive instances by the total number of instances. So it all depends more or less on the size of the world. And if the world is very large, as our world is compared with the instances that we actually do observe, then this quotient, as a measure of the probability of the generalisation, will be approximately zero.

So my thesis is that probability doesn’t get you anywhere, that if you try to apply it you get probabilities indistinguishable from zero for well-established natural laws, instead of getting probabilities approaching l.

I conclude from this that the whole idea of using probability as an inductive measure was a mistake. I don’t say that you cannot say in some other sense that a theory is very probable, but only that ‘probable’ would then mean something different from how it is used in the calculus of probability.

Now I see that our time is up. So let me just close by saying that my own outlook is very different. Inductivists say that it is inconceivable that we should start without observations. But I

would almost like to put it the other way round and say ‘It is inconceivable that we can start with observations’. My own view is that the aim of science is to explain—very simple, to explain. To

explain what? To explain whatever has become a problem for us to understand.

Whenever we ask ourselves ‘Why is it so?’ or ‘Why does it happen in just this way?’, the it in these questions—whatever it may be—has for some reason become a problem for us. The aim of scienceis then to solve this problem by giving an explanation. And we attempt to solve these problems, in the main, by conjecture and refutation—by guess-work. Or, if you like, by trial and error. How do we learn? We learn from our mistakes. How do we gain knowledge? By making mistakes and learning from them. Our guesses are our trials, and we learn from our errors. That is perhaps the simplest way to sum up my view of the whole matter. How do we not gain knowledge? By making mistakes and not learning from them. So scientific method and nonscientific method both make mistakes. The difference is that the one learns from his mistakes and the other doesn’t. The other, on the contrary, dies with his mistakes before he corrects them. And all of this other stuff is like whether you drink whisky or coffee.

Lecture 2