Is there any knowledge so certain that no reasonable person could doubt it? This is one of the most difficult questions that can be asked.

So let’s answer it, with the help of my friend Bertrand Russell.

He’s a mathematician, philosopher, and one of the greatest minds of the 20th century.

(The following is a historically accurate conversation with Russell based on his “Problems of Philosophy”. Citations are included so you can see the original context for each reply.)

Russell: Are you sure that the table in front of you is real?

DKB: I’m pretty sure it is.

Russell: What color is it?

DKB: Light brown.

Russell: What about the fact that the parts reflecting light look brighter? If people look at the table from different angles, they’ll see different colors.1

DKB: Sure there are differences that people see, but it’s objectively a specific color which we can measure scientifically.

Russell: Which color is the objective one? If the lighting in the room changes, the color changes. In a dark room, there’s no color at all. If someone’s wearing blue shades, they see a different color entirely.2

It seems to me that the color isn’t inherent in the table, but dependent on the table, the person viewing it, and the way light falls on it.

When we talk about the color of the table in daily life, we mean the color it will seem to have to a normal person, from an average point of view, in average light conditions. But other colors that appear in other conditions have just as much of a right to be considered real.

To avoid favoritism, we have to deny that the table itself has any particular color.

Let’s use the phrase “sense data” to refer to things immediately known to us via sensation, like colors and sounds.3

The real table, if it exists, we will call a “physical object”. And we’ll call the collection of all physical objects “matter”.4

So what we really want to know is, does matter exist? And if so, what is its nature?

DKB: If we go down this rabbit hole, then can’t I doubt my knowledge of everything?

Russell: Descartes, the founder of modern philosophy, asked the same question. He invented the method of systematic doubt.5

He would believe nothing that he didn’t see clearly to be true. Whatever he could doubt, he would doubt, unless he saw a reason for not doubting it.

He imagined that there could be a deceitful demon, presenting unreal things to his senses. That meant he had to doubt everything he experienced.

But he couldn’t doubt his own existence. Because if he didn’t exist, then no demon could deceive him. If he doubted, he must exist. Thus his own existence was an absolute certainty to him, leading to the famous “I think, therefore I am.”.

We can be certain of our own existence, and our subjective sense data, but is there something else that isn’t sense data, which persists when we leave the room?

DKB: There must be something right? Even though different people see the table slightly differently, they all see something fairly similar. So there must be some physical object that exists.

Russell: That argument assumes that other people besides you exist, which is the entire issue we’re trying to resolve. Other people are represented to me by sense data, so if I have no reason to believe in physical objects independent of my sense data, I have no reason to believe that other people exist.6

DKB: So it’s possible that I’m the only one that exists, and all the physical objects around me are created in my mind?

Russell: Well, we can’t definitively prove that it isn’t all in your head, but there’s no reason to believe that’s the case. The simpler hypothesis is the common-sense view that there really are objects independent of us.7

Consider the example where a cat appears in one part of the room, then at a later moment it’s in another part. It’s natural to assume that the cat moved from one place to another, passing over a series of intermediate positions.8

If it is merely a set of sense data in your head, it can’t have ever been in any place where you didn’t see it. So you’ll have to believe that the cat didn’t exist while you weren’t looking, but suddenly sprang into being in a new place.

It’s a lot simpler to believe that the cat exists independently of us.

Either way, it’s not by argument that we originally start believing in an independent external world. This belief is ready in us as soon as we begin to reflect. It’s what we can call an “instinctive belief”. Since this belief doesn’t lead to any difficulties, and simplifies our explanation of things, there seems to be no good reason to reject it.9

DKB: That’s kind of a weak argument. You’re saying that just because we naturally believe it, and it makes things simpler, that it’s true?

Russell: The argument is definitely weaker than we would want it to be, but it is typical of many philosophical arguments. All knowledge must be built upon our instinctive beliefs. If those are rejected, then nothing is left. We have to start our quest for knowledge somewhere.10

Of course it is possible that all or any of our beliefs may be mistaken, and we should have some element of doubt about them. But we can’t have a reason to reject a belief except when it comes into conflict with some other belief.11

DKB: Alright, so we instinctively believe that physical objects and matter exist. There is some real table, though it might not look like what our sense data gives us.

I don’t see why we can’t also assume that physical objects are basically the same as the sense data we get. The colors we perceive through sense data are similar from many points of view. So maybe the “real” color is an average color between all the possible points of view. Does that make sense?

Russell: While that theory can’t be definitively refuted, it can be shown to be baseless. The color we see depends only on the nature of the light waves striking the eye, and is modified by the air in between the object and the eye, as well as the manner in which the light is reflected.12

The color we see is not just a property of the object. So it is quite gratuitous to assume that physical objects have colors. There’s no justification for that hypothesis.

DKB: Is there anything we can know about the nature of physical objects then?

Russell: Well we know we have our sense data, but that data is limited. It might be possible to know things about physical objects if we could draw inferences from our sense data somehow.13

We need some general principle that enables us to come up with valid inferences. I can reveal the principle everyone uses by asking a simple question.

Do you think the sun is going to rise tomorrow?

DKB: Of course. It’s risen every day in the past, and there’s no reason to believe it won’t rise tomorrow. We know the laws of physics, and we know the motion of the earth and the sun.

Russell: Every day of its life, the thanksgiving turkey gets more data to prove that human beings are benevolent creatures that provide food and shelter. Right when it has the most confidence that it’s safe based on past data, it gets shoved into the oven.14

DKB: I mean, sure, the earth could get hit by a giant asteroid which knocks it out of orbit or something. And then maybe the sun won’t rise. But the laws of physics would still be working as expected.

Russell: You still have the same problem though. Just because the laws of physics have worked in the past, does that mean they will always work in the future?15

The only reason for believing the laws of physics will remain in operation is that they have operated this way for a long time. But do any number of cases of a law being fulfilled in the past give evidence that it will be fulfilled in the future?

If not then we have no grounds whatsoever for expecting the sun to rise tomorrow, or for expecting the bread we eat at our next meal not to poison us.

DKB: Well that’s not good…

Russell: Unfortunately we can’t get true certainty. The most we can hope for is that the more often things are found together, the more probable it becomes that they will be found together another time. This is called the principle of induction.16

DKB: And how do we know the principle of induction is true?

Russell: All arguments which predict the future on the basis of past experience assume the principle of induction. We can never use experience to prove the principle of induction, because that would be a circular argument.17

So we have to either accept the principle of induction on the grounds of its intrinsic evidence, or give up on predicting the future. Everything in our lives depends on the principle of induction, but we can’t prove that it’s true.

DKB: If we can’t prove that it’s true, then what reason do we have to believe in it?

Russell: We need some number of self-evident logical principles before any argument becomes possible. Once we assume some of them are true, others can be proven.18

Take for example the following logical principle: “anything implied by a true proposition is true”.19

This principle is involved in everything. Whenever one thing we believe is used to prove something else, this principle is relevant. If anyone asks us “Why should I accept the results of valid arguments based on true premisses”, we can only answer by appealing to our principle.

The truth of this principle is impossible to doubt, and it’s so obvious that it seems trivial. But it’s not trivial, because it shows that we may have indisputable knowledge that is not derived from sense objects.

DKB: But where do these self-evident principles come from? It’s not innate in us as human beings. We still have to learn them somehow.

I feel like the usual flow is that we have experiences, then come up with general laws based on those experiences.

Russell: It is true that babies aren’t born with an innate knowledge of these principles. So we don’t use the word innate, and prefer the phrase “a priori” to refer to this kind of knowledge.20

All knowledge is caused by experience. But for a priori knowledge, the experience that makes us think of it doesn’t actually prove it. Experience merely directs our attention to the a priori knowledge, so that we see its truth without requiring any proof from experience.

DKB: In that case, how is a priori knowledge even possible? How can we know some underlying principle when we haven’t examined all the possible instances of it?

Russell: To understand that, let’s take a step back into ancient greek philosophy. Plato originally tackled a related problem, and came up with his “theory of ideas”.21

Let’s consider the notion of justice. If we want to know what justice is, we can look at various just acts and see what they all have in common. They will all have some common nature found in just things and nothing else. This common nature will be justice itself. This pure essence is what Plato calls an “idea” or “form”.

The truly real world for Plato is the world of ideas. Whatever we can say about things in the world of sense, we can only succeed in saying that they participate in some ideas, which constitute their character.22

But idea is a loaded word, so let’s use the word “universal” instead of “idea” to capture what Plato meant. Whatever we interact with through sensation is called a “particular”. A universal is anything which may be shared by many particulars.23

DKB: Are universals just something we make up in our minds then? They clearly aren’t physical things.

Russell: Universals are not merely mental. Consider the proposition Canada is north of Mexico. We don’t cause the truth of the proposition by knowing it. The part of the earth’s surface where Canada is would be north of the part where Mexico is, even if there were no human beings to know about north and south, and even if there were no minds at all in the universe.24

But the relation “north of” doesn’t seem to exist in the same sense that Canada and Mexico exist. There is no place or time where we can find the relation “north of”. It is neither in space nor in time. Neither material nor mental. Yet, it is something.25

Let’s only call things “existing” when they are in the world of time. Thoughts, feelings, minds, and physical objects exist. But universals don’t exist in this sense. We can say that they have being. “Being” as opposed to “existence” is timeless. The world of universals is a world of being.26

DKB: We’re starting to get a little mystical here. A timeless world of being makes me think of the way that Hindus describe the Unmanifested, or the way that Taoists describe the Tao. It’s that world beyond shape and form that all religions seem to talk about.

Russell: It does feel somewhat mystical, but the basis of the theory is in logic. Anyway, now we can return to your question about how a priori knowledge is possible. The answer is that all a priori knowledge deals exclusively with the relations of universals.27

A priori knowledge is not the same as knowing a specific instance of a priori knowledge. We can know a priori that two and two are four. But we cannot know a priori that if Alicia and Akari are two, and Edward and Alphonse are two, then Alicia, Akira, Edward and Alphonse are four. We can’t understand the proposition at all because we don’t know that there are such people. That can only be known by experience.

The statement two and two are four deals exclusively with universals, and may be known by anyone acquainted with the universals concerned.

DKB: To have a priori knowledge, we have to know at least a few instances of it from experience so we can observe the general principle right?

Russell: No, we can know a general proposition even if we don’t know a single instance of it.28

For example, we know that multiplication tables contain every product below 100. So take the following proposition: “All products of two integers, which have never been and never will be thought of by any human being, are over 100”. The truth is undeniable, yet we can never give an instance of it, by definition.

Our knowledge of physical objects depends on this possibility of general knowledge where no instance can be given. The same applies to our knowledge of other people’s minds, and any other thing which we can’t know directly.29

DKB: Okay, but what if something that seems self-evident to us turns out to be false? Our knowledge is based on these intuitive general principles, but some of our principles may turn out to be garbage.

Russell: People like to think that everything we believe should be capable of proof. Most people would say that a belief that you can’t give a proper argument for is an unreasonable belief. But in reality, if we keep asking “why”, we end up at some self-evident general principle like induction, which we have no way of proving.30

You are onto something though. Self-evidence has degrees. Truths of perception and some principles of logic have the very highest degree of self-evidence. Memories have diminishing self-evidence as they become remoter and fainter. The truths of logic and mathematics have less self-evidence as they become more complicated. Judgments of intrinsic ethical or aesthetic value have some self-evidence, but not much.31

Degrees of self-evidence are important in the theory of knowledge. When we find a conflict between propositions, we can keep the more self-evident one.32

DKB: So do we have to compare all the things we believe and look for conflicts? Is that how we finally solve this problem of knowledge?

Russell: The majority of what we call knowledge is actually “probable opinion”, because we can never be absolutely certain of it.33

In order to deal with this sea of probable opinion, we can use the test of coherence, which is basically what you’re alluding to. A body of individually probable opinions, if they are mutually coherent, become more probable than any one of them would be individually.

This is how many scientific hypotheses acquire their probability. They fit into a coherent system of probable opinions, and become more probable than they would be in isolation.

But even though this test increases probability, it never gives us absolute certainty. The mere organization of probable opinion will never, by itself, transform it into indisputable knowledge.

We can organize our instinctive beliefs and their consequences, modifying or abandoning beliefs until they don’t clash, forming a harmonious system. Through this method we can arrive, on the basis of accepting as our sole data what we instinctively believe, at an orderly systematic organization of our knowledge.34

Even though the possibility of error remains, its likelihood is diminished by the interrelation of the parts, and the critical scrutiny we have undertaken.

When we go down rabbit holes like this, we can’t always get definitive answers, but at least we have the power to ask questions which increase the interest of the world.35

This type of inquiry shows the strangeness and wonder lying just below the surface, even in the most common things of everyday life.

1

“To make our difficulties plain, let us concentrate attention on the table. To the eye it is oblong, brown, and shiny, to the touch it is smooth and cool and hard; when I tap it, it gives out a wooden sound. Any one else who sees and feels and hears the table will agree with this description, so that it might seem as if no difficulty would arise; but as soon as we try to be more precise our troubles begin. Although I believe that the table is ‘really’ of the same colour all over, the parts that reflect the light look much brighter than the other parts, and some parts look white because of reflected light. I know that, if I move, the parts that reflect the light will be different, so that the apparent distribution of colours on the table will change. It follows that if several people are looking at the table at the same moment, no two of them will see exactly the same distribution of colours, because no two can see it from exactly the same point of view, and any change in the point of view makes some change in the way the light is reflected.”

2

“To return to the table. It is evident from what we have found, that there is no colour which pre-eminently appears to be the colour of the table, or even of any one particular part of the table—it appears to be of different colours from different points of view, and there is no reason for regarding some of these as more really its colour than others. And we know that even from a given point of view the colour will seem different by artificial light, or to a colour-blind man, or to a man wearing blue spectacles, while in the dark there will be no colour at all, though to touch and hearing the table will be unchanged. This colour is not something which is inherent in the table, but something depending upon the table and the spectator and the way the light falls on the table. When, in ordinary life, we speak of the colour of the table, we only mean the sort of colour which it will seem to have to a normal spectator from an ordinary point of view under usual conditions of light. But the other colours which appear under other conditions have just as good a right to be considered real; and therefore, to avoid favouritism, we are compelled to deny that, in itself, the table has any one particular colour.”

3

“Let us give the name of ‘sensedata’ to the things that are immediately known in sensation: such things as colours, sounds, smells, hardnesses, roughnesses, and so on. We shall give the name ‘sensation’ to the experience of being immediately aware of these things. Thus, whenever we see a colour, we have a sensation of the colour, but the colour itself is a sense-datum, not a sensation.”

4

“The real table, if it exists, we will call a ‘physical object’. Thus we have to consider the relation of sensedata to physical objects. The collection of all physical objects is called ‘matter’. Thus our two questions may be restated as follows: (1) Is there any such thing as matter? (2) If so, what is its nature?”

5

“Descartes (1596–1650), the founder of modern philosophy, invented a method which may still be used with profit—the method of systematic doubt. He determined that he would believe nothing which he did not see quite clearly and distinctly to be true. Whatever he could bring himself to doubt, he would doubt, until he saw reason for not doubting it. By applying this method he gradually became convinced that the only existence of which he could be quite certain was his own. He imagined a deceitful demon, who presented unreal things to his senses in a perpetual phantasmagoria; it might be very improbable that such a demon existed, but still it was possible, and therefore doubt concerning things perceived by the senses was possible. But doubt concerning his own existence was not possible, for if he did not exist, no demon could deceive him. If he doubted, he must exist; if he had any experiences whatever, he must exist.”

6

“Now in so far as the above considerations depend upon supposing that there are other people besides ourselves, they beg the very question at issue. Other people are represented to me by certain sense-data, such as the sight of them or the sound of their voices, and if I had no reason to believe that there were physical objects independent of my sense-data, I should have no reason to believe that other people exist except as part of my dream.

7

“There is no logical impossibility in the supposition that the whole of life is a dream, in which we ourselves create all the objects that come before us. But although this is not logically impossible, there is no reason whatever to suppose that it is true; and it is, in fact, a less simple hypothesis, viewed as a means of accounting for the facts of our own life, than the common-sense hypothesis that there really are objects independent of us, whose action on us causes our sensations.”

8

“The way in which simplicity comes in from supposing that there really are physical objects is easily seen. If the cat appears at one moment in one part of the room, and at another in another part, it is natural to suppose that it has moved from the one to the other, passing over a series of intermediate positions. But if it is merely a set of sense-data, it cannot have ever been in any place where I did not see it; thus we shall have to suppose that it did not exist at all while I was not looking, but suddenly sprang into being in a new place.”

9

“Of course it is not by argument that we originally come by our belief in an independent external world. We find this belief ready in ourselves as soon as we begin to reflect: it is what may be called an instinctive belief. We should never have been led to question this belief but for the fact that, at any rate in the case of sight, it seems as if the sense-datum itself were instinctively believed to be the independent object, whereas argument shows that the object cannot be identical with the sense-datum. This discovery, however—which is not at all paradoxical in the case of taste and smell and sound, and only slightly so in the case of touch—leaves undiminished our instinctive belief that there are objects corresponding to our sense-data. Since this belief does not lead to any difficulties, but on the contrary tends to simplify and systematize our account of our experiences, there seems no good reason for rejecting it.”

10

“The argument which has led us to this conclusion is doubtless less strong than we could wish, but it is typical of many philosophical arguments, and it is therefore worth while to consider briefly its general character and validity. All knowledge, we find, must be built up upon our instinctive beliefs, and if these are rejected, nothing is left.”

11

“It is of course possible that all or any of our beliefs may be mistaken, and therefore all ought to be held with at least some slight element of doubt. But we cannot have reason to reject a belief except on the ground of some other belief.”

12

“Such a theory is perhaps not capable of being definitely refuted, but it can be shown to be groundless. To begin with, it is plain that the colour we see depends only upon the nature of the light-waves that strike the eye, and is therefore modified by the medium intervening between us and the object, as well as by the manner in which light is reflected from the object in the direction of the eye. The intervening air alters colours unless it is perfectly clear, and any strong reflection will alter them completely. Thus the colour we see is a result of the ray as it reaches the eye, and not simply a property of the object from which the ray comes. Hence, also, provided certain waves reach the eye, we shall see a certain colour, whether the object from which the waves start has any colour or not. Thus it is quite gratuitous to suppose that physical objects have colours, and therefore there is no justification for making such a supposition. Exactly similar arguments will apply to other sense-data.”

13

“But if we are to be able to draw inferences from these data—if we are to know of the existence of matter, of other people, of the past before our individual memory begins, or of the future, we must know general principles of some kind by means of which such inferences can be drawn.”

14

“The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.”

15

“Of course it might be doubted whether we are quite certain that there is nothing outside to interfere, but this is not the interesting doubt. The interesting doubt is as to whether the laws of motion will remain in operation until tomorrow. If this doubt is raised, we find ourselves in the same position as when the doubt about the sunrise was first raised.

The only reason for believing that the laws of motion will remain in operation is that they have operated hitherto, so far as our knowledge of the past enables us to judge. It is true that we have a greater body of evidence from the past in favour of the laws of motion than we have in favour of the sunrise, because the sunrise is merely a particular case of fulfilment of the laws of motion, and there are countless other particular cases. But the real question is: Do any number of cases of a law being fulfilled in the past afford evidence that it will be fulfilled in the future? If not, it becomes plain that we have no ground whatever for expecting the sun to rise tomorrow, or for expecting the bread we shall eat at our next meal not to poison us, or for any of the other scarcely conscious expectations that control our daily lives.”

16

“It must be conceded, to begin with, that the fact that two things have been found often together and never apart does not, by itself, suffice to prove demonstratively that they will be found together in the next case we examine. The most we can hope is that the oftener things are found together, the more probable it becomes that they will be found together another time, and that, if they have been found together often enough, the probability will amount almost to certainty. It can never quite reach certainty, because we know that in spite of frequent repetitions there sometimes is a failure at the last, as in the case of the chicken whose neck is wrung. Thus probability is all we ought to seek.”

17

“The inductive principle, however, is equally incapable of being proved by an appeal to experience. Experience might conceivably confirm the inductive principle as regards the cases that have been already examined; but as regards unexamined cases, it is the inductive principle alone that can justify any inference from what has been examined to what has not been examined. All arguments which, on the basis of experience, argue as to the future or the unexperienced parts of the past or present, assume the inductive principle; hence we can never use experience to prove the inductive principle without begging the question. Thus we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future.”

18

“The above principle is merely one of a certain number of self-evident logical principles. Some at least of these principles must be granted before any argument or proof becomes possible. When some of them have been granted, others can be proved, though these others, so long as they are simple, are just as obvious as the principles taken for granted.”

19

“The logical principle is as follows: ‘Suppose it known that if this is true, then that is true. Suppose it also known that this is true, then it follows that that is true.’ When it is the case that if this is true, that is true, we shall say that this ‘implies’ that, and that that ‘follows from’ this. Thus our principle states that if this implies that, and this is true, then that is true. In other words, ‘anything implied by a true proposition is true’, or ‘whatever follows from a true proposition is true’.

This principle is really involved—at least, concrete instances of it are involved—in all demonstrations. Whenever one thing which we believe is used to prove something else, which we consequently believe, this principle is relevant. If any one asks: ‘Why should I accept the results of valid arguments based on true premisses?’ we can only answer by appealing to our principle. In fact, the truth of the principle is impossible to doubt, and its obviousness is so great that at first sight it seems almost trivial. Such principles, however, are not trivial to the philosopher, for they show that we may have indubitable knowledge which is in no way derived from objects of sense.”

20

“It would certainly be absurd to suppose that there are innate principles in the sense that babies are born with a knowledge of everything which men know and which cannot be deduced from what is experienced. For this reason, the word ‘innate’ would not now be employed to describe our knowledge of logical principles. The phrase ‘a priori’ is less objectionable, and is more usual in modern writers. Thus, while admitting that all knowledge is elicited and caused by experience, we shall nevertheless hold that some knowledge is a priori, in the sense that the experience which makes us think of it does not suffice to prove it, but merely so directs our attention that we see its truth without requiring any proof from experience.”

21

“The way the problem arose for Plato was more or less as follows. Let us consider, say, such a notion as justice. If we ask ourselves what justice is, it is natural to proceed by considering this, that, and the other just act, with a view to discovering what they have in common. They must all, in some sense, partake of a common nature, which will be found in whatever is just and in nothing else. This common nature, in virtue of which they are all just, will be justice itself, the pure essence the admixture of which with facts of ordinary life produces the multiplicity of just acts. Similarly with any other word which may be applicable to common facts, such as ‘whiteness’ for example. The word will be applicable to a number of particular things because they all participate in a common nature or essence. This pure essence is what Plato calls an ‘idea’ or ‘form’. (It must not be supposed that ‘ideas’, in his sense, exist in minds, though they may be apprehended by minds.) The ‘idea’ justice is not identical with anything that is just: it is something other than particular things, which particular things partake of. Not being particular, it cannot itself exist in the world of sense. Moreover it is not fleeting or changeable like the things of sense: it is eternally itself, immutable and indestructible.”

22

“The truly real world, for Plato, is the world of ideas; for whatever we may attempt to say about things in the world of sense, we can only succeed in saying that they participate in such and such ideas, which, therefore, constitute all their character.”

23

“The word ‘idea’ has acquired, in the course of time, many associations which are quite misleading when applied to Plato’s ‘ideas’. We shall therefore use the word ‘universal’ instead of the word ‘idea’, to describe what Plato meant.”

24

“Consider such a proposition as ‘Edinburgh is north of London’. Here we have a relation between two places, and it seems plain that the relation subsists independently of our knowledge of it. When we come to know that Edinburgh is north of London, we come to know something which has to do only with Edinburgh and London: we do not cause the truth of the proposition by coming to know it, on the contrary we merely apprehend a fact which was there before we knew it. The part of the earth’s surface where Edinburgh stands would be north of the part where London stands, even if there were no human being to know about north and south, and even if there were no minds at all in the universe.”

25

“Hence the relation ‘north of’ is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something.”

26

“We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where ‘being’ is opposed to ‘existence’ as being timeless. The world of universals, therefore, may also be described as the world of being.”

27

“Thus the statement ‘two and two are four’ deals exclusively with universals, and therefore may be known by anybody who is acquainted with the universals concerned and can perceive the relation between them which the statement asserts. It must be taken as a fact, discovered by reflecting upon our knowledge, that we have the power of sometimes perceiving such relations between universals, and therefore of sometimes knowing general a priori propositions such as those of arithmetic and logic. The thing that seemed mysterious, when we formerly considered such knowledge, was that it seemed to anticipate and control experience. This, however, we can now see to have been an error. No fact concerning anything capable of being experienced can be known independently of experience. We know a priori that two things and two other things together make four things, but we do not know a priori that if Brown and Jones are two, and Robinson and Smith are two, then Brown and Jones and Robinson and Smith are four. The reason is that this proposition cannot be understood at all unless we know that there are such people as Brown and Jones and Robinson and Smith, and this we can only know by experience. Hence, although our general proposition is a priori, all its applications to actual particulars involve experience and therefore contain an empirical element.”

28

“The other point is more interesting, and of more philosophical importance. It is, that we may sometimes know a general proposition in cases where we do not know a single instance of it. Take such a case as the following: We know that any two numbers can be multiplied together, and will give a third called their product. We know that all pairs of integers the product of which is less than 100 have been actually multiplied together, and the value of the product recorded in the multiplication table. But we also know that the number of integers is infinite, and that only a finite number of pairs of integers ever have been or ever will be thought of by human beings. Hence it follows that there are pairs of integers which never have been and never will be thought of by human beings, and that all of them deal with integers the product of which is over 100. Hence we arrive at the proposition: ‘All products of two integers, which never have been and never will be thought of by any human being, are over 100.’ Here is a general proposition of which the truth is undeniable, and yet, from the very nature of the case, we can never give an instance; because any two numbers we may think of are excluded by the terms of the proposition.”

29

“We can give instances of the associated sense-data, but we cannot give instances of the actual physical objects. Hence our knowledge as to physical objects depends throughout upon this possibility of general knowledge where no instance can be given. And the same applies to our knowledge of other people’s minds, or of any other class of things of which no instance is known to us by acquaintance.”

30

“There is a common impression that everything that we believe ought to be capable of proof, or at least of being shown to be highly probable. It is felt by many that a belief for which no reason can be given is an unreasonable belief. In the main, this view is just. Almost all our common beliefs are either inferred, or capable of being inferred, from other beliefs which may be regarded as giving the reason for them. As a rule, the reason has been forgotten, or has even never been consciously present to our minds. Few of us ever ask ourselves, for example, what reason there is to suppose the food we are just going to eat will not turn out to be poison. Yet we feel, when challenged, that a perfectly good reason could be found, even if we are not ready with it at the moment. And in this belief we are usually justified.

But let us imagine some insistent Socrates, who, whatever reason we give him, continues to demand a reason for the reason. We must sooner or later, and probably before very long, be driven to a point where we cannot find any further reason, and where it becomes almost certain that no further reason is even theoretically discoverable. Starting with the common beliefs of daily life, we can be driven back from point to point, until we come to some general principle, or some instance of a general principle, which seems luminously evident, and is not itself capable of being deduced from anything more evident.”

31

“One important point about self-evidence is made clear by the case of memory, and that is, that self-evidence has degrees: it is not a quality which is simply present or absent, but a quality which may be more or less present, in gradations ranging from absolute certainty down to an almost imperceptible faintness. Truths of perception and some of the principles of logic have the very highest degree of self-evidence; truths of immediate memory have an almost equally high degree. The inductive principle has less self-evidence than some of the other principles of logic, such as ‘what follows from a true premiss must be true’. Memories have a diminishing self-evidence as they become remoter and fainter; the truths of logic and mathematics have (broadly speaking) less self-evidence as they become more complicated. Judgements of intrinsic ethical or aesthetic value are apt to have some self-evidence, but not much.”

32

“Degrees of self-evidence are important in the theory of knowledge, since, if propositions may (as seems likely) have some degree of self-evidence without being true, it will not be necessary to abandon all connexion between self-evidence and truth, but merely to say that, where there is a conflict, the more self-evident proposition is to be retained and the less self-evident rejected.”

33

“What we firmly believe, if it is neither knowledge nor error, and also what we believe hesitatingly because it is, or is derived from, something which has not the highest degree of self-evidence, may be called probable opinion. Thus the greater part of what would commonly pass as knowledge is more or less probable opinion.

In regard to probable opinion, we can derive great assistance from coherence, which we rejected as the definition of truth, but may often use as a criterion. A body of individually probable opinions, if they are mutually coherent, become more probable than any one of them would be individually. It is in this way that many scientific hypotheses acquire their probability. They fit into a coherent system of probable opinions, and thus become more probable than they would be in isolation. The same thing applies to general philosophical hypotheses. Often in a single case such hypotheses may seem highly doubtful, while yet, when we consider the order and coherence which they introduce into a mass of probable opinion, they become pretty nearly certain. This applies, in particular, to such matters as the distinction between dreams and waking life. If our dreams, night after night, were as coherent one with another as our days, we should hardly know whether to believe the dreams or the waking life. As it is, the test of coherence condemns the dreams and confirms the waking life. But this test, though it increases probability where it is successful, never gives absolute certainty, unless there is certainty already at some point in the coherent system. Thus the mere organization of probable opinion will never, by itself, transform it into indubitable knowledge.”

34

“Hence, by organizing our instinctive beliefs and their consequences, by considering which among them is most possible, if necessary, to modify or abandon, we can arrive, on the basis of accepting as our sole data what we instinctively believe, at an orderly systematic organization of our knowledge, in which, though the possibility of error remains, its likelihood is diminished by the interrelation of the parts and by the critical scrutiny which has preceded acquiescence.”

35

“Philosophy, if it cannot answer so many questions as we could wish, has at least the power of asking questions which increase the interest of the world, and show the strangeness and wonder lying just below the surface even in the commonest things of daily life.”

DKB Show

The Limits of Human Knowledge