(DESCARTES TO KANT; EDDINGTON)
We have already noticed how knowledge is gained by establishing relations between an inner process of understanding in our private minds and the facts of that public outer world which is common to us all. As Plato pointed out, the use of a common language is based on the supposition that such relations can be established by all of us.
In the period we have been considering, science claimed only one source of knowledge of the facts and objects of the outer world, namely the impressions they make on the mind through the medium of the senses. Yet the untrustworthiness of the senses had been one of the commonplaces of philosophy from Greek times on, and if the same facts and objects of the outer world made different impressions on different minds, where did science stand? If we trusted to individual sense-impressions, we could never get beyond the position described by Protagoras (c. 481–411 B.C.): ‘What seems to me is so to me, what seems to you is so to you’; each individual would become his own final arbiter of truth, and there could be no body of objective knowledge. Six centuries before Christ, in the earliest days of Greek philosophy, Thales of Miletus had urged the importance of gaining a substratum of facts, independent of the judgment of individuals, on which a body of objective knowledge could be built.
These difficulties are non-existent to the modern physicist, who can trust his instruments to give absolutely objective and unbiased information, but they loomed large when there were no instruments beyond the unaided human senses. To avoid them, Plato argued in the Theaetetus (c. 368 B.C.), we must distinguish between what the mind perceives through the senses and what it apprehends of itself by thinking. Such concepts as number and quantity, sameness and difference, likeness and unlikeness, good and bad, right and wrong do not enter our minds through our senses, but reside permanently in our minds. And as concepts such as these provide the formal element in all true knowledge, it follows that this does not come from our sensations, but rather from the judgments our minds pass on our sensations.
Plato elaborated this into an argument that the human mind is equipped from birth with a set of forms or ideas which exist in it independently of the objects of the outer world. These latter serve as a sort of raw material for the impress of the forms, so that each object becomes a sort of meeting-place for a number of forms. A red square brick, for instance, is a lump of this raw material stamped with the impresses of the forms of redness, squareness and brickiness. When we declare that a particular object is a red square brick, we mean that in our judgment this particular piece of matter fits into these three forms. We may of course be mistaken; seen in a different light, the object may appear of some colour other than red, measured against a set-square it may prove to be far from square, and hit with a trowel it may prove not to be a brick at all.
On such grounds as these, Plato maintained that we have sure and certain knowledge only of the forms and their relations; our knowledge of the objects of the outer world consists at best of fleeting impressions and shifting opinions. In the matter of reality and certainty, the ideas which reside permanently in our minds, namely the forms, may claim precedence over ideas put there temporarily by objects we perceive with our senses: it is in this world of permanent ideas which exists outside space and time, the world sub specie aeternitatis, that truth alone can dwell.
This train of thought retained existence of a sort through the dark ages of philosophy; it figured prominently, although in a modified form, in the philosophies of St Thomas and the scholastics, and finally reappeared, still further modified, in the philosophy of Descartes.
The ideas of Plato, the forms, had been ideas of qualities or properties; he supposed that these were inborn in our minds, as though for instance they were memories carried over from a previous existence. The ideas of Descartes, on the other hand, were ideas of facts, or propositions as we should now call them. He thought they were innate in a sense rather different from that of Plato; the mind was not born with these ideas inside it, but with a predisposition to acquire them as soon as it came into contact with the world. ‘I called them innate in the same sense in which we say that generosity is innate in certain families and certain diseases such as gout or gravel in others—not that the infants of those families labour under those diseases in the womb of the mother, but because they are born with a certain disposition or faculty of contracting them.’
Leibniz subsequently challenged this, arguing that all ideas are innate in this sense, but that they only mature into actual thought when they have been developed by the growth of knowledge. The mind at birth is not a clean sheet of paper, but rather an unworked block of marble, in which there is already a latent structure of veins; this will to some extent determine the form the marble will assume when the sculptor chisels it into shape.
Others differed still more widely from Descartes, and in the period we now have under consideration we find the philosophers divided, broadly speaking, into two camps—the rationalists, who maintained that the highest truth resides in our own minds and so is to be discovered by reason, and the empiricists, who thought that truth resides outside our minds and so is only to be discovered by observation and experiment on the world outside.
The rationalists, headed by Descartes, argued that all knowledge obtained through direct observation of nature was suspect because it comes through the senses, and such knowledge can notoriously be deceptive, as all kinds of hallucinations and dreams show. Descartes added that even knowledge obtained by mathematical proof may be deceptive—first, because mathematicians have often been wrong, and second, because we can never be certain that an omnipotent God may not have decreed that we should be deceived even in the things we think we know best. In this way the rationalists discredited, even if they did not dispose of, practically the whole of scientific knowledge—it came from tainted sources. They proposed replacing it by the store of knowledge which, as they believed, was to be derived from pure contemplation.
Descartes claimed that his innate ideas, representing knowledge which came from ‘the clear vision of the intellect’, must necessarily be true. The fact that he could clearly and distinctly conceive something in his mind—as, for instance, the existence of God—was for him a sufficient proof of its truth. Others claimed that, inborn in the human mind, there are a number of ready-made principles or faculties by the recognition and skilful use of which it must be possible to discover truths about the universe, just as Euclid was able to discover geometrical truths from a few axioms, the truth of which was obvious. Kant went so far as to claim that it ought to be possible in this way to construct a ‘pure science of nature’, which should be independent of all experience of the world, and therefore uncontaminated by the errors and illusions of observation. A very similar claim has again been put forward in recent years by Eddington (p. 72, below).
Kant attempted a reasoned discussion of this question in his famous Critique of Pure Reason. He reminds us of Plato when he says that a phenomenon, or object of perception, contains both substance and form; the substance produces the effect in the mind of the percipient, while the form enables us to allocate the phenomenon to a wider class. The substance of a phenomenon comes to us as the result of an experience of the world, or, in Kant’s terminology, a posteriori; but the form, which is already in our minds lying in wait for the substance, comes to us a priori—i.e., previously to, and independently of, all actual experience of the world.
Relations between a priori concepts which are such that they can be known without any appeal to experience will constitute a body of knowledge ‘altogether independent of experience, and even of all sensuous impressions’. Such knowledge Kant described as a priori knowledge, in contradistinction to empirical or a posteriori knowledge, which has its sources in experience. A priori knowledge, then, came direct from heaven through the gates of ivory, and so was in every way superior to knowledge discovered through experiment, by observation, or even (according to Descartes) by mathematical demonstration, all of which came only through the gates of horn. A priori knowledge was necessarily applicable to every possible experience, whereas empirical knowledge, which was known only as the result of limited experience or observation, could make no such claims.
Also a priori knowledge was applicable to every possible universe, and not only to this one—for we can distinguish this universe from other possible universes only by observation, and once we do this our knowledge ceases to be a priori. Thus in claiming a priori knowledge, we claim to know enough of the ultimate nature of things to be able to say what kinds of universe a Creator could have created, and what kinds He could not have created. Kant’s claim that a ‘pure science of nature’ is possible in principle involves just this claim. Like every other claim to a priori knowledge, it not only denies the omnipotence of God, but also claims to have detailed knowledge of His limitations. It is a high claim to make for the human intellect.
In opposition to this, the empiricists held that in general knowledge comes from experience alone, so that the only way to discover the facts about the universe is to go out into the world and search for them. Most empiricists were nevertheless willing to concede that certain truths could be known by intuition or through demonstrations based on intuitions.
Locke and Hume, the two most prominent of the empiricists, were in agreement that the truths of pure mathematics could be known in this way, as also are most modern philosophers, as for instance Whitehead and Russell. But J. S. Mill held the opposite view, maintaining that the laws of arithmetic embodied generalizations derived from observations of actual objects, while geometry dealt merely with idealizations of objects of experience—we could not imagine a mathematical point, line, or triangle unless we had first made the acquaintance of their imperfect representations in the outer world. Locke thought that not only the truths of pure mathematics but also the existences of God and ourselves and the truths of morality ought to be admitted to the class of intuitive truths.
The whole question is obviously largely one of words. As regards the truths of morality, for instance, the question at issue is whether God could have made a world in which a different morality would have been ‘true’. And surely the answer depends at least as much on what we mean by morality and truth as on what we know about morality and truth.
In general, however, the empiricists held firmly to the principle that knowledge about the outer world must come from the outer world, and so can be acquired only by observation and experiment. As this is precisely the method of science, it might have been expected that those philosophers who were also scientists, or were of a scientific turn of mind, would be found in the camp of the empiricists, while those of a mystical or religious turn of mind would be found among the rationalists.
Actually almost the exact opposite was the case. I suppose the four most distinguished advocates of rationalism were (in chronological order) Descartes (1596–1650), Spinoza (1632–1677), Leibniz (1646–1716) and Kant (1724–1804). Two of these four names are among the very greatest in mathematics. Descartes was not only the father of modern philosophy but also of modern mathematics, being, amongst other things, the inventor of analytical geometry, while Leibniz shares with Newton the honour of having created the differential calculus, and incidentally anticipated Einstein by maintaining that space and time consist only of relations, in opposition to the Newtonian view that they are absolute.
Kant can make no claims comparable with these, and yet we should remember that astronomy and physics had interested him more than philosophy in his earlier years; according to Helmholtz, he only turned from science to philosophy, at the age of thirty-one, because there were no facilities for scientific research in his University of Königsberg. And he gave scientific lectures regularly to the end of his academic career, and wrote on a variety of scientific subjects, such as earthquakes, lunar mountains and the possibility of changes in the revolution of the earth. Most of his scientific work has long been forgotten, but he was the first to suggest the true nature of the external galaxies—clusters of myriads of stars—and he has the not inconsiderable distinction of having propounded one of the first theories of the evolution of the solar system. Besides introducing these evolutionary ideas into astronomy, he was one of the earliest of biological evolutionists. In his Anthropology he declares in favour of all animals being descended from a common ancestor, although he does not include humanity in this statement—possibly because of its dangerous religious implications. Still, he suggests that man must have changed fundamentally in the course of time, adding that in some future natural revolution orang-outangs might acquire not only human form, but also the organs of speech and the use of intelligence. He once wrote that he was ‘thinking many things, with the clearest conviction and to his great satisfaction, which he would never have the courage to say’. Prof. Paneth has suggested that one of these things may well have been that what could happen to orang-outangs and chimpanzees in the future might also have already happened in the past. Engraved on his tomb at Königsberg are words from the end of his Critique of Practical Reason—‘Two things fill the mind with ever new and increasing admiration and awe, the oftener and more steadily we reflect on them; the starry heavens above and the moral law within.’ The order is significant.
Spinoza can advance no claims to scientific distinction, although his thought is obviously often guided by mathematical and scientific knowledge.
Against this, none of the more prominent of the empiricists—Francis Bacon (1561–1626), Locke (1632–1704), Berkeley (1685–1753) and Hume (1711–1776)—had any special scientific attainments; Berkeley wrote an ‘Essay towards a new theory of Vision’, but its scientific value is not great.
The reason for this rather strange division of forces may have been in part that those who understood science best were also most acutely conscious of its anti-religious implications. But the true line of demarcation between the two schools of thought was geographical. The Continentals, with their love of abstract ideas, can claim all the rationalists, while the British, with their love of practical investigation, claim the empiricists, the four just mentioned being English, English, Irish and Scottish respectively.
The debate as to whether genuine a priori knowledge exists need hardly concern us; the question which matters for our present discussion is not whether such knowledge exists, but the much simpler question of whether, if it exists, it is important. To this it seems possible to give a negative answer without appealing to anything more recondite than the well-known principle that the proof of the pudding is in the eating. Of course we must ourselves be judge and jury, since it is an obvious impossibility for a man who does not claim infallibility to convince one who does that he is wrong. But even if I cannot persuade my cook that her puddings are bad, I can still dismiss her from my service.
The main reason which seems to call for an adverse judgment on alleged a priori knowledge is that it has often been proved false by the subsequent advances of science.
As examples of the kind of knowledge of which the truth was claimed to be obvious a priori may be taken:
‘The same thing cannot at once be and not be.’
‘Nothing cannot be the efficient cause of anything.’
‘The liberty of our will is self-evident.’
‘Everything that happens is predetermined by causes according to fixed law.’
Descartes gives the first three of these, describing the last of them as ‘a truth which must be reckoned as among the first and most common notions which are born with us’. With any reasonable use of language, it is obviously in contradiction with the fourth, which is taken from Kant, so that a priori knowledge begins to discredit itself by its contradictions even before the evidence of science has been called.
Nothing would be gained by trying to analyse these statements in detail, but one general remark at once suggests itself. It is surely improbable, on principle, that these or any similar statements can express absolute truths when stated without qualification in the crude bald forms permitted by common language. Such words as thing, cause, liberty and predetermined mean nothing definite until they have been defined. If we are free to supply our own definitions, we shall probably be able to find a sense in which all the propositions will be true, and a sense in which all will be untrue; or we may be able to find a group of cases in which they are true and a group in which they are untrue. Thus they do not present universal truths so much as topics for debate, the question at issue being the limits or conditions within which each is true. Stated in the uncompromising terms permitted by common language, the propositions merely prejudge questions on which philosophy has broken its teeth through the ages.
Other pieces of alleged a priori knowledge were of a more scientific kind, and these are of more interest to our present discussion. We may take two examples from Descartes:
(a) the sum of the three angles of a triangle is 180°,
(b) divisibility is comprised in the nature of substance, or of an extended thing, and three from Kant:
(c) space has three dimensions,
(d) between two points there can be only one straight line,
(e) in all changes of phenomena, substance is permanent, and the quantity thereof in nature can be neither increased nor diminished.
Kant describes (c) and (d) as principles ‘which are generated in the mind entirely a priori’, and (e) as a piece of knowledge which ‘deserves to stand at the head of the pure and entirely a priori laws of nature’.
As soon as we try to discuss these propositions in the light of modern science, we again feel the need of precise definitions of the terms used. Thus (a) and (d), which are geometrical in their nature, are true in the kind of space which is defined by the so-called ‘axioms’ of Euclid—Euclidean space, as it is usually called—but not in the curved space in which the planets are now usually pictured as moving. Did then Descartes and Kant intend their propositions to refer to Euclidean space, or to this possibly more real curved space? The answer is almost certainly that they were thinking of Euclidean space. In Descartes’ time no other kind of space was contemplated. In Kant’s time, other kinds were under consideration, but Kant held that Euclidean geometry was ‘true’ in a sense in which other geometries were not, although admitting that he could not prove this—because the axioms of Euclid could be denied without any inconsistency or contradiction. Thus we can see now, although Descartes and Kant could not, that their supposed a priori knowledge cannot claim to be applicable to any objective space of the outer world, but only to private worlds of their own. In so far as they thought that their a priori knowledge applied to the real world, they were more wrong than right.
Kant’s proposition (c)—that space has three dimensions—is in a different class; it is hard to see how it can claim to be a priori knowledge. For every mathematician knows that it is just as easy, as an abstract exercise, to imagine a space of one, two or four dimensions as one of three. If, then, a new-born baby knows that the space of the outer world has three dimensions, this must be because he has already been peeping at the outer world, or has otherwise made its acquaintance; his knowledge is empirical and not a priori.
It is much the same with the two remaining propositions, which are of a more physical nature. In (b) Descartes tells us that divisibility is a property of substance or of an extended thing, but fails to tell us what he means by substance or thing. Actually of course divisibility is a property of an elephant or a sandstorm, but not of a photon or an electron; but Descartes does not give any definition of a thing which will include elephants but exclude electrons. In (e) Kant tells us that substance is permanent, but fails to define substance. He does, however, say that his statement is tautological, which seems to imply that he would define substance as that which is permanent, in which case the statement tells us something about Kant’s use of words, but still nothing about the objective world. Since Kant’s time physicists have found that substantial electrons and other material particles may dissolve into, and also be created out of, insubstantial radiation. Even if these phenomena had not been observed, we now know that there is, in principle, no permanence in substance; it is mere bottled energy, and possesses no more inherent permanence than bottled beer, although it is of course true that under the physical conditions prevailing on our particular planet, matter may be regarded as very approximately permanent.
It is a natural transition from this to a reflection of a very general kind, which proves to be of the utmost importance for our discussion of the bearings of science on philosophy. The human race first became acquainted with the properties of matter in the special forms they assume under the physical conditions prevailing on our planet. In the same way, the laws of nature first became known to our race in the restricted form of laws applicable to the behaviour of objects comparable in size with human bodies, the reason of course being that only objects of these sizes could be studied without elaborate instrumental aid. In such studies time was usually measured in seconds or minutes and length in inches or yards, while nothing ever moved much faster than a galloping horse.
But with the instrumental aid now at its disposal, science can study phenomena in which times are measured, maybe in fractions of a millionth of a millionth of a second, maybe in thousands of millions of years; the lengths involved may be small fractions of a millionth of a millionth of an inch, or they may be millions of millions of miles, while the objects concerned may move at a millionth part of a snail’s pace or at a million times the speed of an aeroplane.
Surveying these immense ranges as a whole, we find that ordinary human activities occupy a fairly central position in the scheme of the universe; the world of man lies just about half-way between the world of the electron and the world of the nebulae. It also occupies only an excessively minute fraction of the whole range between electrons and nebulae. The smallest piece of matter we can feel, see or handle without instrumental aid still contains millions of millions of millions of atoms and electrons, while even the smallest of the planets stands in about the same relation to the largest piece of matter we can move with our unaided bodies.
Elaborate studies made with instrumental aid have shown that the phenomena of the world of the electron do not in any way form a replica on a minute scale of the phenomena of the man-sized world, and neither are these latter a replica on a minute scale of the phenomena of the world of the nebulae. As we leave the man-sized world behind us, and proceed either towards the infinitely great in one direction or towards the infinitely small in the other, the laws of nature seem at first sight to change, not only in detail but in their whole essence.
More careful scrutiny discloses that the apparent change is illusory; actually the same laws prevail throughout the range, but different features of these laws become of preponderating importance in different parts of the range. A soap-bubble obeys precisely the same law of gravitation as a cannon-ball, and also precisely the same law of air-resistance. Clearly, then, we can combine these two laws into a single law which must govern the motion of soap-bubble and cannon-ball equally. But if we let the two objects fall together from the leaning tower of Pisa, their motion will seem to be governed by entirely different laws. The reason is that gravitation is all important for the cannon-ball, while air-resistance is all important for the soap-bubble.
In the same way, all objects are governed by the universal laws of physics, but one aspect of these laws is all important for the electron, another for man-sized objects, and yet a third for the movements of the nebulae. These three departments of the universal scheme of law are so different that we are justified in thinking of them as constituting three distinct and separate sets of laws with a different pattern of events in each.
This is a fact of tremendous importance to philosophy as a whole. Its immediate importance to our present discussion is that it opens up two new worlds in which to test the alleged a priori knowledge of the rationalists. If this knowledge is found to be true in the two new worlds, the question of whether it is genuine a priori knowledge must still remain unanswered. If, however, it is found to be untrue in either or both of the new worlds, its claim to be genuine a priori knowledge is obviously discredited—the a priorists have told us that the Creator could not make a world in such and such a way; we study the world of the electron or nebula and find that He has done so already. Thus the alleged a priori knowledge can only be empirical knowledge of the man-sized world.
Now when the actual intuitions of the rationalists are tested in these two new worlds, we find that those which were of a scientific nature are frequently not true for the two new worlds which science has just opened for us; they are only true for the man-sized world which was familiar to the rationalists because it did not need elaborate instrumental aid for its exploration. For instance, three of the examples of a priori knowledge just given ought, as a preliminary step towards the truth, to be amended to read:
‘The sum of the three angles of a triangle is 180°, so long as the triangle is not of astronomical size.’
‘Divisibility is comprised in the nature of substance, so long as the object in question is not of the smallness dealt with in atomic physics.’
‘Substance is permanent, so long as we experiment only to the degree of accuracy possible for eighteenth century physics.’
No philosopher seems to have had an inkling, either a priori or otherwise, of the need for these or any similar reservations until modern physics arrived to point it out. The plain fact seems to be that when a rationalist, guided by his experience of the world, but subject to the scientific limitations of his day, could only imagine things being one way, he confidently announced that they were that way and had to be that way, describing his knowledge as a priori. Now that recent scientific investigations and discussion have opened up new worlds to the imagination, we can think soberly of possibilities that would have seemed sheer absurdities to Descartes and Kant. Not only can we imagine them, but we know that many of them find their counterparts in the actual world, and tell us that the supposed a priori knowledge of the rationalists was erroneous. Kant tells us that there are two infallible tests for true a priori knowledge—necessity and strict universality. The supposed scientific knowledge of the a priorists fails conspicuously under both tests, and this failure of their scientific intuitions naturally discredits their non-scientific intuitions. But knowledge of a mathematical kind requires further investigation.
While philosophers may have differed as to the possibility of obtaining a priori knowledge about the world of physics, they have been in very general agreement—apart from Descartes (p. 34) and J. S. Mill (p. 36)—that abstract knowledge of a mathematical kind could be obtained through purely mental processes, without any appeal to experience of the world, so that such knowledge can be truly a priori. They would have claimed this knowledge to be true in all possible worlds; it would be a knowledge of facts which it was beyond the powers of the Creator to vary. Thus it could tell us nothing about the properties of our particular world, as distinguished from those of other possible worlds which might have been created.
We have cited three instances of supposed a priori knowledge of this kind, all three being geometrical in their nature, but the progress of science has shown that they all three fail to qualify as true knowledge of the physical world.
Now that science is actively concerned with non-Euclidean geometries, philosophers have become chary of finding examples of a priori knowledge in geometry, and are more inclined to look to arithmetic or simple algebra. The proposition that two and two make four is frequently cited in this connection, although its precise content is seldom stated, so that we feel that the first need is for definitions and explanations. The simple question is: Could God have made a world in which two and two did not make four? and however much or little we may claim to know about the Creator, it is obvious that, before we can discuss this, we must know what the two and two are which form the subject of the proposition. Are they things which exist in reality or in our minds? Are they numbers or objects? And in the latter event, what kind of objects?
If the two and two refer to mere numbers, then the proposition is concerned with simple counting, and its content would seem to be a definition of the term four. We count two and then another two, and this brings us to a number to which we must give some sort of a name. The proposition tells us to call it four, although we might equally call it something else, such as quatre or vier; as indeed many people do. Clearly there can be no question of a priori knowledge here.
Obviously then, the proposition must be interpreted as referring to real physical objects. It tells us that if we take two objects of any, but the same, kind, and add to them two more objects of still the same kind, we shall then have a collection of four objects in all—not that we shall have taken four in all, for this would bring us back to mere counting, but shall have four objects under our observation as the result of doing something other than counting. The child is shown that when two apples are placed in juxtaposition with two other apples the result is a collection of four apples; he sees that the same is true of fingers or counters or pennies, and then jumps to the conclusion that it is true of everything we can imagine, as for instance bananas or sea-serpents or unicorns. The knowledge about the apples or fingers is admittedly empirical, but this merely serves to pull a trigger; what is claimed as a priori knowledge is that we may generalize from apples and fingers to sea-serpents and unicorns.
If this is the true content of the proposition, does it not merely provide another instance of incomplete or ill-considered knowledge being labelled a priori? For the generalization (which is the essence of the proposition) proves to be permissible for some classes of objects and for some circumstances, but only for some. It is impossible to say whether it is true in any particular case without detailed knowledge of the case, and such knowledge from its nature can never be a priori. We cannot say what two sea-serpents and two sea-serpents make until we know what a sea-serpent is, and this cannot be a priori knowledge. A sea-serpent is often said to be a cloud of birds; do then two sea-serpents when placed in juxtaposition with two more make four sea-serpents, or do they make one big sea-serpent, or perchance two or three? And what about two raindrops meeting two more on the window-pane? If two negatives make a positive, while two positives also make a positive, what results from adding two negatives to two more? Clearly the proposition is applicable only to objects which retain their identity through the process of physical addition, and we cannot know a priori whether any particular class of objects possesses this property or not. Of recent years mathematicians have studied algebras in which two and two make numbers other than four, perhaps two or one or even zero; such algebras do not of course apply to mere numbers, but to operations, processes or events. Before we can assert that two objects plus two objects make four objects, we must find a definition of object that will exclude such things, and clearly this cannot be inborn in us as a priori knowledge.
Kant did not discuss the proposition that 2 + 2=4, but the proposition that 7 + 5 = 12. He described this as a synthetical a priori proposition (p. 49), meaning that a special addition with fingers was needed to pull the trigger in his mind, and suggest the truth of the general proposition. But he does not define 12, or specify the and 7, other than fingers, to which the proposition is supposed to apply.
A better example would perhaps have been the proposition that 5 × 7=7 × 5, for this at least does not require a definition of 12, or even of 5 and 7, since it is equally true if 5 and 7 are replaced by undefined numbers or numerical quantities p and q. The proposition then states that the product pq is equal to the product qp; in other words, when we multiply p and q together, the order in which we take them is a matter of indifference. This is obviously so if p and q denote pure numbers, but before we can assent to the general proposition, p and q must be defined with some care. Mathematicians now employ algebras, which they describe as non-commutative, in which pq is not the same thing as qp; these are found to be specially applicable to the sub-atomic world. In most of the problems which arise for discussion in the man-sized world, p and q have such meanings that pq is equal to qp, but in the world of the electron this is not so. We may conjecture that a denizen of the world of the electron might vigorously challenge the general proposition that pq=qp, insisting that it was true only under very special conditions (p. 157, below).
Thus a large part of our mathematical knowledge proves on examination to be more empirical, at least in its application, than is evident at first sight, or than appears to have been suspected by the a priorists. We may say that a general proposition, such as that 2 + 2=4 can be true in either of two ways—either a posteriori or a priori. It is not true for objects in the outer world unless these conform to certain conditions. These conditions cannot even be stated, still less applied, without some knowledge of the outer world, so that when the proposition is applied to real objects, it obviously represents a posteriori knowledge; we first test whether the proposition is true for the class of objects under consideration, and the proposition then merely gives back to us the knowledge we have previously put into it. But the proposition can also be applied to classes of objects we imagine in our minds in such a way that they satisfy the conditions necessary for the proposition to be true. When used in this way, the proposition contains pure a priori knowledge, but it can never tell us anything about the outer world—only about the imaginings of our own minds.
For instance the proposition 2 + 2=4 as applied to apples is a posteriori because we call on our experience of the world to assure us that apples retain their individual identities through the process of adding. But as applied to unicorns, it is a priori because the unicorn is a creature of our imagination, which we imagine to retain its identity through the process of adding.
We see that when mathematical propositions are applied to objects in the a posteriori manner, they can supply no knowledge about the outer world beyond that we have previously put into them, while when they are applied in the a priori manner, they can give us no knowledge at all about the outer world—ex nihilo nihil fit.
There is, nevertheless, a wide range of abstract mathematical knowledge which can be derived by purely mental processes, without introducing any knowledge of the world outside. The clearest instance of such knowledge is to be found in the properties of pure numbers or numerical quantities, as expressed in arithmetic or ordinary algebra, but we must notice that even here we have to assume that numbers and measurable quantities exist. For example, we can show by purely mental processes, and without calling on our experience of the outer world at all, that if a is a pure number, then (a+1) × (a−1) is always less than a2—for example, 8 × 6 is less than 72. In the same way, it can be discovered that 8, 9 and 10 are composite numbers (i.e. numbers obtained by multiplying smaller numbers together), while 7 and 11 are primes (i.e. non-composite numbers).
Such facts involve no knowledge or experience about the particular world in which we live (unless we regard the existence of measurable quantity as an empirical fact), but, in so far as they have to do with worlds at all, are true of all worlds which could be either built or imagined. In whatever way this or any other world is constructed, 7 must be a prime, and, just because of this, the primeness of 7 can never tell us anything about the special structure of our particular world; no bridge can be built between the two. The same is true of all the discoveries of the pure mathematician; they are universal in the sense that they would be true in any world, and so cannot tell us anything about the special properties of this particular world.
Indeed any knowledge which is truly a priori must, as Kant says, be universal, and so can tell us nothing about our particular world. Let us imagine a totally uneducated man being told he was going to be sent to Procyon. He would not know whether Procyon was a prison or a gin-palace, an island or a star. But he would know just as much about Procyon as our unaided a priori knowledge can tell us about the universe we live in, and if he tried to construct a ‘pure science of Procyon’, his efforts would be no more futile or misguided than those of Kant to construct a ‘pure science of nature’. In such ways we see that there can be only one possible source of knowledge as to the special properties of our own world, namely experiment and observation; there is only one method of acquiring such knowledge, namely the method of science.
As the admission of this obvious truth would have undermined Kant’s whole position, he made two attempts to evade it; they are quite distinct, although he does not seem to have realized this.
In the first he claimed to be in possession of a special kind of a priori knowledge—synthetic a priori knowledge as he called it—which conveyed knowledge about our particular world.
In the second he claimed in effect that our physical knowledge is not knowledge about the world, but about the workings of our own minds—not knowledge of the world we perceive, but of our mode of perception of the world. Let us consider these two attempts at escape in turn.
We have already had an example of Kant’s synthetic a priori knowledge in the proposition that 7+5=12. A more characteristic instance is the proposition that ‘all bodies are heavy’. In discussing this, Kant first cites the proposition that ‘all bodies are extended’ as a typical piece of a priori knowledge which was, in his judgment, obvious apart from all experience of the world. He then says that, after encountering extended bodies in the actual world, we find that they are heavy as well as extended. Adding this new fact to his previous knowledge he arrives at the proposition that ‘all bodies are heavy’.
He considers that all the propositions of arithmetic, and many of the principles of physics, are of the synthetic a priori type. As instances he selects the conservation of matter (pp. 40, 41) and Newton’s third law of motion (p. 108), expressing them in the words ‘In all changes of the material world, the quantity of matter remains unchanged’ and ‘In all communications of motion, action and reaction must always be equal’.
Science can of course have nothing favourable to say about this. On Kant’s own admission, he only knows of heaviness through observing it in the actual world, and this immediately removes it from the category of a priori knowledge—synthetic a priori is seen to be merely a new, and question-begging, name for a posteriori. In the instance just quoted, Kant’s claim is in effect to know of the existence of gravitation, but if he could know of this, why did he not also know of electric attractions and repulsions? Would he have known a priori that two objects similarly electrified would not attract but repel one another?
In such ways Kant persuaded himself that this supposed a priori knowledge provided definite and certain information about the actual universe. Claims of this kind at once raise questions such as the following:
(1) If a priori knowledge does not come from our experience of the world, whence does it come? The rationalists claimed to have a priori knowledge that everything must have a cause; what, then, is the cause of a priori knowledge itself?
(2) If a priori knowledge does not come from our knowledge of the world, how can it tell us anything about the world? How does it happen that, when we step out into the world, we find this world conforming to our a priori knowledge? If Kant or Eddington succeeded in constructing a whole universe out of such knowledge, on what grounds would he expect the actual universe to conform to his predictions?
Kant saw very vividly the difficulties presented by this and similar questions, and this led him to fall back on his second line of defence, developing a set of ideas as to the precise meaning of which philosophers themselves are not altogether in agreement. Indeed there is every justification for wondering whether Kant altogether understood them himself. Sixteen years after the publication of the Critique of Pure Reason, Kant’s doctrines were making a great turmoil in Germany; university professors were being forbidden to lecture on them and one at least was forced to resign for venturing to disagree with him. This moment was chosen for asking Kant to say which of his commentators had best grasped his meaning. In reply, Kant indicated a certain Schultze, the author of an elementary explanation which seems to have over-elaborated the easier parts of Kant’s philosophy at painful length, while dismissing the more difficult parts in a few words which were demonstrably wrong. Thus the problem of discovering what Kant had been trying to say remained unsolved, as it still is to-day. James Ward tells us that no fewer than six different formulations of Kant’s philosophy were current in the years 1865 to 1878.
Although no one can say precisely what Kant meant to convey, I hope what follows will be found to express an average view as to his meaning, in so far as it affects the problems before us.
To the first of the two questions stated above—If a priori knowledge does not come from our experience of the world, whence does it come?—Kant’s answer seems to be that a priori knowledge comes from the inherent constitution of the human mind. Just as the human body is built in a certain way, with two eyes and two ears and other specific organs which perform specific functions, so the human mind is built in a certain way, with specific faculties which perform specific functions. It is to these faculties that we must look for the sources of our a priori knowledge. They sift out the sense-data with which our senses continually deluge our minds, allowing some to slip through unheeded while retaining others.
Out of such as are retained, the mind creates its own picture of the external world. As a result of the sifting action of the mind, certain laws and regularities emerge to which all our perceptions conform. If we run a miscellaneous collection of potatoes over a sieve of one-inch mesh, we know that any potato-pattern left on the sieve will conform to at least one law—every one of its ingredients will be more than one inch in diameter. This law is not obeyed by potatoes in general, nor by the miscellaneous collection of potatoes that were passed over the sieve; it is a law thrust on to the potatoes by the selective action of the sieve, and expresses a property of the sieve rather than of potatoes. Kant suggests that those laws of nature which we know (as he thought) a priori are thrust on to the perceived world by a selective action of the human mind, which thus acts as a lawgiver to nature; a priori knowledge merely specifies the conditions to which phenomena must conform if they are to be perceived.
Possible modes of selection can perhaps be illustrated by two simple analogies. Light is a blend of constituents of different wave-lengths. If we pass the light through a spectroscope, the different constituents are separated out, and we observe a spectrum of colours ranging from red, through orange, yellow, green to blue and violet—the colours of the rainbow. Outside the limits of this spectrum all looks dark, yet if a thermometer is placed in the dark region beyond the red, the mercury begins to rise, showing that beyond the reddest of visible radiation there is an invisible radiation; it is in fact the infra-red heat-radiation. Beyond the violet at the other end of the spectrum there is another region in which our eyes can see nothing, but in which certain salts phosphoresce, showing that here too there is radiation which is invisible to our eyes. This is the ultra-violet radiation; out beyond this we come to X-radiation, and further still to the γ-radiation emitted by radioactive substances.
Our instruments reveal a continuous spectrum of rays ranging from long radio waves to short γ-rays, the wave-lengths of these extremes standing in a ratio of about twenty thousand million million to one. By contrast, the extremes of radiation that our eyes can see have wave-lengths standing in a ratio of only about two to one. Thus of the whole range of radiation known to us through our instruments, only one part in ten thousand million million is perceptible to our eyes—an infinitesimal fraction of the whole.
The restriction of our vision to so minute a part of the whole spectrum acts as a sieve to our perceptions. All sorts of radiation fall on the retina, but this is sensitive only to a small part of what it receives; it forwards such radiation, and only such, to the mind for its attention. The mind might draw the inference that all radiation lies between the red and the violet. On Kant’s view this would correspond to the a priori knowledge claimed by the rationalists, and it may be noticed that, in so far as the analogy is sound, the only inference to be drawn is that a priori knowledge is wholly untrustworthy.
It is the same with sound. Our ears are sensitive only to sounds the pitch of which lies within about ten octaves, out of the infinite range which can occur in nature. If we took the data provided by our unaided sense-organs at their face-value, we might claim to know that all sounds lay within a range of ten octaves.
Such is the way in which the physical sieves of our sense-organs work. A simple analogy may explain how our mental sieves may work. The night sky exhibits a confused mass of stars which might be sorted into constellations in many ways. The Greeks, with their minds accustomed to run on legend and romance, sorted the stars out into figures of heroes and their accompanying animals; the more prosaic Chinese saw the same groups of stars as quite commonplace animals. But there are also stars in the southern sky which the Greeks had never been able to see, because their travels were confined to the northern hemisphere. When the navigators of a later age explored the southern seas, and first saw these stars, they did not see them as groups of new heroes and animals. The age of such fancies had passed, and the explorers left it to their prosaic astronomers to group the new stars in the forms of triangles, clocks, telescopes, and so on; they chose these because their practical minds were accustomed to thinking of such things. The division of the stars into constellations tells us very little about the stars, but a great deal about the minds of the earliest civilizations and of the mediaeval astronomers.
Kant thinks that it is in such ways as these that our minds sort out the phenomena of nature. The outer world provides us with a confused mass of impressions which our minds might sort out in many ways. They choose one particular way because they are constituted in one particular way; other types of mind might choose other ways. The laws we deduce from our a priori knowledge or reasoning merely represent habits of thought embedded in our own minds. These habits of thought form blinkers, restricting the free vision of our minds. But the mind, not recognizing its own limitations, proceeds to attribute these limitations to nature itself. Thus, in Kant’s own words, ‘reason only perceives that which it produces after its own design’, ‘objects conform to the nature of our faculty of perception’ and ‘we know a priori of things only what we ourselves put into them’.
Kant described this as his Copernican revolution. When no further progress seemed possible to an astronomy which supposed that the sun revolved round the astronomer, Copernicus cleared up the situation by supposing that the astronomer revolved round the sun. Kant thought that he had removed the difficulties of a priori knowledge in a similar way—if our minds conformed to the phenomena they perceived, our knowledge could not be a priori; we must therefore (so Kant thought) make the phenomena conform to our minds.
If this were the true significance of a priori knowledge, it would of course tell very little about nature—only something about our own minds. Our knowledge would not be of the structure of the universe without, but of the structure of our minds within. Here, then, we have the answer to our second question—If a priori knowledge does not come from our knowledge of the world, how can it tell us anything about the world? The answer is that it cannot; it can only tell us about the structure of our own minds.
All this throws a vivid light on the different methods of science and philosophy. Kant proposed in effect that we should base our knowledge of things on something that ‘we ourselves put into them’; the scientist is anxious to eradicate just this something, knowing that it is not knowledge of the outer world at all.
The ‘sieves’ which Kant attributed to the human mind are fourteen in number. First of all come two which he describes as ‘Forms of Perception‘—these are merely space and time. Then come twelve others, which could well be described as ‘Forms of Understanding’, although Kant preferred to describe them as ‘Pure Conceptions of the Understanding’ or ‘Categories’, this latter term being borrowed from Aristotle.
We want ultimately to bring Kant’s views on space and time into relation with present-day science; for this reason we may conveniently proceed at once to discuss space and time in rather general terms.
As present-day science knows, the words space and time admit of many interpretations. Four distinct meanings may be discerned for each, those for space being approximately as follows:
Conceptual space is primarily the space of abstract geometry. It has no existence of any kind except in the mind of the man who is creating it by thinking of it, and he may make it Euclidean or non-Euclidean, three-dimensional or multi-dimensional as he pleases. It goes out of existence when its creator stops thinking about it—unless of course he perpetuates it in a text-book.
Perceptual space is primarily the space of a conscious being who is experiencing or recording sensations. We feel an object, and our sense of touch suggests to us that it is of a certain shape and size; we see a collection of objects, and our vision suggests to us that these objects stand in certain relations to one another. We find that we can reconcile these and all other suggestions of our senses by imagining all objects arranged in a threefold ordered aggregate which we then call space. This is perceptual space, created for himself by a man experiencing sensations, and it goes out of existence as soon as his sensations cease. For a one-eyed man, or one viewing objects so remote that his binocular vision conveys no idea of distance, perceptual space is two-dimensional—at least so long as no sense other than seeing is involved. Thus the ancients located the fixed stars on the two-dimensional surface of a sphere. As soon as near objects are viewed by a normal man, so that binocular vision is employed, or as soon as objects are seen to move one behind another, or as soon as senses other than seeing are employed, a third dimension of perceptual space instantly springs into being.
Physical space is the space of physics and astronomy. Conceptual space and perceptual space are both private spaces, the one being private to a thinker, and the other to a percipient. Science finds, however, that the pattern of events in the outer world is consistent with, and can be explained by, the supposition that material objects are permanently located in, and move about in, a public space which is the same for all observers, apart from a complication introduced by the theory of relativity to which we shall return later (p. 63). Disregarding this complication for the moment, we may say that this public space is physical space.
Absolute space is the particular type of physical space which Newton introduced to form the basis of his system of mechanics (p. 108, below), and remained in general scientific use throughout the period between Newton and Einstein. When we say that a train has moved 10 miles nearer to King’s Cross, we mean that it has moved a distance of 10 miles along the pair of rails along which it is running to King’s Cross, as, for instance, from milestone 105 to milestone 95. In the same interval of time, the earth—carrying this pair of rails with it—may have carried the train 100 miles to the east by its daily rotation around its axis, and may have moved 10,000 miles in its yearly orbit round the sun, while the sun, dragging the earth along with it, may have moved 100,000 miles nearer to the nearest star and 1,000,000 miles farther away from a distant nebula. All these motions are equally real and equally true, but all are relative only to some other moving body.
The sequence might go on indefinitely, but Newton imagined that it did not. He thought that the remotest parts of the universe were occupied by vast masses which might provide fixed points of reference from which to measure motion, while themselves providing standards of absolute rest, although he qualified this by remarking ‘it may be there is no body really at rest, to which the places and motions of others can be referred’. At a later period, space was supposed to be filled with a jelly-like ether, and this again was thought to provide a standard of absolute rest until it was abolished by the coming of the theory of relativity. Assuming that such standards existed, Newton described the space in which measurements were made from them as absolute space; this, he said, ‘in its own nature and without regard to anything external, always remains similar and unmoveable’. He contrasted it with perceptual space, which he described as relative space—‘some moveable dimension or measure of absolute space which our senses determine’.
In a precisely similar way we may discern four distinct meanings for time; there are a conceptual time, a perceptual time, a physical time and an absolute time.
Conceptual time is the time of theoretical dynamics, and of all abstract attempts to study change and motion. Like conceptual space it exists only in the mind of a thinker. He usually makes it one-dimensional, but not always. Dirac, for instance, found it convenient to measure time by a q-number, which amounts to supposing that time has as many dimensions as we please to assign to it.
Perceptual time records the flow of time for any single percipient. Thus it is related to the consciousness of a particular individual, and goes out of existence as soon as this individual loses consciousness. Experience shows that the acts of perception of every percipient lie on a single linear series—in other words, they come one after another. Thus perceptual time is one-dimensional.
Physical time is the time of the active world of physics and astronomy. Like physical space it is public, in contrast with conceptual and perceptual time, which are private. Again disregarding complications introduced by the theory of relativity, science finds that the pattern of events is consistent with the supposition that all events can be arranged uniquely in a single linear sequence, the position on this sequence determining the time. This still permits of an infinite number of ways of measuring the time, so that a convention must be introduced as to how the actual measure is made. We agree to select some motion which repeats itself regularly, such as that of the earth in its orbit, to form a ‘clock’, and let each repetition of this motion count as a unit of time—in this case a year. But as this unit is too large for most practical purposes, other regularly repeating motions must be found, such as the oscillations of a pendulum or the vibrations of a crystal, which repeat many times in a year, and these provide the units needed for ordinary life and for scientific investigations in which time is involved.
Absolute time is the counterpart of absolute space. We have just seen how a ‘clock’ can be devised to give a consistent measure of time at any one point of space. The problem of synchronizing clocks in different parts of space is a different problem, to which we shall return later. If light travelled with infinite speed, it would be as simple, in principle, to synchronize distant clocks as it is to set our watches by Big Ben. Newton, disregarding the finite speed of travel of light, assumed that this could be done, and that a universal time ‘flows equably, and without regard to anything external’ throughout the universe. This we describe as absolute time.
There can be no serious difficulty in understanding the meaning of conceptual and perceptual space and time, for they are our own creations. They exist in our individual consciousnesses, and go out of existence when these consciousnesses cease to function. But a variety of views can be held, and have been held, as to the true significance of physical space and time.
Science has usually adopted a realist view of the world of nature, assuming that our perceptions originate in a stratum of real objects—stars, bricks, atoms, etc.—which exist outside, and independently of, our minds. If our minds go out of existence or cease to function, the stars, bricks and atoms continue to exist, and are still capable of producing perceptions in other minds. On this view, space and time have just as real existences as these material objects; they existed before mind appeared in the world, and will continue to exist after all mind has gone.
But philosophy has pointed out that other views are possible. We can have no knowledge except self-knowledge; what is in our minds we know, but what is outside we can only conjecture. And our conjectures may be erroneous. The mentalist or idealist philosophies suppose that there is no stratum outside all mind having an existence of its own in the way the realists suppose; consciousness is fundamental in the world, and the supposed real objects which produce our perceptions are creations either of our own or of some other minds (p. 196). There is no reason to attribute a higher degree of reality to space and time than to the objects we locate in space and time, so that these also become mental creations. Conceptual and perceptual space and time are now as real as anything there is, while physical space and time become attempted mental generalizations of these realities—in strong contrast to the realist view which makes physical space and time the realities, while conceptual and perceptual space and time are mere reflections of, and abstractions from, these realities.
The first modern to discuss the nature of space and time was Nicholas of Cusa (1401–1464). He held that space and time are products of the mind, and so are inferior in reality to the mind which has created them. In contrast with this purely philosophical conclusion, Giordano Bruno (1548–1600), discussing space and time in their astronomical aspects, argued that the words ‘above’ and ‘below’, ‘at rest’ and ‘in motion’ become meaningless in the world of eternally revolving suns and planets which know of no fixed centre. Thus all motion is relative—as Einstein subsequently convinced the world—and absolute space and time must be figments of the imagination. Leibniz (1646–1716) held very similar opinions, believing that space and time exist only relative to objects and not in their own right; space is merely the arrangement of things that co-exist, and time the arrangement of things that succeed one another. All these thinkers, then, reduced space and time simply to conceptual and perceptual space and time; physical space and time had no real existence, and absolute space and time did not come into the picture at all.
In opposition to them all came Isaac Newton (1642–1727), tacitly assuming that space and time were no mere dependents on consciousness but existed in their own rights, and introducing the hypothesis that absolute measures of space and time were possible, at least in principle.
Kant began his discussion of space and time by asking the questions: What are space and time? Are they real existences? Or are they merely relations between things? And in this case, would these relations belong to the things even though the things should never be perceived, or do they belong only to things when these are perceived—i.e. are they contributions of the perceiving mind?
He made no distinction between the different kinds of space and time that we have mentioned, identifying them all with perceptual space and time. His general view was that space has no real existence of its own, but is supplied by our minds as a framework for the arrangements of objects, so that it is only from the human point of view that we can speak of space, the extension of objects and so forth. ‘Space is not a conception which has been derived from outward experience; it is a necessary representation a priori, which serves for the foundation of all external perceptions.’ Time again is not an empirical conception and has no real existence of its own, but whereas space serves for the representations of external perceptions, time serves for the representations of internal perceptions—‘ the perception of self and of our internal states’.
Kant tries to justify these views in the discussion of his ‘first antinomy’. By an antinomy Kant means a pair of more or less contradictory assertions, each of which seems to be proved by disproving the other. In his own words, we originate a conflict of assertions ‘not for the purpose of finally deciding in favour of either side, but to discover whether the object of the struggle is not a mere illusion, which each strives in vain to reach but which would be no gain even when reached’. ‘Perhaps after [the combatants] have wearied more than injured each other, they will discover the nothingness of their cause of quarrel, and part good friends.’ A new set of ideas which reconciles the combatants is described as a solution of the antinomy. It may or may not be true; its truth is established only if it can be shown to provide a unique solution of the antinomy, but not otherwise—a point which Kant overlooks.
Kant’s first antinomy consists in brief of the assertions that it is impossible to imagine either that
(a) the world had a beginning in time, and is also limited in space,
or (b) the world had no beginning in time, and has no limits in space.
The reasons he gives for dismissing both alternatives as absurd seem entirely unconvincing to a modern scientific mind. There is, of course, no justification for tying up an infinity of space with an infinity of time in the way that Kant does. Mathematicians have investigated the properties of universes in which space is finite but time infinite, and no logical inconsistency has so far been detected in the concept. It is, however, quite simple to discuss time and space separately.
In opposition to alternative (b), Kant argues that any quantity must be regarded as the synthesis of a succession of separate unit quantities. For example, a mile must be regarded as the length of 1760 yardsticks put end to end. If, then, the quantity is infinite, the synthesis can never be completed; this, he says, is the true definition of infinity. Hence ‘it follows, without possibility of mistake, that an eternity of actual successive states up to a given (the present) moment cannot have elapsed, and that the world must therefore have a beginning.’
In this argument the words ‘can never be completed’ are obviously ambiguous. We want to know who or what can never complete them, why he should want to, and whether he wants to complete them in his imagination or in some sort of reality; until this information is given us, the argument is simply a meaningless collection of words.
Apart from this, the argument fails because a quantity can be regarded in other ways than as a succession of units. Must we always think of a mile as 1760 yards? Why this rather than as a succession of eight furlongs? And why either rather than as just one mile? Yet as soon as we concede the last possibility, the bottom drops out of Kant’s argument, since we need only increase our unit pari passu with the length of space or time to be measured. Even though our finite lives may be too short to imagine eternity as a succession of hours or years, we can still think of it as one eternity.
In opposition to (a), on the other side of the antinomy, Kant argues that if the world had a beginning in time, there must have been a previous void time in which there was no world. But there can be no reason for anything beginning in a void time, since ‘no part of any such time contains a distinctive condition of being, in preference to that of non-being’. Thus the world cannot have had a beginning.
This argument fails through assuming that time would necessarily go further back than the beginning of the world. This has not been the usual view of philosophy. Plato, for instance, said that time and the heavens came into being at the same instant; Augustine wrote, ‘Non in tempore, sed cum tempore, finxit Deus mundum’, while Kant himself tells us that time does not subsist of itself, but is ‘the form of the internal state, that is, of the perceptions of ourselves and of our state’. But if time is in ourselves, and we in the world, then time must be in the world, and it is a petitio principii to argue as though the world were in time.
After adducing arguments of somewhat similar type for space, Kant proposes the solution that space and time have no real existences, but are only forms of human perception. As they are, then, creations only of the human mind, we are free to imagine alternative (a) at one moment and (b) at the next, if we so wish; the two assertions of the antinomy become no more contradictory than the uses of a Mercator Projection and a stereographic projection in map-making, and we are free to use whichever serves our purpose best. But even if Kant’s arguments were sound, we should be under no obligation to accept his proposed ‘solution’ of the antinomy, for he does not even attempt to prove that it is the only possible solution.
Three general reflections on the problem of space and time will perhaps not be out of place here in view of their bearing on Kant’s doctrines of space and time.
Light takes time to travel through space, a fact which does not appear to have been known to Kant, although it had been discovered by the Danish astronomer Roemer as far back as 1675. Jupiter has a number of moons which circle round it with the same regularity as that of the moon round the earth. When the precise period of revolution of any moon of Jupiter has been found, it would seem to be a simple matter to draw up a time-table of its future movements. Roemer made such a time-table, and discovered that the moons did not keep to it; they seemed to get late and run behind their scheduled time whenever Jupiter was at more than its average distance from the earth, and to be ahead of time when Jupiter was at less than its average distance. He found, however, that all the observations could be explained by supposing that light travelled through space at a uniform finite speed; the apparent irregularities of Jupiter were then explained by the variations in the time which light took to travel from the planet to the earth. The truth of this explanation was established beyond all doubt when Bradley discovered the phenomenon of aberration in 1725.
This shows that space and time are not totally independent of one another as Kant and many others seem to have imagined; on the contrary there must be a fairly intimate connection between them.
The theory of relativity has revealed the nature of this connection. Newton supposed that all objects could be located in his absolute space, and that all events, wherever they occurred, could be assigned positions uniquely and objectively on an ever-flowing stream of absolute time. These assumptions provided him with an approximation which was good enough for his purpose, and fitted in with the scientific knowledge of the seventeenth century. Subsequent investigation has shown that they are inadequate to explain the passage of light and the behaviour of objects moving at a speed comparable with that of light. The physical theory of relativity suggests, although without absolutely conclusive proof, that physical space and physical time have no separate and independent existences; they seem more likely to be abstractions or selections from something more complex, namely a blend of space and time which comprises both.
It is of course always possible to take any two things of not too dissimilar nature, and blend them into a single unity which shall comprise both. Before the advent of the theory of relativity, no one could have imagined that space and time were sufficiently similar in their natures for the result of blending them together to be of any special interest. Yet such a blend has proved to be of outstanding importance for the understanding of physics.
Any ordinary three-dimensional space may be regarded as hung around a framework of three perpendicular lines, these indicating three perpendicular directions in the space, as for instance East-West, North-South and up-down. The surveyor is accustomed to treat his perceptual space in this way, and the mathematician treats his conceptual space in the same way, except that he replaces the three perpendicular directions of the surveyor by purely mental abstractions which he usually denotes by Ox, Oy and Oz. Now let us imagine the surveyor’s perceptual space sliced into horizontal layers of infinite thinness—much as a skilled chef will cut a round of beef into infinitely thin slices. Any single slice, contemplated by itself, forms a mere horizontal plane which possesses extension in the East-West and North-South directions, but not in the up-down direction. If we imagine these various slices now to be laid back, one above the other, in their original positions and then welded together, we shall have reconstituted the original three-dimensional space. We may say that, in performing this last operation, we have welded verticality on to horizontality and obtained something different from either, namely a three-dimensional space.
Let us now imagine these two-dimensional slices replaced by the perceptual three-dimensional spaces of some individual A at successive instants of his experience. Let us take all these perceptual spaces, and place them contiguous to one another in their proper order. As they are to be contiguous and not overlapping, we must imagine them all placed in a four-dimensional space before we can do this. If we now imagine them welded together, they will form a four-dimensional continuum which we may describe as the space-time unity for the individual A. It is a conceptual space of four dimensions, and as it is constructed out of the perceptual three-dimensional spaces of a single individual A, we might reasonably expect it to be private and subjective to this individual.
We can create a second space-time unity out of the perceptual spaces of some second individual B, which we might expect to be private and subjective to the individual B. The theory of relativity shows that the two space-time unities we have constructed in this way will be identical for A and B, and so also of course for any other percipients C, D, E, ... as well. In other words the space-time unity which we build up out of private perceptual spaces of a single individual proves to be public, and so objective. Space and time separately are private, but the blend of the two is public.
We cannot speak of right-hand and left-hand directions in ordinary space, since the right hand and left hand do not belong to space, but to an observer in space; the division of space into right-hand and left-hand is meaningless except relative to a particular observer. In the same way, we cannot speak of space and time in the space-time unity—space and time do not belong to the space-time unity, but to an observer in it. But it is the body of the observer that we want, and not his mind; a laboratory equipped with cameras and various instruments of measurement would serve our purpose just as well.
Two observers who always keep close together will have the same perceptual space, but if they are moving at different speeds, and so changing their relative positions, they will have different perceptual spaces. The theory shows that these different perceptual spaces are to be obtained by taking cross-sections of the space-time unity in different directions. In other words, each percipient divides up the public space-time unity into space and time in his own individual way, the mode of division depending on his speed of motion.
In the same way, to use a rather imperfect analogy, a cannon-ball may be conceived as having any number of different diameters, all pointing in different directions. It would be inaccurate to speak of any one of these as the height of the ball, to the exclusion of the others. Each one has an equal claim to be regarded as the height, and can indeed be made the height by turning the cannon-ball the right way up. But so long as the cannon-ball enters into no kind of relation with other objects, such terms as height, width and length are meaningless. In the same way, time and space are meaningless when applied to the four-dimensional continuum in the abstract. But, just as, when the ball is placed on a horizontal floor, one particular diameter immediately becomes the height of the ball, so, when we put a particular scientist or observer inside the four-dimensional continuum to measure or explore, one direction immediately becomes identifiable with his time; which particular direction it will be is determined by the precise speed at which this observer is moving.
The question now arises as to whether it is possible to divide up this unity into space and time separately in a way which shall not depend on the circumstances of individual percipients. If such a way can be found, we might identify the space and time so obtained with Newton’s absolute space and time. If such a way is not found, it will not prove that no such way exists, and still less that absolute space and time do not exist; the most we could say would be that they had not so far disclosed themselves. Actually in so far as ordinary physics—i.e. physics on the man-sized scale—is concerned, no such way has so far been found, and it seems highly improbable that it ever can be. For the pattern of events is known with tolerable completeness, and must, it is found, be described in terms of the space-time unity as a whole and not in terms of its separate dimensions, which do not enter into the description at all. This might have been anticipated from the circumstance that nothing less than the unity as a whole is completely objective.
Although physics on the man-sized scale may be unable to disentangle space from time, it is still possible that atomic physics or astronomy—i.e. physics on the scale of the nebulae—may have a different story to tell. Once again, an analogy may help to explain the possibilities.
Let us imagine a race of deep-sea fish, living so far below the surface of the ocean that no ray of sunlight ever reaches them; let them be of precisely the same density as the water in which they live, so that it is just as easy for them to swim up as down; let their semi-circular canals, and any other mechanism they may have for distinguishing directions, be abolished. Such a race of beings will have no means within themselves of distinguishing directions, and if they study physical phenomena, they will find that the laws of optics, electricity, magnetism, etc. make no distinction between the different directions in space. They may then announce that nature treats all directions of space equally. Having no means of disentangling the horizontal from the vertical, they will describe different directions in a purely subjective way. Up and down will not refer to directions determined relative to the earth’s centre, but relative to their own backs and bellies. They will know nothing of an objective north, south, east and west; only of subjective directions, to describe which they may use such words as fore and aft, right and left.
In this analogy, the race of fish represents physicists who study physics on the man-sized scale. The three-dimensional space in which the fishes swim corresponds to the four-dimensional space-time unity of the theory of relativity in which we exist. Man-sized nature has provided no means of dividing this into space and time separately, just as the fishes had found no means of dividing their watery space into horizontal and vertical.
Now suppose that one fish has the enterprise to swim as far as the surface of the sea. He no longer studies nature on the fish-sized scale, but on a world-sized scale. When he does this, he finds a whole range of new phenomena, and amongst them a surface, objective and fixed by nature, which at once determines up-down and horizontal directions in space in a wholly objective way.
There is still a possibility that when we leave man-sized physics for astronomical physics, we may have experiences similar to those of the enterprising fish. The hypothesis that absolute time and space do not exist brings order into man-sized physics, but seems so far to have brought something very like chaos into astronomy. Thus there is some chance that the hypothesis may not be true. Newton thought that the vast masses which occupy the remotest parts of the universe might provide a framework from which to measure absolute rest and motion, and something of the kind seems to be needed if the pattern of events recently revealed by nebular astronomy is to make sense. It may be that before it can make sense, the new astronomy must find a way of determining an absolute time, which it will then describe as cosmical time. The space-time unity will then be divided into space and time separately by nature itself. Apart from this possibility, all observers stand on the same footing, each dividing the space-time unity into his own perceptual space and his own perceptual time.
The foregoing remarks embody the main conclusions of the restricted, or physical, theory of relativity which Einstein put forward in 1905. We must always remember that this theory is a deduction from the observed pattern of events. As the pattern can only be expressed in mathematical terms, the theory of relativity also can only be expressed in mathematical terms. It deals with measures of things, and not with things themselves, and so can never tell us anything about the nature of the things with the measures of which it is concerned. In particular it can tell us nothing as to the nature of space and time.
Nevertheless, as it shows the mathematical measures of space and time to be so intimately interwoven, it seems reasonable to suppose that space and time themselves must at least be of the same general nature. The distinction, which many philosophers besides Kant have drawn, between space and time as forms of perception of external and internal experience is one which can no longer be maintained in respect of physical space and time, although it can for perceptual space and time.
This space-time unity of the theory of relativity figures very prominently in the philosophical system of Alexander (1859–1938), for he supposes that it is the primordial reality out of which all things have evolved. He conjectures that the most primitive, as also the simplest, kind of stuff in the world is pure space-time; out of this various kinds of matter emerge and, gradually rising higher, develop into life, consciousness and Deity in turn. All the continental thinkers whom we have mentioned have seen space and time as creations of mind, but for Alexander mind is a creation of space and time.
As a final remark, science and the various materialistic philosophies have proceeded for centuries on the supposition that all objects and all events, and indeed the whole universe, can be arranged in space and time. Quite recently science has found that such an arrangement is inadequate. The rays of light, waves of sound, and the various other messengers that bring us information as to the happenings of the outer world may quite properly be regarded as travelling in space and time; such a representation is self-consistent, makes sense, and gives a rational account of our perceptions. But we shall see below that we are hardly free to depict the events which despatch these messengers as happenings in space and time; such an interpretation does not make sense or lead to a rational view of the universe. We find there is something in reality which does not permit of representation in space and time. Thus space and time cannot contain the whole of reality, but only the messengers from reality to our senses.
Besides the two forms of perception—space and time—which we have just discussed, Kant’s fourteen mental sieves comprised twelve categories, or ‘forms of understanding’. There is no need to enter into any detailed discussion of these categories, for while eleven of the twelve may or may not be of interest to logic, they are of no interest to science. One only makes any kind of contact with science, and this is the category of Causality; Kant thinks that our minds are so constituted that we see all sequences of events in terms of the cause-effect relation.
Categories figure in a somewhat different manner in other philosophical systems. Aristotle regarded them as forms of structure, not of the mind but of the world. For Hegel they are forms of thought in the Absolute mind, while Alexander returns to the Aristotelian conception of categories as forms of the world itself.
Up to the present, the conclusions of philosophy have all been reached by minds which have all been of one type—the human type—contemplating their perceptions of one and the same world. So long as there is only one type of mind contemplating one world, there can obviously be no means of deciding whether Kant’s forms of perception and understanding result from the structure of the world or from the structure of the mind which perceives the world.
But we have seen (p. 43) how science has just presented us with two new worlds. The world of modern science can be divided into three fairly distinct divisions—a man-sized world in the centre flanked by the minute world of atomic physics on the one side, and the vast-scaled world of astronomy on the other. The same laws of nature prevail in all three divisions, but different aspects of them assume prominence in each, to the almost complete exclusion of all other aspects, so that we may almost regard the three divisions as three different worlds, with different sets of laws in each. But the human minds which study them are the same in each case, and so must contribute the same modes of thought to the study of each.
The two new worlds have already provided us with a testing ground for a priori knowledge. If this really represented some inborn quality of the mind, we should have found its assertions true in all worlds; actually most of them prove to be true only in that world which we can see and study without instrumental aid. We accordingly concluded that such knowledge was found in the human mind, not because it was born there, but as a sort of sediment left by the flow of experience of the man-sized world through our minds. Residence in the worlds of the electron or of the nebulae would have left quite a different sediment in our minds, which those of us who were rationalists would then have announced as a priori knowledge.
A test of a similar kind can be applied to Kant’s theory of knowledge. For, as we shall see in subsequent chapters, those forms of perception and understanding which are of scientific interest—namely causality and the possibility of representation in space and time—prevail in the man-sized world, but not in the small-scale world of atomic physics which we know only through instrumental study. If they really were contributions of the human mind to nature, they would be contributed to all three worlds equally. But as they are not contributed to all three worlds, we may conclude that they are not inborn modes of human thought. Again, they would seem to be ingrained rather than inborn, not so much laws that we thrust on nature as laws that we—with our limited knowledge of the world—have allowed nature to thrust on us. We think everything can be located in space and time because the world that we perceive with our unaided senses seems to admit of location in space and time; the reason is not that things are so located, but that the messengers from them to our sense-organs travel through space and time (p. 139). In the same way we think we see cause and effect running through everything, because the phenomena of the man-sized world seem to conform to a law of causality; here again the reason is not that they do so conform, but that they obey statistical laws which produce an impression of causality on our coarse-grained organs of perception (p. 131). Our experiences of our man-sized world create in our minds habits of thought which take causality and space-time representation for granted. We cannot imagine anything else because we have never experienced anything else.
If this is so, Kant’s forms of perception and understanding are not so much blinkers which restrict our knowledge of the outer world as lenses which condense our knowledge. But the knowledge they condense is knowledge only of the man-sized world, being crystallized experience of this world alone; denizens of the underworld of atoms and electrons would have had other experiences, and a Kant of this underworld, even though endowed with a mental constitution just like our own, would have produced other categories and other forms of intuition.
In any case, it is probably fair to say that all that modern philosophy retains of Kant’s theories on this subject is the possibility of certain forms of thought—whether inborn or ingrained hardly matters—causing our particular type of mind to select what it does rather than something else. Our own minds contribute something to the nature they study—a view, incidentally, which dates back to Nicholas of Cusa and the fifteenth century.
Even this remnant means little, unless we concede the possibility of a priori knowledge about the external universe. Kant’s whole theory was an ad hoc structure designed to remove an obvious difficulty about a priori knowledge, and if a priori knowledge passes away, the need for, and to some extent the importance of, this theory passes away with it.
At the same time a priori knowledge was itself, in a sense, an ad hoc structure designed to help metaphysics in its self-appointed task of championing the doctrines of theology. It can hardly have mattered much to Descartes or Kant whether they knew that the sum of the three angles of a triangle was 180° by having proved it in their minds, or having measured it with their instruments, or having seen it by the clear light of reason. Their primary interest was in the question of principle; they wanted to be able to claim that they were possessed of knowledge which was unchallengeable because it had not reached them through the deceitful gates of the senses. And the kind of knowledge they wanted to claim was not knowledge about triangles, but about GOD, FREEDOM and IMMORTALITY. They wanted for instance to be able to say that, science or no science, the will was free because they saw it to be so by the clear vision of their intellects.
With the passing of this special phase of philosophy, a priori knowledge lost its special significance, and, apart from mathematical knowledge, few philosophers have much to say in its favour to-day; at least it is generally conceded that it is of little consequence. Yet, just when a priori knowledge has become discredited in philosophy, an attempt has been made to revive it in physics.
We have seen how Kant thought that we ought to be able to build up a ‘pure science of nature’ solely by the use of the a priori knowledge inborn in our minds. This amounted to claiming that the world could only be of one kind—or rather could only appear in one way to us, with our minds constituted as they are. Keeping our minds as they are, the Creator could not have made the world appear different to us from what it does.
Sir Arthur Eddington also thinks that we ought to be able to build up what we may describe as a pure science of nature from a priori knowledge, but he thinks of this a priori knowledge as epistemological rather than as inborn. In other words, he thinks we should find logical inconsistencies in reaching any other conclusions about the physical world than those which the physicists have actually reached from centuries of toil in their laboratories. It should be explained that this claim applies only to the general laws of nature and not to individual objects in nature, and also that when Eddington speaks of nature, he is concerned only with nature as it appears to us, and not with an objective nature outside ourselves.
The general point of view will best be understood in terms of a specific example.
We have already seen that, if light travelled with infinite velocity, it would be a simple matter in principle to synchronize all the clocks in the universe. The method would be as simple as that of setting our watch by Big Ben, and we could call in the help of telescopes as needed. But, as light does not travel with infinite velocity, we cannot synchronize distant clocks in this way; we must allow for the time light takes to travel from one clock to another, and the theory of relativity has made it clear that the synchronization of distant clocks, if it could be achieved at all, would call for a far more elaborate technique than looking through telescopes at distant clocks.
In the years 1887–1905 a great number of experiments were performed for another purpose, any one of which might have resulted in the discovery of such a technique. But none of them did, and it is now generally accepted that the synchronization of distant clocks is an impossibility. It is not impossible in the sense in which it is impossible to fly an aeroplane at 1000 miles an hour—i.e. because our technical skill is not yet sufficiently advanced—but rather in the sense in which it is impossible to fly an aeroplane to the moon—i.e. because, as observation has shown, nature provides us with nothing on which we can get a hold, no resistant medium to support our aeroplane. The main result of the physical theory of relativity is usually expressed in the form that it is impossible to determine an absolute velocity in space, but might almost (not quite) equally well be expressed in the form that it is impossible to synchronize distant clocks.
As a matter of historical fact, this conclusion was reached as a generalization from a very large number of experiments. Let us, however, imagine a race of beings who know without experimenting that it is impossible ever to synchronize distant clocks—to avoid cumbersome repetition, let us agree to describe them as asynchronists. These beings would not dream of performing the whole set of experiments just mentioned, because their innate convictions would tell them the results without. If they had a Kant, he would describe this knowledge as a priori knowledge. If they had a Descartes, he would point out that this knowledge, being independent of all experience, could claim a higher degree of certainty than if it had been derived from a finite number of experiments, any generalization from which might be negatived by further experiments.
Now Eddington claims, in brief, that we are ourselves asynchronists, that we have knowledge in our minds as to the impossibility of synchronizing distant clocks. Like Kant he describes this knowledge as a priori—‘knowledge we have of the physical universe prior to actual observation of it’; like Descartes he claims for it a higher degree of certainty than can be possessed by knowledge derived from experiment—‘generalizations that can be reached epistemologically have a security which is denied to those that can only be reached empirically’. This a priori, or epistemological, knowledge is not confined to asynchronism; this is merely a somewhat trivial example. Again like Kant (p. 35), Eddington believes that all the laws of nature that are usually classed as fundamental can be foreseen wholly from epistemological considerations’ and further that ‘not only the laws of nature but also the constants of nature can be deduced from epistemological considerations, so that we can have a priori knowledge of them’. It follows that ‘an intelligence unacquainted with our universe, but acquainted with the system of thought by which the human mind interprets to itself the content of its sensory experience, should be able to attain all the knowledge of physics that we have attained by experiment. He would not deduce the particular events and objects of our experience, but he would deduce the generalizations we have based on them.’
Thus for Eddington knowledge of this fundamental kind results from the constitution of our minds, which are thus once again rehabilitated as law-givers to nature in the Kantian sense. We need never have built physical laboratories, except to study matters of detail; it would have been better to have delved into our own minds, where we should have found the results of all the fundamental experiments of physics, together with the values of the fundamental constants of physics. Eddington goes on to remind us that whatever is accounted for epistemologically is ipso facto subjective; it is demolished as part of the objective world’. Fundamental physics, then, tells us something about our own minds, but nothing about the outer world. To use one of Eddington’s own metaphors: ‘When science has progressed the furthest, the mind has but regained from nature what the mind has put into nature. We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the footprint, and lo! it is our own.’
Eddington’s claim that the fundamental laws of physics can be foreseen epistemologically would carry more conviction if he could himself establish any one of them, even the simplest, epistemologically—in other words, if he could show that there would be a logical inconsistency in believing the laws to be different from what they are. This he never does.
It seems improbable that he ever could, for surely to speak of establishing any fact of science by epistemology alone involves a contradiction in terms. Epistemology has only one tool in its armoury. This is pure logic, and before it can be applied to a scientific fact, we must define the scientific objects about which the fact is stated. We can only do this by calling upon knowledge which has been obtained empirically. In so doing we pass beyond the realm of a prior knowledge, and our discussion ceases to be purely epistemological.
To illustrate by a concrete case, Eddington believes it is possible to establish epistemologically that the mass of the proton must be 1847 times that of the electron. Clearly, though, he must be careful to avoid proving at the same time that the mass of the apple is 1847 times that of the orange; if his argument proves this, we shall feel suspicious of it. He can escape this danger by defining his electrons and protons in a way which makes it clear that they are not apples and oranges. Actually he neglects to do this, with the result that, in so far as his proof of the 1847 ratio is epistemological, it is equally applicable to apples and oranges.
Of course Eddington is entirely justified in assuming that we know what he means by electrons and protons, but what about the visiting intelligence from another universe? Will he not be in the position of the lecturee who said the lecturer had explained beautifully how the astronomers discovered the sizes and temperatures and masses of the stars, but had forgotten to explain how they found out their names? He will not know the difference between an apple and an electron until we tell him, and before we could do this, we should have to acquaint him with whole masses of laboratory knowledge, and epistemology would be left far behind. For the visitor is supposed to be acquainted only with our system of thought, and can it be seriously maintained that this includes the knowledge that the world is made up of similar fundamental particles of two and only two kinds? So far from being an innate part of our mental equipment, this is a hypothesis that did not even enter science until a few years ago (and incidentally left it again, very hurriedly, a few years later).
It is in fact necessary to build a bridge between the abstractions of epistemology and the actualities of observed phenomena; without this epistemology is left up in the air, and cannot know what it is talking about. Kant did this by introducing his synthetic a priori knowledge; Eddington does it by withdrawing his claim that his a priori knowledge is ‘knowledge that we have of the physical universe prior to actual observation of it’, and writing instead that ‘to the question whether it can be regarded as independent of observational experience altogether, we must, I think, answer: No’. But this admission obviously weakens his position enormously; his natural laws are no longer foreseen ‘wholly from epistemological considerations’, but only from a mixture—in unknown and unknowable proportions—of these and observation, which means simply observation combined with sound reasoning. And surely this is just the ordinary procedure of all science. Eddington’s laws, being no longer reached by pure epistemology, must renounce their claims to pure subjectivity, and to ‘a security that is denied to those [laws] that can only be reached empirically’. They become ordinary scientific laws, obtained in the ordinary scientific way, and the only question is whether the mathematics is right or wrong.
A simple test case is provided by the finite velocity of light. We introduced Eddington’s philosophy, as he himself has done, by considering the impossibility of synchronizing distant clocks. The reason why such synchronization is impossible is that light does not pass instantaneously from place to place. Those, then, who believe it is possible to prove all the fundamental laws of nature from epistemological considerations, ought to find it possible to prove in this way that the velocity of light is finite—i.e. they ought to be able to point to some logical inconsistency involved in the idea of light travelling with an infinite velocity. Eddington, however, merely dismisses the question with the statement that it is absurd to think of the speed of light as innnite—as absurd, he says, as to think of it as hexagonal or blue or totalitarian.
So long as we look at the question from the purely epistemological point of view—forgetting all that experience has taught us about space, time and propagation—it is hard to find anything absurd in the idea of instantaneous propagation. Prof. A. Wolf writes that ‘down to the seventeenth century [the velocity of light] had usually been regarded as infinite, and Kepler, and perhaps also Descartes, seem to have held this view. Descartes...believed that light was not a moving substance, nor a motion at all, but a tendency to motion, or a thrust exerted by the luminous body: and he supposed that this thrust, being incorporeal, required no time for its propagation.’ In the same way, most people still think of the thrust of an iron bar as an example of instantaneous propagation. Newton and his contemporaries took it for granted that gravitation was propagated instantaneously; it was over a century later that the alternative possibility of a finite speed of transmission was first considered by Laplace—not because it seemed inherently probable to him, but because he wished to leave no avenue unexplored which might solve the mystery of the moon’s acceleration. And when the first observational evidence (p. 63) of the finite speed of light was produced by Roemer, it was hailed as a sensational new discovery—not as confirmation of something that had been known all the time as a matter of course. Indeed, for a time it was rejected by many of Roemer’s contemporaries who continued to believe in the infinite velocity of light.
All this seems to show that there can be nothing epistemologically absurd in the idea of an infinite velocity of propagation.
Even if it could be conceded that we have a priori knowledge that light travels with only finite velocity, it would still be a long step further to the fundamental postulates of the theory of relativity, and Eddington claims these also as a priori knowledge. Sixty years ago physicists were almost unanimous in imagining space to be filled with an ether through which waves travelled at the finite speed of 186,000 miles a second. This constituted a perfectly self-consistent scheme, it made sense, and explained all the phenomena as then known, so that, so far as epistemological considerations went, it was entirely eligible as a possible explanation of the phenomena ; it had to be abandoned only because experiment decided against it. If these experiments had turned out otherwise than as they did—and we can easily imagine them doing so—this scheme would probably still have prevailed. This of itself gives a sufficient proof that no epistemological arguments compel the abandonment of this scheme, whence it follows that none can require the acceptance of the opposite scheme, which is that of the theory of relativity. Indeed as this latter scheme is purely a generalization from the results of a large number of experiments, there is still a possibility in principle—although not much probability—that further experiments may still be found to compel its abandonment.
There is an alternative way of regarding the matter which would seem to be more true to the facts.
Borrowing a simile from Poincaré, we have already compared the construction of a science to the building of a house. Our stones are a collection of facts of observation. Just because nature is rational, we find that these can be made into something other than a mere shapeless pile; they show definite regularities, and so can be fitted together to form a house with definite characteristic features.
It will be possible to describe these characteristic features in simple terms which will evoke a ready response in our minds; indeed we can describe them in terms of ideas which are already in our minds and familiar to our minds. They are familiar, not because we are familiar with the general laws of physics, but because we are familiar with special and restricted instances of them; it is of such that our daily lives are made up. We may, for instance, say that the house shows no unnecessary ornamentation (Occam’s razor) or no cracks (conservation laws). The ideas of ornamentation and cracks are not innate in our minds, but have been acquired from experience in very special small corners of the world.
Now the design of this house is nothing other than the pattern of events which it is the aim of physics to discover. The physicist finds—after sweat and toil in the laboratory—that this pattern of events shows features like those we have attributed to our house. There is no doubt that a great part (and perhaps all) of the fundamental facts of physics can, when once they have been discovered empirically, be summed up in general statements which seem very simple and intelligible to us because we are familiar with detailed instances of them. These can often (perhaps always) be expressed in the form of what E. T. Whittaker has called Postulates of impotence’, these asserting ‘the impossibility of achieving something, even though there may be an infinite number of ways of trying to achieve it’. It is, for instance, impossible to get mechanical work out of matter which is at a lower temperature than the surrounding objects, and impossible ever to measure an absolute velocity in space. These two postulates of impotence contain practically the whole contents of thermodynamics and of the physical theory of relativity respectively.
Hence, as Whittaker has remarked, ‘It seems possible that, while physics must continue to progress by building on experiments, any branch of it which is in a highly developed state may be exhibited as a set of logical deductions from postulates of impotence, as has already happened to thermodynamics. We may therefore conjecturally look forward to a time in the future when a treatise on any branch of physics could, if so desired, be written in the same style as Euclid’s Elements of Geometry, beginning with some a priori principles, namely postulates of impotence, and then deriving everything else from them by syllogistic reasoning.’
These principles would not of course be a priori in Kant’s sense of ‘pre-observation’; they would be very much a posteriori, being the highly concentrated extracts of immense masses of observations. But we ean imagine a scientist pondering over their simplicity until they became endowed in his eyes with a quality of ‘inevitableness’, and he would begin to regard them as laws of thought. In a sense they would have become laws of thought for him.
This, we may conjecture, is what Eddington has done. And, just as the true nature of Kant’s supposed categories of thought is disclosed by experiments on the atomic world, which show that causality and space-time representation no longer prevail there, so at any time a new experiment may show that Eddington’s supposed a priori principles are mere mental sediments left over from actual experience of the world. Indeed to some extent the discovery of positrons has done this already.
Our discussion seems to bring us back to the age-old conclusion that if we wish to discover the truth about nature—the pattern of events in the universe we inhabit—the only sound method is to go out into the world and question nature directly, and this is the long-established and well-tried method of science. Questioning our own minds is of no use; just as questioning nature can tell us truths only about nature, so questioning our own minds will tell us only truths about our own minds.
The general recognition of this has brought philosophy into closer relations with science, and this approach has coincided with a change of view as to the proper aims of philosophy. The ancient philosophers pursued their studies in the hope of finding a lantern which should guide their feet along the best path in their journey through this life, the philosophers of the seventeenth and eighteenth centuries in a fixed determination to find evidence that this journey ended in a life to come. This humanistic tinge has taken a long time to disappear, but has almost done so in recent years; philosophy has become less concerned with ourselves and more concerned with the universe outside ourselves. It is now recognized that, in Bertrand Russell’s words: ‘Man on his own account is not the true subject-matter of philosophy. What concerns philosophy is the universe as a whole; man demands consideration solely as the instrument by means of which we acquire knowledge of the universe.... We are not in a mood proper to philosophy so long as we are interested in the world only as it affects human beings; the philosophic spirit demands an interest in the world for its own sake.’
This may seem to suggest that philosophy should have not only the same methods but also the same aims and also, broadly speaking, the same field of work as science. But the distinction mentioned at the beginning of the present chapter still holds good. The tools of science are observation and experiment; the tools of philosophy are discussion and contemplation. It is still for science to try to discover the pattern of events, and for philosophy to try to interpret it when found.