In philosophy ever since the time of Pythagoras therehas been an opposition between the men whose thought wasmainly inspired by mathematics and those who were moreinfluenced by the empirical sciences. Plato, Thomas Aquinas,Spinoza, and Kant belong to what may be called themathematical party; Democritus, Aristotle, and the modernempiricist from Locke onwards, belong to the opposite party.In our day a school of philosophy has arisen which sets towork to eliminate Pythagoreanism from the principles ofmathematics, and to combine empiricism with an interest inthe deductive parts of human knowledge. The aims of thisschool are less spectacular than those of most philosophersin the past, but some of its achievements are as solid asthose of the men of science.
The origin of this philosophy is in the achievements ofmathematicians who set to work to purge their subject offallacies and slipshod reasoning. The great mathematicians ofthe seventeenth century were optimistic and anxious for quickresults; consequently they left the foundations of analyticalgeometry and the infinitesimal calculus insecure. Leibnitzbelieved in actual infinitesimals, but although this beliefsuited his metaphysics it has no sound basis in mathematics.Weierstrass, soon after the middle of the nineteenth century,showed how to establish the calculus without infinitezimals,and thus at last made it logically secure. Next came GeorgCantor, who developed the theory of continuity and infinitenumber. "Continuity" had been, until he defined it, a vagueword, convenient for philosophers like Hegel, who wished tointroduce metaphysical muddles into mathematics. Cantor gavea precise significance to the word, and showed thatcontinuity, as he defined it, was the concept needed bymathematicians and physicist. By this means a great deal ofmysticism, such as that of Bergson, was rendered antiquated.
Cantor also overcame the long-standing logical puzzlesabout infinite number. Take the series of whole numbers from1 onwards; how many of them are there? Clearly the number isnot finite. Up to a thousand, there are a thousandnumbers; up to million, a million. Whatever finite numberyou mention, there are evidently more numbers than that,because from 1 up to the number in question there are justthat number of numbers, and then there are others that aregreater. The number of finite whole numbers must, therefore,be an infinite number. But now comes a curious fact: Thenumber of even numbers must be the same as the number of allwhole numbers. Consider two rows:
1, 2, 3, 4, 5, 6, ....
2, 4, 6, 8, 10, 12, ....
There is one entry in the lower row for every one in the toprow; therefore the number of terms in the two rows must bethe same, although the lower row consists of only half theterms in the top row. Leibnitz, who noticed this, thought ita contradiction, and concluded, though there are infinitecollections, there are no infinite numbers. Georg Cantor, onthe contrary, boldly denied that it is a contradiction. hewas right; it is only an oddity.
Georg Cantor defined an "infinite" collection as onewhich has parts containing as many terms as the wholecollection contains. On this basis he was able to built up amost interesting mathematical theory of infinite numbers,thereby taking into realm of exact logic a whole regionformerly given over to mysticism and confusion.
The next man of importance was Frege, who published hisfirst work in 1879, and his definition of "number" in 1884;but in spite of the epoch-making nature of his discoveries,he remained wholly without recognition until I drewattention to him in 1903. It is remarkable that, beforeFrege, every definition of number that had been suggestedcontained elementary logical blunders. It was customary toidentify "number" with "plurality". But an instance ofnumber is a particular number, say 3, and an instance of 3is a particular triad. The triad is a plurality, but theclass of all triads - which Frege identified with the number3 - is a plurality of pluralities, and number in general,of which 3 is an instance, is a plurality of pluralities ofpluralities. The elementary grammatical mistake ofconfounding this with the simple plurality of a given triadmade the whole philosophy of number, before Frege, a tissueof nonsense in the strictest sense of the term "nonsense".
From Frege's work it followed that arithmetic, and puremathematics generally, is nothing but a prolongations ofdeductive logic. This disproved Kant's theory thatarithmetical propositions are "synthetic" and involvea reference to time. The development of pure mathematics fromlogic was set forth in detail in Principia Mathematica, byWhitehead and myself.
It gradually became clear that a great part of philosophycan be reduced to something that may be called "syntax",though the word has to be used in a somewhat wider sense thanhas hitherto been customary. Some men, notably Carnap, haveadvanced the theory that all philosophical problems arereally syntactical, and that, when errors in syntax areavoided, a philosophical problem is thereby either solved orshown to be insoluble. I think this is an overstatment, butthere can be no doubt that the utility of philosophicalsyntax in relation to traditional problems is very great.
I will illustrate its utility by a brief explanation ofwhat is called the theory of descriptions. Bya "description" I mean a phrase such as "The presentPresident of the United States", in which a person or thingis designated, not by name, but by some property which issupposed or known to be peculiar to him or it. Such phraseshad given a lot of trouble. Suppose I say "The goldenmountain does not exist" and suppose you ask "What is it thatdoes not exist?" It would seem that, if I say "It is thegolden mountain", I am attributing some sort of existence toit. Obviously I am not making the same statement as ifI said, "The round square does not exist". This seemed toimply that the golden mountain is one thing and the roundsquare is another, although neither exists. The theory ofdescriptions was designed to meet this and otherdifficulties.
According to this theory, when a statement containinga phrase of the form "the so-and-so" is rightly analyzed,the phrase "the so-and-so" disappears. For example, take thestatement "Scott was the author of "Waverley". The theoryinterprets this statement as saying:
"One and the only one man wrote Waverley, and that man wasScott". Or, more fully:
"There is an entity c such that the statement 'x wroteWaverley' is true if x is c and false otherwise; moreover cis Scott".
The first part of this, before the word "moreover", isdefined as meaning: "The author of Waverley exist (or existedor will exist)". Thus "The golden mountain does not exist"means:
"There is no entity c such that 'x is golden andmountainous' is true when x is c, but not otherwise".
With this definition the puzzle as to what is meant when wesay "The golden mountain does not exist" disappears.
"Existence", according to this theory, can only be assertedof descriptions. We can say "The author of Waverley exists",but to say "Scott exists" is bad grammar, or rather badsyntax. This clears up two millennia of muddle-headless about"existence", beginning with Plato's Theateus.
One result of the work we have been considering is todethrone mathematics from the lofty place that it hasoccupied since Phythagoras and Plato, and to destroy thepresumption against empiricism which has been derived fromit. Mathematical knowledge, it is true, is not obtained byinduction from experience; our reason for believing that 2and 2 are 4 is not that we have so often found, byobservation, that one couple and another couple together makea quartet. In this sense, mathematical knowledge is still notempirical. But it is also not a priori knowledge about theworld. It is, in fact, merely verbal knowledge. "3" means "2+ 1" and "4" means "3 + 1". Hence it follows (though theproof is long) that "4" means the same as "2 + 2". Thusmathematical knowledge ceases to be mysterious. It is all ofthe same nature as the "great truth" that there are threefeet in a yard.
Physics, as well as pure mathematics, has suppliedmaterial for the philosophy of logical analysis. This hasoccurred especially through the theory of relativity andquantum mechanics.
What is important to the philosopher in the theory ofrelativity is the substitution of space-time for space andtime. Common sense thinks of the physical world as composedof "things" which persist through a certain period of timeand move in space. Philosophy and physics developed thenotion of "thing" into that of "material substance", andthought of material substance as consisting of particles,each very small, and each persisting throughout all time.Einstein substituted events for particles; each event had toeach other a relation called "interval", which could beanalyzed in various ways into a time-element andspace-element. The choice between these various ways wasarbitrary and no one of them was theoretically preferable toany other. Given two elements A and B, in different regions,it might happen that according to one convention they weresimultaneous, according to another A was earlier than B, andaccording to yet another B was earlier than A. No physicalfacts correspond to these different conventions.
From all this it seems to follow that events, notparticles, must be the "stuff" of physics. What has beenthought of as a particle will have to be thought of as aseries of events. The series of events that replaces aparticle has certain important physical properties, andtherefore demands our attention; but it has no moresubstantiality than any other series of events that we mightarbitrary single out. Thus "matter" is not part of theultimate material of the world, but merely a convenient wayof collecting events into bundles.
Quantum theory reinforces this conclusion, but its chiefphilosophical importance is that it regards physicalphenomena as possibly discontinuous. It suggests that, in anatom (interpreted as above), a certain state of affairspersists for a certain time, and then suddenly is replacedby a finitely different state of affairs. Continuity ofmotion, which had always been assumed, appears to have beena mere prejudice. The philosophy appropriate to quantumtheory, however, has not yet been adequately developed.I suspect that it will demand even more radical departuresfrom the traditional doctrine of space and time than thosedemanded by the theory of relativity.
While physics has been making matter less material,psychology has been making mind less mental. We had occasionin a former chapter to compare the association of ideas withthe conditioned reflex. The latter, which has replaced theformer, is much more physiological. (This is only oneillustration; I do not wish to exaggerate the scope of theconditioned reflex). Thus from both ends physics andpsychology have been approaching each other, and making morepossible the doctrine of "neutral monism" suggested byWilliam James's criticism of "consciousness". The distinctionof mind and matter came into philosophy form religion,although, for a long time, it seemed to have valid grounds.I think that both mind and matter are merely convenient waysof grouping events. Some single events, I should admit,belong only to material groups, but others belong to bothkinds of groups, and are therefore at once mental andmaterial. This doctrine effects a great simplification in ourpicture of the structure of the world.
Modern Physics and physiology throw a new light upon theancient problem of perception. If there is to be anythingthat can be called "perception", it must be in some degree aneffect of the object perceived, and must more or lessresemble the object if it is to be a source of knowledge ofthe object. The first requisite can only be fulfilled ifthere are causal chains which are, to a greater or lessextent, independent of the rest of the world. According tophysics, this is the case. Light-waves travel from the sun tothe earth, and in doing so obey their own laws. This is onlyroughly true. Einstein has shown that light-rays are affectedby gravitation. When they reach our atmosphere, they sufferrefraction, and some are more scattered than others. Whenthey reach a human eye, all sorts of things happen whichwould not happen elsewhere, ending up with what we call"seeing the sun". But although the sun of our visualexperience is very different from the sun of the astronomer,it is still a source of knowledge as to the latter, because"seeing the sun" differs from "seeing the moon" in ways thatare causally connected with the difference between theastronomer's sun and the astronomer's moon. What we can knowof physical objects in this way, however, is only certainabstract properties of structure. We can know that the sun isround in a sense, though not quite the sense in which what wesee is round; but we have no reason to suppose that it isbright or warm, because physics can account for its seemingso without supposing that it is so. Our knowledge of thephysical world, therefore, is only abstract and mathematical.
Modern analytical empiricism, of which I have beengiving an outline, differs from that of Locke, Berkeley, andHume by its incorporation of mathematics and its developmentof powerful logical technique. It is thus able, in regard tocertain problems, to achieve definite answers, which havethe quality of science rather that of philosophy. It has theadvantage, as compared with the philosophies of thesystem-builders, of being able to tackle its problems one ata time, instead of having to invent at one stroke a blocktheory of the whole universe. Its methods, in this respect,resemble those of science. I have no doubt that, in so faras philosophical knowledge is possible, it is by suchmethods, many ancient problems are completely soluble.
There remains, however, a vast field, traditionallyincluded in philosophy, where scientific methods areinadequate. This field includes ultimate questions ofvalue; science alone, for example, cannot prove that it isbad to enjoy the infliction of cruelty. Whatever can beknown, can be known by means of science; but things whichare legitimately matters of feeling lie outside itsprovince.
Philosophy, throughout its history, has consisted of twoparts inharmoniously blended; on the one hand a theory as tothe nature of the world, on the other an ethical orpolitical doctrine as to the best way of living. The failureto separate these two with sufficient clarity has been asource of much confused thinking. Philosophers, from Platoto William James, have allowed their opinions as to theconstitution of the universe to be influenced by the desirefor edificacion: knowing, as they supposed, what beliefswould make men virtuous, they have invented arguments, oftenvery sophistical, to prove that these beliefs are true. Formy part I reprobate this kind of bias, both on moral and onintellectual grounds. Morally, a philosopher who uses hisprofessional competency for anything except a disinterestedsearch for truth is guilty of a kind of treachery. And whenhe assumes, in advance of inquiry, that certain beliefs,whether true or false, are such as to promote good behavior,he is so limiting the scope of philosophical speculation asto make philosophy trivial; the true philosopher is preparedto examine all preconceptions. When any limits are placed,consciously or unconsciously, upon the pursuit of truth,philosophy becomes paralyzed by fear, and the ground isprepared for a government censorship punishing those whoutter "dangerous thoughts" - in fact, the philosopher hasalready places such a censorship over his own investigations.
Intellectually, the effect of mistaken moralconsiderations upon philosophy has been to impede progress toan extraordinary extent. I do not myself believe thatphilosophy can either prove or disprove the truth ofreligious dogmas, but ever since Plato most philosophers haveconsidered it part of their business to produce "proofs" ofimmortality and the existence of God. They have found faultwith the proofs of their predecessors - Saint Thomas rejectedSaint Anselm's proofs, and Kant rejected Descartes' - butthey have supplied new ones of their own. In order to maketheir proofs seem valid, they have had to falsify logic, tomake mathematics mystical, and to pretend that deep-seatedprejudices were heaven-sent intuitions.
All this is rejected by the philosophers who make logicalanalysis the main business of philosophy. They confessfrankly that the human intellect is unable to find conclusiveanswers to many questions of profound importance to mankind,but they refuse to believe that there are some "higher" wayof knowing, by which we can discover truths hidden fromscience and the intellect. For this renunciation they havebeen rewarded by the discovery that many questions, formerlyobscured by the fog of metaphysics, can be answered withprecision, and by objective methods which introduce nothingof the philosopher's temperament except the desire tounderstand. Take such questions as: What is number? What arespace and time? What is mind, and what is matter? I do notsay that a method has been discovered by which, as inscience, we can make successive approximations to the truth,in which each new stage results from an improvment, nota rejection, of what has gone before.
In the welter of conflicting fanaticisms, one of the fewunifying forces is scientific truthfulness, by which i meanthe habit of basing our beliefs upon observations andinferences as impersonal, and as much diverse of local andtemperamental bias, as is possible for human beings. To haveinsisted upon the introduction of his virtue into philosophy,and to have invented a powerful method by which it can berendered fruitful, are the chief merits of the philosophicalschool of which I am a member. The habit of careful veracityacquired in the practice of this philosophical method can beextended to the whole sphere of human activity, producing,wherever it exists, a lessening of fanaticism with anincreasing capacity of sympathy and mutual understanding. Inabandoning a part of its dogmatic pretensions, philosophydoes not cease to suggest and inspire a way of life.