Logic, Mathematics and

Knowledge of NatureBY HANS HAHN (TRANSLATED BY ARTHUR PAP)1

EVEN A CURSORY glance at the statements of physics showsthat they are obviously of a very diverse character. There arestatements like "if a stretched string is plucked, a tone is heard"or "if a ray of sunlight is passed through a glass prism, then acolored band, interspersed with dark lines, is visible on ascreen placed behind the prism," which can be tested at anytime by observation. We also find statements like "the suncontains hydrogen," "the satellite of Sirius has a density ofabout 60,000," "a hydrogen atom consists of a positivelycharged nucleus around which a negatively charged electronrevolves," which cannot by any means be tested by immediateobservation but which are made only on the basis of theoreticalconsiderations and likewise are testable only with the help oftheoretical considerations. And thus we are confronted by theurgent question: what is the relationship between observationand theory in physics—and not just in physics, but in sciencegenerally. For there is but one science, and wherever there isscientific investigation it proceeds ultimately according to thesame methods; only we see everything with the greatest clarity 1 This contribution comprises the first four sections of the pamphlet "Logik,Mathematik und Naturerkennen," published in Vienna in 1933 as the secondvolume of the series entitled "Einheitswissenschaft." The last two sectionsof Hahn's pamphlet which are omitted do not deal with the nature of logicalor mathematical propositions.

in the case of physics, because it is the most advanced, neatest,most scientific of all the sciences. And in physics, indeed, theinteraction of observation and theory is especially pronounced,even officially recognized by the institution of specialprofessorships for experimental physics and for theoreticalphysics.Now, presumably the usual conception is roughly speaking thefollowing: we have two sources of knowledge, by means ofwhich we comprehend "the world," "the reality" in which weare "placed": experience, or observation on the one hand, andthinking on the other. For example, one is engaged inexperimental physics or in theoretical physics according toone's using the one or the other of these sources of knowledgein physics.Now, in philosophy we find a time-honored controversy aboutthese two sources of knowledge: which parts of our knowledgederive from observation, are "a posteriori," and which derivefrom thinking, are "a priori"? Is one of these sources ofknowledge superior to the others, and if so, which?From the very beginning philosophy has raised doubts aboutthe reliability of observation (indeed, these doubts are perhapsthe source of all philosophy). It is quite understandable thatsuch doubts arose: they spring from the belief that sense-perception is frequently deceptive. At sunrise or at sunset thesnow on distant mountains appears red, but "in reality" it issurely white! A stick which is immersed in water appearscrooked, but "in reality" it is surely straight! If a man recedesfrom me, he appears smaller and smaller to me, but surely hedoes not change size "in reality"!Now, although all the phenomena to which we have been refer-ring have long since been accounted for by physical theories,

so that nobody any longer regards them as deceptions causedby sense-perception, the consequences which flow from thisprimitive, long discarded conception still exert a powerfulinfluence. One says: if observation is sometimes deceptive,perhaps it is always so! Perhaps everything disclosed by thesenses is mere illusion! Everybody knows the phenomenon ofdreams, and everybody knows how difficult it is at times todecide whether a given experience was "real life" or "a meredream." Perhaps, then, whatever we observe is merely a dreamobject! Everybody knows that hallucinations occur, and thatthey can be so vivid that the subject cannot be dissuaded fromtaking his hallucination for reality. Perhaps then, whatever weobserve is only a hallucination! If we look throughappropriately polished lenses, everything appears distorted;who knows whether perhaps we do not always, unknowingly,look at the world as it were through distorting glasses, andtherefore see everything distorted, different from what it reallyis! This is one of the basic themes of the philosophy of Kant.

But let us return to antiquity. As we said, the ancients believedthat they were frequently deceived by observation. But nothingof this kind ever happened in the case of thought: there wereplenty of delusions of sense, but no delusions of thought! Andthus, as confidence in observation got shaken, the belief mayhave arisen that thinking is a method of knowledge which isabsolutely superior to observation, indeed the only reliablemethod of knowledge: observation discloses mere appearance,thought alone grasps true being.This, "rationalistic," doctrine that thinking is a source ofknowledge which surpasses observation, that it is indeed theonly reliable source of knowledge, has remained dominant

from the climax of Greek philosophy until modern times. Icannot even intimate what peculiar fruits matured on the tree ofsuch knowledge. At any rate, they proved to haveextraordinarily little nourishing value; and thus the "empiricist"reaction, originating in England, slowly gained the upper hand,supported by the tremendous success of modern naturalscience—the philosophy which teaches that observation issuperior to thought, indeed is the only source of knowledge:nihil est in intellectu, quod non prius juerit in sensu; in English:"nothing is in the intellect which was not previously in thesenses."But at once this empiricism faces an apparently insuperabledifficulty: how is it to account for the real validity of logicaland mathematical statements? Observation discloses to meonly the transient, it does not reach beyond the observed; thereis no bond that would lead from one observed fact to another,that would compel future observations to have the same resultas those already made. The laws of logic and mathematics,however, claim absolutely universal validity: that the door ofmy room is now closed, I know by observation; next time Iobserve it it may be open. That heated bodies expand, I knowby observation; yet the very next observation may show thatsome heated body does not expand; but that two and two makefour, holds not only for the case in which I verify it by countingI know with certainty that it holds always and everywhere.Whatever I know by observation could be otherwise: the doorof my room might have been open now, I can easily imagine it;and I can easily imagine that a body does not expand on beingheated; but two and two could not occasionally make five, Icannot imagine in any way what it would be like for twice twoto equal five.

The conclusion seems inevitable: since the propositions oflogic and mathematics have absolutely universal validity, areapodeictically certain, since it must be as they say and cannotbe otherwise, these propositions cannot be derived fromexperience. In view of the tremendous importance of logic andmathematics in the system of our knowledge, empiricism,therefore, seems to be irrevocably refuted. To be sure, in spiteof all this the older empiricists have attempted to found logicand mathematics upon experience. According to them we nowbelieve that something must be this way and cannot beotherwise simply because the relevant experience is so old andthe relevant observations have been repeated innumerabletimes. On this view, therefore, it is entirely conceivable that,just as an observation might show that a heated body does notexpand, two and two might sometimes make five. This isalleged to have escaped our notice so far because it happenswith such extraordinary rarity, like finding a piece of four-leafed clover which for superstitious people is a sign of goodluck, an occurrence which is not so very rare—how much morepromise of fortune would there be in the discovery of a casewhere two and two make five! One can safely say that oncloser sight these attempts to derive logic and mathematicsfrom experience are fundamentally unsatisfactory, and it isdoubtful whether anybody seriously holds this view today.Rationalism and empiricism having thus, as it were, sufferedship-wreck—rationalism, because its fruits lacked nourishingvalue, empiricism, because it could not do justice to logic andmathematics —dualistic conceptions gained the upper hand,with the view that thinking and observation are equallylegitimate sources of knowledge which are both indispensableto our comprehension of the world and play a distinctive role in

the system of our knowledge. Thought grasps the most generallaws of all being, as formulated perhaps in logic andmathematics; observation provides the detailed filling of thisframework. As regards the limits set to the two sources ofknowledge, opinions diverge.Thus it is, for instance, disputed whether geometry is a priori ora posteriori, whether it is based on pure thinking or onexperience. And the same dispute is encountered in connectionwith the most fundamental physical laws, e.g. the law ofinertia, the laws of the conservation of mass and energy, thelaw of attraction of masses: all of them have already beenacclaimed as a priori, as necessities of thought, by variousphilosophers—but always after they had been established andwell confirmed as empirical laws in physics. This was bound tolead to a skeptical attitude, and as a matter of fact there isprobably a prevalent tendency among physicists to regard theframework which can be grasped by pure thinking as being aswide and general as possible, and to acknowledge experienceas the source of our knowledge of everything that is somehowconcrete.The usual conception, then, may be described roughly asfollows: from experience we learn certain facts, which weformulate as "laws of nature"; but since we grasp by means ofthought the most general lawful connections (of a logical andmathematical character) that pervade reality, we can controlnature on the basis of facts disclosed by observation to a muchlarger extent than it has actually been observed. For we knowin addition that anything which can be deduced from observedfacts by application of logic and mathematics must be found toexist. According to this view, the experimental physicistprovides knowledge of laws of nature by direct observation.

The theoretical physicist thereafter enlarges this knowledgetremendously by thinking, in such a way that we are in aposition also to assert propositions about processes that occurfar from us in space and time and about processes which, onaccount of their magnitude or minuteness, are not directlyobservable but which are connected with what is directlyobserved by the most general laws of being, grasped bythought, the laws of logic and mathematics. This view seems tobe strongly supported by numerous discoveries that have beenmade with the help of theory, like—to mention just some of thebest known—the calculation of the position of the planetNeptune by Leverrier, the calculation of electric waves byMaxwell, the calculation of the bending of light rays in thegravitational field of the sun by Einstein and the calculation ofthe red-shift in the solar spectrum, also by Einstein.Nevertheless we are of the opinion that this view is entirelyuntenable. For on closer analysis it appears that the function ofthought is immeasurably more modest than the one ascribed toit by this theory. The idea that thinking is an instrument forlearning more about the world than has been observed, foracquiring knowledge of something that has absolute validityalways and everywhere in the world, an instrument forgrasping general laws of all being, seems to us whollymystical. Just how should it come to pass that we could predictthe necessary outcome of an observation before having madeit? Whence should our thinking derive an executive power, bywhich it could compel an observation to have this rather thanthat result? Why should that which compels our thoughts alsocompel the course of nature? One would have to believe insome miraculous pre-established harmony between the course

of our thinking and the course of nature, an idea which ishighly mystical and ultimately theological.There is no way out of this situation except a return to a purelyempiricist standpoint, to the view that observation is the onlysource of knowledge of facts: there is no a priori knowledgeabout matters of fact, there is no "material" a priori. However,we shall have to avoid the error committed by earlierempiricists, that of interpreting the propositions of logic andmathematics as mere facts of experience. We must look out fora different interpretation of logic and mathematics.

IILet us begin with logic. The old conception of logic isapproximately as follows: logic is the account of the mostuniversal properties of things, the account of those propertieswhich are common to all things; just as ornithology is thescience of birds, zoology the science of all animals, biology thescience of all living beings, so logic is the science of all things,the science of being as such. If this were the case, it wouldremain wholly unintelligible whence logic derives its certainty.For we surely do not know all things. We have not observedeverything and hence we cannot know how everythingbehaves.Our thesis, on the contrary, asserts: logic does not by anymeans treat of the totality of things, it does not treat of objectsat all but only of our way of speaking about objects; logic isfirst generated by language. The certainty and universalvalidity, or better, the irrefutability of a proposition of logicderives just from the fact that it says nothing about objects ofany kind.

Let us clarify the point by an example. I talk about a well-known plant: I describe it, as is done in botanical referencebooks, in terms of the number, color and form of its blossomleaves, its calyx leaves, its stamina, the shape of its leaves, itsstem, its root, etc., and I make the stipulation: let us call anyplant of this kind "snow rose," but let us also call it "helleborusniger." Thereupon I can pronounce with absolute certainty theuniversally valid proposition: "every snow rose is a helleborusniger." It is certainly valid, always and everywhere; it is notrefutable by any sort of observation; but it says nothing at allabout facts. I learn nothing from it about the plant in question,when it is in bloom, where it may be found, whether it iscommon or rare. It tells me nothing about the plant; it cannotbe disconfirmed by any observation. This is the basis of itscertainty and universal validity. The statement merelyexpresses a convention concerning the way we wish to talkabout the plant in question.

Similar considerations apply to the principles of logic. Let usmake the point with reference to the two most famous laws oflogic: the law of contradiction and the law of the excludedmiddle. Take, for example, colored objects. We learn, bytraining as I am tempted to say, to apply the designation "red"to some of these objects, and we stipulate that the designation"not red" be applied to all other objects. On the basis of thisstipulation we now can assert with absolute certainty theproposition that there is no object to which both the designation"red" and the designation "not red" is applied. It is customaryto formulate this briefly by saying that nothing is both red andnot red. This is the law of contradiction. And since we havestipulated that the designation "red" is to be applied to some

objects and the designation "not red" to all other objects, wecan likewise pronounce with absolute certainty the proposition:everything is either designated as "red" or as "not red," which itis customary to formulate briefly by saying that everything iseither red or not red. This is the law of the excluded middle.These two propositions, the law of contradiction and the law ofthe excluded middle, say nothing at all about objects of anykind. They do not tell me of any of them whether they are redor not red, which color they have, or anything else. Theymerely stipulate a method for applying the designations "red"and "not red" to objects, i.e. they prescribe a method ofspeaking about things. And their universal validity andcertainty, their irrefutability, just derives from the fact that theysay nothing at all about objects.The same is to be said of all the other principles of logic. Weshall presently return to this point. But first let us insert anotherconsideration. We have previously maintained that there can beno material a priori, i.e. no a priori knowledge about matters offact. For we cannot know the outcome of an observation beforethe latter takes place. We have made clear to ourselves that nomaterial a priori is contained in the laws of contradiction and ofexcluded middle, since they say nothing about facts. There arethose, how-ever, who would perhaps admit that the nature ofthe laws of logic is as described, yet would insist that there is amaterial a priori elsewhere, e.g. in the statement "nothing isboth red and blue" (of course what is meant is: at the same timeand place) which is alleged to express real a priori knowledgeabout the nature of things. Even before having made anyobservation, they say, one can predict with absolute certaintythat it will not disclose a thing which is both blue and red; andit is maintained that such a priori knowledge is obtained by

"eidetic insight" or an intuitive grasp of the essence of colors.If one desires to adhere to our thesis that there is no kind ofmaterial a priori, one must somehow face statements like"nothing is both blue and red." I want to attempt this in a fewsuggestive words, though they cannot by any means do fulljustice to this problem which is not easy. It surely is correctthat we can say with complete certainty before having madeany observations: the latter will not show that a thing is bothblue and red—just as we can say with complete certainty thatno observation will yield the result that a thing is both red andnot red, or that a snow rose is not a helleborus niger. The firststatement, however, is not a case of a material a priori anymore than the second and third. Like the statements "everysnow rose is a helleborus niger": and "nothing is both red andnot red," the statement "nothing is both blue and red" saysnothing at all about the nature of things; it likewise refers onlyto our proposed manner of speaking about objects, of applyingdesignations to them. Earlier we said: there are some objectsthat we call "red," every other object we call "not red," andfrom this we derive the laws of contradiction and excludedmiddle. Now we say: some objects we call "red," some otherobjects we call "blue," and other objects again we call "green,"etc. But if it is in this way that we ascribe color designations toobjects, then we can say with certainty in advance: in thisprocedure no object is designated both as "red" and as "blue,"or more briefly: no object is both red and blue. The reason whywe can say this with certainty is that we have regulated theascription of color designations to objects in just this way.We see, then, that there are two totally different kinds ofstatements: those which really say something about objects,and those which do not say anything about objects but only

stipulate rules for speaking about objects. If I ask "what is thecolor of Miss Erna's new dress?" and get the answer "MissErna's new dress is not both red and blue (all over)," then noinformation about this dress has been given to me at all. I havebeen made no wiser by it. But if I get the answer "Miss Erna'snew dress is red," then I have received some genuineinformation about the dress.Let us clarify this distinction in terms of one more example. Astatement which really says something about the objects whichit mentions, is the following: "If you heat this piece of iron upto 800°, it will turn red, if you heat it up to 1300°, it will turnwhite." What makes the difference between this statement andthe statements cited above, which say nothing about facts? Theapplication of temperature designations to objects isindependent of the application of color designations, whereasthe color designations "red" and "not red," or "red" and "blue"are applied to objects in mutual dependence. The statements"Miss Erna's new dress is either red or not red" and "MissErna's new dress is not both red and blue" merely express thisdependence, hence make no assertion about that dress, and arefor that reason absolutely certain and irrefutable.The above statement about the piece of iron, on the other hand,relates independently given designations, and therefore reallysays something about that piece of iron and is for just thatreason not certain nor irrefutable by observation.The following example may make the difference between thesetwo kinds of statements particularly clear. If someone were totell me: "I raised the temperature of this piece of iron to 800°but it did not turn red," then I would test his assertion; theresult of the test may be that he was lying, or that he was thevictim of an illusion, but perhaps it would turn out that—

contrary to my previous beliefs—there are cases where a pieceof iron heated to 800° does not become red-hot, and in thatcase I would just change my opinion about the reaction of ironto heating. But if someone tells me "I raised the temperature ofthis piece of iron to 800°, and this made it turn both red and notred" or "it became both red and white," then I will certainlymake no test whatever. Nor will I say "he has told me a lie," or"he has become the victim of an illusion" and it is quite certainthat I would not change my beliefs about the reaction of iron toheating. The point is—it is best to express it in language whichany card player is familiar with—that the man has revoked: hehas violated the rules in accordance with which we want tospeak, and I shall refuse to speak with him any longer. It is asthough one attempted in a game of chess to move the bishoporthogonally. In this case too, I would not make any tests, Iwould not change my beliefs about the behavior of things, but Iwould re-fuse to play chess with him any longer.To sum up: we must distinguish two kinds of statements: thosewhich say something about facts and those which merelyexpress the way in which the rules which govern theapplication of words to facts depend upon each other. Let uscall statements of the latter kind tautologies: they say nothingabout objects and are for this very reason certain, universallyvalid, irrefutable by observation; whereas the statements of theformer kind are not certain and are refutable by observation.The logical laws of contradiction and of the excluded middleare tautologies, likewise, e.g., the statement "nothing is bothred and blue."

And now we maintain that in the same way all the other lawsof logic are tautologies. Let us, therefore, return to logic once

more in order to clarify the matter by an example. As we said,the designation "red" is applied to certain objects and theconvention is adopted of applying the designation "not red" toany other object. It is this convention about the use of negationwhich is expressed by the laws of contradiction and of theexcluded middle. Now we add the convention—still taking ourexamples from the domain of colors—that any object which iscalled "red" is also to be called "red or blue," "blue or red,""red or yellow," "yellow or red," etc., that every object which iscalled "blue," is also called "blue or red," "red or blue," "blueor yellow," "yellow or blue," etc., and so on. On the basis ofthis convention, we can again assert with complete certaintythe proposition: "every red object is either red or blue." This isagain a tautology. We do not speak about the objects, but onlyabout our manner of talking about them.If once more we remind ourselves of the way in which thedesignations "red," "not red," "blue," "red or blue," etc. areapplied to objects, we can moreover assert with completecertainty and irrefutability: everything to which bothdesignations "red or blue" and "not red" are applied, is alsodesignated as "blue"—which is usually put more briefly: if athing is red or blue and not red, then it is blue. Which is again atautology. No information about the nature of things iscontained in it, it only expresses the sense in which the logicalwords "not" and "or" are used.Thus we have arrived at something fundamental: ourconventions regarding the use of the words "not" and "or" issuch that in asserting the two propositions "object A is eitherred or blue" and "object A is not red," I have implicitly alreadyasserted "object A is blue." This is the essence of so-calledlogical deduction. It is not, then, in any way based on real

connections between states of affairs, which we apprehend inthought. On the contrary, it has nothing at all to do with thenature of things, but derives from our manner of speakingabout things. A person who refused to recognize logicaldeduction would not thereby manifest a different belief frommine about the behavior of things, but he would refuse to speakabout things according to the same rules as I do. I could notconvince him, but I would have to refuse to speak with him anylonger, just as I should refuse to play chess with a partner whoinsisted on moving the bishop orthogonally.What logical deduction accomplishes, then, is this: it makes usaware of all that we have implicitly asserted—on the basis ofconventions regarding the use of language—in asserting asystem of propositions, just as, in the above example, "object Ais blue" is implicitly asserted by the assertion of the twopropositions "object A is red or blue" and "object A is not red."In saying this we have already suggested the answer to thequestion, which naturally must have forced itself on the mindof every reader who has followed our argument: if it is reallythe case that the propositions of logic are tautologies, that theysay nothing about objects, what purpose does logic serve?The logical propositions which were used as illustrationsderived from conventions about the use of the words "not" and"or" (and it can be shown that the same holds for all thepropositions of so-called prepositional logic). Let us, then, firstask for what purpose the words "not" and "or" are introducedinto language. Presumably the reason is that we are notomniscient. If I am asked about the color of the dress worn byMiss Erna yesterday, I may not be able to remember its color. Icannot say whether it was red or blue or green; but perhaps Iwill be able to say at least "it was not yellow." Were I

omniscient, I should know its color. There would be no need tosay "it was not yellow": I could say "it was red." Or again: mydaughter has written to me that she received a cocker-spaniel asa present. As I have not seen it yet, I do not know its color; Icannot say "it is black" nor "it is brown"; but I am able to say"it is black or brown." Were I omniscient, I could do withoutthis "or" and could say immediately "it is brown."Thus logical propositions, though being purely tautologous,and logical deductions, though being nothing but tautologicaltransformations, have significance for us because we are notomniscient. Our language is so constituted that in assertingsuch and such propositions we implicitly assert such and suchother propositions—but we do not see immediately all that wehave implicitly asserted in this manner. It is only logicaldeduction that makes us conscious of it. I assert, e.g., thepropositions "the flower which Mr. Smith wears in hisbuttonhole, is either a rose or a carnation," "if Mr. Smith wearsa carnation in his buttonhole, then it is white," "the flowerwhich Mr. Smith wears in his buttonhole is not white." PerhapsI am not consciously aware that I have implicitly asserted also"the flower which Mr. Smith wears in his buttonhole is a rose";but logical deduction brings it to my consciousness. To be sure,this does not mean that I know whether the flower which Mr.Smith wears in his buttonhole really is a rose; if I notice that itis not a rose, then I must not maintain my previous assertions—otherwise I sin against the rules of speaking, I revoke.

III

If I have succeeded in clarifying somewhat the role of logic, Imay now be quite brief about the role of mathematics. Thepropositions of mathematics are of exactly the same kind as thepropositions of logic: they are tautologous, they say nothing atall about the objects we want to talk about, but concern onlythe manner in which we want to speak of them. The reasonwhy we can assert apodeictically with universal validity theproposition: 2 + 3 = 5, why we can say even before anyobservations have been made, and can say it with completecertainty, that it will not turn out that 2 + 3 = 7, is that by "2 +3" we mean the same as by "5"—just as we mean the same by"helleborus niger" as by "snow rose." For this reason nobotanical investigation, however subtle, could disclose that aninstance of the species "snow rose" is not a helleborus niger.We become aware of meaning the same by "2 + 3" and by "5,"by going back to the meanings of "2," "3," "5," "+," andmaking tautological transformations until we just see that "2 +3" means the same as "5." It is such successive tautologicaltransformation that is meant by "calculating"; the operations ofaddition and multiplication which are learnt in school aredirectives for such tautological transformation; everymathematical proof is a succession of such tautologicaltransformations. Their utility, again, is due to the fact that, forexample, we do not by any means see immediately that wemean by "24 x 31" the same as by "744"; but if we calculatethe product "24 x 31," then we transform it step by step, insuch a way that in each individual transformation we recognizethat on the basis of the conventions regarding the use of thesigns involved (in this case numerals and the signs "+" and "x")what we mean after the transformation is still the same as what

we meant before it, until finally we become consciously awareof meaning the same by "744" as by "24 x 31."To be sure, the proof of the tautological character ofmathematics is not yet complete in all details. This is a difficultand arduous task; yet we have no doubt that the belief in thetautological character of mathematics is essentially correct.There has been prolonged opposition to the interpretation ofmathematical statements as tautologies; Kant contested thetautological character of mathematics emphatically, and thegreat mathematician Henri Poincare, to whom we are greatlyindebted also for philosophical criticism, went so far as toargue that since mathematics cannot possibly be a hugetautology, it must somewhere contain an a priori principle.Indeed, at first glance it is difficult to believe that the whole ofmathematics, with its theorems that it cost such labor toestablish, with its results that so often surprise us, should admitof being resolved into tautologies. But there is just one littlepoint which this argument overlooks: it overlooks the fact thatwe are not omniscient. An omniscient being, indeed, would atonce know everything that is implicitly contained in theassertion of a few propositions. It would know immediatelythat on the basis of the conventions concerning the use of thenumerals and the multiplication sign, "24 x 31" is synonymouswith "744." An omniscient being has no need for logic andmathematics. We ourselves, how-ever, first have to makeourselves conscious of this by successive tautologicaltransformations, and hence it may prove quite surprising to usthat in asserting a few propositions we have implicitly alsoasserted a proposition which seemingly is entirely differentfrom them, or that we do mean the same by two complexes ofsymbols which are externally altogether different.

IV

And now let us be clear what a world-wide difference there isbetween our conception and the traditional—perhaps one maysay: platonizing—conception, according to which the world ismade in accordance with the laws of logic and mathematics("God is perennially doing mathematics"), and our thinking, afeeble reflection of God's omniscience, is an instrument givento us for comprehending the eternal laws of the world. No! Ourthinking cannot give insight into any sort of reality. It cannotbring us information of any fact in the world. It only refers tothe manner in which we speak about the world. All it can do isto transform tautologically what has been said. There is nopossibility of piercing through the sensible world disclosed byobservation to a "world of true being": any metaphysics isimpossible! Impossible, not because the task is too difficult forour human thinking, but because it is meaningless, becauseevery attempt to do metaphysics is an attempt to speak in a waythat contravenes the agreement as to how we wish to speak,comparable to the attempt to capture the queen (in a game ofchess) by means of an orthogonal move of the bishop.

Let us return now to the problem which was our point ofdeparture: what is the relationship between observation andtheory in physics? We said that the usual view was roughlythis: experience teaches us the validity of certain laws ofnature, and since our thinking gives us insight into the mostgeneral laws of all being, we know that likewise anythingwhich is deducible from these laws of nature by means of

logical and mathematical reasoning must be found to exist. Wesee now that this view is untenable; for thinking does not graspany sort of laws of being. Never and nowhere, then, canthought supply us with knowledge about facts that goes beyondthe observed. But what, then, should we say about thediscoveries made by means of theory on which, as we pointedout, the usual view so strongly relies for its support? Let us askourselves, e.g., what was involved in the computation of theposition of the planet Neptune by Leverrier! Newton noticedthat the familiar motions, celestial as well as terrestrial, can bewell described in a unified way by the assumption that betweenany two mass points a force of attraction is exerted which isproportional to their masses and inversely proportional to thesquare of their distance. And it is because this assumptionenables us to give a satisfactory description of the familiarmotions, that he made it, i.e. he asserted tentatively, as anhypothesis, the law of gravitation: between any two masspoints there is a force of attraction which is proportional totheir masses and inversely proportional to the square of theirdistance. He could not pronounce this law as a certainty, butonly as an hypothesis. For nobody can know that such is reallythe behavior of every pair of mass points— nobody canobserve all mass points. But having asserted the law ofgravitation, one has implicitly asserted many otherpropositions, that is, all propositions which are deducible fromthe law of gravitation (together with data immediatelyderivable from observation) by calculation and logicalinference. It is the task of theoretical physicists andastronomers to make us conscious of everything we implicitlyassert along with the law of gravitation. And Leverrier'scalculations made people aware that the assertion of the law of

gravitation implies that at a definite time and definite place inthe heavens a hitherto unknown planet must be visible. Peoplelooked and actually saw that new planet—the hypothesis of thelaw of gravitation was confirmed. But it was not Leverrier'scalculation that proved that this planet existed, but the looking,the observation. This observation could just as well have had adifferent result. It could just as well have happened thatnothing was visible at the computed place in the heavens—inwhich case the law of gravitation would not have beenconfirmed and one would have begun to doubt whether it isreally a suitable hypothesis for the description of theobservable motions. Indeed, this is what actually happenedlater: in asserting the law of gravitation, one implicitly assertsthat at a certain time the planet Mercury must be visible at acertain place in the heavens. Whether it would actually bevisible at that time at that place, only observation coulddisclose; but observations showed that it was not visible atexactly the required position in the heavens. And whathappened? They said: since in asserting the law of gravitationwe implicitly assert propositions which are not true, we can-notmaintain the hypothesis of the law of gravitation. Newton'stheory of gravitation was replaced by Einstein's.It is not the case, then, that we know through experience thatcertain laws of nature are valid, and—since by our thinking wegrasp the most general laws of all being—therefore also knowthat whatever is deducible from these laws by reasoning mustexist. On the contrary, the situation is this: there is not a singlelaw of nature which we know to be valid; the laws of nature arehypotheses which we assert tentatively. But in asserting suchlaws of nature we implicitly assert also many otherpropositions, and it is the task of thinking to make us conscious

of the implicitly asserted propositions. So long, now, as theseimplicitly asserted propositions, to the extent that they areabout the directly observable, are confirmed by observation,these laws of nature are confirmed and we adhere to them; butif these implicitly asserted propositions are not confirmed byobservation, then the laws of nature have not been confirmedand are replaced by others.