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      This is that mathematics which distinguishes only two differentvalues, and is of great importance for logic.

      This is the system which has a scale of values of only twodegrees. Since these may be identified (in an application of thispure mathematical system) as thetrueand thefalse, this system callsfor somewhat elaborate study as a propaedeutic to logic.

      Such are number, multitude, limit, infinity, infinitesimals,continuity, dimension, imaginaries, multiple algebra, measurement,etc. My former contributions, though very fragmentary, have attractedattention in Europe, although in respect to priority justice has notbeen done them. I bring the whole together into one system, defendthe method of infinitesimals conclusively, and give many new truthsestablished by a new and striking method.

      My work in this direction is already somewhat known, althoughvery imperfectly. One of the learned academies of Europe has crowneda demonstration that my definition of a finite multitude agrees withDedekind's definition of an infinite multitude. It appears to me thatthe one is hardly more than a verbal modification of the other. I amusually represented as having put forth my definition as a substitutefor Dedekind's. In point of fact, mine was published six years beforehis; and my paper contains in very brief and crabbed form all theessentials of his beautiful exposition (still more perfect as modifiedby Schröder). Many animadversions have been made by eminent men uponmy remark, in theCentury Dictionary, that the method ofinfinitesimals is more consonant with then (in 1883) recent studies ofmathematical logic. In this memoir, I should show precisely how thecalculus may be, to the advantage of simplicity, based upon thedoctrine of infinitesimals. Many futile attempts have been made todefine continuity. In the sense in |209| the calculus, no difficultyremains. But the whole of topical geometry remains in an exceedinglybackward state and destitute of any method of proof simply becausetrue continuity has not been mathematically defined. By a carefulanalysis of the conception of acollection, of which no mathematicaldefinition has been yet published, I have succeeded in giving ademonstration of an important proposition which Cantor had missed,from which the required definition of a continuum results; and afoundation is afforded for topical geometry, which branch of geometryreally embraces the whole of geometry. I have made several otheradvances in defining the conceptions of mathematics which illuminatethe subject.

      I shall be glad to place early in the series so unquestionable anillustration of the great value of minute analysis as this memoir willafford. The subjects of corollarial and theorematic reasoning, of themethod of abstraction, of substantive possibility, |358| and of the methodof topical geometry, of which I have hitherto published mere hints,will here be fully elaborated.

      [This memoir] will examine the nature of mathematical reasoning.Logic can pass no judgment upon such reasoning, because it is evident,and as such, beyond all criticism. But logic is interested instudying how mathematical reasoning proceeds. Mathematical reasoningwill be analyzed and important properties of it brought out whichmathematicians themselves are not aware of.

      I have hitherto only published some slight hints of mydiscoveries in regard to the logical processes used in mathematics. Ifind that two different kinds of reasoning are used, which I |210|distinguish as thecorollarial and thetheorematic. This is a matterof extreme importance for the theory of cognition. It remainsunpublished. I also find that the most effective kind of theorematicdemonstration always involves the long despised operation ofabstraction, which has been a common topic of ridicule. This is theoperation by which we transform the proposition that "Opium putspeople to sleep" into the proposition that "Opium has a soporificvirtue". Like every other logical transformation, it can be appliedin a futile manner. But I show that, without it, the mathematicianwould be shut off from operations upon lines, surfaces, differentials,functions, operations—and even from the consideration of cardinalnumbers. I go on to define precisely what it is that this operationeffects. I endeavor in this paper to enumerate, classify, and definethe precise mode of effectiveness of every method employed inmathematics.

      No science of logic is needed for mathematics beyond that whichmathematics can itself supply, unless possibly it be in regard tomathematical heuretic. But the examination of the methods ofmathematical demonstration shed |91| extraordinary light upon logic, suchas I, for my part, never dreamed of in advance, although I ought tohave guessed that there must be unexpected treasures hidden in thisquite unexplored ground. That the logic of mathematics belonged tothe logic of relatives, and to the logic of triadic, not of dyadicrelations, was indeed obvious in advance; but beyond that I had noidea of its nature. The first things I found out were that allmathematical reasoning is diagrammatic and that all necessaryreasoning is mathematical reasoning, no matter how simple it may be.By diagrammatic reasoning, I mean reasoning which constructs a diagramaccording to a precept expressed in general terms, performsexperiments upon this diagram, notes their results, assures itselfthat similar experiments performed upon any diagram constructedaccording to the same precept would have |92| the same results, and expressesthis in general terms. This was a discovery of no little importance,showing, as it does, that all knowledge without exception comes fromobservation.

      At this point, I intend to insert a mention of my theory ofgrades of reality. The general notion is old, but in modern times ithas been forgotten. I undertake to prove its truth, resting on theprinciple that a theory which is adapted to the prediction ofobservational facts, and which does not lead to disappointment, isipso facto true. This principle is proved in No. 1. Then my proof ofgrades of reality is inductive, and consists in often turning aside inthe course of this series of memoirs to show how this theory isadapted to the expression of facts. This might be mistaken forrepetitiousness; but in fact it is logically defensible, and it alsohas the advantage of leading the reader, step by step, to thecompre|93|hension of an idea which he would not be able to grasp at once,and to the appreciation of an argument which he could not digest atone time. I will not here undertake to explain what the theory is indetail. Suffice it to say that since reality consists in this, that areal thing has whatever characters it has in its being and its havingthem does not consist in its being represented to have them, not evenin its representing itself to have them, not even if the characterconsists in the thing's representing itself to represent itself;since, I say, that is the nature of reality, as all schools ofphilosophy now admit, there is no reason in the nature of reality whyit should not have gradations of several kinds; and in point of fact,we find convincing evidences of such gradations. It is easy to seethat according to this definition the square root of minus 1 possessesa certain grade of |94| reality, since all its characters except only thatof being the square root of minus one are what they are whether you orI think so or not. So when Charles Dickens was half-through one ofhis novels, he could no longer make his characters do anything thatsome whim of a reader might suggest without feeling that it was false;and in point of fact the reader sometimes feels that the concludingparts of this or that novel of Dickens is false. Even here, then,there is an extremely low grade of reality. Everybody would admitthat the word might be applied in such cases by an apt metaphor; but Iundertake to show that there is a certain degree of sober truth in it,and that it is important for logic to recognize that the reality ofthe Great Pyramid, or of the Atlantic Ocean, or of the Sun itself, isnothing but a higher grade of the same thing.

      But to say that the reasoning of mathematics is |95| diagrammatic isnot to penetrate in the least degree into the logical peculiarities ofits procedure, because all necessary reasoning is diagrammatic.

      My first real discovery about mathematical procedure was thatthere are two kinds of necessary reasoning, which I call thecorollarial and the theorematic, because the corollaries affixed tothe propositions of Euclid are usually arguments of one kind, whilethe more important theorems are of the other. The peculiarity oftheorematic reasoning is that it considers something not implied atall in the conceptions so far gained, which neither the definition ofthe object of research nor anything yet known about could ofthemselves suggest, although they give room for it. Euclid, forexample, will add lines to |96| his diagram which are not at all required or suggested by any previous proposition, and which the conclusionthat he reaches by this means says nothing about. I show that noconsiderable advance can be made in thought of any kind withouttheorematic reasoning. When we come to consider the heuretic part ofmathematical procedure, the question how such suggestions are obtainedwill be the central point of the discussion.

      Passing over smaller discoveries, the principal result of mycloser studies of it has been the very great part which an operationplays in it which throughout modern times has been taken for nothingbetter than a proper butt of ridicule. It is the operation ofabstraction, in the proper sense of the term, which, for example,converts the |97| proposition "Opium puts people to sleep" into "Opium hasa dormitive virtue". This turns out to be so essential to the greaterstrides of mathematical demonstration that it is proper to divide alltheorematic reasoning into the non-abstractional and theabstractional. I am able to prove that the most practically importantresults of mathematics could not in any way be attained without thisoperation of abstraction. It is therefore necessary for logic todistinguish sharply between good abstraction and bad abstraction.

      It was not until I had been giving a large part of my time forseveral years to tracing out the ways in which mathematicaldemonstration makes use of abstraction that I came across a fact whicha mind which had not been scrutinizing the facts so closely |98| might haveseen long before, namely, that all collections are of the nature ofabstractions. When we pass from saying, "Almost any American canspeak English", to saying "The American nation is composed ofindividuals of whom the greater part speak English", we perform aspecial kind of abstraction. This can, I know, signify little to theperson who is not acquainted with the properties of abstraction. Itmay, however, suggest to him that the popular contempt for"abstractions" does not aim very accurately at its mark.

      When I published a paper about number in 1882, I was alreadylargely anticipated by Cantor, although I did not know it. I howeveranticipated Dedekind by about six years. Dedekind's work, althoughits form is admirable, has not influenced me. But ideas which I havederived from Cantor are so mixed up with ideas of my own that I couldnot safely undertake to say exactly where the line should be |99| drawnbetween what is Cantor's and what my own. From my point of view, itis not of much consequence. Like Cantor and unlike Dedekind, I beginwith multitude, or as Cantor erroneously calls it, cardinal number.But it would be equally correct, perhaps preferable, to begin withordinal number, as Dedekind does. But I pursue the method ofconsidering multitude to the very end, while Cantor switches off toordinal number. For that reason, it is difficult to make sure that myhigher multitudes are the same as his. But I have little doubt thatthey are. I prove that there is an infinite series of infinitemultitudes, apparently the same as Cantor'salephs. I call the firstthe denumerable multitude, the others the abnumerable multitudes, thefirst and least of which is the multitude of all the irrationalnumbers of analysis. There is nothing greater than these but truecontinua, which are not multitudes. I cannot see that Cantor has evergot the conception of a true continuum, such that in any |100| lapse of timethere is room for any multitude of instants however great.

      I show that every multitude is distinguished from all greatermultitudes by there being a way of reasoning about collections of thatmultitude which does not hold good for greater multitudes.Consequently, there is an infinite series of forms of reasoningconcerning the calculus which deals only with a collection of numbersof the first abnumerable multitude which are not applicable to truecontinua. This, it would seem, was a sufficient explanation of thecircumstance that mathematicians have never discovered any method ofreasoning about topical geometry, which deals with true continua.They have not really proved a single proposition in that branch ofmathematics.

      Cayley, while I was still a boy, proved that metrical geometry,the geometry of the elements, is nothing but a special |101| problem toprojective geometry, or perspective. It is easy to see thatprojective geometry is nothing but a special problem of topicalgeometry. On the other hand, since every relation can be reduced to arelation of serial order, something similar to a scale of values maybe applied to every kind of mathematics. Probably, if the appropriatescale were found, it would afford the best general method for thetreatment of any branch. We see, for example, the power of thebarycentric calculus in projective geometry. It is essentially themethod of modern analytic geometry. Yet it is evident that it is notaltogether an appropriate scale. I can already see some of thecharacters of an appropriate scale of values for topical geometry.

      My logical studies have already enabled me to prove somepropositions which had arrested mathematicians of power. Yet Idistinctly disclaim, for the present, all pretension to having beenremarkably successful in dealing with the heuretic |102| department ofmathematics. My attention has been concentrated upon the study of itsprocedure in demonstration, not upon its procedure in discoveringdemonstrations. This must come later; and it may very well be that Iam not so near to a thorough understanding of it as I may hope.

      I am quite sure that the value of what I have ascertained will beacknowledged by mathematicians. I shall make one more effort toincrease it, before writing this second memoir.

      I now pass to a rough statement of my results in regard to theheuretic branch of mathematical thought. At the outset, I set up formyself a sort of landmark by which to discern whether I was making anyreal progress or not. Cayley had shown, while I was, as a boy, justbeginning to understand such things, that metric geometry, thegeometry of theElements, is nothing but a special problem inprojective geometry, or perspective, and it is easy to see thatprojective geometry is nothing but a special problem in topicalgeometry. Now ma|130|thematicians are entirely destitute of any method ofreasoning about topical geometry. The 25th proposition of the 7thbook of theÉléments de Géométrie of Legendre, which is strictly allthat is known of the subject except some extensions of it, of whichthe chief is Listing's census-theorem, was demonstrated with extremedifficulty by Legendre, having exceeded the powers of Euler. Reallythe proof is not satisfactory, nor is Listing's. The simpleproposition that four colors suffice to color a map on a spheroid hasresisted the efforts of the greatest mathematicians. If, then,without particularly attending to that proposition or to topicalgeometry, I find that my studies of the method of discovering heureticmethods leads me naturally to the desired proof of the map-problem, Ishall know that I am making progress. From time to time, as Iadvanced, I have tried my hand at that problem. I have not |131| proved ityet, although the last time I tried I thought I had a proof, whichcloser examination proved to contain a flaw. Since then, I have made,as it seems to me, a considerable advance; but I have not been inducedto reexamine that subject, as I certainly should do if I were quiteconfident of being able to solve it with ease. I have, however,applied my logical theory directly with success to the demonstrationof several other propositions which had resisted powerfulmathematicians; and I have greatly improved upon Listing's theory; sothat I am confident that what I have found out is of value; and Ibelieve the same method has only to be pushed a little further tosolve the map-problem.

      I can show that numbers, whether integral, fractional, orirrational, have no other use or meaning than to say which of twothings comes earlier, which |133| later, in a serial arrangement. To ask,How much does this weigh? is answered as soon as we know what thingsamong those which concern us it is heavier than [and] what it islighter than. A system of measurement has no other purpose than that;and it appears to be the best artificial device for that purpose.

      But all relations whatever can be reduced to relations of serialorder; so that every mathematical question can be looked upon as ametrical question in a broad sense; and perhaps the best and readiestway to get command of a branch of mathematics is to find what systemof measurement is best adapted to it. Thus, the barycentric calculusapplies to projective geometry [considered as] a sort of measurement;and in fact modern analytic geometry results from just thatapplication. But it evidently labors under the difficulty of notbeing a sufficiently flexible and well-adapted system of measurement.Hermann Schubert'sCal|133|culus of Geometry gives some hint of what iswanted.