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

      This is the system which has a scale of values of only two degrees. Since these may be identified (in an application of this pure mathematical system) as the true and the false, this system calls for 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 attracted attention in Europe, although in respect to priority justice has not been done them. I bring the whole together into one system, defend the method of infinitesimals conclusively, and give many new truths established by a new and striking method.

      My work in this direction is already somewhat known, although very imperfectly. One of the learned academies of Europe has crowned a demonstration that my definition of a finite multitude agrees with Dedekind's definition of an infinite multitude. It appears to me that the one is hardly more than a verbal modification of the other. I am usually represented as having put forth my definition as a substitute for Dedekind's. In point of fact, mine was published six years before his; and my paper contains in very brief and crabbed form all the essentials of his beautiful exposition (still more perfect as modified by Schröder). Many animadversions have been made by eminent men upon my remark, in the Century Dictionary, that the method of infinitesimals is more consonant with then (in 1883) recent studies of mathematical logic. In this memoir, I should show precisely how the calculus may be, to the advantage of simplicity, based upon the doctrine of infinitesimals. Many futile attempts have been made to define continuity. In the sense in |209| the calculus, no difficulty remains. But the whole of topical geometry remains in an exceedingly backward state and destitute of any method of proof simply because true continuity has not been mathematically defined. By a careful analysis of the conception of a collection, of which no mathematical definition has been yet published, I have succeeded in giving a demonstration of an important proposition which Cantor had missed, from which the required definition of a continuum results; and a foundation is afforded for topical geometry, which branch of geometry really embraces the whole of geometry. I have made several other advances in defining the conceptions of mathematics which illuminate the subject.

      I shall be glad to place early in the series so unquestionable an illustration of the great value of minute analysis as this memoir will afford. The subjects of corollarial and theorematic reasoning, of the method of abstraction, of substantive possibility, |358| and of the method of 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 in studying how mathematical reasoning proceeds. Mathematical reasoning will be analyzed and important properties of it brought out which mathematicians themselves are not aware of.

      I have hitherto only published some slight hints of my discoveries in regard to the logical processes used in mathematics. I find that two different kinds of reasoning are used, which I |210| distinguish as the corollarial and the theorematic. This is a matter of extreme importance for the theory of cognition. It remains unpublished. I also find that the most effective kind of theorematic demonstration always involves the long despised operation of abstraction, which has been a common topic of ridicule. This is the operation by which we transform the proposition that "Opium puts people to sleep" into the proposition that "Opium has a soporific virtue". Like every other logical transformation, it can be applied in a futile manner. But I show that, without it, the mathematician would be shut off from operations upon lines, surfaces, differentials, functions, operations—and even from the consideration of cardinal numbers. I go on to define precisely what it is that this operation effects. I endeavor in this paper to enumerate, classify, and define the precise mode of effectiveness of every method employed in mathematics.

      No science of logic is needed for mathematics beyond that which mathematics can itself supply, unless possibly it be in regard to mathematical heuretic. But the examination of the methods of mathematical demonstration shed |91| extraordinary light upon logic, such as I, for my part, never dreamed of in advance, although I ought to have guessed that there must be unexpected treasures hidden in this quite unexplored ground. That the logic of mathematics belonged to the logic of relatives, and to the logic of triadic, not of dyadic relations, was indeed obvious in advance; but beyond that I had no idea of its nature. The first things I found out were that all mathematical reasoning is diagrammatic and that all necessary reasoning is mathematical reasoning, no matter how simple it may be. By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, assures itself that similar experiments performed upon any diagram constructed according to the same precept would have |92| the same results, and expresses this in general terms. This was a discovery of no little importance, showing, as it does, that all knowledge without exception comes from observation.

      At this point, I intend to insert a mention of my theory of grades of reality. The general notion is old, but in modern times it has been forgotten. I undertake to prove its truth, resting on the principle that a theory which is adapted to the prediction of observational facts, and which does not lead to disappointment, is ipso facto true. This principle is proved in No. 1. Then my proof of grades of reality is inductive, and consists in often turning aside in the course of this series of memoirs to show how this theory is adapted to the expression of facts. This might be mistaken for repetitiousness; but in fact it is logically defensible, and it also has the advantage of leading the reader, step by step, to the compre|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 at one time. I will not here undertake to explain what the theory is in detail. Suffice it to say that since reality consists in this, that a real thing has whatever characters it has in its being and its having them does not consist in its being represented to have them, not even in its representing itself to have them, not even if the character consists in the thing's representing itself to represent itself; since, I say, that is the nature of reality, as all schools of philosophy now admit, there is no reason in the nature of reality why it should not have gradations of several kinds; and in point of fact, we find convincing evidences of such gradations. It is easy to see that according to this definition the square root of minus 1 possesses a certain grade of |94| reality, since all its characters except only that of being the square root of minus one are what they are whether you or I think so or not. So when Charles Dickens was half-through one of his novels, he could no longer make his characters do anything that some whim of a reader might suggest without feeling that it was false; and in point of fact the reader sometimes feels that the concluding parts of this or that novel of Dickens is false. Even here, then, there is an extremely low grade of reality. Everybody would admit that the word might be applied in such cases by an apt metaphor; but I undertake 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 of the Great Pyramid, or of the Atlantic Ocean, or of the Sun itself, is nothing but a higher grade of the same thing.

      But to say that the reasoning of mathematics is |95| diagrammatic is not to penetrate in the least degree into the logical peculiarities of its procedure, because all necessary reasoning is diagrammatic.

      My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the corollarial and the theorematic, because the corollaries affixed to the propositions of Euclid are usually arguments of one kind, while the more important theorems are of the other. The peculiarity of theorematic reasoning is that it considers something not implied at all in the conceptions so far gained, which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it. Euclid, for example, will add lines to |96| his diagram which are not at all required or suggested by any previous proposition, and which the conclusion that he reaches by this means says nothing about. I show that no considerable advance can be made in thought of any kind without theorematic reasoning. When we come to consider the heuretic part of mathematical procedure, the question how such suggestions are obtained will be the central point of the discussion.

      Passing over smaller discoveries, the principal result of my closer studies of it has been the very great part which an operation plays in it which throughout modern times has been taken for nothing better than a proper butt of ridicule. It is the operation of abstraction, in the proper sense of the term, which, for example, converts the |97| proposition "Opium puts people to sleep" into "Opium has a dormitive virtue". This turns out to be so essential to the greater strides of mathematical demonstration that it is proper to divide all theorematic reasoning into the non-abstractional and the abstractional. I am able to prove that the most practically important results of mathematics could not in any way be attained without this operation of abstraction. It is therefore necessary for logic to distinguish sharply between good abstraction and bad abstraction.

      It was not until I had been giving a large part of my time for several years to tracing out the ways in which mathematical demonstration makes use of abstraction that I came across a fact which a mind which had not been scrutinizing the facts so closely |98| might have seen long before, namely, that all collections are of the nature of abstractions. When we pass from saying, "Almost any American can speak English", to saying "The American nation is composed of individuals of whom the greater part speak English", we perform a special kind of abstraction. This can, I know, signify little to the person who is not acquainted with the properties of abstraction. It may, 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 already largely anticipated by Cantor, although I did not know it. I however anticipated Dedekind by about six years. Dedekind's work, although its form is admirable, has not influenced me. But ideas which I have derived from Cantor are so mixed up with ideas of my own that I could not safely undertake to say exactly where the line should be |99| drawn between what is Cantor's and what my own. From my point of view, it is not of much consequence. Like Cantor and unlike Dedekind, I begin with multitude, or as Cantor erroneously calls it, cardinal number. But it would be equally correct, perhaps preferable, to begin with ordinal number, as Dedekind does. But I pursue the method of considering multitude to the very end, while Cantor switches off to ordinal number. For that reason, it is difficult to make sure that my higher multitudes are the same as his. But I have little doubt that they are. I prove that there is an infinite series of infinite multitudes, apparently the same as Cantor's alephs. I call the first the denumerable multitude, the others the abnumerable multitudes, the first and least of which is the multitude of all the irrational numbers of analysis. There is nothing greater than these but true continua, which are not multitudes. I cannot see that Cantor has ever got the conception of a true continuum, such that in any |100| lapse of time there is room for any multitude of instants however great.

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

      Cayley, while I was still a boy, proved that metrical geometry, the geometry of the elements, is nothing but a special |101| problem to projective geometry, or perspective. It is easy to see that projective geometry is nothing but a special problem of topical geometry. On the other hand, since every relation can be reduced to a relation of serial order, something similar to a scale of values may be applied to every kind of mathematics. Probably, if the appropriate scale were found, it would afford the best general method for the treatment of any branch. We see, for example, the power of the barycentric calculus in projective geometry. It is essentially the method of modern analytic geometry. Yet it is evident that it is not altogether an appropriate scale. I can already see some of the characters of an appropriate scale of values for topical geometry.

      My logical studies have already enabled me to prove some propositions which had arrested mathematicians of power. Yet I distinctly disclaim, for the present, all pretension to having been remarkably successful in dealing with the heuretic |102| department of mathematics. My attention has been concentrated upon the study of its procedure in demonstration, not upon its procedure in discovering demonstrations. This must come later; and it may very well be that I am 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 be acknowledged by mathematicians. I shall make one more effort to increase it, before writing this second memoir.

      I now pass to a rough statement of my results in regard to the heuretic branch of mathematical thought. At the outset, I set up for myself a sort of landmark by which to discern whether I was making any real progress or not. Cayley had shown, while I was, as a boy, just beginning to understand such things, that metric geometry, the geometry of the Elements, is nothing but a special problem in projective geometry, or perspective, and it is easy to see that projective geometry is nothing but a special problem in topical geometry. Now ma|130|thematicians are entirely destitute of any method of reasoning about topical geometry. The 25th proposition of the 7th book of the Éléments de Géométrie of Legendre, which is strictly all that is known of the subject except some extensions of it, of which the chief is Listing's census-theorem, was demonstrated with extreme difficulty by Legendre, having exceeded the powers of Euler. Really the proof is not satisfactory, nor is Listing's. The simple proposition that four colors suffice to color a map on a spheroid has resisted the efforts of the greatest mathematicians. If, then, without particularly attending to that proposition or to topical geometry, I find that my studies of the method of discovering heuretic methods leads me naturally to the desired proof of the map-problem, I shall know that I am making progress. From time to time, as I advanced, I have tried my hand at that problem. I have not |131| proved it yet, although the last time I tried I thought I had a proof, which closer examination proved to contain a flaw. Since then, I have made, as it seems to me, a considerable advance; but I have not been induced to reexamine that subject, as I certainly should do if I were quite confident of being able to solve it with ease. I have, however, applied my logical theory directly with success to the demonstration of several other propositions which had resisted powerful mathematicians; and I have greatly improved upon Listing's theory; so that I am confident that what I have found out is of value; and I believe the same method has only to be pushed a little further to solve the map-problem.

      I can show that numbers, whether integral, fractional, or irrational, have no other use or meaning than to say which of two things comes earlier, which |133| later, in a serial arrangement. To ask, How much does this weigh? is answered as soon as we know what things among those which concern us it is heavier than [and] what it is lighter 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 serial order; so that every mathematical question can be looked upon as a metrical question in a broad sense; and perhaps the best and readiest way to get command of a branch of mathematics is to find what system of measurement is best adapted to it. Thus, the barycentric calculus applies to projective geometry [considered as] a sort of measurement; and in fact modern analytic geometry results from just that application. But it evidently labors under the difficulty of not being a sufficiently flexible and well-adapted system of measurement. Hermann Schubert's Cal|133|culus of Geometry gives some hint of what is wanted.