Here's the first instalment of the fourth Lowell Lecture of 1903. My transcription is of two manuscripts, Robin numbers 466 and 467. 467 is identified on its first page as Lecture 4, about the gamma part of existential graphs; but 466 is a bit of a puzzle, as it starts off "Ladies and Gentlemen," ends abruptly, doesn't connect coherently with 467, and doesn't seem to fit anywhere in the sequence. But it's interesting enough to be worth posting serially, I think. So that's where "Lowell 4" will begin. The MS is at https://www.fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-466-467-1903 -lowell-lecture-iv/display/13956
Gary f. Ladies and Gentlemen: Mathematics is the science which draws necessary conclusions. Such was the definition given first by my father as early as 1870. At that day, when the new mathematics was in its infancy, the novelty of this definition was disconcerting even to the profoundest mathematicians; but today nobody would propose a definition differing much from that. The only doubt that should entertain is whether we ought not to recognize as a part of mathematics, what is certainly a most important part of the mathematician's business, the formation of the assumptions on which his reasoning is to be based. Some of the mathematicians who have the most deeply studied the fundamentals of their science have even gone so far as to pronounce mathematics to be a branch of logic. Dedekind is one of these whose two little books published in one volume in translation by the Open Court Company, which is doing so much for American culture, I should strongly recommend to your attention. The fact that so profound a mathematician can hold this opinion is a sufficient justification of my devoting several lectures of this short course to a study of the nature of mathematics. I do not quite agree with Dedekind, myself; and I will tell you why presently. The question of whether mathematics was a branch of logic was the subject of careful discussion between my father and me at the time be had his definition under consideration. But first I had better notice an objection which will seem weighty to superficial minds. Namely, it will be said that much necessary reasoning is not at all mathematical. On that I take direct issue. Eminent jurists, moralists, and philosophers can be found whose powers of reasoning are famous, and who yet declare that they have no head for mathematics. This is, in part, a delusion owing to bad instruction which has given rise to such an aversion to everything that seems mathematical that as soon as one talks to them of x, y, z, they stop thinking. But what is also true of those persons is that they cannot hold clearly before their minds intricate relations between objects that are almost exactly alike except in respect to abstract relations. But when I ask one of those gentlemen to give me an example of a necessary reasoning that he considers not to be mathematical, it turns out to be one of those that are most readily amenable to mathematical representation, differing only from the reasoning he cannot grasp in its extreme simplicity. But that which conclusively stamps all necessary reasoning as mathematical is that in such reasoning, it makes not the slightest difference whether the premisses express observed facts (as strictly speaking they seldom do) or whether they describe wholly imaginary states of things. The conclusion follows as necessarily concerning the imaginary state of things as it would if that state of things had been observed. This, indeed, is precisely what the necessity of such reasoning consists in. For the purposes of the reasoning, therefore, the premisses are mere assumptions. If they happen to be more, it has nothing to do with the reasoning. Now the only science which deals with pure assumptions regardless of their real truth is mathematics. That is, on the whole, the best definition of mathematics. All necessary reasonings, therefore, are pieces of mathematics. http://gnusystems.ca/Lowell4.htm }{ Peirce's Lowell Lectures of 1903
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