MS 468-471 (1903) - Lowell Lecture V

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But it can assert what it asserts of any graph-replica you please so long as it be equivalent to a given graph replica.

The matter is unsuitable to presentation in a course like this, although it is indispensible that something should be said about it. If I can ever get the funds to print it, I shall be very happy to send copies to all who may care for them.

This matter of the graphs if I could present it would also illustrate the nature of Pure Mathematics, still better than it is illustrated on pages 20 to 22 of the Syllabus distributed last Thursday. Mathematics, in general, is tbe

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science of the logical possibility and impossibility of hypotheses. Upon that definition first put forth by my father Benjamin Peirce in 1870, mathematicians are now pretty unanimously agreed. Pure mathematics differs from mathematics in general in not admitting into its hypotheses any element that does affect their logical possibility or impossibility. Thus, the pure mathematics of graphs, as you see in those pages of the syllabus, says nothing at all about the logical interpretation of graphs but defines them exclusively by their logical relations to one another. So the pure mathematical presentation of

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of graphs of graphs says nothing at all about the graphs considered being representations of graphs but merely defines the graphs representing the sheet of assertion, a cut, its area, a line of identity, and so forth, in terms of their logical relations to one another.

The theory of graphs of graphs rests upon a larger number of independent hypotheses than do most branches of pure mathematics. I divide these hypotheses into those which relate to physical possibilities and necessities and those which relate to permissions and prohibitions. The former class, about thirty in number, are all expressible in alpha and beta graphs.

The latter class which d.o not much exceed half a dozen require, one and all of them, gamma graphs to express them. In particular the broken cut enters into every one, and into some of them with curious complications.

One is surprised to find how extremely complicated are the graphs that express some of these fundamentals of the nature of graphs; and this complication puts into a strong light two merits of this system of existential graphs. For, on the one hand, it shows the extraordinary analytical power of the system that analyzes conditions which seemed on our first

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acquaintance with them to be very very simple but which the system of graphs forces us to see are in reality very complex; while on the other hand, this same complication shows that the system of existential graphs possesses to an extraordinary degree the virtue that belongs more or less to all diagrams that of putting a matter a matter really extremely complicated into a light under which it is fully and adequately represented and yet seems as easy and natural as slipping off a floating log.

Now to Vol 1 p 2.