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or general tool of reasonging, but is only meant for use in the study of logic, where it quite infrequently happens that it is necessary very complicated graph is alpha-possible ,the waste of tie due to this being teh siplest possible, is not of much importance.

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formulate a similar routine but it would be utterly impracticable owing to the stupdendous complications that it would lead to. For example, here is a graph, not particularly intricate:

yet from this graph as a premiss no less than ninety entirely independent conclusions can be drawn, showing the misconception inolved in its popular expression, "the" conclusion "from given premisses. omreover, none of these 90 conclusions uses any part of the assertion of the graph more than once. By repeated making use of teh same assertion, any

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[?]

The short theory of numbers depend upon five premised represented in this graph

(Five graphs drawn with the letters u, x and possibly the number 0 or letter o and red lines)

The red ligatures refer to numbers, the brown ligaturesto any universe such that there is a relation that u may be understood to express that will make the entire graph true.

x is to understood to be replaceable by [mon?] graph whatever without altering the truth of the [?] entire graph.

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1) [lemmas?], and 27 [corollsnes?]. There are also 2 [seh?].At the sme time, I ought to remark that, while the possible cconclusions are innumerable, yet after all premises have been iterated so to as to exhaust all the different ways of using them together with all simpler ways, there will be no more theorems of any particular interest and the branch of mathematics in question may be said to be substantially exhausted. The theory of [uni?] is an instance. The great geometer Charles after he attained to a great age continued to [g?] out by the bushel basket full, such theorems as that (in a plane, text written above 'that') the number of [conies?] that touch five given [conies?] is 3264. I think that when a mathematical theory has nothing [ever?] to discover but such propositions as that it may be said to be.

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[exhauster?]. Every given mathmatical theory must, in that sense, eventually become exhausted. Whether or not new theories of interest will continue to offer new problems or not, I am not prepared today. But to come back to the matter in hand, 4 give you without further preface, the following rule for positively, ascertaining whether or not a given graph is beta possible. The first place [find?] in the graph [every radius?] (regarding 0 as a [lexis?]) of which ([radius?] is within an odd number of [cuts?]) one replica is odly enclosed (i.e. is [within?] a certain number of cuts) The latter [not?] being enclosed for every cut that encloses the former; and call every such pair of spots an adaptible pair. An adaptible pair is said to be [conjoined?]

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part of the graph including every part thatwill be affected by the operation and onlyoperate upon. The new replica so produced.

Operation A. Two oddly enclosed lines of identity in the same area may be joined and an evenly enclosed line of identity may be severedThese operations are irreversible.Operation B. From any line of identity

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simply as propositions and may be general, in the betapart, besides these, individuals, (above which [form?]) an entirely different category enter into the graphs. I now go on to (written above insert 'a preface to') the gamma part of the subject which is by far the most important of the three. (written above 'which') is distinguished by taking account of abstractions.(I shall have to preface it) crossed out) I begin by a remark drawn from [Phenemenologi?] [Ronomenology?] which is the science which describes the kinds of elements that are always present (the phenomenon meaning by the [Phe?] ) whatever is before the mind in any kind of thought , fancy, or cognition of any kind. Everything that you can possibly think involves {these?] kinds of elements. Whence it follows that you

Left Margin: Omit from here (lines drawn after 'of abstractions'). New paragraph

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cannot possibly think of any one of those elements in its purity. The most strenuous endeavors of thinking will leave your ideas somewhat confused. But I think I can help you to see that there [underlined] are [underlined/] three kinds of elements, and to discern what they are like. I begin with that one [illegible insertion] which the rough and tumble of life renders most familiarly prominent. We are, continually bumping up against hard facts. We expected one thing or passively took it forgranted, and had the image of it in our minds. But experience forces that idea into the background, and compels us to think quite differently. You get this kind of consciousness in some approach to purity when you put your shoulder against a door

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and try to force it open. You have a sense of [resistance ??]and at the same time a sense of effort. There canbe no resistance without effort: there can be noeffort without resistance. They are ~~but~~only two waysof describing the same experience. It is a doubleconsciousness. We become aware of our self ~~by~~inbecoming aware of the [not-self ??]. The waking stateis conciousness of reaction; and as the consciousness__itself__ is two-sided, so it has also two__varieties__; namely, ~~not willing~~ action, where ourmodification of other things is more prominent thantheir reaction on us, and perception, where their effecton us is ~~far~~ overwhelmningly greater than our effect on them. Andthis notion of being such as ~~another thing~~other things makes us [~~becomes~~ ??], is such a prominent part of our life, that we conceive

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other things also to exist by virtue of their reactionsagainst each other. The idea of other, of __not__, becomesa very pivot of thought. To this element I give thename of Secondness. But now there are elementsof what is before the mind which do not dependupon other, each of them being such as it ispositively, in itself, regardless of anything else.Such for example is the quality of purple. [Gon ??][hast ??] may cause it to strike us more; but howeverlittle it strikes us, the quality of the purple remainsthe same, peculiar and positive; and we canonly say of it that it is such as it is. We attributesomething analogous to ~~these~~our Qualities offeeling [~~A~~ ??] outward things. We conceive that ahard body, that is to say a body not readily