List, This part of MS 466 explains why logic is a "positive science" while mathematics is not. It's because the logician investigates "what positive facts about the real universe of things and of thoughts it is from which the necessity of the mathematician's reasonings and the validity of other kinds of reasonings depend, and exactly what the nature of that dependence is."

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We already know that the factual truth of its premises is irrelevant to the necessity of the mathematician's reasonings; the conclusions drawn by the latter are "necessary" because they assert no positive facts about the real universe of things, and are therefore immune to the unavoidable fallibility of such assertions. But from the logician's point of view, there must be some implicit rules or laws governing such reasoning, and they cannot be arbitrary assumptions as the premises of the mathematician's argument can be. They must be dependent on "positive facts about the real universe of things and of thoughts" (including facts about the relations between things and thoughts). Moreover, the validity of other kinds of reasonings must depend on those same "positive facts." The mathematician per se is content to rely on the implicit guidance of these "facts," but the logician per se wants to make them explicit, to show what they are. A logician like Peirce is interested in "examining the fundamental nature of mathematics" not because logic is mathematical, but because mathematics must have a logical core, so to speak. If we can show the positive facts which constitute the core of even the simplest mathematics, we will have the key to the validity of reasoning generally - that is, we will know something basic about cognition itself, about how it is possible to learn from experience. The simplest mathematics would have only two values, and says Peirce. "our system of Existential Graphs is precisely an application of the mathematics of two values." He also says that "there would be a Mathematics of a System of Three Values which would not be without utility," but does not venture here to begin development of this would-be Mathematics. So it would seem that system of Existential Graphs - including the Gamma part, which Lecture 4 is supposed to introduce, but which is not mentioned in MS 466 - confines itself to "the mathematics of two values" in its attempt to reveal the core facts on which cognition depends. But maybe that should be posed as a question; and maybe we can ask whether analysis of a System of Three Values could tell us more about cognition than we can learn from a two-valued system. Gary f. From: g...@gnusystems.ca [mailto:g...@gnusystems.ca] Sent: 18-Feb-18 15:45 To: 'PEIRCE L' <PEIRCE-L@list.iupui.edu> Subject: [PEIRCE-L] Lowell Lecture 4.2 Continuing from Lowell Lecture 4.1, https://www.fromthepage.com/jeffdown1/1903-lowell-lectures/ms-466-467-1903-l owell-lecture-iv/display/13956: The only reason I do not agree with Dedekind in making mathematics a branch of logic is that logic is not a science of pure assumptions but is a study of positive truth. The mathematician seeks only to trace out the consequences of his assumptions in the readiest and speediest way. The logician does not care much what the conclusions from this or that system of assumptions may be. What he is interested in is in dissecting reasonings, in finding out what their elementary steps are, and in showing what positive facts about the real universe of things and of thoughts it is from which the necessity of the mathematician's reasonings and the validity of other kinds of reasonings depend, and exactly what the nature of that dependence is. I have already pointed out that the characters that make a system of symbols good for [mathematical] purposes are quite contrary to those which would make it good for logical purposes. The truth is that the mathematician and the logician meet in one department on a common highway. They meet; but one is facing one way while the other is facing just the other way. Each of them, it is true, finds it interesting to turn round occasionally and take a glance in the opposite direction. The mathematician, however, has little or nothing to learn of the logician. Mathematics differs from all the special sciences whether of physics or of psychics in never encountering logical difficulties, which it does not lie entirely within his own competence to resolve. The logician on the other hand has everything to learn from the mathematician. Mathematics is, with the exception of Phenomenology and Ethics, the only science from which he can draw any real guidance. There is therefore every reason in the world why we should not delay examining the fundamental nature of mathematics. The simplest possible kind of mathematics would be the mathematics of a system of only two values. Do we in experience meet with any case in which such mathematics could have any application? I reply that we do. There are just two values that any assertion can have. It is either true when it has all the value it can have, or it is false, and has no value at all. It follows that our system of Existential Graphs is precisely an application of the mathematics of two values. It will give you an idea of what Pure Mathematics is to imagine the Existential Graphs to be described without any allusion whatever to their interpretation, but to be defined as symbols subject to the fundamental rules of transformation. Namely, 1st, Graphs are figures drawn on a surface and composed of any finite number including zero of each of these 3 kinds of elements: 1st oval cuts of which no two intersect, 2nd heavy lines, and 3rd spots each with definite places each of which is at an extremity of a heavy line. 2nd, Any graph within an even number of cuts can be erased and any within an odd number of existing cuts can be inserted. 3rd, Any graph can be iterated or deiterated provided the iterated or deiterated replica is not outside of any cut that the other is inside of. 4th, Two cuts one inside the other with nothing between except heavy lines passing from within the inner to outside the outer, can either be made or destroyed anywhere. 5th, A heavy line of which both ends connect with lines inside a cut which lines do not connect spots connected inside the cut can anywhere be erased or inserted. >From those assumptions everything universally true of existential graphs could be deduced, and that would be the Pure Mathematical Treatment. I have defined Mathematics in general as the science which draws necessary conclusions and which formulates the assumptions from which such conclusions can be drawn. I now define Pure Mathematics as that Mathematics which leaves its assumptions entirely indeterminate in respects which have no bearing upon the manner in which they can be combined to produce conclusions. The Pure Mathematics of the System of Two Values would leave us free to regard the Graphs as representing anything for which their fundamental transformations would hold good. In like manner there would be a Mathematics of a System of Three Values which would not be without utility and which has been in some measure developed. The theory of numbers furnishes partial developments of the mathematics of every system having a finite multitude of values. http://gnusystems.ca/Lowell4.htm }{ Peirce's Lowell Lectures of 1903

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