List, Here is a simple illustration with explanations of degenerate forms of conic curves: http://www.open.edu/openlearnworks/mod/page/view.php?id=43857
There are two interesting features of this analogy. The first is that there are continuous transformations between the various curves and their degenerate forms (point, line, pair of lines). As such, the degenerate forms are limiting cases between the kinds of curves. Another interesting feature is that, considered topologically, the surfaces characterized by taking such curves as the horizon in a perspective system that is taken to be a part of a larger projective surface are different kinds of surfaces. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________________ From: Clark Goble [[email protected]] Sent: Wednesday, December 16, 2015 3:01 PM To: Peirce-L Subject: Re: [PEIRCE-L] signs, correlates, and triadic relations On Dec 16, 2015, at 1:48 PM, Helmut Raulien <[email protected]<mailto:[email protected]>> wrote: Degenerateness, I think, is a relation too. So, something may be (regarded for) degenerate, if you look at it as a mode. Because degeneracy is a trait of modes. But if you look at the same thing regarding it for a sign (a triadic sign), then degeneracy is not something you can assign to it. And anything can be interpreted for a triadic sign. It is the point of view that makes it. Anyway, I think, that "degenerate" is merely a Peircean technical term, and has nothing to do with the opposite of "to generate". Subsumption or classification has to do with generation and inheritance: This is a one-way-affair, in which there is only generation, but never a degeneration. In compositional hierarchy you may say, that something complex is made of less complex things, and ok, you may substitute "less complex" with "degenerate", but that also has nothing to do with the opposite of "to generate". All in all, I merely wanted to say, that I do not like the term "degenerate", because it leads to nothing but astray. I may be completely off here but doesn’t the use of degeneracy as a term arise from geometry for Peirce? So a pair of parallel lines is a degenerate conic (as opposed to the curves that are usually generated by a conic section) Thus, the whole book being nothing but a continual exemplification of the triad of ideas, we need linger no longer upon this preliminary exposition of them. There is, however, one feature of them upon which it is quite indispensable to dwell. It is that there are two distinct grades of Secondness and three grades of Thirdness. There is a close analogy to this in geometry. Conic sections are either the curves usually so called, or they are pairs of straight lines. A pair of straight lines is called a degenerate conic. So plane cubic curves are either the genuine curves of the third order, or they are conics paired with straight lines, or they consist of three straight lines; so that there are the two orders of degenerate cubics. Nearly in this same way, besides genuine Secondness, there is a degenerate sort which does not exist as such, but is only so conceived. The medieval logicians (following a hint of Aristotle) distinguished between real relations and relations of reason. A real relation subsists in virtue of a fact which would be totally impossible were either of the related objects destroyed; while a relation of reason subsists in virtue of two facts, one only of which would disappear on the annihilation of either of the relates. Such are all resemblances: for any two objects in nature resemble each other, and indeed in themselves just as much as any other two; it is only with reference to our senses and needs that one resemblance counts for more than another. Rumford and Franklin resembled each other by virtue of being both Americans; but either would have been just as much an American if the other had never lived. On the other hand, the fact that Cain killed Abel cannot be stated as a mere aggregate of two facts, one concerning Cain and the other concerning Abel. Resemblances are not the only relations of reason, though they have that character in an eminent degree. Contrasts and comparisons are of the same sort. Resemblance is an identity of characters; and this is the same as to say that the mind gathers the resembling ideas together into one conception. Other relations of reason arise from ideas being connected by the mind in other ways; they consist in the relation between two parts of one complex concept, or, as we may say, in the relation of a complex concept to itself, in respect to two of its parts. This brings us to consider a sort of degenerate Secondness that does not fulfill the definition of a relation of reason. Identity is the relation that everything bears to itself: Lucullus dines with Lucullus. Again, we speak of allurements and motives in the language of forces, as though a man suffered compulsion from within. So with the voice of conscience: and we observe our own feelings by a reflective sense. An echo is my own voice coming back to answer itself. So also, we speak of the abstract quality of a thing as if it were some second thing that the first thing possesses. But the relations of reason and these self-relations are alike in this, that they arise from the mind setting one part of a notion into relation to another. All degenerate seconds may be conveniently termed internal, in contrast to external seconds, which are constituted by external fact, and are true actions of one thing upon another. (CP 1.365 (1890))
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