If you want to explain to me how the synthetic continuum is in fact recovered fully by category theory, I would be very grateful. But can you do that? — apokrisis
I guess you disagree with Zalamea about this? If so, why? — aletheist
Now you are right, I'm just trying to learn what this means. But your unwillingness to explain anything of your jargon-filled posts says something about you.
Is it time for me to say fuck you to you again? I've had enough. Fuck you. — fishfry
Peirce usually distinguished vagueness (1ns) from generality (3ns). "Perhaps a more scientific pair of definitions would be that anything is general in so far as the principle of excluded middle does not apply to it and is vague in so far as the principle of contradiction does not apply to it." — aletheist
That is not how I understand it, unless by "constrained possibility" you mean the actually possible as opposed to the logically possible. — aletheist
Then you are the one who responded with the first insult, alleging that he does not understand category theory. — aletheist
And yet where does the principle of identity sit as actual individuation if vagueness and generality are the apophatic definition of the PNC and the LEM? — apokrisis
So category theory seeks an analytic foundations whereas semiosis seeks a synthetic one. — apokrisis
I find that to be the first insult here. I gave a full answer and I get back no useful reply. — apokrisis
If you or fishfry want to enlighten me otherwise, be my guest. — apokrisis
But don't keep attacking me personally instead of addressing the actual ideas I have attempted to put out there. — apokrisis
I see nothing insulting about pointing out a discrepancy between what you wrote here and what is claimed in a paper that you recommended. — aletheist
Have I ever attacked you personally, in this thread or elsewhere? — aletheist
...it will just be the two of us trading thoughts about our favorite philosopher. I was hoping for much more than that... — aletheist
everything actual is indeterminate to some degree — aletheist
Sorry to repeat myself, but would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context — aletheist
Have I ever attacked you personally, in this thread or elsewhere? — aletheist
Yep. You are doing that right now too. — apokrisis
And so you would rather chase me off now. — apokrisis
everything actual is indeterminate to some degree — aletheist
Yes. And so does that now suitably define 2ns or actuality as that to which the principle of identity does not apply? (And can you find the quote where Peirce said that?) — apokrisis
I was just trying to moderate a dispute between two of my favorite PF participants. — aletheist
Ah, good point. Where Peirce said what I said, what you said, or both? — aletheist
I expected fishfry to tell me where I was wrong about category theory vs semiotics in his own words ... — apokrisis
He has now told me to f*** off. And you seem to think he is right to do so. — apokrisis
I'm not aware that Peirce ever made this point about identity. — apokrisis
Do I need to rebuke him to demonstrate my impartiality? — aletheist
How would you formulate the principle of identity such that it would not apply to the actual, because nothing that exists is determinate with respect to every predicate? Does it apply to 1ns and 3ns, such that its inapplicability is a distinguishing feature of 2ns as you seem to be suggesting? — aletheist
Why not just do much less rebuking all round and focus on dealing with the substance of any post. — apokrisis
Generality is defined by its contradiction of LEM. Vagueness is defined by its contradiction of PNC. So it would be neat if actuality or 2ns were contradicted by (thus apophatically derivable from) the remaining law of thought. — apokrisis
Used in logic in two closely connected senses. (1) According to the more formal of these an individual is an object (or term) not only actually determinate in respect to having or wanting each general character and not both having and wanting any, but is necessitated by its mode of being to be so determinate ...
(2) Another definition which avoids the above difficulties is that an individual is something which reacts. That is to say, it does react against some things, and is of such a nature that it might react, or have reacted, against my will ...
... whatever exists is individual, since existence (not reality) and individuality are essentially the same thing; and whatever fulfills the present definition equally fulfills the former definition by virtue of the principles of contradiction and excluded middle, regarded as mere definitions of the relation expressed by "not." As for the principle of indiscernibles, if two individual things are exactly alike in all other respects, they must, according to this definition, differ in their spatial relations, since space is nothing but the intuitional presentation of the conditions of reaction, or of some of them. But there will be no logical hindrance to two things being exactly alike in all other respects; and if they are never so, that is a physical law, not a necessity of logic. — CP 3.611-613
Does that make any more sense? — aletheist
Well my view is that the laws of thought are designed to make the world safe for predicate logic - reasoning about the concretely particular or actually individuated. So the three laws combined - or rather three constraints - secure this desirable form of reasoning in a suitable strait-jacket. — apokrisis
Again, this is somewhat of a departure from conventional Peirceanism. I employ the logic of dichotomies (as it is understood from the vantage of hierarchy theory) where definite actuality or 2ns is emergent from the interaction of constraints and free or vague potential. So 2ns comes last in a sense (though this is no contradiction of Peirceanism, just making something further explicit). — apokrisis
If you want to explain to me how the synthetic continuum is in fact recovered fully by category theory, I would be very grateful. But can you do that? — apokrisis
A slightly different way to make sense of the situation is to think of mathematical objects as types for which there are tokens given in different contexts. This is strikingly different from the situation one finds in set theory, in which mathematical objects are defined uniquely and their reference is given directly. Although one can make room for types within set theory via equivalence classes or isomorphism types in general, the basic criterion of identity within that framework is given by the axiom of extensionality and thus, ultimately, reference is made to specific sets. Furthermore, it can be argued that the relation between a type and its token is not represented adequately by the membership relation. A token does not belong to a type, it is not an element of a type, but rather it is an instance of it. In a categorical framework, one always refers to a token of a type, and what the theory characterizes directly is the type, not the tokens. In this framework, one does not have to locate a type, but tokens of it are, at least in mathematics, epistemologically required. This is simply the reflection of the interaction between the abstract and the concrete in the epistemological sense (and not the ontological sense of these latter expressions.)
A major development in the refounding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry, also known as smooth infinitesimal analysis (SIA). Based on the ideas of the American mathematician F. W. Lawvere, and employing the methods of category theory, smooth infinitesimal analysis provides an image of the world in which the continuous is an autonomous notion, not explicable in terms of the discrete ... We observe that the postulates of smooth infinitesimal analysis are incompatible with the law of excluded middle of classical logic.
To start we mention that the main attraction of this theory for followers of Peirce beyond the simplification of mathematical practice is in the circumstance that the law of excluded middle does not hold for the points on this extended real line. Hence, there is a strong sense in which points merge together so that they are no longer distinct individuals. Peirce often said that traditional laws of logic, like the law of excluded middle or the law of contradiction, do not apply to points which are merged together in the continuum. The most important parts of the line in synthetic geometry are infinitesimal "linelets" surrounding each point ...
... the consistency of synthetic geometry's infinitesimals is established by formulating it inside topos theory, a subbranch of the theory of categories whose logic is solidly intuitionistic. Thus, we should not expect the law of excluded middle to hold for the objects we can construct with topos theory.
It appears that Peirce would favor this version of a theory of infinitesimals over that of Robinson because it satisfies more of his desiderata. First, the infinitesimal intervals surrounding every point and whose image under a function are linear are the true parts from which the line is built. The points lying on the extended real line are not true atomic elements from which the line is built considering that the law of the excluded middle cannot be used to distinguish between points with infinite precision. We should consider points as the potential elements of the line which are welded together ... It seems that the only way in which Peirce could be disappointed about this model is that it is a projective theory of geometry, one which needs a straight line as a fundamental part of the geometry. It would be very interesting to explore topos theory to see if there are any restrictions on how many points can be placed on the extended real line of synthetic geometry in order that Peirce's desire to fit any cardinality of points on a line can be satisfied within this model of infinitesimals.
How can that be satisfactory in a philosophical sense? If you can divide the point on one of its sides, why can't the next cut divide it to its other side, leaving it completely isolate and not merely the notion of an end point of a continua? — apokrisis
I don't see a problem. Nor does one appear when we make a second cut at >2. We now have three pieces: <2, 2, >2. — Banno
2 is the boundary between all that is less than two and all that is greater than two — Metaphysician Undercover
2 is not included in what is less than two. — Banno
What he can't understand is logical definion in terms of itself-. — TheWillowOfDarkness
IN short there is no such thing as "self-definition"; what you actually mean is 'self-identity'. — John
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.