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

Come now - this is not @fishfry's own claim, but Zalamea's, in the very paper that you recommended. The later discussion linking category theory with Peirce's continuum is on pages 38-40. I guess you disagree with Zalamea about this? If so, why? Again, I do not know much about category theory (yet), so I personally have no opinion one way or the other and am open to persuasion.

I guess you disagree with Zalamea about this? If so, why? — aletheist

I just said why. If fishfry thinks I was wrong, then I am genuinely interested to know on what grounds.

I hardly have a settled view here. And I don't have time right at the moment to re-read Zalamea more closely on this particular point. But I do welcome further discussion ... and not just nitpicking in place of honest rebuttal.

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

What was I saying about instability?

I don't claim special expertise in category theory. But I think I know enough to know from your description that you know even less.

So I tried to explain my own point of view. I offered you the chance to rebut that from your current close reading of Zalamea. At which point - and I can't say I'm surprised - you explode in anger because you are not in the position to do so.

But never mind.

Again, @fishfry simply pointed out something that Zalamea claims in the paper that you recommended, which is contrary to your own comments. I took him to be asking you for an explanation of this particular discrepancy, not nitpicking or in any way asserting that Zalamea is right and you are wrong, since he is still in the process of digesting the paper. Then you are the one who responded with the first insult, alleging that he does not understand category theory. If you had left out that one unnecessary sentence, I suspect that a more fruitful exchange would have ensued - one that I would still very much like to see happen.

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

Yep. I cite that brilliant insight most days. 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?

Peirce starts the discussion. It remains to be concluded.

That is not how I understand it, unless by "constrained possibility" you mean the actually possible as opposed to the logically possible. — aletheist

I don't think so. The actually possible is the counterfactually possible and so the logically possible.

Perhaps what you find confusing here is that I am striving to wed all this to actual physics (hence pansemiosis). So the missing factor is materiality or energetic action. The mathematical/logical view is all about form or structure - constraints in an abstract Platonic sense. And so that leaves out the material principle that ultimately must "breathe fire into the equations".

So physics too tends to leave actual materiality swinging in the wind of its formal endeavours. One finds the animating principle of a "material field" having to be inserted into the "theories of everything" by hand in an ad hoc way.

It is a really big and basic problem. Physics just gets too used to talking glibly of degrees of freedom (like mathematicians talk of points on a line) without having an account of their developmental history (and thus developmental mechanism).

So that is why I am focused on the two senses in which "pure possibility" get routinely confused in the history of metaphysics. And I don't think Peirce sorts it out in fully transparent fashion - even if he did get it and was trying to articulate that.

Then you are the one who responded with the first insult, alleging that he does not understand category theory. — aletheist

Well the facts are I gave a lengthy explanation of how I see the connection between category theory and semiotics, then fishfry came back with no other answer but "Zalamea appears to contradict you".

I find that to be the first insult here. I gave a full answer and I get back no useful reply.

And yes, I in fact avoided answering on the category theory point initially because I thought I might spare fishfry's blushes. His enthusiasm for Zalamea seemed hyperbolic and his thumbnail account of category theory quite naive.

As I say, I don't claim to be expert on category theory. I've given it a good try and for me it just doesn't compute. I get its general sense I think, but I end up feeling that it is in the end pretty sterile and useless - for the purposes of generalised metaphysics.

If you or fishfry want to enlighten me otherwise, be my guest. But don't keep attacking me personally instead of addressing the actual ideas I have attempted to put out there. I've no issue with those being kicked as hard as you like.

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

The actual is that which is neither vague nor general by these definitions - both principles apply to it. The tricky part is that this notion of absolute singularity is strictly ideal - everything actual is indeterminate to some degree; no existing object definitively possesses or lacks every predicate.

So category theory seeks an analytic foundations whereas semiosis seeks a synthetic one. — apokrisis

Sorry to repeat myself, but would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context - i.e., as characterizations of the kind of foundation that a particular theoretical approach seeks? That might help me understand why you apparently disagree with Zalamea on whether category theory helps us recover the Peircean notion of a continuum.

I find that to be the first insult here. I gave a full answer and I get back no useful reply. — 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.

If you or fishfry want to enlighten me otherwise, be my guest. — apokrisis

I have already acknowledged (more than once) that I do not know enough about category theory (yet) to say anything at this point. I was hoping to learn more about it from the two of you.

But don't keep attacking me personally instead of addressing the actual ideas I have attempted to put out there. — apokrisis

Have I ever attacked you personally, in this thread or elsewhere? At this point, I am just annoyed that you seem to have driven off @fishfry, who I thought was making helpful contributions to the discussion. If it now devolves into "Peircian exegetics," as @SophistiCat thinks it already has done, then it will just be the two of us trading thoughts about our favorite philosopher. I was hoping for much more than that when I started the thread.

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

Sigh. It was the failure to reply in kind. I made substantial points I believe. It is then tiresome to be told to go read what the paper says rather than have those points replied to.

Have I ever attacked you personally, in this thread or elsewhere? — aletheist

Yep. You are doing that right now too.

...it will just be the two of us trading thoughts about our favorite philosopher. I was hoping for much more than that... — aletheist

Oh what a disaster. And so you would rather chase me off now. Hilarious.

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?)

Sorry to repeat myself, but would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context — aletheist

Reductionist vs holistic, causally closed vs causally open, externalist and transcendent vs internalist and immanent, etc, etc.

Have I ever attacked you personally, in this thread or elsewhere? — aletheist

Yep. You are doing that right now too. — apokrisis

Really? That was not my intention at all. I was just trying to moderate a dispute between two of my favorite PF participants.

And so you would rather chase me off now. — apokrisis

What? Why would that be your response? My whole objective was to bring everyone back to the table; I have no desire to chase anyone off.

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

Ah, good point. Where Peirce said what I said, what you said, or both?

I was just trying to moderate a dispute between two of my favorite PF participants. — aletheist

Where is the dispute as such? I expected fishfry to tell me where I was wrong about category theory vs semiotics in his own words, not assign me further homework and file a further essay for his delectation.

He has now told me to fuck off. And you seem to think he is right to do so. Champion.

Ah, good point. Where Peirce said what I said, what you said, or both? — aletheist

I'm not aware that Peirce ever made this point about identity. And I'm not even sure that was the point you intended. But it is the point that now leaps out at me as a very neat extension of the Peicean line of thought. If it is unclaimed, one might even write a paper about it.

I expected fishfry to tell me where I was wrong about category theory vs semiotics in his own words ... — apokrisis

He never said that you were wrong. He merely said that Zalamea said the opposite of what you said.

He has now told me to f*** off. And you seem to think he is right to do so. — apokrisis

What gave you that idea? I thought that was also unfortunate and unnecessary. Do I need to rebuke him to demonstrate my impartiality?

I'm not aware that Peirce ever made this point about identity. — apokrisis

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?

Do I need to rebuke him to demonstrate my impartiality? — aletheist

Why not just do much less rebuking all round and focus on dealing with the substance of any post.

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

What are you talking about.

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.

So it is not the job of 2ns to make the principle of identity true. Instead, it is how identity can be derived as a limit on the actuality of 2ns in line with the vagueness of 1ns and the generality of 3ns that would be of interest.

Why not just do much less rebuking all round and focus on dealing with the substance of any post. — apokrisis

This is probably good advice, and I will try to heed it going forward.

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

Sorry to nitpick, but is "contradiction" the right word here? In accordance with the Peirce quote, we started out using "inapplicability," which seems more appropriate to me. So your hypothesis, as I understand it, is that actuality/2ns is defined by the inapplicability of the principle of identity; and I am still wondering which particular formulation of it you have in mind, since there are several. For example, Peirce did say that "Leibniz's 'principle of indiscernibles' is all nonsense" (CP 4.311). In fact, in his definitions of "individual" for Baldwin's Dictionary of Philosophy and Psychology (1911), he wrote the following.

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

My other point was that if vagueness/1ns is defined by the inapplicabiity of the principle of contradiction, then actuality/2ns and generality/3ns must be subject to it; and if generality/3ns is defined by the inapplicability of the principle of excluded middle, then vagueness/1ns and actuality/2ns must be subject to it. Likewise, if actuality/2ns is defined by the inapplicability of the principle of identity, then vagueness/1ns and generality/3ns must be subject to it. Otherwise, each characteristic is not distinctive of its corresponding category after all.

Does that make any more sense?

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.

If x is x, and x is not not-x, and x is either x or not-x, then that seems to remove all wiggle room for constructing a logical tale founded in brute atomistic particulars.

So it was unconscious semiotics that produced the laws of thought. Their triadicity was no accident as indeterminism of three kinds had to be sealed off.

Then Peircean semiotics tells the inverse story. Instead of determinate actuality or identity being foundational - the first law of the three - it becomes instead the final outcome secured via the other two.

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).

Anyway, the principle of identity becomes the last thing to be secured. As I described it earlier, the habit of 3ns must arise in a way that knocks all the sharp corners off the variety that is 2ns, reducing it to the law-bound regularity that limits every reactive dyad to being as boringly repetitive and mechanical as possible. So 2ns secured is 2ns once lively spontaneity now turned dully persistent. Or effete matterial habit.

So that would be why Peircean 2ns is not obedient to the principle of identity. At least on its first appearance (before it gets tamed by 3ns). In the beginning, any damn reaction is possible. There is no stable identity in the sense that you don't even have things which could be assured of being the same as their previous selves if ever they were to reappear again. 2ns in its purity is maximally non-identical. But once incorporated into 3ns, it gets tamed. It becomes as identical or self-repeating as possible.

So it goes beyond simply "not applying". It cannot apply because it comes from a contradicting direction of thought. It is holism contradicting reductionism.

The logic of the particular starts with particularity being treated as already secured. Peircean semiosis stands in exact contrast saying that is precisely what has to be secured by way of completed 3ns. Only then is 2ns properly constrained to have reliable identity.

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

Yes, I agree. What remains unclear to me is what it means to say that the principle of identity does not apply to something. Zalamea helpfully formalizes the principles of vagueness and generality on page 21 of his paper; he describes them as failures of distribution of the principles of contradiction and excluded middle, respectively. Is there an analogous way to formalize the principle of identity and/or its failure, which would show what you have in mind here?

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

I likewise dissent from the most common interpretation of Peirce's cosmogony, in which 1ns came first (so to speak), then 2ns, and finally 3ns. I think that 3ns, with "its really commanding function," is the most fundamental - "the clean blackboard" as a continuum of two dimensions representing the original one that had "some indefinite multitude of dimensions." A chalk mark then represents the spontaneous introduction (1ns) of a brute discontinuity (2ns), but the mark is not really a line - it is a surface whose own continuity is parasitic on that of the underlying blackboard. Only after developing habits (3ns) lead multiple chalk marks to persist and aggregate into "whiteboards" that represent Platonic worlds of possibility (1ns) does the final step occur, when "this Universe of Actual Existence" (2ns) comes about as "a discontinuous mark" on one of those whiteboards.

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 certainly cannot do that - at least, not yet - but I just came across one possible clue in the SEP article on "Category Theory."

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 continuum (such as a line) is a type and its parts are tokens, which are instances of it (smaller lines) rather than members or elements of it (discrete points).

Another possible clue in the SEP article on "Continuity and Infinitesimals."

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.

Another possible clue from Timothy Herron's 1997 paper, "C. S. Peirce's Theory of Infinitesimals."

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.

Any thoughts on these passages, @fishfry?

I was interested in this until Apo became so Snarky. It was a good topic.

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

This seemed to be where things went astray. My own comprehension of both of mathematics and philosophy is left puzzled; what's the problem for Apo?

Take the rationals, and make a cut at <2. 2 stays on on side, all the numbers less than 2 on the other. One side contains 2, while the other might approach 2, but by the very fact of the cut, never reaches it.

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.

Nor is a problem introduced when considering continuity. My simple understanding is that a line is continuous if it is differentiable. Well, the limit of <2 as it approaches 2 is 2. It does not seem problematic.

So explain it to me, rather than abusing each other.

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. But what is 2?

2 is the boundary between all that is less than two and all that is greater than two — Metaphysician Undercover

So what. 2 is not included in what is less than two. It is included in what is not less than 2.

2 is not included in what is less than two. — Banno

Nor is it include in what is greater than two, it is included in what is not greater than two. If it's not included in what's less than two, and not included in what's greater than two, then what is it? I know what you'll say, it's 2.

What is the negation of "less than 2"?

It's 2.

Apo's philosophy is ignorance of self-definition. What he can't understand is logical definion in terms of itself-. For him, everything must be logically defined in terms of something else, so it's not enough for to be 2.

Seeing without being shown.

But it always worries me when I agree with you. :-|

The purported logical definition of something "in terms of itself" is an empty formulation. This is due to a conflation of identity with definition. Of course something is defined as being itself in the logical sense; but this is not the same as being defined in terms of itself, because even to say that something is defined as being itself is meaningful only in terms of understanding that to mean that it is not defined as anything else, so even that supposedly 'pure' definition is not free of everything else. Entities can only be defined in terms of their properties and relations; that is they are defined precisely in terms that are the terms of qualities that are anything but the thing being defined..

IN short there is no such thing as "self-definition"; what you actually mean is 'self-identity'.

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

"Self-definition", or "self-identity", whatever you want to call it, what do you mean by this?