I will take your word for it, but my understanding is that the precise nature of the relationship between Peirce's continuum and nonstandard analysis is still not fully settled. — aletheist

You can take my word for it that .999... = 1 is a theorem of nonstandard analysis. But actually you don't need to take my word, I provided a proof above. The nonstandard reals are a model of the first-order theory of the reals. .999... = 1 is a theorem of that theory, hence it's true in any model of that theory. That's Gödel's completeness theorem, as opposed to his more famous incompleteness theorem. If you have a syntactic proof of a statement from some axioms, that statement is true in every model of those axioms,. The converse holds as well. If you don't have a proof, then there are models where it's true and models where it's false.

Please ask for more detail if this proof isn't clear (and you care one way or the other).

I'm afraid I don't know anything about Peirce's continuum but I will certainly try to educate myself in the near future. But remember, any field containing infinitesimals must be incomplete. Does Peirce know his continuum has holes in it? It's logically necessary.

On an unrelated meta-note, it would not be good to allow this discussion to degenerate into a .999...-fest. That's a tremendous distraction.

Does Peirce know his continuum has holes in it? It's logically necessary. — fishfry

Peirce operates at a deeper level of generality so his continuum would be "holey" in the sense of being fundamentally indeterminate.

That is, either the judgement of "continuous" or "discrete" would be determinations imposed (counterfactually) on pure possibility. So the presence or absence of holes is a matter of vagueness once one drills down that far into the metaphysics of existence.

This is of course the philosophical view. Mathematics ignores it for its own pragmatic reasons. Although one can wonder - as Peirce did - what kind of maths might be founded on a logic of vagueness.

So when it comes to Peirce's notion of the continuum, there is an ambiguity as he was both trying to cash out some mathematics from a crisp notion of the continuous (as a determination counterfactual to the usual presumption of numerical descreteness) and also taking "the principle of continuity" as a general metaphysical stance which was in turn an irreducibly triadic relation - where the discrete and the continuous are merely the logically dichotomous limits of a determination, and what is being so determined by this semiotic act is the more fundamental ground of pure possibility that he dubbed Firstness or Vagueness.

If you read up on Peirce's synecheism - as his model of continuity - it gets clearer. The "continuity" then is of the systematic "constraints plus freedoms" kind that I employ.

If you read up on Peirce's synecheism - as his model of continuity - it gets clearer. The "continuity" then is of the systematic "constraints plus freedoms" kind that I employ. — apokrisis

I started reading the Zalamea paper, Googled around, and found a pdf of his awesome book Synthetic Philosophy of Contemporary Mathematics. I'm enthralled. A mathematician who actually understands the conceptual revolution that happened in math in the second half of the twentieth century; and writes brilliantly and clearly about philosophy. And he's written a lot about Peirce as well. So at the moment I'm coming to Peirce by way of Zalamea's exposition of the mathematical philosophy of Grothendieck. This is sublime. I truly thank you for this reference.

Different point ... when you talk about points coming in and out of existence, that reminds me of the intuitionists. Which I regard as a somewhat mystical strain of thought. For them, real numbers come into existence when they are thought of or needed or used. In the modern incarnation, a number comes into existence when it's computed. Intuitionism is coming back into style.

Now here is my question.

* It's my understanding that the intuitionistic real line has fewer points than the standard real line. After all only countably many reals are computable, for the reason that there are only countably many Turing machines. Where the standard real line has noncomputable numbers, the intuitionistic line has holes.

* It's also my understanding that the hyperreal line (this is the only nonstandard model of the reals I know anything about) has MORE numbers than the standard reals. After all the hyperreals have all the real numbers, plus a cloud of infinitesimals around each one. (I've heard these are Leibniz's monads, but I don't know anything about Leibniz, being more of a Newton fan. Maybe that explains a lot :-))

* So my question is: Is Peirce a restrictionist, squeezing the noncomputables out of the standard reals and only creating reals when they pop into his intuition; or is he an expansionist, blowing wispy clouds of infinitesimals onto the real line?

Different point ... when you talk about points coming in and out of existence, that reminds me of the intuitionists. Which I regard as a somewhat mystical strain of thought. — fishfry

Yep. But all foundational approaches end up mystical in philosophy of maths. Is Platonism any less bonkers?

So yes, this is rather like intuitionism. But pragmatism/semiotics brings out the fact that maths works by replacing the "thing in itself" with its own system of signs.

So the numbers are conjured out of the mist of the continuum - which seems too magical or social constructionist. Standard thinking would insist either the numbers are "really there" in determinate fashion, or that the only alternative is that they are a "complete fiction" - an arbitrary invention of the free human imagination.

However the whole point of Peirce - as managing to resolve the tortuous dilemmas of Kant, Descartes and all the way back to the Miletians vs the Stoics - is that it is itself metaphysically fundamental that reality is organised by its own sign relations.

So number would have to be plucked out of the indeterminate continuum via acts of localising constraint. It is the trick of being able to make them appear "at will" which is the very nature of their existence (exactly as quantum theory needs the classical collapse - the system of symmetry breaking constraints - which reduces the indeterminacy of the wavefunction to some actually determinate outcome).

Where the standard real line has noncomputable numbers, the intuitionistic line has holes. — fishfry

And my answer already is that the Peircean continuum would have the third alternative of vagueness - irreducible and thus inexhaustible uncertainty or indeterminism.

That was the point of my question to Tom. Even the number 1 should really be understood as a claim about a convergence to a limit. It is really 1.000.... with every extra decimal place adding a degree of determinancy, yet still always leaving that faint scope for doubt or indeterminism. The sequence must surely return zeroes "all the way down". But then it can't ever hit bottom. And yet neither is there a warrant to doubt that if it did, it would still be returning zeroes.

So to properly characterise this state of indeterminate possibility, we must call it something else than "continuous" or "discrete".

* So my question is: Is Peirce a restrictionist, squeezing the noncomputables out of the standard reals and only creating reals when they pop into his intuition; or is he an expansionist, blowing wispy clouds of infinitesimals onto the real line? — fishfry

I can only speak for the spirit of Peirce, given I'm not aware of him ever answering such a question. And as I say, the general answer on that would be that if there is ever any sharp dichotomy - like your restrictionist vs expansionist - then the expectation is that both are a dichotomisation or symmetry-breaking of something deeper, the perfect symmetry that is a vague potential. Together, they would point back even deeper to that which could possibly allow them to be the crisp alternatives.

So you can see that talk of clouds of virtual infinitesimals is trying to speak of a vagueness. Except rather than the clouds obscuring anything more definite, they are the thing itself - the indefiniteness from which all determination can then spring.

Likewise intuitionism notes the magic by which numbers can be conjured up as concrete signs from imagined cuts across an imagined line. And that makes the whole business seem arbitrary. But now Peircean semiotics explains that because an apparatus of determination is needed even in nature (if nature is to bootstrap itself into concrete being).

So as I say, the continuum represents the (definite) potential for as many numeric distinctions as we might wish to find, or have a good use for. And semiotics - the triadic theory of constraints - is then a universal account of the apparatus of determination. The way to determine things is not arbitrary at all. There is only just the one way that reality permits. And maths - quite unconsciously - has picked up on that.

Zalamea spells that out with his story of the evolution of the reals. A hierarchical series of constraints was needed to squeeze numbers out of the continuum - winding up finally with Cauchy convergence as the promise "if we could compute all the zeros, we could know that 1 is actually 1 and not just close enough for practical purposes".

So there is little point asking about Peirce's philosoph of maths without understanding the logic and metaphysics that motivated his particular approach.

If you are arguing over which pole of some dichotomy to choose, you are completely misunderstanding what Peirce would be trying to say. Peirce is always saying look deeper. This is actually a trichotomy - the irreducible triadicity of a sign relation.

...the whole point of Peirce - as managing to resolve the tortuous dilemmas of Kant, Descartes and all the way back to the Miletians vs the Stoics - is that it is itself metaphysically fundamental that reality is organised by its own sign relations. — apokrisis

Just struggling a bit with how 'sign relations' come into the picture outside of biology.....

Just struggling a bit with how 'sign relations' come into the picture outside of biology..... — Wayfarer

Biology, that's interesting. I thought sign relations were some kind of postmodern talk I don't know anything about other than that Searle thinks Derrida is full of sh*t. That much I know.

Well, I think Apokrisis' background is biological sciences. 'Biosemiotics' is the discipline that is descended from Pierce. There's a particular scientist by the name of Howard Pattee who is a key figure. Google the phrase 'epistemic cut' if you're interested. (Incidentally, agree with Searle on that count.)

Just struggling a bit with how 'sign relations' come into the picture outside of biology..... — Wayfarer

Well biology is lucky. It is just damn obvious that life (and mind) are irreducibly semiotic in their nature. (And ironic that physicists like Schrodinger and Pattee were the first to really get it, letting the biolog,ists know what they ought to be looking for in terms of central mechanisms).

And now the speculative extension of that would be physiosemiosis - or pansemiosis as the most inclusive metaphysical position.

So right back at you physics! It turns out that you are a branch of "information science" too.

I thought sign relations were some kind of postmodern talk I don't know anything about other than that Searle thinks Derrida is full of sh*t. — fishfry

PoMo is full of shit because it is based on Saussurean semiotics rather than Peircean. So it is dyadic, not triadic.

Well of course nothing wrong with Saussure if you want a simple and lightweight introduction. But it is alcopops compared to fine wine.

t is just damn obvious that life (and mind) are irreducibly semiotic in their nature. — apokrisis

In the beginning was the word, eh? ;-)

The word plus the vagueness it could organise.

So the ancient Greeks got it. The peras and aperas of the Pythagoreans. The logos and flux of Heraclitus. The formal and material causes of Aristotelean hylomorphism.

Or really in the beginning there was the light. And someone said let there be word. :)

There was the vagueness that would be utterly patternless and directionless action. And someone said that's a little boring. Let's tweak it with some contrast. Let's add some constraints to give it some light and dark. Let's create a little story about differentiated being.

You can take my word for it that .999... = 1 is a theorem of nonstandard analysis. But actually you don't need to take my word, I provided a proof above. — fishfry

I have no reason to doubt that you are correct about this. Thanks for another helpful clarification, especially since @tom chose for some reason not to provide one.

That is, either the judgement of "continuous" or "discrete" would be determinations imposed (counterfactually) on pure possibility. — apokrisis

In this context, do you basically see continuity as 3ns, discreteness as 2ns, and possibility as 1ns?

The "continuity" then is of the systematic "constraints plus freedoms" kind that I employ. — apokrisis

Given that existence is 2ns, do you generally prefer to characterize 3ns as "constraints" and 1ns as "freedoms"?

In the beginning was the word, eh? — Wayfarer

The word plus the vagueness it could organise. — apokrisis

Peirce: "If we are to explain the universe, we must assume that there was in the beginning a state of things in which there was nothing, no reaction and no quality, no matter, no consciousness, no space and no time, but just nothing at all. Not determinately nothing. For that which is determinately not A supposes the being of A in some mode. Utter indetermination. But a symbol alone is indeterminate. Therefore, Nothing, the indeterminate of the absolute beginning, is a symbol. That is the way in which the beginning of things can alone be understood." (EP 2:322; c. 1904)

There was the vagueness that would be utterly patternless and directionless action. — apokrisis

Peirce: "In that state of absolute nility, in or out of time, that is, before or after the evolution of time, there must then have been a tohu-bohu of which nothing whatever affirmative or negative was true universally. There must have been, therefore, a little of everything conceivable." (CP 6.490; 1908)

Genesis 1:2: "The earth was without form and void (Hebrew tohu wa bohu) ..."

And someone said that's a little boring. Let's tweak it with some contrast. Let's add some constraints to give it some light and dark. Let's create a little story about differentiated being. — apokrisis

Someone? Now you are just teasing me.

Peirce, describing the author of Genesis 1:2-5: "His tohu bohu, terra inanis et vacua is the indeterminate germinal Nothing. His Spiritus Dei ferebatur super aquas is consciousness. His Lux is the world of quality. His fiat lux is an arbitrary reaction. His divisit lucem a tenebris is the recognition of the necessary duality. His vidit Deus lucem quod esset bona is the waking consciousness. Finally; his factumque est vespers et mane, dies unus is the emergence of Time." (NEM 4:138; c. 1898)

And a better paper on the Peircean project is probably... http://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf — apokrisis

So at the moment I'm coming to Peirce by way of Zalamea's exposition of the mathematical philosophy of Grothendieck. — fishfry

I was delighted to learn this evening that Zalamea has agreed to a "slow read" of this very paper via the Peirce-L e-mail list in the near future. If you are interested in joining that conversation, or even just monitoring it, I can keep you posted on the details as they are worked out. He hopes to participate himself, although likely to only a limited extent, because he apparently intends to go into seclusion over the next couple of years in order to focus on writing a lengthy new monograph on Grothendieck.

In this context, do you basically see continuity as 3ns, discreteness as 2ns, and possibility as 1ns? — aletheist

Yep. So that does conflict with some of Peirce's apparent definition of 1ns as brute quality (with its implications of already being concrete or substantial actuality).

But that is a constant tension as to speak of vagueness, we are already reifying it as some kind of bare material cause - an Apeiron. And Peirce never actually delivered a logic of vagueness in a way that would save us having to read between the lines of his vast unpublished corpus.

So continuity or synechism itself is 3ns - but 3ns that incorporates 2ns and 1ns within itself. So 3ns is literally triadic and incorporates as "continuity" the very things that you might want to differentiate - like the discrete and the vague.

I'm sure you get this critical logical wrinkle that makes Peircean semiotics so distinctive (and confusing). This is the way he avoids the trap of Cartesian division. 3ns incorporates all that it also manages to make different.

So 1ns (in a misleadingly pure and reified sense) is vagueness (a certain unconstrained bruteness of possibility - as in unbounded fluctuations).

Then 2ns is really 2(1)ns in that action meets action to become the dyad of a reaction. Something definite and descrete has now happened in the sense that there is some event that could leave a mark. (It takes two to tango or share a history of an interaction).

Then 3ns is really 3(2(1))ns. If there is something about some random dydaic interaction that sticks, a habit can form - which in turn starts to round the corners of any local instants of dyadic interaction being produce by the spontaneity of naked possibility.

So 3ns is habit, which is constraint. And constraint transforms even 1ns to make it far more regular and well behaved. It winds up a substantial looking stuff following then disciplined laws of action and reaction which in turn speak to the establishment of global lawfulness.

Thus the triadic intertwining that is 3(2(1))ns is justified as the inevitable outcome of the very possibility of a mechanism of development. And vagueness can change character as a result. Potentiality gets replaced by (actualised) possibility - which is more the kind of notion of possibility you get from Aristotlean being and becoming, for instance. And certainly the kind of possibility imagined by standard statistics.

(Of course, Peirce twigged that too. That was why he was working on a theory of propensity.)

Given that existence is 2ns, do you generally prefer to characterize 3ns as "constraints" and 1ns as "freedoms"? — aletheist

In terms of the standard categories, I would map them as necesssity, actuality and possibility. So 3ns is necessity, 2ns is actuality, and 1ns is possibility.

Constraints and freedoms is then a dyadic framing which gets into the tricky area I just mentioned. But it does connect to Aristotelean causality in that it makes sense of habit as standing for top-down formal and final cause - the 3ns that shapes the 1ns into the 2ns that is best suited for perpetuating the 3ns.

And then freedom is fundamentally the utter freedom of 1ns - the unconstrained. But then in practical terms, it must get transmuted into the actualised freedom of constrained 2ns. It must be a possibility that is fruitfully limited - and so the kind of actual substantial variety that Aristotelean becoming, or probability spaces, standardly talk about.

So the synechic level is 3ns - pure constraint. And the tychic level is 1ns - pure freedom. Then 2ns is the zone in between where the two are in interaction - one actually shaping the other to make it the kind of thing which in turn will (re)construct that which is in the habit of making it.

So "real freedom" is 2ns because it is action now with the shape of a purpose (the actual Aristotelean understanding of efficienct cause as Peirce understood - and see Menno Hulswit's excellent books and papers on this issue - http://www.commens.org/encyclopedia/article/hulswit-menno-teleology )

And again, as I say, this is really confusing because everything is so intertwined with Peirce (or any other true holism). But once you get used to it, it all makes sense. :)

And I expect you already get most of this. But just in case, that is a summary of why the answer is not so straightforward.

So that does conflict with some of Peirce's apparent definition of 1ns as brute quality (with its implications of already being concrete or substantial actuality). — apokrisis

No, I understand his 1ns in itself to be quality as possibility, or unembodied quality; medad rather than monadic predicate. Anything brute and/or actual is 2ns.

So continuity or synechism itself is 3ns - but 3ns that incorporates 2ns and 1ns within itself. — apokrisis

Agreed, 3ns involves 2ns and 1ns, and 2ns involves 1ns.

So 1ns (in a misleadingly pure and reified sense) is vagueness (a certain unconstrained bruteness of possibility - as in unbounded fluctuations). — apokrisis

Again, Peirce did not use "bruteness" to refer to 1ns, only 2ns.

So 3ns is necessity, 2ns is actuality, and 1ns is possibility. — apokrisis

That is certainly one manifestation of the categories. Others include quality/fact/law, spontaneity/reaction/habit, and feeling/action/thought. I also think that 3ns is often conditional necessity, rather than absolute necessity.

... see Menno Hulswit's excellent books and papers on this issue ... — apokrisis

Yes, I have read a bunch of his stuff, although it has been a while.

But once you get used to it, it all makes sense. — apokrisis

It definitely takes some getting used to, but is well worth the time and effort.

And I expect you already get most of this. But just in case, that is a summary of why the answer is not so straightforward. — apokrisis

Thanks, it definitely helps me map the terminology that you tend to use around here to Peirce's own.

the irreducible triadicity of a sign relation. — apokrisis

Well when you put it THAT way it's totally CLEAR. LOL.

I see that Peirce has some jargon associated with him. From Googling around I think being triadic is what a mathematician would call ternary, a relation that inputs three objects and outputs T or F. Like equality is binary, it inputs (5,3) and outputs F, inputs (2,2) and outputs T. A ternary relation takes three inputs. Does that have anything to do with this?

If you can briefly explain some of these technical terms it would help.

I wanted to mention that at one point Zalamea basically says that Peirce is doing category theory, or category theory is Peircean. Now I still don't know what Peirce is about, but this is a fantastic connection. I've had some exposure to category theory. Not much but enough to know that modern math is done very differently than anything you see as an undergrad math major. Equations are replaced by arrow diagrams. It's a very different point of view. I've always understood category theory to be loosely related to structuralism. We no longer care what things are made of, we care about their relationships to other things; and about very general patterns in those relationships that tie together previously unrelated areas of math.

Why have I never heard of Peirce before? I've been in lots of online discussions about the philosophy of math, and I've heard the name but never knew he anticipated the math of the future in some deep way.

ps ... I'm randomly reading sections in Zalamea and I come to this: "The triadic Peircean phenomenology ..."

To read that phrase used by an author who speaks so knowledgeably about Grothendieck ... this is breathtaking. Why isn't Zalamea famous? He doesn't even have a Wiki page. This guy has moved the philosophy of math forward fifty years.

I still don't know what triadic Peircean phenomenology is. Can this be explained simply?

From Googling around I think being triadic is what a mathematician would call ternary — fishfry

Not really. Although ternary logic is something like it in fleshing out the strict counterfactuality of 0/1 binary code by introducing a middle ground indeterminate value - the possibility to return a value basically saying "um, not too sure either way".

So it is about arity, which ought to be familiar as a concept. But I could have as well said trichotomic or triune as triadic. It is the threeness that is the distinction that matters.

So really triadic just means not dyadic. Instead of two things in relation, we are talking about the higher dimensionality of three things all relating. And that is irreducibly complex as each thing could be changing the other thing that is trying to change the third thing which was changing the first thing.

In other words, we are dealing with the instability that makes the three body problem or the Konigsberg bridge problem so difficult to compute. One can't caculate directly as none of the values in a complex relation are standing still. Associativity does not apply. Thus you have to employ a holistic constraints satisfaction strategy. You approach the limit of a solution by perturbation. Jiggle the thing until it seems to have settled into its lowest energy or least action state.

I'm guessing this is all familiar maths and so demonstrates what a vast difference it makes to go from the two dimensional interactions to a metaphysics which begins with the inherent dynamical instability of being a relation in three dimensions hoping to find some eventually settled equilibrium balance.

If you get that, then triadic then points towards the mathematical notion of a hierarchy. The best way to settle a complex relation into a stable configuration is hierarchical order. That is the three canonical levels of a global bound, a local bound, and then the bit inbetween that is their interaction.

So reverting to the classical jargon, necessity interacting with possibility gives you actuality. Or constraints, by suppressing chaos, give you definiteness.

So two key points there. Threeness is about irreducible dynamism and thus intractable complexity. Computation in the normal sense - the one dependent on associativity - instantly collapses and other constraints-based or peturbative techniques must be employed.

Then threeness is the link across to hierarchy theory - reality with scale symmetry. Now Peirce himself was not strictly a hierarchy theorist. But once you have studied hierarchy theory, then immediately you can see how Peirce was talking about the same thing from another angle.

And that is indeed how I entered this story - from hierarchy theory as very important to theoretical biology at a time when the connection to Peircean semiotics was being made about 15 years ago.

Again, Peirce did not use "bruteness" to refer to 1ns, only 2ns. — aletheist

Yes, I realise. But my point was that he actually talks about 1ns in misleadingly brute terms. For instance when he makes the analogy with being infused with the pure experience of red. The very idea of a psychological quality is already too substantial sounding to my ear. Too material and passive.

I've had some exposure to category theory. Not much but enough to know that modern math is done very differently than anything you see as an undergrad math major. Equations are replaced by arrow diagrams. It's a very different point of view. — fishfry

My only exposure to category theory (so far) is Zalamea's paper, which I am in the process of rereading because I suspect that it will make even more sense to me now than it did a few months ago. Peirce was a strong advocate of diagrammatic reasoning, but he did not confine the term "diagram" to visual representations; rather, a diagram is any sign that embodies the significant relations among the parts of its object. An algebraic equation is a diagram in this sense, but it is not as "iconic" as a geometrical sketch. This is precisely why Peirce developed the Existential Graphs, a diagrammatic system of logic whose three versions are equivalent to standard propositional logic (Alpha), first-order predicate logic (Beta), and certain kinds of modal logic (Gamma). He hoped that diagrammatic reasoning would be a means by which mathematics could overcome the limitations of the discrete and better account for true continuity.

I've always understood category theory to be loosely related to structuralism. We no longer care what things are made of, we care about their relationships to other things; and about very general patterns in those relationships that tie together previously unrelated areas of math. — fishfry

Several commentators on Peirce have suggested that his philosophy of mathematics was very similar to modern structuralism; e.g., this paper by Christopher Hookway, and this one by Paniel Reyes Cardenas.

Why have I never heard of Peirce before? — fishfry

Ah, the perennial question whenever someone discovers him for the first time. My reaction was exactly the same. Best I can tell, the key factors are:

• He never wrote any books compiling his insights, just lots of articles and tens of thousands of pages of unpublished manuscripts.
• His only academic position was as a part-time lecturer on logic at Johns Hopkins for a few years, so he mostly toiled in relative obscurity.
• Both of the above likely stem from his generally cantankerous demeanor and somewhat scandalous (for its time) personal life.

I still don't know what triadic Peircean phenomenology is. Can this be explained simply? — fishfry

Peirce believed that there are exactly three universal categories that are present in every phenomenon, and in order to avoid specific associations that might be too narrow, he preferred to call them simply Firstness (1ns), Secondness (2ns), and Thirdness (3ns). As I mentioned above, there are various ways to differentiate them - possibility/actuality/necessity, quality/fact/law, spontaneity/reaction/habit, feeling/action/thought. Logically, they correspond to monadic/dyadic/triadic relations; Peirce postulated, and Robert Burch proved (much later), that each of these is irreducible to the others, while all relations of tetradic or higher adicity can be reduced to triadic ones.

The very idea of a psychological quality is already too substantial sounding to my ear. — apokrisis

Peirce acknowledged this - as soon as we talk or even think about a color or other quality, it is no longer 1ns in itself.

Are your comments directed at any particular person or post? — aletheist

My comments were directed at your OP and some following posts. It seemed to me that your dissatisfaction with Cantorian mathematical theories of continuity stemmed from the idea that according to these theories the continua are composed of discrete points - a seeming contradiction. But it's not about composition - it rarely is.

When wondering about what a thing really is, asking "what is it made of?" is a good way to proceed in many common-place situations. For instance, if you find that something is made of wood and not wax, that is going to tell you quite a bit about that thing's properties. But this intuition often trips up people when more subtle questions are asked. In mathematics, and to a large extent, in science, the question "what is it made of" is often unproductive and misleading, as it is in this case.

Anyway, I see that this discussion has long since turned to Peircian exegetics, which interests me not at all, so I'll bow out.

I wanted to mention that at one point Zalamea basically says that Peirce is doing category theory, or category theory is Peircean. — fishfry

I don't really see that myself as category theory seeks a closed structure preserving relation whereas semiosis is open ended both in being grounded in spontaneity and hierarchically elaborative. The spirit seems quite different as even though Peirce appears to be proposing rigid categories (and indeed goes overboard in turning his trichotomy into a hierarchy of 66 classes of sign), essentially the whole structure is quite fluid and approximate - more always a process than a structure as such.

So category theory seeks an analytic foundations whereas semiosis seeks a synthetic one. One is about the tight circle of a conservation principle where you can move about among different versions of the same thing without information being lost (the essential structure always preserved), while the other is an open story about how information actually gets created ... from "nothing".

They may still relate. But probably as Peirce telling the developmental tale of how any exact structure can come to be, and then category theory as a tale of that developed general structure.

So perhaps a connection. But coming at it from quite different metaphysical directions. So foundationally different as projects.

I have to say that I have a somewhat negative view of category theory because it seems to add so little to the practice of science. In particular, two rather brilliant people - Robert Rosen in mathematical biology and John Baez in mathematical physics - have tried to apply it in earnest to real world modelling (life itself, and particle physics). Yet the results feel stilted. Nothing very fruitful was achieved.

By contrast, semiosis just slots straight into the natural sciences. It makes instant sense.

Category theory is dyadic and associative - which is not wrong but, to me, the flattened mechanical view of reality. It is structure frozen out of the developmental processes from which - in nature - it must instead emerge as a limit.

Then semiosis is the three dimensional and dynamical view of reality - organic in that it captures the further axis which speaks to a fundamental instability of nature, and hence the need for emergent development of regulating structure.

The switch from a presumption of foundational stability to foundational instability is something I want to emphasise. That is the Heraclitean shift in thought. Regularity has to emerge to stabilise things. And yet regularity still needs vague or unstable foundations. The world can't be actually frozen in time.

And this connects back to models of the continuum. The mathematician wants to have a number line that can be cut - and the cut is stable. The number line is a passive entity that simply accepts any mark we try to make. It is a-causal - in exactly the same mechanical fashion that Newton imagined the atomism of masses free to do their causal thing within the passive backdrop of an a-causal void.

But Peirceanism would say the opposite. The number line - like the quantum vacuum - is alive with a zero point energy. It sizzles and crackles with possibility. On the finest scale, it becomes impossible to work with due to its fundamental instability.

And regular maths seems to understand that unconsciously. That is why it approaches the number line with a system of constraints. As Zalamea describes, the strategy to approach the reals is via the imposition of a succession of distinctions - the operations of difference, proportion and then finally (in some last gasp desperation) the waving hand of future convergence.

So maths tames the number line by a series of constraining steps. It minimises its indeterminism or dynamism, and looks up feeling relieved. Its world is now safe to get on with arithmetic.

But the Peircean revolution is about seeing this for what it really is. Maths just wants to shrink instability out of sight. Peirce says no. Let's turn our metaphysics around so that it becomes an account of this whole thing - the instability that is fundamental and the semiotic machinery that arises to tame it. Maths itself needs to be understood as a semiotic exercise.

So that would be where semiotics stands in regards to category theory. It is the bigger view that explains why mathematicians might strive to extract some rigid final frozen closed sense of essential mathematical structure from the wildly tossing seas of pure and unbounded possibility.

I would note the interesting contrast with fundamental physics where the crisis is instead quantum instability. In seeking a solid atomistic foundation, at a certain ultimate Planck scale, suddenly everything went as pear-shaped as could be imagined. Reality became just fundamentally weird and impossible.

But that is too much hyperventilation in the other direction. Just looking around we can see the fact that existence itself is thoroughly tamed quantum indeterminacy. The Universe as it is (especially now that it is so close to its heat death) is classical to a very high degree. So all that quantum weirdness is in fact pretty much completely collapsed in practice. Instability has been constrained by its own emergent classical limit (its own sum over possibility).

So where maths is too cosy in believing in its classicality, physics is too hung up on its discovery of basic instability. Both have gone overboard in complementary directions.

Semiosis is then the metaphysics that stands in the middle and can relate the determinate to the indeterminate in logical fashion. Especally as pansemiosis - the nascent field of dissipative structure theory - it is the quantum interpretation that finally makes sense.

Hot damn! ;)

Peirce acknowledged this - as soon as we talk or even think about a color or other quality, it is no longer 1ns in itself. — aletheist

You've been reacting to the word "brute" and missing the reason I applied it.

There is still this tension when trying to look back at talk of freedom, indeterminism, instability, or whatever, from the vantage point of 3ns.

Possibility comes in two varieties - 1ns and 2ns. Firstness is unconstrained possibility and secondness is constrained possibility. So 1ns is more like the notion of pure potential, and 2ns more like the ordinary notion of statistical probabilty (or even a propensity).

So while Peirce may have truly understood vagueness (and I'm not so sure that he did for some particular reasons), his routinely quoted descriptions of it are too much already bounded and precise. If you mention the quality of red, you are already making people think of other alternative colour qualities like purple or green. So there is a fundamental imprecision in his attempts to talk about firstness that then ought to motivate us to attempt to clarify the best way to talk about something which is admittedly also the ultimately ineffable.

Others have noted this too.

Firstness is more or less indeterminate or determinate, not more or less vague or precise; only with Peirce's category of Thirdness can we speak of vagueness versus precision (and then there's also vagueness versus generality).

http://www.paulburgess.org/triadic.html

So that is why - rather paradoxically it might seem - I approach the modelling of vagueness by treating it as a state of perfect symmetry. Meaning in turn, an unbounded chaos of fluctuations that is the purest possible form of "differences making no difference" - that being the dynamical and teleological definition of a symmetry.

Ie: If we have to resort to concrete talk any time we speak about the indefinite, well let's make that bug a feature. Let's just be completely concrete - as in calling the wildest chaos the most unblemished symmetry.

And the reason for making that backward leap into deepest thirdness is so that firstness can become maximally mathematically tractable. We can apply the good stuff of symmetry and symmetry breaking theory to actually build scientific theories and go out and measure the world.

So I didn't talk about this tension over the definition of the idea of "possibility" lightly. I actually don't believe Peirce finished the job. He did not leave us with a mathematical model of vagueness, even if he was pointing in all the right directions.

Hot damn! ;) — apokrisis

In the Zalamea paper on Peirce's continuum, Zalamea says on page 8:

"As we shall later see, this synthetical view of the continuum will be fully recovered
by the mathematical theory of categories, in the last decades of the XXth century."

"This" above refers to Peirce's concept of the continuum. So Zalamea's understanding of category theory seems radically different from yours.

Thanks for your comments and clarification.

Is it? Like when I say that category theory might recover the stablised image of the synthetic in the limit?

It seems you don't understand either category theory (at a philosophical level) or semiosis and are just seeking to nitpick with contradictory sounding quotes.

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?

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

Would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context?

Possibility comes in two varieties - 1ns and 2ns. Firstness is unconstrained possibility and secondness is constrained possibility. So 1ns is more like the notion of pure potential, and 2ns more like the ordinary notion of statistical probabilty (or even a propensity). — apokrisis

That is not how I understand it, unless by "constrained possibility" you mean the actually possible as opposed to the logically possible. Peirce sometimes even distinguished continuous potentiality as 3ns ("indeterminate yet capable of determination in any special case") from pure/ideal possibility as 1ns ("incapable of perfect actualization on account of its essential vagueness"). I associate propensity more with 3ns as habit than with 2ns as brute actuality. Ultimately it depends on the particular type of analysis that we are doing, since all three categories are part of every phenomenon to some degree.

Firstness is more or less indeterminate or determinate, not more or less vague or precise; only with Peirce's category of Thirdness can we speak of vagueness versus precision (and then there's also vagueness versus generality).

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."

It seems you don't understand either category theory (at a philosophical level) or semiosis and are just seeking to nitpick with contradictory sounding quotes. — apokrisis

Your erudition seems to have overtaken your common sense and your manners. You are incapable of explaining, only insulting. I'm using individual quotes as an alternative to copy/pasting pages and chapters of Zalamea. His entire thesis is that category theory has resurrected Peircean synthesis. 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.