• Continuity and Mathematics
    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.
  • Continuity and Mathematics
    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.
  • Continuity and Mathematics
    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.
  • Continuity and Mathematics
    everything actual is indeterminate to some degreealetheist

    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 contextaletheist

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

    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.
  • Continuity and Mathematics
    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.
  • Continuity and Mathematics
    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?
  • Continuity and Mathematics
    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).

    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.
  • Continuity and Mathematics
    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! ;)
  • Continuity and Mathematics
    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.
  • The Implication of Social Contract on Social Relations
    So you already dismiss the alternative that the social relations are the source of the personal individuation? The capable individual is what society in fact has in mind?
  • Continuity and Mathematics
    From Googling around I think being triadic is what a mathematician would call ternaryfishfry

    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.
  • Continuity and Mathematics
    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 - )

    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.
  • Continuity and Mathematics
    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.
  • Continuity and Mathematics
    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.
  • Continuity and Mathematics
    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.
  • Continuity and Mathematics
    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.