I'm guess that you were baffled by the OP's talk of both a general social contract and a variety of more particular sub-contracts.

Don't bother answering. I've already lost interest in your failed attempt at pedantry.

So you agree with me. Cool.

But an individual doesn't agree with many institutions from his society. Take me for example. There's many institutions, cultural trends, etc. which are very dominant, and yet I don't agree with, and I don't want to see perpetuated. — Agustino

And so you demonstrate how entrenched an intolerance for difference can be. You really think yours should be the only institution handing out the subcontracts. You believe deeply in genericity. It just troubles you that your version has so little general hold.

how is it that the individual must perpetuate the agenda of the institutions by having more people that will perpetuate the institutions? — schopenhauer1

But if the institutions do shape the individual, then why wouldn't the individual - in at least a general way - not want that to continue? In wanting that from the institution, the individual is simply saying, if we are to have more, let them be like me. What would or could possess the individual to have a different desire.

Even nihilism and anti natalism are subcontracts or local institutions. They shape mindsets. And those individuals - yourself for instance - certainly seem to want to create more of just the same mind. So why do you perpetuate that agenda? Don't you find it logical as it ensures the longevity of your particular institution and increases thus the likelihood of ever more of you?

(Of course if this subcontract involves a quick suicide or a conscious failure to breed, then it will soon be a forgotten trope - defined by its production of the generically incapable.)

If you say that evolution has created humans that have minds that want to promote survival through a certain cultural means, then this is simply restating the idea that institutions are perpetuated, you are just throwing in the word survival which is essentially the same thing at the species level, but not addressing the fact that it is still begging-the-question as to why keep the institutions going. — schopenhauer1

It's not question begging. It is logical for the individual to want more of much the same. Then evolution makes sure that sameness is tracking whatever actually works.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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! ;)

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.

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?

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.

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.

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.

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.

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.

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.

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.

But does 1 = 1.000... in your book?

Or is there some reason why we don't have to treat it as a convergent limit as well?

↪aletheist It's you. — tom

Ha. Tom really isn't keeping up with the argument.

I might even agree with that whole post, if I am understanding it correctly. — aletheist

Yep. SophistiCat is talking the language of constraints.

The values ARE equal. There is NO difference between them. — tom

So by 1, do you really mean 1.000... ? ;)

Geometrical objects are nothing except relations, that's why they are actually impossible in the real world - for example a line has no thickness — Agustino

If you read what I said, I did say that maths is the projection of images of perfection on to the imperfect world of experience.

So the difference in approach would be that biologists think in terms of constraints, development and semiosis - all the good top down stuff.

Thus a line is understood not as a construction of points but a constraint on a freedom. The 1D line is the limit of the 2D plane. So it is not an issue of how thick it might be. It is about how thin it has managed to develop. It is not an issue of a relation that connects two points, but the degree to which more generalised states of relating (of which the two dimensions of a plane are merely the start) have been suppressed.

Constraints speak to an apophatic or negative space approach to existence - even the existence of geometric relations. And that in turn requires a machinery of context or memory. Or semiotically speaking, habits of interpretance which could fix geometrical relata - such as "a line" - as a sign of something mathematically concrete.

So that is what biology brings to the table. An innate understanding of constraints based, or semiotic, thinking. You can still arrive at the classical Euclidean image of geometry, but from exactly the opposite end of the spectrum of causal process. ie: not starting with atomistic construction.

Charles Sanders Peirce likewise took strong exception to the idea that a true continuum can be composed of distinct members, no matter how multitudinous, even if they are as dense as the real numbers. — aletheist

Getting back to the Peircean conception of continuity, what comes through in that paper for me is the Gestalt nature of his argument. From the recognition of imperfect nature we can jump to a knowledge of what perfect nature would be like. If we see a fragment of counting, we can leap to the whole that would be the continuum. If we see a rough drawn triangle in the sand, we can leap to the ideal that would have perfect triangular symmetry.

So the general mental operation here is that the very imperfection of things in the world is in itself the springboard to an understanding of the what perfection would then antithetically look like. We only have to look around to already start to see the ideal.

And so that is then really saying that to recognise something as a broken symmetry is the start of seeing the symmetry that could have got broken. Thus insight is abductive. We see through the imperfections to find the symmetry that could permit them as its potential blemishes.

So the continuum, as a number line, is a symmetry - a translational symmetry. And it can be blemished (cut or marked at points) with infinite possibility. The infinite or perfect symmetry of the continuum is what reciprocally permits an infinity of possible symmetry breakings. That is, absolutely any mark - no latter how slight or infinitesimal already is a blemish on perfection. The absoluteness of the one (there is only one way to be perfectly symmetrical) is in complementary fashion the guarantor for unlimited potential breakings of that symmetry. Just anything could muck up the continuity and create a discontinuity.

So this all gets cashed out in the ultimate notion of symmetry and symmetry breaking. That would be the mathematically general view. We gain knowledge of Platonic abstracta by noticing that shapes or patterns or relations in the world have imperfections that could be eliminated to produce versions with higher symmetry. So our job is then to eliminate all imperfections until we arrive at the symmetry limit - the absolute perfection that is a state where difference finally ceases to make a difference.

Take a triangle and seek its most perfectly regular form. You have to arrive at an equilateral triangle. There is no other choice with fewer differences that make a difference.

So this is topological thinking (topology being the discovery of geometrical symmetry by letting connecting relations "flow" under a least action principle). The continuum as a numberline is this kind of flow towards a symmetry limit. It is the imagining of a perfection that can then be disturbed by the slightest imperfection.

But still, this is rather a little too conventional to be completely Peircean. Perfect symmetry here is being described as if it were a static and eternal kind of state. But Peirce believes in action or spontaneity too. So really a symmetry state actually is an unbounded rustling of fluctuations - like the quantum vacuum with its zero point energy. It is alive and active - but at equilibrium. It is a symmetry in the deeper sense that difference is unbound, but difference can't make a difference.

I've illustrated this in the past by talking about the relativity of rotational and translational symmetry in Newtonian mechanics. Spin an unmarked disc and you can't tell if it is even spinning, let alone in what direction or how fast. Any actual rotation is a difference that makes no difference to the perfection of the disc. And this active (or inertial) form of symmetry has crucial consequences for physical reality - as known from Noether's theorem and the conservation of energy principle.

So anyway, the continuum is the kind of perfect symmetry which can thus reciprocally accept an infinity of potential imperfections. It can be marked in an unlimited number of ways ... by acts like assigning an order to a sequence of numbers.

And this is essentially a top-down or constraints-based logic, not the usual bottom-up constructive view (where lines are a bunch of points glued together).

That is, you can pin down the location of some number by a succession of limitations - such as determining what might bound it to either side. That of course also leaves the remaining identity of the part thus contained still fundamentally indeterminate - a fragment of the continua awaiting its further determination. However that is not a big concern because you can still define any number you like with as much precision as you choose.

In principle one could count for ever, or calculate ever more decimal places for pi. But for purely pragamatic reasons, indifference will rightfully kick in once your purposes have been sufficiently served. And symmetry is itself defined by the arrival at differences that don't make a difference.

I have never and I repeat never said that I think mathematics has the slightest thing to say about reality. — fishfry

But that would be just as bad from my point of view because no one could deny the "unreasonable effectiveness" of maths.

In my own lifetime, it has been a shock the inroads that maths has made on "chaos". I remember reading Thom's SciAm paper on catastrophe theory as an undergrad and thinking this sounds neat - but all a bit out there. Then a trickle became a flood.

So the issue for me is that clearly maths does say something deep about metaphysics. Yet then - something which theoretical biologists have particular reason to be alert to - maths also fails to supply the whole story (for the reasons I've outlined at some length).

Therefore I am neither a Platonist nor a social constructionist when it comes to foundational issue. However in the end I am quite Platonist in believing maths is no accident. It describes the inevitable structure of any reality. Which again is the controversial Peircean Metaphysical position - the idea that existence itself can be conjured into being as a matter of mathematical necessity (the actual maths being the scientific project still in progress of course).

Anyway, if you are interested in the "alternative view" of the connection between mathematical models of infinity and the reality of such models, then a good book is Robert Rosen's Essays on Life Itself.

What I am saying is that they, as are all symbols, are an absolute awful representation of the experience itself. — Rich

But the point of semiosis is to get away from that very notion that either cognition or experience are "representational" - data displays in the head. Re-presentation doesn't fly at any level for mind science. It just leads to homuncular regress. That is why the idea of sign relations has so much more to recommend it.

You could make the same argument for your brain's neural codes. You could complain that changes in firing rates of the ganglion cells in your eyeballs are an "absolutely awful representation" of electromagnetic radiation. The colour red is nothing like what is really happening in the physical world.

But that would be obviously silly. And so in the end is any complaint about semiosis being deficient in representing the "thing in itself" ... even the phenomenal thing in itself. Because semiosis - of which mathematics is our most refined example - was never about re-presenting anything in the first place. Instead it is all about structuring our working relationship with the world.

The implication being 'people like this only recite what they're learned in class, they're not really capable of philosophy'. — Wayfarer

Remind me next time you go around accusing folk of Scientism. I too will get all PC on your arse.

Meanwhile note that the fair implication of what I wrote is not that the toms and fishfrys are incapable of doing philosophy, but that they are complacently accepting an institutional reason for their particular philosophical stance. As that is what I in fact said.

And coming on here - as a philosophical forum - it is fair enough that this gets criticised.

Again, on what grounds precisely?

Telling someone to fuck off is simply vulgar. — Wayfarer

Fishfry started it. And I am keeping the joke going to make a serious point.

My initial remark was mild - talking of "toms and fishfrys" in a generalised fashion. You are now calling that "brutal" and "personal". I am illustrating to you what brutal and personal actually sounds like - using fishfry's own escalatory terminology.

So again, justify your case if you think you have one.

Sure didn't read like that to me. Seemed a pretty brutal put-down, actually. — Wayfarer

So this is honestly your idea of a brutal put-down?...

So this is why the Toms and Fishfrys are so content with what they learn in class. To the degree that philosophy can still make a feeble groaning complaint about incompleteness, they feel utterly justified not to care.

Well you can just go fuck off. ;)

But this is an online forum, not an academic debate. Oops, putting it that way is likely to have the opposite effect of what I intend ... :s — aletheist

Yeah. So there are a variety of threads - many purely social. But clearly you are hoping - like me - for a properly scholarly discussion with references on request and something philosophically meaningful at the heart of it.

And perhaps it is because I have focused on interdisciplinary matters that I am used to people calling each other out on their institutionally embedded presumptions. But I dunno. It seems a basic hygenic principle in the sciences at least.

Look, you and fishfry are two of my favorite participants here, so I hope that we can all dispense with any unpleasantness and get on with what has already been a fascinating and helpful thread, at least from where I sit. — aletheist

Thanks. And I'm fine with that. As I've tried to point out to fishfry, I wasn't really attacking him personally but the institutionalised way of thought he was representing.

It is like how you (as far as I can tell) take a deeply theist reading of Peirce, and I the exact opposite. But at the end of the day, the philosophy itself seems strong enough to transcend either grounding. So I can argue even angrily against any hint of theism while also conceding that it is still "reasonable" in a certain light. (I mean there has to be for there to be something definite in any consequent argument.)

The answer is in the concept of "occupies a place." If we view the real numbers as specifying locations on a line, and we stop talking about points, perhaps things are less muddled. I agree with you that nobody knows how a line of dimension 1 can be made up of points of dimension 0. But math has formalisms to work around this problem. Would you at least agree that if math hasn't answered this objection, it's been highly successful in devising formalisms that finesse or bypass the problem? — fishfry

Of course I agree that maths is highly successful. But what you call finessing, I am calling being studiedly indifferent. So yes - a thousand time yes - maths has developed spectacular calculational machinery. But then - because it has replaced reality with a mechanical image of reality - it fails equally spectacularly when it tries to "do metaphysics" from within its virtual Platonic world.

Getting back to the physics of numberlines, I would point out that what has gone missing in the imagining is the idea of action - energy, movement, materiality. So we can mark a location (in the spacetime void) and it just sits there, inert, eternal, unchanging ... fundamentally inactive. That is the mathematical mental picture of the situation in toto.

However why couldn't this marked location dance about, appear and vanish, erupt with all sorts of nonsense ... rather like an actual mathematical singularity?

So what we point at so confidently as a point in a void could be a dancing frentic blur - a vagueness - on closer inspection. We say it has zero dimensions, and all the properties so entailed, but how do we know that a location exists with such definiteness? And why is modern physics saying that in fact it cannot (following Peirce's logical/metaphysical arguments to the same effect).

I'll forego the opportunity to start a pissing match here and you earned a lot of good will with me for pointing me to the Zalamea paper. But really, was this necessary? Maybe I should just say fuck you or something. Would that serve any purpose? — fishfry

You are very sensitive. I apologise if you have feelings that are easily hurt. But is this my problem or your problem?

I'm used to a robust level of discussion in academic debate. One hopes that others will try to knock seven shades of shit out of one's arguments. And then afterwards, everyone shakes hands and go gets a drink at the bar.

So you are welcome to be as rude to me as you like. Water off a duck's back. But what I am looking for from you is a genuine counter-argument, not a solipsistic restatement of your position ... or as I said, a restatement of a particular institutional view that is widely held for the pragmatic reasons I've previously stated.

Perhaps there's a sort of Heisenberg uncertainty between truth and precision. What we can say truthfully is imprecise; and what we can say precisely isn't true. — fishfry

This is rather the point of Peircean semiotics. We deal with reality by replacing it with a system of completely definite signs. And mathematics is simply the most powerfully universal method of imposing a system of sign on our perceptions of reality.

So yes, again the way maths organises itself institutionally is completely pragmatic (under the proper Peircean definition). It is exactly how you go about modelling in as principled a fashion as possible.

But the philosophical irony is that it is all about replacing reality with a model of reality. We tell reality to lose all its imprecision, vagueness, indeterminacy, etc. We are just going to presume that it might be a bit of a hot mess, yet what reality itself really wants to do is be completely crisp, definite, determinate ... mechanical. So our job is then to see reality in terms of its "own best version of itself".

We don't feel guilty about treating reality as being Platonically perfect, properly counterfactual, fully realised, because ... hey, that's what reality is striving to be. The fact that it always falls shorts, never arrives at its limits, is then something to which we studiously avert our eyes. It is a little embarrassing that reality is in fact a little, well, defective. The poor sod doesn't quite live up to its own ambitions. But we generously - in our modelled reality that replaces the real reality - simply ignore its shortcomings and marvel at the perfection of the image of it that lives in our imaginations.

What I am trying to draw attention to here is how we take reality for more than it actually is, and not only is that socially pragmatic (good for the purposes of building perfect machines) but it feels even psychologically justified, as we spare reality's own blushes. We know what it was trying to achieve.

However eventually we will have to turn around and deal with reality as it actually is, not our Platonic re-imagination of it. Which is where Peircean semiotics - as the canonical model of a modelling relation - can make a big difference to metaphysics, science and maybe even maths.

When we mark a point on a line, we introduce a discontinuity. — aletheist

That is the flipside of this. Wholes must exist to make sense of parts. But those wholes must crisply exist and not be indeterminate. And those only crisply exist to the extent they are constructed as states of affairs. Thus crisp parts are needed too, leading to the chicken and egg situation that a logic of vagueness is needed to solve.

So the discrete vs continuous debate is doomed to circular viciousness unless it can find its triadic escape hatch. And this is where semiotics really has its merit. It introduces the hierarchical world structure - the notion of stabilising memory (or Peircean habit) - by which the part (the event, the point, the locality, the instance) can be fixed as a sign of the deeper (indeterminate) generality.

That is, there is the "we" who stand outside everything and produce the cuts - make the marks that point - to the degree they satisfy "our" purposes.

And this is all the Cantorian model of the reals does. It produces a tractable notion of the discrete vs the continuous to the degree we had some (mathematical) purpose.

The Zalamea article puts this nicely in stressing how the mathematical approach to the protean concept of continuity proceeds by "saturating" degrees of constraint. It starts with the bluntly assumed discontinuity of the naturals. Then tightens the noose via the successive operations of the notions of "a difference", "a proportion", "a convergence to a limit". The gaps between numbers gets squeezed until they finally seem to evaporate as "that to which there is a mark that points".

That is, the gaps are rendered infinitesimal in a way that they truly do become the (semiotic) ghosts of departed quantitiies. They become simply a sign that points vaguely over some imagined horizon ... the mathematical equivalent of the old maps indicating the edge of the world as "here be dragons". Once we get to the convergence that is the real numbers in their unfettered multiplicity, maths is left pointing to its own act of exclusion and no longer at anything actually real.

As I have said, that is fine for maths given its purposes. It is itself a tenet of pragmatism that finality defines efficiency. Models only have to serve their interests and so - the corollary - they also get to spell out their limit where their indifference kicks in.

Cantorian infinity is just such an example of the principle of indifference. Actual continuity has been excluded from the realm of the discrete ... to the degree that this historical vein of mathematical thought could have reason to care.

So this is why the Toms and Fishfrys are so content with what they learn in class. To the degree that philosophy can still make a feeble groaning complaint about incompleteness, they feel utterly justified not to care. They are trained within a social institution that had a purpose (hey guys, lets build machines!) and the very fact of having a definite purpose is (even for pragmatists) where an equally sharp state of indifference for what lies beyond the purpose to be fully justified.

Unfortunately for scientific purposes, the world isn't in fact a machine. We know that now. But while mathematics is groping for a sounder foundations - see category theory - it hasn't really got to grips with the new semiotic principles that would be a better model of reality than the good old machine model of existence.

You can't subdivide a point and a point has no sides. It's sophistry to claim otherwise. If we can all agree on anything, it's that a point has no sides. — fishfry

So you defined a point as a howling inconsistency - the very thing that can't exist? The zero dimensionality that somehow still occupies a place within a continuity of dimensionality?

Philosophy can't even get started here if you are happy with sophistry by axiomatic definition.

So yes, the properties of a continua with zero dimensionality would have to be as you describe. But then that simply defines your notion of a point either as a real limit (a generalised constraint - thus a species of continuiity) or as a reductionist fiction (a faux object that you inconsistently treat as existing in its non-existence).