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never actually delivered a logic of vagueness in a way that would save us having to read between the lines of his vast unpublished corpus.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).

When you divide a line at a point, the point stays with one segment and not the other. As someone trained in math, it's hard for me to understand how this answer isn't satisfactory — fishfry

How can that be satisfactory in a philosophical sense? If you can divide the point on one of its sides, why can't the next cut divide it to its other side, leaving it completely isolate and not merely the notion of an end point of a continua?

And a better paper on the Peircean project is probably...

http://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf

Continuity can only be relative to discreetness (at least in actualised existence). That is, continuity Is defined by the lack of it other. So even spacetime as generalised dimensionality would be only relatively continuous. And that is what physics shows both with the quantum micro scale and also the relativistic macro scale (where spacetime is "fractured" by he event horizons of its light one structure).

Can you (or anyone) supply some of relevant Bergson and Pierce links that would shed light on the relation between the mathematical real numbers and the philosophical idea of the continuum? — fishfry

You are better off asking aletheist that as that is his argument. And I am certainly no Bergsonite.

I am not convinced that this is true. Two of Peirce's major objectives for philosophy were to make it more mathematical (by which he meant diagrammatic) and to "insist upon the idea of continuity as of prime importance." Surely he must have considered these efforts to be complementary, rather than contradictory. — aletheist

I think Rich is right that maths is generally premised on the notion of atomistic constructability and so is anti-continuity in that sense. (And that is not a bad thing in itself as constructionist models - even of continuity - have a useful simplicity. Indeed, arguably, it is only by a system of discrete signs that one can calculate. And signs are themselves premised on understanding the world in terms of symbolic discontinuities of course - signs are no use if they are vague.)

So then the holistic reply to this routine mathematical atomism would be a countering mathematics of constraints - of pattern formation calculated via notions of top-down formal and final cause. And that is damn difficult, if not actually impossible.

This would be why Peirce felt his diagrammatic logic was so important. Like geometry and symmetry maths, it tries to argue from constraints. Once you fence in the possibilities by drawing a picture with boundaries, then this is a way to "calculate" mathematical-strength outcomes.

So yes, there is no reason why a construction-based maths should not be complemented by a constraints-based maths. And arguably, geometry illustrates how maths did start in that fashion. Symmetry maths is another such exercise.

However to progress, even these beginnings had to give way to thoroughly analytic or constructive techniques. Topology had to admit surgery - ways that cut apart spaces could be glued back together in composite fashion - to advance.

So that is at the heart of things here. For a holist, it is obvious reality is constraints-based. So regular maths is "wrong" in always framing reality in constructivist terms. And yet in the end maths is a tool for modelling. We actually have to be able to calculate something with it. And calculation is inherently a constructive activity.

So while we can sketch a picture of systems of constraints - like Peirce's diagrammatical reasoning - that is too cumbersome to turn into an everyday kind of tool that can be used by any schoolkid or universal turing machine to mechanically grind out results.

Of course, that kind of holistic reasoning is also then absolutely necessary for proper metaphysical level thinking, and diagrammatical reasoning can be used to advance formal arguments in that way. You have probably seen the way Louis Kauffman has brought together these kinds of thoughts, recognising the connections with knot theory, as well as Spencer-Brown's laws of forms. And I would toss hierarchy theory into that mix too.

So construction rules the mathematical imagination as tools of calculation are the desired outcome of mathematical effort.

While that doesn't make such maths wrong (hey, within its limits, it works I keep saying), it does mean that one should never take too much notice of a mathematician making extrapolations of a metaphysical nature. They are bound to be misguided just because they hold in their hands a very impreessive hammer and so are looking about for some new annoying nail to bang flat.

I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. — aletheist

Again this is an example of rationally seeking a way for the part to speak for the whole. What can't be achieved via actualisation can be supported by appeal to the existence of a local property - in this case, not bijection but a quick demonstration that any nameable number implies in its own syntactic construction a number immediately larger (or immediately smaller).

Tom is also employing this local syntactic property.

So yes, bijection seems more abstract a level of definition because it maps maths to maths rather than maths to physics (ie: syntactic spaces where time is still part of the deal - as in saying any time you name a number, the next higher number awaits). But still, the general mathematical tactic is the same - seek a local property that constructive principles will guarantee stands for the truth of the whole. And thus, the very nature of this tactic reveals the deeper questionable presumptions that metaphysics would be interested in.

It is the idea that reality is perfectly constructible that is questioned by a synechetic or holistic point of view.

But then even a simple holism falters - the idea of the continuum being instead " the foundational". The continuum is that to which an infinity of cuts can be made. If a division is possible, another one right next to it ... but spaced by the infinitesimal of some continua ... must be possible. So simple holism is simply the inverse problem. Although - like division as an arithmetic operation - there is an advantage that at least it is being flagged that there is a more primitive presumption about there being in fact a pre-existent whole (that gets cut or divided).

So simple holism brings out the fact that simple constructionism is presuming an infinite empty space that can be filled by an unbounded act of counting. The standard atomistic approach presumes its numerical void waiting to be filled. And even bijection just illustrates the presumed existence of this numerical void as a waste disposal system that can swallow all arithmetical sequences. You can toss anything into the black hole that is infinity and it will disappear without a splash.

So the simplest view treats infinity as the void required by atomic construction. The next simplest view treats infinity as a continuum - a whole that is in fact an everything, and so able to be infinitely divided.

Then obviously - as usual - there is the properly complex view where instead of an atomistic metaphysics of nothingness, or even the partial holism of a reciprocal everythingness, we arrive at the foundational thing of a vagueness as that deepest ground which can be divided towards this reciprocal deal of numerical construction vs numerical constraint, the filling of a numberless void vs the breaking of a numberful continuum.

Of course, none of this deep metaphysics need trouble those only concerned with ordinary maths. They can believe that Cantor fixed everything for atomistic construction and the story ends there.

But deep metaphysics makes the argument that the very act of trying to cut is what produces the divided that appears to either side. The continuum arises because it is cuttable. Which like the Chesire Cat's grin, sounds really weird to those only used to everyday notions of logic or causality where something - either everything or nothing - has to be the starting point or prime mover for any chain of events.

f you think to yourself, "The natural numbers, the integers, and the rational numbers are examples of foozlable sets," you will not confuse yourself or others by shifting the meaning of a technical term to its everyday meaning. — fishfry

I think the issue here has been metaphysical - so neither everyday, nor mathematical. Although the mathematics of course has to have some grounds for finding its own axiomatic base "reasonable".

So the Zeno paradox is about a particular difficulty between a mathematical operation and the world we might want it to describe. The math seems to say one thing, our experience of the world another.

Bijection is great. It replaces the need for a global act of quantification (demonstrating an example of infinity by showing a sequence is measurably unbounded) with a local demonstration of a quality (if bijection works for this little bit of a sequence, then that property ensures the infinite nature of the whole). So bijection doesn't do away with the notion of counting or a syntactic sequence. But it does extract a local property that rationally speaks for the whole.

No problems there.

And then we get back to the metaphysics on which even the mathematical intuitions are founded. Which was the issue the OP broaches and which you are side-tracking.

Here is the text box definition, pulled from one of my statistics course books. — Jeremiah

That would be why probability ranges from 0 to 1 then? Categorical differences are measured relatively in fact?

Positivity doesn't exist unless it's in the company of negativity. So if we're just dealing with 2 dimensions, left and right are negatives of one another. It's more complicated if we add that third dimension. — Mongrel

Symmetry broken simply is symmetry broken on just a single scale. So it is easily reversed. There is no real separation of what just got separated and so there is nothing stopping a distinction immediately erasing itself.

That is what literally happens with "positivity and negativity" when it comes to fundamental particles. They pop out of the quantum vacuum in opposing pairs (as the conservation laws derived from symmetry mandate) and then annihilate so fast that physics ends up calling them virtual.

To get a persistent symmetry breaking requires a "third dimension" - a breaking over scale that creates an effective state of separation or asymmetry. Stuff has to be put far enough apart from itself so it can do something else while it takes its time to - by the end - just annihilate.

With our actual Universe, there is a complex charge asymmetry built in because "raw matter" could fall into several different local symmetry-breaking arrangements. You could have the quarks with their eight-fold way that left a sufficient excess of positive protons. Then you had the leptons which - after an entanglement with the further symmetry-breaking of the Higgs fields - eventually left a sufficient excess of negative electrons.

So right there - in a series of complicated symmetry breakings that turned out to have the makings of an actual asymmetry - you have an illustration of reality being a something because it got separated across scale (thermal scale, as heat all this asymmetric residue and you can return it to its Big Bang equilibrrium where all particles are simply virtual fluctuations of a vanilla force).

I've still going to have to think about what you've said more, do a little reading on the subject before I get back to you. — Wosret

No problem. I understand it is a dense issue. But as SX indicates, we can deal with actual similarity and difference in the world with an apparent intuitive ease that belies the underlying metaphysical complexity.

And that complexity is what gets revealed as soon as we instead start to ask how a difference comes to make a difference. That question is like finding the loose end of a woolly jumper and beginning to pull.

Well, right, left, up down are all positive things. "not-me", "not-us" and "not-shit" are not, and could really conceivably be anything at all except for me, us, and shit.. — Wosret

This illustrates particularity. We seem to start metaphysics with a brute something. There is the positivity of some concrete proposition - that is then either true or false.

But look closer and you can see here that the brute somethingness points to an "otherness" of two possible kinds - the more general, or the more vague.

All the not-As might accounted for as by a concrete generality of some constraint that then defines the nature of what may count as a certain genus of particular. It might be the "me" that is a subset of the "we", or the "shit" that is defined in contrast to the undigested banquet.

Or the not-A might simply refer to the indeterminism that is by contrast the generalised lack of such a determining context. Or in other words, it refers to the freedom or contingency that is also an equally inescapable aspect of reality. It might be the random seeming collection of "me, apples, tanks, galaxies". The "other" being spoken of via the logical construction of "not-A" could be just every kind of stuff. So just, in semantic effect, a vagueness.

Thus once more, a complex triad is revealed at the heart of conventional monistic thought.

From the particular - viewed as some brute substantial particular - you can talk about the "other", the not-A, as either the vague or the general. So that is something to be further specified in any attempt to apply logic to ontology.

Peirce made the difference clear enough. He argued that the law of the excluded middle does not apply to generality, while it is the principle of non-contradiction that fails to apply to the vague.

So within the (triadic) laws of thought, this important distinction between generality and vagueness is perfectly well defined (along with particularity as being that which to all three laws of thought then do apply).

You mentioned the relevance of transversing the Planck scale. And while I applaud taking the physical facts seriously, in fact any exactness of location results in a complementary uncertainty about momentum (or equivalently, duration).

So if you talking about a physical continuity on the Planck scale, your attempt to mark the first location would already then have your fixed point transversing the whole distance to its resulting destination.

It is like the way a photon is said to experience no time to get where it is going. Travelling at c means the journey itself is already described by a vector - a ray rather than a succession of points.

So in the real world, locating your starting point is subject to the uncertainty relation. The Planck scale is the pivot which prevents you reaching your goal of exactitude by diverting all your measurement effort suddenly in the opposite direction. In effect you so energise the point you want to measure that it has already crossed all the space you just imagined as the context that could have confined it.

Zeno definitely does not apply in quantum physical reality.

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous. — aletheist

MU is right that it has to be more complex than that. Talk of actually counting smuggles in the necessity of the maker of the infinesimal divisions or Dedekind cuts.

For there to be observables, there has to be an observer. Or for the semiotician, for there to be the signs (the numeric ritual of giving name to the cuts), there has to be a habit of interpretance in place that allows that to be the ritualistic case. Which is why the number line itself is just a firstness or vagueness. In the ultimate analysis it is the raw possibility of continua ... or their "other", the matchingly definite thing of a discontinuity.

So infinity and infinitesimal describe complementary limits - one is the continuum limit, the other the limit on bounded discreetness, the limit of an isolate point.

Thus counting presumes an observer then able to stand inbetween. The counter can count forever because the counter also determines the cuts that pragmatically "do no violence" to the metaphysics, at least as far as the counter is concerned.

My point is thus that an observerless metaphysics is as obtuse as an observerless physics, or theory of truth, or observerless anything when it comes to fundamental thought.

Great. The essential thing is not to be scared of complexity.

Metaphysical analysis always arrives at dichotomous contrasts. Logical intelligibility itself demands a world divided into what is vs what is not. The problem is that this has to work for both sides of the equation. So the "what is not" has to be still something else itself - whatever it is that can make the "what is" what it is.

So analysis sounds like it demands the resolution of a monadic outcome - the arrival at the fundamental via the rejection of all that is superficial, or contingent, or emergent, etc. Yet the fact is the dichotomy - the dyadic relation - is irreducible. You can't have any notion of the "what is" in the absence of the complementary notion of it being precisely "that which is not what it is not".

So there is a doubled or recursive negation at work. Monadicity can only arrive at itself via the denial of its own denying. The essentially self defeating nature of monadic metaphysics is thus revealed. It others othering and thus falls into inconsistency even with itself.

Thus the dichotomy forms the irreducible basis of intelligible existence. It both finds the natural divisions of being, and relates them as each other's other. Each is the others limit.

Having established that, we also establish that we are thinking in active and developmental terms, not passive and brutely existence ones. Existence is revealed as having a necessary history - as divisions must both arise and terminate. Which is where you get the thirdness or triadicity that is the ultimately irreducible metaphysical state. So yes, what is fundamental is not twoness, let alone oneness, but threeness. Three is the number of actual complexity.

A further point is that to cash all this out in terms of some actual world requires a global state of asymmetry - or scale symmetry. That is, if reality is constructed by a symmetry breaking of pure possibility, then this breaking must happen freely and completely across all available scales of being.

In terms of cosmological theory, the results must be homogenous and isotopic - invariant with the scale of observation. That is why fractal maths are found everywhere where nature is at its most simplest. Zoom in or zoom out, the fractal world looks always exactly the same. And that is because the dichotomy or distinction being expressed is being expressed fully over all possible scales. It is the same damn thing - the same damn seed asymmetry - absolutely everywhere.

The Koch triangle shows this in its fractal generator, which is the simple asymmetry of natural log2/ natural log3 (or fractal dimension of .63).

To unpack this, the Koch triangle fractal is a line divided into three and then the middle segment sprouting the two sides of a further triangular bump. So a line buckles in the simplest imaginable fashion. That gives you the seed ratio - the 3/2. And then the natural log simply forces the growth of that act of buckling over every possible scale. You thus have two exponential actions in a constantly specified balance. The result is a mathematical model of perfectly complete asymmetry - or rather, the emergence of a new axis of scale symmetry, a fractal dimension that stands in the middle of two bounding extremes of action (between the flatness of the line that gets radically broken, and then the curvature of the buckling that is a departure from the now radical thing which is to be instead flat and "a line all the same with itself").

So it may seem a bit of excursion to talk about the maths of fractals. On the other hand, it is a fact that the new maths of complexity (fractals, scalefree networks, universality, criticality, etc) gives a picture of reality that is precisely the kind of irreducibly triadic metaphysics I just described.

So don't expect monadic metaphysics to be right. Expect the dichotomies or symmetry breakings that point to the broken symmetries or equilibrated outcomes that are then in turn their natural "scale symmetry" limits (or, the same thing, their states of asymmetric or hierarchical final order).

Ie: The maths of complexity has vindicated this irreducibly triadic vision of nature during the past 40 years.

Similarity and difference are a metaphysical dichotomy. So each is defined in terms of being not the other. Or rather, in practice as the breaking of a symmetry, the least like each other as possible by each being as far apart as possible as states of being or categories.

In being two poles being differentiated, then brings in the further thing which is the vagueness or firstness that they divide. They are both crisply actual - as limits - of what was the purely possible.

This furthere "in reference to" also manifests (confusingly) in the crisply divided outcome. The world that emerges between two opposed limits (here the similar and the different) is iteself everywhere some mixture or equilibrium balance of the two categories. So the world itself does sit in the middle - with this concrete mixture of states being found to be the same blend over all observable scales.

So that is the basic set up - for all metaphysical dichotomies. They speak to the firstness that is their common vague origin (the symmetry that got broke) as well as the thirdness which is their own completely mixed state of being - the further thing of having become broken in the limit and arriving at an equilibrium balance.

Of course it may sound crazy to talk of similarity and difference as being themselves united and divided. Or instead, that they are united in initial vagueness, then concretely divided by a logical symmetry breaking, and then reunited by the emergent symmetry of being as mixed together as they can possibly be, is the feature here. The developmental trajectory involved of firstness, secondness and thirdness describes itself in terms of itself.

Anyway, it means that for there to be a world, similarity and difference must be a division concretely respected over all scales of differentiation (and hence integration).

Now we can get down to the detail of the mechanism.

SX makes the standard semiotic point that to be an actual difference, a difference has to make a difference. So difference itself is divided into the meaningful vs the meaningless, the signal vs the noise, the teleological vs the contingent. Thus now we do bring in the active or causal nature of being.

The alternative view is that existence is a passive brute fact. It has no reasons. Difference or similarity has no meaning. It is all just arbitrary labels for a world that has no developmental story and thus no reasons for its apparently definite state of organisation.

But here I have described a developmental or process metaphysics where existence is an emergent equilibrium state where change keeps changing, but by the end further change can make no difference. It is like a new pack of cards. Once the deck is well shuffled, continued shuffling makes no effective difference. It does make a difference to the exact order, but now such differences are a matter of indifference. When the deck is as random as possible, it can't be made more random.

So yes, this all seems now a rather mindful or psychological kind of metaphysics. Similarity and difference are relative judgements that are about differences that make a difference (in breaking the symmetry of a state of similarity which is another word now for a state of indifference).

But again, that bug is really a feature. It brings minds or observers firmly within the metaphysics of actual being. It unites epistemology and ontology in making meanings and thus purposes part of the world.

The final twist to bring an organic or pansemiotic metaphysics into focus is then understanding the triadic relation in causal terms as the hierarchical contrast between constraints and freedoms. One is top down causality, the other acts in causally bottom up fashion.

So similarity is enforced on natural possibility by general constraints. Worlds as states form constraining contexts. They limit free possibility in particular ways. And all objects or events thus limited are the same in that fashion. They all participate in that particular form.

But then difference stil exists. That is what freedom means. Spontaneous and unconstrained in some regard. So accidents and contingency are also fundamental in this organic picture of nature. They too exist over all scales of being. (The statistics of fractals or power laws being the signature of actual natural systems for this reason.)

So now we have that triadic set up. There are general constraints. And there are particular freedoms. Then there is the rule of indifference in operation that marks the emergent boundary where now further differences fail to make a difference to the general state of things - which, dichotomously, also then defines the differences that do make a difference.

So if enforcing similarity is the telos of a constraint, then that also means that eventually the world becomes equilibrated - like a well shuffled deck - and so apparently only composed of a whole bunch of accidents. The differences that don't make a difference become the apparent ground of being because they are what get left once the development of a world has arrived at the dichotomous satisfaction of it's own symmetry breaking desires.

Contingency rules when organisation has had its say. Existence is a bunch of indifference (a heat death) in the end.

But the story of how it gets to that fate is the bit that is metaphysically interesting.

A true continuum is infinitely divisible into smaller continua; it is not infinitely divisible into discrete individuals. — aletheist

The story in a nutshell. Points are a fiction here. The reality being modelled is the usual irreducibly complex thing of a vector - a composite of the ideas of a location and a motion...

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

....and the corollary is that what is being counted is not points but (Dedekind) cuts. The numbers count the infinite possibility for creating localised and non-moving discontinua.

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

The cut bounds the continua in question. So the continua has already been "traversed" in the fact there is this first cut. You are then asking how near the other end of the cut continua can be brought in the direction of the first cut in question. The answer is that it can be brought arbitrarily close. Infinitesimally near.

So you are creating difficulties by demanding that continua be constructed by sticking together a sequence of points. However there is no reason the whole story can't be flipped so that we are talking about relative states of constraint on a continuity - or indeed, an uncertainty - when it comes to the possibility of some motion, action, or degree of freedom.

Yep, simple isn't it. If you actually break things apart, they are no longer in a relation.

Again, close reading will show that I stress that this is about "directions" and "extents", and so the intrinsic relativity of a logical dichotomy is presumed. Your pretence otherwise is just trolling.

Don't pretend to be so dim. Maximising the separation is night and day different from breaking the connection.

Are your close reading skills really as challenged as you pretend?

(Of course, living beings can't actually ignore the world. They must live in it. But the point here is the direction of the desires. Rationalism got the natural direction wrong - leading to rationalist frustration and all its problems concerning knowledge. Pragmatism instead gets the direction right and thus explains the way we actually are. There is a good reason why humans want to escape into a realm of "fiction" - and I'm including science and technology here, of course. As to the extent we can do that, we become then true "selves", the locus of a radical freedom or autonomy to make the world whatever the hell we want it to be.) — apokrisis

Get back to me when you want to discuss what I actually said and not what you are pretending I said.

If you want to discuss this seriously, define madnesss properly.

Are you talking paranoia or bipolar mania or what? A primary symptom of schizophrenia is a breakdown of perceptual predictability. So a loss of control over experience rather than a gain.