Continuity and Mathematics
In the lengthy thread on "Zeno's paradox," the following (lightly edited) comments prompted an interesting side discussion.
Quoting Rich
Quoting Rich
Quoting Rich
Along similar lines, Philip Ehrlich wrote that Paul du Bois-Reymond "attacked the [arithmetical, rather than geometrical] Cantor-Dedekind philosophy of the continuum on the ground that it was committed to the reduction of the continuous to the discrete, a program whose philosophical cogency, and even logical consistency, had been challenged many times over the centuries." 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.
I am starting this new thread in the hope that more people will chime in on the matter. A few questions come to mind to get the ball rolling.
Thanks in advance for your input.
Quoting Rich
Does mathematics actually model a continuum? I don't think so. If it did, it wouldn't lead to so many paradoxes and incorrect descriptions of experiences. Mathematics, I believe, provides rough models of discrete, measurable actions, which in themselves are practical for certain applications, but are also quite distant from experiences.
Quoting Rich
As far as I can tell, mathematics is totally reliant on the discrete, and because of this limitation constantly makes philosophical/ontological errors. Admission of this major limitation would allow philosophy to move ahead. As long as philosophers are pinned to mathematics, paradoxes will continue to confound.
Quoting Rich
So concretely, a discrete approach cannot uncover the nature of a continuous ontological reality. Other approaches must be used, and unfortunately current mathematics is simply not equipped. It is only adequate for discrete approximate measurements and predictions of non-living matter. It cannot be used to understand the nature of a continuous universe.
Along similar lines, Philip Ehrlich wrote that Paul du Bois-Reymond "attacked the [arithmetical, rather than geometrical] Cantor-Dedekind philosophy of the continuum on the ground that it was committed to the reduction of the continuous to the discrete, a program whose philosophical cogency, and even logical consistency, had been challenged many times over the centuries." 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.
I am starting this new thread in the hope that more people will chime in on the matter. A few questions come to mind to get the ball rolling.
- Is contemporary mathematics inherently discrete, such that it is incapable of accurately capturing the philosophical/ontological notion of real continuity?
- If so, what specific errors and misconceptions have resulted (and propagated) from thinking otherwise?
- Is there any way that mathematics could evolve going forward that would enable it to deal with continuity more successfully?
- If so, what are some specific alternatives? (e.g., Peirce's approach of diagrammatic reasoning)
- Is it possible to determine whether there are any real continua vs. everything (including space and time) being discrete?
- If so, how should we go about it?
- How (if at all) does this issue affect the traditional debate between nominalism and realism regarding universals/generals?
Thanks in advance for your input.
Comments (339)
As I asked in the OP, is it possible to determine whether there are any real continua vs. everything (including space and time) being discrete? If so, how should we go about it?
There is no a priori way of determining anything about reality: hence Zeno's paradox is solved.
There are some conjectures in physics about a granularity of space and time, but there is absolutely no evidence that such a state of affairs exists. According to our deepest, most fundamental and rigorously tested theories, we inhabit a space-time continuum.
Your insinuation that the real numbers cannot somehow model the physical continuum is rather odd.
Is there an a posteriori way of determining whether there are any real continua vs. everything (including space and time) being discrete?
Quoting tom
As I stated in the other thread:
Quoting aletheist
As a structural engineer, I analyze continuous things using discrete models (i.e., finite elements) all the time, and it works just fine for that purpose. Of course, I also apply safety factors to the results, since I am not interested in having the underlying theory falsified.
Can you explain (so that a philosophical simpleton like me could understand it) how mathematics has failed to successfully deal with continuity? Modern topology and real analysis have been wildly successful in dealing with continuity, at least in the practical sense of physics and engineering. Just ask any freshman who's had to slog through epsilons and deltas :-)
Of course one might argue that math hasn't solved the philosophical problems, but math isn't philosophy. It's like asking why physics hasn't solved the problem of tooth decay. That's the job of dentistry, not physics. Can't blame math for not solving the problems of philosophy. Although in fact it has been incredibly successful in doing so. We DO have a satisfactory theory of continuity in math.
I read the Peirce article you linked and my impression was that math has clarified all these confusions. 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.
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
That is really what my first question is asking. @Rich seems pretty convinced, but I am still trying to make up my own mind, especially since I adhere to Peirce's broad definition of mathematics as the science of drawing necessary conclusions about ideal states of affairs. You mentioned topology specifically, and that is precisely where Peirce turned during the last years of his life.
Quoting fishfry
Peirce disagreed with this; he argued that when you divide a line at a point, there are now two points - one goes with each segment. This is because the line does not consist of points and cannot be divided into points; only smaller and smaller lines. Between any two actual points marked on a line, there is an inexhaustible supply of potential points, because it is only when we mark them that they exist at all.
That is indeed a terrific paper, but it gets pretty technical and might be tough to follow for someone not already acquainted with Peirce's thought. Thanks for posting the link, though.
A point is that which has no part. Euclid was right about that. 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.
I'll read your link. I actually do agree that there is something unsatisfactory about the set theoretical view of the continuum. But at least it's clear and sensible. "Good sense about trivialities is better than nonsense about things that matter." Something I read once on a sign outside a math professor's office. 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.
ps ... Glanced at the link. Large cardinals, category theory, sheaf theory. Now this looks like an interesting read. Thanks much.
Right, and a continuum is "that of which every part has parts of the same kind," so obviously a continuous line cannot contain any points. When we mark a point on a line, we introduce a discontinuity.
In regards to penetrating the nature of nature, I prefer Bergson's and Bohm's approach which is the use of intuition, or otherwise described to use the mind to penetrate the mind. At least in my life, this already had been highly successful and progress continues. At least I don't believe I'm a computerized robot.
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).
Good point. No pun intended. 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?
Perhaps, but it seems to me that we have then already conceded that the real numbers do not and cannot constitute a true continuum. They are now just labels that we have assigned to particular locations along the line, not parts of the line itself.
I agree with that. Calling real numbers locations seems to avoid a lot of the philosophical issues about the nature of points.
I've started reading the Zalamea paper and I'm spend some time with it. I looked him up, he's a mathematician who's familiar with modern set theory and other foundations (not just the Cantorian set theory of the 1880's but actual contemporary practice) and also the philosophical ideas of the continuum. It's good stuff.
Agreed, and you probably understand a lot more of the non-Peircean content than I do. :)
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.
Your characterization of me is quite unfair. One, to lump me in with Tom, whose erroneious and confused mathematical misunderstandings I've refuted and corrected numerous times already; and two, to claim that I "feel utterly justified not to care."
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? I've spent years online, you think I don't know how to return a gratuitous insult? What were you thinking here?
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.
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.
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).
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
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.
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.
Quoting 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.)
Agreed, and likewise. I certainly do not believe that one must share all of a thinker's presuppositions and commitments in order to understand his/her thought and make fruitful use of it. For example, Peirce's theism was far less traditional and institutional than my own, which is probably why you and others can run with him quite a long way without invoking God at all. I have benefited from many of your insights, even if you are completely wrong about this one very crucial detail. :D
Sure didn't read like that to me. Seemed a pretty brutal put-down, actually.
So this is honestly your idea of a brutal put-down?...
Well you can just go fuck off. ;)
Quoting 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.
OK, maybe I picked the wrong pejorative but it's at the least, highly patronising. The implication being 'people like this only recite what they're learned in class, they're not really capable of philosophy'. You could have made the substantative point without resorting to blatant ad hominems. Anyway, I will keep out of it, I can see I am only further derailing what was a good thread. Sorry for barging in.
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.
Yet, for some reason, philosophers have allowed mathematical symbols to replace that which they barely represent in nature. It is interesting that Bergson used music as his analog for duration as opposed to others, such as Einstein, who extraordinarily have given ontological value to some equations. With this act we received, not a better understanding of nature, but instead we get the paradox of time travel, a sure sign that things have gone awry. Paradoxes are a red flag for any one examining a particular line of thought.
But without that representation, you would not be able to play that particular piece of music at all, unless it happened to be one that you composed yourself and memorized. The goodness of any representation is inextricable from its purpose. Musical notation has proven to be an excellent way to transmit musical ideas from one person to another, across space and time.
Quoting Rich
Again, it depends on one's purpose. For understanding certain relations and making predictions accordingly, mathematical symbols and equations are very good representations of nature. Some of them are even quite beautiful in their simplicity and elegance.
Quoting Rich
Sound in general is a good analog for duration, since there can be no sound at all in a timeless instant. (There can be no light, either, but somehow that is not as intuitive; probably because we have static photographs that remain meaningful.) A collection of sounds only qualifies as music because of the relations among them over time.
As for predictive value, mathematics is extremely limited but if we focus only on that which it can approximate, e.g. the behavior of some non-living matter within extremely limited constraints, then we walk away thinking that science had the power of a god. In practice though, the approximations that v are given by equations are notable, but in the scheme of things hardly make a dent in the uncountable number of events that one experiences in life. We must put mathematical equations in proper perspective and not get carried away by them. Otherwise they just become yet another idol.
It is trivially true that no representation reproduces its object in every respect, and the purpose of musical notes - and mathematical symbols/equations - is obviously not to represent "the experience itself."
Quoting Rich
Agreed, but we also should not undervalue them, either.
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.
My problem! LOL. I haven't read anything after this morning. No harm no foul. For what it's worth, I've often wondered about the relation between the mathematical formalisms that allow you to add up infinitely many 0-dimensional points to get a line of nonzero length. We can integrate dx from 0 to 1 and the answer is 1, we can even teach that to high school students. But it's really the most mysterious equation in the world.
I don't happen to know much about what philosophers have said about this, other than some very nodding acquaintance with intuitionism which I find murky in the extreme. I did try to study free choice sequences once and gave up.
So I'm ignorant but not apathetic. I don't know but I do care. And yes I'm way too sensitive for my own good.
Quoting apokrisis
What? I'm basically in agreement with you. Maybe you can tell me what you think my thesis is. You might be misunderstanding me. I'm totally baffled by this remark. That's why I was so startled by your saying I don't care. Nothing I've written has supported that conclusion unless I'm expressing myself very poorly.
Is accusing people of not caring part of the academic give and take?
Quoting apokrisis
Just baffled. When a mathematical point has needed to be made, I've made it. I have never and I repeat never said that I think mathematics has the slightest thing to say about reality. I'm often a formalist. I don't think set theory necessarily applies to the real world. I said that earlier, maybe in the other thread. Why do you think I'm maintaining otherwise?
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.
Our fundamental theories are based on the continuum of space-time. What more do you want?
Quantum mechanics may be a theory that yields discrete observables, but the theory itself, and the dynamicas happen on the continuum.
Quoting aletheist
I hope you noticed that computers can't even instantiate the reals.
How does it fail to provide the whole story? If "things" don't really exist, and every "thing" is a network of relationships, then mathematics would describe the world, because mathematics has no objects - it's pure relationality. Think about geometry. A line in geometry isn't constituted in-itself, but always in reference to all other possible geometrical constructs - indeed to understand what a line is, one must ultimately understand all of geometry. Geometrical objects are nothing except relations, that's why they are actually impossible in the real world - for example a line has no thickness. They are constituted by the whole system - it is their interrelationships which constitute them, and in the end give them the properties they have. So "things" are illusory - reality is fundamentally relation, not thing. Scientific knowledge is merely useful, but not true, because the truth would have to be the Whole of reality, and every part of reality would be in its totality determined by this Whole. That's why we can get better and better approximations for everything, but we can never be exact, because we - as a part of the Whole - can never know the Whole completely - there will always be a residue of uncertainty.
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.
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.
That's the point. Yet, that is exactly what was done with Relativity. This is what Bergson objected to. Time in the Relativity equations and space-time represented by intervals has no ontological basis. They are just symbols for measurements convenience. As such, relativity did not describe the nature of nature. What's more, not only is scientific time at odds with experience, Relativity itself is internally inconsistent (accelerated systems do not exhibit reciprocity), but also introduces all sorts of paradoxes. This would be an example of how science has used mathematical equations to disrupt reasonable philosophical thought that seeks to free itself from mathematical symbols. The other example being the bottom of discrete particles.
An analytic continuum (real numbers), or a true continuum (Peircean)?
Quoting tom
How is that significant in the context of this thread?
The real continuum of the real numbers. You know parameters like "t" and "x" are real numbers, unless they are complex numbers.
Right - the real numbers constitute an analytic continuum, but not a synthetic continuum; i.e., a true continuum in the Peircean sense, which cannot be represented by numbers at all. As he put it, "Breaking grains of sand more and more [even infinitely] will only make the sand more broken. It will not weld the grains into unbroken continuity."
Breaking sand "infinitely" yields tiny bits of sand? Please!
Breaking the real numbers "infinitely" yields what?
In order to get what we intuitively understand as a continuous line (for example), you need to build up some mathematical structure, such as ordering and neighborhoods. You won't get that just from a point, the structure is global and independent of the properties of individual points or their aggregates. (By the way, we keep saying "points", but topology is agnostic about what those elementary entities are: in fact, they can be anything, such as functions, for example.) So it is really the structure of the continuum that makes it what it is, and this focus on "points" is misguided. Or I should say a structure, because our intuitive requirements for a continuum can be realized with multiple mathematical structures, some of them isomorphic, some not.
It obviously does not yield an unbroken continuum.
Quoting tom
More real numbers, all of which are distinct; again, it obviously does not yield an unbroken continuum. Put another way, the set of real numbers has a multitude or cardinality, which is exceeded by its power set; but a true continuum exceeds all multitude or cardinality. It is not composed of parts, it can only be divided into parts, all of which can likewise be divided into more and smaller parts of the same kind.
Which real number is bigger?
1 or 0.999...
Are your comments directed at any particular person or post?
And by the way, which real number is bigger:
1 or 0.999...
According to you, or Peirce?
We typically treat the two values as equal, but arguably there is an infinitesimal (non-zero) difference between them. As you might have guessed, the Peircean continuum is non-Archimedean.
I doubt it, since I have been clearly saying all along that a true continuum is not made up of "individual points or their aggregates." I might even agree with that whole post, if I am understanding it correctly. In any case, we might as well wait for @SophistiCat's own answer.
The values ARE equal. There is NO difference between them.
You cannot create a new number by adding an infinitesimal quantity.
Ha. Tom really isn't keeping up with the argument.
Quoting aletheist
Yep. SophistiCat is talking the language of constraints.
Quoting tom
So by 1, do you really mean 1.000... ? ;)
As it happens, .999... = 1 is a theorem even in nonstandard analysis. This is easily shown. The hyperreals are a model of the first-order theory of the reals; and .999... = 1 is a theorem of that theory. You see we can reason logically about infinitesimals and we just debunked the claim that the hyperreals (a particular flavor of non-Archimedean field) invalidate .999... = 1.
Another problem with infinitesimals as a model for the continuum is that any field containing infinitesimals must necessarily be topologically incomplete. There are Cauchy sequences that don't converge. Any real line containing infinitesimals is shot full of holes. This is fairly easy to prove.
Last year I was in one of those endless and moronic .999... debates (I have a firm personal policy NOT to get involved in those, but on that one occasion I broke my own rule and needless to say ended up regretting it). As part of that debate I got tired of always hearing about the hyperreals so I went and learned a little about them. I can tell you for a fact that they will not help anyone's argument that .999... is anything other than 1, if you give those symbols their standard meaning. Of course if you change the interpretation of the symbols you can have .999... = 47 if you like. That's another point that's generally missed. If you interpret the symbols using their standard technical meaning, then .999... = 1. There's just no question about it. The only question is how you're allowed to manipulate the symbols.
I'm not saying that the .999... deniers don't have a philosophical point or two. I'm just saying that although .999... deniers often mention the hyperreals, the hyperreals don't help their argument. .999... = 1 is just as true in the hyperreals as it is in the reals.
I am actually much less familiar with other non-Archimedean systems such as the surreals, but the formalization of the surreals is fairly murky. I've never seen anyone claim that .999... is one thing or another in the surreals. Maybe I'll see if I can look that up.
I am neither a mathematician nor a philosopher, but that statement seems consistent with the claim that the real numbers do not qualify as a true continuum in Peirce's sense, since they skip over those infinitesimal intervals. If you disagree, I would sincerely appreciate an explanation why I am mistaken about this. Have you read the Zalamea paper, or at least its first chapter that spells out the essential properties of a Peircean continuum?
Quoting fishfry
I will take your word for it, but my understanding is that the precise nature of the relationship between Peirce's continuum and nonstandard analysis is still not fully settled.
Wow! I'm out!
Or is there some reason why we don't have to treat it as a convergent limit as well?
You can take my word for it that .999... = 1 is a theorem of nonstandard analysis. But actually you don't need to take my word, I provided a proof above. The nonstandard reals are a model of the first-order theory of the reals. .999... = 1 is a theorem of that theory, hence it's true in any model of that theory. That's Gödel's completeness theorem, as opposed to his more famous incompleteness theorem. If you have a syntactic proof of a statement from some axioms, that statement is true in every model of those axioms,. The converse holds as well. If you don't have a proof, then there are models where it's true and models where it's false.
Please ask for more detail if this proof isn't clear (and you care one way or the other).
I'm afraid I don't know anything about Peirce's continuum but I will certainly try to educate myself in the near future. But remember, any field containing infinitesimals must be incomplete. Does Peirce know his continuum has holes in it? It's logically necessary.
On an unrelated meta-note, it would not be good to allow this discussion to degenerate into a .999...-fest. That's a tremendous distraction.
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.
I started reading the Zalamea paper, Googled around, and found a pdf of his awesome book Synthetic Philosophy of Contemporary Mathematics. I'm enthralled. A mathematician who actually understands the conceptual revolution that happened in math in the second half of the twentieth century; and writes brilliantly and clearly about philosophy. And he's written a lot about Peirce as well. So at the moment I'm coming to Peirce by way of Zalamea's exposition of the mathematical philosophy of Grothendieck. This is sublime. I truly thank you for this reference.
Different point ... when you talk about points coming in and out of existence, that reminds me of the intuitionists. Which I regard as a somewhat mystical strain of thought. For them, real numbers come into existence when they are thought of or needed or used. In the modern incarnation, a number comes into existence when it's computed. Intuitionism is coming back into style.
Now here is my question.
* It's my understanding that the intuitionistic real line has fewer points than the standard real line. After all only countably many reals are computable, for the reason that there are only countably many Turing machines. Where the standard real line has noncomputable numbers, the intuitionistic line has holes.
* It's also my understanding that the hyperreal line (this is the only nonstandard model of the reals I know anything about) has MORE numbers than the standard reals. After all the hyperreals have all the real numbers, plus a cloud of infinitesimals around each one. (I've heard these are Leibniz's monads, but I don't know anything about Leibniz, being more of a Newton fan. Maybe that explains a lot :-))
* So my question is: Is Peirce a restrictionist, squeezing the noncomputables out of the standard reals and only creating reals when they pop into his intuition; or is he an expansionist, blowing wispy clouds of infinitesimals onto the real line?
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).
Quoting 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".
Quoting 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.
Just struggling a bit with how 'sign relations' come into the picture outside of biology.....
Biology, that's interesting. I thought sign relations were some kind of postmodern talk I don't know anything about other than that Searle thinks Derrida is full of sh*t. That much I know.
Well 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.
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.
In the beginning was the word, eh? ;-)
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 have no reason to doubt that you are correct about this. Thanks for another helpful clarification, especially since @tom chose for some reason not to provide one.
In this context, do you basically see continuity as 3ns, discreteness as 2ns, and possibility as 1ns?
Quoting apokrisis
Given that existence is 2ns, do you generally prefer to characterize 3ns as "constraints" and 1ns as "freedoms"?
Quoting apokrisis
Peirce: "If we are to explain the universe, we must assume that there was in the beginning a state of things in which there was nothing, no reaction and no quality, no matter, no consciousness, no space and no time, but just nothing at all. Not determinately nothing. For that which is determinately not A supposes the being of A in some mode. Utter indetermination. But a symbol alone is indeterminate. Therefore, Nothing, the indeterminate of the absolute beginning, is a symbol. That is the way in which the beginning of things can alone be understood." (EP 2:322; c. 1904)
Quoting apokrisis
Peirce: "In that state of absolute nility, in or out of time, that is, before or after the evolution of time, there must then have been a tohu-bohu of which nothing whatever affirmative or negative was true universally. There must have been, therefore, a little of everything conceivable." (CP 6.490; 1908)
Genesis 1:2: "The earth was without form and void (Hebrew tohu wa bohu) ..."
Quoting apokrisis
Someone? Now you are just teasing me.
Peirce, describing the author of Genesis 1:2-5: "His tohu bohu, terra inanis et vacua is the indeterminate germinal Nothing. His Spiritus Dei ferebatur super aquas is consciousness. His Lux is the world of quality. His fiat lux is an arbitrary reaction. His divisit lucem a tenebris is the recognition of the necessary duality. His vidit Deus lucem quod esset bona is the waking consciousness. Finally; his factumque est vespers et mane, dies unus is the emergence of Time." (NEM 4:138; c. 1898)
Quoting fishfry
I was delighted to learn this evening that Zalamea has agreed to a "slow read" of this very paper via the Peirce-L e-mail list in the near future. If you are interested in joining that conversation, or even just monitoring it, I can keep you posted on the details as they are worked out. He hopes to participate himself, although likely to only a limited extent, because he apparently intends to go into seclusion over the next couple of years in order to focus on writing a lengthy new monograph on Grothendieck.
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.)
Quoting 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.
No, I understand his 1ns in itself to be quality as possibility, or unembodied quality; medad rather than monadic predicate. Anything brute and/or actual is 2ns.
Quoting apokrisis
Agreed, 3ns involves 2ns and 1ns, and 2ns involves 1ns.
Quoting apokrisis
Again, Peirce did not use "bruteness" to refer to 1ns, only 2ns.
Quoting apokrisis
That is certainly one manifestation of the categories. Others include quality/fact/law, spontaneity/reaction/habit, and feeling/action/thought. I also think that 3ns is often conditional necessity, rather than absolute necessity.
Quoting apokrisis
Yes, I have read a bunch of his stuff, although it has been a while.
Quoting apokrisis
It definitely takes some getting used to, but is well worth the time and effort.
Quoting apokrisis
Thanks, it definitely helps me map the terminology that you tend to use around here to Peirce's own.
Well when you put it THAT way it's totally CLEAR. LOL.
I see that Peirce has some jargon associated with him. From Googling around I think being triadic is what a mathematician would call ternary, a relation that inputs three objects and outputs T or F. Like equality is binary, it inputs (5,3) and outputs F, inputs (2,2) and outputs T. A ternary relation takes three inputs. Does that have anything to do with this?
If you can briefly explain some of these technical terms it would help.
I wanted to mention that at one point Zalamea basically says that Peirce is doing category theory, or category theory is Peircean. Now I still don't know what Peirce is about, but this is a fantastic connection. I've had some exposure to category theory. Not much but enough to know that modern math is done very differently than anything you see as an undergrad math major. Equations are replaced by arrow diagrams. It's a very different point of view. I've always understood category theory to be loosely related to structuralism. We no longer care what things are made of, we care about their relationships to other things; and about very general patterns in those relationships that tie together previously unrelated areas of math.
Why have I never heard of Peirce before? I've been in lots of online discussions about the philosophy of math, and I've heard the name but never knew he anticipated the math of the future in some deep way.
ps ... I'm randomly reading sections in Zalamea and I come to this: "The triadic Peircean phenomenology ..."
To read that phrase used by an author who speaks so knowledgeably about Grothendieck ... this is breathtaking. Why isn't Zalamea famous? He doesn't even have a Wiki page. This guy has moved the philosophy of math forward fifty years.
I still don't know what triadic Peircean phenomenology is. Can this be explained simply?
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.
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.
My only exposure to category theory (so far) is Zalamea's paper, which I am in the process of rereading because I suspect that it will make even more sense to me now than it did a few months ago. Peirce was a strong advocate of diagrammatic reasoning, but he did not confine the term "diagram" to visual representations; rather, a diagram is any sign that embodies the significant relations among the parts of its object. An algebraic equation is a diagram in this sense, but it is not as "iconic" as a geometrical sketch. This is precisely why Peirce developed the Existential Graphs, a diagrammatic system of logic whose three versions are equivalent to standard propositional logic (Alpha), first-order predicate logic (Beta), and certain kinds of modal logic (Gamma). He hoped that diagrammatic reasoning would be a means by which mathematics could overcome the limitations of the discrete and better account for true continuity.
Quoting fishfry
Several commentators on Peirce have suggested that his philosophy of mathematics was very similar to modern structuralism; e.g., this paper by Christopher Hookway, and this one by Paniel Reyes Cardenas.
Quoting fishfry
Ah, the perennial question whenever someone discovers him for the first time. My reaction was exactly the same. Best I can tell, the key factors are:
Quoting fishfry
Peirce believed that there are exactly three universal categories that are present in every phenomenon, and in order to avoid specific associations that might be too narrow, he preferred to call them simply Firstness (1ns), Secondness (2ns), and Thirdness (3ns). As I mentioned above, there are various ways to differentiate them - possibility/actuality/necessity, quality/fact/law, spontaneity/reaction/habit, feeling/action/thought. Logically, they correspond to monadic/dyadic/triadic relations; Peirce postulated, and Robert Burch proved (much later), that each of these is irreducible to the others, while all relations of tetradic or higher adicity can be reduced to triadic ones.
Peirce acknowledged this - as soon as we talk or even think about a color or other quality, it is no longer 1ns in itself.
My comments were directed at your OP and some following posts. It seemed to me that your dissatisfaction with Cantorian mathematical theories of continuity stemmed from the idea that according to these theories the continua are composed of discrete points - a seeming contradiction. But it's not about composition - it rarely is.
When wondering about what a thing really is, asking "what is it made of?" is a good way to proceed in many common-place situations. For instance, if you find that something is made of wood and not wax, that is going to tell you quite a bit about that thing's properties. But this intuition often trips up people when more subtle questions are asked. In mathematics, and to a large extent, in science, the question "what is it made of" is often unproductive and misleading, as it is in this case.
Anyway, I see that this discussion has long since turned to Peircian exegetics, which interests me not at all, so I'll bow out.
I 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! ;)
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.
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.
In the Zalamea paper on Peirce's continuum, Zalamea says on page 8:
"As we shall later see, this synthetical view of the continuum will be fully recovered
by the mathematical theory of categories, in the last decades of the XXth century."
"This" above refers to Peirce's concept of the continuum. So Zalamea's understanding of category theory seems radically different from yours.
Thanks for your comments and clarification.
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?
Would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context?
Quoting apokrisis
That is not how I understand it, unless by "constrained possibility" you mean the actually possible as opposed to the logically possible. Peirce sometimes even distinguished continuous potentiality as 3ns ("indeterminate yet capable of determination in any special case") from pure/ideal possibility as 1ns ("incapable of perfect actualization on account of its essential vagueness"). I associate propensity more with 3ns as habit than with 2ns as brute actuality. Ultimately it depends on the particular type of analysis that we are doing, since all three categories are part of every phenomenon to some degree.
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."
Your erudition seems to have overtaken your common sense and your manners. You are incapable of explaining, only insulting. I'm using individual quotes as an alternative to copy/pasting pages and chapters of Zalamea. His entire thesis is that category theory has resurrected Peircean synthesis. Now you are right, I'm just trying to learn what this means. But your unwillingness to explain anything of your jargon-filled posts says something about you.
Is it time for me to say fuck you to you again? I've had enough. Fuck you.
Come now - this is not @fishfry's own claim, but Zalamea's, in the very paper that you recommended. The later discussion linking category theory with Peirce's continuum is on pages 38-40. I guess you disagree with Zalamea about this? If so, why? Again, I do not know much about category theory (yet), so I personally have no opinion one way or the other and am open to persuasion.
I 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.
What was I saying about instability?
I don't claim special expertise in category theory. But I think I know enough to know from your description that you know even less.
So I tried to explain my own point of view. I offered you the chance to rebut that from your current close reading of Zalamea. At which point - and I can't say I'm surprised - you explode in anger because you are not in the position to do so.
But never mind.
Again, @fishfry simply pointed out something that Zalamea claims in the paper that you recommended, which is contrary to your own comments. I took him to be asking you for an explanation of this particular discrepancy, not nitpicking or in any way asserting that Zalamea is right and you are wrong, since he is still in the process of digesting the paper. Then you are the one who responded with the first insult, alleging that he does not understand category theory. If you had left out that one unnecessary sentence, I suspect that a more fruitful exchange would have ensued - one that I would still very much like to see happen.
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.
Quoting 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.
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.
The actual is that which is neither vague nor general by these definitions - both principles apply to it. The tricky part is that this notion of absolute singularity is strictly ideal - everything actual is indeterminate to some degree; no existing object definitively possesses or lacks every predicate.
Quoting apokrisis
Sorry to repeat myself, but would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context - i.e., as characterizations of the kind of foundation that a particular theoretical approach seeks? That might help me understand why you apparently disagree with Zalamea on whether category theory helps us recover the Peircean notion of a continuum.
I see nothing insulting about pointing out a discrepancy between what you wrote here and what is claimed in a paper that you recommended.
Quoting apokrisis
I have already acknowledged (more than once) that I do not know enough about category theory (yet) to say anything at this point. I was hoping to learn more about it from the two of you.
Quoting apokrisis
Have I ever attacked you personally, in this thread or elsewhere? At this point, I am just annoyed that you seem to have driven off @fishfry, who I thought was making helpful contributions to the discussion. If it now devolves into "Peircian exegetics," as @SophistiCat thinks it already has done, then it will just be the two of us trading thoughts about our favorite philosopher. I was hoping for much more than that when I started the thread.
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.
Quoting aletheist
Yep. You are doing that right now too.
Quoting aletheist
Oh what a disaster. And so you would rather chase me off now. Hilarious.
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?)
Quoting aletheist
Reductionist vs holistic, causally closed vs causally open, externalist and transcendent vs internalist and immanent, etc, etc.
Quoting apokrisis
Really? That was not my intention at all. I was just trying to moderate a dispute between two of my favorite PF participants.
Quoting apokrisis
What? Why would that be your response? My whole objective was to bring everyone back to the table; I have no desire to chase anyone off.
Quoting aletheist
Quoting apokrisis
Ah, good point. Where Peirce said what I said, what you said, or both?
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.
Quoting 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.
He never said that you were wrong. He merely said that Zalamea said the opposite of what you said.
Quoting apokrisis
What gave you that idea? I thought that was also unfortunate and unnecessary. Do I need to rebuke him to demonstrate my impartiality?
Quoting apokrisis
How would you formulate the principle of identity such that it would not apply to the actual, because nothing that exists is determinate with respect to every predicate? Does it apply to 1ns and 3ns, such that its inapplicability is a distinguishing feature of 2ns as you seem to be suggesting?
Why not just do much less rebuking all round and focus on dealing with the substance of any post.
Quoting 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.
This is probably good advice, and I will try to heed it going forward.
Quoting apokrisis
Sorry to nitpick, but is "contradiction" the right word here? In accordance with the Peirce quote, we started out using "inapplicability," which seems more appropriate to me. So your hypothesis, as I understand it, is that actuality/2ns is defined by the inapplicability of the principle of identity; and I am still wondering which particular formulation of it you have in mind, since there are several. For example, Peirce did say that "Leibniz's 'principle of indiscernibles' is all nonsense" (CP 4.311). In fact, in his definitions of "individual" for Baldwin's Dictionary of Philosophy and Psychology (1911), he wrote the following.
[quote=CP 3.611-613]Used in logic in two closely connected senses. (1) According to the more formal of these an individual is an object (or term) not only actually determinate in respect to having or wanting each general character and not both having and wanting any, but is necessitated by its mode of being to be so determinate ...
(2) Another definition which avoids the above difficulties is that an individual is something which reacts. That is to say, it does react against some things, and is of such a nature that it might react, or have reacted, against my will ...
... whatever exists is individual, since existence (not reality) and individuality are essentially the same thing; and whatever fulfills the present definition equally fulfills the former definition by virtue of the principles of contradiction and excluded middle, regarded as mere definitions of the relation expressed by "not." As for the principle of indiscernibles, if two individual things are exactly alike in all other respects, they must, according to this definition, differ in their spatial relations, since space is nothing but the intuitional presentation of the conditions of reaction, or of some of them. But there will be no logical hindrance to two things being exactly alike in all other respects; and if they are never so, that is a physical law, not a necessity of logic.[/quote]
My other point was that if vagueness/1ns is defined by the inapplicabiity of the principle of contradiction, then actuality/2ns and generality/3ns must be subject to it; and if generality/3ns is defined by the inapplicability of the principle of excluded middle, then vagueness/1ns and actuality/2ns must be subject to it. Likewise, if actuality/2ns is defined by the inapplicability of the principle of identity, then vagueness/1ns and generality/3ns must be subject to it. Otherwise, each characteristic is not distinctive of its corresponding category after all.
Does that make any more sense?
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.
Yes, I agree. What remains unclear to me is what it means to say that the principle of identity does not apply to something. Zalamea helpfully formalizes the principles of vagueness and generality on page 21 of his paper; he describes them as failures of distribution of the principles of contradiction and excluded middle, respectively. Is there an analogous way to formalize the principle of identity and/or its failure, which would show what you have in mind here?
Quoting apokrisis
I likewise dissent from the most common interpretation of Peirce's cosmogony, in which 1ns came first (so to speak), then 2ns, and finally 3ns. I think that 3ns, with "its really commanding function," is the most fundamental - "the clean blackboard" as a continuum of two dimensions representing the original one that had "some indefinite multitude of dimensions." A chalk mark then represents the spontaneous introduction (1ns) of a brute discontinuity (2ns), but the mark is not really a line - it is a surface whose own continuity is parasitic on that of the underlying blackboard. Only after developing habits (3ns) lead multiple chalk marks to persist and aggregate into "whiteboards" that represent Platonic worlds of possibility (1ns) does the final step occur, when "this Universe of Actual Existence" (2ns) comes about as "a discontinuous mark" on one of those whiteboards.
I certainly cannot do that - at least, not yet - but I just came across one possible clue in the SEP article on "Category Theory."
A continuum (such as a line) is a type and its parts are tokens, which are instances of it (smaller lines) rather than members or elements of it (discrete points).
Any thoughts on these passages, @fishfry?
This seemed to be where things went astray. My own comprehension of both of mathematics and philosophy is left puzzled; what's the problem for Apo?
Take the rationals, and make a cut at <2. 2 stays on on side, all the numbers less than 2 on the other. One side contains 2, while the other might approach 2, but by the very fact of the cut, never reaches it.
I don't see a problem. Nor does one appear when we make a second cut at >2. We now have three pieces: <2, 2, >2.
Nor is a problem introduced when considering continuity. My simple understanding is that a line is continuous if it is differentiable. Well, the limit of <2 as it approaches 2 is 2. It does not seem problematic.
So explain it to me, rather than abusing each other.
2 is the boundary between all that is less than two and all that is greater than two. But what is 2?
So what. 2 is not included in what is less than two. It is included in what is not less than 2.
Nor is it include in what is greater than two, it is included in what is not greater than two. If it's not included in what's less than two, and not included in what's greater than two, then what is it? I know what you'll say, it's 2.
What is the negation of "less than 2"?
It's 2.
Apo's philosophy is ignorance of self-definition. What he can't understand is logical definion in terms of itself-. For him, everything must be logically defined in terms of something else, so it's not enough for to be 2.
But it always worries me when I agree with you. :-|
IN short there is no such thing as "self-definition"; what you actually mean is 'self-identity'.
Quoting John
"Self-definition", or "self-identity", whatever you want to call it, what do you mean by this?
Willow often says that things are "defined in themselves". I am not exactly sure what is meant by that, but I assume that it is an attempt to express the principal of identity; which is that a thing is identical with itself, and with no other.
Another possibility is that it comes from Willow's understanding of Spinoza, who says that a thing is either conceived through itself or through another. Only substance is, according to Spinoza, conceived through itself. Modes are conceived through their relations to other modes and, ultimately, through substance. So existence is of the essence of substance, but existence is not of the essence of modes. So, for example I can be conceived as not existing, but substance (God) cannot be so conceived. So my essence does not involve existence but Gods' essence does. Anyway this is getting away from the topic of the thread. I am thinking about starting a thread on Spinoza's understanding of substance and modes, but I'm not sure I will have the time.
This would be what Peirce's secondness challenges. Uniqueness would still be defined relatively. Inidividuation or identity is a difference that makes a difference. So - following Aristotle - the substantial is has some particular matter and some particular form. That is, it stands in relative contrast to the absolutely vague and the absolutely general.
This secondness or substantiality then shows itself in the sharp possibility of a reaction. One thing can react with definiteness to another thing. We have the dyad of some relation. We have a difference that is distinctive as part of a context and so can go on to be remembered as changing its developing history. We have the uniqueness of some difference that actually made a difference to the whole.
So 2ns is dyadic reaction. Actuality is being defined in terms of a difference that makes a difference. This is quite in contrast to a tautology where the actual is simply a difference. It is not about what the world can see and remember as a concrete happening - a unique event that produce some further change. It is simply presuming the existence of some thing as that which is "the same as itself" - absolutely secure in its difference from other things without further determination. Nothing has to be actually shown or remembered by way of the demonstration of some reaction.
Thus it is not hard to see secondness contradicting identity in a big way. Actuality is about some relative change that is definite due to a context. A thing must be reacting to at least one other thing.
And then when does such an interaction ever exhaust all properties. If I bump into a car in the pitch dark, I have some idea perhaps of an encounter with something metal and solid. But is it a Porsche or a Fiat. Might it just be a lamppost?
So identity is only approached in the limit by the dyad of actual interaction. And Newtonianism said at the end of the day, the limit itself dissolved into the purely relative. In space, am I drifting away from the rocket, or is the rocket moving away from me? The mechanics of the situation are fundamentally reversible or symmetric. We can't any longer use local differences as the guide to what is actually the case. The identity of individuals can't be arrived via the exhaustion of their observables, even if it can be approached with arbitrary closeness in principle.
So 2ns switches things in flipping it so that actuality is not real in the sense of a limit state having been concretely achieved - the usual classical notion of identity. Instead a limit is a limit - the place that "exists" in the apophatic sense of never actually being arrived at. So 2ns is substantiality approaching its exhaustive limit, not substantiality in and of itself, nothing further needing securing.
I think I get what you mean when you say that the principle of identity does not apply to the actual - it is a limit that existing things can approach, but never fully achieve. But in what sense, then, is this distinctive of 2ns, in the same way that the inapplicability of the principles of contradiction and excluded middle are distinctive of vageness/1ns and generality/3ns, respectively? Again, can you state and/or formalize exactly what you mean by the principle of identity in this context?
Any thoughts on the excerpts that I posted as possible clues to why Zalamea claims that the synthetic continuum is recovered fully by category theory via synthetic differential geometry or smooth infinitesimal analysis?
Yes, Spinoza's concept of substance is contradictory to Aristotle's concept. Spinoza denies that there can be many finite substances and contends that there can be only one infinite substance.
Vagueness and generality are defined as not being constrained in two of the ways that actual particulars are constrained. So materiality can be vague and not substantial. Form can be general and likewise not substantial.
Although given 3ns, these categorical distinctions are themselves all just aspects of the single triadic, irreducibly complex, sign relation. So that is why forms or universals can be real but not actual. And more unusually, the same is being said of the material principle. Materiality (as the vagueness of pure tychic spontaneity) is also a real potential, but not actually substantial 2ns (as it lacks yet the regularisation of habitual form or 3ns).
You see here that I go back and forth between hylomorphism and semiotics as of course the two are essentially the same metaphysical scheme with semiotics doing the better job of explaining the "how" by its foregrounding the mechanical role played by the sign relation in producing a world of suitably "deadened" substance.
Anyway 2ns would stand in relation to the law of identity as this same kind of protest - I am not constrained by that constraint which is said to be required to produce the brutely particular.
So 2ns instead talks about the deeper process that produces the brute particular. It points to the materiality and the formality, the vagueness and the generality, that have to be in interaction to produce actual substantial events, or differences that make a difference. 2ns treats actuality as what you get in the limit (with full 3ns). So actuality does apply to 2ns ... in the limit. But then 2ns is thus not actuality as brutely conceived by the law of identity. It is completely contextual once you step back to see the full 3ns scheme of things.
And this becomes more acceptable if we choose our intuition pumps more carefully and stop imagining reality in already presumptively Newtonian terms - like billiard balls rattling about on green baize.
What happens when two clouds collide? Where does any one cloud stop and start? What is the definite shape of any cloud? What is the physical logic of cloudy objects?
Clouds surely have actuality - we talk about them enough. But really, the law of identity fails to apply in a big way. And we can now specify the nature of that failure in the language of fractal maths. The contextuality of identity stands completely exposed these days.
I mean even Cantor was on the right track without really understanding it.
Have you checked - https://en.m.wikipedia.org/wiki/Cantor_set
Yep. The problem with Spinoza is that he was right about there having to be a "One", but wrong in conceiving of that basic materiality as a singular substance rather than as the vagueness of unbounded action. So it is material cause ... in its most insubstantial form. So action utterly lacking in form or purpose. An everythingness that is a singular being only because we call its fundamental disunity or lack of direction a single property or characteristic.
Vagueness is the canonical many. And when the question is asked of how many manys there are, the answer that comes back is "I am only counting the one".
I said:
Quoting Banno
The law of identity expressed in what way, either verbally or formally (or both)? What exactly is this constraint with which 2ns "refuses" to comply? Would "contextuality" be a good descriptive term for this characteristic, as the second member of a trichotomy with vagueness and generality? What about "substance" to go along with matter and form?
The more I think about it, the more I really like "contextual" as a candidate for the 2ns counterpart to "vague" for 1ns and "general" for 3ns. If I am tracking with you properly, it expresses the specific kind of indeterminacy that is characteristic of the actual, especially as manifested in Peirce's semeiotic concept of the index. Consider this passage:
As a first cut: A sign is objectively contextual, in so far as, leaving its interpretation indeterminate, it relies on some aspect of the actual situation to complete the determination. "That house is on fire." "What house?" "That one over there."
Peirce used a very similar example to illustrate the indexical nature of pronouns, which "call upon the hearer to use his powers of observation, and so establish a real connection between his mind and the object; and if the demonstrative pronoun does that - without which its meaning is not understood - it goes to establish such a connection; and so is an index. The relative pronouns, who and which, demand observational activity in much the same way, only with them the observation has to be directed to the words that have gone before" (CP 2.287, c. 1893). Proper names are also indices, and rely entirely on the interpreter already being familiar with whom or what they reference.
Now back to the other passage:
As a first cut: The contextual might be defined as that to which the principle of identity does not apply. This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place.
As I noted before, Zalamea - right after quoting the same passage - formalizes generality and vagueness as "failures of distribution" of the principles of excluded middle and contradiction, respectively. (Robert Lane provides a helpful explanation of the important differences between these principles, as defined and used by Peirce, and the modern laws of excluded middle and non-contradiction.) He associates the general with the universal quantifier and the vague with the existential quantifier, so it seems like the contextual should be associated with a singular proposition. After wrestling with his notation for a while - I am still not sure that I am interpreting it correctly - I came up with these formulations:
As a first cut: If x is contextual, then it is not necessarily true that under all circumstances, x = x.
What do you think?
Yeah. But anyone can wiki the set theoretic definition. Keep up.
I don't think it is essential to arrive at one perfect word. Peirce called them one, two, and three precisely because the same basic triadic relation could have its many manifestations.
But if vagueness is the best term for 1ns, and generality the best for 3ns, then another term for 2ns (after hierarchy theory) would be specificity.
Now the issue is that you think that 2ns needs to be contradicted by (or there be a failure of distribution concerning) the law of identity. So somehow 2ns itself should mean the opposite of the individual, the specific, the determinate,
But the Peircean triad actually wants to give the particular its real place in the scheme of things. So we don't need to contradict identity itself to contradict the principle of identity.
I mean 2ns looks the most like the regular reductionist notion of the atomistically and mechanically determinate - in simply being Newtonian action and reaction.
That is why I said the contradiction lies in the genesis of specificity. Peirceanism says it is a contextual deal. The laws of thought say it is brutely tautological. So the opposition is there between the holism and atomism, but Peirceanism would still call 2ns "actuality" or one of its synonyms, like particular, local, substantial, specific, determinate, individual, etc. The difference is that what the laws of thought presume as the brute foundations - nominalistic identity - Peirceanism shows to be the emergent final product.
Or x = not not-x is true. That employs the context to derive the specificity via a dichotomy.
Check out the Spencer-Brown's laws of form. Or Kaufmann's note on Peirce's sign of illation - http://homepages.math.uic.edu/~kauffman/Peirce.pdf
No, the two concepts aren't contradictory in any way. They are actually compatible. It is true that Aristotle means something different by Substance than Spinoza, however, the two concepts (their meanings) are not contradictory, but complementary. Substance in Spinoza is that which cannot be conceived as not existing, and which must be conceived through itself. There is only one element of Aristotle's metaphysics which fits this description - and there is only ONE of them - the Prime Mover. So Substance in Spinoza is NOT Substance in Aristotle, but rather Prime Mover. Hence the two definitions of Substance aren't even incompatible to begin with.
I agree, but I think that it will be helpful to clarify the distinctions that we are trying to draw if we can assign a term to 2ns that goes along with "vague" for 1ns and "general" for 3ns.
Quoting apokrisis
I am not very familiar with hierarchy theory, but I know that you refer to it a lot, and I might do some reading about it once I finish my current exploration of category theory and smooth infinitesimal analysis. Although I can see how "specific" is an antonym for both "vague" and "general," it strikes me as too much of a synonym for "determinate," such that the principle of identity would apply. Furthermore, we can "specify" something by description, rather than requiring an index to pick it out within a particular context.
Quoting apokrisis
I agree, especially since part of Peirce's point in describing vagueness/1ns as the inapplicability of PC and generality/3ns as the inapplicability of PEM is to define 2ns as that to which both PC and PEM do apply. However, he also wrote those two definitions of "individual" that I quoted a while back (CP 3.611-613, 1911). The first requires determinacy with respect to every general character, and thus - as he wrote elsewhere (see below) - can only be an ideal limit; while the second makes individuality a matter of reaction, and therefore existence. Both effectively deny the identity of indiscernibles, the first by virtue of the different "hecceities" that two distinct individuals must have, and the second because no two reacting things can have the same spatial (or, I would add, temporal) relations.
The latter is what I had in mind when I suggested as an example of contextuality, "This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place." I was also thinking of this passage:
This is the basis on which I have elsewhere suggested "extreme realism" as the view that reality consists entirely of generals, or at least that everything real is general to some degree. In other words, there are no absolute individuals/singulars that are determinate in every conceivable respect. Hence "general" here encompasses both the "positive generality" of 3ns as "conditional necessity" and the "negative generality" (i.e., vagueness) of 1ns as "the merely potential," since they both constitute forms of indeterminacy.
I remain intrigued by the prospect of identifying a third kind that pertains uniquely to 2ns, and still think that "contextuality" is the most promising candidate. It is that which precludes any individual from being absolutely singular, because it cannot strictly satisfy the principle of identity while occupying different places at different times and/or being referenced from different points of view.
[quote="apokrisis;57694]"That is why I said the contradiction lies in the genesis of specificity. Peirceanism says it is a contextual deal. The laws of thought say it is brutely tautological.[/quote]
I guess you have been talking mainly about how the genesis of individuality/determinacy/identity is contextual, rather than brute, because it involves the ongoing interaction between vague freedoms and general constraints. I have been talking mainly about how the nature of individuality/determinacy/identity is contextual, rather than absolute, because nothing is exactly the same as anything else, including itself.
Quoting apokrisis
I am not sure what you mean here. Could you please elaborate?
As a second cut: If x is contextual, then it is not necessarily true that under all circumstances, x = ¬¬x.
This version appeals to me because it seems to parallel better the "failures of distribution" of the other two principles for the vague and the general. Peirce stated within his first definition of "individual" that "the principles of contradiction and excluded middle may be regarded as together constituting the definition of the relation expressed by 'not'" (CP 3.612, 1911). Hence if one or the other does not apply, negation is left undefined. The same is true if this formulation of the principle of identity does not apply, and it also eliminates the (classical) logical equivalence of the other two principles.
That sounds paradoxical but makes the point that actuality - as the 2ns of substantial events - is the emergent outcome of the two real causes of being, the material potential of 1ns and the formal constraint of 3ns. So actuality is not real in being merely an effect, not a cause.
Zalamea p23 highlights that points are the limits on actuality - and so actual actuality is an unreal possibility in being completely or Platonically ideal. Existence can approach but not reach the perfection of discontinuous actualisation that the principle of identity demands.
Zalamea also highlights the irreducible mutuality that is thus at the heart of spatiotemporal existence when he goes on to talk about Peirce foreshadowing a modern desire for possibilitia surgery techniques.
For me - coming at this from a more physical perspective where energy or action must be made properly part of any world geometry model - you can understand energy in terms of spatiotemporal curvature. So now we can understand the continuum - that blackboard that is already determinate in being definitely dimensional - in terms of its own more primal pre-geometry.
Briefly, you have the two things going on as a reaction. We start with the unboundedness of disconnected curvature - a roil of hot spacetime indeterminate fluctuation. A chaos of directions all erasing each other. That is, an infinity of scraps of hyperbolic curvature. Space as energetic action is buckled maximally at every point and so curves apart from itself to lack all actual connection. The only continuity is this sea of rupture. 2ns exists only as the reactions that are immediate hyperbolic divergence - fluctuation dyads that are breaking apart as soon they connect, leaving behind no history or memory, no 3ns of some context of continuity.
But in the very fact of chaotic or locally hyperbolic curvature, you then have the latent possibility of a constraint to flat and simply connected Euclidean space. If only 1ns could be cooled and its wild curvature could start to join up to share a common story - each point or rupture be flattened just enough for a history of ongoing relations to start to form. Ie: the birth of 3ns as now a telos. Euclidean flatness could become the thermal goal. If definite dimensionality begins to form - like the three dimensions of the universe in which it concretely expands - then you can establish the feedback loop that drives the primal chaos towards the flat connectedness of a true continuum state.
So a world gets born by starting with unbounded freedom. From every possibility of a point or locale there is the possibility of a momentum or curvature. You just get these two complementary things together in the pregeometry as a necessity. If there is a locale as spontaneous fluctuation of pure possibility or 1ns, then it come automatically with the equally phantom possibility of its motion or action. And given no restrictions or bounds as yet on that other possibility, it would have to be as unrestricted as possible - hence hyperbolic curvature, or Planck scale divergence.
But then given also we are presuming there must in fact also be interaction or constraint going on between these extremophile locales, these unbounded fluctuations (a reasonable conclusion as it must be the case as otherwise we could not be here to question its existence) then it only takes a little bit of interacting to provide a little more persistent for the fluctuations, the ruptures, to start to cool and start to line up in flatter fashion.
Given such an interaction - a state of 3ns - constraints would provide a generalised flattening force, while the vagueness with its unbounded curvature would provide the energy to be disposed of as a developing extent of spacetime. We start with a lump blob of energy - a wodge of fluctuations going off in disconnected directions. Then like a ball of pastry, it gets rolled flat and spread very thin. It tells the story of a Big Banf becoming a Heat Death via an asymptotic story of self-equilibrating cooling and expanding. At the Heat Death, the fluctations or local curvature is almost completely dissipated, leaving just a Euclidean actuality of a maximally cold, dark, even, and perfectly connected void.
So if we are to get deeply physical about the mathematical continuum, we have to wind the story back even pre-dimensionally or pre-geometry. Peirce's blackboard analogy talks of an infinity of flat dimensions. But even vaguer would be an infinity of hyperbolically curved fluctuations that lack all connection or communication, and so flatness can become part of their telos if flatness is also a latent possibility of that pre-geometric beginning.
And it must be because any curvature at all is already speaking to the otherness that would be flatness and connectedness instead.
This is a nonsensical objection.
The two concepts are contradictory because Spinoza conceives substance as one and infinite and Aristotle conceives substance as plural and finite. Aristotle does not conceive the prime mover as substance at all, as you admit, nor as having any attributes other than being prime mover, so the two conceptions of Spinoza's substance and Aristotle's prime mover are also obviously incompatible.
Peirce's definition of "real" is that which has characters regardless of what anyone thinks about it. He came to realize by about 1896 that all three categories are real in this sense, which is why Max Fisch characterized him as a "three-category realist" from that point until the end of his life. Consider also this passage:
Of course, here we once again encounter the notion that existing things are "absolutely determinate," which - as we have been discussing - is really (pun intended) just an ideal limit. Strictly speaking, nothing "exists" in this sense. Is that what you are getting at by suggesting that the actual is the not real?
Quoting apokrisis
Again, please elaborate. How does the principle of identity demand discontinuous actualization?
I already covered this where you first made your objections about my mention of an unresolved "tension".
If 3ns is the constrained totality, then 1ns and 2ns stand constrained by it as first, the 1ns of in fact unconstrained possibilty, and then second, the 2ns of now constrained or determinate possibility.
So actuality from this standpoint is simply regularity of spontaneity. Energy or fluctuation has become so ordered by global law or habit as to be fixed in its dimensionality and thus completely determinate and countable as a physical degree of freedom.
This is where we actually are in quantum cosmology. We can count the total number of physical degrees of freedom in the visible universe at its Heat Death - there are 10^122.
So in including energy in the physical picture closes it, turns the apparently open or infinite into a tale of the inherently finite.
This means the continuum is "grainy" under a quantum gravity "theory of everything" view of the Universe. But grainy doesn't mean definite or determinate discontinuity - as in the points of a line, or the pixels of an image. It means that the necessary duality in terms of the forces of integration and differentiation - the constraining generality of global 3ns and constructing actuality of local 2ns - are already present in germinative fashion in the vague potentiality of 1ns.
So again, 1ns is unconstrained possibility. 2ns is constrained possibility (local actual constructive freedom). And 3ns is the constraint of both kinds of possibility.
So then the actual only has character to the degree that reality has thought about it. That fits.
What materialists call the actual is only that which physical pan-semiosis has secured as some persisting mark of being.
If 3ns is real, it is real because it can't be wished away. Likewise the ultimate tychism of 1ns is real in the same fashion.
But 2ns is only real in that it is the emergent result of the other foundational reals - the actual causes of actual being. So it is not itself really real in being the product of pansemiotic "thought", or the universal growth of reasonableness. Matter is effete mind as they say. Or in other words, any material event could have been thought otherwise. Newtonian mechanics was always about inserting ourselves and our desires into reality. We want to discover the "hard facts" of atomistic events so as to then be able to rearrange the machinery of existence for our own convenience.
There is much to ponder here, but you still have not explained - at least, not in a way that "clicks" for me - what you mean when you say that the principle of identity (x = x, or perhaps x = not not-x) demands discontinuous actualization by employing the context to derive specificity via a dichotomy. If you could spell this out, I would be grateful.
Again, as already said, semiosis goes further. It defines indiscernability as a pragmatic issue - the principle of indifference that underlines probability theory.
So indiscernability is not ontic, but epistemic. If the Universe has a purpose, then that in itself creates a boundary, an event horizon, where it will cease to sweat the detail. It meets that purpose and then everything beyond that is a matter of generalised indifference.
So maths is hung up on radical openness. Counting seems something that extends to infinity because the very definiteness of any first step seems to already to guarantee the openness of that. And then the radicalness of that openness leads to a desperation to also produce a matching closure. Philosophy of maths ties itself into knots to discover global bounds on unbounded construction - the 2ns of already determinate degrees of freedom.
But semiosis already comes with the closure to match the openness. If there is openness due to their being a purpose, then there is a closeness in the way that also creates the possibility of its own satisfaction.
Thus at the level of 1ns, the continuum is neither an open or closed set. It is a clopen set - https://en.wikipedia.org/wiki/Clopen_set
Or as I said earlier, a scalefree situation like a fractal. If you are seeking either points or their contextual neighbourhoods, they exist with perfect evenness across every possible scale of being (and so, like a fractal, there is radical openness). And yet at the same time, the perfection of that evenness is a scale symmetry or the definiteness of actual closure. You can use a single number to capture the exact (symmetry broken) dimensionality of the resulting structure. The Cantor set for instance has an "angle" of ln(2)/ln(3) or ? 0.631.
So note Peirce argues here by implication for the pragmatic principle of indifference. If we are now talking cosmology, it is the Universe that is indifferent to any difference that doesn't make a difference in being beyond the needs of its genenralised purpose.
Of course the theist will run into real difficulties here because beyond the Universe is some further 3ns of a god with a mind. Peircean semiotics was heading so nicely in the direction of pure self-manifestation on the logical grounds that "nothingness is impossible". And then the theist has to do a sharp 180 when this dangerous final answer eventually does show itself. 1ns can't be the true initial conditions of existence as Peirce's own logic makes necessary. And from that balking at the final hurdle, the whole point of the metaphysics falls apart. Transcendentalism re-enters to claim its false dominion.
Anyways...
So this is dichotomistic reasoning. The general and the particular are defined via their mutual opposition. And so they both only exist in a relative fashion. Nothing ends up real - except to the degree its own reality is secured by a sufficient distancing from its apophatically reality-erasing "other".
Thus you have not fully surrendered to the essence of semiotics. Everything is ultimately emergent, immanent, a matter of relative development. Even the universals that are real under semiotics are only historically realised - ideal only relative to their own developed degree of expression.
So any law is not exceptionless. It may approach the Platonic ideal, but it can't arrive there. That makes even laws or 3ns emergent and not really real (in that wishes can change things still - as we humans demonstrate in defying laws like universal gravitation with wings and aeroplane engines.)
Thus when we are talking about the mirror that are the three laws of thought and the semiotic triad, it is this kind of immanent mutuality we would seek on each level. And in your language, you keep wanting to talk about generality or vagueness in unsuitably specific terms. You are treating the categories as hard and existent, not developmental and relative to the business in hand.
So as I say, there are three laws of thought that constrain naked possibility so that only brute substantial actuality appears left. The world is a nominalistic totality of facts, a cause-transcending state of affairs. We only then need a mechanical mode of reasoning - the atomism of predicate logic - to make complete sense of existence.
But then Peirceanism is the counter to that - the counter that doesn't destroy the particular, but instead flips the perspective to show how it is the irreducibly emergent and not the brutely foundational.
So that then is how generality and vagueness emerge as the "other" of the LEM and PNC.
Generality denies the LEM by saying that particularity is always contexted by a purpose. So middles have to be excluded to the general satisfaction of some constraint. There has to be a semantic act of judgement, an act of measurement.
But then that purpose can be satisfied often quite easily in practice. So constraint is self-limiting in its essence. There is always a point beyond which even the universal law doesn't in fact give a bugger. So indiscernability arises for that self-interested reason.
The 1ns of vagueness then stands against the "2ns" of the PNC. With generality vs the LEM, it was 3ns against 3ns - top level against top level. But now we have an odd mixing of levels that seems confusing.
However it seems to work out right. Where the PNC talks in terms of crisp possibility, vagueness says no, crispness is only relative to its "other" of the vague. So what the laws of thought treat as having simple existence - bivalence - semiotic shows to be an emergent property that has to erupt via a fluctuation of vague suggestiveness, followed up by the solidification of established 3ns, or universalised habit.
Then we arrive at the 1ns of the laws of thought - the principle of identity. This then is made to stand in dichotomous contrast to what? It stands in apophatic relation to 2ns/actuality ... whatever that really means in Peirce-speak.
So in fact we already find this a well traversed issue in Leibniz's doctrine of indiscernibles. But where the reductionist thinks that the differences that make a difference are atomistically unbounded - there is no reason why we could ever in principle cease the pursuit of further detail, chase down the last decimal of the expansion of pi until we are exhausted - the Peircean system offers principled relief. We can stop when the differences cease to matter to our over-riding purpose. And the same goes for the Universe (whose primary telos is thermalisation as far as we can discern).
So the laws of thought presume the brute existence of the indiscernible difference that secures the principle of identity. And Peirceanism flips this to say indiscernability kicks in at the point where some 3ns ceases to have a reason to care, and so 1ns is left undisturbed.
Thus indiscernability describes some prevailing state of equilibrium where there is enough 3ns to create generalised order, and enough 1ns to generate freedoms to be regulated. 2ns as actuality is the fractal balance where we can put a definite number on the real actions of integration vs differentiation (flattening and curving) going on.
So again, 2ns in Peirceanism is about the emergence of crisp possibility or determinate degrees of freedom. And this then stands "other" to the 1ns of the laws of thought in denying their assertion that crisp possibility or determinate degrees of freedom are instead the brute foundational facts in nature.
The laws of thought in turn try to dismiss vagueness as merely semantic, not ontic. Any vagueness in logic is due to informal issues like measurement error ... or indifference to the finer facts, an insufficiency in taking care.
So quite neatly, the laws of thought also employ the same dichotomistic othering. They just take for granted what semiotics demands be tracked via some developmental history reflecting a "growth of reasonableness".
This of course is what I deny. There is only relativity, never the absolute.
However, vagueness can't be absolute either. And generality can approach its own limit asymptotically. So certainly the Universe can be expected to wind up as totally generic as possible - the shorthand description of its ultimate Heat Death.
[edit:] So where I agree is that it is "constraints all the way down". There is no foundation of actual material being, just always a constraint acting to suppress free variety. Reality is thus always contextual (while remaining also irreducibly tychic). And if that's what you mean here, then I guess that is extreme realism in deny non-emergent or transcendent reality to brute particulars.
Quoting aletheist
Huh? X is being made its own context. That is the tautology here. The assertion is being made that the context is crisply existent too.
The upshot then is that the statement is true only to the degree that either term is true. But also each is true in direct proportion to the truth of its "other". So as least we do have a definite form of relative truth in play. Each term is as true as the other, even if neither might never be a perfect truth ... and that also is stated in the reciprocality of the relation. If either one was perfectly true, it would negate its other - erase it from existence and so lose the other as that to which anything was being related. So it is a limit statement - with each limit having to have the positive existence of its other (in familiar yin-yang fashion).
So yes, in the end, it is a statement of contextuality. But contextuality "absolutely divided". So that is why 2ns is about emergent particularity or individuation and the 1ns of identity is about the brute facticity of the same. The triadic scheme makes the middle the last thing to emerge from an organic process of constraint on absolute possibility. Monadic reductionism presumes it is the first brute fact to get the mechanics of construction started.
Quoting aletheist
Yep. I have said that there must be this coupling of constraints and unbounded freedom that then leaves the third thing of determinate possibility - material degrees of freedom. So it is a pincer movement to arrive at where reductionist ontology wants to start itself.
Quoting apokrisis
Which is what, in your view? According to Peirce, "the universe is a vast representamen, a great symbol of God's purpose, working out its conclusions in living realities … The Universe as an argument is necessarily a great work of art, a great poem - for every fine argument is a poem and a symphony - just as every true poem is a sound argument" (CP 5.119, 1903, emphasis added). The dynamic object of this sign is God Himself, and its immediate object is His purpose, the development of reason - i.e., the growth of our knowledge about God and His creation. As an argument, the interpretant is its conclusion, the living realities that the universe is constantly working out.
Quoting apokrisis
My interpretation is that 3ns is the true initial condition of reality - which is prior to both possibility (1ns) and existence (2ns) - as Peirce's own cosmology makes necessary (Ens necessarium).
Quoting apokrisis
This is one point at which I am having consistent trouble tracking with you. I understand 2ns in Peirceanism to be about brute reaction/resistance, the absence of freedom (1ns) and reason/purpose (3ns).
Quoting apokrisis
How is this inconsistent with my suggestion that everything is general (i.e., indeterminate) to some degree? Perhaps I just need to clarify that this is generality in the broad sense, both negative (vagueness) and positive. It is a corollary of the thesis that all three categories are present and irreducible in every actual phenomenon.
Quoting apokrisis
By "here," do you mean the standard application of the principle of identity as x=x and/or x=not-not-x? Who is making the assertion "that the context is crisply existent too"?
Quoting apokrisis
Which statement? Which terms? I want to make sure that I clearly understand what you are saying here. Also, how would you fill in the blank with some formalized version of the principle of identity?
While I am at it, do you agree or disagree with my other "first cut" definitions of "contextual" that parallel what Peirce wrote about "vague" and "general"?
When we seek the truth, differences never cease to matter. A difference, by its very nature, as a difference, is a difference, and therefore it must be treated as a difference. If one adopts the perspective that a difference may be so minute, or irrelevant, that it doesn't matter, and therefore doesn't qualify as a difference, then that person allows contradiction within one's own principles ( a difference which is not a difference), and the result will be nothing other than confusion.
If x is not not-x, then it just seems straightforward that it is claiming its identity apophatically. I don't get that there can be any difficulty.
I am what I am because I am not what I am not.
But the difference is that now I have made the natural relativity of the question of identity explicit. I can suggest some feature as a characteristic that is "me" - like I am male. Because I'm not female. Then I can start to measure my maleness in terms of its,distance from femininity. Like perhaps I'm really hairy, so more truly male. Or perhaps I've got shapely legs, so that seems more ambiguous.
A silly example but it illustrates the principle. To state I am male is not very helpful. To state that I'm not female is to anchor the statement in a Universe of relative measurement. The context is now made fully part of the deal.
Quoting aletheist
But contextuality leaves it open whether the further possibility is 1ns or 2ns. It could a future condtional (the coming battle with the Persian fleet) or it could be some event already fixed by a determination (what will I discover when I finally check my ticket for the lottery drawn last week?).
So contextuality is simply 3ns or the generality of a constraint. And possibility divides into two kinds - that already determined by the past yet simply unknown or unmeasured, and then the true spontaneity of an undetermined future. Newtonianism talks about the first. Quantum theory is our best model for handling the latter.
Quoting aletheist
Following the second law of thermodynamics, I put it quite simply - to dissipate vagueness.
This is of course a very weak kind of telos from a theist point of view. But there you are. The world exists because vagueness proved to be intrinsically unstable. To the degree it existed, it already contained the possibility of regulation that could organise it to turn it into a crisp nothingness - a Heat Death void.
And throw all the Peirce quotes at me that you like. Semiosis starts with 1ns and so is radically at odds with any conventional transcendent monotheism - any 3ns notion of a higher purpose or creating mind or pre-existent harmony.
Quoting aletheist
Why should any reaction be determinate in itself? Two things may collide and bounce. But how do you know which hit which, or who came off worst? You need some kind of fixed backdrop to close the story - give it a context against which the elements of the reaction can be measured. So that was what Newton's three laws of motion were about (and the triadicity was hardly an accident).
As I say, once we talk about 3ns or generality as constraint or purpose or law, then 2ns becomes the constraint of 1ns and hence the determinate thing of constrained possibility ... or a material degree of freedom.
Any point particle has six degrees of freedom - three directions of translational symmetry or straight line inertial motion, and three directions of rotational symmetry, or inertial spin. So Newton captured this fact that the fixing of a Euclidean flat backdrop then left these irreducible degrees of freedom. Constraint could stop everything but these last, now crisply definite, forms of local symmetry breaking. A rolling ball or spinning top - in a frictionless world - would remain in motion without change forever.
So Newtonianism is about a set of absolute freedoms. And thus also the corollary of absolute constraints.
Of course then along came relativity to demonstrate all this classical definiteness was relativistically contextual and quantumly indeterminate. That is why Peirce gets credit for foreseeing the physical revolutions about to come.
Mind you, if you claim that everything actually does matter to you, excuse me if I think that is patent bullshit. Does it make any difference to you if I wear a red or blue shirt tomorrow? Do you need that to be another determinate fact ... or do you believe in free will in contradiction to your what you just posted?
So the idea is that the context of x is not-x, and defining the identity of x as not-not-x recognizes this, rather than making it a contextless tautology? "x is x" does not apply to the contextual, but "x is not-not-x" does apply as an apophatic alternative?
Quoting aletheist
Quoting apokrisis
Did something get accidentally deleted from your post? I do not see how your response here addresses my question.
Quoting apokrisis
Indeed, "frame of reference" has been in my mind throughout our discussion of contextuality.
Your understanding seems not incorrect.
If we are talking about identity, and the overriding purpose is, that we want to know the truth about the matter, then of course every difference matters. That's no bullshit, it's reality. If we allow that some differences do not matter, then we allow that two distinct things can have the same identity. Since giving two distinct things the same identity is a mistake, then in relation to identity, there is no such thing as a difference which does not matter.
You claim that the "identity of indiscernibles" cannot be upheld, but this is only supported by the claim that some differences do not matter. If we allow into our principle of identity, the notion of "some differences do not matter", then we have a compromised law of identity. Failure to hold fast to strong logical principles allows vagueness to creep into the logic. Such vagueness hinders our ability to determine the truth. Therefore, if our purpose is to determine the truth, we must uphold the principle of identity to the strongest of our capacities, and assume that every difference matters.
Your "Peircean flip" is this act of compromise. It takes identity from the particulars of the individual, and loses it into the vagueness of the general. By claiming some differences don't matter, you claim that we can disregard accidentals to focus on what is essential, so what is identified is a generality. But since the principal purpose of identification is to identify the particular, distinguishing it from other similar things, you negate the capacity to fulfill this fundamental purpose of identity, with that process, the flip.
I think Spinoza did not conceive of substance as "basic materiality", but rather as infinite activity, but in any case he certainly did not use the language of "vagueness". He did write somewhere in the Ethics that God is the efficient cause of all things; which amounts to saying that substance is the efficient cause of every modification of its attributes. Spinoza says that substance/ God has "infinite attributes" (which I think should be taken in both the quantitative and the qualitative senses) of which we can know only two: thought and extension. I guess this means that substance must also be considered the material cause of all things, but Spinoza does not allow of any final cause. Substance has no purposes, beyond the purposes of its modes.
As to the question of the oneness of substance, Spinoza wrote (quoted from Hegel or Spinoza by Pierre Macherey: page 103) :
"Nothing can be called one or single unless some other thing has first been conceived, in relation to it, as having the same definition (so to speak) as the first. But since the existence of God is his essence itself, and since we can form no universal idea of his essence, it is certain that he who calls God one or single shows either that he does not have a true idea of him, or that he speaks of him improperly."
Wow, that's your counterargument? You think I am going to be frustrated by not be able to figure out specific things which I believe are actually knowable in principle? Meanwhile, you deny the fundamental principles of logic, which might be used to figure these things out, and you satisfy yourself with the claim that such things can't be known. Relax, you're absolutely right, of course they can't be known, when you deny yourself the capacity to know them. Ha, ha, ha, humour me some more, it relieves me of my frustration. So, go ahead then, pour yourself a nice drink, and congratulate yourself, you seem to have convinced yourself that you already know all that it is possible for you to know.
God has infinite attributes. To only be one would be a contradiction with God's very nature. It would be like saying: "God only has thought" or "God only has a bookcase." God's nature is always to be more than one, to never have "only a" and have everything all at once. In this respect, God is discrete. Not a "One" of a distinct and separate state, but a defined whole of many without end or beginning.
Being of infinite attributes, God cannot be accounted for by giving the singular. One gets caught neither here nor there if they try. Does God end at the computer? At the book case? In thought? In the death of the sun? No. God is infinite. God cannot be said to begin or end at any point. It's anything but vague.
The vague account comes out of trying to treat God as a singular "One." It takes the obvious truth that God has many singular attributes and try to account for God through them. In the presence of a single attribute, mode or semiotic expression, someone claims to have discovered God, after all it belongs to God. Only they can't say what God is because the singular attribute, mode or semiotic expression is clearly not enough to be God, so they claim God must be vagueness that belongs to or underpins the singular attribute, mode or semiotic expression.
In truth, they have not discovered God at all. They have confused what belongs to God (singular attribute, mode, a semiotic expression) for God itself, taken the singular that points towards God and suggested it amounts to comprehending God. God gets morphed into this "fabric of vagueness" which is prior in causality rather than realised as the discrete infinite that's true regardless of time.
There looks a good article on exactly this topic - Firstness, Evolution and the Absolute in Peirce’s Spinoza by Shannon Dea - http://files.bloodedbythought.org/texts/On%20Peirce/Dea-44.4-Peirces%20Spinoza.pdf
I'll give it a read, but put the link here in case you want to consider it too.
OK, thanks for the clarification, Willow. :)
Thanks for linking that article, apo. It looks very interesting.
Some excerpts....
So the Peirce initially thought Spinoza didn't get firstness, then later wanted to change his mind...
So the true Aristotle got it, the scholastics didn't. And Spinoza was lucky in being uninfluenced...
So here, from logical considerations, Peirce describes the trajectory from a hot Big Bang to a cold Heat Death about 50 years before science confirmed it....
And here Peirce employs non-Euclidean geometry to model the various metaphysical cum cosmological alternatives in explicit fashion. He really was running rings around absolutely everyone....
In your face MU. :)
But then the paper struggles to identify any grounds by which the Spinozean absolute is Peircean firstness in another guise. It shows Peirce thought Spinoza wasn't fooled by his own "Euclidean" concreteness. But the reasons for being so charitable are not then adduced in any convincing fashion.
Dea says this...
Then offers a sketch....
Yep. But now Spinoza in his own words - where it gets less convincing....
So we seem to end up with the claim that Spinoza defined substance in untraditional fashion - not the formed secondness of hylomorphism or even Peirceanism - but as the pure potentiality of a vagueness.
Yet I don't think that adds up. At best, Spinoza might have dimly realised the need for pure potential (he was Aristotelean), but still made the mistake of thinking Firstness was still some kind of "material stuff", hence already in the hands of thirdness or the habits of form/purpose.
And this is not surprising given a theistic goal where the stuff of existence must be an expression of some mind's meaning - even immanently.
So summary is that Peirce certainly said by the end that Spinoza seemed a fellow triadicist. But so far no evidence to show that Spinoza really got it at a deep and explicity level. The similarity stems mainly from being a developmental process philosopher trying to make an immanent conception of the divine work out. So any logical properties of the metaphyics are consequences of that general orientation - that goal! - not well worked through arguments that carried the day in their own right.
Oh, that's a gas, coming from the one who's reply to my post was: nature's going to frustrate you.
If you're at all serious, then address the points of my post, and quit making a joke of yourself by saying that I'm the one who's not being serious.
In case you've forgotten, I'm waiting to hear justification for overruling the "identity of indiscernibles" with the claim that some differences don't matter. Obviously, if overlooking these differences allows you to deny the "identity of indiscernibles", then they do matter.
The very thing of a purpose defines its own epistemic boundaries - the point at which differences don't make any difference. And if you can't follow that argument, then that's your problem.
In Spinoza’s philosophy, the actual world (causality, determinism) is not opposed to possible worlds(possibilities, truths of what might occur). The one “inevitable” outcome (i.e. the future that will exist) is true alongside the possibility of every outcome. Possibility is always true for Spinoza. At any point, the world may be just about anything. Even though the sun rose this morning, and this is a necessary truth (the “inevitable,” the one future that occurs), every possibility where the sun doesn’t rise (e.g. it “pops” out of existence, the Earth’s orbit or rotation stops, etc.,etc.) is also true-- the didn't occur, but it's still true they are possible. "Potential" never ceases for Spinoza. With every state that occurs, event that is caused to thing that exists, there is the potential to be otherwise.
Realising the necessity of potential, Spinoza also points out potential cannot be "firstness." Why? Well, because it never begins nor ends. There is no time where a cut between a "firstness" of potential and the "secondary" or "tertiary" of actuality can be made. Potential is just as true for any time. It cannot be that which ends to form actual and discrete states of the world. The world of today must have just as much potential as any quantum foam of the distant past.
Spinoza understood the need for pure potential more than anyone else. He realised it must be beyond "firstness" (or "secondary" or "tertiary" ), finally realising potential's poisonous grip on metaphysics, where it thought to be something a force (i.e. a final cause) must "add" to the world for anything to make sense.
What matters to someone is always a function of that person's purposes. Surely you can agree that some differences matter to you more than others; and since everyone has finite resources (including time), we have to prioritize which differences - as well as which similarities - are significant enough within a given context to warrant our focused attention.
Quoting Metaphysician Undercover
Read more carefully - in the comment that you referenced, did not say anything about a difference not being a difference; he was talking about a difference not making a difference. Do you see the difference (pun intended)? There are times when a difference really is so minute, or irrelevant, that it does not matter in that context; i.e., it is not worth taking into account, given one's purposes. It still is a difference, but it does not make a difference. I care not one whit about the color of paint that is going to be applied to a steel beam when I am analyzing it by means of a mathematical model to determine whether it can carry the forces that the building code says it must be able to withstand as part of an actual structure. The architect has a different purpose, and therefore might have a different stance - although typically he or she just wants to hide everything above the ceiling anyway.
Quoting Metaphysician Undercover
Why would this always be a mistake? Standardization and mass production are all about minimizing unimportant differences, such that we can treat different things as effectively identical. When I select a particular section for that beam, I am counting on the fact that it is irrelevant which mine produced the iron ore, which cars and washing machines provided the scrap metal, which mill melted all of that together to make the steel, which service center stored it after rolling, which fabricator assembled it, or which erector installed it. None of those differences make a difference in the finished product, as long as it meets certain minimum specifications - i.e., there are no differences that would make a difference - and that is a good thing!
Quoting Metaphysician Undercover
But what if it turns out that vagueness is a fundamental and ineliminable aspect of reality? What if the truth is that vagueness constitutes an actual limitation on our ability to determine the truth? In that case, your dogmatic insistence on assuming that every difference matters hinders your ability to determine the truth about vagueness.
Quoting Metaphysician Undercover
The very act of distinguishing one thing from other things already involves neglecting differences that do not make a difference. Why do we pick out this chair or that table or this book or that door as individual objects, rather than always and only referencing them at a molecular, atomic, or even quantum level? Because the difference between one particle and those adjacent to it within the object is irrelevant to our purpose in picking out that object as a single object. You do this all the time, but it comes so naturally that you do not realize it. No one is capable of paying attention to every single difference among phenomena, because there are far too many of them to do so - even just within your field of vision during the passing of one second.
If that is truly Spinonza's view then at least we can cross him off the list. ;)
Quoting TheWillowOfDarkness
It would be nice if you could support your claims with references or quotes for once. This sounds wildly made-up.
And yet the traditional/classical conception of God is that He is absolutely simple; His attributes are not discrete in the way that you seem to be suggesting.
Quoting TheWillowOfDarkness
With all due respect, that seems rather ... vague to me.
Quoting TheWillowOfDarkness
This seems like a case where Peirce's attempt to use generic terminology for his categories may have been misleading. They are not called 1ns, 2ns, and 3ns because they always and only come about in that order; on the contrary, my interpretation of his cosmology is that in the hierarchy of being, 3ns is primordial relative to the other two. In any case, 1ns/possibility does not "end" where 2ns/actuality "begins," they are both - along with 3ns/necessity - indispensable and irreducible ingredients of ongoing existence.
For the purpose of understanding the nature of nature, we need precision otherwise we miss the boat.
It is alright to say that a book, for practical purposes, has the same identity before and later. But it is more precise to say that the book has changed and continuously changes so that it is never is the same in duration. After all, we are trying to understand nature and not simply make arguments.
I am surprised that you would say this, considering that we started the thread with your comments to the effect that discrete mathematics cannot properly represent the continuity of nature. Precision is a matter of measurement, and measurement is a matter of discrete mathematics; but the continuous is indeterminate.
Quoting Rich
I made this point earlier; the contextuality of actuality entails that it is not necessarily true that this object from one point of view, or at one place and time, is identical to this object from another point of view, or at another place and time. Perhaps we can agree that it is more precise (in your sense) to recognize the imprecision (in my sense) of reality.
The purpose, as you yourself, described in that passage, is identity. If you are prepared to say, that two things with the exact same identity, are not in fact the exact same thing, (according to the identity of indiscernibles), because of some differences which do not matter, then you only defeat the purpose of identity, which is to distinguish one thing from another.
Quoting aletheist
Correct, and whether or not a difference matters, depends on the context. And the context in which this was stated was in relation to a point where the "identity of indiscernibles", is supposed to no longer be relevant.
Quoting apokrisis
But in this situation there is no such thing as a difference which doesn't make a difference. Consider identity A and identity B. If these two identities are the same, then according to the principle of identity of indiscernibles, they are one and the same thing. If you deny that principle of identity, and say A and B are really not the same thing, because of some difference between them which does not matter, and is therefore not part of the identity, (the identity being one and the same), then how is it true to say that this difference does not matter? It is only by claiming that there is a difference between them, which does not matter, that you can say they are two distinct things, rather than necessarily one and the same thing, as stipulated by the "identity of indiscernibles". So it is false that this difference does not matter, because it is the only difference which makes them two distinct things.
Therefore, apokrisis' claim, from Peirce, is that two distinct things can have the very same identity, if we allow that there are differences which do not matter. But of course these differences really do matter, because these are the differences whereby we distinguish the two things as distinct. And it is simple contradiction to say that these differences do not matter.
Quoting aletheist
The point is, that with respect to the principle of identity these minimal differences are the differences which really are important. If you do not respect this fact, then you allow that all mass produced items are in fact, the very same entity, because you are insisting that they are identical. My car is the same object as your car, because they are mass produced and identical. Your desire is to claim that the factors which differentiate them (the differences of the particular) do not actually differentiate them, and identify them as distinct, as those differences are unimportant. So you will claim that they have the very same identity, yet you will also claim that they are two distinct objects. They are distinct objects not by being different though, because those differences don't matter, they have the same identity. What would justify the claim that they are different then? Or is it the case that my car and your car might really be the very same object?
The purpose of the law of identity is so that we can distinguish one object from another, and come to know that object as the thing it is. To claim that we can overlook some minor differences such that numerous objects may have the same identity only defeats this purpose. We simply deny ourselves the capacity to tell these objects apart.
Quoting aletheist
You should consider that perspective as rather nonsensical. Even if vagueness is real and fundamental, we will not know this until it is proven. And we cannot prove its reality without identifying it.
Quoting aletheist
Yes of course, distinguishing one thing from another usually involves neglecting differences which do not matter. But here, we are in the context of the principle of the identity itself, the idenity of indiscernibles. So we are, according to that defined context, dealing with things which appear to be the same. We can conclude, as apokrisis implies, that we cannot tell them apart, because they have the very same identity, yet they are not really the very same thing, due to some differences which do not really matter. Or, we can uphold the principle of the identity of indiscernibles, and conclude that if they are distinct entities, then there must be real differences, which matter, by which we can tell them apart.
But measurements are imprecise and cannot ever be precise which is exactly the point of this thread. It is impossible to stop anything (continuity and continues flow are embedded in nature)in order to retrieve precise measurements. Measurements are always approximations and it is why measurements specifically and mathematics in general (because of its discrete nature) are very poor tools for understanding nature. Straying from this understanding ultimately will always render a poor understanding of nature, the worse one of which by far is turning humans into number machines.
Quoting aletheist
Agreed. In any discussion of nature one should recognize continuous flow and change, which brings us back to the OP. Discrete and nature are like oil and order - so don't try to mix them up.
Peirce appears to face the problems which are associated with the idea widely accepted in mathematics, that a continuity is divisible. But instead of following through to where his investigations lead, and coming to the proper conclusion, that a true continuity is truly indivisible, he compromises and proposes this principle of a difference which does not matter. That is a mistake. The proper decision would have been to accept the metaphysical principle that continuity is indivisible, because to divide it proves it to be discontinuous, regardless of what mathematicians want to do.
The problem with this representation, is that 2 here is not part of the continuity, it is a point of difference, differentiating one part from the other.. There is a supposed continuity which goes right through 2. If we divide that continuity at 2, then 2 is not part of the continuity, but a point of difference. A continuity cannot have a point of difference because this would make it discontinuous.
This sentence makes no sense to me. Differences that do not matter enable us to treat two things that are not identical as if they were identical, for a particular purpose; this is the opposite of claiming that two identical things are not, in fact, the same thing. If our purpose is to distinguish two things, then obviously more differences will matter.
Quoting Metaphysician Undercover
Again, this is backwards. The point is not to claim that there is a difference that does not matter in order to distinguish two things that are really identical, it is to treat two things as identical because the real differences between them do not matter within the context of a particular purpose.
Quoting Metaphysician Undercover
Again, it depends entirely on our purposes, which depend entirely on the context. For the most part, the difference that makes a difference in this example is that one car is yours and the other is mine - a human convention, not something intrinsic to the objects themselves. If they were sitting side-by-side on a dealer's lot - same year, make, model, trim, colors, options, condition, mileage, price, etc. - then there would be no differences that make a difference, until you (arbitrarily) choose which one to buy.
Quoting Metaphysician Undercover
It defeats that particular purpose, but it can be useful for other purposes. By acknowledging that the law of identity has a particular purpose, rather than being an absolute and intrinsic feature of the universe regardless of the context, you are effectively agreeing with the point that we have been discussing.
I continue to be skeptical of your claim that all mathematics is inevitably discrete. In the last several decades, category theory - as a more general alternative to (discrete) set theory for the foundations of mathematics - has facilitated developments like synthetic differential geometry or smooth infinitesimal analysis, which show great promise for more faithfully representing the continuous as continuous.
He does? Where? Please cite his writings to support this claim. Did Aristotle also employ this notion, since he likewise held that a continuum is (infinitely) divisible, though actually undivided?
Quoting Metaphysician Undercover
No one is disputing that actually dividing a continuum introduces a discontinuity. However, that discontinuity is not there until we break the continuity by that very act of division.
Quoting Metaphysician Undercover
Again, citations please. As far as I can tell, you have no clue about what Peirce had to say regarding these matters.
Quoting Metaphysician Undercover
Indeed, but what you still refuse to acknowledge is that a continuum does not contain any points at all.
Could you give an example of how you "actually" divide a continuum, and "introduce a discontinuity"?
Correct, now reflect on what you have said. If our purpose is to identify things, which is what we are discussing here, identity, then allowing that there are differences which do not matter, defeats our purpose. This is because it allows two things which are not identical to be treated as if they are identical. This mean, explicitly, that we will confuse one with the other, and we will fail in our efforts at identity.
Quoting aletheist
Let me reiterate, the particular purpose which we are discussing is identity, identification. Under no circumstances would we want to treat what we know as "two distinct things" as identical, when our purpose is identification, because this explicitly defeats the purpose.
Quoting aletheist
Sure, it is useful to treat distinct things as similar, for certain purposes, such as generalization, and so consider that some differences do not matter. But in doing this, we respect the fact that similarities do not render two distinct things as the same, we simply produce a generalization. So when we overlook differences which do not matter, we do this with the intent of looking at things as similar, not with the intent of looking at distinct things as having the same identity. We overlook differences which do not matter, for the purpose of saying that two things are similar, not for the purpose of identity, or saying that two things are the same.
Quoting aletheist
The act of dividing something demonstrates that the thing divided is not continuous. The claim that it was continuous prior to being divided needs to be justified. If a continuity can be divided then the logical conclusion is that it cannot consist of indivisible parts, this would deny continuity, so it is necessarily infinitely divisible. To prove that the thing divided was in fact continuous then, requires that it be divided infinitely. This produces an infinite regress, and is a simple denial of the fact that it is impossible to divide something infinitely. The fact that it is impossible to divide something infinitely implies that it is impossible to divide the continuous. Therefore the thing which you divide, was never continuous in the first place and the continuous is actually indivisible.
Quoting aletheist
Isn't that exactly what I said? A continuum necessarily has no points, and this is why it is inherently indivisible. You, on the other hand, by assuming that a continuum is divisible, assume that there are points of possible division. If we deny that there are any points to the continuum, then it is necessarily indivisible because there are no points where it could potentially be divided. How do you propose that the continuum is divisible if there are no points of possible division?
Quoting aletheist
I've read enough Peirce, and secondary sources, to know what he was talking about. If you think that what I said is wrong, then please correct me with more accurate information, I would welcome a chance to upgrade my understanding.
Well, you can certainly partition the continuum of the Reals wherever you wish.
Only if by "identify" you mean "distinguish." What I thought we were discussing was whether the relation of identity itself is absolute or contextual (more below). An object is not even strictly identical to itself from different points of view, or at different times and places. However, we often treat it as the same object from different points of view, and at different times and places, because doing so suits most of our purposes. Perhaps we agree on this and can move on.
Quoting Metaphysician Undercover
No, the act of dividing something that was continuous causes it to become discontinuous. Not surprisingly, we disagree on whether the infinite divisibility of a line renders it discontinuous, even if it is not actually divided. I am never going to convince you that "x-able" does not entail "actually x-able," and you are never going to convince me that the two are necessarily equivalent; so we might as well just agree not to waste each other's time by going down that road yet again.
Quoting Metaphysician Undercover
You made assertions about Peirce's views, so the burden is on you to show that you accurately restated them. The specific language of "a difference that does not make a difference" comes from @apokrisis, not Peirce; and he did not bring it up "to support the proposition that a continuity is divisible," he was talking about identity within existence as contextual, rather than absolute.
Quoting apokrisis
Quoting apokrisis
If I offered you the choice between two McDonalds cheeseburgers, would it make a difference which one you picked?
If there are differences that don't make a difference, then there are differences that do. And on that logical distinction would hang the pragmatic definition of a principle of identity.
You may insist on your own unpragmatic definition. It would be interesting to hear what it might be. How does difference end for you? What makes something finally "all the same" for your impractical point of view?
What does "dividing" even mean? If you mean that you can take an interval of the Reals, well we learned how to do that at school, so we can do it. We can even write down an expression for taking (countably) infinite number of (finite) intervals of the Reals. So it is actually and trivially x-able.
I leave it as an exercise to the reader to ascertain whether the intervals must be open or closed.
The only way to understand nature is to fully and completely remove symbolism from the investigation. One must explore music, light, motion, thought, consciousness, dreams, sound, etc. directly. One must use consciousness to directly explore itself and penetrate it deeply.
Admittedly, without telepathic communication available to us, for discussion purposes we must resort to symbolic metaphors that in some way describe the continuous flow (I use the ocean and the symphony as my metaphors) always avoiding any addition of symbols that might allow for discreteness. Much of your arguments fully depend on creating discrete, which may be practical under many circumstance. But when discussing the nature of nature, discrete symbolism is more than impractical, it unleashes all kinds of paradoxes which are sure fire red flags that another mode of analysis is required.
I agree with you on this. However, mathematics is not always and only a matter of numbers or other discrete entities. Geometry, especially topology, is an obvious example.
Quoting Rich
Except that symbolism is part of reality, so fully and completely removing it would limit one's overall understanding just as much as focusing on it to the exclusion of the other kinds of experiences that you mention.
I have disagreed with you on this point previously, and clearly showed you that identity is not the same as identification, and yet you continue to repeat this mistaken thought. Things are not identified by means of their identity, that is absurd; they are identified because they stand out, and they stand out on account of their differences from, and similarities to, other things.
If course we must rely on symbolism to communicate, since mind to mind communication is not available, but before such communication is performed, one must first probe nature directly and then admit in any use of metaphors that the metaphors are incomplete.
The Dao that is named is not the Dao.
Pretty good though from the ancients.
As I keep telling you, mathematics does not rely totally on manipulation of the discrete; or at least, mathematics need not rely totally on manipulation of the discrete. The problem is that since the late 19th century, mathematics has largely relied on the manipulation of the discrete, because it has been grounded primarily in set theory. In recent decades, category theory has emerged as a viable alternative that is more general and much more compatible with the concept of continuity. This is evident from subsequent developments like synthetic differential geometry and smooth infinitesimal analysis.
But why do you presume the job of the mind is to see reality "as it is"? That makes no evolutionary sense.
So pragmatism/semiosis is a realistic theory of the epistemic modelling relation we have with the world (and then ontologically - the surprising bit - that reality has with itself so as to indeed form "itself").
This flips everything around. Now the job of the symbolising mind is to take as little note of the actuality of the world as possible. To the degree the mind has detached itself from brute actuality, then it is starting to see the world only in terms of its future possibilities.
So for consciousness - as attentive level processing - less is more. The goal is to reduce awareness of the surrounding to the least amount of detail necessary to make successful future predictions, and thus to be able to insert oneself into the world as its formal and final cause. We gain control in direct proportion to our demonstrable ability to ignore the material facts of existence.
This is why science is the highest form of consciousness. It reduces awareness of the world to theories and measurements. We have an idea that predicts. Then all we have to do is read a number off some dial.
The fact that reality might be continuous is the reason why psychological mechanisms evolved to extract semiotic discreteness from it - a tale of distinct signs. The mind is designed to zero in on some single telling point of view in any moment - to attend. And in doing that, everything else can be ignored as noise. The world outside the focus of attention is simply ... vague.
My point is that you, like MU, are arguing from a particular set of presumptions. There is this wrong idea that the mind should see everything exactly as it really is. But that is illogical in evolutionary terms - in terms of the principles of modelling. The mind wants to do the exact opposite - transcend the world, so as to gain the power to re-imagine the world.
Of course the world still exists in brute continuous fashion. It has its recalcitrant being that ultimately acts as a constraint on our desires. Yet that doesn't mean we should just give in and give up. The highest state of consciousness is the one that is most semiotically developed - the best able to impose its own reality on reality through the creativity of a sign relation.
What? Have you not been paying attention? The continuum was discovered via set theory!
Bateson must get the actual credit - http://www.informationphilosopher.com/solutions/scientists/bateson/
You need to get out more. The mathematical world is larger than just set theory - https://en.wikipedia.org/wiki/Continuum_(topology)
The continuum made of glued together points is complimented by the continuum made of glued together relations.
And the true continuum of Peirce (or Thom, or Brouwer) goes beyond that duality in being the source of that duality.
The continuum was not discovered via set theory, it was (and still is) modeled using set theory. Real numbers merely constitute an analytic continuum; they do not form a true continuum as defined by Peirce - as well as duBois-Reymond, Brentano, Brouwer, and many others.
Science demonstrates its control over existence in everything that in fact makes your own modern existence possible.
So your cry of protest here could hardly sound more feeble.
MU's thoughts indeed form an undivided continuum. :)
If I may interject here, it seems to me that the job of the mind (or any organ for that matter) is to provide the organism the necessary nutrients to survive. In the case of the mind (or the brain depending on how you see the relationship between the two), paired with the sense organs, provides the organism a valuable nutrient - information.
Information, of course, needs to be accurate. The mind needs to be able to predict future outcomes, and it does this through trial-and-error learning, habitual behavior and unconscious memory. If everything was perfectly known, there would be no need for a mind. No thinking would be required. Thinking is the process in which we evaluate different sorts of information and construct a path of action. If we wanna go the psychoanalytical meta-psychological route, then consciousness is the (painful) method in which the unconscious satisfies its endless depth of want and need in a temporal world of insufficiency. Without this inherent ambiguity and uncertainty, there wouldn't seem to be any reason for an organism to expend energy on a representation of the world for the sake of representing the world. Certainly the mind cannot be an easy thing to maintain.
So apo is right in that for biological organisms, less tends to be more. Efficiency is what's up. But of course the mind has to be modelling the world somewhat accurately, otherwise theories like apo's wouldn't even make sense themselves. Here we have Plantinga's argument against naturalism, which in my opinion fails but certainly provokes discussion and refinement of naturalism.
Quoting apokrisis
The tricky part is to figure out that balance between seeing too little and seeing too much. Curiosity as much as ignorance is a source of many problems. A cultural domino effect.
Quoting apokrisis
Why just science, though? Why not soccer? Surely goalies reduce their awareness of the world to the game, its rules and the movement of the players and the trajectory of the ball. Why isn't this the highest form of consciousness?
To denote science (or anything else) as the "highest" form of consciousness is sort of ambiguous in my opinion. Higher than what? What measuring system are we using here?
If anything I would have to say philosophy is the "highest" form of thought, since it deals with abstract concepts in a purely possible modality. Or, hell, even just daydreaming.
Or meaningful in fact.
And this is the semiotic point. Information is not noise but a message when it is a sign connected to our desires.
So there are two views of information. One is that it is a material difference. That makes even noise countable as bits of information.
The other says differences have to make a difference. And that happens when there is a context of purpose which interprets a difference in terms of its meaning. A signal now also has a message.
There is a science of these things you know...
Quoting darthbarracuda
I often drive long stretches of road with no conscious memory of a lot of pretty technical and dangerous actions. We are designed to automate our awareness of the world so that we can do everything at the most habitual and inattentive level possible.
So attention and thought are reserved for dealing with the unpredicted, the novel - the things we hope to turn into habits in the future.
That is why the old are wise. They know all the answers already. Correct thought appears effortless.
Quoting darthbarracuda
No. Let's not go into the bogus science/dressed-up romanticism of psycho-analysis.
Quoting darthbarracuda
You just don't get the nuances correctly yet.
The whole notion of "re-presentation" is a psychological fallacy. The mind - as a modelling relation - wants "efficiency" in always knowing the shortest path between its desires and their fulfilment. So it is that shortest path which fills awareness, not the totality of all the world's facts.
Quoting darthbarracuda
I defined it - going the furthest in reducing awareness of reality to a matter of signs - that is, the theory we create and then the numbers we read off our instruments.
The soccer goalie does just the same in the end. Success or failure is ultimately read off a score board ticking over - the measurement of the theory which is the rules of a game.
One-nil, one-nil, onnneee-nilllll-ahh! Comes the happy chant of the home crowd.
Quoting darthbarracuda
You are forgetting the role of measurement. Ideas must be cashed out in terms of impressions.
Science is the metaphysics that has proven itself to work. It is understanding boiled down to the pure language of maths. And so measurements become actually signs themselves, a number registering on an instrument.
That may be true, but what we are discussing is continuity itself, as an identified thing. This is like if we were discussing 'red" as an identified thing, not the objects which are red. You refer to "something that was continuous". So you are deferring now to an underlying substance, a thing which is continuous, and you are saying that this substance could be continuous, or it could be divided to be discontinuous. It's analogous to if we were talking about a liquid. It is incorrect to say that a liquid could become a solid, because it is the underlying substance, water, H2O, which changes form, from being a liquid to being solid. The property of being liquid is negated, for the property of being solid. Likewise, if something continuous is divided, and becomes discontinuous, it is the underlying substance which changes its form from being describable as continuous to being describable as discontinuous.
What we are inquiring into is not the nature of the underlying substance, but what it means to be continuous, and what it means to be discontinuous.
Quoting aletheist
Now you are talking about the infinite divisibility of a line. But if that line has substantial existence, as if it were written on a paper or something like that, it is impossible that it is actually infinitely divisible. We could only cut up the paper into so many pieces. So I assume that you are talking about an ideal line, in the mind, and assuming that the ideal line is infinitely divisible.
This is our substance now, an ideal line, and I will assume that it is continuous. How is that ideal line divisible? If you divide it up into sections it is no longer the ideal line which it was. Either the idea is of a continuous undivided line, or it is an idea of a discontinuous divided line. It cannot be both because this is contradictory. It makes no sense to say that the ideal continuous line is divisible, because you don't actually divide it. You replace the idea of a continuous line with the idea of a divided line, each having a different definition. You do not divide the ideal continuous line in your mind, so it makes no sense to say that this ideal line is divisible. The line on the paper is divisible, but it makes no sense to say that it is infinitely divisible.
The conclusion is that it is nonsense to talk about "the infinite divisibility of a line".
Quoting apokrisis
Yes of course it would make a difference. The one I chose would be the one that I eat, the one I didn't choose would not be eaten by me. Do you think the difference between being eaten and not being eaten is not a difference?
Quoting apokrisis
In the case of the identity of indiscernibles, as I explained, every difference matters. We cannot claim to have properly established identity unless every difference is accounted for.
If in some circumstances, we can establish a usable identity without accounting for all of the differences, then that is fine for those circumstances. But as a formal "principle of identity", upon which one would base a logical structure, it is unacceptable to allow that there are differences which do not make a difference, because this allows that there may be two distinct things which have the same identity.
Quoting apokrisis
You are going in the wrong direction here. It seems like you have some sort of backward notion of identity. The purpose of a principle of identity is to ensure that each thing has it's own identity, that it is identifiable as the thing which it is. The goal here is not to make things "all the same", it is to make every thing different, thereby allowing that everything is identifiable as the thing which it is, and not confused with anything else. That is why there will be no success to any principle of identity which does not seek to determine every last difference.
Quoting John
I really don't see your point. A thing's identity is established according to its individuation, and this means its difference from other things. And, it is identified according to its identity. So I really do not know what you mean when you say that things are not identified by means of their identity. It doesn't make sense. What division are you trying to create between "identity", and "identify"?
But how did it make a difference to you that you ate one and not the other? And how even did it make any difference to the world, if the world had any discernible interest in the matter.
So - as has been repeated ad nauseum by both me an altheist now - it is not that there isn't a difference, but there needs to be a difference that makes a difference ... which is the difference that makes a difference in this discussion.
You are talking about meaningless differences and claiming they are now meaningful. But you can't say why that would be so (except you would then be able to count some points being scored on your private anti-apokrisis metre ... Bing! MU scores another (own) goal!)
A thing' s identity is not the same as what it is identified as. In our previous conversation, the example was Pluto, which had at an earlier time been identified as a planet. It's easy to see that its identity cannot logically be the same as was it is identified as, because Pluto is the entity which had previously been identified as a planet and is now not identified as a planet. Pluto's identity is not affected by what it happens to be identified as, in other words. Identity is always a matter of mere logical stipulation, and it is never actually realized, which I think is pretty much in accordance with something apokrisis had said earlier about the relation between identity and actuality.
If that were true, then you would not be arguing with me, because it is simply a fact that - going back at least to Aristotle - "continuous" means being infinitely divisible though actually undivided. In any event, this is what I mean by continuous, and your insistence on your idiosyncratic definition is not going to change that.
Quoting Metaphysician Undercover
That is apparently your conclusion. Mine is that it is pointless (pun intended) to talk about anything related to this topic with you.
And those later objections have been swept aside. Cantor was the first to rigorously define the continuum in 1870s and all the dissenters have been forgotten.
I think you'll find that Peirce got into the act somewhat later than Cantor, after being inspired by Cantor. And, in the history of Real analysis, set theory, etc, Peirce is a dead-end. Cantor's ideas have been extended and developed, Peirce's have been abandoned.
Developments such as category theory, nonstandard analysis, and synthetic differential geometry or smooth infinitesimal analysis reflect dissatisfaction in some quarters with the dominant paradigm. Only time will tell whether any or all of these supplant set theory and its progeny.
Quoting tom
He was indeed inspired by Cantor, but he also achieved some of the same results and reached some of the same conclusions at least semi-independently. In the end, he became disenchanted with Cantor's whole approach; as @Rich has been emphasizing, you cannot adequately represent true continuity with something that is discrete.
Quoting tom
And yet, here we are, discussing them. There has been quite a revival of interest in Peirce's ideas, both in philosophy and in mathematics, over the last couple of decades. As @apokrisis likes to point out, he was far ahead of his time in many ways, and we are only now catching up to him.
You really do live in your own private Idaho. Underground like a wild potato.
So the first mark (or cut of the line) anchors the second mark by becoming the memory - the context in which a difference can now make a difference.
This introduces the direction of time, and hence energy and dissipation, into axiom-level mathematical thinking. The past is spent (gone to synechic 3ns), and yet the future is still open (still primal or tychic 1ns).
So the continuum does have this neighbourhood property - this extra hidden dimension - which is its memory. The first cut becomes the context for a second cut (and together they underwrite an endless repetition of cuts). And this is where counting and even ordinality gets justified. Effort has to be spent in constructing a history of what has happened. But the future extends to infinity and beyond - underwritten by its own past success.
In appreciating the intuitionistic approach to the continuum, we can see what the set theoretic approach simply freezes out and takes for granted. The Cantorian infinity is timeless and effortless - and thus patently unreal on that score.
If maths wants to speak more truly of nature, we can see how memory and action must be added back into the mix.
The point is the difference is meaningful, no matter how much you might pretend otherwise-- a thing's identity is not found in what it is to you (i.e. your experience of it, semiotics, your "epistemic cut" ), but rather itself. There is a difference ( "This one dies, not the other" ) no matter if you care about it. Your generality is a myth, a dishonest story you tell yourself to eliminate subjects in the contexts of your "practical" concerns.
How is "itself" to be "found" if not in experience and thought?
That's the trick-- it is found in experience and thought... but it is not the experience and thought.
Consider the knowledge of you posting on this forum. How would anyone know about this? Well, there are only their experiences, be they of the empirical world (i.e. observations of your posts) or experiences of logical comprehension (e.g. their identity, your identity, the logical discintion of a forum and posts, etc.). We really do know objects themselves-- else we would be merely talking about ourselves rather than other things or people.
But... as apo demonstrates here, our experiences are their own. They are distinct from the thing-itself, such that, on some occasions, we make mistakes about things. Apo's experience shows these two states to be identical, even though they are not. He does not know the identity of these states itself. His experience has missed it.
Which is exactly the problem with the traditional/classical God. There is no way an infinite of attributes can belong to such a being. Such a God is simple and empty. No thing belongs to them because it would mean owning the discrete. The continuum of God is left with other at all, merely an empty set that has nothing to do with the world.
Spinoza's point is both the continuum and it's members are discrete-- the former as the continuum itself, the latter as each particular member. To be a continuum or category is discrete, not to lack identity or discintion. The infinite is its own discrete truth-- it's the infinite itself, a set which contains endless discrete members.
Only because you don't recognise the infinite as its own thing. It seems "vague" because you are trying understand it though a finite lens. You want us to say what is in the world or what it does to the world, how does it manifests in our observations of the world.
The point is the infinite is not vague at all. It means a set without beginning nor end, a distinction and identity all of its own, which is never any member of itself.
A hierarchy is an order. No doubt he is talking about logical terms, but hierarchy or order is an inherently a conception of the finite and time. It results in a leakage into causality (as we see in apo's argument which treat "vagueness" as the origin of the states of our reality), where 2ns (actuality) is considered to be born of 3ns (semiotics, necessity) and 1ns (possibility).
Possibility is necessary. In Peirce's terms, it is also 3ns, along with logical distinction forms (semiotics), possibility is necessary. The hierarchy collapses. Since possibility is necessary and infinite, it never begins or ceases. It is just as "primordial" as semiotics. Logical distinction has always been. Possibility has always been. Neither came first. Peirce's triad collapses into the necessary (semiotics, possibility) and the contingent (actual states).
Furthermore, the hierarchy between the necessary and contingent does not make sense. For a state to be actual, it takes more than the presence of either semiotic discintion(i.e. form, meaning) and possibility. If I say: "There is a logical form of me being the US president and the possibility of me being the US president," it doesn't birth the actual state of me as the US president. Only an actual state can do that. In terms of definition, actual state are self-defined, not given by the necessity of semiotic and possibility. Actuality becomes just as "primordial" as semiotics or possibility.
If I am to be president of the US, there can only be a concurrence of possibility, semiotics and actuality. Only when those are all at once, of themselves, am I US president. There can be no hierarchy.
Going back to this point now. The difference which Peirce claims does not matter, which enables him to divide the continuity, is the difference in order. Consider the example of dividing at 2. On one side of 2 the order goes lesser, and on the other, the order goes greater. And there is a place, occupied by 2. Peirce considered time to be a primal continuum, and dividing time creates this same problem with order. Any division in time creates this same issue of a different order on each side of the division, and a place occupied by the inserted divider.
Here's the reference you requested:
https://books.google.ca/books?id=iy76kUCZYb0C&pg=PA88&lpg=PA88&dq=peirce+divisibility+of+continuity&source=bl&ots=tyuF0tOcKH&sig=BeofMNRZHX7Uu_58YstLlE4Bk5g&hl=en&sa=X&ved=0ahUKEwj99tn0h7TSAhWb8oMKHciMAOsQ6AEIOzAF#v=onepage&q=peirce%20divisibility%20of%20continuity&f=false
The issue is well explained in BK. 6 of Aristotle's Physics. After stipulating that anything continuous, including time, is divisible, and necessarily infinitely divisible, he proceeds to determine "the present" as indivisible. Then he describes a "primary when" as indivisible also. This creates a problem, because these indivisibles which are used in the act of dividing, are inconsistent with the infinite divisibility which has been assigned to the continuous time. To divide the continuity requires that there is an "indivisible" within the divisible. Failure to reconcile these two, the continuous and the individisible, produces infinite regress in all change and motion.
By those preceding principles, all change and motion must be infinite. But infinite regress in change and motion is contrary to the doctrine that all change is finite, that change is from something to something, and motion is from here to there. So at the end of the book 6 it is demonstrated that all motion except circular motion, is in fact, finite. Now we must reflect back on the primary assumption, that continuity is infinitely divisible, because this assumption produces the unacceptable conclusion that all change and motion is infinite.
Quoting apokrisis
How is this relevant? If I chose one over the other, for a reason, then it made a difference to me. If I flip a coin, then it doesn't make a difference. But all this is just a distraction, because we are discussing identity, so the issue is whether or not there is a difference between the two. Clearly, this makes a difference to me, because if there is really no difference between them, they are one and the same, and I have no choice.
Now, the issue with continuity, and divisibility, is whether or not there is a difference between what is on the two sides of any proposed division. If part A is identical, the same as part B, then there is no problem with division, we keep dividing infinitely. However, if part A is really identical to part B, then in what sense can we say that they are two distinct parts. They are, by the identity of indiscernibles, one and the same. But if we claim that there is a difference between part A and part B, in what sense were they ever continuous in the first place? There would be a change, a discontinuity, between a and B.
Quoting apokrisis
The difference is the difference which makes one burger #1, and the other burger #2. If there was no difference between them, then by the identity of indiscernibles, I would have no choice, there would be only one burger.
Quoting John
If I understand you correctly, we are assuming an object which has the identity "Pluto". Now, let's say that we remove this object from sight, and then we bring an object, which also has the identity "Pluto". We want to know if they are both the same object, so we consider differences. Unless we consider all possible differences, we cannot jump to the conclusion that they are both the same object.
Quoting aletheist
OK, this is what you mean by "continuous". Now are you ready to face the problems with this definition? That's what I've been trying to bring to your intention, there are problems inherent within this definition. First, address the issue of my last post. It is impossible that anything divisible is infinitely divisible. That's what I explained in the last post. Do you agree? If not, why? If so, then you need to change your definition. Either the continuous is not divisible at all, or the continuous is divisible but not infinitely so.
If you could define meaningful in a meaningful fashion here - ie: in a way that makes a quantifiable difference - then there might be something to talk about.
So how do you define meaningful exactly?
We could never possibly take account of every difference, and even if we actually had taken account of every difference we would have no way of knowing we had, because it would always be possible that there could be differences we had missed.
So identity is always something stipulated, not something logically proven or empirically demonstrated.
I agree the things are not our experiences and thoughts about them. This is tautologously true. But from that it does not follow that the things have any existence which is indentical to their experienced and conceived existence, independently of their being experienced and conceived.
It took me a while, but I finally figured out Zalamea's notation, and thus his point about "failures of distribution": the concepts of vagueness and generality are manifested in the non-distributive nature of the universal quantifier for disjunction and the negated existential quantifier for conjunction in classical predicate logic.
Of course, the first expressions correspond to the standard laws of non-contradiction and excluded middle, respectively; and they are equivalent to each other if double negation elimination is permitted. The second expressions are also equivalent to each other, but regardless of whether double negation elimination is permitted; and rather than spelling out that they do not follow from the first expressions, Zalamea simply presents them as non-tautologies. As such, vagueness means that "both P and not-P are possible," while generality means that "neither P nor not-P is necessary." No big revelation here - in fact, Zalamea says this quite plainly - but I am only now finally connecting the dots.
Although the first expressions do not entail the second ones, it does work the other way around - the first expressions follow from the second ones. If only P (or not-P) is possible - i.e., P (or not-P) is necessary - then LNC and LEM are trivially true. Now, what you were saying about identity is that it is likewise a non-tautology when it comes to the actual as unavoidably contextual. I wonder if it then makes sense to say this:
This is parallel to the other two in that the first expression requires double negation elimination and follows trivially from the second one. So perhaps contextuality means that "P is whatever is distinguished from not-P."
So MU, you quote Peirce in a way that directly contradicts you and directly supports me.
Interesting argumentative strategy.
This ties computation to physical actuality in a helpful fashion here.
Memory or history is irreversible symmetry breaking created by some expenditure of energy. So 3ns can be neatly defined in information theoretic terms as the erasure of 1ns. While 2ns is a still locally reversible state - the dynamics can be read off in either direction until fixed by a 3ns context.
Hence the continuum does model time as memory and action. The past is 3ns - the information fixed by irreversible acts of information erasure. What was possible as 1ns is now decided with counterfactual definiteness one way or the other.
The present is then 2ns, the instant when there is just an event that could be read in either direction. Which is action, which reaction? All we know is that there is an event - a symmetry breaking - that could be about to be fixed as something definite and 1ns erasing in the 3ns memory of a developing history.
Then the future is 1ns or the vague. It is open possibility. It is the freedom waiting to be dissipated by acts that steadily rob the system of the energy to locally distinctive rather than globally generic.
So the continuum can't just be freely divided or counted without limit. Computation has been defined by Landauer in terms which spell it out as a game of diminishing returns. The Brouwerian requirement that the actual numberline needs a memory (context being primal) means that constructing the memory is dissipative. It costs to erase possibility.
In our universe, this is captured with complete precision by quantum mechanics. There is a holographic limit on computation. Try to do too much of it and the resulting heat would even melt spacetime, turn it locally into a black hole.
Of course maths can simply ignore all these issues - imagine numberlines as spatial things with no time, no memory, no action, no dissipation.
But while that might make a paradise for Hilbert, mathematical physics might believe that it wants mathematical conceptions much more in line with reality as it is observed. Which is where a Peirce comes in.
Or formally, each is the other's context. As in the logic of a dichotomy - that which is mutually exclusive and jointly exhaustive.
So mathematically, it is a reciprocal or inverse relation.
X = 1/not-x. And not-x = 1/x.
This is why I say the degree each secures the other contextually is strictly relative. Each is only as precise as its alter ego allows it to be. If not-x is vaguely defined, then x remains just as vague too.
This ties identity directly to the strength of the answering context. And so put together, it allows for a controlled way to go from 1ns to 3ns via 2ns. You have every degree of mutual definition on the way from one limit (the 50/50 completely vague state of not knowing which is which), to the 1/0 limit which would represent absolute counterfactuality or completely secured identity).
But why should I see this as the "height" of consciousness? Are you saying that this is consciousness at its most effective, as a well-oiled cog in the dynamic of the world?
Quoting apokrisis
I mean, soccer isn't the only example available. What about artists who paint pictures blind or compose pieces deaf? Or the taxicab driver who doesn't need to look at the speedometer to know how fast he's going? Or the laborer who pounds stakes in the ground in a perfect repetition?
Quoting apokrisis
So, Hume? mkay
Quoting apokrisis
It's really not that romantic, though. What if the instrument isn't working properly? What if you messed up in the calculation? What looks like understanding can easily be an error propagating through a system.
You say it boils down to the pure language of mathematics. Yet surely not all science rests on mathematics. Unless you wanna go all Meillassoux on us.
What is discrete in the Reals? What aspect of the Reals is being inadequately represented by this discrete thing?
I do not completely agree. Identity is also something which we stipulate that an object has, whether or not we are capable of determining it. We do stipulate the identity of the thing, as you say, when we say it is "X" or whatever. This seems to be the way that apokrisis speaks of identity, we give a thing an identity relevant to the purposes at hand.
On the other hand though, we say that a thing has its own identity, it is what it is, independently of our efforts to identify it. This is the basis of Aristotle's law of identity, a thing is the same as itself. This puts the real identity of a thing within the thing itself, rather than the identity which we stipulate. Apokrisis appears to be saying that there is no use in assuming such a principle of identity. But I think it is very important. It is important because if we cannot identify every little difference, as you say, we will still respect that those differences are there even though we incapable of identifying them. Therefore we have respect for a difference between the identity we stipulate and a thing's true identity. And of course, respect for the possibility of mistake.
From the other perspective, the thing's identity is the identity which we give it, regardless of any other differences. So this does not take into account the fact that we might be mistaken when we say that one instance is the same as another. For example, we may say that instance X and instance Y are two occurrences of the same object. This serves our purpose, so we have no reason to doubt that. Therefore we conclude that there is a continuity of existence between them X and Y, and they are the same object. The continuity of existence is true by the fact that we have identified them as such and this identification serves our purpose. Only if we allow that the object has an identity proper to itself, independent of the identity we stipulate for pragmatic purposes, do we allow that we may have made a mistake in this determination.
Of course it supports you, that's the point. My claim is that this is Peirce's mistake. And, if you follow it, it is also your mistake. The mistake is to say that if the difference between two things is infinitesimal, then the two things are the same. Clearly, there is a stated recognition of difference, which indicates a recognition that the two are not the same. Then the claim is that since the difference is infinitesimal, we can just say that the two are the same. It's blatant contradiction. We recognize that the two are different, but we're just going to overlook that fact, and say that they are the same, because the difference is so small.
This is called infatuation with winning an argument vs. truly interested in understanding nature. Translation: The difference is small but big enough for me to admit that I will lose the argument, so let's ignore it.
Quoting Metaphysician Undercover
That would be the same Book VI of Aristotle's Physics that I quoted at some length in the thread on "Zeno's paradox," which you immediately dismissed because "Aristotle says many different things in many different places, often contradicting himself." Since you brought it up here, let me quote it again:
So that which is continuous must be divisible into parts, and those parts cannot themselves be indivisible; in fact, they must be infinitely divisible, because they are likewise continuous.
Quoting Metaphysician Undercover
Yes, but he already resolved this paradox in Book IV:
The indivisible present is not a part of time, because time does not consist of indivisible instants; since it is continuous, it is infinitely divisible into durations that are likewise infinitely divisible into durations. An indivisible point is not a part of a line, because a line does not consist of indivisible points; since it is continuous, it is infinitely divisible into lines that are likewise infinitely divisible into lines. Peirce's insight was that time cannot be divided into durationless instants, only into infinitesimal durations; likewise, a line cannot be divided into dimensionless points, only into infinitesimal lines. We can mark time with indivisible instants, such as "the present" or "the primary when" that corresponds to the completion of a change; and we can mark a line with indivisible points. However, those instants are not parts of time, just as those points are not parts of the line.
Numbers are intrinsically discrete; and it is not a matter of whether this discrete thing adequately represents the real numbers, but whether it adequately represents true continuity.
Vagueness means that "both P and not-P are possible."
Generality means that "neither P nor not-P is necessary."
Contingency means that "either P or not-P might not be actual."
But surely you are aware that the set of real numbers is complete?
Quoting Metaphysician Undercover
You seem uncertain that this is my actual position for some reason. Is that because you know you're just making up things I never would say?
Example of continuous: duration (time) as it is actually experienced by consciousness.
Example of discrete: 1 ft., 2, ft, 3 ft etc.
Example of continuous: space as we actually experience it as memory.
What is the first number after 0?
What is the first number after 0 according to Peirce?
What is the first number after 0 according to mathematics - i.e. Cantor/Dedekind/Cauchy et al?
If you don't know, just admit it!
In Peirce's terms, the real numbers between 0 and any arbitrarily small but finite value form a collection with an abnumeral multitude, which has an even larger power set (in Cantor's terms). However, the potential points between any two actual points marked on a truly continuous line exceed all multitude, and thus have no power set.
So to complete the pattern, 2ns would need to be characterised by a failure of distribution of the principle of identity. And you are saying 2ns is really to be labelled contingency rather than actuality?
I can't really agree with your framing here as my point was that P is only truly actualised to the degree that not-P (as its generic 3ns context) is also actualised.
Returning to my original remark about a "tension", vagueness is pure contingency while 2ns is constrained or contextualised contingency. That is, 2ns is about actualised degrees of freedom - a degree of freedom being a determinate direction of action, or an existent with a predicate.
So it is confusing to call 2ns contingency when 1ns is usually regarded as the maximally contingent. Really, 2ns is contingency limited, regulated, contextualised.
My argument has been that the principle of identity makes a claim that a thing is the same as itself by definition - it appears no context or larger relation is needed and no contingency or uncertainty could be involved.
The laws of thought make identity the concrete and completely uncontingent starting point for then reasoning about the particular. Particularity is claimed as an atomistic fact and so off logic can merrily trot to derive its further two laws.
Peirceanism then stands against that with its holism. Now the concrete particularity of identity - the category of the actual - is instead the emergent intersection of the possible and necessary, the local potential and a wider context of constraint.
So actuality or 2ns becomes the transition zone. It reflects the mixing of the polar extremes of being - vague possibility and crisp generality. Or total freedom vs total constraint. Actuality is actualisation - the process of coming to be framed by the limits on being. A developmental arc is being described that (for me, if not for you) goes from vague 1ns to generic 3ns via the concrete foothold or symmetry breaking which is bald 2ns - a difference that can make a difference in that it does serve to construct, or at least continue to reinforce, some large state of 3ns habit.
The emergent nature of 2ns or the concretely particular is what makes for ambivalence. We are talking about actuality - but the concreteness is secured by the 3ns it anticipates. Context is what gives the particular its definite character, what allows it to be seen and remembered as an occasion that is the same as itself/different from aught else. And this in turn means the particular has been sharply formed by the pruning away of all unnecessary possibilities. Identity is arrived at apophatically.
So there is a pattern to be completed. The law of identity ought to have an exact apophatic definition in the "true meaning" of 2ns, or actuality.
And what does identity presume most? It presumes brute existence instead of emergent development. It presumes a pure state rather than a mixed state. It presumes it stands at the beginning rather than arriving at the end.
Yet then "actuality" in semiosis requires the wholeness of 3ns (the 3ns that incorporates the 2ns and 1ns). So the identification of 2ns as actuality - or better yet, actualisation - has to be understood in that light.
Thus in terms of your logical formalisation - "contingency means that "either P or not-P might not be actual" - it seems to me rather that we are talking not about contingency but about actualisation. So the category of contingency reduced to its deterministic minimum by the constraint of a generality - ie: a freedom that has a direction.
You simply seem to be re-stating the fact that the PNC does not apply to the vague (the vague being the radically contingent).
So Vagueness means that "both P and not-P are possible." = Vagueness means that "either P or not-P might not be actual."
Although, as I say, vagueness defined directly is the degree to which P and not-P are co-jointly not actual. This captures the anticipated 3ns which is the further rule that actuality is irreducibly contextual. So it takes a matching degree of P and not-P for actualisation. And a matching absence of P and not-P for there to the maximum indistinctness or lack of identity.
In vagueness, P is indistinguishable from not-P. In actualiity, they are as distinct from each other as possible. And in generality, that actualised counterfactuality is not merely a one-off event but a habit, a law, a routine state of affairs, an irreversible fact of history.
You need to try harder to keep up with the thread. You are still wanting to construct your numbers, and yet the point being made is that the continuum needs to be cut or divided - which is an act of primal constraint, not construction.
So every cut of the continuum must leave behind a continua that is capable of being cut again. Thus every naming of some "first number" must allow the naming of yet still earlier numbers ... as the continuous can never be computationally erased. Constraint isn't just subtraction or negative addition. It is what it says, a limitation marking continuity. And the divisibility of the continuum is inexhaustible. Every named number - in attempting to cut a part of the line away from the whole - still leaves a bounded line segment.
So the answer in terms of a constraints-based understanding of number is just obvious.
And if your own constructive viewpoint actually could account for the numberline, then you would be granting a zero dimensional point some actual size. Which is why the paradox implicit in your constructive viewpoint was also the bleedingly self-evident since Zeno first put stylus to wax.
Sure, an uncountable infinity of real numbers exist within any finite interval, but you can't identify them and can't distinguish them.
None of these numbers, except a measure zero fraction, can be represented physically in any way - they are non-computable. The only reason you can tell they are there is because you know, from the properties of the continuum, that they must exist.
Your claim that indistinguishable numbers are individual is simply a contradiction.
That is fine, I was just playing around with another angle. I like how you stated this.
Quoting apokrisis
I finally found where Peirce did make a trichotomy with vagueness (1ns) and generality (3ns), but not with a third type of indeterminacy; rather, his third term was determination (2ns).
Nevertheless, consistent with your approach, something is determinate only to the extent that it is distinguished from its context. Identity in this sense, as (?x)(Px = ¬¬Px), defines double negation elimination, rendering LNC and LEM equivalent for anything to which it applies.
Are you now taking up the argument that MU always insists on making? Pure mathematics has nothing to do with what is actual, physical, or computable.
Quoting tom
That is all it takes for them to fail to qualify as a synthetic/true continuum - "that they must exist" as (individual) numbers.
Quoting tom
All numbers are distinguishable in principle. That is part of what it means to be a number.
Yes, I agree that we do impute individual identity to entities in the absence of complete knowledge of the said entities, and that is the point of the 'Pluto' example.
And I also agree that we think that identity as being independent of our thinking of it. So, logically, the entities have their identity completely independently of our imputations of identity. In fact it is more or less taken for granted that we impute identities, whether correctly or incorrectly, on account of the actually independent identities of entities. But we are thinking all this from the standpoint of our imputations and their logical corollaries.
It is assumed that we are led to impute identity by the independent nature of things, and that our imputations therefore reflect, or better express, the independent nature of things. But we cannot give an account of that process of expression from the independent things to our expression of them, because we are 'inside' our expression of them. We thus cannot say how identities could 'be' in an absolutely independent actuality. This is why it has been said that identities are in God or substance, and that our logical expressions of identity are finite expressions of infinite identity.
There are no numbers in nature. Numbers are symbols that we share with each other for some practical applications. All of mathematics for that matter is a bunch of highly limited symbols, which are forever changing, depending the application. It is simply a tool. Nothing more. And certainly should never be used or recognized as anything more that that.
If someone wishes to understand nature, as a philosopher might (but not necessarily), one needs to observe nature directly and experience it directly via music, art, sports, etc. Direct experience is what is required not a constrained set of incomplete and inadequate symbols.
I think I'll never see,
A poem as lovely as a tree.
This is philosophy of nature.
Ah. Direct experience. Good luck with that. :)
Actually, it wasn't Peirce I quoted, it's a book entitled "The Continuity of Peirce's Thought", by Kelly A. Parker.
Quoting aletheist
OK, now I think we satisfactorily understand each other's terms, that we can approach the problem. As Aristotle indicated, the continuity (continuum), is divided by the means of the indivisible point. But the continuum, if it is divisible, must be infinitely divisible, and therefore cannot consist of any indivisible points. The indivisible point would produce a discontinuity Do you agree with me, that this is a problem? When we divide the line, or divide time between past and future, it is not that we insert a point into the line, or insert "the present" into time, we assume that these points of potential division are within the line or within time itself, and we utilize these points for division. Once we remove the present, or the indivisible moment, from time, there is no apparent means for dividing time. That is why Peirce turns to the infinitesimal. So I think you agree with me, that it is contradictory, that the indivisible point is within the continuously divisible continuum, and the continuum cannot be divided in this way.
Quoting aletheist
Now Peirce's proposal is that the continuum consists of infinitesimal durations, like you say. But what happens when we divide time in Peirce's way, is that we lose an infinitesimal piece of the order. So according to the book I quoted there is an infinitesimal difference in the order between part A, and part B. The Peircean procedure is to say that this difference doesn't matter, and claim that the end of part A is the same as the beginning of part B, despite the acknowledgement that there is an infinitesimal difference between these two.
To claim that two things are the same when it is stated that there is a difference between them, is contradiction. So all that Peirce has done, is replaced the contradiction of having indivisible points within the divisible continuum, with another contradiction of saying that two different things are the same.
In case you are not understanding, refer to the example of "2". If "2" represents an indivisible point, within the infinitely divisible continuity, we have a contradiction. There cannot be an indivisible 2 in the infinitely divisible continuum. Now, if "2" represents an infinitesimal part of the continuum, then there is an infinitesimal difference in value between less than 2 and greater than 2. To claim that the highest value of "less than 2" is the same as the lowest value of "greater than 2", when it is stipulated that 2 is an infinitesimal part of the order, is also contradictory. So neither of these proposals, the indivisible point, nor the infinitesimal point, represent an acceptable resolution to the problem of dividing the continuity, they both involve contradiction.
What I tried to explain in the other thread, is that to truly resolve this problem we need to turn to some deeper metaphysical principles. Parmenides placed much importance in the principle of non-contradiction. He proposed a unity, one, continuous, indivisible, whole. In this instance the continuous is indivisible. In Aristotle's physics, and hylomorphism, the continuity of existence is provided for by the existence of matter. Matter is what persists, continues existing, through time. This is temporal continuity, which matter gives us. It is the form of an object which changes, and which is divisible, not the matter which is indivisible and continuous. In modern science, mathematics, with all its operations of additions and divisions, is applied to forms, it is not applied to the matter itself.
In reply to the op, I submit to you, that it is the mathematician's desire to represent the continuum as divisible, which is what leads to the afore mentioned problems, and ultimately contradiction. A true continuum must be as Parmenides describes, whole and indivisible. The claim by mathematicians, that the continuum is divisible can only result in infinite regress, and contradiction. That is because the continuum can only be understood as indivisible. Defining it as divisible is what causes the problems. To start with the assumption that it is divisible, is to start with a contradictory premise.
But semiotics transcends physics because it can imagine its marks as having zero dimensionality. So we have to recognise the computational aspect of this too.
For the hardware of the computer, a bit - the state of switch - is a purely physical thing. It has materiality and thus a cost involved in switching it back and forth. Eventually it will even wear out.
Yet then the same switch, the same bit, can also be a symbol, a sign, within a software's system of interpretance. The programmer can encode some model (representing a purpose) in a syntactical structure (a logical form), then run it on the machine. The switch flips back and forth, doing its entropic or material thing. Meanwhile - in a place with zero material constraints, as the hardware doesn't care if what it computes is meaning or noise - a system of signs does its thing, crunches away to some symbolic end.
So when talking about a mathematical model of the continuum, we have to allow for this fundamental distinction between the real world (which is materially dissipative) and the sign world (which can pretend what it likes, so long as it costs the hardware nothing extra to switch in one direction instead of the other).
Thus in the real world, cutting a material line quickly gets messy. Our knife eventually gets too blunt and starts mushing when the cuts are getting fine. And there is no such thing as an infinitely sharp blade.
But in the imagined world of maths - Hilbert's paradise - we can imagine infinitely sharp blades and cuts made ever finer with no issue about the cuts getting mushed or vaguer and vaguer.
Yet while there are two worlds - matter vs sign - in semiotics they are also in mutual interaction. So that gives you the third level of analysis that would be a properly semiotic one ... where sign and matter are in a formal, generically-described, relation. Or pragmaticism in short. The triadicity of a sign relation.
And that is when we can ask about a third, deepest-level, notion of the continuum - one in which the observer, or "memory" and "purpose" are fully part of the picture. It is no longer just some tale about either material cuts or symbolic marks - a bare tale of observables.
My mistake, I did not follow the link to check your source; everything that you wrote right before the quote implied that it was directly from Peirce himself, so why would I think otherwise? I guess I should have known better. Parker's book was the first one that I read about Peirce, and it is quite good; have you actually read the whole thing? I doubt it, since your link indicates quite clearly that you went Googling for "Peirce divisibility of continuity" and then went with the first reference that came up. Tell you what, read Parker's whole book - or better yet, read some actual Peirce - and then get back to me if you still think that an infinitely divisible continuum is somehow inherently contradictory.
Quoting Metaphysician Undercover
How is this different from what I have been saying all along - that there are no indivisible points in a truly continuous line? Why do you suddenly claim to agree with me now, after arguing with me about it all this time? What changed your mind?
Quoting Metaphysician Undercover
Remember, to say that something continuous (like time) is infinitely divisible is NOT to say that we can actually divide it, without breaking its continuity.
Quoting Metaphysician Undercover
It is not necessarily a contradiction - I am the same person that I was yesterday, and also different; almost any object that I observe is the same object now that it was a minute ago, but also different. Regardless, the claim in this case is that two things are indistinct, but distinguishable; and this is clearly NOT a contradiction.
Quoting Metaphysician Undercover
You are still stuck on the idea of points. Infinitesimals are NOT points of ANY kind, they are extremely short lengths of line. As for your example, all numbers are intrinsically discrete; so the number 2 is an indivisible, not an infinitesimal. Think of it this way - what are the "parts" of the number 2? Mind you, I am not referring to smaller numbers that can be added up to reach 2, but the number 2 itself, as a single "point" on the real number "line." As I have stated over and over, in this thread and others, a true continuum is that which has parts, ALL of which have parts of the same kind. The number 2 cannot be a part of any continuum, because the number 2 itself does not have any parts!
OK, but we need to relate semiotics to a continuity.
Quoting apokrisis
This doesn't negate the problem. If the continuity is cut with indivisible points, like Aristotle suggested, or if it is cut with infinitesimal points as Peirce suggests, the result is contradiction, like I explained.
Quoting apokrisis
The issue with continuity, or the continuum, is whether or not it is something real, or just imaginary. If it is real, we need to describe what type of existence it has. Is it of the nature of matter, or of sign, or of the interaction? To tell me that it has to do with memory and purpose suggests that it is just imaginary. That was my first reply to the op. It is possible that continuity is just imaginary, fictional, and if so then it really doesn't matter how mathematics relates to it.
Quoting aletheist
As I said, I've read enough Peirce and secondary sources already to know what he's talking about. I made a statement about how Peirce deals with dividing the continuum, utilizing the principle of "the difference that doesn't matter". You did not believe me that this was true of Peirce's division, and asked for a reference. So I googled the subject and found someone else who made the same statement about Peirce, Parker. Are you suggesting that Parker misunderstands Peirce as well?
If you have a different opinion about how Peirce claims that the continuum is divisible, then bring it forward, so we can discuss it. If not, then why not just accept what Parker and I have said, and discuss that. Unless you can point to how my understanding of Peirce is really a misunderstanding, why do you insist that I should read more Peirce. I've exposed Peirce's mistake, so why should I inquiry further down that mistaken road? If you believe that Peirce's procedure is not mistaken, then present your argument.
Quoting aletheist
This is where we've always agreed, that saying a continuity is divisible with indivisible points, is a mistake. Where we proceed from there, is in different directions. I claim that a true continuity is indivisible, you claim that it is divisible by Peirce's means.
Quoting aletheist
Here, you are describing yourself in terms of continuity, such that you are the same person despite having changed. And of course this is not contradictory, because of that assumed continuity. Aristotle assumes the existence of "matter" to justify that continuity. But in the case of Peirce's claims, the continuity is distinctly broken. That is what Peirce is doing, dividing the continuity. So you cannot refer to a continuity between the two different things, to justify any claim that they are the same here, despite being different, because the continuity between them is what has been divided.
Quoting aletheist
I was using "2" only as an example. It was supposed to represent an infinitesimal value. If you say that it can't, we can use something else to represent the infinitesimal value. Let's use X. X represents an infinitesimal value. We have a continuous order, and we divide it at X. The value on one side of X is not the same as the value on the other side of X, because X signifies an infinitesimal value. Therefore there is an infinitesimal difference from one side of X to the other. To say that the two values, on either side of X are the same, is contradictory, because we've already stated that there is an infinitesimal difference between them. You cannot deny the contradiction by claiming that they are still the same in the way that you are still the same person after changing an infinitesimal amount, because the identity of your person is justified by an assumption of continuity. With X, the continuity is what has just been divided.
There is a very real problem, which Plato demonstrated in numerous dialogues, with working in philosophy using stipulated definitions. What happens, is that if the stipulated definition of the word is not consistent with how the word is actually being used in society, then when we try to find the real existence of the thing referred to by the word, we get lost, incapable of finding that object, mislead by the stipulated definition. A very good example is found in the Theaetetus. The participants in the dialogue approach "knowledge", with the preconceived notion (stipulation) that knowledge excludes falsity, and mistake. So when they proceed to look at all the different ways in which knowledge could exist, in actuality, they are stymied because none of these is capable of excluding mistake. So it appears like there is no such thing as knowledge in the world, because no human process can exclude the possibility of mistake. At the end of the dialogue, Socrates points out that perhaps the problem is that they have approached knowledge with the wrong stipulation. They themselves were wrong to approach "knowledge" with this preconceived notion, because the thing which we actually call "knowledge" in the real world, doesn't exist like this, knowledge doesn't exclude the possibility of mistake.
This, aletheist, is what I think you are doing with "continuity". You approach continuity with a stipulated definition. And this stipulation is impeding your ability to understand what continuity really is. And just like Plato's stipulated meaning of "knowledge", it doesn't matter how many thousands of others have utilized this stipulation, you just blindly follow them down a mistaken pathway. This is why Plato introduced the dialectical method. This method allows us to approach a word like "continuity" without stipulations as to what that word means, and analyze its usage to find out what it really means.
First of all, you keep referring to "a continuity" as if it were a thing. Continuity is a property, not a thing; a continuum is a thing that has the property of continuity - i.e., being continuous.
Take a blank piece of paper and draw a line with arrows at both ends, then draw a series of five equally spaced dots along the line between the arrows. Mark each dot with a numeral from 0 to 4. The drawing itself is not a continuous line with points along it, it is a representation - a sign - of a continuous line with points along it; the latter constitutes the sign's object. What we come to understand by observing (and perhaps modifying) the drawing is the sign's interpretant. All signs are irreducibly triadic in this way - the object determines the sign to determine the interpretant; the sign stands for its object to its interpretant.
There are three ways that a sign can be related to its object. In simple terms, an icon represents its object by virtue of similarity, an index by virtue of an actual connection, and a symbol by virtue of a convention. As a whole, the drawing is an icon; specifically, a diagram, which means that it embodies the significant relations among the parts of its object. Individually, the drawn line and dots are also icons of a continuous line and points, respectively; but they are symbols, as well, because we conventionally ignore the width and crookedness of a drawn line, as well as the diameter and ovalness of a drawn dot, since they are intended to represent a one-dimensional line and a dimensionless point. The arrows at the ends of the drawn lines are likewise symbols, conventionally suggesting the infinite extension of the line in both directions. The numerals labeling the dots are indices, calling attention to them and assigning an order to them as an actual measurement of the drawn line. They are also symbols, conventionally representing the corresponding numbers.
What can we learn about continuity from this diagram? We marked five points with dots and assigned numerals to them. Are those dots parts of the line? No, they are additions to the line; we did not draw any dots while drawing the line itself, we came back and drew them later. Likewise, any point along a continuous line is not part of the line; it cannot be, because a continuum must be infinitely divisible into parts that are themselves infinitely divisible. A point is indivisible, so it is not part of the the line; it is added to the line. The middle dot is labeled with the numeral "2". Numbers are like points; they are indivisible. The number 2, which is represented by both the numeral "2" and the dot that it designates, does not have parts. Therefore, no collection of numbers, no matter how dense, forms a true continuum - not even the real numbers, although they serve as an adequate model (i.e., representation) of a continuum for most analytical purposes.
Quoting Metaphysician Undercover
As I keep having to remind you, everything in pure mathematics is "imaginary" - ideal, hypothetical, etc. The question of whether there are any real continua is separate from the question of what it means to be continuous. We have to sort out the latter before we can even start investigating the former.
Quoting Metaphysician Undercover
This still reflects deep confusion about infinitesimals. They are not "points," and they are not "values." In our diagram, they are extremely short lines within the continuous line, indistinct but distinguishable for a particular purpose.
Quoting Metaphysician Undercover
If you divide it at X, then X is not an infinitesimal, it is a dimensionless point; and by dividing at X, we have agreed that you introduce a discontinuity - you no longer have a continuum. Rather than division, think instead about zooming in on a truly continuous line with an infinitely powerful microscope. No matter how high the magnification, you would never see any gaps in the line; but you would also never find any place along the line where it would be impossible to divide it by introducing a discontinuity in the form of a dimensionless point. This is precisely what it means to be continuous - undivided, yet infinitely divisible.
Quoting Metaphysician Undercover
When it comes to "what continuity really is," there is no "fact of the matter" - it is a mathematical concept, so we can define it however we like. We can then go on to determine whether anything in reality - time, space, motion, etc. - satisfies that definition. This is why it is a mistake to define the possible only in terms of the actual, or the (presumed) actualizable; you block the way of inquiry by ruling out certain kinds of hypotheses before fully explicating them and subsequently examining reality to see whether it conforms to them.
Quoting Metaphysician Undercover
What makes you think that I have any interest at all in how the words "continuum," "continuity," and "continuous" are used in ordinary language? On the contrary, my interest is in a particular concept, not a particular terminology. Telling me that my definition is "wrong" is ultimately beside the point.
This is philosophy, it is common, and accepted practise to refer to particular abstracted properties as things. We take a concept such as "blue", "infinite", "continuity" and treat it as a thing. In this way we analyze what it means to have that property. So when I talk about "a continuity", as a thing, I am talking about what it means to have that property of being continuous.
If we are talking about a thing which is continuous, and calling it "a continuum", then we are not talking about what it means to be continuous, we are talking about that thing which has been deemed continuous, the continuum. What happens if we were mistaken in designating that thing as continuous? Then if we come to an understanding about "continuity" according to that particular thing designated as a continuum, we will actually be misunderstanding "continuity". This may be the root of our difference, I want to talk about what it means to be continuous (continuity), you want to talk about a particular thing which is continuous (a specific continuum).
Don't you think that we need to first determine what it means to be continuous, before we can designate a particular thing as being a continuum? The problem I am having, is that you already presume to know what it means to be continuous, so you presume that you can move along and designate something as a continuum. I don't agree that you know what it means to be continuous, because your stipulated definition results in contradiction.
Quoting aletheist
OK, I agree with your terminology here, but what is this "object" you refer to here. The line on the paper is a sign which represents an object. I assume that the object is an ideal. This ideal is a line which has the property of being continuous. From my perspective, we need to determine what it means to be continuous, (what it means to be a continuity) if we want to understand this ideal, the line which has the property of being a continuum.
Quoting aletheist
To me, this is irrelevant because you are talking about how continuity is signified, not what it means to be continuous.
Quoting aletheist
I agree here, the dots are not part of the line, in this representation, they are added signifiers.
Quoting aletheist
Now I think you are making an unwarranted assumption. You are assuming that a continuum must be infinitely divisible. Let's drop that assumption for now, and wait until it is proven necessary, before we are forced to accept it as a logical necessity. We have an assumed continuous line, with no reason to deny that there are points assumed as part of that continuum. The line is composed of contiguous points, and this produces a continuity of parts. It is only when we want to make the point a defined ideal point, saying that it is zero dimensional, and indivisible, that the nature of the point becomes inconsistent with the nature of the line and therefore we are forced to conclude that the point is not a part of the line. But we need to respect the fact that just because the ideal point is inconsistent with the nature of the line, this does not mean that the continuous line is infinitely divisible.
It is only when we assume an ideal line, that we can designate it as being infinitely divisible. But we have choices when defining the ideal line. We might also define the ideal line as continuous. My argument is that these two are incompatible. If we define the ideal line as continuous, it is impossible that it is infinitely divisible, and if we define it as infinitely divisible it is impossible that it is continuous.
So, you are talking about an ideal line. Now you need to define your ideal line. You cannot define it as continuous and also infinitely divisible until you demonstrate that these two are compatible. I've already demonstrated to you that they are incompatible. No matter how you attempt to show that the continuous is infinitely divisible the result is contradiction, so I suggest you choose a new way to define your ideal line.
Quoting aletheist
The issue here is not one of simple imagination. I agree that pure mathematics, and ideals are imaginary. The issue is whether we can produce this image, the concept of continuity, the idea of what it means to be "continuous", while avoiding contradiction. If we cannot produce an ideal continuity without contradiction, then the ideal of "continuity" is logically unsound, fictitious, impossible, false. And this type of ideal is one which should be rejected.
Quoting aletheist
I was referring to "values" because we were using "time" as our continuum. the same objection holds with your continuous line. If we remove a short section of line, we have not divided the continuous line, we have removed a section. No matter how infinitesimally small that short section is, you have removed a section, so this must be respected. you have created your division by removing a section. If it's a finite, continuous line, you cannot remove an infinite number of infinitesimal section, so you do not have infinite divisibility, in this way.
Quoting aletheist
You do not seem to understand that if the line is continuous as you describe, inserting a dimensionless point does not divide it. The dimensionless point is completely invisible, it does not itself divide the line. And for all you know there could be an infinite amount of such dimensionless points already along the line, you wouldn't see them no matter how much magnification. So inserting a dimensionless point does nothing to divide the line, there could already be an infinite number of them there, and the line still be undivided. But you are not talking about an ideal line here, it is an observed line. If we are talking about an ideal line, one might refer to the existence of those dimensionless points to deny that the line is truly continuous. But this depends on how one understands "continuous".
Quoting aletheist
That's not true at all. You cannot define a mathematical term in any way that you want. It must be defined in a way which is consistent with the existing conceptual structure. The problem here is that "continuity" is not a mathematical concept as you claim, it is rooted in the ontology of Parmenides. That, I thought was the topic of your op, whether the ontological concept of continuous could be consistent with mathematical concepts. Mathematicians want to apply mathematics to assumed continuums, so they give "continuity" a definition which is consistent with what they desire to do with the mathematics. But this definition ends up being inconsistent with ontological understandings of "continuity".
Quoting aletheist
Why are you not interested to know whether or not your definition is inherently contradictory? Your op clearly demonstrates that you are interested in the relationship between the discrete, the continuous, and mathematics. When someone demonstrates to you that your definition of "continuity" actually involves contradiction, why say that's "beside the point", I am only interested in my particular concept regardless of what you think? You should either defend your concept, or accept the demonstration of contradiction and move on toward a different conception. But to just keep stipulating your definition, and reasserting that this is the only concept of "continuity" which I am interested in, is rather pointless.
Our posts seem to have gotten long and convoluted,. So I'm going to get right to the point of where I think our difference of opinion lies, and if you're willing perhaps we could work it out. If you do not want to, don't reply to the last post, but consider this point.
We agree to the difference between a line on a paper, and the ideal line. Also, the line on the paper is divisible, we can cut the paper, or do whatever is necessary to divide it. We cannot divide it infinitely though, that is impossible.
But ideals have a different type of existence from things in the physical world. Our difference is with respect to the ideal line. I do not believe that the ideal line is divisible. Once divided, it would no longer be a line, it would be two lines, and two lines is different from one line. This cannot be accepted as an ideal, because it allows that you can do something to the ideal line, divide it, which would make it no longer a line. You cannot negate ideals in this way. Allowing that the line can be divided allows that you can make the ideal line what it is not. And in the realm of the ideal, i.e., what is and is not, this is contradiction. So to allow that the ideal line is divisible, is to allow within the ideal, contradiction.
Why are you so adamant about imposing your terminology on any discussion here? I was just trying to improve clarity by distinguishing "continuity" as the property of being continuous from "continuum" as any object (real or imagined) that possesses that property. It really does not matter what words we use, it is the concepts that are at issue.
Quoting Metaphysician Undercover
Having quoted The Princess Bride in one thread already today, I might as well do so again: "You keep saying that word; I do not think it means what you think it means." I know you believe that my/Peirce's concept of continuity results in contradiction, but contrary to your repeated assertions, you have yet to demonstrate this. Instead, you keep revealing over and over that you still have not properly grasped the concept, no matter how many different times and in how many different ways I have tried to express it.
Maybe this is because I have done a terrible job of explaining myself. Maybe it is because you cannot get past your own alternative concept of continuity. Maybe it is because you cannot set aside your dogmatic insistence that "x-able" must always and only mean "actually x-able." Whatever the reason, we just keep going around and around, wasting each other's time and energy. So rather than respond directly to your other comments, I will simply quote a somewhat lengthy passage from Peirce that summarizes the matter to my satisfaction. These are marginal notes that he wrote by hand in his personal copy of the 1889 Century Dictionary, next to its definition of "contintuity," which he himself had provided to its editor in about 1884.
Notice that infinite divisibility, by itself, is not sufficient to make something continuous; I suspect that I may not have made this clear previously. However, the definition that I have invoked most often - that which has parts, all of which have parts of the same kind - is exactly what Peirce presented here (twice), attributing it to Kant and providing (in my opinion) a convincing case for it. The parts of a continuum, most notably infinitesimals, are not definite; once we create them by the very act of defining them, we have broken the continuity.
Quoting Metaphysician Undercover
"Once divided" is a different situation from "divisible." Once divided, the line is indeed no longer continuous; but as long as it remains continuous, the line is both infinitely divisible and undivided. Obviously "divisible" in this context does not mean "capable of remaining continuous after being divided," as you seem to be taking it.
Actually, from your quoted passage, Peirce is dismissing infinite divisibility as the defining characteristic of continuity, and that's why I said earlier, he was on the right track. However, as I said, I don't think he follows through with the principles he implies, he compromises, and this is his mistake.
So we are back to where we were on the last thread. Remember, I pointed out that to define a continuity as a relationship of parts is itself contradictory. To say it consists of parts is to say that it is has separations, is broken, discontinuous. If we talk about space or time as a continuity, an unbroken whole, what justifies the claim that the unbroken whole consists of parts? As Peirce indicates, If it is continuous, it cannot consist of "definite parts". Peirce claims that the mere act of defining the parts breaks the continuity, I am more strict than that, adhering firmly to the underlying principles. My claim is that even to say that it consists of parts, is to state a contradiction. And if it cannot be said to consist of parts, it cannot be divisible.
Quoting aletheist
The point though is that the line cannot be divided unless it is no longer what it is said to be, a line. To divided it once is to deny that it remains a line, so clearly it cannot be infinitely divided. Therefore when you say that the line is divisible, you must mean something other than capable of being divided. What do you mean then by divisible?
The point is contiuum is a thing. Indivisble things made up of seperate finite states. We can cut a number line anywhere because its infinite particular members. Whether we cut at 2, 3,50,12445,9765564 or788956765677, we select a member of the contiuum.
But how does the contiuum change when we do this? It doesn't. 3 still comes after 2 in the contiuum of the number line. And so on in both directions. In selecting 3, we haven't destroyed the contiuum of the number line at all. We've just stop talking about it because we are interested in a particular member.
The notion of dividing the contiuum is a red herring. It confuses what we are talking about (3) with everything for that moment. We confuse ourselves into thinking the contiuum has been lost or divided when we are talking about one of it members. In truth, it's still there, infinite and undivided, just as it was before we were talking about one of its members.
The continuum is its own object, not merely a sum of every finite member.
How do you deal with facts like, for example, a motor vehicle consists of parts, it is divisible, it can be disassembled and its various parts sent all over the world?
Quoting John
The continuum which we are talking about is an ideal. I do not deny that things like cars are continuous in our common manner of using continuous, and these things are divisible, like the line on the paper is divisible. But I deny that these things are infinitely divisible.
Remember, I pointed out that this is false, because the concept of separate/broken/discontinuous is not necessary to the concept of parts. In any case, as Peirce stated (and you also quoted), "a continuum, where it is continuous and unbroken, contains no definite parts" (emphasis added). Therefore, its parts are indefinite; or as I have said about infinitesimals, indistinct (but distinguishable).
Quoting Metaphysician Undercover
But Peirce never said that a continuum consists of parts, as if you could somehow build up a continuum from them; and I certainly have never said such a thing, either. In fact, I have said exactly the opposite (emphasis in original):
Quoting aletheist
I even said it (about Peirce) in the OP (emphasis added):
Quoting aletheist
The continuum is the more basic concept here, not its parts. You cannot assemble a continuum from its parts, you can only divide it into parts; and once you have done so, even just once, it is no longer a continuum.
Quoting Metaphysician Undercover
A continuous line is divisible if there is no location along it where it is incapable of being divided; but again, once it is divided, it is no longer continuous. So a continuum is undivided but infinitely divisible; you keep focusing on the second characteristic, to which you object, and losing sight of the first, with which you agree. You seem to want to define a continuum as undivided and indivisible, but these are the properties of a discrete point, not a continuous line.
In that sense, the motor vehicle doesn't consist of parts. One doesn't have a motor vehicle when they have a wheel or dashboard. A whole is not a sum of parts. It's its own thing.
That's why one cannot use the loss or absence of parts to identify when there is no longer a car. Do I no longer have a car if I lose a door? What about a steering wheel? An engine? Most of the time those are still cars, only missing a part they are expected to have. Sometimes a car has hardly anything at all-- consider someone building a car who refers to a half finished frame as "their car".
As stated in the OP and several times since then, the real number line is not a true continuum as defined by Peirce, nor is anything else that consists of discrete members - even if there are infinitely many of them.
Quoting TheWillowOfDarkness
That is the gist of what I have been saying all along. It is a top-down concept, not a bottom-up one.
No, if something is known to contain no definite parts, the logical conclusion is that it contains no parts. That it contains "indefinite parts" is illogical. What could it mean for a thing to contain parts but these parts are indefinite?
Quoting aletheist
The quote from Pierce is: "I think we must say that continuity is the relation of the parts of an unbroken space or time." This clearly implies that parts are a necessary aspect of the continuity. It doesn't make sense turn this around, and say that the parts only come about through division.
But let's assume that we can do this. Let's say that the continuity has no parts, that is consistent with what I say, I think that to define continuity as containing parts is contradictory. So, how do the parts come about? We cannot define the continuity as having parts, we've already denied that. So if we "define into existence" some parts, these are not parts of the continuity, they are parts of something else. Don't you agree?
Quoting aletheist
There is no point along the continuous line where it is capable of being divided. We already determined, and agreed that there are no points on the continuous line, that would be contradictory. Therefore it is impossible that there is a point on the line where it is capable of being divided, as this assumes a point on the line. So your claim that the continuous line is divisible, cannot be supported in this way.
But that's exactly the problem. Such a "real contiuum" is meaningless. It's a set without any members-- an infinite of nothing at all.
Pierce fails to grasp the nature of contiuum here. Supposedly, when finite members belong to a contiuum (e.g. a number line), it cannot be said to be a proper contiuum. He treats having finite members as if it meant a beginning and end, as if it meant the given contiuum wasn't infinte.
This isn't true. A number line is really infinite and a genuine contiuum. It doesn't begin or end. It is indivisible. (picking out a number from the line doesn't destroy the line).
Pierce makes the mistake of treating such a conntuim as if it were made, in a bottom up manner, by the finite members that belong to it. It is not. The infinite set of finite members is its own. Pierce fails to recognise it is a conntuim because he's still stuck trying to account for the infinte by the finite.
You no longer have a car when you no longer have something that can function as a car.
If and when you ever come to understand this, you will then finally understand what Peirce and I mean by a true continuum.
Quoting Metaphysician Undercover
You are now equivocating on "point." I intentionally did not use that word. Any location along a continuous line is a potential point, but it only becomes a point if we mark it as such - for example, by dividing the line at that location.
A true continuum is not a set or collection at all; it does not consist of discrete members.
Quoting TheWillowOfDarkness
Who is Pierce? If you mean Peirce, it is clear from your comments - especially this one - that you are not familiar with his thought at all.
In a sense yes-- at some point I will reject I have a car-- but what is that point? Clearly, not because it's missing a part (I get a replacement part for the car). Nor is it driving function (I get the car fixed). It's not even a question of having a working car yet (I've built the body of my car).
As a whole, the car is not defined by its parts.
Really? You going to play this game of picking on typos and ignoring the criticism made? I know very well how Peirce defines a true contiuum.
The point is this is mistaken. He's failed to understand that a set or collection can be infinite, that it is not defined in a bottom-up manner as a sum of its parts. Consisting of finite members does not mean being finite.
Yes, actually I was wrong to say that you no longer have a car when you no longer have something that functions as a car, because what you have then is a broken-down car. The problem is that when the point is reached due to subtraction of parts, or, say, corrosion, that a car ceases to be a car, whether functioning or broken down, is not precisely definable. But then nothing about the finite is precisely definable, which means that it is actually in-finite, and is modeled as finite only for practical purposes. Finitude is real only insofar as our terms of conception are discrete.
Exactly the point, yours and Peirce's concept of true continuum is incoherent and will never be understood. As I said, Peirce was proceeding in the proper direction, but didn't follow through. Instead of adhering to his logical conclusion, that defining parts into a continuum negates its essence as a continuum, and therefore a continuum is necessarily an indivisible whole, he follows his desire to have a divisible continuum. And this produces the incoherent notion of an indefinite part, the part which doesn't exist as a part until it is defined, but this definition renders it as an individual whole itself, rather than as a part because the thing which it was supposed to be a part of is negated by the so-called part's very existence. If he would have only adhered to the logic, he would have discovered Parmenides' indivisible, continuous, whole, and there would have been no need for this incoherent "indefinite part".
It one is truly interested in understanding nature, one has only to ask is there any experience of any sort in their life on this earth that gives them any reason to adopt a position that space or time can partitioned (and what in heaven's name is left in between??), and that mathematics in any symbolic form can possibly ever model nature that all experiences have shown is an indivisible whole? This is the only question that needs to be addressed insofar as the OP is concerned. After that, what every other symbol ever created (a car?) means. As one might guess, the meaning of each symbol lies in the beholder. And such disagreements create discussion, but gets us no closer to understanding the meaning of duration and space.
Apparently not, given your subsequent comments.
Quoting TheWillowOfDarkness
He understood all of that extremely well. His point was that consisting of members - whether finite or infinite - means being discrete, rather than continuous.
Consisting of finite members means being discrete. Consisting of in-finite members means being continuous.
Your failure to understand it does not render it incoherent. I understand it, I just seem to be unable (so far) to explain it in a way that you will accept. Is this, in the end, the substance of our disagreement here? If you were to wake up tomorrow and decide that the notion of an indefinite part makes sense to you after all, would you have any other objections remaining?
I was talking about the quality, not the quality of the members. I wrote "in-finite" rather than "infinite", to hopefully avoid that misunderstanding. Finite entities are finite only in the sense that we conceive them as such. Of course in saying that, I don't mean to suggest that entities are infinitely large, either! Infinitely large is itself an incoherent notion, because the notion of largeness implies the idea of 'body' which is itself incoherent without the idea of boundaries.
In reviewing Fernando Zalamea's paper, "Peirce's Continuum: A Methodological and Mathematical Approach," I came across his explanation of what we have been discussing, which are the properties of a true (Peircean) continuum that he calls reflexivity and inextensibility. Maybe what he has to say will convey the whole idea better than my previous attempts.
So Zalamea seems to agree with you that a continuum is not divisible, and even quotes Parmenides to that effect; but he also agrees with me (and with Peirce) that it nevertheless has parts, which cannot be points because those do not themselves have any parts. Instead, he calls them "neighborhoods," and he provides an accompanying illustration that is very similar to my microscope example:
We have already agreed that a continuum does not consist of points, that it is undivided, and that it is indivisible in the sense that once it is divided, it is no longer continuous. It seems, then, that the last hurdle - as I have already suggested - is your insistence that a continuum cannot have parts of any kind, grounded in your rejection of indefinite parts, such as infinitesimals or Zalamea's "neighborhoods." Do you concur with this assessment?
Perhaps it could make sense to me, but to say that a part is indefinite would be to say that this part is unintelligible, it cannot be known. Unless it can be demonstrated that such a part actually exists, why would I accept this assumption? To arbitrarily designate something as unknowable is contrary to the philosophical nature of human beings, which is the desire to know.
Instead, I know of a much more reasonable way to understand the ideal continuity, and that is as indivisible. Maintaining that the ideal continuity is indivisible, avoids this problem of having to assume unintelligible parts. So I see a reasonable approach, which is that the ideal continuity is indivisible, and an unreasonable approach, which is that the ideal continuity consists of unintelligible parts. So until it is demonstrated to me that there is something wrong with this notion that the ideal continuity is indivisible, why would I be inclined toward an unreasonable ideal, which contains unintelligible parts?
Quoting aletheist
Yes I think I concur. But there is still another option which we haven't yet explored, and this is what I stated in my first post, that it is possible that an ideal continuity is just a fiction. The only way I can conceive of an ideal continuity is as Parmenides did, as an indivisible whole, without parts. However, it is possible that the only real continuities, are those physical, spatial entities which can be divided, but not divided infinitely. The fact that we divide continuous things, objects, wholes which we can cut up, inclines us to believe that there is some sort of thing (space or time for example) which may be capable of being divided infinitely. From this, we create the notion of an ideal continuity, one which could be divided infinitely. But this idea proves to be logically unsound, and we are forced toward the realization that the only ideal continuity is the indivisible whole, having no parts. Now we started with the fact that a continuous thing is divisible, and have ended with an opposing conclusion concerning the ideal continuity, that it is indivisible. Is it the case that we need to include divisibility in the ideal continuity? Perhaps we are just chasing an impossible dream, and an ideal continuity is just a logical impossibility. Maybe continuity is just not a real thing, and that's why it's impossible to understand.
There isn't a conflict. Ideal continuity is present. Any infinite can't be divided such that it said to begin or end. We don't divide continuous things, such as objects and wholes at all. Our "cutting" of a whole is merely picking out something specific. It doesn't affect continuity. If I pick out a rock, it doesn't make the whole of the world go away. The whole remains, uncut and indivisible, no matter how many times we might suggest we separate it, an all together different object to the individual states we pick out.
A whole has or is no parts, even as parts belong to it. When we pick out a part, the whole remains and is undivided (and is indivisible).
The mistake people make is thinking wholes as defined by parts in the first instance, such selecting a part would somehow divide and destroy the whole. It hides the indivisible nature of whole from us and sees us misread the wholes we do encounter as failed continuity.
This is precisely the question at hand. If we indeed are able to separate duration or space into parts, what the heck is left in-between?
What is happening is space is being reformed and duration is indivisibly evolving. I have absolutely no idea how anyone can possibly understand a space or a unit that is in any manner being spliced up as mathematics would have it. It is all a just symbolism which is being taught in schools. It's not really happening - is it? Is space and time really being partitioned. Does everything stop an infinite amount of times so Achilles can't move - because he can't leap across non-space??
I think indefiniteness or indeterminacy is certainly part of what I mean when I say that all actual entities are in-finite. I guess what I am saying is that entities are finite only in principle, or in other words in terms of our spatio-temporal conceptions of them.
The other side of it is that the identities of entities are not realized in our experience of them, but rather are formulated in purely conceptual terms. What is formulated is finite only insofar as it is as formulated. This is the misleading aspect of Spinoza's notion of substance; it is conceived as being actually, and not merely formally, infinite, but the conception of it, as formulated,is obviously finite. On the other hand our conceiving of it, prior to its formulation, is in-finite.
I suspect I will be accused by someone of producing word salad here, but at least I know what I mean. :)
I don't think I agree with this part of your statement. An object, as a whole, is something specific. The wholeness, the unity of the object, is something real, and something which is destroyed when we divide that object. Dividing an object is a real activity which has a real effect on the world, one object becomes two objects, and this is a considerable difference. It is not simply a case of choosing some specific parts, over the whole, it is a case of making those apprehended parts, into wholes themselves, through the act of dividing.
So we must have respect for what really happens in division, and that is that one whole becomes two wholes. And we know that it is different to be a whole than it is to be a part, because unity is attributed to the whole, not the part. So when we divide a whole, we take the unity away from it, and give unity to the parts. It is not the case that the unity of the whole is given to the parts, the unity of the whole is destroyed, and a new unity is given to each of the parts. A unity, or continuity is never divided, it simply has a beginning and an end.
If we associate continuity with the whole, with the unity, then each time we divide something, we destroy a continuity, and create new continuities. From this perspective, continuities have temporal beginnings and ends, so we need to be able to determine a cause of continuity. If nothing comes from nothing, there must be some actuality, some act, which causes the beginning of any particular continuity. Likewise, there must be an act which causes the end of a particular continuity.
Quoting TheWillowOfDarkness
I agree that this would be the ideal continuity, an infinite continuity, without beginning or end. We could assign this continuity to "the present" of time, as a working premise, and then we apprehend the whole of existence as one infinite continuity. But when we look at the nature of individual continuities, as I described above, we see that they all begin and end, so by induction, it is probably the case that there is no such ideal continuity, the infinite continuity.
This may be where mathematics misleads metaphysics or ontology, by allowing the possibility of the infinite. It is necessary that we allow infinity in mathematics in order that we can understand the most vast expanse which is possible. We cannot imagine the most vast expanse, because we don't know what it is, so we must allow that mathematics is limitless in order that the most vast expanse can be apprehended. But if we allow that the infinite has real existence within the world which we are trying to apprehend, then we deny ourselves the capacity to understand that world.
Absolutely not. When a wave in an ocean transforms in two or more or even dissolves in the ocean, no continuity is lost whatsoever.
Forms of substance are nothing more than waves in the fabric of the universe. They are just more solid by degrees. How does one break continuity in the universe, in space, in duration? With a very fine knife? Exactly how fine? Finer than Planck's constant? Continuity can never be broken. It can only be reformed, as waves reform in oceans.
Which is the reason that mathematics is constantly giving a awful picture of nature. It's layering is own self-induced picture of the universe onto a universe. It's like putting a cardboard full of holes over a picture. It gives an incomplete and quizzical view of the picture itself. If you want to look at the picture and understand the picture, look at the picture itself, not through lots of holes in the cardboard.
Of course there's a loss of continuity. The description of the wave as one wave applies no longer, and the description of two waves applies. Therefore the continuity of the one wave ends, and the two distinct continuities begin. You might be thinking that the continuity of energy is not lost, but if you are talking about the energy involved, this is something different than talking about the waves involved. The continuity of energy is a different continuity than the continuity of the wave.
Quoting Rich
As I explained in my post, it's quite obvious that continuity is broken, this we know. It is not broken in the sense of being divided though, it begins and it ends. But this is not "ideal continuity" this is the continuity of existence of various different things, they have beginnings and endings. And as I said, I don't think we know the cause of continuity. What you call "reforming" of the same continuity, is really the beginnings and endings of various different continuities. We must allow that there are various different continuities to account for the fact that there are various different, individual objects.
Are you arguing that there is no such thing as individual objects?
The description changes, not the intrinsic continuity. And yes, everything is ultimately energy with differing substantially imbued into the fabric of the universe. As with mathematics, descriptions (for communication purposes only) is symbolic. Symbols are not that which is being described. Just because I describe two different events in my life does is constantly starting and stopping. Duration is continuous when observed directly. Symbolics only are necessary for communication or as a tool for manipulation. The important point is that no continuity is ever lost and no symbolic, which is intrinsically formed by individual units can possibly capture this continuity. This thread is basically about the ability for symbolics to adequately describe continuity. It can't. In fact, the description they yield is pretty much totally contrary to experience. The waves never, ever, ever break the continuity of the ocean. The objects are formed and reformed out of the continuity.
The problem is, that according to the laws of logic, when the description changes the continuity which was described, ends, and a new continuity begins. That is why the laws of logic are not compatible with "becoming", and becoming is not compatible with continuity. So logic forces us toward discrete units, states of existence, and becoming occurs somehow in between. The position aletheist was arguing, the one derived from Peirce, takes continuity from the individual things with beginnings and endings, and hands continuity to the "becoming", the spatial temporal continuum. But this may render the laws of logic as useless.
Quoting Rich
The symbol must symbolizes something, and intelligibility depends on the assumption that what is symbolized remains the same. This is the continuity which I am referring to. The description, being symbols, describes something, and what it describes must remain the same. So that which is being described, remains the same, as a continuity. If the description is no longer applicable, then the state intended to be described, has changed, and that continuity no longer exists.
Quoting Rich
The problem with this perspective is that what exists between the described states, is activity, becoming. And since becoming is change, it cannot be understood as continuity, which is a remaining the same. That is why we have had so much difficulty in this thread describing space and time as continuous. These are the principles of flux, and flux is contrary to the continuity of being. I have now opted to describe individual existing objects as wholes, continuities. So "continuity" must be used to refer to each described state which continues to exist without change, and "becoming" is something other. Therefore continuity is always being lost into becoming, as things change.
Quoting Rich
What is described by the symbols, what the symbols refer to is something which doesn't change in time, therefore the thing described is a temporal continuity. But if we want to describe "continuity" itself, what it means to be continuous, that is something different from what the symbols refer to, it is this unchangingness. There is nothing "contrary to experience", about describing what it means to be continuous, as a state which doesn't change in time. We observe all sorts of objects to be like this. We also observe that these continuities can be ended, as is the case when we divide up an object, and when an object is constructed, a continuity begins. The aspect of reality which is difficult to describe with symbols is the "becoming" the means by which these various continuities which may be described, begin and end. That is because our symbols are intended to have a static, fixed meaning, while becoming is a changing.
There are no states in nature. Everything is continuously evolving. Problems arise when attempting to describe this and replacing the existence with the symbolics. For this reason I do not utilize discrete symbolics, but rather I attempt to utilize imaginative metaphors such as the ocean and the waves. Problems in communication and description should not interfere with the actual experience. For the most part, Bergson also used metaphors in his books. Metaphors are good because the macro reveals itself in the micro and vice-versa.
Not really. To say that a continuum has no definite parts just means that it does not have any distinct, discrete, or indivisible parts. With this qualification, I might even be willing to grant that a continuum has no parts at all, as long as it remains undivided. After all, we agree that the act of dividing a continuum breaks its continuity; so what "infinitely divisible" means in this context is that if we start dividing a continuum, we will never reach the point (literally) of reducing it to an indivisible part. In other words, a continuum is indivisible in the specific sense that if it were divided into parts, and thus made discontinuous, then none of those parts would be indivisible. What do you think?
Quoting Metaphysician Undercover
Again, whether there are any real continua is a separate question from what it means to be continuous. Per my definition, something that can be divided, but not divided infinitely (as described above), is not continuous. In any case, I think that the first thing to establish is whether space and time are themselves continuous. If not - if they are discrete - then presumably all spatio-temporal entities are also discrete. However, if we establish that they are continuous, then we can investigate whether anything within space and time is also continuous.
This should be interesting, finding anything that is within space and duration (real time) that is not continuous.
The only candidates for non-continuity that I can imagine are the unconscious state (including death), and the sleep state. Perhaps dreams themselves have duration, but certainly of a different type, which symbolically is almost impossible to describe. But then again it appears that even within a sleep state, and I would include daydreaming in this, duration seems to pop in and out. Consciousness seems to have an ability to exit memory when it is unconscious. Interestingly, Bergson had very little to say on this rather perplexing switch in states. It is sort of a mini birth/death cycle. Without question, mathematics cannot be applied to this activity.
I disagree. I can look around my room and describe the positioning of the objects, and this will stay the same until it is changed, therefore it is a state that naturally persists. That's what Newton's first law describes, the state which things are in, will persist until a force causes that to change.
Quoting aletheist
Yes, that is what I was arguing, if the parts are not distinct, or discrete, it doesn't really make sense to think of them as parts. So why not just say that the continuum has not parts?
Quoting aletheist
I think that the ideal continuum cannot be divided at all, because it has no parts. And to give it parts would deny its existence as a continuum, so the ideal continuum must be indivisible. Therefore it makes no sense to talk about dividing an ideal continuum. But now I've been talking to the others about real existing continuums, and these are physical objects which display the quality of continuous existence. We can divide an object, or change an existing state, but in doing so we end its continuity, and start new continuities, as the divided parts are now objects which display continuous existence. But I believe these objects are not infinitely divisible. Infinite divisibility seems like an impossibility.
Quoting aletheist
This is the same type of point that we are always disagreeing on. If we want to know what it means to be continuous, we need to refer to real things which are continuous. You seem to think that we can just stipulate the meaning of a word, regardless of whether there are any real examples of this. What good does that do us? If we want to know what it means to be continuous, we need to look at real examples of continua and determine what they have in common.
Quoting aletheist
OK, if we are going to venture into this subject we need to agree on some fundamental principles in order that we can understand each other. I believe that there are real existing things in the world, physical objects. And I also believe that we have concepts of space and time, and that these concepts have been produced to help us deal with, and understand the physical objects. So if we are to understand what the concepts of space and time refer to, we must relate them to physical objects, because that is where these concepts are abstracted from, the assumed existence of objects. They are not abstracted from the observations of real space, and real time. We cannot start to talk about space and time, as if they are real things in the world, because they are just concepts derived from our study of objects. Therefore if we assume that there is a real space, and a real time, because we have concepts of these, our understanding of these things is limited by our understanding of objects, so that any misunderstanding of objects which we have, will also manifest as a misunderstanding of space and time. Our only means to understanding real space, and real time, is through our understanding of objects.
So if we assume that discrete objects are continuous, as things, perhaps beings, we can proceed to analyze how this relates to space and time. First, I would suggest that an object appears to be discrete in relation to other objects. They may overlap, as gravity overlaps, and substances overlap in solutions, but the objects appear to be generally discrete in space. Therefore space appears to be discrete. However, objects appear to obtain their continuity from having continuous temporal existence. So we might consider that time is continuous.
To describe to you what I mean by the indivisibility of the continuous, consider that time is continuous. We can mark points in time, and durations in time all over the map. any where we want. But these are marked on the map, they are not within time itself. These points and segments are not part of time itself, they are part of the grid, the map, or marking system, which is independent of time itself.
Actually, it is constantly changing. Some quite overtly others very subtly. But everything is constantly changing in one manner or another. Energy never stands still. Heraclitus was right and my guess is that he intuited it. If you were correct, then a whole new problem is created, like how does all quanta stop long enough, in concert with each other, to create your state. That would be interesting.
That which cannot be divided at all is an individual, not a continuum - e.g., a point rather than a line. There has to be a way to distinguish these two concepts. What would you call something that satisfies the following definition of a continuum? That which has potential parts, all of which would have parts of the same kind, such that it could be divided (but would then cease to be continuous), and none of the resulting parts would ever be incapable of further division.
Quoting Metaphysician Undercover
This just seems completely backwards to me. How can we identify any real examples of continua without first defining what it means to be continuous? What interests me is whether there is anything real that satisfies my definition of continuity, even if you want to call it something else. That means beginning with space and (especially) time as the framework for our phenomenal experience - examining whether they seem to exhibit the characteristics that I mentioned. If so, we can then move on to whatever spatio-temporal objects we deem the best candidates for also being continuous in that sense.
The whole doesn't get divided in instances where we cut up an object. In such an instance, we are destroying a particular state of the world. When we cut a carrot, we don't target the whole. The knife doesn't split a whole into two halves, such there is a division of the whole.
If I try and say: "Here is half the whole carrot," my statement is incohrent. Since the whole is indivisible, I can't split it such that I have half the whole here and the other half of the whole over there.
In a sense we could say I destroy the whole. In cutting, I take a state expressing an infinite of continuity out of the world. Where one the whole was expressed in the world in front of me, now it is only done so in logic. There's never a split in the whole though, such that we end up with seperate parts of it. We are only destroying an object which expesses the whole.
Following on, this also means particular continuities don't have a beginning or end. Yes, any given object has a start and end, but this is not the unity expressed by it. Whether we are talking about a rock, a person or bacteria, it doesn't take existence for them to be whole-- imagined objects are no less whole than existing ones. In the birth and death of states, there only presence in time, as divided moments. It is only those divided moments, expressing a whole, which are lost and formed. Wholes themsleves are neither created or destroyed.
Impossible to do so this. Everything is in a constant change as much as at the edges as anywhere else, where we have clouds not edges.
I'm not talking about energy here, I'm talking about the physical things in the room. Clearly something must be staying the same, as I can describe the room, write everything down, and then come back later, to find that it is the same. You want to describe the room in terms of energy, but that is a completely different description. If there is an incompatibility between the two descriptions, then this indeed is a problem, and there might be the need to work out some principles to reconcile this.
Quoting aletheist
An individual is a physical object and it is divisible (the name "individual" is misleading). It is also a continuum, that's how we can call it a whole, because of its continuity, as one unit. A point may be an ideal individual, being dimensionless it is indivisible, but we don't generally call a point an individual. The difference is that "individual" refers to a physical object, but "point" refers to an ideal.
Quoting aletheist
There is probably more than one contradiction in this description, but I'll try to sort it out. This is a collection of discrete individuals. Being described as consisting of parts indicates that it is discrete. A "potential part" meaning nothing more than a potential parting, indicating that the point of potential division exists within the so-called continuum. What you have described is a physical object which is capable of being divided. That each part will again be created of parts, infinitely, creates the contradiction we already discussed. That all of the parts are of the same kind will probably result in a contradiction, as well, if we follow it through to analyze what this means. But how we might resolve "each part is of the same kind" depends on how we might resolve infinite divisibility. The two contradictions would play into each other, how one is resolved would depend upon how the other is resolved. To begin with, we could recognize that what you have described is a physical object, and it is highly unlikely that any physical object has all parts of the same kind.
The problem though, which results in the contradictions, is that you take as your premise, your starting point, an individual thing, a continuum, which is divisible, and this is a description of a physical object, and then you try to turn it into an ideal. You cross categories. We haven't found a way to have an ideal continuum, except as Parmenides' indivisible whole, so when you start with a divisible continuum, you are starting with a physical object, not an ideal. But then you want to assign ideal qualities to this physical object, such as infinitely divisible, and having all parts of the same type..
Quoting aletheist
This is the point I described to you earlier, concerning Platonic dialectics. The words "continuous", and "continuity", are commonly used. We look and see what kind of things are described by these names, and see what they have in common, why people call them continuous, and from here we can say what it means to be continuous. We have already determined that it is impossible that there is something real which satisfies your definition of "continuous", because it is contradictory. So we can conclude that somehow, along the way, some people got mixed up, and produced a faulty definition of continuous, and this got accepted and used. Peirce worked at determining the faults, and attempted, with no success in my opinion, resolution.
Just like In my example of Plato's Theaetetus, "knowledge" was defined as necessarily being true, i.e. excluding the possibility of falsity. But all the instances of knowledge in the world, what people were referring to as "knowledge", could not be shown to necessarily exclude falsity. So this was a faulty definition. It defined an "ideal" knowledge, but knowledge as it exists, and what we call "knowledge", doesn't have that ideal character. So when we have confused concepts like "continuous", we need to straighten things out by referring to how the word is used, we cannot just accept a definition which has been shown to be defective.
You want to take "continuous", and give it an ideal definition which has already been shown to be contradictory. The reason this definition is contradictory can be understood like this. "Continuous" already has an ideal definition, as described by Parmenides, indivisible, whole. It also has a definition which we use to refer to physical things, a whole which is divisible. The two are clearly incompatible, but some people have wanted to create a single definition, which encompasses both, the ideal indivisible whole, and the physical divisible whole. So they compromise in one way or the other.
You might notice that in the above description, the two definitions of "continuous" do have something in common. They both refer to a "whole". The ideal continuum is an indivisible whole, and the physical continuum is a divisible whole. So they do have a point of compatibility, and if we want to produce a definition which encompasses both, we should start with this, "a continuum is a whole". Do you agree with this definition, a continuum is a whole, whether it is an ideal continuum or a physical continuum?
Quoting TheWillowOfDarkness
Yes, I can, in principle, agree with this. We don't actually divide the whole, we destroy it, cause its existence to end. And in doing so though, we cause the beginning of existence of the wholes which we have created, what we call the parts, as they are now actually wholes. We do end up with separate parts, but each part is not a part, it is itself a whole. Prior to cutting the object, we can describe our intended action in terms of "parts", describing the parts we will cut, which will each become separate wholes after the act of cutting. In this way we can say that any whole is indivisible, just like the ideal whole of Parmenides, because we never really divide the whole, we just destroy it. The physical object wholes, are describable in terms of parts but these wholes, physical objects, have a beginning and an ending to their existence, and this is why we can describe them as parts, unlike Parmenides' ideal whole. Parmenides' whole which is defined as without a beginning or an end cannot be described in terms of parts, because this would imply that it could end.
Quoting TheWillowOfDarkness
I don't follow your logic here though. If an object has a start and an end, doesn't this imply necessarily that the continuity of that object is broken? How can you assume that the continuity continues through the end, or prior to the beginning of the object. Let's say that prior to an object's physical existence there is an intended existence, an idea, plan, formula, or blueprint for that object, and after the object's physical existence there is the memory of that object. Aren't these two distinctly different from the physical object itself? Isn't there a break in the continuity between the plan and the physical object, and between the physical object and the memory of it? This being the difference between being in a mind and being independent of a mind.
They are one and the same. It is a continuum. The is no discontinuity between that which physical and that which creates it. I am bewildered at how you are able to separate the two. If we aren't energy, then what are we? And what do believe surround us? The energetic form is simply moving within an energy field as a wave moves in water. There is no separation.
To describe a thing, and to describe the activity of a thing, is two distinct description. So I am bewildered at how you do not recognize this. To describe my car as a physical object, and to describe what my car is doing, is two very distinct descriptions. You can insist all you want, that there is no difference between the description of the car, and the description of what the car is doing, but that doesn't change the fact there is a difference between these two.
As a discussed earlier, the problems involved in accurate descriptions is not at issue here, e.g. how to describe things that are separate from their environment. Usually approximate descriptions are sufficient for practical purposes. What is at issue is a precise description of the nature of nature. Physical objects, as normally understood, in themselves have degrees of substantiality, e.g. energy, air, water, fire, humans, rocks etc.. The continuum between it all cannot be broken though substantially most certainly can be sensed to a certain degree.
As I have repeatedly made clear, I am discussing mathematics here, which has to do with ideal states of affairs; I am not saying anything whatsoever about physical objects. As for "individual," if you look into its etymology, you will find that it has the same root as "indivisible"; one is a noun, the other is an adjective, but they originally meant the same thing - much like "continuum" and "continuous." Nevertheless, since "individual" has come to have a different meaning in common usage, and this seems to be an obstacle for you, we can set that term aside for the sake of clarity and simply substitute "indivisible" as a noun. Restated accordingly: There has to be a way to distinguish a continuum (such as a line) from an indivisible (such as a point).
Quoting Metaphysician Undercover
There is nothing contradictory about my/Peirce's definition, and if you are going to keep insisting that there is, we might as well call yet another impasse and go our separate ways. I get that you disagree with me/Peirce on all this, but I have addressed each of your objections, even if you remain unsatisfied with the result. I have to wonder if you keep saying this over and over because you are still trying to convince yourself.
Quoting Metaphysician Undercover
No, a continuum per my/Peirce's definition is not a collection of individuals at all, and having potential parts clearly does not entail that it is discrete; it would only become discrete if it were somehow divided into indivisible parts. But by my definition, it cannot be so divided; therefore, not only is it not discrete, it is not even potentially discrete. A true continuum cannot be composed of discrete elements, and it also cannot be decomposed into discrete elements. We can only introduce indivisible points along a continuous line, and those points are not parts of the continuous line itself.
I still suspect that this right here is what you are perceiving as contradictory, perhaps because you are locked into the standard rules of classical logic. As Peirce explained, "continuity is simply what generality becomes in the logic of relatives" (CP 5.436, 1904), and "anything is general in so far as the principle of excluded middle does not apply to it" (CP 5.448, 1905). Therefore, the principle of excluded middle does not apply to that which is continuous; and this is all that it means to say that a continuum has only indefinite or potential parts. Intuitionist logic does not uphold excluded middle - or, for that matter, double negation elimination - and thus may be better suited for reasoning about true continuity than classical logic. Excluded middle does not apply universally in smooth infinitesimal analysis, although it does hold for particulars; this is one reason why it seems like a promising candidate for mathematically modeling true continuity.
But we've already determined that mathematics refers to discrete units. So as soon as you describe something as a continuum we are not dealing with mathematics, and therefore I cannot assume that we are dealing with ideal states of affairs. You want an ideal continuum, so that perhaps you can establish a compatibility with mathematics, but this requires that you can define "continuity" in a way which is not contradictory. I can, "continuity" refers to an indivisible whole, and this is consistent with mathematics.
Quoting aletheist
Sure, they are different, the point is defined as non-dimensional, and the line has a specified dimensionality.. As non-dimensional, the point is purely ideal. The "line" in its definition is purely ideal, but it describes a spatial extension so what is described is not completely ideal. To be divisible, it requires this spatial extension, and this means that to be divided it requires extension outside the mind. It is only by means of this non-ideality, looking at the physical thing which the description describes, do we get divisibility. So when you say that a line is a divisible continuum, you are appealing to a non-ideal line, a physical representation to say that it is divisible. It is by means of this assumed spatial extension, which makes it non-ideal, that we say it can be divided. If there is a truly ideal line, it exists by definition only, and cannot be divided because then it would not be a line, it would be a line segment or something like that. The truly ideal line cannot be divided.
Quoting aletheist
If you insist, then have fun with your crossing back and forth from the ideal to the physical thing, with all the contradictions which that entails. Look, you want to be able to divide the ideal line, but that's clearly impossible, and contradictory. The line is defined as a specific form of spatial extension, and you think that because of this spatial extension you should be able to divide it. But it's an ideal, you can't divide it, because that would render it other than its definition, and this is contradictory. So you must face the logical conclusion that the line as an ideal, is an indivisible entity. What distinguishes it from the point is to be found in its definition, of a particular form of spatial extension. But that is its definition only, it doesn't make it have real spatial extension, such that you can divide it, it's an ideal.
Quoting aletheist
OK, so you say that a continuum "cannot be so divided". Why do you keep insisting, in a contradictory way, that a line is infinitely divisible. In one sentence you'll say that a continuum is infinitely divisible, then you insist that you are not contradicting yourself, and then you say "it cannot be so divided". If it cannot be "so divided", then how is it divided? You talk as if there is some magical way of dividing something which doesn't actually involve dividing it. What could that be? Thinking of something continuous as being divisible doesn't actually divide it, nor does it make it divisible. Unless it can actually be divided, it is false to say that it is divisible.
Quoting aletheist
Why are the ideal points not part of the ideal line? This is completely consistent with common geometrical principles, and consistent with mathematics as well. It is only your desire to do the impossible, define a divisible continuity, which makes you reject the standard definition that a line is a collection of points.
Quoting aletheist
As I said, I do not agree with the way that Peirce dismisses logical principles. It is unwarranted. He does this in order to compromise, where compromise is unnecessary. As I've been trying to explain, we can stick to the principles which keep the ideal separate from the non-ideal, and proceed toward a much more comprehensive understanding, than Peirce's compromised understanding. It should be evident from the above passage, that Peirce's move only plunges us into an unnecessary vagueness, by failing to maintain the difference between that which has parts, and that which does not have parts. Thus he compromises the principles with "indefinite" parts.
The first four questions that I posed in the OP were as follows.
Based on the ensuing discussion, my answers are yes; see most of MU's posts; yes; and category theory, in particular smooth infinitesimal analysis. So while I do think that mathematics in accordance with the arithmetic/Dedekind-Cantor/set-theoretic paradigm is intrinsically discrete, I deny that all mathematics is constrained to refer only to discrete units. We certainly have not determined otherwise in this thread.
Quoting Metaphysician Undercover
One more time: I have always and only been dealing with mathematics and ideal states of affairs throughout this thread. Your unwillingness or inability to think of a continuum in mathematical/ideal terms is your own limitation, not mine.
Quoting Metaphysician Undercover
Which is exactly what I have done. You have not demonstrated otherwise; you just keep asserting it over and over, apparently expecting a different result. Where have I claimed that both P and not-P are true at the same time and in the same respect?
Quoting Metaphysician Undercover
Complete and utter nonsense.
Quoting Metaphysician Undercover
Is your reading comprehension that poor, or are you just being obtuse? What I stated is that a continuum cannot be divided into parts that are themselves indivisible; so it can be divided into parts that are themselves divisible, although once that happens it is no longer a continuum.
Quoting Metaphysician Undercover
Because points are indivisible, and a continuous line cannot be divided into parts that are themselves indivisible. Try to keep up.
Quoting Metaphysician Undercover
As I said, you are locked into the standard rules of classical logic, which are very useful for most purposes, but not for understanding the nature of true continuity. Failure of excluded middle is not a contradiction in all viable forms of logic.
Quoting Metaphysician Undercover
I suppose it was inevitable that we would end up right back here, at square one. Cheers.
Any object has a start and end... but these can only be finite states. They are never a whole in the first place.
There is a break between any object and a plan about an object. A plan, as an object, is a different finite state. I make a plan-- it's a state that begins and then ends. Then I have a created object, another state that begins and ends; a plan has ended, the planned state begun.
Niether of these objects are a whole, either of the plan or the object. The wholes are indivisble and so remain untouched. My plan doesn't suddenly become "not whole" because it began and ended. Nor does the created object. The point is starts and ends do not amount to a breaking of a contiuum. If one could break a contiuum, it not indivisible. The infinite nature of a contiuum means it must be unaffected by beginnings and ends.
Now what I've been trying to explain, is that dealing with continuity is not an issue of developing different, or better, mathematical principles, it is an issue of getting a proper definition of "continuity" one which renders real continuity intelligible to mathematics. What I've been trying to demonstrate is that your idea of continuity, or definition of "continuity" does not properly represent real continuity, and this is why mathematics has difficulty with your sense of "continuity".
Quoting aletheist
There is no such thing as a mathematical continuity, you are making that up. We can apply mathematics toward analyzing or understanding something which is assumed to be continuous, a continuum, but this does not mean that the mathematics itself is a mathematical continuity. The only mathematical continuity there is, is the consistency of all mathematical principles together, if there were no contradictions, which would form one continuous whole, that is called "mathematics".
Quoting aletheist
You have claimed that a continuum is both divisible and not divisible.
Quoting aletheist
You circle around the contradiction, without addressing it, as if by circling it it will be obscured. You have a continuous thing which you call a continuum. It is a continuum because it is undivided, by definition. It is necessarily undivided, or else it cannot be said to be a continuum. If it is necessary that the continuum is undivided, then it is not possible to divide it. It is indivisible by that definition. Clearly, it is contradictory to say that a continuum can be divided. You seem to think that by the power of you statement alone, "a continuum can be divided into parts", it actually can be divided into parts. It's blatantly contradictory, and you're just in denial of that fact.
Look at it this way. There is a thing which is being called continuous. Because of this it may be called a continuum. That thing must remain undivided or else it is no longer a continuum. Let's say that something can act on that continuum and change it into something else. This is what you call "dividing it", and this is the premise for your claim that a continuum is divisible. Something can act on the continuum and change it into something other than a continuum. But this, acting on the continuum, is not "dividing it" in the mathematical sense of division, it is a change, which constitutes going from continuum to not-continuum.
By your premise, that this is "dividing" the continuum, in the mathematical sense of "dividing", you come up with the idea that the two parts produced are mathematically equivalent to the continuum. But this is not the case at all, in reality. And this is why it is wrong to assume that when a continuous thing is divided, it can be divided infinitely. When a continuous thing is divided, what is produced is two things which are not mathematically equivalent to the original continuum. So we must respect a difference between mathematical division, and "dividing" a continuum, which is not really an act of division at all, in the sense of mathematical division. What we call "dividing" the continuum, is really annihilating that continuum to produce something new. What is produced may be a number of new continuums. But it is wrong to believe that the new continuums produced are mathematically equivalent to the original continuum.
Quoting aletheist
But this is plainly wrong. The continuous line, the one which exists on the paper can be cut up, but eventually there will be parts that are indivisible, too small to cut. The ideal line is defined as consisting of a succession points, and is therefore not continuous, it is discrete. You seem to have no respect for the fact that the ideal line is defined as consisting of points, and is therefore discrete, because you want to work with a continuous line. By ignoring this reality, you put yourself into your contradictory position.
Quoting aletheist
Yes, standard logic is very good for understanding the nature of true continuity, as was demonstrated by Parmenides. The problem is that you define "continuity" in some absurd, contradictory way, so you must resort to some absurd logic in an attempt to understand this absurd notion of continuity. All you get is lost in vagueness. What is required to understand this subject is firm adherence to fundamental principles.
Quoting TheWillowOfDarkness
Why would you say that an object is not a whole? Sure it is not a whole in the perfect sense, like in the sense of a unity of everything is a the perfect whole, but by its own right as an individual unity, can't we say that it's a whole?
Quoting TheWillowOfDarkness
How would you define "whole" then? To be a whole, doesn't it suffice just to be a unity? A unity doesn't need to be a perfect unity in order to be a whole. So numeral such as 5, 8, 12, signify wholes, but since they are each not the complete whole of all the numbers, nor the primary unity, 1, they are not perfect in their wholeness.
Quoting TheWillowOfDarkness
Do you agree that a continuum is a whole? And do you agree that there are wholes, continua which are less than perfect in their nature? If an object which is a unity, a whole, ceases to exist, isn't that the end of that particular continuum? But if that object is described as part of a larger, more perfect whole, then that larger, more perfect continuity would persist, and the annihilation of that smaller whole, which was really just a part of the more perfect whole, would just be a slight change to that more perfect continuum.
Quoting TheWillowOfDarkness
You seem to be assuming that all continua are ideal, infinite. But I don't see why you shouldn't consider that any existing object, as a whole, a unity, is a continuum. And surely these objects can be annihilated, so the continuum which is that object must be broken. I wouldn't call this dividing the continuum though, as I explained to aletheist, because "divide" implies a mathematical division. That's why I believe that a continuum must be capable of beginning and ending, but this is not properly called dividing.
That would be news to mathematicians.
Quoting Metaphysician Undercover
Not at the same time and in the same respect, hence no contradiction.
Quoting Metaphysician Undercover
It is not possible to divide it and still have a continuum.
Quoting Metaphysician Undercover
Dividing it is precisely what causes it to change from a continuum to a non-continuum.
Quoting Metaphysician Undercover
I have never said any such thing.
Quoting Metaphysician Undercover
You are clearly not paying attention at all.
I didn't.
I said the beginning and end, or any other point of an object, was not the whole of an object. I never said an object wasn't whole. By it's own right, as an individual entity, a whole, we can say it's a whole. In other words, there are only perfect wholes and one is expressed by every single object.
The point is it does suffice just be a unity. Since this is the case, the problem you present is nothing more than a red-herring-- 5, 8, 12 all have their own unity, as does 1 and the set of real numbers. All are compete and indivisible. When picks out a number from the set of real numbers, it doesn't make the set of real numbers divided. When we divide 8 by 2, it doesn't undo the unity of 8. And so on. They are all perfect in their wholeness.
What you suggest as a problem is just a category error, a mistaken assumption that unity is given by other things.
Yes.
No, all wholes or continua are perfect.
No, the unity of an object does not exist in the first place and has no end, so there is no end to the particular continuum-- hence the dead and fictional are still whole, despite being broken apart or never existing.
Impossible, all wholes are perfect. "More perfect" is an oxymoron. Perfection means "the best."
First, before any description, one must penetrate deeply with all faculties. Forget about all the symbolic tools learned in school. And once one behind to fanthom the actual experience, then and only then it's one ready to attempt to create a metaphor that may describe that experience.
Labeling with descriptions are useless. What is useful, is watch the ocean and the waves and observe closely what is actually happening as forms come and go in seamless never ending pattern of becoming and going. No states, no division, no difference between the whole and the parts, no way to divide, no way to say this is where it begins and this is where it ends, yet it is all there.
This would even go deeper, and it certainly shares much with Daoism. If the nature that we are connected with is all continuity and waves, is there an opposite of Singularity from which it all began - the Dao?
I know it's news to mathematicians, that's what we've been arguing. Many mathematicians believe in the contradiction of an infinitely divisible continuity. But mathematicians specialize in mathematics, not ontology. So it's more likely that a metaphysician would know these principles better than a mathematician
Quoting aletheist
You don't seem to understand what an ideal is. Ideals are timeless truths. "The same time and in the same respect" is irrelevant to an ideal, 2+2=4 regardless of what time it is. We cannot attribute to a thing, that it is continuous, and that it is not continuous at the same time, by the law of non-contradiction. We agree with this. But as you keep saying, we are concerned with the ideal here. The ideal continuum is indivisible regardless of what time it is. To say that the ideal continuum might at some time be divided is like saying that 2+2 might at some time not equal 4. To say that the ideal continuum is divisible is contradiction, false. That's the problem here, you want to compromise the ideal, with principles of temporal existence, so that the thing which is called a continuum can be at one time not-divided, but at a later time divided, thus it is divisible. But ideals are timeless principles they are not things which can change. So all you do is introduce contradiction into a timeless ideal. A continuum, when the thing which is continuous is ideal, rather than a physical object, is necessarily indivisible. If it's a physical object, it's divisible but not infinitely divisible.
Quoting aletheist An ideal is a timeless truth. And that a continuum cannot be divided is such an ideal. So are you arguing that there will be a time after the ideal continuity is divided and then there would no longer be such an ideal? But since an ideal is a timeless truth, if there will be a time when there is no longer an ideal continuity, then there must not be an ideal continuity even now.
Quoting aletheist
An ideal cannot change to what it is not. That's the thing with an ideal, it must always remain the same. 4 will always be 4, it cannot change to non-4. A point will always be a point, it will not change to a non-point. A circle will always be a circle, a square always a square. These things, ideals, do not change. It doesn't make sense to say that the ideal continuity will change to be non-continuous. You are just committing category error, trying to introduce characteristics of physical existence, "change", into the ideal. If you want to talk about a continuum which can change to a non-continuum, then we are talking about a physical object, not an ideal. And physical objects cannot be divided infinitely.
Quoting aletheist
Oh I'm paying attention, you're just not listening to reason, continually making the same unreasonable assertions over and over again.
Quoting TheWillowOfDarkness
Sorry then, I misunderstood.
Quoting TheWillowOfDarkness
Not a red-herring, just a misunderstanding. I was trying to make a point to you, but I guess I didn't realize that you already agreed with that point.
Quoting TheWillowOfDarkness
But I still think that unities have a beginning and an ending, therefore they must be caused, ("given" by something else). You seem to think that all unities are infinite, but I don't see any examples of infinite unities, and I don't see how anything other than the ideal unity could be infinite.
Quoting TheWillowOfDarkness
We create objects, which are unities, continuums, we bring them into existence, and annihilate them, so I don't see the basis for your claim that a particular continuum has no end.
Quoting Wolf
I don't quite get what you mean by "world of zero". What do you mean by this, and, all we find is "+1 & -1 of discrete objects"?
Quoting Rich
Are you saying that descriptions are absolutely false? If not, then there must be some truth to a description. Just because it doesn't describe every aspect of the scene which it is describing, doesn't mean that it is false. So if a description describes some things which are unchanging during a period of time, then don't you think that there are some aspects of reality which are unchanging during that period of time, corresponding to the description?
Quoting Rich
Concentrating on the active parts of the world, and complete ignoring the things which are passive, or unchanging, is no better of a way to produce an ontology than concentrating on the unchanging aspects and ignoring the activities.
I am only talking about mathematics in this thread, not ontology; maybe you should start your own thread on "Continuity and Ontology."
Quoting Metaphysician Undercover
I am only talking about ideal states of affairs in this thread, not "ideals"; creations of mathematical imagination, not "timeless truths."
Quoting Metaphysician Undercover
Once again - pot, kettle, black.
Then what is that continuous thing you are always referring to as a continuum, which you are attempting to understand with mathematics?
Descriptions are necessarily limited, inaccurate, imprecise, and provide no avenue to understand the nature of nature in themselves. They are simply a tool for communication which may or may not help two explorers to better understand. To this end, I have always felt metaphors to be far more helpful.
Have you ever tried describing duration in words or mathematics? Just your own memory as it flows continuously and unceasingly and never stops evolving. You should try it. Direct observation of what you are suggesting is possible. Stop duration, create a state, and describe it while still observing your efforts to describeit in the same duration. with such an attempt you should witness the impossibility of what you are suggesting as should anyone who believes that mathematics, words, logic, or any symbol is adequate to describe the nature of experience in duration.
A single example?
I don't understand how a metaphor is a better means for understanding the nature of nature than a description is.
Quoting Rich
OK, this is how I describe duration. I recognize a difference between past and future by means of memory and anticipation. This gives me a sense of being present. As I am aware of being present, I notice that things are changing while I am present, and I can refer to duration through describing these changes which occur.
Quoting Rich
So at the same time that I am noticing changes, which enable me to describe duration, I also notice things which are not changing. I can describe these things as not changing, for the entire duration of the change which I am describing.
Quoting Rich
So for example, I have the blueprint for the layout of my kitchen, and this is a description of the things which are not changing in my kitchen. It describes where the cupboards, counters, the sink, the stove, and the fridge are, and this is a static description which persists and remains true through this day. I can stand in my kitchen, frying eggs in the frying pan, and this is a change which I can focus on to give me a sense of duration, because it always takes about the same amount of time to fry the eggs. But at the same time I can notice that the location of all the cupboards, counters, sink, stove and fridge, remain static, in the same place, throughout that duration of time.
Metaphors are more of a holistic, active image that two people may share. Something like one picture is worth a thousand words.
Quoting Metaphysician Undercover
I do not mean give a brief definition of duration. I am suggesting that you actual describe an actual experience in duration as you are describing it. This provides an actual observation of your own duration and the impossibility for you to describe it. I am asking for a more direct experience.
Quoting Metaphysician Undercover
If you are attempting to describe your own duration directly you may notice that your act of describing is melting into what you are trying to describe. There is no "state" . There is a continuous flow of one into the other. Observe closely that one memory that you are attempting to describe is flowing directly into the description itself, continuously and unceasingly. It cannot be stopped long enough for you to describe it. In other words, your act of describing is within that which your are attempting to describe.
I am not asking you to describe some past memory, which will be as complete as you may remember and subject to change, I am asking you to describe duration as you are experiencing it.
Quoting Metaphysician Undercover
So, this is s metaphysical viewpoint that I cannot argue because it is something you believe very strongly. However, if I was to be put in the same kitchen, I would observe everything changing on the macroscopic level (the dust in the air, the deterioration in the wood, your life itself, the ink on the paper), and at the microscopic level (the energy of all quanta). This is why I say, philosophers need to be constantly exercising their observation skills via the arts. I first learned of the skill in the art of observation when I studied photography many years ago. A philosopher must always be exercising and refining the art of observation.
If that's not an example of my experience of duration then I don't know what you are asking. To me it's an example of my experience of duration. What more are you asking for?
Quoting Rich
As I said already, I don't deny that some things are changing, but I also don't deny that some things are staying the same. You, for some reason seem to be intent on denying that there are some things around you which are staying the same in time, and that we can describe these things, and notice that the things described remain the same.
Quoting Rich
If you cannot see that there are things around you which remain the same through a duration of time, then I don't think that you are very good at the art of observation.
I believe that the earth is moving, but that fact is irrelevant to the fact that the layout of my kitchen remains the same. That's the point, we have to have respect for what is staying the same, as well as what is changing.
It is quite clear to me that everything is changing in one manner or another all the time. There is nothing I can say or do to convince you of this. It must come to you by your own personal observations. Until then, your metaphysics will be determined by your belief that some things don't change some of the time. For me, it is an impossible chasm to cross until I observe something that is not changing. At this same point, physics itself will come crashing down along with my own personal beliefs.
If this were true, then how do you explain the fact that the layout of things in my kitchen is the same still as it was ten years ago? Everything is in its proper position. It's easy for you to assert that everything is changing, even if what you say is false, because people state untruths all the time. So if you want me to take you seriously, you should be prepared to explain to me how there are all these relationships around me which appear to stay the same. I can take a tape measure, and measure things, minute after minute, hour after hour, day after day, and show you that they are staying the same. Do you really believe that the distance between my fridge and my stove changes from one minute to the next? Why are you so convinced that these things aren't really staying the same for any length of time? If you could explain to me how the distance between the fridge and the stove changes from one minute to the next, when my tape says that it stays the same, then perhaps I might believe you.
It's all about observation. You may not notice the changes, but even a casual observer most certainly would.
Observational skills improve with constant practice.
I would need to see some proof of that. I have a hard time believing that the movements of electrons, which have a very tiny fraction of the mass of an atom, changes everything.
Quantum motion
"...quantum mechanics, the underlying physical rules that govern the fundamental behavior of matter and light at the atomic scale, state that nothing can quite be completely at rest."