It is possible that "continuity" is just an imaginary notion, a fiction conjured up by the human mind. If this is the case, then there is really no need to model continuity. So if it is true that mathematics can only model the discrete, this is not a problem if continuity is just a fiction anyway.

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?

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

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? — aletheist

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.

There is no a priori way of determining anything about reality ... — tom

Is there an a posteriori way of determining whether there are any real continua vs. everything (including space and time) being discrete?

Your insinuation that the real numbers cannot somehow model the physical continuum is rather odd. — tom

As I stated in the other thread:

Of course mathematics can and does model a continuum. However, the accuracy and usefulness of such a model depend entirely on its purpose, and that is what guides the modeler's judgments about which parts and relations within the actual situation are significant enough to include. — 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.

Is there any way that mathematics could evolve going forward that would enable it to deal with continuity more successfully? — aletheist

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.

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

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

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

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

Can you explain (so that a philosophical simpleton like me could understand it) how mathematics has failed to successfully deal with continuity? — fishfry

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.

When you divide a line at a point, the point stays with one segment and not the other. — 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.

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

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.

If you can divide the point on one of its sides ... — apokrisis

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.

A point is that which has no part. — fishfry

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.

The simplest way to understand mathematics failure to deal with experience are the unresolvable internal conflicts within mathematics, e.g. infinities, division by zero, paradoxes and the appeal to illusions to explain concrete, real life experiences. Unfortunately, much of philosophy had fallen into similar quagmire in order to cling to some fanciful notions that discrete symbols (whether it be words or mathematical) can in some fashion be used to understand the nature of a fluid and ever changing nature. To say that science and mathematics has failed in the realm of understanding nature would be casually polite. It is simply a contradictory mess. But then again, such playful endeavors are in itself part of nature and can be understood as such.

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.

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

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

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

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

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? — apokrisis

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?

If we view the real numbers as specifying locations on a line, and we stop talking about points, perhaps things are less muddled. — fishfry

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.

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. — aletheist

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.

I've started reading the Zalamea paper and I'm spend some time with it ... It's good stuff. — fishfry

Agreed, and you probably understand a lot more of the non-Peircean content than I do. :)

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

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

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

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

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

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

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

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

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

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

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

So this is why the Toms and Fishfrys are so content with what they learn in class. — apokrisis

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I'm used to a robust level of discussion in academic debate. — apokrisis

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.

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

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

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

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

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

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

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. — apokrisis

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

I've tried to point out to fishfry, I wasn't really attacking him personally... — apokrisis

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

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

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

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

Well you can just go fuck off. ;)

I would call it brutal. Telling someone to fuck off is simply vulgar. Anyway, carry on, I'm glad to see Fishfry has joined here, he has completely different interests to myself and I find what he says interesting, even though most of it goes over my head.

Again, on what grounds precisely?

Telling someone to fuck off is simply vulgar. — Wayfarer

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

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

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

What is this, a 'meta-argument'? An argument about an argument? Anyway, it was this:

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.

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.