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

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

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

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

As I just finished practicing piano, I observed that the notes that I was playing as I read them off paper was can awful representation of the music that I was creating. So are mathematical symbols and equations an awful representation of nature. Yes, notes have value (for some, not all) but to understand music one must penetrate it with the consciousness - that is what the great musicians and teachers proclaim. The notes are empty, lifeless representations of the music artists create. This so are mathematical symbols.

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.

As I just finished practicing piano, I observed that the notes that I was playing as I read them off paper was can awful representation of the music that I was creating. — Rich

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.

So are mathematical symbols and equations an awful representation of nature. — 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.

It is interesting that Bergson used music as his analog for duration ... — 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.

I am not suggesting that musical notes are not useful. What I am saying is that they, as are all symbols, are an absolute awful representation of the experience itself. Music itself is transmitted via experience. Notes have limited symbolic value and have absolutely no resemblance to music itself. Such also is the nature of mathematics and words. They cannot replace or be mistaken for the actual experience.

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.

I am not suggesting that musical notes are not useful. What I am saying is that they, as are all symbols, are an absolute awful representation of the experience itself. — Rich

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."

We must put mathematical equations in proper perspective and not get carried away by them. — Rich

Agreed, but we also should not undervalue them, either.

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

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

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

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

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

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.

what I am looking for from you is a genuine counter-argument, — 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?

a restatement of a particular institutional view that is widely held for the pragmatic reasons I've previously stated. — 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?

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

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

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

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

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

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

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

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.

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. — aletheist

I hope you noticed that computers can't even instantiate the reals.

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). — apokrisis

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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." — aletheist

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.

Our fundamental theories are based on the continuum of space-time. — tom

An analytic continuum (real numbers), or a true continuum (Peircean)?

I hope you noticed that computers can't even instantiate the reals. — 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."

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." — aletheist

Breaking sand "infinitely" yields tiny bits of sand? Please!

Breaking the real numbers "infinitely" yields what?

It seems to me that you are laboring under a simplistic mereological and atomistic understanding of topology. In topology a line is not just a bunch of points that are put side by side, which indeed sounds wrong - how can you get a one-dimensional object from any number of zero-dimensional objects thrown together? Of course you can't, and that's not how it works.

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.

Breaking sand "infinitely" yields tiny bits of sand? Please! — tom

It obviously does not yield an unbroken continuum.

Breaking the real numbers "infinitely" yields what? — 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.

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. — aletheist

Which real number is bigger?

1 or 0.999...

Are your comments directed at any particular person or post?

It's you.

And by the way, which real number is bigger:

1 or 0.999...

According to you, or Peirce?

Which real number is bigger? 1 or 0.999... — tom

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.

It's you. — tom

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.

↪aletheist It's you. — tom

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

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

Yep. SophistiCat is talking the language of constraints.

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

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

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. — aletheist

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.

You cannot create a new number by adding an infinitesimal quantity. — tom

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?

As it happens, .999... = 1 is a theorem even in nonstandard analysis. — 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.

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. — aletheist

Wow! I'm out!

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

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