No, you claimed the reals can be disordered and made discrete. — tom

Here is what he actually said.

The reals in their usual order are a continuum. They can be reordered to be discrete. — fishfry

He then added this, which is basically the same point that I made.

If by continuum we mean a particular philosophical idea of a continuous space, then the mathematical real numbers may or may not satisfy a philosopher. If by continuum we mean the standard mathematical real numbers, then we are being circular. Certainly the standard real numbers are not a proper model of the intuitionistic continuum. — fishfry

So I would still like to see the alleged proof that the real numbers form a true continuum as Peirce defined it, which (as I understand it) is similar but not identical to the intuitionistic continuum. I highly doubt this - especially if, in fact, the real numbers can be reordered to be discrete.

As far as I can tell, mathematics is totally reliant on the discrete and because of this limitation constantly makes philosophical ontological errors. — Rich

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

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

I think you have presented to most essential issue. I understand the dilemma, but understand that Bergson had truly mastered all mathematics of his time but never sought to use it in any of his writings. One cannot use a fatally flawed approach to arrive at more knowledge no matter how difficult it is to admit to these flaws.

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.

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

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

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

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

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

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

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

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

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

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

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

So concretely, a discrete approach cannot uncover the nature of a continuous ontological reality. — Rich

I agree with this; what I question is whether mathematics is totally reliant on the discrete. As I have indicated previously, I have in mind Peirce's broad definition of mathematics as the science of drawing necessary conclusions about ideal states of affairs. In fact, toward the end of his life, he largely moved away from describing continuity using set theory - although he employed the term "collection" - and embraced topology instead.

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. — Rich

You will likely appreciate this quote that I just came across from Philip Ehrlich, referring to Paul duBois-Reymond, another late-19th-century mathematician who wrestled with the concept of continuity.

... he attacked the 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.

I suspect that Peirce would have endorsed this criticism wholeheartedly.

So regular maths is "wrong" in always framing reality in constructivist terms. And yet in the end maths is a tool for modelling. We actually have to be able to calculate something with it. And calculation is inherently a constructive activity. — apokrisis

Agreed. This is what I have in mind when I cite the famous quote by George Box: "All models are wrong, but some are useful." In my own field of structural engineering, Mete Sozen has posed the question: “Is an exact analysis of the approximate model an approximate analysis of the exact structure?” An affirmative answer is a fundamental yet subtle presupposition of modern practice - one that is easily (and often) overlooked, and not always correct.

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

Right - because although diagrammatic reasoning is deductive, it requires the retroductive steps of creating, augmenting, and manipulating an icon that embodies the significant relations among its parts within a suitable representational system. This can only be done successfully by a well-prepared mind that has developed the right kind of judgment through (mathematical) experience.

one should never take too much notice of a mathematician making extrapolations of a metaphysical nature. They are bound to be misguided just because they hold in their hands a very impreessive hammer and so are looking about for some new annoying nail to bang flat. — apokrisis

I'll certainly take that to heart :-)

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

So I would still like to see the alleged proof that the real numbers form a true continuum as Peirce defined it, which (as I understand it) is similar but not identical to the intuitionistic continuum. — aletheist

Oh I'm in no position to do that. I'm not familiar with the philosophical thinking on the continuum. You did note that I questioned whether philosophers accept the real numbers as the correct model of the continuum. I didn't claim that the real numbers form a true continuum, someone else did. I'm aware that there are philosophical objections but I don't know much about them.

Quite interesting.

The OP substantially calls to question the whole premise of using discrete to describe the continuous. As the OP describes, by discarding the discrete, one quickly resolves the paradox. Similar paradoxes can be similarly addressed with this approach. But first, discrete must be discarded in the realm of ontology.

Please identify a natural number or integer that is not capable of being counted. — aletheist

That's a very simple question to answer. The highest number is the one that's not capable of being counted.

Instead, what I am arguing is that it is possible in principle to count all of the natural numbers (and integers) because it is possible in principle to count up to and beyond any particular natural number (or integer). — aletheist

Again, you are saying that because it is possible to count any particular number, it is therefore possible to count all the particular numbers, and this is known as the fallacy of composition.

Pretty much all of Bergson's works revolves around around this subject. It is best to start at the beginning and work forward as he refined his ideas and criticism taking particular notice of his critiques of Relativity Theory in his later works. For a more modern take (though incomplete) Stephen Robbins explains Bergson's ideas his Youtube videos which are easily found. Bergson meticulously describes his reasoning though you may from time to time choose to zigzag between papers on his writings and his writings to better understand his thought process. Gunter did a good job.

As far as I can tell, mathematics is totally reliant on the discrete and because of this limitation constantly makes philosophical ontological errors. — Rich

This is exactly the problem with the Zeno paradox of the op. Zeno's premise is that space is continuous. Then he introduces mathematics to deal with this assumed continuity. It fails. The conclusion which should be drawn, is that mathematics is incapable of dealing with the continuous.

If we move to the ontological implications, then if there is no such thing as a real continuity there is no problem. But if there is a real continuity then we have a problem, if we are trying to understand that continuity mathematically. So either space and motion are discrete, and there will be no problem to understand them with mathematics, or they are continuous, in which case they cannot be understood with mathematics.

And if we assume a real continuity of any sort, then we should not expect to be able to understand it mathematically.

100% agree.

Here's a question then. Do you think that there is such a real thing as a continuity, and if so what would be its nature? Remember the premise, it cannot be understood with mathematics.

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

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

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

I am not familiar with Bergson, and Peirce (not Pierce) is notoriously difficult to get one's arms around, because he never managed to write a book on philosophy - just lots of articles, and tens of thousands of pages of unpublished manuscripts. I usually recommend the two volumes of The Essential Peirce, but for this particular subject, several chapters in Philosophy of Mathematics: Selected Writings are more pertinent; likewise his 1898 lecture on "The Logic of Continuity," which is the last chapter in Reasoning and the Logic of Things.

As for secondary literature, chapter 4 of Kelly Parker's book, The Continuity of Peirce's Thought, might be a good place to start. Benjamin Lee Buckley's book, The Continuity Debate, based on his dissertation (as Lisa Keele), summarizes and compares the views of Cantor, Dedekind, duBois-Reymond, and Peirce; however, I take exception to some aspects of how Peirce's position is described.

Yes, there is such thing as continuity and we experience it quite concretely as duration (real time). To better understand life and the universe one must use the arts. Bergson uses the quality of musical sounds flowing into each other as one way to understand memory (the foundation of life) and duration as we experience it.

I personally am spending my later years of life immersed in music, art, dance, Eastern meditative practices such as Tai Chi, to expand my toolset for greater knowledge. Without experience in the arts, I do not believe one is properly equipped to understand life.

But first, discrete must be discarded in the realm of ontology. — Rich

Just curious, on this basis do you reject nominalism - the view that reality consists entirely of singulars - in favor of realism? Peirce did; he eventually described himself as an "extreme scholastic realist," affirming the reality of generals as continua.

Yes. I am quite a realist myself. It is all real but being experienced differently. A dream, therefore is as real as anything else. There is no illusion, just different experiences or forms of memory.

Wikipedia he does a reasonable job on Bergson, duration, and continuity.

https://en.m.wikipedia.org/wiki/Duration_(philosophy)

It is a reasonable starting point for further study.

Yes, there is such thing as continuity and we experience it quite concretely as duration (real time). — Rich

This is questionable though. We can understand time as discrete units, or we can understand time as a continuity. We can also understand it as some kind of composition of both. What if real time, which we are experiencing, consists of discrete units, and it is just the brain and living systems which are creating the illusion of continuity? I tend to think that the only real continuity is the existence of the soul itself, and the soul, during the act of experiencing, renders the appearance of time as continuous, to make it compatible with its own existence, and therefore intelligible to the lower level living systems. Now, as highly developed life forms, we have developed mathematics, which will allow us to understand the true nature of time, as discrete, but we must get beyond the way that time is presented to us by our lower level living systems, (i.e, that intuitive impression of time) to be able to understand time mathematically.

The highest number is the one that's not capable of being counted. — Metaphysician Undercover

And which number would that be? I asked you to identify it, not describe it.

Look, we have been using at least three different definitions of "countable" in this thread:

• The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
• My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count.
• The notion of "actually countable," which requires it to be possible to finish counting.

You have made it quite clear by now that you reject the first two, but that does not render them false or contradictory - just different from yours.

And which number would that be? I asked you to identify it, not describe it. — aletheist

There is no highest number, that's what makes the set of natural numbers uncountable. If I could identify the highest number, we could count to it, and count all the numbers. I cannot, and nor can you, or anyone else, and so the natural integers remain uncountable, as they always will be.

The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count.
The notion of "actually countable," which requires it to be possible to finish counting.


You have made it quite clear by now that you reject the first two, but that does not render them false or contradictory - just different from yours. — aletheist

I accept the first one, but that defines "countable" relative to the natural numbers, so it is insufficient to tell us whether or not the natural numbers, are countable. It is a definition used to judge things in relation to the natural numbers so it cannot be used to judge the set of natural numbers itself. We need a definition which we can apply to see whether or not the set of natural numbers itself is countable.

The second definition of countable, your definition, makes no sense. "There is no particular largest value beyond which it is logically impossible to count." If this were the case, then no subsets of integers would be countable, because each of these has a particular largest value.

The third is the obvious choice as a definition to apply in order to determine whether the natural numbers are countable. It would be false to say that something which is not capable of being counted is countable. Therefore we can conclude that the set of natural numbers is not countable.

A dream, therefore is as real as anything else. — Rich

I am inclined to subscribe to how Peirce addressed this.

'Real' is a word invented in the 13th century to signify having Properties, i.e. characters sufficing to identify their subject, and possessing these whether they be anywise attributed to it by any single man or group of men, or not. Thus, the substance of a dream is not Real, since it was such as it was, merely in that a dreamer so dreamed it; but the fact of the dream is Real, if it was dreamed; since if so, its date, the name of the dreamer, etc. make up a set of circumstances sufficient to distinguish it from all other events; and these belong to it, i.e. would be true if predicated of it, whether A, B, or C Actually ascertains them or not. The 'Actual' is that which is met with in the past, present, or future.

It would be false to say that something which is not capable of being counted is countable. — Metaphysician Undercover

Only if it were true that we must be able to finish counting something in order to call it "countable." Your definition requires this; mine does not.

Therefore we can conclude that the set of natural numbers is not countable. — Metaphysician Undercover

And yet set theory explicitly says otherwise.

And yet set theory explicitly says otherwise. — aletheist

Some theories are false. It's very hard to convince the people who believe in false theories, that they are false. That's life. It appears like your set theory, if it really is as you describe, relies on the fallacy of composition. You should investigate this, and if the theory is as you describe, quit believing in it so strongly, because it's false. Or else it isn't as you describe, then you should develop a better understanding of what the theory really says.

Therefore we can conclude that the set of natural numbers is not countable.
— Metaphysician Undercover

And yet set theory explicitly says otherwise. — aletheist

You are each using different definitions. This is the fallacy of ambiguity. Surely we need not argue about this any more. I have humbly offered the word foozlable as standing in for the set theoretic definition, because it carries no semantic baggage from any common meaning.

It's no different than a doctor examining you and finding your condition "unremarkable." That might be an insult in daily life but it's the best news you can get in a doctor's office. Surely you understand this. It's a medical term of art.

That said, the question of whether the natural numbers can be "counted" in any meaningful sense of the word -- stipulating that technical conditions in formal set theory are not necessarily meaningful -- is a good one.

I don't think you can count past 200. You'd get bored. You can't count to a zillion. You just can't. It couldn't be done in the age of the universe. If counting is an activity that takes place in time, then a finite universe doesn't give you enough time to count any more than some finite number. There are 10^80 hydrogen atoms in the universe. That's a very small natural number. You can't count it.

What can it mean to count to 10^80? Mathematically you can count a set if you can order-biject it to a natural number, or in a more general context to some ordinal. Sure, you can count 10^80 that way. But that's just set theory. [Note that counting via numbers involves finding an order-preserving bijection, not just a bijection].

If you want to say that it means something other than formal set theory to count to 10^80, I'd like you to tell me what it is.

Myself I tend to be a formalist. I like set theory but I don't think it actually means anything. I don't think there are sets in the real world. It's the famous singleton problem. There is no such thing as an apple and "the set containing an apple." There's madness down that road.

I'm fascinated to read a little on Wiki about Bergson. "Bergson is known for his influential arguments that processes of immediate experience and intuition are more significant than abstract rationalism and science for understanding reality." I actually agree with that point of view. I'm opposed to scientism. I've got a lot of reading to do. I'm afraid I can't pick up those many volumes that have been suggested, but I will definitely Google around.

It appears like your set theory, if it really is as you describe, relies on the fallacy of composition. — Metaphysician Undercover

Apparently you are in such a big hurry to reply that you are not even bothering to pay attention to what I actually post. In this case, you are mixing up the first two definitions that I so carefully spelled out. The first one, which directly quoted @fishfry, is the one from set theory - not my set theory, but standard set theory - and if it helps, we can substitute "foozlable" as he just suggested (again). The second one - the one that I assume you are still criticizing - has nothing to do with set theory at all, as @fishfry helpfully pointed out a while ago. We simply disagree on whether "countable" always and only entails the ability to finish counting; you say yes, I say no.

If counting is an activity that takes place in time, then a finite universe doesn't give you enough time to count any more than some finite number. There are 10^80 hydrogen atoms in the universe. That's a very small natural number. You can't count it. — fishfry

I appreciate the sensibility that you have tried to bring to this discussion, but I still have to comment on this (again). I agree that neither you, nor I, nor anyone else can actually count that many things. However, this is irrelevant to what we have been discussing, since mathematics is the science of drawing necessary conclusions about ideal states of affairs. An immortal being in an eternal universe could actually count that many things (and more), and this is basically what I mean when I say that they are countable in principle.

I've got a lot of reading to do. I'm afraid I can't pick up those many volumes that have been suggested, but I will definitely Google around. — fishfry

Yeah, I kind of opened the fire hose on you there; sorry about that. There is a handy online dictionary of Peirce quotes on a lot of different topics; going through a few of the most relevant terms might at least give you a taste of his thought on these matters.

Please locate that quote of mine, I can't find it and don't remember saying it. — fishfry

The reals in their usual order are a continuum. They can be reordered to be discrete. — fishfry

If the reals can be disordered, just do it. Can't be more than a few lines?

Take any real number, leave a gap, state where the gap is, and place it between any two other real numbers.

Look, we have been using at least three different definitions of "countable" in this thread:

The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count.
The notion of "actually countable," which requires it to be possible to finish counting. — aletheist

That's not right.

Everyone who is not being deliberately obtuse understands what countable means - it means you can count elements of the set. No one, unless they are being deliberately obtuse, thinks that this fact has any bearing on whether anyone would be willing to embark on counting all the members of a very large or even infinite set.

Now, will someone pleas count the number of reals that exist in the range (0, 0.0000000000000001)?