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.

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.

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.

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because he never managed to write a book on philosophy - just lots of articles, and tens of thousands of pages of unpublished manuscripts.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.

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.

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.

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.

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.

Does mathematics actual model a continuum? I don't think so. — Rich

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.

The real numbers have been proved to for a continuum, even in the Peirceian sense. — tom

He certainly did not think so. Could you please point me to the proof? Note, I acknowledge that the real numbers serve as a useful mathematical model of a continuum.

Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable. — Metaphysician Undercover

Please identify a natural number or integer that is not capable of being counted. You are still conflating the notion of counting with the notion of being finished counting.

But it is false to claim that the entire infinite set is countable in principle, what is countable is finite subsets. — Metaphysician Undercover

Only if "countable" means "capable of being finished counting," which is not how the term is defined within mathematics, nor how I have ever used it in this thread.

This is a textbook case of the fallacy of composition. — Metaphysician Undercover

Only if I were arguing that it is possible in principle to count all of the natural numbers (and integers) merely because it is possible in principle to count some of the natural numbers (and integers). 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). Again, there is no largest value that is countable in this sense, so they must all be countable in this sense.

However, note that - per @fishfry's helpful clarification - this argument of mine has nothing to do with the technical definition of "countable" (or "foozlable"), which pertains only to sets and is easily proved to be a property of the natural numbers (and integers), but not the real numbers.

Cantor proved the reals constitute a continuum. — tom

As he defined "continuum," yes. However, Peirce argued (and I agree) that the real numbers still do not conform to our common-sense notion of a true continuum as "that of which every part has parts of the same kind," such that it cannot be understood as a collection of individuals, no matter how dense. A truly continuous line can be infinitely divided into smaller and smaller lines, but never into points. As you have noted, the real numbers form a multitude larger than that of the rational numbers; but a true continuum exceeds all multitude.

Once you conflated the technical meaning of countable with its every day meaning -- a logical fallacy -- the thread lurched off on a very unproductive tangent IMO. — fishfry

I am not the one who took us down that road by repeatedly insisting that "countable" must always and only mean the same thing as "actually countable." On the contrary, I carefully maintained the distinction between these two concepts throughout.

It is far from clear that "given enough time" you could count to any specified value. If time itself is part of the universe, then you will run out of time between the Big Bang and the heat death of the universe. — fishfry

Now you are the one conflating the mathematical with the actual. Any natural limitations on my ability as a human being to count up to very large numbers, whether there was a Big Bang, whether there will be a heat death of the universe, and how much time will have passed in between these two posited events are all completely irrelevant to the discussion. We are drawing necessary conclusions about an ideal state of affairs, so actuality (including time) has nothing to do with it.

You have just conflated counting up to some big finite number with counting ALL the natural numbers. — fishfry

I have done no such thing. I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers).

Your point is that this is not part of the relevant technical definition of "countable." I have accepted that clarification, and even thanked you for it, so I do not understand why you keep harping on it. Even so, there is presumably a reason why the standard term is "countable" and not "foozlable" or anything else.

If it helps you feel any better, it took me about two solid years of reading Peirce - including two complete passes through both volumes of The Essential Peirce and a considerable amount of secondary literature - before I started thinking that I was finally understanding what he wrote. It was totally worth the time and effort, though.

Countability as defined in mathematics simply has nothing at all to do with the everyday meaning of the ability to be counted ... If you counted, in the sense of saying out loud "one, two, three ..." the natural numbers, starting at the moment of the Big Bang, at the rate of a number per second; or ten numbers, or a trillion -- you would not finish before the heat death of the universe ... You can't count the natural numbers in the every day meaning of the word. — fishfry

I have acknowledged this repeatedly - the natural numbers (and integers) are not actually countable, in the sense that someone or something could ever finish counting them. However, they are all countable in principle, in the sense that there are no natural numbers (or integers) that are uncountable; given enough time, someone or something could count up to and beyond any arbitrarily specified value. As I said before (with sincere gratitude), you have stated more accurately what I meant all along.

We cannot imagine anything that we have not already experienced in the past. — Samuel Lacrampe

Charles Sanders Peirce offered a similar argument for the reality of his universal categories of Firstness (quality/feeling/possibility), Secondness (reaction/will/actuality), and Thirdness (habit/thought/necessity), as well as his synechism - the doctrine that there is real continuity.

Whatever unanalyzable element sui generis seems to be in nature, although it be not really where it seems to be, yet must really be in nature somewhere, since nothing else could have produced even the false appearance of such an element sui generis. For example, I may be in a dream at this moment, and while I think I am talking and you are trying to listen, I may all the time be snugly tucked up in bed and sound asleep. Yes, that may be; but still the very semblance of my feeling a reaction against my will and against my senses, suffices to prove that there really is, though not in this dream, yet somewhere, a reaction between the inward and outward worlds of my life.

In the same way, the very fact that there seems to be Thirdness in the world, even though it be not where it seems to be, proves that real Thirdness there must somewhere be. If the continuity of our inward and outward sense be not real, still it proves that continuity there really be, for how else should sense have the power of creating it? — Reasoning and the Logic of Things, pp. 161-162

Utmost is the issue that science way over steps it's bounds when it begins to replace everyday experiences with symbolic equations and declaring the equations to be more real. — Rich

Yes, we must always keep in mind that such equations are models - or in Peirce's terminology, diagrams - which embody only the parts and relations within the actual situation that someone has deemed to be significant. Consequently, they are only as "accurate" as this judgment on the part of the modeler and the underlying assumptions of the selected representational system, including its transformation rules.

No I'm afraid you are still missing my point. — fishfry

No, I get it, you just stated more accurately what I meant all along. :)

It sounds like we are on the same page here. As Charles Sanders Peirce put it, citing his father:

I do not know that anybody struck the true note before Benjamin Peirce, who, in 1870, declared mathematics to be 'the science which draws necessary conclusions' ... the essence of mathematics lies in its making pure hypotheses, and in the character of the pure hypotheses which it makes. What the mathematicians mean by a 'hypothesis' is a proposition imagined to be strictly true of an ideal state of things. In this sense, it is only about hypotheses that necessary reasoning has any application; for, in regard to the real world, we have no right to presume that any given intelligible proposition is true in absolute strictness.

The assumption that space is divisible is a matter of convenience, it does not reflect experience. — Rich

I should add that with this simple observation, that space is indivisible, all mathematical theories about nature that rely on mathematical divisibility of space, automatically lose all ontological meaning. — Rich

What exactly do you mean when you assert that "space is indivisible"? Are you merely saying (as I do) that space is continuous, rather than discrete - i.e., it cannot be divided into dimensionless points, only smaller and smaller three-dimensional spaces? Or do you have something else in mind?

They are not synonymous, but infinite is by definition not countable. — Metaphysician Undercover

Again, incorrect. You evidently have a rather idiosyncratic personal definition of "infinite." My dictionary provides several widely accepted definitions, and none of them state or imply that it means "not countable." Besides, as I keep noting, the concept of being "countably infinite" is well-established and well-understood within mathematics.

But you should not claim that you can make the two compatible by saying that one refers to an actuality and the other to a potentiality, because this is not the case. — Metaphysician Undercover

I have never claimed that our different definitions of "countable" are compatible. I have simply demonstrated that my definition is not contradictory, and that yours is simply wrong, at least within mathematics. Whether something is actually possible is completely irrelevant when dealing with ideal states of affairs, which is all that pure mathematics ever does.

How does this imply that all the natural numbers are countable? — Metaphysician Undercover

Because you can always keep counting beyond any arbitrary finite value; i.e., you cannot identify a single natural number or integer that is uncountable. Obviously, if there are no uncountable natural numbers or integers, then all of the natural numbers and integers are countable.

Every number you count has a larger number, therefore it is impossible that all of the natural numbers are countable. — Metaphysician Undercover

Now you seem to be confusing "countable" with the idea of being finished counting. This is not what "countable" means within mathematics, either. That larger number is just as countable as the one that you already counted; and so is the next larger number; and so on, ad infinitum - which is the whole point.

I think you really believe that it is possible to count infinite numbers, because this statement seems to be an attempt to justify this. — Metaphysician Undercover

I have stated plainly (and repeatedly) that I believe this to be logically possible, but not actually possible.

No, the fact is that you cannot count an infinite set, that's what "infinite" means ... The point I made earlier is that there is actually no difference between the countable infinity and the uncountable, as "infinite", they are the same. — Metaphysician Undercover

Incorrect; "uncountable" and "infinite" are not synonyms in mathematics, since there are countable infinities and uncountable infinities. This is a fact, not an opinion.

"Countable" is just a name, as @fishfry explained, it has no other meaning. — Metaphysician Undercover

All words are just names, with no other meanings than how people use and understand them. Mathematicians use and understand "countable" in a very specific way. You do not have to like it, but it is silly to continue insisting otherwise.

What is different is the thing which we are attempting to count, one is a continuity the other discreet units. The continuity cannot be counted, the discrete units can. — Metaphysician Undercover

Not quite, since even the real numbers are still discrete despite being uncountable; they thus form a pseudo-continuum. A true continuum is "that of which every part has parts of the same kind" (Peirce), so it can never be divided into discrete individuals. For example, a truly continuous line can be divided into infinitely many smaller (continuous) lines, but never into (discrete) points.

Thanks for the excellent clarification/explanation.

A set is defined as countable if it can be put into bijection with the natural numbers. — fishfry

Right, this is all that I meant when I said that the natural numbers are countable by definition. I agree with you that we can then subsequently prove that the integers are also countable.

It's a mistake to think that countability has anything to do with the ability to be counted. — fishfry

Right, this is all that I meant when I said that "countable" is not the same concept as "actually countable." However, I agree with @tom that "you can count members of a countably infinite set"; again, there is no largest natural number or integer beyond which it is (logically or actually) impossible to count, so all of the natural numbers and integers must be countable.

If you now introduce a principle, and say that this principle states that the infinite is countable, such that you can say "it is possible in principle to count them", all you have done is introduced a contradictory principle. — Metaphysician Undercover

There are really two basic principles here:

• All counting is by means of the natural numbers; therefore, the natural numbers are countable.
• Any set that can be arranged in one-to-one correspondence with the natural numbers, such as the integers, is also countable.

There is nothing contradictory about either of these principles; in fact, together they constitute the very definition of what it means for something to be countable within mathematics. The fact that both the natural numbers and the integers are infinite is completely irrelevant. Think of it this way - it is logically (and actually) impossible to identify a particular integer beyond which it is logically (or actually) impossible to count. If all integers up to any arbitrary finite value are countable, but there is no largest countable integer, then all of the integers must be countable.

No. counting all the integers is not logically possible, it is impossible. — Metaphysician Undercover

One more time: it is logically possible, but actually impossible. You claim to know the difference, but your responses keep indicating otherwise.

That's what infinite means, that it is impossible to count them all, you never reach the end. It is such by definition. — Metaphysician Undercover

That is obviously not what infinite means within mathematics, since the natural numbers and integers are very explicitly defined as countably infinite. You can rail against this terminology all you want, but it will not change the fact that there is no contradiction in saying that the integers are countable as that concept is defined within mathematics.

See, you say that no one can actually count them, yet it has been proven that it is possible in principle to count them. — Metaphysician Undercover

Yes, and there is no contradiction at all in saying this - unless you insist on conflating "someone can actually count them" with "it is possible in principle to count them," thus refusing to acknowledge that they are NOT the same concept. Counting all of the integers is logically possible, but actually impossible. Infinitely dividing space is logically possible, but actually impossible. Creating a perfect circle is logically possible, but actually impossible. Pure mathematics is the science of drawing necessary conclusions about ideal states of affairs; the actual has nothing to do with it.

No I don't see the difference, and you've already tried to explain, but all you do is contradict yourself. — Metaphysician Undercover

Show me one genuine contradiction in any of my previous posts, without conflating "countable" (as defined in mathematics) with "actually countable." They are two different concepts.

To say that there is a difference between actually countable and potentially countable is nonsense. — Metaphysician Undercover

To say that there is no difference between actually countable and potentially countable is simply incorrect. Do you really not understand the distinction between the actual and the potential? between the nomologically possible and the logically possible?

But just because it's called "countable" doesn't means it's actually countable. You seem to believe that it actually does mean that it's countable. — Metaphysician Undercover

Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference?

I'd rather a smaller world view which distinguishes fact from fiction, than a larger world view which doesn't distinguish fact from fiction. — Metaphysician Undercover

And I would rather have a worldview that does not make the mistake of treating that which is real as fictional just because it is not actual.

It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. — Metaphysician Undercover

It appears very obvious to me that you do not understand the accepted meaning of the word "countable" and, more fundamentally, the distinction between logical possibility and nomological possibility. It is possible in principle to count all of the integers or all of the rational numbers, even though it is not actually possible (as far as we know) for a human being, a machine, or any other physical thing to do so.

No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity? — Metaphysician Undercover

No one is talking about spatial magnitude, and talking about numbers does not entail talking about quantities. Your worldview is too small because it limits the real to the actual and the finite.

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

You believe that something is possible (potentially doable) though it is actually impossible to do it. — Metaphysician Undercover

No one is talking about doing anything. To say that something is infinitely divisible does not mean that a human being is actually capable of infinitely dividing it. It means that it is possible in principle to divide it infinitely.

You seem to be confusing "infinite divisibility" and "infinitely divided". — Michael

No, you are the one with that confusion, as I have stated before. Space only has to be discrete if it is infinitely divided, not merely infinitely divisible.

Space is actually infinitely divisible and potentially infinitely divided. — Michael

This illustrates your muddled thinking perfectly. "Divisible" means "potentially divided."

The point is that, as with the example of a clock hand, the very act of moving from one point to another can be considered to be an act of counting. — Michael

It can be, but it does not have to be. Your whole argument hinges on insisting that the very act of moving from one point to another must be considered to be an act of counting, and that this counting must include every single point corresponding to a rational number in the interval. Just because we can model it that way does not entail that it actually is that way.

You're still making the same mistake. It is false to say that space is potentially infinitely divisible unless it actually is. — Metaphysician Undercover

You're still making the same mistake. It is false to say that space is potentially infinitely divisible only if it actually is.

This indicates that you have a deep misunderstanding of the concept of "potential". — Metaphysician Undercover

No, it indicates that I have a different understanding of the concept of potential. We have previously established in other threads that you and I have a fundamental disagreement about this, so there is really no point in discussing it further here.

Motion is logically impossible but physically actual. And so the first of MadFool's suggestions seems correct; our logic is faulty. — Michael

That which is physically actual must be logically possible; and so it is only your logic that is faulty here, because you insist on applying the logic of finite/discrete mathematics to a problem that involves infinity/infinitesimals. Peirce said it well - "Of all conceptions Continuity is by far the most difficult for Philosophy to handle."

The problem is that the logic of continuous motion is incoherent, hence motion isn't continuous. — Michael

The problem is that the logic of discrete motion is incoherent, hence motion isn't discrete.

Like I said, we are at an impasse. Cheers.

We agree that it is not possible for us to plot actual coordinates at distances that correspond to all of the rational numbers. We thus agree that it is not possible for us to measure actual motion at that level of granularity.

You then draw the further conclusion - unwarranted, in my view - that the motion itself cannot occur at that level of granularity, such that space itself must be discrete. This has the consequence that all actual motion must involve somehow "jumping" from one discrete location to another, without ever occupying any of the space in between.

Since I find this patently absurd, I affirm instead that space must be continuous. Our inability to measure infinitesimal distances does not entail that they do not exist; objects can and do traverse infinitely many of them while moving from one arbitrarily defined coordinate to another. However, as @Svizec just explained, each such transit occurs in an infinitesimal interval of time. In the end, objects actually move finite distances in finite times, and that is what we can and do observe and measure.

Given that it has occupied an infinite number of prior locations in succession, it has completed an infinite series of events. — Michael

No. The motion from one potential location to the next is not a discrete event. Only the motion from one actual location (i.e., arbitrarily defined coordinate) to the next is a discrete event. We can only define a finite number of distance coordinates, so we can only measure motion in discrete units. However, the motion itself is continuous between those discrete coordinates that we use to measure it.

Your position is the one with logical problems from my point of view. How can something "jump" from one discrete location to another without ever occupying the space in between? This is pure nonsense to me.

For example, the first coordinate would be the one at 1 Planck length. The second coordinate would be the one at 2 Planck length. And so on. But at no point does it pass through the coordinate at 0.5 Planck length or at 1.5 Planck length. — Michael

This is a good example to show why we are at an impasse. Your claim is that all actual objects that actually move go from one Planck length coordinate to the next without ever occupying any intermediate locations. My claim is that all actual objects that move occupy infinitely many intermediate locations between any two arbitrary coordinates, even if the interval between them is one Planck length. You thus take the Planck length to be a limit on actual events themselves, while I take it to be a limit only on our measurement of actual events.

Does it then follow that [X] and [Y] are the same region of space? It does not appear so to me. — Arkady

Well, it does appear so to me. So we have contradictory intuitions, which just goes to show that intuitions are not infallible guides to truth.

Edit: Wait, I see it now. You said that [X] and [Y] are adjacent regions of space, not adjacent locations in space. So I understand that two regions can have zero separation between them, yet not be the same region. But what we were discussing was whether two (dimensionless) locations can have zero separation between them, yet not be the same location.

In this scenario, the machine performs a count by moving to a different point in space. So there is no fundamental difference between moving and counting. — Michael

No, all we can say is that there is no fundamental difference between measuring movement/distance and counting, which I have acknowledged all along. Measurement is not a prerequisite for motion.