I put "adjacent" in quotation marks for a reason. No two points on a continuum - potential or actual - are strictly adjacent. Like I said, between any two points - potential or actual - there are potential points that exceed all multitude.

Okay, but there can be two points?

There can be two actual points. Again, potential points are indistinguishable unless and until actualized.

Sure. Isn't there something numerical about the two actual points?

Yes - I already acknowledged that two actual points on a truly continuous line are individual, and not numerically identical. What I denied is that there is anything individual or numerical about potential points.

Sure. So for one, that's a difference between actual and potential points, right?

Yes. Please get to the point if you can, I need to call it a night.

Right. So that's what I was referring to by "separate" for one. As I said above: "they're not the same in every respect, including numerically." If there's a difference, they're not the same in every respect, and that's all I meant by "separate."

Okay, but "separate" implies something that is obviously incompatible with true continuity, which is why I could not give an immediate and simple answer to your initial question.

Getting back to my question, then - how could space and time (or anything else) be truly continuous, rather than discrete, under nominalism?

Okay, but "separate" implies something that is obviously incompatible with true continuity, which is why I could not give an immediate and simple answer to your initial question. — aletheist

I didn't mean anything like that by that term, though.

Of course, I wouldn't say that I really understand Peircean "continuity" talk in this context. It really makes very little sense to me.

Re your question, for example, you said this earlier: "If space and time are truly continuous, then they exceed all multitude of individual locations and instants." I haven't the faintest idea what "exceeding all multitude" might refer to. Individual locations and instants I don't have a problem with--it's simply a way of referencing extensions/extensional relations (in the case of space) or change-oriented/motion-oriented relations (in the case of time). I don't have any idea what we might be talking about re "exceeding or not exceeding a multitude" of individual locations and instants.

I also have no idea what anything like a mathematical/geometrical analogy (is it an analogy in Peirce's view? I don't know) of lines, points, etc. would have to do with the idea of universals and particulars, potentials and actuals, etc. in Peircean philosophy.

It might be helpful to remind folks that I'm not a realist on mathematics (or mathematical objects etc.), by the way. So I don't think that anything we refer to in mathematical terms pegs anything real. Mathematics on my view is a social and subjective psychological construction, a language we invented for talking, in the most abstract context, about how we think about relations. I do think that on a very rudimentary level that some of the relations we base mathematics on are real relations that we experience empirically, but "based on" doesn't mean "the same as" (think of how The Texas Chain Saw Massacre is "based on" the real-life story of Ed Gein)--the real relations in questions are not actually mathematical relations. Mathematics is our invented language only. And most of mathematics is a thought-based extrapolation of the based-on-but-not-the-same-as rudimentary relations that we experience.

Do space and time, as a whole, consist entirely of the actual aggregate of individual locations and instants? This is analogous to saying that a line, as a whole, consists entirely of the actual aggregate of individual points. The line with points serves as a diagram, because it embodies the significant relations of its object - in this case, space and/or time with individual locations and/or instants.

Well, first I don't think that the idea of points makes a lot of sense aside from a fuzzy abstract concept--that's because the idea of a "zero dimensional" something can't be a very exact idea in my opinion.

Re the empirical question, I've commented a couple times that I don't think we know whether extensional relations or change/motion relations are discrete or continuous . . . assuming the question even makes sense as an empirical question. If it does (I'd have to think about it a lot more to assess whether it makes sense as an empirical question), I don't think it's something we ever could know, and I don't know why it would make a difference either way.

Re lines and points, I'm not of the opinion that it actually makes logical sense to say that a line is comprised of zero-dimensional points.

The dimensionality of points (or lack thereof) is not relevant to the diagram. Think of a marked number line instead. If the marks correspond to all of the integers, then it is obvious that those discrete marks do not comprise the continuous line, because there are rational numbers (i.e., fractions) that can be marked between them. If the marks correspond instead to the rational numbers, then it is still obvious that those discrete marks do not comprise the continuous line, because there are irrational numbers that can be marked between them. If the marks correspond instead to the real numbers, then most mathematicians since Cantor and Dedekind have held that those discrete marks DO comprise the continuous line.

However, Peirce disagreed, calling this only a "pseudo-continuum" because the real numbers cannot be placed in one-to-one correspondence with all of the (potential) marks on a truly continuous line; this is what it means for them to exceed the multitude of the real numbers. He went on to argue that a true continuum is such that there is "room" for any multitude of (potential) individuals between any two (actual) individuals; this is what it means for them to exceed all multitude. He attributed this concept to Kant - that "a continuum is precisely that, every part of which has parts, in the same sense"; i.e., the parts of a continuum are not points, marks, or other individuals, but are themselves continua.

The nominalist must reject the reality of any true continuum, because it cannot be reduced to a collection of discrete individuals - it has no ultimate parts - and the nominalist does not believe that anything is real except discrete individuals. That is why Peirce said, "Thus, the question of nominalism and realism has taken this shape: Are any continua real?" Of course, as he also acknowledged at the beginning of the same lecture, "Of all conceptions Continuity is by far the most difficult for Philosophy to handle."

The nominalist must reject the reality of any true continuum, — aletheist

First, the stuff above this comment--your first two paragraphs--is about mathematical thinking and/or the conventions of mathematical thinking. It's a sort of game we play with abstract thinking about relations. What does that have to do with anything that's not itself mathematical thinking? In other words, what does that have to do with what space and time are like?

At that, by the way, I'm not sure the games being played re infinities really make much sense, especially when we're making claims about one-to-one correspondence, etc. Also, if a "multitude" refers to an infinity(?) of potential numbers in between two "actual numbers," how the heck would we "exceed" that? That seems incoherent to me.

Nominalism doesn't say anything about "ultimate" parts. That's not what it's about. Nominalism isn't about discrete versus continuous--again, assuming that that distinction is anything but a game we're playing with abstract thinking about relations. Re Peirce saying "Thus, the question of nominalism and realism has taken this shape: Are any continua real," I'd say that he doesn't seem to know what he's talking about. I'd need to be convinced that it's not just nonsense. This is related to what I said earlier: "I think folks of [this sort of] cultural stature . . . often get a break simply because of that cultural stature, where I think that a lot of their work should be fit for the garbage bin aside from it being a matter of historical curiosity (and sometimes entertaining because it's so ridiculous). We could do with more iconoclasm and less reverence. Plato, Aristotle, etc. were just guys with ideas and biases etc. like the rest of us."

First, the stuff above this comment--your first two paragraphs--is about mathematical thinking and/or the conventions of mathematical thinking ... What does that have to do with anything that's not itself mathematical thinking? — Terrapin Station

All forms of reasoning depend upon necessary reasoning, and all necessary reasoning is mathematical reasoning, and all mathematical reasoning is diagrammatic reasoning. It tells us all that we can ever know about hypothetical states of things; i.e., whatever is logically possible.

Also, if a "multitude" refers to an infinity(?) of potential numbers in between two "actual numbers," how the heck would we "exceed" that? That seems incoherent to me. — Terrapin Station

It has taken me a while to wrap my head around it, and I may or may not have explained it properly. If you are interested, besides Peirce's own writings about it, you could look into Cantor's theory of cardinal numbers, which Peirce considered to be a somewhat erroneous notion of multitude.

Nominalism isn't about discrete versus continuous ... — Terrapin Station

It is about individual versus general, or particular versus universal - right? Discrete versus continuous is another expression of the same contrast. If only (discrete) individuals are real, then nothing is really continuous (general).

I'd need to be convinced that it's not just nonsense. — Terrapin Station

Sure, and I am not likely ever to convince you - especially since I am still at the stage of trying these ideas out as a working hypothesis, and seeing how far I can take them.

It is about individual versus general, or particular versus universal - right? Discrete versus continuous is another expression of the same contrast. — aletheist

No, that's a different idea. Say that space were continuous (whatever the distinction would amount to empirically--again, I'm not at all convinced that the distinction is coherent in this realm). That in no way implies that continuous space isn't solely particular continuous space. Use the line analogy, where a line is continuous. Well, it's that particular line. Same for a plane, etc.

As I understand it, nominalism holds that only individuals are real; therefore, space and time must consist of discrete locations and instants, respectively, rather than being truly continuous in the sense that I have been trying to describe. Are you suggesting instead that space and time are individuals - or rather, that space-time as a whole is an individual? If so, how would that square with your definition of time as change? I thought that your view was that each instant of time - each discrete change - introduces a new particular.

In any case, the approach to universals that I have been exploring ultimately entails that everything is continuous; there is no such thing as an individual continuum. This goes back to the thesis that a continuum is "that, every part of which has parts, in the same sense."

Wikipedia has a very straightforward explanation of the issue that fuels the distinction:

"In metaphysics, a universal is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For example, suppose there are two chairs in a room, each of which is green . . ."

The issue is not anything about whether particulars and/or universals are discrete or continuous.

Imagine that everything is continuous. Well, under nominalism, then, all particulars are continuous. If there's only one continuous existent, then under nominalism there's just that one particular existent. The only thing that makes a difference for universalism versus nominalism is whether we're saying that there are properties that somehow obtain where they can be identically instantiated, multiple times, in numerically distinct instances. In other words, re green, whether we're saying that green can be identically instantiated multiple times. That's the issue. Nothing about discrete versus continuous is the issue.

I thought that your view was that each instant of time - each discrete change - introduces a new particular. — aletheist

Each change or motion results in the "thing" in question being different/non-identical to what it was. That's a given if we're talking about change or motion--it's change after all. I'm not saying anything about whether the changes or motions are discrete or continuous. That doesn't make a difference. The same thing would go for both discrete and continuous changes/motion.

Imagine that everything is continuous. Well, under nominalism, then, all particulars are continuous. — Terrapin Station

This is incoherent to me. Particulars cannot be continuous; anything that is truly continuous can only be general. In Peirce's words, "Generality, then, is logically the same as continuity."

The only thing that makes a difference for universalism versus nominalism is whether we're saying that there are properties that somehow obtain where they can be identically instantiated, multiple times, in numerically distinct instances. — Terrapin Station

This is not what Wikipedia actually says - not in what you quoted, and not anywhere else in the same article. One of the points that I have been trying to make all along is that a property does not have to be identically instantiated multiple times in order to be a universal; hence the whole notion that a universal is an inexhaustible continuum of potential properties, only some of which are ever actualized in particulars.

In other words, re green, whether we're saying that green can be identically instantiated multiple times. — Terrapin Station

I am saying that the green in one chair is not identical to the green in the other one, no matter how closely the two colors may resemble each other. Nevertheless, they are two different actualizations of the same continuum - the universal, green. Likewise, the two chairs are obviously not identical; but they are two different actualizations of the same continuum - the universal, chair.

Each change or motion results in the "thing" in question being different/non-identical to what it was. — Terrapin Station

I am suggesting that this requires a discrete step, since each change or motion constitutes the actualization of a new individual.

This is not what Wikipedia actually says - not in what you quoted, and not anywhere else in the same article. — aletheist

This frankly suggests to me that you can't understand what you read. I don't know how else to explain it.

This is incoherent to me. Particulars cannot be continuous; anything that is truly continuous can only be general. In Peirce's words, "Generality, then, is logically the same as continuity." — aletheist

The point is that the two don't have anything to do with each other. A universal is a property that can be instantiated in multiple particulars. A particular is what exhibits a(n instantiation of a) property. It has nothing to do with anything being continuous or discrete.

I am saying that the green in one chair is not identical to the green in the other one, no matter how closely the two colors may resemble each other. — aletheist

Then per the conventional terminology and understanding of this issue, you're denying that there are universals. This is the case no matter what you personally call it.

I am suggesting that this requires a discrete step, since each change or motion constitutes the actualization of a new individual. — aletheist

Would you say that there can't be change or motion if time and/or space aren't discrete?

So even a particular form is "all essence".

...

So all form is tolerant of accidents to some degree. And particularity arises from generality by narrowing the definition of the accidental - making it also more particular. Or crisper. — apokrisis

I don't see how this can work, because the essence of the particular is that it is a material object, while the essence of a universal is that it is an abstract generality. You seem to want to remove the matter from the particular, making it purely form, or essence, but this actually denies the true essence of the particular, which is the matter. So even when we allow that a particular has a distinct form, we remove that form from the particular in abstraction, so we do not have the true particular form. That particular form is proper to the object itself, and cannot be abstracted to exist within the mind. This takes it away from the matter which is essential to it. If the desire is to allow that this form is somehow distinct from the matter, we must do this by other principles.

Yet it contradicts dialectical reasoning to not accept that there must be the unintelligible for there to be the intelligible. It can make no sense to claim the one except in the grounding presence of its other. So as soon as you commit to crisp intelligibility, you are committed to its dichotomous other - vague unintelligibility - as a necessity. — apokrisis

I don't see the basis for this claim. You are simply asserting that all things have a dichotomous other or else that thing is unintelligible. But the essence of the particular is that it is other, but not in the sense of a dichotomous other, as opposition, because it is still in some sense the same as the things which it differs from. And this is why its form may partake in universals, by being the same (in some sense) as the things which it is other from. So I think your claim that the only type of "other" which is intelligible is a dichotomous "other", is unjustified, because difference as a type of "other" which is not a dichotomous "other" is in fact intelligible.

So the principle of dialectical reasoning which you assert here will render the particular as unintelligible, and this is contrary to the philosophical mindset. We want all things to be considered intelligible, thereby denying the possibility of the unintelligible. Therefore all things are the same, in the sense that they may be classed as intelligible, but the difference between them, which makes them other, is not itself unintelligible, because as each one is different, they are by this designation of "different" all the same. So the same principle which makes every particular thing intelligible as a particular, also makes the difference between them intelligible, such that there is no such thing as the unintelligible.

But you need vagueness to make its inverse an intelligible possibility. The difficulty is then to represent this in some fundamental metaphysical framework. — apokrisis

That's not the case though. We do not need unintelligibility or vagueness to make intelligibility possible. Vagueness itself is intelligible. "Difference", which is essential to, and inherent within the material particular, and constituting the vagueness of matter, is itself intelligible. It is intelligible because it is itself a sameness. And since it is the most universal of all properties, the inherent difference, which is proper to all material particulars, it is actually the most highly intelligible. So the very thing which appears to us as other than intelligible (therefore unintelligible), because it is the basis for difference rather than the intelligible similarities which produce universals, is really the most intelligible because it is the most universal similarity. This is matter itself, it gives us the appearance of vagueness and unintelligibility, but it is really the most highly intelligible of all because it is consistently the same, as different.

.

This frankly suggests to me that you can't understand what you read. I don't know how else to explain it. — Terrapin Station

And here I thought we were having a pleasant, respectful conversation despite our evident disagreements. I simply pointed out that you added the word "identically."

A particular is what exhibits a(n instantiation of a) property. It has nothing to do with anything being continuous or discrete. — Terrapin Station

It does if a property is itself a real continuum and its instantiations are discrete individuals actualized on it. This is an (admittedly non-standard) attempt at explaining how universals might work.

Would you say that there can't be change or motion if time and/or space aren't discrete? — Terrapin Station

No, just that space and time cannot consist entirely of individual locations and instants.

It does if a property is itself a real continuum — aletheist

What would you say it amounts to (what would you say it "means") for a property to be a real continuum?

I have already explained it as best as I can at this point.

Well, take sphericity for example, Is the idea that it would be continuous as a property the idea that there's not just one thing that counts as the property, but a range of "more or less spherical"? That would be my guess, but I'm just guessing based on common usages of the terms.

That is one aspect; there is a continuum of shapes that are roughly spherical, including the actual shape of the earth. Even for genuine spheres, there is a continuum of potential sizes, since any individual sphere has a particular diameter.

What would make them the same property in that case, though?

The whole point is that a universal is not an individual. People talk about the earth, the moon, basketballs, soccer balls, baseballs, marbles, etc. as "spherical" even though none of these - in fact, no actual thing at all - is perfectly spherical. Likewise, there is no paradigmatic chair or shade of green, either; just a range of things that qualify as chairs, and a range of colors that qualify as green.

So re "universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things," you'd say there are no such things, then, right?

It depends on what exactly we mean by "repeatable or recurrent entities." I would say that there are no individuals that can be instantiated or exemplified by many particular things, but there are continua that can be instantiated or exemplified by many particular things. Each such instantiation or exemplification is a different manifestation of the same continuum.

Do you think that there is any significance to the fact that Peirce preferred the term "general" to "universal" for this notion?