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."

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.

Fair enough, thanks for clarifying.

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?

Some people would see final cause in these instances that you mention, and most of those people would be religious people, and attribute this intention to the Will of God. — Metaphysician Undercover

If you require every final cause to be identical to some intention in some mind, then I agree that this is the only approach that works; but since it effectively presupposes theism, obviously non-theists will reject it out of hand.

Physicalists for example claim that all instances of causation are reducible to efficient causation. This is the basis for determinism. So they don't see final cause here at all. — Metaphysician Undercover

My understanding is that @Terrapin Station is a physicalist, and yet he is arguing for final causes along lines similar to what I have been saying. I think that it is important to be able to talk about final causation without relying on human and/or divine intentions.

I am not as interested in the content of your argument as I am in trying to sort out its logic. What you seem to be saying is that any truth about the world is necessarily a truth that depends on the world. You may be right, but you have yet to offer an argument to demonstrate it, since this is the first time (as far as I can tell) that you have clearly articulated this additional premiss.

In addition, I am not sure that anyone seriously claims that all truth is exclusively mind-dependent in the way that you have sought to establish here. As a couple of us have pointed out, if all minds are world-dependent, and all truth is mind-dependent, then there is no contradiction in acknowledging that all truth is (also) world-dependent.

What we do is assume intention, or purpose, as final cause, without claiming to know the source of that intention. — Metaphysician Undercover

But we do not normally assume intention, or purpose, as final cause in cases where no human is involved. Again, a seed in the ground or a ball at the top of an incline has a final cause, regardless of whether any intelligent agent (i.e., God) wills it to be so.

However, in theology they want to go beyond this, to account for the existence of intention in general, as it appears to be a very unusual (unnatural) form of causation. — Metaphysician Undercover

Intention may seem "unusual" or "unnatural" to some, but final causes are quite common and natural, as Aristotle observed. That is why I do not think it is accurate to equate the two.

Sorry, you lost me again. What is contradictory about "All truth depends on the mind" being true about the world? What happened to your key word "exclusively"?

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

Okay, got it. All truth depends on the mind. No truth depends on the world. Therefore, some mind does not depend on the world.

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.

Again, what exactly do you mean by "exclusively"? How could there be minds that are not world-dependent?

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

You lost me. What two propositions are you claiming to be contradictory?

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.

Please provide the diagram for "All truth is exclusively mind-dependent." In particular, please clarify the particular significance that you are attaching to the word "exclusively." Be sure to include the world in the diagram so that I understand how you are thinking it relates to truth and minds.

No, there is nothing "numerical" about potential points on a truly continuous line. Every part of a true continuum is itself a true continuum. Between any two "adjacent" potential points, there are potential points that exceed all multitude.

Again 'Truth is exclusively dependent upon minds"
Is not logically equivalent to the statement
"Truth is dependent upon mind and upon the world"
One has a logical connective and the other does not. — m-theory

No one is claiming that they are logically equivalent. You seem to think that they are contradictory, but they are not. "All dogs are exclusively mammals" and "All dogs are mammals and animals" are not logically equivalent, and also not contradictory, given the additional premiss, "All mammals are animals." Again, the only way that your two propositions are contradictory is if we add the premiss, "No minds are part of the world."

Hence the diagram is wrong/contradictory with the labels you have applied to the variables used in the diagram. — m-theory

The labels that I applied were G=World, F=Minds, and phi=Truth. All truth is exclusively mind-dependent, and all minds are part of the world; i.e., world-dependent. Therefore, all truth is (also) world-dependent.

Using the term people denotes the set containing both men and women which leads to contradiction. — m-theory

"This product is for people" is vague regarding the quantification of "people." I read it more naturally as implying "This product is for some people," which does not contradict "This product is exclusively for women." As I said, it was only when you changed it to "This product is for all people" that it did create a contradiction.

So the statement "Truth is dependent upon the mind and truth is dependent upon the world" looks like this. — m-theory

The Venn diagram for "truth is dependent upon the mind" (by itself) is identical to your diagram, except that the portion of the circle for "World" that is outside the circle for "Mind" is omitted. Hence the conjunction does not set up a contradiction; both statements can indeed be true. If we say instead (as you did above) that "truth is exclusively dependent upon minds," and then add the premiss (as suggested by ) that "minds are part of the world," we get his diagram (Gs=World, Fs=Mind, phi=Truth). In this case, "truth is dependent upon the world" is necessarily true; more precisely, "truth is exclusively dependent upon part of the world." The only way I can see to make the two statements genuinely contradictory is to say that "truth is exclusively dependent upon minds" and "minds are not part of the world."

There is also no contradiction between "this product is exclusively for women" and "this product is for people." This becomes clearer if we change the second premiss to "this product is for some people." There would only be a contradiction if the second premiss instead was (as you rewrote it above) "this product is for all people."

It seems to me that the whole concept of numerical identity only applies to determinate individuals. Hence we can say that no two actual points on a truly continuous line are numerically identical, but it makes no sense even to ask whether an actual point is numerically identical to any potential (i.e., indeterminate) point on the same line, or whether two "adjacent" potential points are numerically identical. The latter are contiguous, and therefore indistinguishable, yet potentially different; the laws of non-contradiction and excluded middle do not apply unless and until individuals are actualized.

... this is the theological argument which introduces God as the source of telos in natural things. Aristotle did not seek the source of telos, he just affirmed that it was there. — Metaphysician Undercover

We are not discussing "the source of telos in natural things," we are discussing what that telos is itself. While I am a theist, it seems problematic to me to require the existence/reality of God in order for natural things to have final causes. It also seems highly dubious that Aristotle himself would have endorsed such a view.

Aren't the potential points separate from the actual points? — Terrapin Station

No, the potential points are continuous with the actual points. That is why I had such a hard time with both of the options that you presented. However, any two actual points are "separate" from each other in the sense that there are potential points exceeding all multitude between them.

No--I don't think it would make any difference if space and time were discrete rather than continuous. — Terrapin Station

If space and time are truly continuous, then they exceed all multitude of individual locations and instants. But nominalism holds that only individuals exist, and only that which exists is real; hence it entails that if space and time exist and are real, then they must be discrete, consisting of distinct individual locations and instants.

The logical relationship between a general property and its individual instantiations is the same as that between a truly continuous line and the discrete points that can be marked on it. Note that the line does not consist of these points, no matter how many of them are marked; rather, between any two actual points, the continuum includes potential points exceeding all multitude, all of which are also real. Likewise, a general property is not merely the collection of its individual instantiations.

Now - do you agree that nominalism requires space and time (and everything else) to be discrete, rather than continuous?

In theology, the intention (Will) of God is assigned to such cases of final cause. — Metaphysician Undercover

This is a Philosophy Forum, not a theology forum. You are effectively conceding that there are no final causes apart from willing agents, which - as I understand it - was not Aristotle's own position.

The point is to only focus on the logical relationship at the moment. — Terrapin Station

The logical relationship of what, exactly? And why do I have to answer your question before you address mine?

No, I'm offering a reasonable interpretation of "final cause", as intention, exactly as it is described by Aristotle. — Metaphysician Undercover

A seed in the ground or a ball at the top of an incline does not have any intentions, yet each has a final cause - the full-grown plant and coming to rest at the bottom of the incline, respectively.

As long as we accept that properties have ontogeneral status and particulars that exhibit properties have ontogeneral status, there are only two logical possibilities: only properties "in" particulars have ontogeneral status or properties separate from particulars have ontogeneral status, too ... So do you agree that there are only those two possibilities? — Terrapin Station

I honestly have no desire to "haggle," but if "we accept that properties have ontogeneral status," are we adopting realism and rejecting nominalism? If not, I cannot understand what you mean by "ontogeneral status." As I said before, I am suggesting that properties "in" particulars are actualizations of real continua of potential properties.

What about my question - do you agree that nominalism requires space and time (and everything else) to be discrete, rather than continuous?

As long as we accept that there are properties and particulars that exhibit properties, there are only two possibilities: there are only properties "in" particulars or there are also properties separate from particulars. — Terrapin Station

I am suggesting that both of these statements are correct, but "there are" means two different things. Properties only exist in particulars, but properties are real - i.e., they have another mode of being - apart from particulars. Each actual property is an individual, but all potential properties are contiguous parts of a true continuum that exceeds all multitude.

On that note - do you agree that nominalism requires space and time (and everything else) to be discrete, rather than continuous? Otherwise, they would not consist of individual locations and moments.

Thanks, that was indeed a good read.

Realists on universals claim that properties are identical instantiations of a separately existing universal like this: [diagram] It's important to note that realists on universals are saying that the instantiations are identical, that is "one and the same," in a complete sense, of the universal at hand. Literally, they're not two separate things, but the same thing somehow multiply instantiated. — Terrapin Station

Reading Paul Forster's book, Peirce and the Threat of Nominalism - thanks to for bringing it to my attention - is helping me to get a better handle on why this presentation did not sit right with me. It might be an accurate description of Platonism, but - as I have pointed out before - there are other forms of realism that do not claim that universals are "separately existing," even while holding them to be real by a different mode of being.

More to the point that we were discussing, it is not necessary for the realist to claim that the instantiations of a universal are numerically identical. Instead, they are different actualizations of the same inexhaustible continuum of possibilities. The nominalist, on the other hand, insists that what we call universals are sets or collections of discrete individuals. This is why Peirce said that "the question of nominalism and realism has taken this shape: Are any continua real?" It also gets at why I was suggesting that space and time would have to be discrete on the nominalist view - each instance of "here and now" has to be an individual.

This whole line of discussion began because asserted that "the cause is necessarily prior (temporally) to the effect," and then challenged anyone who disagreed to "describe a type of causation which is not like that." I responded by pointing out that final causation "is often subsequent (temporally) to the effect," and that "most efficient causation is really simultaneous with the effect." We have been quibbling ever since over whether the final cause is the desire/goal/intention/purpose to achieve a future state of affairs, or that future state of affairs itself. My whole point was simply that final cause is always about the future. In that sense, the "cause" is temporally subsequent to the "effect"; the end is temporally subsequent to the means.

Would it make sense for any baseball team (even the Cubs) to announce today that its goal, intention, or purpose is to win the 2016 World Series? If not, why not?

If you want to speak of INFINITE possibilities then that is a real possibility. — intrapersona

That is not how it works. Infinite possibilities do not entail that anything and everything is a real possibility. There are infinitely many possible triangles, but none of them have four or more sides.

Peirce would say that there are two tokens of one type. One type is the same as one universal per my earlier comments. — Terrapin Station

Indeed, "the" was an illustrative example that Peirce employed more than once in his writings. Before coming up with type/token, he used legisign/replica for the same basic distinction.

And the issue is whether that type is something real, something that's identically instantiated in the two "tokens." — Terrapin Station

I am not convinced that "identically" is necessary here. It seems like two tokens of the same type could be different actual points on the same continuum of potential points. But I am still reading and thinking about all of this.

That's why universals are best understood as constraints. — apokrisis

Yes, this brings to mind Peirce's use of "determination" in the sense of constraint.

But it is not a problem if we roll the history of the Cosmos back to when the particle was in such a hot and dense world that it simply existed as a fleeting fluctuation in a generic vanilla field - where effectively no identity was yet locked in. — apokrisis

And this is reminiscent of his cosmogony, which begins with a continuum of vague potentiality that eventually actualizes our existing universe.

... I don't know what the heck Peirce is on about really ... — Terrapin Station

I have been reading stuff by and about Peirce for two solid years, and only recently started to feel like I was finally really getting it. I was warned beforehand that this is how it would go, and for me it has been worth the effort, but it has definitely been a slog at times.

Thanks, but that seems like a lot of material to digest. I am looking for a relatively concise and neutral summary of how the various terms are typically defined and distinguished, preferably online.

Which comes first in time, the end or the means? I am not referring to any thought about the end, or the desire for the end, or the decision to adopt a particular end as a goal; I am talking about the end itself.

There's one universal for "spherical." And the whole gist of universals is then that particular substances that are spheres exihibit the universal "spherical." — Terrapin Station

I still hesitate at this description, because I am contemplating the alternative that a universal (or a general) is not one single "item" exhibited by multiple particular things, but rather a continuum. In Peirce's words, "Thus, the question of nominalism and realism has taken this shape: Are any continua real?"

... generality is opposed to particularity, and not singularity ... Singularity (and it's natural 'pairing', universality, which is in turn not generality) cuts across the general-particular dichotomy, such that a general regime may itself be particular. — StreetlightX

See, I understand universal and general on the one hand as being opposed to particular, singular, and individual on the other. That is why I keep asking for clarification of the terminology. Nothing personal, but I cannot just take your word for it; I need some references.