OK Banno. What would you say "to be determined" is? This in the ontological sense rather than the psychological. — javra

It has a bunch of uses, which we might set out one by one, but which change and evolve over time - like all such words. Trying to capture it with something like

Determinacy specifies a process: namely a process via which states of being are obtained. These obtained states of being are defined by the limits or boundaries set by determinants or, in the case of indeterminacy, the absence of these. — javra

...should fall flat; but unfortunately it just leads to long threads that say precious little.

In proposing and taking on the OP there's a wilful rejection of clear thinking in favour of confusion that is the antithesis of mathematics.

Oh boy, here we go again . . . :roll: — jgill

This is the right response.

It has a bunch of uses, which we might set out one by one, but which change and evolve over time - like all such words. — Banno

Aright. What use do you take it to presently hold in the notion of causal determinism in particular? If you find that it holds different uses in this context, I'm more than happy to listen.

Is there any property in taxicab geometry analogous to curvature? — Srap Tasmaner

Taxicab Fun

What makes them countable if they are completely devoid of any boundaries? — javra

I don't know what you're talking about. Provide an illustration, please. Show me how you count them, 1,2,3,4,5,...

What use do you take it to presently hold in the notion of causal determinism in particular? — javra

I don't have an opinion; that's rather my point. If you think there is such a use, then it is over to you to provide a coherent account of how and why.

I don't know what you're talking about. — jgill

Nor I.

I don't know what you're talking about. — jgill

Nor I. — Banno

and ...

I don't have an opinion [on what determinacy is]. — Banno

I can only interpret this as implying that to you causal determinism is meaningless or nonsensical, as is its notion of determinacy.

But you're still butting in as the measure of all that can be understood.

OK, then.

If we admit that a line is not ontically determinate, then I suppose it's ontically indeterminate. I think your problem lies in equating "indeterminate" with "vague". One may draw part of the line in a specific location, just not the line in its entirety. Is this what you're looking for?

Is this what you're looking for? — Real Gone Cat

Thanks for the offer.

Unfortunately, not to my satisfaction as expressed, no.

Can not two points in a plane (with the plane itself determined by a multitude of points) determine a unique line, this as offered? In which case, the line here then has determinants and is thereby not indeterminate (i.e., undetermined). An indeterminate line so far makes little sense to me, as it would not be determined by determinants (here, namely, by points).

... as it is, been sitting on my own ass a little too long today. Going to take a break.

I can only interpret this as implying that to you causal determinism is meaningless or nonsensical, — javra

Well, see this. So that's not quite right, and certainly does not follow from what I said.

But you're still butting in as the measure of all that can be understood. — javra

It's what I do. You asked where you went wrong. The answer is that your OP needs substantial clarification.

Once you have a line, whether any other point in the plane is on it or not can be determined; it becomes an absolute yes/no question. Within a plane, every point is either on the line, above it or below, so the line perfectly bifurcates the plane. (Not for nothing, but given a line and a point, you figure out its relation to the line using a mathematical construction called a 'determinant'.)

There's also a sense in which a line, like any other function, gives a perfectly clear answer to how a segment of it can be extended: go on exactly like this.

It's altogether very well-behaved, and as sharply defined as, say, a triangle or some other sort of figure.

With regard to a segment, are you saying this segment is determinate in that it is finite and indeterminate in that it has infinite points in it?

Can not two points in a plane (with the plane itself determined by a multitude of points) determine a unique line, this as ↪Srap Tasmaner offered? In which case, the line here then has determinants and is thereby not indeterminate (i.e., undetermined) — javra

By Jove, you're getting there! Two points do indeed determine a unique line segment joining those points, But there are lots of line segments including and extending beyond this initial segment, aren't there? The big Kahuna here is extending this segment infinitely in both directions.

agreed
this is what I'm questioning.

In attempts to simplify the reason for this thread:

We have two well-established concepts: that of determinacy (such as can be found in the notion of causal determinism), on one hand, and that of “non-metaphysical" (aka, countable, mathematical) infinity (such as can be found in a geometric line of infinite length), on the other. On their own, both concepts are cogent (to most folks, at least). However, when attempting to define infinity (which describes a certain state of affairs) via determinacy (which describes how a certain state of affairs comes to be), inconsistencies emerge.

(Non-metaphysical) Infinity can thus either be:

a) determined, hence determinate
b) undetermined, hence indeterminate
c) neither (a) nor (b)

If determinate, then you run into problems such as given by

If indeterminate, then this directly contradicts the fact that, for example, a geometric line can be determined by geometric points … as well as having properties specified by once so determined

---------

Seeing how I’m having a hard time in even getting people to understand the problem, my only current conclusion regarding this problem is that it’s so dense that I needn’t concern myself with it when specifying metaphysical possibilities of determinacy.

Thanks for the input.

... and that of “non-metaphysical" (aka, countable, mathematical) infinity (such as can be found in a geometric line of infinite length), — javra

Um, the points of a line may be put into one-to-one correspondence with the set of real numbers, which Cantor proved to be uncountably infinite in 1874. In fact, the points in a tiny line segment are uncountable.

I'm unsure why you're hung up on causal determinism. Do you think two points in the plane cause a line to be? I.e., the line was not there before? How else is countable infinity determinate? Because the act of counting gives us the set in its entirety? (OK, try it - count to infinity. We'll wait.)

Seeing how I’m having a hard time in even getting people to understand the problem, my only current conclusion regarding this problem is that it’s so dense that I needn’t concern myself with it when specifying metaphysical possibilities of determinacy. — javra

You seem genuinely interested in the topic. Depending on your math background, you could try to find a source that discusses the concept of infinity in math that you can use to begin to understand it. You could google texts on Set Theory for beginners, or find nice presentations on Youtube (this might be a good starting point).

Um, the points of a line may be put into one-to-one correspondence with the set of real numbers, which Cantor proved to be uncountably infinite in 1874. In fact, the points in a tiny line segment are uncountable. — Real Gone Cat

You seem to be asking me to explain a commonly established attribute. If you’d bother to check the link to “infinity” I posted you’d find the following:

1. (uncountable) endlessness, unlimitedness, absence of a beginning, end or limits to size.
2. (countable, mathematics) A number that has an infinite numerical value that cannot be counted. — https://en.wiktionary.org/wiki/infinity

The definitions can of course be questioned, but they are commonly established, at the very least as best approximations of, as Banno would say, the term’s usage.

I’ll again try to explain. A metaphysical infinity has absolutely no limits or boundaries. Due to this, it cannot be discerned as a unit: it is immeasurable in all senses and respects and hence, when ontically addressed (rather than addressed in terms of being a concept) it is nonquantifiable. As a thought experiment, try to imagine two ontically occurring metaphysical infinities side by side; since neither holds any delimitations (be these spatial, temporal, or any other) how would you either empirically or rationally discern one from the other so as to establish that there are two metaphysical infinities? In wordplay games, we can of course state, “two metaphysical infinities side by side” but the statement is nonsensical. More concretely, ontic nothingness, i.e. indefinite nonoccurrence - were it to occur (but see the paradox in this very affirmation: the occurrence of nonoccurrence, else the being (is-ness) of nonbeing) - is one possible to conceive example of metaphysical infinity. Can one have 1, 2, 3, etc., ontic nothingnesses in any conceivable relation to each other? (My answer will be “no” for the reasons just provided regarding metaphysical infinity. However, if you believe this possible, please explain on what empirical or rational grounds.)

With that distinction hopefully out of the way, you can then have limitlessness or unboundedness that applies to a certain aspect of what nevertheless remains a unit. That which is limitless or unbounded about the unit cannot be measured of counted to completion - this as I've previously mentioned. The unit itself - which is a unit only because there are limits or boundaries which so delimit it - can however be counted. A geometric line does not have limiteless or unbounded width; its width holds a set limit or boundary, namely that of zero width. Because of this, one can quantify and thereby count geometric lines on a plane as individual units.

It bears note that I’m not arguing for a novel concept here. As I pointed out to jgill, these are established notions: you have dictionary definitions such as those provided by Wiktionary and SEP entries on infinity in reference to this.

I'm unsure why you're hung up on causal determinism. — Real Gone Cat

It was given as one possible concrete example of ontic determinacy, primarily on account of all-knowing people such as Banno not getting the context of the usage of the term "determinacy". But no, causal determinism does not hold the only conceivable type of determincay: there can be already established notions of non-causal determinacies, this as @Srap Tasmaner illustrated in this post.

You seem genuinely interested in the topic. — Real Gone Cat

Yea, I am. And as opposed to what? (a rhetorical question)

Ontic determinacy, or the condition of being ontically determined, specifies that which is determined to be limited or bounded in duration, extension, or some other respect(s) - this by some determining factor(s), i.e. by some determinant(s). — javra

Every object is bounded in its identity, that is, it has a boundary that differentiates the object from what it is not. Does "ontically determinate" mean having such a boundary? Then it doesn't seem important whether the object is in some way infinite.

The definitions can of course be questioned, but they are commonly established — javra

I warned you this would be trouble.

The usual way of using these words in mathematics is pretty straightforward. 'Countable' means there is a one-to-one correspondence between the set you have and a subset of the natural numbers, maybe all of them. So either finite, or 'countably infinite' like the natural numbers. We're talking about sets where you can write down the members in a list, even if that list goes on forever. 'Uncountable' is for bigger infinite sets. The real numbers, to start with, cannot be written down in a list that goes on forever, no matter how clever you are.

Obviously countable is nicer to deal with, because you can use algorithms that iterate (or recurse) their way through a list and you know that will get you not to the end but as far as you'd like to go.

(Also: Zeus could write out all the natural numbers in a finite amount of time just by doing the next one faster each step; not even Zeus could write out the real numbers in a finite amount of time. Lists are friendlier, even when they don't terminate.)

The definitions can of course be questioned — javra

I'll say. Go deeper: Countable

The unit itself - which is a unit only because there are limits or boundaries which so delimit it - can however be counted. A geometric line does not have limiteless or unbounded width; its width holds a set limit or boundary, namely that of zero width. Because of this, one can quantify and thereby count geometric lines on a plane as individual units. — javra

An infinite line is a line, therefore, I suppose, a "unit". But they can't be counted since the points in the Euclidean plane cannot be counted and so pairs of these points - defining lines - cannot be counted.

I too wonder how a continuum makes up something discrete

Every object is bounded in its identity, that is, it has a boundary that differentiates the object from what it is not. Does "ontically determinate" mean having such a boundary? Then it doesn't seem important whether the object is in some way infinite. — litewave

If I understand you right, yes, every individual cognition as identity is delimited from other cognitions and hence bounded. Yes, and this holds true for the concept of metaphysical infinity as well - in direct contrast with that supposedly ontic occurrence that the concept of metaphysical infinity specifies.

Ontically occurring metaphysical infinity is devoid of any ontic identity for it has no boundaries via which such an ontic identity can be established. Nothingness, for one conceivable example of such, can be identified by us on grounds of being different from somethingness, so to speak. There thereby is a conceptual boundary between nothingness and somethingness via which nothingness can be identified. But on its own, where this to be possible, nothingness would hold no ontic identity - for an identity would be something.

As to your conclusion, thanks for offering. I’ll think about it some.

I warned you this would be trouble. — Srap Tasmaner

There is such a thing as equivocation between two or more meanings or usages of a term, right? I repeatedly described countability in its non-mathematical sense of “able to be counted; having a quantity”. As does the Wiktionary definitions previously posted.

(Also: Zeus could write out all the natural numbers in a finite amount of time just by doing the next one faster each step; not even Zeus could write out the real numbers in a finite amount of time. Lists are friendlier, even when they don't terminate.) — Srap Tasmaner

Are the infinities of natural numbers and of real numbers two different infinities? Or are they the same nonquantifiable infinity?

An infinite line is a line, therefore, I suppose, a "unit". But they can't be counted since the points in the Euclidean plane cannot be counted and so pairs of these points - defining lines - cannot be counted. — jgill

In other words, “countable” can only hold the valid usage in its mathematical senses when addressing things such as lines. Therefore, the concept of there being “2 lines” is … invalid and nonsensical. This as all mathematicians know, in contrast to the stupidity of common folk.

And I must take my own head out of my own tunnel-visioned ass in order to realize this.

Got it.

By no means in agreement, but I got it.

I too wonder how a continuum makes up something discrete — Gregory

Yea. That appears to roughly sum up the issue.

A metaphysical infinity has absolutely no limits or boundaries. Due to this, it cannot be discerned as a unit: it is immeasurable in all senses and respects and hence, when ontically addressed (rather than addressed in terms of being a concept) it is nonquantifiable. As a thought experiment, try to imagine two ontically occurring metaphysical infinities side by side; since neither holds any delimitations (be these spatial, temporal, or any other) how would you either empirically or rationally discern one from the other so as to establish that there are two metaphysical infinities? In wordplay games, we can of course state, “two metaphysical infinities side by side” but the statement is nonsensical. More concretely, ontic nothingness, i.e. indefinite nonoccurrence - were it to occur (but see the paradox in this very affirmation: the occurrence of nonoccurrence, else the being (is-ness) of nonbeing) - is one possible to conceive example of metaphysical infinity. Can one have 1, 2, 3, etc., ontic nothingnesses in any conceivable relation to each other? (My answer will be “no” for the reasons just provided regarding metaphysical infinity. However, if you believe this possible, please explain on what empirical or rational grounds.) — javra

I don't wish to be mean, but this strikes me as complete word salad. The Wayans brothers (In Living Color) used to do a skit where two self-educated street preachers have a nonsensical conversation by stringing together unrelated words. Reminds me of that.

The bolded sentence ends in a question mark, so I assume its a question. And it seems to represent the crux of your argument. Can somebody translate?

In other words, “countable” can only hold the valid usage in its mathematical senses when addressing things such as lines. Therefore, the concept of there being “2 lines” is … invalid and nonsensical. — javra

You have reached an absurd conclusion. Of course there can be "two lines". Any finite collection of lines is clearly countable. And there are countable infinite collections of lines such as all lines parallel to the x-axis that pass through y= 1, 2, 3, ....

What are not countable are all lines in the plane.

I too wonder how a continuum makes up something discrete — Gregory

As the seconds tick by in the continuous flow of time we have minutes and hours which are "discrete".

I was just about to post this very same response.

Each line consists of an uncountable infinity of points, each plane consists of an uncountable infinity of lines, and 3D space consists of an uncountable infinity of planes. But two points are always discrete, two lines are always discrete, and two planes are always discrete (except where they may intersect).

Ontically occurring metaphysical infinity is devoid of any ontic identity for it has no boundaries via which such an ontic identity can be established. — javra

So an infinite line has no ontic identity?

So an infinite line has no ontic identity? — litewave

An infinite line is not metaphysical infinity. An infinite line is infinite only in length, not in width. Whereas metaphysical infinity would be infinite in length, in width, and in all other possible manners.

There is such a thing as equivocation between two or more meanings or usages of a term, right? I repeatedly described countability in its non-mathematical sense of “able to be counted — javra

Except (a) you want specifically to talk about mathematical infinities, and there's prior art there you might as well become familiar with; and (b) the mathematical usage of 'countable' is actually something a lot like 'able to be counted', because listable.

I think what's throwing the discussion off is that we don't normally talk about the cardinality of a line except when we're considering it as a collection of points, the continuum, which is not countable. But that's not really measuring its length, different deal. If you have an infinite ruler marked off in centimeters, you'll be counting again.

Are the infinities of natural numbers and of real numbers two different infinities? — javra

Yes. The cardinality of the set of natural numbers is aleph-0; the cardinality of the set of real numbers is aleph-1, aleph-0 raised to the aleph-0 power. It is not known whether there is a size in between, but I think most mathematicians think not. Could be wrong.

The unit itself - which is a unit only because there are limits or boundaries which so delimit it - can however be counted. A geometric line does not have limiteless or unbounded width; its width holds a set limit or boundary, namely that of zero width. Because of this, one can quantify and thereby count geometric lines on a plane as individual units. — javra

An infinite line is a line, therefore, I suppose, a "unit". But they can't be counted since the points in the Euclidean plane cannot be counted and so pairs of these points - defining lines - cannot be counted. — jgill

In other words, “countable” can only hold the valid usage in its mathematical senses when addressing things such as lines. Therefore, the concept of there being “2 lines” is … invalid and nonsensical. — javra

You have reached an absurd conclusion. Of course there can be "two lines". Any finite collection of lines is clearly countable. And there are countable infinite collections of lines such as all lines parallel to the x-axis that pass through y= 1, 2, 3, ....

What are not countable are all lines in the plane. — jgill

All emphasis mine. Um. Okay. I certainly don't understand what your stance is on whether or not infinite lines are countable. But I'm glad others like Real Gone Cat can make sense of your writing.

Do we start with the discrete and then divide it, or are the divisible parts already there?? Continuity and discretenes seem to assume each other. Trying to put one first leads to infinite regress