Of Determinacy and Mathematical Infinities

javra

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).

A mathematical infinity (in contrast to metaphysical infinity) is not limited or bounded in only certain respects and is thereby countable (metaphysical infinity is not limited or bounded in all possible respects and is thereby uncountable - and I won’t be addressing this concept). A geometric line, for example, is limitless in length but not in width (technically, it has 0 width, which is a set limit or boundary). An infinite set, as another example, is limitless in terms of how many items of a certain type it contains but is limited or bounded in being a conceptual container of these definite items. And, of course, one can have quantities of both infinite lines and infinite sets.

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.

Mathematical infinity specifies a state of being. This state of being is defined by the lack of limits or boundaries.

That then sets the stage for this question:

-- Can mathematical infinities (e.g., geometric lines, infinite sets, and so forth) be ontically determinate?

If yes, wouldn’t that then imply that mathematical infinites are unchanging, unvarying, fully set, and fully fixed (as they at least to me so seem to be) but, then, further entail that they have all their possible limits or boundaries determined? But then this sounds like a blatant contradiction: a mathematical infinity has all possible limits or boundaries set and, at the same time and in the same respect, does not have all its possible limits or boundaries set (for at least some of its possible limits/boundaries will be unset in so being in some way infinite).

If no, wouldn't that then imply that mathematical infinities are indeterminate, hence ontically vague? Yet this directly contradicts the precise demarcations which define mathematical infinities such as that of a geometric line’s length.

Or, else, are mathematical infinities to be taken as beyond the dichotomy of determinacy/indeterminacy - such that, for one example, in a system of causal determinacy all mathematical infinities are taken to be beyond the causally deterministic system? But, then, this wouldn't be causal determinism by definition.

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Anyone have any idea of where the aforementioned goes wrong? My best current guess is that mathematical infinites can be determinate but that this determinacy sets the limits of that which is not limited … yet this seems so convoluted a notion that to me it currently borders on being nonsensical.

Gregory

↪javra

Maybe mathematical infinities only make sense in relation to the metaphysically infinite

javra

Maybe mathematical infinities only make sense in relation to the metaphysically infinite — Gregory

How so? As background: I've read a little on Cantor's "Absolute Infinity" and find it to be a nice poetic conception. But I don't see any relevant connection between, for example, a geometric line and the attributes which Cantor said of God - aka of Absolute Infinity - e.g., that of being "the supreme perfection".

More to the point: how would this in any way clarify whether mathematical infinities are determinate, indeterminate, or neither?

To be more explicit on this matter, metaphysical infinity, in being perfectly devoid of all possible limits and boundaries (as defined in the OP), would then be perfectly undetermined in all respects. As such, it would then be perfect indeterminate (aka, nondeterminate). Yet the infinite length of a geometric line is definite, and so I take it in at least some meaningful way determinate; but, then, if it is determinate this brings back the issue addressed in the OP of apparent contradiction in regard to limits (contradictions which don't occur for metaphysical infinity on account of it being perfectly indeterminate).

[BTW, to my way of thinking, one can conceptualize metaphysical infinity as perfect being (i.e., God) just as readily as perfect nonbeing (i.e., nothingness ... as in, "why is there something rather than nothing") - this though the two are polar opposites. And neither have been either empirically or rationally evidenced to be to the satisfaction of most. Whereas at least some mathematical infinities - like the irrational number pi (whose decimal expansion is infinite despite the sequence of its decimals being, by all apparent accounts, determinate) - do occur in nature, or at least can influence our reality as though they do. But discussion of metaphysical infinity to me appears to enter a whole different ballpark than what the OP is asking.]

Real Gone Cat

Yet the infinite length of a geometric line is definite, — javra

Can you elaborate? Do you mean that the line is measurable?

I know so little about math, but I'm always eager to learn.

Srap Tasmaner

I know so little about math, but I'm always eager to learn. — Real Gone Cat

Uh huh

jgill

A mathematical infinity (in contrast to metaphysical infinity) is not limited or bounded in only certain respects and is thereby countable — javra

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

javra

Yet the infinite length of a geometric line is definite, — javra

Can you elaborate? Do you mean that the line is measurable?

I know so little about math, but I'm always eager to learn. — Real Gone Cat

I'm in no way a mathematician; not my personal forte. Wanted to be forthright about that. But sure on elaboration of my philosophical reasoning regarding the matter:

As I tried to point out in the OP: a geometric line is defined by an uncurved infinite length of zero width. Its length's expansion in both directions is not limited or bounded, yes. Its length is then of itself immeasurable. But its width and shape is subject to fully set limits or boundaries, thereby endowing the geometric line with a definite uncurved length. Devoid of this definite state of being brought about by fully set limits or boundaries - namely, of having zero width and a straight length - we wouldn't be able to discern it as a geometric line. Hence, as with all other mathematical infinities I currently know of, a geometric line is not perfectly infinite in all respects but only infinite in some respects while being finite in others. Due to its finite aspects, we hold a definite idea of what a geometric line is (it then becomes measurable in this sense; else expressed, it becomes countable).

javra

↪jgill

Mea, in terms of what you've quoted, here's some reference: https://plato.stanford.edu/entries/infinity/#InfiPhilSomeHistRema

OP's question is one of whether mathematical infinity - your field I take it - is determinate, indeterminate, or neither?

Srap Tasmaner

countable — javra

Just don't say that. It has a specific meaning in mathematics, and the length of a line is not countable in that sense.

Doesn't matter to whatever you're saying. Carry on.

javra

and the length of a line is not countable in that sense. — Srap Tasmaner

I specifically said the length is immeasurable. If one can discern the quantity of lines specified, then lines as a whole are indeed countable. Or would you disagree with what I actually said?

jgill

OP's question is one of whether mathematical infinity - your field I take it - is determinate, indeterminate, or neither? — javra

Set theorists and foundations people might be interested in such distinctions, but for me infinity simply means unbounded. Going back millennia to study what the ancients thought is of historical interest, but there has been progress since then. If that's your goal, then don't even mention mathematics. Once you do you are out of your depth. Not criticising, just fact.

javra

Set theorists and foundations people might be interested in such distinctions, but for me infinity simply means unbounded. — jgill

Is a geometric line - a maths concept - to you unbounded in all possible respects?

Besides, again, the issue is one of whether such unbounded things that we discern via definite demarcations are determinate, indeterminate, or neither.

Srap Tasmaner

↪javra

Ah, I see, you meant countable as a unit, as a line. Sure.

javra

Ah, I see, you meant countable as a unit, as a line. — Srap Tasmaner

Right, and its countable as a unit on account of having some limits or boundaries via which it can be so distinguished.

If we're in agreement on this, cool. :smile:

Srap Tasmaner

↪javra

So for your question about the determinateness of mathematical infinities, you would say here that a line is I guess 'determinate enough' that we can pick it out as an object?

jgill

If one can discern the quantity of lines specified, then lines as a whole are indeed countable. Or would you disagree with what I actually said? — javra

So, if I have a countable collection of lines, they are countable? I suppose that's a step in the right direction.

So for your question about the determinateness of mathematical infinities, you would say here that a line is I guess 'determinate enough' that we can pick it out as an object? — Srap Tasmaner

If it's in a countable collection that would seem to be the case. But if a line is the shortest distance between two points, it could depend upon the metric you are using. For example, in the taxicab metric the shortest distance between two points is greater than in the Euclidean metric.

javra

↪Srap Tasmaner

That's where I'm currently stuck. It feels like I'd equivocating between apples and oranges. Yet both determinacy/indeterminacy and finitude/infinitude are defined by the ontic presence or absence of limits/boundaries.

So, a geometric line for example, once its placed on a geometric plane it is - in one sense - fully determinate. Its direction and figure are fully fixed in place. No variance; no vagueness. It is not as though the geometric line is semi-fixed or semi-vague. Yet in a different sense, it is only semi-limited or semi-bounded: being infinite only in length (but not in figure or, as a one-dimensional object, in width).

Yet this doesn't make it semi-determinate in the sense of being only partially fixed, or set.

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BTW, not naval gazing. I'm trying to address three metaphysical possibilities in regard to determinacy - namely, that of being a) completely determinate, b) completely nondeterminate, or the possibility of c) being only partly determined by determinants (i.e., of being semi-determinate). And my stumbling block is that by defining determinacy as I did in the OP (i.e., having limits or boundaries set by one or more determinants), I run into this stubborn paradox of having to differentate semi-determinacy from what I've so far termed "mathematical infinities" ... which are, again, only partly infinite in some respect while yet being finite in others.

I'm assuming this is somewhat dense, but there's the background to the OP and an answer to your question.

javra

So, if I have a countable collection of lines, they are countable? I suppose that's a step in the right direction. — jgill

What makes them countable if they are completely devoid of any boundaries? So might staying on topic be another step in the right direction.

Srap Tasmaner

↪jgill

Is there any property in taxicab geometry analogous to curvature?

Maybe the average distance of the intersections at which you turn from the impossible direct route. Not sure what the point would be, but it's interesting to think of taxicab routes as approximations of the direct routes that are unavailable. (Or vice versa.)

Srap Tasmaner

↪javra

Hmmm. I was hoping you'd say you were okay with this example so we could compare it to another that you feel differently about.

Does it help at all to look at how mathematics handles this? Vaguely similar questions do arise in mathematics.

So, for instance, we say any two points in a plane determine a unique line. But if we go up a dimension, thus allowing that third coordinate to vary without bound, two points are not enough to pick out a single plane, and there are infinitely many planes that contain the line they determine. You need one more point, not on the line, to uniquely determine a plane.

Just an example. Mathematics does sometimes directly address how determinate its objects are, at least in this sort of sense, whether there's a unique solution, finitely many, infinitely many, etc.

Is this sort of determinateness any use to you?

Real Gone Cat

its width and shape is subject to fully set limits or boundaries, thereby endowing the geometric line with a definite uncurved length. — javra

So width is length?

And what is "uncurved" length?

Still eager to learn.

Real Gone Cat

determinacy/indeterminacy and finitude/infinitude are defined by the ontic presence or absence of limits/boundaries — javra

I would like a better definition of determinacy. You seem to be implying that the line is determinate because the line exists in its entirety in the plane. Is this correct?

Banno

Anyone have any idea of where the aforementioned goes wrong? — javra

Well, the problem starts here:

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

...with the supposition that any of this makes sense.

Real Gone Cat

...with the supposition that any of this makes sense. — Banno

Oh, Banno. You're ruining our fun.

javra

Just an example. Mathematics does sometimes directly address how determinate its objects are, at least in this sort of sense, whether there's a unique solution, finitely many, infinitely many, etc.

Is this sort of determinateness any use to you? — Srap Tasmaner

So far I don't find it being of use to alleviate the issue. Thanks for the input, though. What you say addresses determinacy in the sense of "that determined has its limits or boundaries set by one or more determinants". I'm so for robustly in favor of this definition.

Yet the same issue results: if a unique line is so determined by any two points on a plane (in the sense just provided above) how does one then commingle this same stipulation with the fact that that which is being so determined is - at the same time and in some respect - infinite and, hence, does not have some limits or boundaries in any way set by its determinants. Here, the two point on a plane do not set the limits or boundaries of the line's length - despite setting the limits or boundaries of the unique line's figure and orientation on the plane. Again, once the line is so determined by the two points on a plane, it is fully fixed or fully set; hence, fully determined in this sense. But going back to the offered definition above, this would imply that "the line, aka that determined, has it limits or boundaries fully set by one or more determinants" - which it does not on account of being of infinite length.

javra

So width is length? — Real Gone Cat

no

And what is "uncurved" length? — Real Gone Cat

a straight extension in space

I would like a better definition of determinacy. — Real Gone Cat

see my latest post for the definition also mentioned in the OP

You seem to be implying that the line is determinate because the line exists in its entirety in the plane. Is this correct? — Real Gone Cat

no

Oh, Banno. You're ruining our fun. — Real Gone Cat

Let him play! As an self proclaimed anti-philosophy philosopher enamored with Witt, he's into games. :wink:

javra

...with the supposition that any of this makes sense. — Banno

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

Banno

↪Real Gone Cat

Should we go into it in more detail?

that which is determined to be limited or bounded in duration, extension, or some other respect(s) — javra

...so it's anything that is bound, presumably meaning anything with a boundary, an edge. So ask what is excluded here - can you think of something that does not have a boundary? So ontic determinacy includes whatever you want. And off we go.

he's into games — javra

Here we have games without frontiers. So we get things such as

Mathematical infinity specifies a state of being. This state of being is defined by the lack of limits or boundaries. — javra

But of course infinities are bounded - the odd numbers are infinite yet do not include the even numbers, and so on.

So what we have here is the now ubiquitous stringing of words together out of context, the pretence of rational discourse, sitting on a hollow foundation.

javra

Should we go into it in more detail? — Banno

Please do. Answer this question:

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

Real Gone Cat

↪javra

If a line (not a line segment) is ontically determinate, I assume you can draw it in its entirety. No?

I can't. Can you?

javra

If a line (not a line segment) is ontically determinate, I assume you can draw it in its entirety. No?

I can't. Can you? — Real Gone Cat

OK, I get that. Tis why I've started the thread. But then, would you say that it is instead indeterminate? Neither determinate nor indeterminate?

Of Determinacy and Mathematical Infinities

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