It would be helpful if the philosophy of mathematics was ungraded to address the problem of abstract concepts. — Mark Nyquist

I don't know what 'ungraded' means there, but there many many library shelves worth of articles and books in the philosophy of mathematics on the subject of abstractions and concepts vis-a-vis questions of existence and truth.

It seems some hold a magical view of how mathematics is physically done. — Mark Nyquist

Who?

But then, what on earth would “the ‘non-mathematical’ countably of infinity” signify to a general audience?! — javra

Indeed! You are the one who claims to represent a layman's non-mathematical notion. It's a safe bet that no one unfamiliar with set theory or upper division mathematics has any notion of all of a countably infinite set. There is no layman's notion of this. So it's silly trying to represent it. — TonesInDeepFreeze

You again blatantly misunderstand what I was saying.

Curious to see if you might comprehend what I’ve been intending from the commencement of this thread if I were to use the rather pompous term “numeration”:

One can numerate geometric lines and infinite sets. Therefore, these and like infinities are capable of being numerated. As in 2 infinite lines or 2 infinite sets.

In contrast, the infinity of - for one example - a complete nothingness cannot be numerated for, if there were such a thing (linguistic problems in so saying aside), the infinity referenced would have no limits by which to be discerned nor, for that matter, would there occur any sentient being to psychologically delimit or define its presence.

In a similar vein, a “non-mathematical numeration” is a conceptual contradiction, this because to numerate is a mathematical faculty of mind.

Also, “countability” as it is defined in mathematics cannot occur in the complete absence of numeration - and can be viewed as a specialized format of numeration.

(Yes, though, “to numerate” is defined as “to count”.)

And although this thread is not intended to debate the properties of infinities as defined by mathematics,

“mathematical” notion of “countability” — javra


Whatever your questions about it, it would be best to start with knowing exactly what it is.

df. x is countable iff (x is one-to-one with a natural number of x is one-to-one with the set of natural numbers).

As far as I can tell, that is different from the everyday sense, since the everyday sense would be that one can, at least in principle, finish counting all the items, but in the mathematical sense there is no requirement that such a finished count is made. — TonesInDeepFreeze

No, not in my neck of the woods. The everyday sense would be that one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish. This then addresses the notion of a "potential infinity" as first coined by Aristotle - in contrast to the Aristotelian notion of "actual infinity" which Cantor played a major role in making mainstream in part via use of the one-to-one correspondence you address.

In regard to this, from a previous post:

If there are no infinite sets, then there is no set of all the integers nor set of all the reals.

But the observation about them could still hold in the sense of recouching, "If there is a set of all the integers and a set of all the reals, then the cardinality of the former is less than the cardinality of the latter.'

Moreover, in any case, even without having those sets, we can show that there is an algorithm such that for every natural number, that natural number will be listed; but there is no such algorithm for real numbers. — TonesInDeepFreeze

So, when the conceptual grouping (to not irk mathematicians by saying "set") of all natural numbers is taken to be a potential infinity it is still taken to be an infinity - else an infinite grouping - just not one that claims to be complete or else whole. Here, one can contrast the conceptual grouping of all natural numbers with - to keep thing as simple as possible - with the conceptual grouping all natural numbers that are even. There will be a one-to-two correspondence between them: for every one even natural number in the grouping of even natural numbers there will be two natural numbers in the grouping of all natural numbers. When both groupings are taken to be compete wholes, then the grouping of even natural numbers will contain a lesser cardinality than (more precisely, half the cardinality of) the grouping of all natural numbers contains - with both groupings yet being infinite. But when both groupings are taken to be never-complete, then for ever one item added to one grouping there will likewise be one item added to the other, and this without end. Such that one cannot compare the cardinality of infinities in each grouping, other than by affirming that they are both infinite in the same way.

Actual infinities can nowadays be very easily expressed and manipulated - and, so, have become of great mathematical use. But that it makes sense to conceive of any infinity composed of discrete items as "actual" rather than as "potential" (this in Aristotle's usage of these terms within this context - rather than what we understand by these term today) is not something that, for example, is amicable to mathematical proofs. Opinions can differ. This though, yes, the mathematics which Cantor introduced is nowadays mainstream.

Then again

But then you'd do well to leave mathematics out of it if you don't know anything about the mathematics. — TonesInDeepFreeze

I may not be a mathematician but I can take care of my own bank-account via numerations of various sorts just fine, and still have the occasional leisure to philosophically contemplate issues regarding quantities.

I shouldn't point at anyone but you can raise your hand if it's you.

Wikipedia defines information as an abstract concept so that is a good indicator of common usage and mathematics is the same way.

The problem is abstract concepts exist as brain states in individuals and that part is left out of the definitions.

nonquantifiable infinity — javra

Pleroma?

Pleroma? — Srap Tasmaner

Wasn’t familiar with the Pleroma. Don’t yet know how this is intended but, as Carl Jung’s Gnostic understanding, sure, the Pleroma qualifies as nonquantifiable infinity.

Other possible candidates include certain understandings of God, Moksha, Nirvana, the Ein Sof, Brahman, and what some claim to be the ineffable (as in G-d) … with all of these being traditionally understood as being the form which perfect Being takes. Then, again, there’s the concept of nothingness as the absence of all being, which also qualifies as a possible candidate.

Whether or not any of these concepts are anything else but vacuous is irrelevant to the issue. The issue being that such type of infinity can and has been conceptualized by humans at large for a good sum of human history … and that it differs from types of infinity that can be quantified and thereby numerated.

And, again, this thread was not supposed to be about such type of infinity, but about those infinities that can be numerated. As in two infinite lines on a plane can either intersect or be parallel.

Oh, you've been comparing math to woo all along. Seems like a category error to me. Carry on.

and that it differs from types of infinity that can be quantified and thereby numerated — javra

Sure.

So a line. On the one hand, there's a sort of procedure, which is repeatable, by which you can keep extending a line; there may be more than one way to do that — physically different techniques, for example — but they're all equivalent in the long run, because there's the usual asymmetry here: there is exactly one way to extend a line as a line and an infinite number of ways to extend it otherwise, with curves, angles, gaps, and so on. So we have a pretty strict formal constraint. On the other hand, we want to extend it forever, which requires the procedure to be repeated forever, without constraint.

If you think of the possible figures you could draw in a plane, you're constrained to the plane, but otherwise have complete freedom. If you compress and channel that freedom in a particular way, you can get a line: completely constrained in one dimension, but completely unconstrained in the other.

Is this the sort of thing you had in mind?

Oh, you've been comparing math to woo all along. Seems like a category error to me. Carry on. — Real Gone Cat

That stubborn reading comprehension problem again. Have I not termed the type of non-finitude you address as “woo” as “metaphysical” from the very commencement of this thread?

You don’t strike me as the type of person who takes metaphysical enquiries and topics seriously, hence considering them to be woo. But correct me if I'm wrong.

At any rate, glad to see you find readings such as A Universe from Nothing to be “woo” - despite this notion being proposed by a well-established physicist.

If you think of the possible figures you could draw in a plane, you're constrained to the plane, but otherwise have complete freedom. If you compress and channel that freedom in a particular way, you can get a line: completely constrained in one dimension, but completely unconstrained in the other.

Is this the sort of thing you had in mind? — Srap Tasmaner

Yes. Precisely.

Here understanding "constraints" as being determined or else determining factors (again, when it comes to maths, for one example, this not in causal ways - such as, per previous posts, how determinants can be addressed in maths; e.g., two geometric points can determine a geometric line ... this in non-causal manners).

Fyi, since the begining of this thread, I think I've figured the issue out. In the logical trichotomy of metaphysical possibilities regarding determinacy - namely: a) being completely determined, b) being completely nondetermined, and c) being semi-determined - quantifiable infinities will then be categorized by (c). Importantly though, when regarding quantifiable infinities as specified by maths, this semi-determinacy will always be devoid of causal determinacy.

Unless you find reason to disagree with this generalization regarding the determinacy of such infinities, I think I'm good to go.

Unless you find reason to disagree with this generalization regarding the determinacy of such infinities, I think I'm good to go. — javra

There is one other little hitch though: a line, for example, not only can or may contain all the points in a plane colinear with it (that is, with any two of the points on the line), but it must and does.

Do we still call it freedom, absence of constraint, if you must actualize every open possibility?

I've been speaking of a line as embedded in a plane, because it's simpler to visualize that way, and you can contrast a line to the other possible figures in a plane, but a line is, by itself, simply a dimension. It is in one sense a result of constraining a plane, but in another sense a constituent of an infinite number of planes, whether seen as an infinite collection of zero-dimensional points, or — more importantly here, I think — seen as a formal constituent of the plane, as representing one of its dimensions. And here's the kicker: any line can itself be considered a constraint that partially determines a plane, as can any point.

There is one other little hitch though: a line, for example, not only can or may contain all the points in a plane colinear with it (that is, with any two of the points on the line), but it must and does. — Srap Tasmaner

By definition, of course. This will be the determinate aspect of it. But then

Do we still call it freedom, absence of constraint, if you must actualize every open possibility? — Srap Tasmaner

Even in the concept of "actual" or complete or whole infinity, can every open possibility be actualized?

I'm very open to learning otherwise, by what I currently understand by infinite length is that actualizing every open possibility would entail a limit/boundary/end of open possibilities ... thereby negating its affirmed infinitude. Am I misinterpreting something in the terminology?

I've been speaking of a line as embedded in a plane, because it's simpler to visualize that way, and you can contrast a line to the other possible figures in a plane, but a line is, by itself, simply a dimension. It is one sense a result of constraining a plane, but in another sense a constituent of an infinite number of planes, whether seen as an infinite collection of zero-dimensional points, or — more importantly here, I think — seen as a formal constituent of the plane, as representing one of its dimensions. And here's the kicker: any line can itself be considered a constraint that partially determines a plane, as can any point. — Srap Tasmaner

Hm. Not disagreeing.

I've been intending to keep the topic as simple as possible, but I am personally recognizing at least four distinct types of determinacy: including two which could be here termed "top-down" determinacy or "constraint" (e.g., a line's occurrence can be deemed to of itself concurrently determine the placement of all points that constitute the line) and "bottom-up" determinacy or "constraint" (e.g., two points concurrently determine a line) - neither of which are causal. And via this somewhat simple understanding, things can get complex very quickly - especially when taking into account all four determinacy types I'm entertaining (the other two being causal determinacy and teleological determinacy). But maybe this is neither here nor there.

G-d? Brahman? Pleroma? This isn't woo? Merriam-Webster :

woo-woo : dubiously or outlandishly mystical, supernatural, or unscientific

But whether you value such things is beside the point. I stand by my assertion : it's a category error.

I stand by my assertion : it's a category error. — Real Gone Cat

The category implicitly addressed is that of "infinity". Do tell: how is the distinction between metaphysical infinity and quantifiable/mathematical infinity of itself a "category error" of the concept of infinity?

Your assertion is a bit nonsensical at it stands.

Even in the concept of "actual" or complete or whole infinity, can every open possibility be actualized?

I'm very open to learning otherwise, by what I currently understand by infinite length is that actualizing every open possibility would entail a limit/boundary/end of open possibilities ... thereby negating its affirmed infinitude. Am I misinterpreting something in the terminology? — javra

Well there's a formal out, if you want to take it, and then there are new questions.

The formal out is that in modern logic (Frege's logic, which he developed specifically for formalizing mathematics), "every" is of course no sort of number at all. "Every" indicates a conditional: "Every sperm is holy" says "If something is a sperm, then it is holy." This veers somewhat sharply away from the old treatment of universal generality (from Aristotle and medieval logicians, the square of opposition) in that universals are no longer taken to have 'existential import'; in this case, the existence of sperm is not entailed, and the claim is vacuously true if there are no sperm to be holy or otherwise. (Frank Ramsey was even of the opinion that universal generalities were exactly this, habits or rules of inference, nothing more, and not really quantification in the way people think.)

For our case, "Every colinear point is included" says "If a point is colinear with any two points already included, it's also included." Now that doesn't say, "If a point is colinear with any two points already included, add it"; it looks like a rule for adding points, but instead it claims directly that they are all already there. The rule is the line. You don't really construct the line at all, and then know what you have constructed, but by knowing the rule, know the line.

This is how mathematics makes the infinite comprehensible. No human being will ever have the opportunity to observe a one-dimensional line of any length, much less of infinite length; but any human being is capable of understanding the rule that defines such a line.

Of course, one can say, that's not really infinity; or one can say, that really is infinity and thus no one really understands such a rule, they only know how to work with it formally, as a bit of symbolism. (I think I've now alluded to all the principle schools of the philosophy of mathematics: realism, intuitionism, and formalism, for what that's worth.)

Not sure how this fits your thing, but there it is.

Honestly, @apokrisis is the only guy I know around here who's comfortable with this sort of metaphysics, and I learned the habit of looking for constraints from him. He'll mainly tell you that whatever system you're cooking up is a partial reconstruction of his own, but he'll understand what you're up to. You know the drill.

I do think it might be worth thinking a little more about how dimensions work, because they are so explicitly a matter of adding degrees of freedom, each of which is constrained by what was previously an added degree of freedom. That's a curious pattern. There are weirdnesses we're passing by, like fractals and space-filling curves, but gotta walk before you can run.

Hope some of this has been helpful.

It's a category error because you're judging mathematical notions of infinity by some dubious metaphysical standard. One that is vague at best.

You keep coming back to a line as being "constrained" in one dimension but not another. Are you aware that a plane consists of an uncountably infinite set of lines? And 3D space consists of an uncountably infinite set of planes? Now, by your understanding, is 3D space "constrained"?

Finally, it can be shown that the cardinality of the set of points in 3D space is equal to the cardinality of points in a line. I.e., the line can be mapped onto 3D space (and vice versa). So how is the line constrained again?

Before accusing another of nonsense, try picking up a math book.

This is how mathematics makes the infinite comprehensible. No human being will ever have the opportunity to observe a one-dimensional line of any length, much less of infinite length; but any human being is capable of understanding the rule that defines such a line.

Of course, one can say, that's not really infinity; or one can say, that really is infinity and thus no one really understands such a rule, they only know how to work with it formally, as a bit of symbolism. (I think I've now alluded to all the principle schools of the philosophy of mathematics: realism, intuitionism, and formalism, for what that's worth.)

Not sure how this fits your thing, but there it is. — Srap Tasmaner

Right. In general agreement. Thoughts go back to Cantor's popularization of actual infinities.

As I've previously mentioned, I've learned that this issue - that of how determinacy (or constraint) applies to infinities - is so esoteric (such as to most of the posters on this thread) that I need not concern myself with addressing it directly. For what its worth, I've at least gained an understanding - fallible though it is - regarding the issue which the OP addressed - in part, due to the interactions in this thread.

Honestly, apokrisis is the only guy I know around here who's comfortable with this sort of metaphysics, and I learned the habit of looking for constraints from him. He'll mainly tell you that whatever system you're cooking up is a partial reconstruction of his own, but he'll understand what you're up to. You know the drill. — Srap Tasmaner

Actually, it in fact is a partial reconstruction of Aristotelian causes (predating apokrisis and his system by some time). Instead of addressing these causes as "explanations to why questions", I'm addressing them (in short) as distinct determinacy types.

Hope some of this has been helpful. — Srap Tasmaner

It has, and thanks for it.

It's a category error because you're judging mathematical notions of infinity by some dubious metaphysical standard. — Real Gone Cat

In your mind this sure seems to be the case. In reality as written in all of my posts, I have only differentiated between the two - without in any way judging one by the other.

You keep coming back to a line as being "constrained" in one dimension but not another. Are you aware that a plane consists of an uncountably infinite set of lines? And 3D space consists of an uncountably infinite set of planes? Now, by your understanding, is 3D space "constrained"?

Finally, it can be shown that the cardinality of the set of points in 3D space is equal to the cardinality of points in a line. I.e., the line can be mapped onto 3D space (and vice versa). So how is the line constrained again?

Before accusing another of nonsense, try picking up a math book. — Real Gone Cat

Don’t know why but not answering these questions bothers me. Might be your added in snide insult.

Yes: 3D space is by its very demarcation constrained to three dimensions – rather than to two, one, zero (cf. geometric points), or else more than tree dimensions (cf. the ten dimensions of space in string theory).

I grant my non-mathematician mind doesn’t comprehend how the first sentence entails the second, but yes: lines will still be constrained to individual units that can be numerated. Else we wouldn’t be able to discern them as lines.

A point and any geometric extension are completely dissimilar from each other. It is strange that there is no thing in-between them. A point goes from itself into segments that have as many points as as any 3D object. There is something unintuitive about this and seems to resemble something from nothing

A point and any geometric extension are completely dissimilar from each other. It is strange that there is no thing in-between them — Gregory

If you are thinking of a line segment between points A and B, then philosophically there really is nothing there - its merely a hypothetical path in Euclidean spaces of shortest length from A to B. We draw it with a pen, but that always gives us a two dimensional version having width. Modern math of course says otherwise.

Honestly, @apokrisis is the only guy I know around here who's comfortable with this sort of metaphysics, — Srap Tasmaner

I agree. As a biophysicist/philosopher this is the goto guy. Were we talking about foundations/set theory @Tones is the resident expert. Fellow mathematician, @Real Gone Cat for math in general. For modern or theoretical physics I'm not sure who that would be. @Kenosha Kid qualified, but he has left the room. Speak up, anyone.

The history of "infinity" is over two millennia old and progress over that period was done by philosophers/mathematicians. Were Aristotle to rise from the dust today he would tell philosophical devotees to pay attention to what has been achieved and not to refer to his ancient ideas. Or do you think he would eschew progress?

foundations/set theory Tones is the resident expert — jgill

I'm not an expert.

A point goes from itself into segments — Gregory

No it doesn't. There are points and there are segments between points. A point doesn't "go from itself" to something else.

I shouldn't point at anyone but you can raise your hand if it's you. — Mark Nyquist

I don't have "a magical view of how mathematics is physically done", so you can leave me out.

But I thought you might have some particular mathematicians or philosophers in mind. Or do you have a magical view of people existing that don't actually exist?

So, when the conceptual grouping (to not irk mathematicians by saying "set") of all natural numbers is taken to be a potential infinity it is still taken to be an infinity - else an infinite grouping - just not one that claims to be complete or else whole. Here, one can contrast the conceptual grouping of all natural numbers with - to keep thing as simple as possible - with the conceptual grouping all natural numbers that are even. There will be a one-to-two correspondence between them: for every one even natural number in the grouping of even natural numbers there will be two natural numbers in the grouping of all natural numbers. When both groupings are taken to be compete wholes, then the grouping of even natural numbers will contain a lesser cardinality than (more precisely, half the cardinality of) the grouping of all natural numbers contains - with both groupings yet being infinite. But when both groupings are taken to be never-complete, then for ever one item added to one grouping there will likewise be one item added to the other, and this without end. Such that one cannot compare the cardinality of infinities in each grouping, other than by affirming that they are both infinite in the same way. — javra

That is such an inpenetrable mess that it would be a task to unsort it all. But a couple of points:

If we said that there are half as many even numbers as natural numbers, then we can also say there two-thirds as many even numbers as natural numbers, and three-fourths as many even numbers as natural numbers, ad infinitum. For example:

0 2 1 4 6 3 8 10 5 12 14 7 ...

Between every odd number, there are two even numbers.

Moreover, the notion that there are half as many even numbers as there are natural numbers induces that there are infinite subsets ("groupings" or whatever you call them) of the the natural numbers in smaller and smaller size ad infinitum. We would have "half as many even numbers as natural numbers", "one-third as many multiples of three as natural numbers", "one-fourth as many multiples of four as natural numbers", ad infinitum.

Detractors of set theory are put off by the fact that an infinite set has the same cardinality of certain of its subsets. Okay. But then we would ask, "So what is your axiomatic alternative?" But the notion of infinite subsets of the naturals in infinitely descending chains of smaller and smaller cardinality is itself utterly unintuitive. You see, at least set theory does preserve the most basic and most intuitive notion of even everyday mathematical thought: Sets are the same size if there is a one-to-one correspondence: sheep and counting stones.

/

As to your "quantified mathematical infinites", "metaphysical infinities", etc., I would suggest that instead of getting vocabulary all mixed up with mathematics, you could stipulate terminology such as:

q-infinite for "quantified mathematically infinite"

m-infinite for "javra's personal metaphysical notion of infinite"

etc.

And perhaps you'd be so gracious as to provide crisp definitions of each.

That would at least show some respect to the people reading your posts by allowing it to be clear in which of the different contexts you are claiming.

Might be your added in snide insult. — javra

[said to Real Gone Cat]

Because javra is never snide, you see.

But that it makes sense to conceive of any infinity composed of discrete items as "actual" rather than as "potential" (this in Aristotle's usage of these terms within this context - rather than what we understand by these term today) is not something that, for example, is amicable to mathematical proofs. — javra

It doesn't invite mathematical proof because 'actual' and 'potential' are not mathematically defined terms.

Meanwhile, formal set theory does not use the term 'is actually infinite' but instead plain 'is infinite', which is rigorously defined. And set theory is made of rigorous, formal, objectively verifiable, indeed machine-algorithmically checkable mathematical proofs.

On the other hand, the notion of 'is potentially infinite' has not, as far as I've ever found, been given a formal definition, let alone a system in which it used. Instead, it is an informal notion that is thought to be captured by (but not defined in) certain systems. Though, countenancing adoption of such systems raises questions about how much they can prove of mathematics, their own intuitive strengths and weaknesses, and their complexity in formulation and ease or difficulty in using.

What an exciting thread this has been! Never a dull moment as we delve into two thousand year-old mysteries. :chin:

You again blatantly misunderstand what I was saying. — javra

No, I clearly see what you actually posted. In earlier posts, you mentioned that there is a layman's notion of countably infinite. Then later you asked what that could mean. And I replied to the effect that that question is your problem alone, since indeed there is no layman's notion of countably infinite.

Ask any person at a busy street corner what their notion of counting the infinite is. Here are the three possible answers you will get:

"Huh?"

"You can't count infinity. Everybody knows that."

"I'm just trying to catch a cab here. Do you know any good Thai restaurants uptown?"

/

There's a bunch more written by javra that I'd like to address, but I'm out of time now.

countably infinite — TonesInDeepFreeze

What I have said is "countable infinity" ... not "countably infinite".

Sorry, but I have better things to do that to spend more time in addressing such replies.

Oh come on! How captious can a person get?

'countably infinite' and 'countable infinity' are tantamount to each other.

'countable infinity' though is less apt, since there is no object that is infinity. Rather there are different things that are infinite.

And lately you wrote:

“the ‘non-mathematical’ countably of infinity” — javra

'countably of infinity' is not even English.

To move past your ridiculously captious objection that I said 'countably infinite' (a coherent notion, used as a favor to you) rather than 'countable infinity' (an unclear notion since there is no object that has the name 'infinity'), instead I'll couch using only 'countable infinity':

You said there is a layman's (or whatever synonym of 'layman' you used) notion of countable infinity. Then you asked what "“the ‘non-mathematical’ countably of infinity” would mean to the general audience (in other words, presumably, a layman audience).

And my point stands, there is no layman's notion of 'countably of infinity' (let alone that it's not English), nor layman's notion of 'countable infinity', nor (put better) layman's notion of 'countably infinite'.

/

Sorry, but I have better things to do that to spend more time in addressing such replies. — javra

So your excuse for not dealing with the point that there is no layman's notion of a countable infinity (better put, of countably infinite) is that I used such a slight variation in phrasing.

Meanwhile, there's a trail of other falsehoods, blatant misconceptions and nonsense you've posted, and I explained your errors. As well as your posts in this thread are an impenetrable morass of your ersatz undefined terminology, with various ersatz undefined qualifiers popping in and out and out and in again. Any fault for you not being understood is yours, not your readers.