Again, it's not my sense of what it means, [the description of "rationality" which T Clark previously posted is] what it actually does mean. — T Clark

For the record: Best I could do in finding a good reference for what “rationality” means is this. The SEP section of this article lists seven possible meanings of rationality (without claiming them to be exhaustive), none of which appear to me to coincide with what you claim the official, or formal, meaning of “rationality” is. Particularly, your claim that it be "a systematic search for knowledge and understanding following a formal system such as logic, the rules of which are specified in advance".

If you could provide some reference for the meaning of rationality as you've specified it, I could then learn something new and be more in the know, so to speak.

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ps. If interested, a different SEP article on rationality that touches on the topic of this thread. From the first paragraph of the article:

"This thesis appears to threaten the “rational authority” of morality. It seems possible that acting morally on some occasion might not be a suitable means to an agent’s ends. If so, then according to this thesis, it would not be irrational for her to refuse to act morally on such an occasion."

[...] we never did define what "rational" means back at the beginning. For me, it means a systematic search for knowledge and understanding following a formal system such as logic, the rules of which are specified in advance. — T Clark

To be forthright: First off, as a matter of opinion, we disagree on what the term rational ought to refer to. I for one believe it should be roughly described as “the ability to discern and apply reasons (like causes and motives) and comparisons (with ratios as one example among humans) for the sake of optimally fulfilling goals, be these needs (like physical sustenance so as to maintain physical health) or desires (with improved eudemonia as one example sometimes spoken of by philosophers)”. But, I grant, my definition does not need to be strictly applicable to only humans, and I get that many don’t want to ascribe rationality to any lesser life form. Be this as it may.

An observation based on the quoted definition of “rational”:

So called primitive people that lack rationality as just defined (and which have not been intruded nor in any other significant way influenced by westerners: certain people in the Amazonian forests and Inuit people as two examples) have lived in mutual benefit with their natural environment for as long as they’ve been known to be, resulting in the preservation of a healthy ecology in which they subsist.

We westerners, who as a grouping arguably represent the apex of this rationality as defined, are deteriorating our natural environment to the point of causing the sixth mass extinction, environmental collapse, and our own demise as a people - and this, for the most part, without giving a hoot.

I know there’s bound to be (this from at least some person somewhere), but in assuming no indignantly emotive attempts to rationalize these two just stated facts:

If the cultures in which (your sense of) rationality prevails happen to callously and obliviously bring about the steady obliteration of the inhabitable planet - and, via rational inference, of themselves as a peoples in the process - while those cultures devoid of rationality (as you've defined it) do no such thing, what’s one to make of rationality’s value?

I don’t know, your present definition leaves me with a topsy-turvy feel in this context.

You know what they say: careful what you wish for. — Mww

No regrets so far. Thanks for the reply.

I also view both empiricism and rationalism equally essential for empirical knowledge, or knowledge of the empirical content of our cognitions. But I think we have just as much capacity for pure rational thought in the form of logical relations, which have no empirical content. But, if I want to prove that logical relation, I must subject it to empirical conditions, let Mother Nature be the judge. — Mww

For what its worth, here we differ a little. What you term "pure rational thought" I would understand as (very) abstract thought ... which, as abstraction, is abstracted by us from experience (of the world, of our thoughts' workings, and so forth). As one example, our modern knowledge of formal logic(s) is, to my mind, then governed by a long history of axiomatic stances which more or less correlate with our experience and which, for the most part, have been improved with time; axioms that would themselves not be conceivable in the hypothetical absence of, again, what Hume terms "impressions". Nevertheless, I concur (it at least so far seems) with the idea that at least the most basic aspects of logic of which our reasoning makes use of are not empirically - nor for that matter evolutionarily - developed in us. Instead being, for lack of better phrasing, existentially fixed aspects of the world; existentially fixed aspects we have biologically evolved to make much better use of, via our far more abstract understanding, then any other species of living being known to us.

I know. Lots to potentially disagree with in this point of view. But I'll leave this in even though its not paramount to the discussion. Thanks again for the previous post.

On the right-hand side, we want to take the in-world perspective, and leverage that to define a term in our world, on the left-hand side. In Middle Earth, we want to say, Frodo is a person; in our world, he's a fictional character. Is this the same 'entity' we're talking about? Has it a dual existence, in one 'world' as one sort of thing and in ours as another? Is this no different from saying that chocolate can exist as something yummy for one person and something repulsive for another? — Srap Tasmaner

In thinking this might help, perspective might be the crucial link. “Real” from whose perspective?

Fiction as a genre of story (fantasy, sci-fi, or any other) will intend to successfully present fictional realities, wherein fictional sentient beings are real to themselves and to those other fictional sentient beings with which they interact (as will be their activities and behaviors). We understand that these sentient beings are real from their own perspective, as is the world they communally inhabit - but that all these are fictional from our own perspective, in which we presumably know in advance that these are characters which pertain to fictional realities as presented by real sentient beings.

This play on perspective can then make use of fictions within fictions, such as can be found in “The Neverending Story”. Here, the fictional character who is real from his own perspective immerses himself in a story that is fictional from his own perspective. Complex as this sounds, it is readily understood by the readership of the book at large – which has no problem in empathizing with the fictional character who, in the reality that is real relative to his fictional being, reads what is to him a fictional story.

I in part say this with the understanding that in our modern lexicon “real” and “actual” are taken to be synonyms.

I also say this as one who maintains that our individual first-person point of view is the central reality, or actuality, from which all other realities, or actualities, become discerned by us. Very much including that reality which we take to be equally applicable to all other sentient beings whose first person point of view is as real, actual, as is our own. The latter then being what we term and conceptualize as non-qualified reality proper.

the very faculty of reason is again ascribed to natural impulses, instincts; such that it is as inescapable (and I’ll add, a-rational) as is the natural impulse to breath: A toddler does not reason that one breaths in order to live and thereby breaths; nor does it reason that it is using its faculties of reason to develop its reasoning skills in order to better live; yet it inevitably engages in both activities a-rationally - this, the argument would then go, just as much as we adult humans do. — javra

Overall, a well-thought post. Nothing in it to counter-argue conclusively. That being said, it might be worthwhile to consider the different between reason the faculty, which the infant hasn’t developed, and reason the innate human condition, by which development of the faculty is possible. — Mww

As to your comments on my post, thanks. It can happen now and then. :smile:

Words can be ambiguous. So as to clarify what I had intended: by “faculty of reason” I intended “ability or capacity to reason” rather than “reasoning skills” … equating the former to what you’ve termed “reason the innate human condition”. It then was this “capacity to reason / reason the innate human condition” which was claimed to be a “natural impulse or instinct” in my last post. If its warranted, my bad for lack of clarity in the expression.

But to address an overarching theme in Hume the empiricist that was previously addressed: Take the nonrealistic hypothetical of a human who is completely deprived of all present and past “impressions” as Hume terms them; be these what we moderners term perceptions, memories, the experience of physical pain, or anything other which could quality. I for instance disagree with Hume’s definition of ideas as “faint images” of impression – instead understanding ideas to be concepts and, thereby, abstractions which are a) abstracted from “impressions” and b) are of themselves perfectly devoid of imagery in so being concepts/abstractions. E.g., the idea/concept of animal does not have a “faint image” – and to ascribe an image to this concept (e.g., the image of a cat) is to at the same time exclude a plethora of other possible images that the concept encapsulates (dogs, whales, insects, etc). Neither does the concept of cat, for – for one example – to see the “faint image” of a white cat is to exclude all the different colors which cats can take. Yet, be this as it may, a question for the non-empiricist:

In the absence of all present and past impressions, what reasoning might such a hypothetical human yet engage in? And this via what content?

More concretely, in Kantian terms, to paraphrase, we innately endow our perceptions with time and space. Yet, in the complete absence of all present and past perceptions, is it to be assumed that we’d yet hold the ideas of time as space as contents to reasoning?

(BTW, so it’s said, I personally neither agree with empiricists nor rationalists, instead viewing both experience and reasoning as essential to epistemological content. But I’m here addressing the issue in what I take to be Hume’s favor: where it's argued that reasoning is brought about by impressions - such that there can be no reasoning in the complete absence of impressions and of that which is derived from impressions.)

I’m currently more interested in your point of view regarding these questions than to engage in debate.

Holy Crap, Batman!!! We cannot grant the existence of bodies to sensations, where it belongs as a seemingly “first appearance”, because impressions are not reasonings, but the existence of bodies is granted to ideas, because it is reasoning, but impressions cause those ideas, so….sensation of an object cannot be so low as to be the same as its idea, impression of an object causes our reasoning to an idea of that object……the very reasoning of which we have already been shown we should be skeptical of.

We’ve been granted the very thing we’ve no warrant to trust. The skeptic cannot defend his reason by reason, so how does he defend it, or does he not bother defending the very thing by which he acquires his ideas? — Mww

Addressing your question with the presumption it addresses (non-Cartesian) skeptics in general:

Unless the skeptic is Pyrrhonian - whom I so far gather would claim to suspend all reasoning (though I am very dubious of this being actualizable in practice) - I take the skeptic to not be capable of finding any rational alternative to so trusting. And, due to this reason alone, the skeptic thereby trusts. Despite the mistakes we can on occasion make in our reasoning.

In Hume’s case, the very faculty of reason is again ascribed to natural impulses, instincts; such that it is as inescapable (and I’ll add, a-rational) as is the natural impulse to breath: A toddler does not reason that one breaths in order to live and thereby breaths; nor does it reason that it is using its faculties of reason to develop its reasoning skills in order to better live; yet it inevitably engages in both activities a-rationally - this, the argument would then go, just as much as we adult humans do.

But this issue isn’t one confined to the particular worldview(s) of skeptics. The provision of a reason for the trustworthiness of reason squarely lands one into Agrippa’s trilemma: circularity or reasons (a is so because a; as in: reason's trustworthiness is so because x, y, z, etc ... all of which are to be deemed valid because reason is trustworthy), ad infinitum regression or reasons (which never provides a foundational reason), or axiomatic dogma (which would here translate into “it is so because I/you/they so state”). None of which are deemed rationally satisfactory by most. And, despite this irking a good deal of rationalists among others, no human in the history of mankind has been able to envision any alternative than the three just provided.

But one can abductively infer that reason of itself is a natural impulse in us … whose trustability as impulse can neither be rationally supported not rationally renounced.

In reference to the first quoted paragraph of yours, I’m not claiming to not find problems in Hume’s arguments. But I so far do agree with Hume’s general perspectives on this point, as I so far best interpret them, and as they would likely stand in relation to your question regarding trust: our trust of reason as a faculty can of itself only be instinctive and in this means unavoidable. And to this I’ll add foundationally a-rational (i.e., neither rational nor irrational).

While I can’t support all of the just stated by Hume’s writings, nothing in Hume’s writings regarding reason being an instinct will to my mind contradict this affirmed stance being one that a skeptic can take. To re-quote this, one such writing is from Part III Section XVI of the Treatise:

“To consider the matter aright, reason is nothing but a wonderful and unintelligible instinct in our souls,”

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As an afterthought: One cannot rationally doubt the faculty of reasoning without trust in the very faculty of reasoning one claims to doubt. Which to me only further evidences the claim I've intended to make.

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.

Free speech can be "self-destructive" if you will, because it allows speech against itself, and, more directly, might incite harm or violence — then it becomes real.
As of typing, there are places where such freedom is stomped out to a wretched degree, and other places where it's abused ("weaponized" dis/mal/misinformation, whatever).
Middle grounds? — jorndoe

Words can harm. Intentionally shout “fire” in a crowded gathering so that a stampede results due to the lie, prank, or whatever it might be, such that the stampede leads to people getting trampled on and even dying … and that word was the cause of perceptible, physical harm. (This leaving the emotional harm which words can cause out of the issue. Sometimes serious enough to add to increasing suicide rate, imo) Arguably, in at least some sense regarding intent and outcome, this person is as culpable of the resulting physical harm as they would be if the person were to physically assault those harmed. And yet so falsely shouting is considered perfectly legal in the USA.

Your topic is akin to equally difficult topic of tolerance: when there is tolerance of intolerance, what results is the disappearance of tolerance and the universalization of intolerance among a people.

As to laws, they’re as good as those who make them. Get corrupt or tyrannical lawmakers in charge - say, which are elected by corrupt or tyrannical voters - and laws will be implemented against those who might otherwise be deemed ethical in what they desire to freely speak about. In turn giving more power to the corrupt/tyrannical voices only.

Btw, as a distantly related apropos, a working sustainable democracy is typically envisioned dependent on an informed, educated, and civil society. This being the principal reason why, for example, public education was once introduced. I figure whether a democratic republic remains or perishes is mostly, if not fully, up to the constituents. On whether, for example, individual constituents come to be accepting of, or even in some way emotively endorse, the shouting of “fire” in crowded gatherings when no such fire occurs (so its said, for no sensible reason). In sum, to my mind its an issue of individual and cultural ethics. Laws can only follow suit.

Don’t have a ready solution to the issue. Just chimed in with this post because I do feel the issue’s importance is one worth endorsing.

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.

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.

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.

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.

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.

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.

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.

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.

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.

If you're of a mind, and not burned out on the topic, take another swing at it. It is, after all, a philosophy forum not a math forum. Maybe if you could explain a little more clearly how your problem relates to mathematics without being a mathematical problem, we could make some progress. — Srap Tasmaner

To properly address this, I believe there first ought to be a commonly understood or accepted, philosophical (rather than one pertaining to established schools of mathematics) differentiation between types of concepts regarding the notion of infinity (again, infinity not as its is mathematically defined but as a general, sometimes philosophical, notion: commonly defined by the absence of (non-mathematically defined) “limits” to that addressed).

I don’t know. Once bitten twice shy. This thread’s issue can to my mind be easily overtaken by a broader philosophical issue and possible underlying stance. Namely, one of whether a) mathematics subsumes all reality (such as, for example, by grounding all of physics and, via further inference, thereby all of physicality … this being one example of what could be termed a “neo-Pythagorean” view) or, else, b) reality holds aspects which can be in part and imperfectly modeled by what we humans have devised - over the long course of history you address - as various schools of mathematical thought.

So, does the mathematician’s specialized definition of countability thereby take precedence over what layman understandings (such as the two Wiktionary definitions of infinity previously provided) of countability are? This on account of stance (a). Or else is the so here termed “mathematical” notion of “countability” a specialized understanding that is subsumed by commonly accepted every day notions of countability in general, namely "the capacity to count quantities"? As would naturally be the case in scenario (b).

Must I express “non-mathematical” for every term I use in attempts to define a certain species of concepts regarding infinity?

But then, what on earth would “the ‘non-mathematical’ countably of infinity” signify to a general audience?! To be countable but not mathematical is a bit of a conceptual contradiction.

Yes, I’m a little frustrated, maybe blowing off my own steam. But help me out a bit if you can.

Can we at least mutually understand a differentiation between “quantifiable infinity” which can thereby be addressed in the plural and “nonquantifiable infinity” which cannot thereby be properly addressed in the plural?

All infinities defined by mathematics would then be stereotypical examples of “quantifiable infinity”.

In contrast, the infinity of nothingness would then be one example of “nonquantifiable infinity” - such as when nothingness is conceptualized to have once “existed/been/occurred” devoid of anything. Those who claim the possibility that before the big bang was nothingness (e.g., https://en.wikipedia.org/wiki/A_Universe_from_Nothing) can be ascribed to implicitly make use of such concept of infinity.

If we can’t conceptualize - and then properly term - this differentiation between species of infinity which humans at large can historically conceive of, then I don’t see much point in further addressing the topic of the OP and, by extension, in answering your inquiry.

And this, in part, because “nonquantifiable infinity” (if this term doesn’t get lambasted as well) can only be completely non-determined and thereby completely indeterminate ontically (though, again, it will be a determinate concept). The OP’s inquiry, however, applies only to those infinities that are “quantifiable” and thereby in some way definite due to some demarcation or other occurring in that being addressed. Furthermore, to simplify the variety of such, the OP limits itself only those quantifiable infinities that are themselves made up of discrete quantities (like a geometric line being made up of discrete geometric points, or an infinite set made up of discrete numbers ).

To further complicate matters, then there needs to be a commonly held understanding of what “to determine” signifies (when the term isn’t used to address psychological processes of mind or states that thereby result). There’s again been much criticism of how I’ve attempted to define it (in short, as “to set the limits or boundaries of” - a standard dictionary definition, spelled out by me in greater explicit detail in the OP for an intended greater accuracy); none of this criticism being constructive in offering any alternative definition.

… I’m not hopeful this can work out, but I’ll check back in some time. Thanks, however, for the offer.

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). — javra

If the context is mathematics, [...] — TonesInDeepFreeze

Hey, though I hope I'm wrong, your forgone conclusions regarding me and what I was addressing, your indignation, and your seeing of red is evident to me.

But so it’s said: There are at least two distinct senses of “mathematical”.

I have nowhere stated - nor to my mind insinuated - that I am addressing infinities as they are defined by schools of mathematical thought, i.e. as they are defined by mathematics.

I have instead used “mathematical infinities” in layman’s terms from the get-go, as I initially thought (mistakenly) the OP made clear by the way terms were defined in relation to each other: in the sense of infinities that can be quantified and that furthermore pertain to quantities, and which are thereby, in this sense alone, mathematical. “Mathematical” in the sense that if a bird can count to ten, then this bird holds a respective measure of mathematical skill - despite this bird having no cognizance of any theoretical underpinnings devised by humans for the quantities it can count. In the sense that any appraisal of quantities, such as 1 + 1 = 2, is a mathematical ability that makes use of mathematical notions - irrespective of how these notions are established (hence, with or without the symbols that we humans use to express 1 + 1 = 2).

The thread was in part because of this placed in the category of “General Philosophy” rather than “Logic and Philosophy of Mathematics”.

Nor do I personally take established concepts in mathematics to the foundational cornerstone of what "infinity" at large can signify.

Notwithstanding, for my part, I have learned from this thread not to term quantifiable infinities that pertain to quantities “mathematical” - nor “countable” for that matter. Yes, my bad for not understanding beforehand how use of these terms would be strictly understood by those with a mindset such as your own.

I have no interest in addressing the many comments made in your many posts. And presently intend to let you have the last word, laugh, insult, or what have you.

So the cardinality of the integers is "less than" that of the reals. — jgill

This will be true only when one assumes the occurrence of actual infinities, in contrast to potential infinities. As an easy to read reference: https://en.wikipedia.org/wiki/Actual_infinity From my readings the issue is not as of yet definitively settled - or at least is relative to the mathematical school of thought.

At any rate, the issue of whether infinities (in the plural) are determinate, indeterminate, or neither has dissipated from this thread some time ago. I'm looking to follow suit. Best.

Are you talking about a single infinite line being somehow countable? Like the points on the line?

Or are you talking about the set of all infinite lines being countable?

Neither are countable. Countable means this is #1, this next is #2, the next is #3, etc. It means some sort of algorithm for actually counting.

Maybe you are using the word differently. Like "I can be counting on you to do the best you can." Rather than counting 1, 2, 3, ... — jgill

In presuming that - unlike at least the initial posts of Real Gone Cat - you’re not posting this to have “a good time” at the expense of a poster you assume to be stupid (because laughing at retards is such an admirable trait in today’s world):

I’m sincerely bewildered at the chasm of understanding (your previous unwarranted rudeness aside).

Nowhere did I state either possibility you offer.

Countable in the sense of: one infinite line and another infinite line make up two infinite lines.

Or: the infinity of real numbers and the infinity of natural numbers and the infinity of transfinite numbers make up three numerically distinct infinities. More technically, make up three numerically distinct infinite sets.

As in, the infinity of real numbers is infinity #1, the infinity of natural numbers is infinity #2, and the infinity of transfinite numbers is infinity #3. Each of these three infinities is in turn other than the infinity of surreal numbers, for example, which on this list would be infinity #4, making a total sum of four infinities that have been so far addressed.

As it is I'm wanting to bail out. But on the possibility that your latest post was sincere in its questions, I've answered.

Nothingness cannot have an ontic occurrence since it has nothing to occur, and if there were an infinite God he would be different from other objects, for example from us humans, so he would have a boundary of his identity too. — litewave

OK. So you uphold that the concept (which has been around in the history of mankind for some time) is vacuous. While neither agreeing nor disagreeing with you, I see nothing wrong with that.

Friend, as litewave pointed out, by your own argument, when you name a thing, you are placing a boundary on it. If you can, name two metaphysical identities. Now count them. Two. — Real Gone Cat

As has been typical, there's a reading comprehension problem. By my very own argument, the concept is quantifiable whereas that which the concept refers to is not.

but I think I'm out. — Real Gone Cat

That makes the two of us.

I assume you only recognize one metaphysical infinity, so haven't you counted it? One. — Real Gone Cat

Not to be rude but, in this thread, you've made it a habit to assume things I haven't expressed and most often don't believe.

Whereas metaphysical infinity would be infinite in length, in width, and in all other possible manners. — javra

But such a metaphysical infinity would still have a boundary of its identity because it would be differentiated from what it is not, for example from finiteness or from infinite lines. — litewave

This will be true when it comes to it being a concept (a map of the territory). But it cannot be true of it as an ontic occurrence (as the territory itself). The boundaries you specify would nullify the possibility of its occurrence.

Mind, I've made no claim as to whether or not metaphysical infinity can ontically occur - and am intent on leaving this issue open ended. But - as with a) the infinity of nothingness or b) the infinity of at least certain understandings of God (each being a different qualitative version of what would yet be definable as metaphysical infinity) - it is possible for certain humans to conceptualize its occurrence.

My point was and remains:

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

Yes. — Srap Tasmaner

By being 2 different infinities, they are thereby quantifiable as infinities wherein each individual infinity is demarcated from the other by some limits or boundaries. These infinities are thereby countable (in a non-mathematical sense): 2 infinities.

This cannot be so of metaphysical infinity (for reasons I've become tired of repeating).

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.

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.

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

Yea. That appears to roughly sum up the issue.

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 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?

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.

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)

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.

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 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.

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.

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?

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

...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.

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:

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.