What about, say, is 1+2=3 in all possible worlds? — Jorndoe

If it weren't, you would want to check your change very carefully. X-)

It occurs to me that God would have to follow the rules on this one. I don't know what the implications are for so called omniscience.

What exactly is your question, jorndoe?

I can assume that there can be mathematical systems that simply are incommensurable to each other, yet are totally logical and true. Hence in one 1+2=3, but in another 1+2 is not 3. Now these two seeminly at odds systems can actually go just well together, because usually the error is just to assume that one or the another is an universal axiom of some sort, true in every system/language. There being Euclidiean geometry and non-Euclidean geometry doesn't create a Paradox. These belief that all geometry is Euclidiean is just false.

The above definition of abstract numbers falls back on the more concrete world.

Whether or not it works is perhaps up for debate. Multisets or hypotheticals were mentioned, along with induction and recursion, since otherwise we'd automatically have to assume infinitudes of concretes. From memory (i.e. unreliable :)), Russell (and Frege and a few others) had something to say on this.

Clearly our mathematical systems can work consistently and coherently with infinitudes, it actually makes them (more) useful. Yet, that by itself surely does not imply there are infinitudes in the universe.

But there's the question of Platonism. If these abstract numbers can be coherently defined by deferring to concretes, then Platonism might fall back on concretes as well (as far as numbers go at least). Otherwise, do we somehow arrive at (justify) Platonism automatically?

Technically, a basic abstraction is axiomatic set theories (somewhat analogous to foundationalism over in epistemology), though that by itself wasn't really the intended topic of the opening post.

Are numbers (modally) necessary?

FYI, this thread was prompted due to comments by @Wayfarer, @John and some others, in a parallel thread, which seemed better in a thread of it's own.

What about, say, is 1+2=3 in all possible worlds? — jorndoe

As it is not always true in this world I would have thought it perfectly obvious that it is not. Dealing with numbers as things in themselves is the door to madness. Numbers exist only within the logical system that we call counting, which is a subset of arithmetic which is in turn a subset of mathematics. And as Kant pointed out in responding to the ontological argument any necessity pertaining to numbers is therefore entirely dependent upon the logical system. The cardinality of numbers is not, as we so fondly imagine, derived from the real world at all.

Threeness is not an inherent property of triplets. It is a synthetic property imposed by the observer and entirely mutable. We can just as easily identify triplets as one set, two sets (divided by gender, or handedness, or any other quality) or biliions of cells containing identical DNA or trillions of molecules formed in the crucible of a single womb. How many colours are there in a rainbow? Six? The seven that Newton conveniently saw? As many as you can count? More than you can count? When you add one lump of plasticine to two other lumps of plasticine do you have three lumps of plasticine or one? When you walk one mile south and then one mile north have you travelled 2 miles or none?

See that fellow over there?



If you were to answer "that is 3 fellows", then I'd likely wonder if you were drunk or something. :)
Of course I could just have missed the other 2, but that would just re-emphasize the non-arbitrary nature of the number of fellows over there, wouldn't it?

The semantics of what we count can also be questioned, yet that also seems different.

Abigail and Brittany Hensel are considered 2 persons.



These details do not really seem all that relevant to the numbers, do they?

There being Euclidiean geometry and non-Euclidean geometry doesn't create a Paradox. These belief that all geometry is Euclidiean is just false. — ssu

Right. I tend to agree. Seems @Wayfarer does as well.

If it weren't, you would want to check your change very carefully. X-) — Wayfarer

But maybe @Barry Etheridge doesn't?

As it is not always true in this world I would have thought it perfectly obvious that it is not. — Barry Etheridge

I take it you're not a Platonist @Barry Etheridge?

Dealing with numbers as things in themselves is the door to madness. — Barry Etheridge

Personally, I'm not much of a Platonist. Yet we do this sort of thing all the time. See the 3 cows over there?



We identify, differentiate and predicate them, coming up with a set (or just "those cows over there"), which, in turn, has a property, a cardinality which is 3. If the set both had 3 members, and not 3 members, then the law of identity (or the law of non-contradiction) would be violated, and something similar could be said of each of the 3 cows individually.

Oddly enough perhaps, if we consider a world comprised of just 1 thing (whatever that may be, and assuming that makes any sense), then only "0" and "1" exist in such a world. — jorndoe

That is a (trivial) example of the mathematician's fallacy. Even in a reality that is only "comprised" of "0"s and "1"s, what exists in such a world and how it behaves, is determined by the laws of physics alone.

If the set both had 3 members, and not 3 members, then the law of identity (or the law of non-contradiction) would be violated, and something similar could be said of each of the 3 cows individually. — jorndoe

But the concept of 'set' is just another product of the logic system. That's precisely the point. Within the logical system we know as mathematics the cardinality of the set 'cows in the field' is indeed 3 and necessarily so. But this is an abstract reduction of the real world to concepts which are coherent within mathematics. The reality is that there are not three 'cows' because the logic system demands that all cows are equal and clearly they are not. To give the set 'cows in the field' the cardinality of 3 it is necessary to idealise or average 'cow', to turn it from a living, breathing, unique individual into a cipher.

That this is a problem becomes clear the moment you start to do arithmetic with these cows. Say a will calls for these cows to be divided equally among three beneficiaries. Easy for the logic system. That's a cow each. But the chances are that in real life that's actually a huge inequality. A cow that's 10 years old is not the equal of a cow 2 years old. A cow that produces 10 gallons of milk a day is not the equal of one that produces 6 gallons. A cow that spontaneously aborted its last two calves is not the equivalent of one that's produced 5 perfect calves in a row. And so on.

Moreover, what if the farmers who keep these animals count only pairs as 'a cow' (it was good enough for Noah) and all trade is conducted with that system. This is easily encompassed within a logical system for calculating prices, yields and everything else but now there are only one and a half cows in the field. Immediately the necessity of this sets cardinality being 3 disappears in a puff of logic.

Number is something imposed on reality by the observer in accordance with the logical system under which the numbers are defined (and the demands of the purpose in counting in the first place). It is not a description of reality nor an inherent property of reality.

Actually I don't this clashes with Platonism at all. Platonist Forms surely imply all divisions which might be numbered are by definition illusory. There are not many cows in reality. That's simply a product of our imperfect 'vision'. There is only Cowness. So numbers should not be seen as Forms in themselves because there is no counting in the realm of the real.

Personally, I'm not much of a Platonist. Yet we do this sort of thing all the time. See the 3 cows over there? — jorndoe

Numbers exist only within the logical system that we call counting, which is a subset of arithmetic which is in turn a subset of mathematics. And as Kant pointed out in responding to the ontological argument any necessity pertaining to numbers is therefore entirely dependent upon the logical system. — Barry Etheridge

Barry is on the money. Numbers are part of their own symbolic game and so stand outside the real world (all the better to be able to describe it).

The question is what do numbers best refer to when it comes to reality. And the answer is "the individuated". Or better yet, they count acts of individuation - stressing the fact that nothing in existence comes ready-made as a particular, but must in fact be individuated as a process, and so individuation is always contextual, always a matter of degree, and never absolute in the way counting appears to imply.

Nothing philosophically is demonstrated by pictures of cows, people, cups, or whatever and saying no one could deny that they see x number of individuated beings. This just bypasses all the deep questions at stake.

By our perceptual processes, we make judgments that we see a number of things that are similar enough to fit our rules about counting. We have a theory - about numbering. And we then can measure the world according to our best interpretation of that theory. We can informally decide what looks "near enough" to another cow, another cup, to be named as a further individual of that kind.

The real question then is about how individuation arises in the world itself. How do we describe that?

That is when we have to get into an active view of the world where individuation is a contextual action. We can no longer treat existence as a passive state of affairs. And then also we have to recognise that both necessity and spontaneity can be at work. So strict determinism is out too. An act of individuation could be a fluctuation or a propensity.

The shape of a cup or a cow is highly determined by a memory - a manufacturer's intent that is genetic or human. The individual is shaped in a proscribed fashion with a limited tolerance for contingency.

But the shape of a cloud is far more accidental or emergent. The constraints acting upon its formation are far more probabilistic. That is why it is often very vague - an uncertain judgment - if a cloud has yet separated from the rest, or if the cloud in turn is composed of distinct parts.

Yet importantly - to ontology - cows and clouds are just degrees of difference along this spectrum. Both are formed contextually by some kind of information (with clouds, this is material laws). And both are also subject to spontaneous fluctuation (in cows, we don't count "normal" variations in size, colour or shape).

So we have a few things going on.

The world does seem to naturally come individuated. So numbers as a logical system appears to map on to the world in a naively direct fashion. Our theory (of numbers) can be cashed out unambiguously by our acts of measurement (our acts of counting), ignoring all questions about whether we are counting instances which themselves are a necessary fact of reality, or instead an arbitrary fluctuation, or indeed, some trickier combo of both.

It becomes a blurred issue once we start counting the number of hills on a landscape or measuring the curves of a coastline. Are our criteria for what counts as a geomorphic bump the same as nature's in some rigorous fashion?

So numbers are tantalisingly powerful. They can sum up a lot about reality using very little information. And yet they do that precisely by leaving the reality of reality out - ignoring the whole tricky business of what might result in individuation.

And then less obviously, they are powerful also because they by-pass the question of necessary vs contingent too. We can count accidents of nature right along with nature's necessary facts if we so choose. Numbers really don't sweat the details at all.

Actually I don't this clashes with Platonism at all. Platonist Forms surely imply all divisions which might be numbered are by definition illusory. There are not many cows in reality. That's simply a product of our imperfect 'vision'. There is only Cowness. So numbers should not be seen as Forms in themselves because there is no counting in the realm of the real. — Barry Etheridge

Platonism is normally confused by the fact it does try to include mathematical objects right alongside natural objects in Platonia. There is cowness, and also twoness and triangleness, as perfect ideas.

So the interesting question there becomes what in the end is Platonically special about numbers and mathematical objects in general? What is the further truth that maths seems to capture that is the source of its "unreasonable effectiveness"?

In some sense, maths must capture the limits of what can exist - the limits of acts of individuation that would apply (modally, logically) in any conceivable world. So we are now imagining the most primitive measuring operations - as uncovered via early counting, but more especially geometry - and discovering the forms that must result as their limits.

Counting arises from the fact of being able to point to a succession of things - wag a finger to a sequence of locations as an actual physical act embedded in time and space. Geometry then treats this discovered embedding time and space more generally in moving about it, following straight lines, measuring angles - all the primal measuring operations that are encoded in Euclid's axioms.

So in this unnoticed fashion, physicality - the embodied physicality of being able to "make a measurement" - is exported to a Platonic realm where various perfect objects (or acts of individuation) arise as the limits of the measurable. Geometry imagines the abstract world needed to make these possible kinds of measurement necessarily "true" and not simply contingent or accidental events.

If the world actually is Euclidean, then you get triangles and circles as this world's perfect limiting forms. And our actual physical world is indeed Euclidean - or at least immeasurably close to that on the energy and distance scales we typically measure it on.

Thus does 1+2=3 in any possible modal world? Well we can see that already a particular kind of world is being imagined - one with the kind of global dimensionality that underwrites the acts of measurement which make this a logical necessity. Rulers aren't bent, clocks are not dilated simply due to changes in energy scale. There is no quantum entanglement, no classical collapse, to muddy the sharp possibility of measuring things exactly. The Universe only has its three spatial dimensions, its one temporal direction, and its exactly flat, etc.

So the whole of Platonic maths is merely then just an extrapolation from the starting point of what we seem to be able to measure or individuate. It is an exercise in imagining the kind of world which would make our beliefs about the possibility of some measurement act necessarily true. If the world is that way - for example Euclidean - then we now have this absolutely secure platform by our (otherwise possibly arbitrary) acts of measurement.

But of course, we have learnt the world is not Euclidean. It may not have a deep geometry at all in terms of some certain number and shape of dimensions. It may be utterly contextual or arbitrary if we follow quantum theory down its rabbit-hole.

So all we can really say in the modern era about counting (or algebra) and geometry as logical structures is that they depend on "reasonable" axioms - axioms that encode what seem to be primal acts of measurement. Platonia describes the kind of world that would make some type of measurement "true".

But we've also long since shredded any classical/Euclidean notion of where maths or measurement should come to rest. We are now in the rather advanced situation of trying to imagine what any notion of measurement could look like in any notion of a world. What is the mathematical limit description of that exactly? Does even category theory get there yet?

Anyway, the point is that any Platonic assurance, any sense of strong structure, rests on the unpacking of hidden assumptions built into the axiomatic notion of "the measurement" - the act of individuation, the difference that makes a difference. So if you want to get to the source of things, this has to be the philosophical focus. (As it is in Peircean semiotics for instance.)

I had a very simple epiphany once, which is that natural numbers (1) don't come into, or go out of, existence and (2) are the same for anyone that can count. Realizing that, seemed to me a kind of 'lightbulb moment' - I thought 'aha! This is why the ancients held numerical reasoning in such high esteem'. Because it occurred to me that in this sense numbers have a very different kind of reality to things, which do always go into and out of existence.

Having said that, at the time I had this realisation, I had never formally studied Greek philosophy or even much maths for that matter (in fact I was one of those students terrible at maths and much better at English).

But I've researched the subject and this is how I understand it.

The Platonist intuition about mathematics and geometry was that it provided a type of logical and indeed apodictic certainty which couldn't be found in the sensory domain. But to understand that, you have to appreciate that the ancient philosophers could question 'the reality of sense' in a way that us moderns find difficult. Plato et al seriously considered the idea that the 'world of sense' might be an illusory domain. I think it's very difficult for us to think that, as naturalism is conditioned into us. I think that it was less difficult for the ancients for historical reasons, i.e. their cultural paradigm and level of conscious development was very different (see Julian Jaynes, Owen Barfield.)

In any case, in the Allegory of the Divided Line, Plato gives a summary of the different forms and domains of knowledge, Eikasia, Pistis, Dianioa, and Noesis. But note the comment underneath the table ' It is not enough for the philosopher to understand the Ideas (Forms), he must also understand the relation of Ideas to all four levels of the structure to be able to know anything at all.'

Now there's a lot that could be said about that, but I want to bring out the general principle of Greek rationalism, which is that understanding the underlying 'ratio' and 'logos' provided the philosopher with insight into the principles of things, which actually provided enormous power over them. I think one of the key exemplars of that was Archimedes, who was indeed one of the founders of science and mathematics. But he also devised some astonishing inventions, based on mathematical reasoning, like the means by which he set fire to an invading fleet of ships by focussing sunlight on them with mirrors. (God knows who devised the Antikythera Mechanism.)

The other seminal figure that should be mentioned, the 'ur-scientist', is of course Pythagoras, who among other things discerned the relationships of mathematical ratios and musical harmonies, amongst other acts of genius. (It is well worth going back and reading Bertrand Russell's chapter on Pythagoras in HWT.)

But the key point is that insight into mathematical principles, is insight into a different domain. And the problem we now have is that we have no means of envisaging such a domain, because we are so habitually disposed to locating everything in time and space. The 'formal domain' of laws and numbers and the like, is not anywhere, however. It pertains to the structure of mind-and-world, it is not 'in' the mind or 'in' the world, but precedes and underlies the manner in which the mind interprets sensory experience.

But the key point is that insight into mathematical principles, is insight into a different domain. And the problem we now have is that we have no means of envisaging such a domain, because we are so habitually disposed to locating everything in time and space. — Wayfarer

That's a good summary of the impact of Greek maths on the very creation of a rationalising mindset. And there is indeed the irony that we now mentally inhabit that very world because we "see" it in this dimensional fashion.

We learn maths as kids and grow up seeing we exist within three dimensions surrounded by countable objects, etc. Indeed we actually used the maths to build a carpented world of straight lines, perfect circles, exact right angles - the geometry of the modern house or formal garden. So we have really internalised Platonism. And if you are up to date with maths, you now see the same all over again in terms of the forms of trees, clouds, coastlines, mountain ranges,, and other fractal/dissipative structures.

But I think the key point about Platonia is that it is implicitly "other" to something. And that other is the act of measurement. The implied self who is the observer, waving about measuring instruments like rulers, clocks, balances, etc, in ways that make "perfect sense".

So Platonia was rational paradise. But it made opaque its necessary other - empirical paradise. That has to "exist" to. And that is what has become the target of inquiry in the modern era.

If we could generalise the notion of an "observer", an act of measurement or individuation, with matching rigour, then we would really be closing in on a fundamental view of things. We could finish what the Greeks started.

I'm not entirely sure about that; I think I am inclined to take issue with 'Platonia'. I often notice that in debates about Platonic realism, that there that they founder on this notion of 'where could such a domain be'? As I have tried to explain, I think this is based on a misconception. Or rather, I think it is 'the habit of extroversion' that our culture has drilled into us. The 'empiricist' mindset is such that 'what is real' must have a location in the physical matrix of matter-energy-space-time. So everything we say exists, must be either locatable there, or be shown to have some evidence or consequences in that domain. That is what 'empiricism' means, right?

I have been hanging out briefly on another forum and discussing this point with a diehard materialist, and he simply cannot accept that something can be real in any sense other than being somewhere. 'To be real' is 'to have a location in time and space'. If I ask 'what about abstract ideas', the answer is, 'they're located also - in the mind, which is generated by the brain'. And that is the sense in which they're real. End of story. How they're predictive and so on - 'we're working on that'.

Whereas I have for years argued that numbers (and the like) are real in a different sense to material objects. But I know from long experience that this notion of 'in a different sense' is not acceptable to empiricist thinking. There is only one sense in which something exists, and that is that it is real, and that applies to chairs, apples, real numbers, sentences, snowflakes, or whatever. Whereas fictional or imaginary things don't exist except for in the mind, which is in the brain, which is physical. 'Existence' is, then, univocal, it has precisely one meaning, whereas I think in the Platonic and neo-platonic understanding, existence is hierarchical, with nous and its objects higher, and the senses and their objects, on a lower level. But that philosophy has been rejected by the nominalists and empiricists who won the argument. And history, as they say, is written by the victors.

//I am out of here until end of working day, I'm working on far less abstract stuff.//

I often notice that in debates about Platonic realism, that there that they founder on this notion of 'where could such a domain be'? As I have tried to explain, I think this is based on a misconception. Or rather, I think it is 'the habit of extroversion' that our culture has drilled into us. — Wayfarer

I disagree as it was already an issue in Ancient Greece even if it became both heightened - and nominalistically talk away - in modern times.

At the centre of Western philosophy is already this rock-solid dualistic distinction between observer and observables, the mind and the world. And while I agree there is then the unfortunate tendency to want to place them in different places as realms (the mind has to exist "somewhere"), it is also an inevitable kind of issue that must be resolved.

And it is also inevitable that people either tried to reduce everything back to one world (as in idealism, or instead nominalism), or indeed, started talking about the three worlds of material reality, mental reality and Platonia.

The 'empiricist' mindset is such that 'what is real' must have a location in the physical matrix of matter-energy-space-time. So everything we say exists, must be either locatable there, or be shown to have some evidence or consequences in that domain. That is what 'empiricism' means, right? — Wayfarer

My point here is that the empirical in fact invokes the very notion of "making a measurement". And so it speaks about observers as much as observables. And that is why philosophy has to focus on generalising the very notion of an observer - which Peircean semiosis does and dualistic approaches to "conscious minds" don't.

The kind of empiricism you describe is dualistic - the familiar unwitting dualism of the naive realist.

I have been hanging out briefly on another forum and discussing this point with a diehard materialist, and he simply cannot accept that something can be real in any sense other than being somewhere. 'To be real' is 'to have a location in time and space'. If I ask 'what about abstract ideas', the answer is, 'they're located also - in the mind, which is generated by the brain'. And that is the sense in which they're real. End of story. How they're predictive and so on - 'we're working on that'. — Wayfarer

Exactly. You wind up in dualism unless you can generalise the very thing of "an act of measurement".

But I would add (slightly digressively) that modern materialism would have to say things are real when they have an energy density and thus an energy potential when located in a spacetime frame. So now there is also a "location" in being the unlocated possibility of an action.

That is to say, modern physics is dualistic with its realms so that energy exists in a separate abstract way - the contents are abstracted from the container. And so energy has to be located somewhere that isn't, in the formalism, "somewhere".

Of course the ambition of modern physics is to heal this rift via a theory of quantum gravity - a story of spatiotemporal containers and energetic contents are shaped in mutual interaction.

There is only one sense in which something exists, and that is that it is real, and that applies to chairs, apples, real numbers, sentences, snowflakes, or whatever. Whereas fictional or imaginary things don't exist except for in the mind, which is in the brain, which is physical. — Wayfarer

But speaking for empiricists, you are talking about old-fashioned notions of the empirical - ones that rest on classical presumptions about observers. And modern physics has left that behind (even if it is also as reluctant as hell to let completely go).

Of course I also agree here that the ontic issues can't be solved by the materialists pointing out that the mind is emergent from the brain. That's not a sufficient theory. You would have to have an account that makes sense of such a claim - such as semiosis, where we can see how sign relations or "symbol processing" is indeed both physically instantiated by, yet causally disconnected from, the material dynamics it can arise to regulate.

This is why I point to the centrality of the measurement problem. It has to be solved by both mind science and physical science. It is where we have got to. And oh look, semiotics, biology and thermodynamics all speak right to that.

I think in the Platonic and neo-platonic understanding, existence is hierarchical, with nous and its objects higher, and the senses and their objects, on a lower level. — Wayfarer

You can extract a "ladder of life" story from this - the one that runs from pan-semiosis through bio-semiosis and linguistic-semiosis. So from the simplest self-organised dynamics to the most complex semiotic organisation.

But you instead are endorsing a dualism of mind and world here. Or at least, a divine and world. There maybe a useful difference.

Mind~world dualism treats consciousness as some definite Cartesian substance - a concrete soul stuff. While divine~world can ease you back towards a more pantheistic and immanent rendering of the situation. It starts to sound more like my organic and pansemiotic conception.

But I would say that is simply because divine~world is vaguer - less in your face than hardline Cartesian dualism. On the other hand, it does want to imbue the whole of reality with what is missing from hardline materialist accounts - formal and final cause. So that is where our worldviews usually overlap. That is a common interest arrived at from quite contrasting start points.

Reminds me a bit of 1984:

How many fingers, Winston? — O'Brien

···



— How many cows, Thinker?

The quantity of cows seems real enough to me. Surely there are 3 cows regardless of anyone walking around counting them. Remove the cows, and just "3" (supposedly) remains? Well, not really, but perhaps it's surprising (to some), that there are some structural consistencies among such quantities, regardless of the subjects?

Quoting Philosophy 103 (cheatsheet) at Lander University, just for the usual technicalities:

The quantity of a categorical proposition is determined by whether or not it refers to all members of its subject class (i.e., universal or particular). The question "How many?" is asking for quantity.

The quantity of a standard form categorical proposition determines the distribution of the subject (such that if the quantity is universal, the subject is distributed and if the quantity is particular, the subject is undistributed), and ...

The quantity of cows seems real enough to me. — jorndoe

You will note that this dichotomy of quality~quantity relies on the quality of cowness being real too - otherwise how else do you know that cows are what you are counting?

So naive realism always conceals what it pretends to answer.

If you put the concept of cow all in the counter's mind, how is that realism and not idealism? Nothing secures the truth of the counting except some individual's claim to know what they are doing.

And alternatively if you put cowness out in the world as a further fact, how is this not still idealism? Now you are claiming to "see" an abstract object in some fashion.

So confusion is rife in your naive realism. You are doing nothing yet to improve your situation.

Where do you see naïve realism, @apokrisis?

(I'd ask the cows, but they always chase me off the field, and don't carry a cellphone.)

thanks, splendid reply!

Where do you see naïve realism — jorndoe

In the lack of a reply on the question of how you ontologise quality. The first rule of naive realism is explaining is losing, so just deflect.

In the lack of a reply on the question of how you ontologise quality. The first rule of naive realism is explaining is losing, so just deflect. — apokrisis

?

? — jorndoe

As I say, we can quantify the quality of naive realism by counting the number of deflections we observe when it is faced with evidence of its central epistemic self-contradiction - its cosy presumption of a real self "in here" to observe the real world "out there".

So the count is 2 in this sequence. Are you about to make it 3?

It seems @apokrisis, we have a situation where you could just pretend to be me, and have at it. :D Please go ahead. You mentioned something about realism (not I)?

(I'm not sure where your sudden demand for personal confessions came from, but feel free to jump ahead.)

So are you claiming to be able to count 3 cows as an indisputable fact or not?

If not, what kind of fact is it? If you want to proclaim yourself instead a Pragmatist or whatever, then say so.

If we could generalise the notion of an "observer", an act of measurement or individuation, with matching rigour, then we would really be closing in on a fundamental view of things. We could finish what the Greeks started.

This is simple enough, one is simply required to accept that "Platonia", or number, is appealing to the conception of mental or abstract forms which have a reality(presence) in a divine(transcendent) reality, as well as our reality. In which all forms in that divine reality are as fundamental as number appears to us. Another such form is the form of self or being, which is both present in the divine reality and the world. So in measurement we find the divine witnessing the divine in concert with the person waving rulers around. Hence the fascination of the Greeks with number and Platonia, they can intellectually sense something absolute going on.

Mind~world dualism treats consciousness as some definite Cartesian substance - a concrete soul stuff. While divine~world can ease you back towards a more pantheistic and immanent rendering of the situation. It starts to sound more like my organic and pansemiotic conception.

This is only superficially a duality, there is always a third veiled component, of spirit. Or an imminent divine self, or being. This being is simply a witness, eye, a lense. So we have a triad, world, mind, spirit. Mind, body, spirit. Spirit being as universal as number, but immutable, so is essentially veiled to us. In the absence of specific tutoring to know the spirit for what it is.

Hence the fascination of the Greeks with number and Platonia, they can intellectually sense something absolute going on. — Punshhh

It's that numbers and ratios are permanent, not subject to change, constant, and only visible to the intellect. So they're in a different category - a higher order - to what is mutable, visible and inconstant. That was also part of the Pythagorean tradition (which might well have originated with the Egyptians) and which Plato also was heir to. (And, one of the main reasons, in my view, why the scientific revolution occured in Europe and not India or China.)

(Have a look at the book reviews of an Amazon reader who goes under the name of Johannes Platonicus. There's a wealth of information about such ideas, and what some of the better sources are. If only there were more time.....)

I think the biggest problem with mathematics is that we have put as it's basis a practical use of mathematics and that is counting. Math evolved from counting... with natural numbers, and then went on. Yet natural numbers and counting, computability, isn't everything in math.

Yes, natural numbers are so obvious, something that we use to answer practical and real World questions or problems and give us a tool for mapping better than the system of separating things into "none, single, many". Yet the problem is that it simply cannot be the foundation of absolutely everything in Mathematics. And as Mathematics is logical and a rigorous system, the starting point with counting and finite numbers makes then the system in it's entire reach a bit confusing.

I was at the old site and still am a believer that actually infinity is a number. We just don't have the correct definition of a number. Because infinity and it's counterpart, which we avoid by talking about limits, are so useful and so obvious in mathematics that the former shouldn't be just taken as an axiom an left there. (And I do believe that there's an absolute infinity, actually)

Mathematics itself is abstract. Use of math typically isn't, even if it can be too.

I was at the old site and still am a believer that actually infinity is a number. We just don't have the correct definition of a number. Because infinity and it's counterpart, which we avoid by talking about limits, are so useful and so obvious in mathematics that the former shouldn't be just taken as an axiom an left there. (And I do believe that there's an absolute infinity, actually)

Interesting, I do play with the idea of an actual infinity from time to time, it always ends with with some kind of fractal. Although when I contemplate Brahman, as defined in Hinduism I can conceive of that kind of infinite in the sense of infinitely transcendent.

Are you thinking of something multidimensional, or transcendent in some way?

Thanks, I don't seem to have much time for reading at the moment. To much time spent painting, or restoring my house.

Are you thinking of something multidimensional, or transcendent in some way? — Punshhh

Not actually, in a way that kind of thinking comes once we have the solutions.

I'm thinking about it from the standpoint of basically "What Cantor didn't get" or what is missing here that we cannot define it. More like starting from that there's some truly awesome theorems, even laws, that would tell us immediately what our problems have been since ancient times. Now I'm not saying I know the answer, but I assume that there's something here to be found out.

Because, and here I have to thank the old PF site for advice, Cantor actually did think about Absolute Infinity, he didn't reject it. He couldn't define it, couldn't fathom it and basically came to the conclusion that it was something that only God knew. As a deeply religious man who did see mathematics from a religious viewpoint, it seems to have been very important to him. Since not much actual rigorous math (or set theory) was said by Cantor on the absolute, then it has been forgotten.

Now how would absolute infinity go along with the Cantorian system of cascading larger and larger infinities is a bit confusing, but basically set theory has been a way to contain the paradoxes of infinity right from the start. But then again, we have the Continuum Hypothesis, hence our understanding of the infinite isn't perfect. And then there are the paradoxes. Just look at how many axioms in ZF are there to avoid the paradoxes. In my view to separate the paradoxes of the infinite (as sometimes are called Cantor's Paradox) and Russells Paradox as being something totally different doesn't get the whole picture. From my opinion, a paradox in mathematics (or logic) isn't a problem to be solved, but actually an answer to be understood.

If the term number, which is an mathematical object used for counting and measuring, would also incorporate objects that do define an unique quantity that simply is incommensurable to the ordinary numbers (and the quantities they refer to), then "the least" and the "the most" or "everything" could go as... a number. Because we do use them, but we simply don't talk of them as numbers, but for example limits. Or have rules to infinity like below:



Above all, the idea of these kind of numbers goes totally against the idea of everything being derived from the natural numbers in some way or another. Because if you first create a number system, start with natural numbers, then add rational numbers etc, these kind of numbers would be very confusing.