Abstract numbers
There are various ways to construct the non-negative whole numbers, {0, 1, 2, 3, ...}, from other primitives: Set-theoretic definition of natural numbers (Wikipedia article).
Defining the abstract number "7", would (at least) give properties, operations, etc, that are common to every set with that cardinality, be it 7 meters, 7 fingers, 3 hydrogen atoms and 4 helium atoms, 7 planets or other Solar system bodies, ... Here abstract means a specific quantity regardless of the particular unit or metric, just "7".
This has led to sometimes defining the abstract number "7" as the collection of every set with that cardinality (e.g. 7 meters over to the neighbors door, 5 fingers on my left hand and my 2 ears, the 3 hydrogen atoms and the 4 helium atoms in that experiment, the 4 inner planets plus the Moon and the Sun and the International Space Station, ...).
"7" = {S | S is a set, |S| = 7, with 7 as per the construction}
Every such set has a unique cardinality in common, which is what we mean by the abstract number "7". (Notice the vague similarity with disquotationalism.)
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. That is, unless we accept multisets, where the 1 thing could be counted more than once, or unless we introduce hypotheticals in some sense. This may have implications for considering abstract numbers modally necessary or not, and may have implications for Platonism.
(OK, we know abstract numbers, let's go back to ordinary notation.)
Are numbers (modally) necessary? What about, say, is 1+2=3 in all possible worlds?
By finitism there is a definite largest number, and not just any number, the largest number.
For the sake of argument, let's restrict our counting to, say, {0,1,2,3,4,5,6,7,8,9,10}. I can count my fingers, but I appear to have a non-existing number of teeth. I'm in luck while measuring the number of meters over to my neighbor; it's about 5 meters; unfortunately, there's no measuring the distance in inches, that would be a non-existing quantity (even in principle).
For some reason, this seems unsatisfactory. And any definite largest number seems ad hoc, arbitrary (not to mention less useful for our mathematics).
The likes of mathematical induction (which is different from inductive reasoning) and recursion (self-reference) exemplify methods that involve an amount of numbers that is not itself such a number, i.e. ?.
Defining the abstract number "7", would (at least) give properties, operations, etc, that are common to every set with that cardinality, be it 7 meters, 7 fingers, 3 hydrogen atoms and 4 helium atoms, 7 planets or other Solar system bodies, ... Here abstract means a specific quantity regardless of the particular unit or metric, just "7".
This has led to sometimes defining the abstract number "7" as the collection of every set with that cardinality (e.g. 7 meters over to the neighbors door, 5 fingers on my left hand and my 2 ears, the 3 hydrogen atoms and the 4 helium atoms in that experiment, the 4 inner planets plus the Moon and the Sun and the International Space Station, ...).
"7" = {S | S is a set, |S| = 7, with 7 as per the construction}
Every such set has a unique cardinality in common, which is what we mean by the abstract number "7". (Notice the vague similarity with disquotationalism.)
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. That is, unless we accept multisets, where the 1 thing could be counted more than once, or unless we introduce hypotheticals in some sense. This may have implications for considering abstract numbers modally necessary or not, and may have implications for Platonism.
(OK, we know abstract numbers, let's go back to ordinary notation.)
Are numbers (modally) necessary? What about, say, is 1+2=3 in all possible worlds?
By finitism there is a definite largest number, and not just any number, the largest number.
For the sake of argument, let's restrict our counting to, say, {0,1,2,3,4,5,6,7,8,9,10}. I can count my fingers, but I appear to have a non-existing number of teeth. I'm in luck while measuring the number of meters over to my neighbor; it's about 5 meters; unfortunately, there's no measuring the distance in inches, that would be a non-existing quantity (even in principle).
For some reason, this seems unsatisfactory. And any definite largest number seems ad hoc, arbitrary (not to mention less useful for our mathematics).
The likes of mathematical induction (which is different from inductive reasoning) and recursion (self-reference) exemplify methods that involve an amount of numbers that is not itself such a number, i.e. ?.
Comments (81)
If it weren't, you would want to check your change very carefully. X-)
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.
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.
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.
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?
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?
Right. I tend to agree. Seems @Wayfarer does as well.
Quoting Wayfarer
But maybe @Barry Etheridge doesn't?
Quoting Barry Etheridge
I take it you're not a Platonist @Barry Etheridge?
Quoting 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.
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.
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.
Quoting 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.
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.)
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.
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 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 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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
[quote=O'Brien]How many fingers, Winston?[/quote]
···
— 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:
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.
(I'd ask the cows, but they always chase me off the field, and don't carry a cellphone.)
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.
?
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?
(I'm not sure where your sudden demand for personal confessions came from, but feel free to jump ahead.)
If not, what kind of fact is it? If you want to proclaim yourself instead a Pragmatist or whatever, then say so.
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.
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.....)
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.
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?
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.
And the reality of numbers? When you broaden the definition of existence to abstract, not material objects, then numbers are real. They are some of the best tools with which we can make a models of the reality around us, so the idea that it's only in our heads, that they are totally fictional, simply isn't so useful.
For the more rigorous about imaginary numbers answer I think you have to ask that from other members here.
And do note that there are many real world problems that are best modelled with using imaginary numbers. And that for me is allways the reality check: if the math, however "unrigorous" or based on non-proven assumptions has uses in the real World, it's likely to correct. Best example of this is the history of the infinitesimal: such a useful concept, but oh, the historical debate around them! However, if you have math that hasn't got any real World applications, then either the use of the math is still to be found or ...there can be something fishy about the premises of the math.
That actually comes to mind with the Cantorian infinities larger than aleph-0/aleph-1. Not much engineering or physics model use the larger aleph infinities, although I remember that someone did say there were some real world applications for them. As I have some background in economics, I am quite sceptical of the "real world" modelling of economic systems. Usually something like engineering tells it better: if the machine works which is based on some math calculations, then likely the math itself is OK.
Regarding absolute infinity, I feel it is an over simplification. This is not to say it is not the case, rather that it's intricacies may be beyond us at this time. The idea of a kind of transcendent continuum is interesting, but primarily within the sphere of being, rather than anything external. I have developed a thought experiment which attempts to illustrate this.
Imagine you are looking at a transcendent being, like the Christ, a bodhisattva, a god*. As you focus, you feel you are only seeing the surface, the external body of them, the real person is further back(metaphorically), under the surface. You refine your conception to peel back the layers a bit. You imagine a purer more transcendent being, on a higher plane, in a higher dimension. But equally present next to you in your plane, in the same body. It is just your conception which has changed.
So you repeat the process, imagining an even higher purer form, perhaps in a divine realm next to God. Again you are seeing exactly the same person, in the same place nothing has changed other than your perception. Then you repeat the process and begin to realise that each time you peel back a layer it curls up and folds back behind the being like a hair on his head. Then you notice there are many thousands of hairs alongside it. The being becomes in your imagination transcendent. Traversing many dimensions, even realms worlds, times. While all the time absolutely present in this one place. You realise that the dimension they inhabit is of another order another kind and for them to step into the manifest world's we find ourselves in is like dipping their toe into the water. You look again and the being is as before before you looked closer, just another person standing next to you, all along it was your conception which had changed.
* I have experienced something along these lines in person with a guru, I once new, which helped me to develop the idea.
An otherwise inconmensurable amount is naturally equal to itself (a correct model of itself), but just how addition and substraction goes, operations which basically come from the finite realm, is not an easy question.
You say an "indeterminate form", yes I can see that, but isn't an infinite quantity also indeterminate? I don't see how it is significantly different. You could add infinity to infinity and get infinity. Can't you then take that same infinity away again and leave the first infinity as it was before. I agree that addition and subtraction might come from the finite realm, so in a sense this is trying to discuss an unknown in another room, you can't see into.
Anyway, the Hindu's describe Brahman as infinitely infinite(I'll look for a reference), so there are plenty of infinites around in there for one to be subtracted without diminishing his omnipotence.
I would point out that I don't use infinity much, I find eternity much more fruitful.
I do feel sorry for Cantor, that he was not able to progress further in the direction of the continuum. I think he was lacking some transcendent insight, which might have helped.
In transfinite arithmetic we have a collection of objects, called 'cardinalities', which is all the numbers, finite and infinite. Then we have a set of operations that can be performed on those cardinalities. These operations are addition, multiplication and binary exponentiation (x goes to 2^x). The first two operations take two inputs and give one output (binary operations). The last one takes one input and gives an output.
There is no operation of subtraction, so to talk of it is meaningless. Trying to talk about it is an example of inappropriate generalisation. That occurs when subset U of a set S has a property P that is not held by all members of set S, and one then asks a question that presupposes that P applies to all of S.
The subset in this case is the set of all finite cardinalities, for which we can define an operation of subtraction. We end up in confusion if we assume that means we can define an operation of subtraction for all cardinalities.
Another analogy: All primates have eyes. Primates are a subset of the kingdom of eukaryotes - organisms whose cells have nuclei. Fungi are eukaryotes. So let us ask ourselves
which is the left eye of a mushroom?
Let's say there is an infinite amount of grains of green sand and there is also an infinite amount of grains of blue sand, both exist. We know that if we theoretically count them as one group infinity + infinity and that we will then have an infinite amount of grains of sand, which we know are green and blue, but which are still seperate, because we are only imagining them as grouped together. Now let's imagine we mix them up so that they are all randomly mixed in together, a set of an infinite amount of grains of sand, of undefined colours. Now we could theoretically sort through this set and put all the green ones in one place and all the blue in another until we are back were we started. So we have subtracted an infinite quantity from an infinite quantity.
I am by the way aware of the problems of applying infinity to existence.
I think that's a really good idea. Why not actually perform the experiment to prove your point? Once you have demonstrated it, mathematicians will have no place to hide, they will have to accept your proof!
If you can't find two literally infinite piles of sand of different colours, why not use the odd and even numbers instead?
Quoting andrewkYou mean here Cantorian set theory and cardinals and ordinals here.
What Tom is talking is about Cantor's most basic findings (with even and odd numbers).
Can this number be counted?
Also, I am aware how many grains there are;
blue grains = green grains = green plus blue grains = (green plus blue grains minus green grains) = infinity.
Yes, but numbers are ideas, so susceptible to human frailty. An alien, or a monkey, can count the grains of sand and can only come to the same conclusion, because they are not ideas.
I didn't say that. What made you think that I did?
Quoting ssuArithmetic with cardinals, not ordinals. See this wolfram page.
Arithmetic with infinite ordinals is also possible, but is different from arithmetic with cardinals. For instance cardinal addition is commutative, but ordinal addition is not.
We have done something, and that something has to do with the folk notion of 'taking away'. But subtraction is a much more precise notion than the folk notion of taking away. To be able to subtract two things they must be members of a set that has a binary operation, which we can call 'addition', such that the elements of that set form an Abelian Group under that operation.
Transfinite cardinals form an Abelian monoid, but not a group, under the binary operation of addition. So it is not possible to define an operation of subtraction for them.
I think this is as far as I can go in abstract mathematics, my interest is more in the direction of maths in the real world or where it is to be found, or relevant in/to existence.
Even better, get your sand and set the monkeys to work!
The set of all grains of sand, whatever colour is finite. The total number of particles in the visible universe is only 10^80!
You appeared to be discussing infinite sets earlier, which behave differently from finite sets.
Quoting Punshhh
If you are interested in reality, you haven't even begun to scratch the surface. Reality takes place in the continuum, where denumerable infinity is not big enough.
I'm all ears, what is infinite about reality?
Reality takes place in the continuum, which is Aleph1or 2^(Aleph0) if you have reason to reject the continuum hypothesis.
Nice idea, but that is a mathematical form, we are talking about life and existence, where is this continuum? And how does it produce these finite things I see before me?
I did point out when I brought this up that I am aware of the difficulties in applying infinity to reality.
Actually, you are not discussing an hypothetical situation, you are discussing a meaningless, impossible, unphysical, imaginary situation, in a different reality with different laws of physics.
Despite your earlier protestation:
Quoting Punshhh
I think you will find the idea of "number" better defined and understood than aliens, counting monkeys, and piles of sand so large that they would create infinite black holes by now.
Quoting Punshhh
The laws of physics take place in the continuum: they are differential equations based on the continuum.
And, there is the point of view that the integers are *not* fundamental, but the continuum is.
I'm interested in these ideas about an existing continuum, is this in the field of mathematics, or astrophysics?
So you are suggesting that number, i.e. Integers are not fundamental. Does this mean that there are places where 1+1 doesn't equal 2?
As I said, hypothetical alien monkeys aren't as well understood as numbers. But you refuse to discuss the odd and even numbers.
Quoting Punshhh
The laws of physics take place in the continuum, what more do you want? You've got literally everything including astrophysics, quantum mechanics and general relativity.
Quoting Punshhh
How does taking the continuum as fundamental entail that nonsense?
Odd numbers = even numbers = odd and even numbers = ( odd numbers and even numbers munis even numbers) = infinity.
There you go, you can subtract infinity from infinity.
You still haven't either described this continuum, or said where it is, in realation to us?
Maybe you should watch this
Quoting Punshhh
With the opening post I tried to account for abstract numbers by falling back on the concrete world.
If we speak of just 3, the abstract number, then it becomes more concrete when we speak of 3 Hollywood celebrities, 3 meters across the yard, ...
Kind of analogous to speaking of hypotheticals, if you will.
If my attempt (at falling back on concretes) works, then we could perhaps remove such numbers from Platonism?
(That was an intended discussion point. Where's Nagase?)
I don't think anyone disputes that axiomatic set theory is abstract.
I suppose there's a question of unwarranted ontologization involved somewhere as well.
Well, you cant account for numbers that way. What you can account for, is what we are able to discover about numbers, which is determined by the laws of physics.
Quoting jorndoe
These aspects of reality permit us to instantiate the number 3, though there are more useful instantiations.
Nevertheless, if we take the fundamental equations of physics as somehow mirroring reality, then the continuum is fundamental, not the integers.
However I am not an idealist as such, so I can take it or leave it. The idea of letting abstract numbers fall back on the concrete world does appeal to me, which is why I am using the analogy of grains of sand. Also I am reluctant to ontologize number too, I see it as a natural quality of manifest existence, but not exclusively in the form we find it.
One of the intended discussion points.
Where does it go awry?
I suppose a conundrum is that I (or whomever, doesn't matter) take abstract numbers just as serious as not taking Platonism serious.
Is measuring quantities, counting or physics examples of instantiating abstract numbers? (If yes, then it seems a kind of Platonism.)
Hilbert's Hotel says differently. ;)
It's called a paradox, but it's not actually contradictory, it just has counter-intuitive implications.
Speaking of infinity, I enjoyed the recent movie, The Man who Knew Infinity, about the tradically short life Indian math prodigy Ramanujan. I thought it was very sensitively made. There's an excellent review here by a reviewer who is also a number theorist, and knows whereof he speaks (and on a very interesting website, too).
One point which interested me was that Ramanujan always claimed that he had been shown his remarkable mathematical discoveries by the goddess Namagiri; in the film, Hardy, (played by Jeremy Irons) is a resolute atheist, along the lines of Bertrand Russell (also depicted in the film), who will have no truck with talk of goddesses, but who is obliged to admit that he has no explanation for Ramanujan's genius. (In a scene towards the end, whilst addressing the Fellows of the Society, he seems to come close to recanting his atheism, but I suspect that this was probably added by the film-makers.)
Anyway well worth seeing, in my view, and relevant to this topic.
I think your problem (and it's not just your problem), in respect of this question, is what is meant by "real". I think the whole tendency of modern (i.e. post-Enlightenment) thinking has been the requirement to ground any account of what is real in the 'empirical domain', that is, to say that it is something that can be located and understood in terms of matter-energy-space-time. (It used to be simply 'matter', but then Einstein came along with the matter-energy equivalence and space-time - so now 'matter' itself is a much more slippery kind of concept.)
But within this kind of master paradigm, everything that exists must be demonstrated with respect to M-E-S-T, as that is the whole scope of what exists. So from a naturalist viewpoint, numbers cannot be said to exist, because there's nowhere for them to reside (not even in the rooms of Hilbert's Hotel). So the naturalist effort (i.e. Lakoff and Johnson) is directed towards providing a naturalistic account of numbers, which generally starts with how 'the ability to count' evolved, and how the mind uses models and metaphors to stand for abstract relationships. So in those naturalistic models, numbers are said to be real, but only in the sense that they model real things accurately, and are grounded in neurological processes which are ultimately physical (as is everything).
Whereas, platonism says that number is real in its own right, that natural numbers are just as real as tangible objects, and measurable effects, albeit in a purely intellectual sense. For instance:
Rebecca Goldstein, Edge interview.
Karl Popper was also a quasi-dualist, i.e. he believed that the domain of ideas was really separate reality. As did Gottlieb Frege (see Frege on the Third Realm, Tyler Burge.)
The problem here is the implied dualism, i.e. there is an 'intelligible domain' , and that is impossible to reconcile with standard-issue naturalism. (I am wondering if both 'ontic structural realism' and bio-semiosis are, in their own ways, attempts to reconcile naturalism with the apparent reality of intelligible objects.)
Quoting Wayfarer
In the opening post I suggested one more concrete account of abstract numbers, which is somewhat different. Whether or not this account holds up, was one of the intended discussion points.
It seems some of the posters find some sort of dualism inevitable...?
The difficulty is, though, that whatever the 'substance' is, that appears as 'mind' from some perspectives, and 'matter' from others, is neither! So, work that one out.
Quoting Wayfarer
The diallelus applies, whether you bring up mind, matter or whatever. Some apparently think that matter is simple, uninteresting, lacking, yet that's false; plenty questions in that area of inquiry.
Perhaps, then, a more interesting question is whether both sides of these apparent dualisms can be accounted for. Not necessarily derive one side from the other, just account for both coherently, contextualize sufficiently (without multiplying entities to no end).
Aristotle was right about mathematics after all; James Franklin; Aeon Essays; Apr 2014
As for the broader 'mind v matter' question: I really think an historical perspective is needed. Much of modern materialist philosophy came from the reactions to, or against, Cartesian dualism: put very crudely, they declared that 'res extensia' was all that was real, 'res cogitans' (and god along with it) becoming a mere 'ghost in the machine'. 'All I see, is bodies in motion', said D'Holbach, another of his compatriots, 'the brain secretes thought as the liver secretes bile', and most materialists since have followed suit. Then, also as a consequence, was the attempt to reify or locate mind as 'substance', or give an account for it in neurobiological terms, as 'what the brain does'. This then became underwritten by evolutionary neuro-biology, i.e. you can account for thought in terms of Darwinian adaption, which is what 'naturalism' often means nowadays.
Whereas, from my point of view, that attitude gives rise to what Bas Van Fraasen has said is a form of false consciousness.
In terms of how "mind" and "matter" are are often used in philosophical discussions, Spinoza's Substance is neither. Those terms are, respectively, usually used to refer to experiences (mind) or objects which manifest some sort of sensory affect with space-time.
Substance is logical. The unity expressed by all, whether it be by a rock floating in space or someone's experience. The truth that the distinct aspects of mind and matter are together, without becoming or accounting for the other.
All things are expressed in both Extension (states of existence) and Mind (meaning/logic). The rock by the river equally expresses extension (it exists) and mind (a logical expression, which is distinct from the presence of the object). Substance (God) is the unity of all extension and mind-- the togetherness of all existing states and meanings. It's Cartesian dualism destroyed. The mind and body are never separate, though they are distinct from each other.
When it comes to Platonia or eternity my intuition is that number is present more in a form of geometry, wherein it is an expression of something more transcendent and fundamental. A kind of divine geometry.
We are "surrounded" by abstractions. It is impossible to explain reality without appeal to abstractions. E.g. we can't explain evolution without referring to abstract replicators. Certain entities, like the perfect circle, or the set of all primes are purely abstract, but that does not mean that we are condemned to Platonism.
Plato's claim is that, since we have only access to imperfect circles, we cannot obtain any knowledge of perfect circles. But, we don't have access to planets either; we only have access to images of planets.
That's an interesting observation. Notice the use of scare quotes, because we can't be literally sorrounded by abstractions, as they're not in physical space. Instead they are, indeed, part of the means by which we explain, or make sense of, the impressions and perceptions that constitute reality. Note that logic is 'the relationship between ideas', and yet logic is foundational to scientific method and its applicability to all kinds of issues.
That's true, but it's not so clear-cut. The principle dialogue about the nature of knowledge is the Theaetetus. It is of note that there is no definite resolution to the question of the nature of knowledge in that debate, instead it is aporetic: the dialogue suggests various answers to the question of 'what is knowledge', but it does come to a conclusion about it.
Have a listen to the two minutes of this lecture beginning at 38:00 - Lloyd Gerson (one of the current academic experts on Platonism) on Aristotle's account of what 'knowing a form' entails:
Some entities are purely abstract, like the set of prime numbers, but many abstractions are physically instantiated, and we are immersed in those. Every cell in our bodies contains abstract information in physical form. Indeed, humans *are* abstractions. Then of course there is our culture, knowledge and technology! If you happen to live in the right place, you might notice that cicadas become noisier every 13 or 17 years *because* these numbers are prime!
While "abstraction" can refer to several things, there are abstract entities that can only be explained in a way that attributes independent existence to them, such as the Natural numbers.
Consider:
1 is a natural number.
Each natural number has precisely one successor.
1 is not the successor of any natural number.
Two natural numbers with the same successor are the same.
Look, no mention of primes, how primes are distributed on very large scales, whether the distribution is "random" or not. Indeed, there is an entire field of mathematics - number theory - devoted to the study of these entities that are so easily defined. I would appear that the natural numbers are complex, autonomous, and therefore real!
I think it means that abstractions are real, autonomous, causal, and as much a part of reality as anything else. Mostly, however, we gain knowledge of them by proving our conjectures, rather than testing them.
Agree with all the above, except to note that humans are rather more like 'instantiations' than 'abstractions'.
Hereunder a snippet from my first post on the old Philosophy Forum (in January 2009). It recaps some of the points explored here.
*what I meant here, was that numbers cannot be said to be 'mind-independent', in the way we say that objects are, because they can only be grasped by a mind.
(Incidentally, one of the books I have read and enjoyed on this theme is Is God a Mathematician? by Mario Livio.)