You clearly did not watch the video.

I tried to account for abstract numbers by falling back on the concrete world. — jorndoe

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.

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. — 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.

I don't think we can deconstruct Platonism, but we don't have to accept it either it's just part of philosophy. If Plato hadn't come up with it, someone else would have, I probably came up with the rudimentary idea myself before I read Plato.

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.

...and your point is?

Well, you cant account for numbers that way. — tom

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.)

If there were an infinite amount of sand, storing it might be problematical, because there wouldn't be room for anything else..... — Wayfarer

Hilbert's Hotel says differently. ;)
It's called a paradox, but it's not actually contradictory, it just has counter-intuitive implications.

'You can check out, but you can never leave'.

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.

Quite telling that Hardy admitted that he couldn't explain the kids math skillz, that clearly necessitates that a divine otherworldly being is responsible. The implication being not that Hardy could come up with no other logically possible explanation, but rather a more plausible one. That is the only thing that renders this suggestion sensible -- but unfortunately also ridiculous. Presumably lots of explanations, whether true false, or ridiculous sounding to people that hold different values and authorities as paramount, would be available to Hardy. He could probably even think of some stupid ideas he'd still think more plausible than magic, but still obviously false.

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.) — Jorndoe

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:

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.

Rebecca Goldstein, Edge interview.

“Platonic-mathematical, physical, and mental – has its own kind of reality, and where each is (deeply and mysteriously) founded in the one that preceeds it (the worlds being taken cyclicly). I like to think that , in a sense, the Platonic world may be the most primitive of the three, since mathematics is a kind of necessity, virtually conjuring its very self into existence. — Roger Penrose

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.)

I know of Ramanujan. Studied some of his mathematics many years ago. One of my (several) heroes. There are many known (child) prodigies in all kinds of areas. Ramanujan was surely one of them.

directed towards providing a naturalistic account of numbers — 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...?

I don't see how dualism can be avoided, although I think of all the various 'standard Western positions' available, the one I'm nearest to is 'dual-aspect monism'. Looking at the Wikipedia article on double aspect theory most of the featured names are on my whitelist, so to speak (although I have never been able to make head or tail of Spinoza.)

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.

Not a whole lot of objections to the attempt at letting abstract numbers fall back on the more concrete world. :-/

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. — 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 it happens, I know Jim Franklin - when I was managing a uni computer store, he used to be one of my customers. I read his paper when it came out, and it's close to the way I see it. Aristotle is, if you like, a critical or a dissident Platonist, so that view is nearer Platonism than naturalism, although I still don't think that his depiction of the way Platonists understand the reality of universals is exactly correct. But as Franklin points out, it is an argument that has been waged for millennia, and maybe will never be resolved. And generally speaking I find a lot more to agree with in his paper, than to disagree with.

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.

Yes, I was thinking of dual aspect monism. I favour the thought that number is something that comes with manifestation, like time, or extension. This is not to say that they do not exist in Platonia, eternity, or other forms of manifestation. But I don't think we can make any hard conclusions about what happens elsewhere, or that we have stumbled upon anything fundamental.

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.

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.) — jorndoe

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.

, does that mean Platonism holds...?

We are "surrounded" by abstractions. It is impossible to explain reality without appeal to abstractions. — Tom

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.

Plato's claim is that, since we have only access to imperfect circles, we cannot obtain any knowledge of perfect circles. — Tom

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:

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. — Wayfarer

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.

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! — Tom

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.

Consider number. Obviously we all concur on what a number is, and mathematics is lawful; in other words, we can't just make up our own laws of numbers. But numbers don't exist in the same sense that objects of perception do; there is no object called 'seven'. You might point at the numeral, 7, but that is just a symbol. What we concur on is a number of objects, but the number cannot be said to exist independent of its apprehension - at least, not in the same way objects apparently do*. In what realm or sphere do numbers exist? Where are numbers? Surely in the intellectual realm, of which perception is an irreducible part. So numbers are not 'objective' in the same way that 'things' are. ...

I started wondering, this is perhaps related to the Platonic distinction between 'intelligible objects' and 'objects of perception'. Objects of perception - ordinary things - only exist, in the Platoniist view, because they conform to, and are instantions of, laws. Particular things are simply ephemeral instances of the eternal forms, but in themselves, they have no actual being. Their actual being is conferred by the fact that they conform to laws. — Jeeprs

*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.)