I would say that number consists in difference and similarity; which can be both perceived and conceived; the former being concrete and the latter abstract. I'm not proposing any absolute division between the concrete and the abstract; because they do partake of one another.

The further point is that the fact ( if it is a fact) that a perception must be (to at least some minimal degree) conceptually mediated, does not entail that a perceptual experience is a concept. Similarly then, an experience of material concreteness is not a concept of material concreteness. — Janus

My argument is that conception and perception would be the two extremes of the one process. To speak of one versus the other is merely to highlight the grounding generality vs the focally individuated particular. You actually have nothing without having both together striking an appropriate balance. So you need three things. The limits of the conceived, the limits of the perceived, and the outcome of that which is then "the experiential state".

That is a holistic description of experiencing as a general psychological process. And that would apply to the business of modelling the world whether it was neurological, linguistic, or mathematical.

But then you introduce the further issue of how the different levels of semiosis might relate - given that they do seem differentiable in principle, but also again are always in action together in any brain trained to understand its world in a "modern" way.

So you want to argue something familiar - the biology does the perception, the sociology does the conception. And there is a rough truth to that. But I am saying it becomes an unacceptably rough description if we want to be thorough-going metaphysical holists. It wants to make absolute a distinction which can't absolutely exist.

My original response was a result of you both accusing me of misplaced concreteness and then refusing to recognise that you yourself were doing just that with your own talk of "materiality" or "tangible objects".

Science is always talking about matter. But it long ago dropped most of what most folk think "matter" means from their everyday linguistic talk about the kind of neurobiological perceptions they experience. Science is now talking about matter as fields, and fields as information. It is galloping along because it doesn't restrict itself to ordinary language descriptions of neurobiological conceptions.

I was pointing out how your own use of the term was still overly concrete in taking the meaning of "material" as just something one could simply point at the world and exclaim: "See, right there before your eyes, just as it looks and feels." To avoid the charge of naive realism, or simple epistemic confusion (they are usually the same thing), you would have to show how you were meaning something more sophisticated.

I would say that number consists in difference and similarity; which can be both perceived and thought about. — Janus

And are difference and similarity properties of physical objects?

If the Platonists had their way, what is 'real' would be the eternal, the changless, and the deathless: the diametric opposite of everything anyone would recognize as real; such is the Platonic hatred of the world and everything and anyone in it. And that people often take this inverted, hollowed out notion of reality for granted is a sad indictment on a certain intellectual poverty that still exists and cries out for eradication.

No, I would say that objects have qualities or characteristics which are different from and similar to other objects. So it is the qualities or the characteristics which are the properties of objects, not difference or similarity per se. In other words I think it is better to think not in terms of qualities constituting difference and/ or similarity as such, but in terms of qualities being different and/ or similar to other qualities.

My argument is that conception and perception would be the two extremes of the one process. — apokrisis

I agree and I say as much in the quote directly above.

No, I would say that objects have qualities or characteristics which are different from and similar to other objects. So it is the qualities or the characteristics which are the properties of objects, not difference or similarity per se. In other words I think it is better to think not in terms of qualities constituting difference and/ or similarity as such, but in terms of qualities being different and/ or similar to other qualities. — Janus

Difference and similarity are properties of relationships then? Your presentation is stuffed full of abstractions. I pointed this out before: they're kind of indispensable.

My own opinion is that form and the object of formation are products of reflection. In very much the same way we separate ourselves out from the world on reflection, we separate universals from particulars. Then from that dismantled arrangement, we try to understand the world. Make sense?

I think the key point here is that saying a number is what it is regardless of what anyone thinks about it is not the same as saying that a number is what it is independently of all thought whatsoever. — Janus

Exactly right; according to Peirce, reality is independent of what any individual mind or finite collection of minds - including, notably, the collection of all actual minds - thinks about it; but reality is not independent of thought in general. As he once put it, "just as we say that a body is in motion and not that motion is in a body, we ought to say that we are in thought and not that thoughts are in us." In fact, another of his definitions is that reality is whatever would be included in the ultimate consensus of an infinite community after infinite inquiry. This is obviously a regulative ideal, not something that could ever actually be achieved.

To try and clarify why I remain unconvinced, consider the author's synopsis of his paper:

If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent from us. Both alternatives challenge mathematical platonism.

So, M is the "platonic world" of mathematical facts. The author observes that if M is too large then it is uninteresting to us, and if it is smaller and interesting then it is not independent of us.

Perhaps I fail to grasp the dilemma, but can't we just accept both the existence of a very large M and also that we are only interested in a small subset of it? I don't see why our interest in only a small subset of M - the part which we find interesting or useful - should falsify the existence or independence of the much larger M.

If we assume from the outset that M is the "platonic world" of mathematical facts - even those which are of no interest to us - then we cannot also say that M might contain only those mathematical facts that we find interesting (and therefore M is not independent from us). And only one of these resembles mathematical platonism.

The key passage is the following:

"Mathematics may be the investigation of structures. But it is not the list of all possible structures: these are too many and their ensemble is uninteresting. If the world of mathematics was identified with the platonic world M defined above, we could program a computer to slowly unravel it entirely, by listing all possible axioms and systematically applying all possible transformation rules to derive all possible theorems. But we do not even think of doing so. Why? Because what we call mathematics is an infinitesimal subset of the huge world M defined above: it is the tiny subset which is of interest for us. Mathematics is about studying the “interesting” structures". (my emphasis)

Note the identification of what is mathematics with what is 'important' to us; or, contrapositively, the exclusion of most of M as that which is not mathematics. Or more starkly still: most of M is not mathematics. It's not that we pick out some of interesting parts of math out of a wider set of math: it is that what we don't pick out is not even considered math. This is the import of Rovelli's metaphor of the sculpture and the stone: the stone really does 'contain' every possible sculpture that could be made from it, but what it contains is a kind of sheer potential, indefinite and undifferentiated such that the stone cannot be identified with 'every possible sculpture'. The stone is not a sculpture in the same way that M is not to be identified with math ("Mathematics... is not the list of all possible structures").

Importantly, this is not an assumption that Rovelli makes: this really is how math is, how it 'works'. So the question is: why does math look the way it does (and not otherwise)? What selection principle was employed to sculpt the indefiniteness of M into what is, in fact and in reality, considered math? What Rovelli essentially points out is that Platonism can provide no such principle, because it specifically divorces math from the practice of mathematical activity, which it considers something of an epiphenomenon, and which thus cannot play any constiutitve role in defining math. This is in contrast, Rovelli points out, to how math actually proceeds, wherein 'interest' provides just the selective principle that is missing from Platonism. And if this is the case, then Platonism cannot possibly be true.

That's the argument at play here; those who think that the paper simply proceeds on the basis of begging the question simply lack any basic comprehension ability.

interest' provides just the selective principle.... — StreetlightX

Phew. Natural selection saves the day, again.

Yep. The universal acid, at work in math no less than animals. The death of God by caustic immolation across whatever asylum one wants to find for him.

So the argument goes:
1. Mathematical platonism is the view that mathematical reality exists by itself, independently from our own intellectual activities.
2. But what we call mathematics - containing only what is important to us - is but "an infinitesimal subset" of mathematical platonism.
3. Mathematical platonism cannot explain which mathematical facts are important to us.
4. Therefore, mathematical platonism is false.

?

This is very clearly false. It conflates the object of a sign with the sign itself. — aletheist

No it does no such conflation. It states the simple fact that it is impossible, that there is the object which corresponds with a particular sign without the existence of that sign. Correspondence requires two things, the sign and the object. Without the sign there is no correspondence, therefore no object which corresponds.

The reality of a character, and the existence of things that possess it, is very clearly independent of any particular system of signs that represent that character and those things. Otherwise, the same claim would apply to the world - i.e., it is absolutely impossible that there was a world before there was the word "world" - which is obviously absurd. — aletheist

Can you not grasp the very simple fact that it is impossible to have "the character which is represented" without the sign which represent? One is clearly prior to the other as "the character which is represented" requires for its existence, the act of representation, and this act is dependent on having a sign which represents.

What you describe as "obviously absurd" is also obviously true. Sometimes truth is stranger than fiction. There is something which the word "world" refers to. Prior to the existence of this word, there was nothing that the word "world" referred to. Therefore there was no world prior to the word world.

However, many of us project, and claim that the thing which is now referred to by "world" had existence prior to the word. This requires a separation between the thing and the word. The thing must be conceived of as independent from the word. But when we conceive of the thing as separate from the word, we deny any necessary relationship between the thing and the word. So there is no logic which allows us to claim that this independent thing, which is conceived of as existing independently from the word associated with it, is actually the thing which is referred by this word. Therefore the claim involved with this projection, that the independent thing is the thing referred to by the word, is unsupported logically. The thing which is assumed to have existed before the word, cannot be proven to be the same thing as the thing referred to by the word. This inability of logic to prove that the independent thing is the thing referred to by the word, is indicative of the simple fact referred to above, that there is no such thing as the thing referred to by the word, without the word.

To distill the argument most concisely, I'd get rid of any reference to interest at all, in order to clearly see the 'negative' import of the argument. So something perhaps like this:

P1. Any account of mathematics would need to explain why mathematics is the way it is.
P2. Mathematical Platonism is the view that there is a world M, that contains all possible mathematical objects and truths.
P3. Mathematics is but "an infinitesimal subset" of any such mathematical reality.
P4. Any account of mathematics would need to explain why P3 is the case, in order to satisfy P1.
P5. Mathematical Platonism has no way to explain why P3 is the case.
C1. Mathematical Platonism cannot satisfy P1.

Ergo, Mathematical Platonism fails to have any explanatory force with respect to mathematics.

I 'excluded' the question of interest because the argument works without it. 'Interest' is Rovelli's effort to provide a positive explanation that he finds lacking in Platonism. The negative argument works without any reference to it. Rovelli weaves both the positive and negatives aspects of the argument together in the paper, but isolating the negative aspect makes the 'argument against Platonism' easier to see, I think. I'm not super confident about my construction of syllogisms (it's not something I'm trained in, and I find it hard to think with them), but I'm happy to hash this out if possible.

I don't know. It just sounds a bit like bemoaning the fact that mathematical platonism is unable to tell us which mathematical facts are interesting to us (or "why mathematics is the way it is"), despite mathematical platonism being the view that mathematical facts exist independently of our intellectual activities.

That's why I removed any reference to interest, so it would be literally impossible to 'sound like' that. Arguments I can deal with. Hearing imaginary noises - not my problem.

Mathematics is the way that it is because of its interest or usefulness to us, surely, so interest is still there in your P1.

That would be begging the question. And in any case, no self-respecting Platonist would agree.

Put it this way: Why should mathematical platonism be entirely invalidated by its supposed inability to account for an infinitesimal subset of itself?

That's not what it can't account for. Read again, I'm not helping you here.

Is it really true to say "all math is quite logical"? Within mathematics in general, there are numerous contradictions such as Euclidean vs. non-Euclidean geometry, imaginary numbers vs. traditional use of negative integers. — Metaphysician Undercover

Have been away (so this is an answer to page 4)

Euclidean and non-Euclidean geometry simply starts from different premises (or should I dare to say axioms). The geometry on a blank paper and the geometry on a sphere are different, but their existence doesn't make one or the other illogical. The only mistake is if you assume that all geometry is, let's say Euclidean (and that the parallel postulate is universal). That argument is wrong, but it doesn't make either geometry illogical. Especially in set theory you can choose your axioms and have different kinds of set theories with different answers, but that in my mind don't make them illogical.

You might argue that it is just different branches of mathematics which employ different axioms, but if one discipline (mathematics) employs contradictory premises, can it be true to say that this is logical? — Metaphysician Undercover

Premises (axioms) can make the math to seem contradictory, but can be totally logical. Only if you prove that something that we call an axiom is actually false, then is the statement simply wrong.

Thank you for so convincingly demonstrating the patent absurdity of nominalism.

It's only a 'conflation' if one assumes from the outset the Platonic position on mathematical objects. — StreetlightX

No, not really. If you begin with the neutral position, it is the one making the argument that begs the question.

It doesn't matter if you're a Platonist or not; the argument is simply bad.

Hi! Would you mind presenting a quick explanation of the argument? I'll pay you in hamburgers.

I don't see how his argument refutes mathematical Platonism at all. It's not even historically accurate for one thing, and his conclusion does not follow from his premises. There has been a lot of cases where seemingly useless mathematics was developed, only for us to discover later on it was quite useful. But, mathematicians, at least pure mathematicians, don't just focus on what is useful and ignore mathematical topics as a result. Applied mathematicians may, but not the purists.

Even if we assume a lot of math is useless, or, let's even assume the vast majority of it is, so what? How does it then follow from this claim that Platonism does not exist? It doesn't.

I find affinities with (late) Wittgenstein's view, which in general I find the most appealing view on math. The emphasis on mathematical practice and on the selection principle are topics which preoccupied W. too. I'm curious if you're familiar with W's view as expounded by Rodych and of your opinion on it — Πετροκότσυφας

Yes! I think basically think that Wittgenstein basically hit the nail on the head with his reflections on math and that everyone else has more or less been playing catch-up ever since (and failing rather miserably, at that!). That said, I say this only having gleaned Witty's position from some selective reading of the Lectures and primarily the work of Bob Clark and Paul Livingston. I read the two papers you linked by Rodych, and while I have minor quibbles (I wouldn't call Witty a finitist - or an 'infinitist', for that matter, insofar as I think his position explodes the terms of that debate - in a productive manner), I really liked the way they tracked Witty's evolving thoughts on math across his work.

But yes, my enthusiasm for Rovelli's paper is partly coloured by the Wittgenstinian hue with which I bring to it.

Except, as I detailed above, it does not.

Hi! Would you mind presenting a quick explanation of the argument? I'll pay you in hamburgers. — frank

It's difficult for me to improve much on @fdrake's summary of Rovelli's argument, earlier in this thread. This is a broadly negative argument, however, that consists in highlighting that the version of mathematical Platonism which Rovelli is targeting incorporates too many items into the set of what Platonists themselves intuitively feel are entitled to be counted as intelligible mathematical patterns. The argument relies on the acknowledgement that the manner in which we sort out the wheat (fruitful mathematical theories) from the chaff (unprincipled and uninteresting sets of axioms) reflects contingent features of our specific form of life. This consideration, supporting the negative argument, fails however, it seems to me, to properly account for the fact that mathematical truths appear to have a grade of necessity (and degree of generality) somehow intermediate between, and qualitatively distinct from both, logical necessity and pure contingency. (I've suggested that Kant, and neo-Kantians such as Sellars, are gesturing towards the right kind of necessity with the concept of synthetic a priori propositons). But that's not what you're asking about. Maybe I'll comment more about this in another post.

We become aware of our own forms of life because we can compare ourselves to people in other cultures or other eras. Taking that idea deeper isn't uncontroversial, though, is it? Aren't we just speculating that there could be sentient beings who see a radically different world from our own?