In defense of a dogma seems like a really fun article. I just started it and I'm impressed by the style, precision and generosity of the argument!

It's reasoning made for and by idiots. — StreetlightX

Well you have to make allowances for the reading age of the audience, who are, after all, only ‘apes on a rock’.

Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. — StreetlightX

I think this is already starting on shaky grounds. What does "exist" mean here? Mathematically it would mean something like "provable" but if we're dealing with a possible world in which everything is provable, we're dealing with the Trivial World. A world without any coherent structure at all. You later ask'

This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder — StreetlightX

And often what mathematicians mean by "mathematical interesting"ness is some set of results that do not entail triviality (every sentence becoming a theorem). Because if some math explodes into everything it loses coherency and thus can't really be analyzed at all. Non-triviality is the baseline for what mathematicians consider a theory worth investigating.

Far from being stable and universal, our mathematics is a
fluttering buttery, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.... The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather." — StreetlightX

I'm not really sure what his argument is supposed to be. Math platonism doesn't say math is universal in the sense he is assuming. I'd say I'm a mathematical pluralist, so I don't find myself committed to any particular math system as a matter of principle, but this doesn't preclude being a math platonist. But the preference of real numbers in the scenario he refers doesn't seem to contradict math platonism unless I'm missing something. It's just a scenario where it's more useful to apply real numbers.

The mistake that animates modern naive empiricism is to explain the phenomenon of convergence -- such as the discovery of laws of nature, or of general logico-grammatical features shared by (most) natural languages -- as a result of the faithful (or approximate) reproduction, as contents of our mental representations, of the structure of an independently existing empirical ('external') reality. — Pierre-Normand

That would naturally flow from representative realism, in that there is a tacit assumption that the mind 'mirrors' or 'represents' the world, which ultimately stems from Locke, and is ubiquitous in empiricism.

One might avoid both mistakes though recognizing that the phenomenon of convergence is a dynamical product of the enactment of the social practice of arguing for or against doing and/or believing things. — Pierre-Normand

However, the cardinal difference between modern and ancient philosophy in this regard, is that modern philosophy is largely underwritten by biological science, which situates h. sapiens along a continuum with other species, and which brackets out any sense of there being either a first cause or final end.

Whereas

For Platonism, the universe is itself alive and filled with living things. Soul is the principle of life. Life is not viewed as epiphenomenal or supervenient on what is non-living. On the contrary, soul has a unique explanatory role in the systematic hierarchy. Though soul is fundamentally an explanatory principle, individual souls are fitted into the overall hierarchy in a subordinate manner. ...

Persons belong to the systematic hierarchy and personal happiness consists in achieving a lost position within the hierarchy. All Platonists accepted the view that in some sense the person was the soul and the soul was immortal. Since perhaps the most important feature of the divine was immortality, the goal or telos of embodied personal existence was viewed as ‘becoming like god’

Lloyd Gerson, What is Platonism?

This, though, opens up a new question - what is 'interesting?' Well, interest simply is in the eye of the beholder: we develop some parts of M and not others because those parts help us do stuff, alot of the time. — StreetlightX

My point was there are reasons to think the structures and relations we use math to model exist in the world independent of us, since they led to us existing. — Marchesk

Yep. The issue is that reality and maths can seem very far apart when one is being understood in terms of the physics of constraints and the other is being viewed as a free grammatical construction. So junk can spew out of a syntax freely and meaninglessly - as in the examples like Borges’ library which Rovelli uses. But then reality and maths can seem fundamentally connected - as is the case of mathematical physics, where Rovelli is one of the leading players. When some kind of intelligible constraint - a suitable selection principle - is applied to the random junk spewing, then that gets closer to making sense of the relation.

To be fair to Plato, he did suggest such a finality. He argued for the Good as an optimisation principle that revealed the truth, beauty and justice of a subsidiary realm of mathematical (and other) forms. So that was of course still rather mystical. But it was at least recognising that something higher - a global constraint - was needed to pick out the most meaningful or fundamental patterns and structures.

Several points. It should be noted that if there turn out to be multiple descriptions of the same thing, then that thing seems to be something beyond the mere descriptions. Rovelli gives several examples. The choice between Euclidean and spherical geometry. Heisenberg's matrix mechanics and Schrodinger's wave mechanics. And as Cartesian geometry showed, in general, algebraic maths and geometric maths offer dual descriptions of everything that seems of mathematical interest. So Rovelli seems to want to use this multiplicity of descriptions to argue for social construction. Yet it also argues strongly that everyone is feeling the same elephant. Like any use of language, it is being constrained by its encounters with a recalcitrant reality beyond.

Then if we consider how maths actually advances, it is largely by the relaxation of constraints. Maths moves up to a higher level by generalising and abstracting. Euclidean geometry gives way to non-Euclidean geometry. Then geometry gives way to topology. So the map of maths - M - is not some flat plane of existence, cluttered mostly with junk with a few bright spots of interest. Instead, maths is itself a hierarchical structure. We rise up out of the clutter of the particular detail by finding some key constraint we can relax. And having gained control over that parameter, we can then add back constraints to toy with different worlds.

Give up Euclid's parallel postulate and the geometry of space can be curved. That curvature can be added back as some positive, negative, or neutral number. So what makes maths interesting is where we find the constraints that are holding things in place - imposing a particular structure or form - and can then twiddle the knobs to discover the "world" the more abstracted description lives in. Keep abstracting and you go past set theory to arrive at category theory - at least according to current wisdom. There is an ur-form at the top of the mathematical hierarchy that speaks to what is Platonically the Good - a basic idea of a relation which is the most generalised possible constraint we can imagine.

So Plato was certainly on to something. Our physical world is not some random junk of accidents. It has an intelligible structure. But the problem with Platonism is the way it suggests a flat plane of forms - where the perfect triangle exists alongside the perfect turd. That problem can be addressed by the addition of a selection principle - a hierarchical story - which does then separate reality into its accidents and its necessities. And in Aristotelian fashion, this is what you get when the material realm becomes the source of accidents or fluctuations - the blind atomistic construction - and the formal realm supplies the downward-acting constraints which are a system's regularities and inevitabilities, its essentials, or universals, or necessities.

So we recognise a triangle as speaking purely of nature's necessities. A three sided polygon is going to have internal angles of pi, or 180 degrees. This is a truth of physical space - at least in a Euclidean setting. But a turd seems mostly a set of physical accidents. A very material construction. A mathematician might find the structuring formal principles that do in fact regulate the shape of any given turd - why a liquid one might behave differently from a more solid one. But Plato's realm of forms is really a space of abstractions. Maths explores the constraints that nature imposes on material accidents - and mostly seeks to abstract them away, because what is interesting is to gain control over them.

Most of the apparent physical constraints we encounter turn out to be accidents of history. The Universe seems Euclidean to us because it has grown so large, flat and cold. So we can generalise away the particular in the world as it seems structured right now, to work back towards to the way a world could possibly be - if we were to add other general material conditions, like it being as small and hot as possible.

The mathematical enterprise is thus about trying to discover the rational forms that structure some set of material accidents. And that dualism has to give way to the triadicism of a hierarchical metaphysics as that exercise gains scientific sophistication. Science speaks of global laws and material constants. Actual substantial existence is then what arises inbetween.

So the turd becomes an entity that arises in some sense because of the shaping laws of dynamical flows - a general model that would be paired to specific material parameters, like a measure of the viscosity of the turd in question. It is this triadic complexity, this hierarchical or systems story, which the debate over Platonism is so insensitive to, but which should already be evident from the Aristotelian version.

Then a final point. The usual view is that either maths is Platonic, or it is purely a social construction. Either it speaks truly of necessity, or it is merely always a cultural accident - reality as described from a basically freely creative human subjective point of view.

But I see nothing wrong with the inbetween position. What we are getting at with mathematical physics at least is the objective point of view - the one from the perspective which would be the Cosmos contemplating its own rational structure.

So that does apply a pragmatic constraint to the enterprise - a view that has an embodied interest. The realm of junk maths is being limited by a viewpoint which speaks to the basic finality of wanting to be embodied. It is maths of that type, structure that can produce that result, which is "true, just and beautiful". We can see the universality in it as we stumble across it.

The maths of symmetry and symmetry-breaking are a good example of that. Likewise statistical mechanics and dissipative structure theory. There are areas of maths that look very organic - because they marry the accidental and the necessary in a way in which global regularity must emerge from local randomness. Structure - as the stabilisation of instability - can develop.

So sure, when maths is understood as just a realm of everything that unconstrained syntax will produce - a Borges library - then it seems to bear no real relation to a reality in which limitation or finitude is apparent everywhere. But when maths is viewed organically - as a language to capture the emergent regularities of pure possibilities - then that is a strong selection principle to sift the wheat from the chaff. We arrive at the structures that matter because they are the most irresistible. Randomness can't erode them, because randomness is in fact constructing them.

I'm sympathetic to that view. Constrained math has a relationship with reality. Aristotle's view was more correct than Plato's. The in-between position seems more reasonable.

I like this:

hat we are getting at with mathematical physics at least is the objective point of view - the one from the perspective which would be the Cosmos contemplating its own rational structure. — apokrisis

Our physical world is not some random junk of accidents. It has an intelligible structure. But the problem with Platonism is the way it suggests a flat plane of forms - where the perfect triangle exists alongside the perfect turd. That problem can be addressed by the addition of a selection principle - a hierarchical story - which does then separate reality into its accidents and its necessities. — apokrisis

Platonism - and Aristotle - both assume an hierarchy, but it's from a top-down, not bottom up, perspective.

'The Platonic view of the world – the key to the system – is that the universe is to be seen in hierarchical manner. It is to be understood uncompromisingly from the ‘top-down’. The hierarchy is ordered basically according to two criteria. First, the simple precedes the complex and second, the intelligible precedes the sensible. The precedence in both cases is not temporal, but ontological and conceptual. That is, understanding the complex and the sensible depends on understanding the simple and the intelligible because the latter are explanatory of the former. The ultimate explanatory principle in the universe, therefore, must be unqualifiedly simple. For this reason, Platonism is in a sense reductivist, though not in the way that a 'bottom-up' philosophy is. It is conceptually reductivist, not materially reductivist. The simplicity of the first principle is contrasted with the simplicity of elements out of which things are composed according to a 'bottom-up' approach. Whether or to what extent the unqualifiedly simple can also be intelligible or in some sense transcends intelligibility is a deep question within Platonism.

....

The hypothesis that a true systematic philosophy is possible at all rests upon an assumption of cosmic unity. This is Platonism's most profound legacy from the Pre-Socratics philosophers. These philosophers held that the world is a unity in the sense that its constituents and the laws according to which it operates are really and intelligibly interrelated. Because the world is a unity, a systematic understanding of it is possible' (Lloyd Gerson, 'What is Platonism')

(Incidentally, whether there is indeed 'cosmic unity' is very much in question in current physics, is it not?)

What we are getting at with mathematical physics at least is the objective point of view - the one from the perspective which would be the Cosmos contemplating its own rational structure. — apokrisis

That also has a precedent in Aristotle - 'The Prime Mover is simply the formal-noetic structure of the cosmos as conscious of itself' [ibid].

Platonism - and Aristotle - both assume an hierarchy, but it's from a top-down, not bottom up, perspective. — Wayfarer

Yep. Constraints act top-down.

But the tricky part - which the maths of hierarchy theory realises! - is that the causality has to go both ways. There is also the upward construction kind of hierarchy. So the story becomes about the synergy between parts and wholes. You have compositional hierarchies that are the bottom-up view, and subsumption hierarchies that are the top-down view. And a stable reality can only emerge when the two are reinforcing each other's existence.

This is the kind of balance of causality that is described by dissipative structure theory - such as Bejan's work on material flows and "constructal law" - https://www.forbes.com/sites/anthonykosner/2012/02/29/theres-a-new-law-in-physics-and-it-changes-everything/#33e609ee618d

For this reason, Platonism is in a sense reductivist, though not in the way that a 'bottom-up' philosophy is. It is conceptually reductivist, not materially reductivist. — Wayfarer

I agree with the quote. But of course calling it "conceptual" is dangerous as it does suggest the mental.

For me, calling it structurally reductive would be better. My goal would be to avoid lapsing into actual mind~matter dualism on this one.

Lloyd Gerson, What is Platonism? — Wayfarer

It's funny that you would mention Gerson's book, since I added it to the Platonism folder in my digital library a few hours ago, having found it thanks to the title of the first chapter: "Was Plato a Platonist?" (which was also the title of the paper by Konrad Rokstad that I referenced earlier).

It's not difficult to explain, although it might be difficult to accept, or to understand. — Wayfarer

Nothing in the quoted passage explains what kind of existence (apart from the 'ideal' existence they have insofar as they are thought by humans) these abstract so-called "objects" enjoy.

Does it follow from the fact that the world is numerable, and of course numerable in the same conventional way for all (a fact that reflects the common culture of human practices), that numbers have some kind of abstract ontological existence independent of human life? This would be an objective abstract existence, (if any sense could be made of that).

You seem to contradict yourself insofar as it is you that is always saying that there is no (concrete) objective reality apart from subjective human experience. If there is no objective concrete reality ( we can at least make sense of the idea of concrete objective existence, as the default tendency to naive realism attests) how much less would there be an objective abstract reality (an idea that we seem to be able to make no sense of at all)?

You often seem to want to claim that the distinction between reality (or being) and existence can explain platonist claims. I think this is untrue, because the terms and the notions they are associated with are more or less interchangeable, even though it is also true that under certain restricted interpretations of them, distinctions can be made. The bottom line, in any question of being, existence or reality, is whether something is concretely real or merely imaginary. That is a real distinction even though the imaginary can in one sense be said to be real, but only insofar as it participates as idea in the concrete act of imagining.

So, for example, the idea of apophatically attempting to dodge the question of God's existence by claiming that he is real but does not exist is a conceptually fraught, perhaps I should go as far as to say intellectually dishonest, strategy. The proper question is whether God's existence is really real (i.e. completely independent of humanity) or really imaginary (completely dependent on humanity)..

(Incidentally, whether there is indeed 'cosmic unity' is very much in question in current physics, is it not?) — Wayfarer

What if what we understand to be "cosmic unity" is due to the fact that everything is 'at bottom' (in terms of the so-called quantum vacuum, say) irreducibly entangled with, and inseparable from, everything else. In that view nature can be it's own designer and creator; no need for any supernaturally transcendent God or abstract (as opposed to virtual) realms of form and number.

I was alerted to the possibility of the distortion by a handful of scattered remarks on Plato versus Platonism by John McDowell. But I haven't pondered much on the historical roots of the distortion, nor do I feel equipped for tracing such roots anywhere earlier than the modern period. — Pierre-Normand

What I understand is that modern-day Platonism is more like Pythagorean idealism. Although the refutation of Pythagorean idealism is commonly attributed to Aristotle, it has been argued that Plato actually laid the grounds for this. Plato worked to expose and clarify all the principles of Pythagorean idealism, and in the process uncovered its failings. I've seen it argued that the Parmenides, though it is quite difficult to understand, serves to refute this form of idealism.

For this reason, Platonism is in a sense reductivist, though not in the way that a 'bottom-up' philosophy is. It is conceptually reductivist, not materially reductivist. The simplicity of the first principle is contrasted with the simplicity of elements out of which things are composed according to a 'bottom-up' approach. Whether or to what extent the unqualifiedly simple can also be intelligible or in some sense transcends intelligibility is a deep question within Platonism. — Wayfarer

I don't understand how you can read this passage - which I agree with almost entirely - and not think to yourself: "gee this Platonism business is just as shittily reductive as the materialist reductionism that I'm always moaning about". But of course you don't care about reductionism - you just want reductionism in the 'right way'.

Nice article. Thanks for sharing it.

I think this is already starting on shaky grounds. What does "exist" mean here? Mathematically it would mean something like "provable" but if we're dealing with a possible world in which everything is provable, we're dealing with the Trivial World. A world without any coherent structure at all. — MindForged

I believe that's exactly the point: M would be entirely trivial. This is the dilemma that the paper poses for Mathematical Platonism: either M is trivial and has no structure whatsoever (and thus largely says nothing at all about our world), or, if M is not trivial in this way, then it cannot be independent from our intellectual activity. In either case Platonism is undermined because if the former, then it has no explanatory power, and if the latter, then it simply isn't Platonism.

I'm not really sure what his argument is supposed to be. Math platonism doesn't say math is universal in the sense he is assuming. I'd say I'm a mathematical pluralist, so I don't find myself committed to any particular math system as a matter of principle, but this doesn't preclude being a math platonist — MindForged

The paper's index of Platonism has to do with the independence of mathematics from human intellectual activity; part of this, in turn, has to do with the modal status of our math: contingent or necessary, and to what degree? Rovelli's answer is a kind of qualified contingency: our math is contingent ("Which tiny piece of M turns out to be interesting for us, which parts turns out to be \mathematics" is far from obvious and universal. It is largely contingent"), but this contingency in turn is premised upon the kind of beings we are, and the kind of things we encounter in the world, along with what we do with them - which lends our mathematics a kind of empirical necessity (Rovelli doesn't use that term, but I think it's appropriate in this context).

In defense of a dogma seems like a really fun article. I just started it and I'm impressed by the style, precision and generosity of the argument! — fdrake

I'm glad you like it. I think it has some relevance to the present topic since what is at issue, in Rovelli's polemics against the Platonic thesis that the domain of interesting mathematical objects might be identified with an ideal universe 'M' allegedly knowable a priori, is the contingency of our constitutive relation to what it is that we justifiably find interesting in mathematics (in such a way that it can so much as count as genuinely mathematical). Strawson and Grice, however, began loosening up, in their response to Quine, the false dichotomy between a prioricity (wrongly construed as a purely epistemic notion) and contingency (wrongly construed as a purely metaphysical notion), long before Kripke. They thereby recover some insight from Kant about the requirement for synthetic a priori propositions. (Much confusion arises, though, from inconsistencies in the use of the analytic/synthetic, a priori/a posteriory, and necessary/contingent pairs of predicates). This false dichotomy is something I'll have a bit more to say about in another post.

What I understand is that modern-day Platonism is more like Pythagorean idealism. Although the refutation of Pythagorean idealism is commonly attributed to Aristotle, it has been argued that Plato actually laid the grounds for this. Plato worked to expose and clarify all the principles of Pythagorean idealism, and in the process uncovered its failings. I've seen it argued that the Parmenides, though it is quite difficult to understand, serves to refute this form of idealism. — Metaphysician Undercover

Thanks very much for those reminders. That's indeed an important thread of the history of ideas to be reminded of. Maybe Aristotle shares some part of the blame for having appropriated (while he doubtlessly improved on some of them) some features of Plato's criticism of idealism while ascribing to Plato himself theses that Plato (or Plato's Socrates) was merely expounding rhetorically. Aristotle's third man argument, if I remember, which is directed by him against Plato, is actually borrowed from Plato, if I remember.

I believe that by the time Aristotle was writing, "Platonism" was already fractured. Because Plato's ideas evolved over his lifetime there was probably never a firm "Platonist" platform. In his "Metaphysics" Aristotle directs his cosmological argument against the Pythagoreans and "some Platonists". He may have considered himself to be a Platonist, at odds with the other Platonists.

The issue which Plato exposed, which is expounded on in Aristotle's division of passive and active aspects of reality, is that Pythagorean idealism assigns to "Ideas" a passive existence, as outlined in the theory of participation. That which is participated in (the Idea) is passive (being eternal), while that which is participating is active. The cosmological argument demonstrates that the passive cannot be prior to the active in an absolute sense, so it is impossible that these passive "Ideas" are eternal. The later Neo-Platonists and Christian theologians, following more closely Plato's later work, "Timaeus", developed a metaphysics whereby "Forms" are active, in a way more consistent with Aristotle.

So sure, when maths is understood as just a realm of everything that unconstrained syntax will produce - a Borges library - then it seems to bear no real relation to a reality in which limitation or finitude is apparent everywhere. — apokrisis

The relation between such an unconstrained world of math and a limited finite world is that the limited finite world is a part of the unconstrained world of math.

The relation between such an unconstrained world of math and a limited finite world is that the limited finite world is a part of the unconstrained world of math. — litewave

... Just like the material of Michelangelo's David was a smaller part of the whole block of marble, which it was carved out from. Would you say that, therefore, the statue already existed as a distinctive part of the whole block independently of Michelangelo's act of carving it out?

Charles Sanders Peirce described himself as an extreme scholastic realist, rather than a Platonist, and the distinction that he carefully made between existence and reality seems pertinent here. Something exists iff it reacts with other like things in the environment; something is real iff it possesses certain characters regardless of what anyone thinks about it. As such, mathematical objects do not exist apart from their concrete representations, but they are nevertheless real. In fact, Peirce defined pure mathematics as the science of reasoning necessarily about strictly hypothetical states of affairs; it deals only with the logically possible, not the actual.

Your comparison isn't completely right. Finite mathematical objects exist both like the uncarved material and the carved material. I suppose you compared the carved material to the representations of mathematical objects in human consciousness.

I think the answer to Rovelli is simple: we never can escape our own subjectivity, we will allways do mathematics from our viewpoint and to get utility from it. Does this then wreck objectivity in mathematics or refute Platonism? In my view, no.

There being rational and transcendental numbers or Euclidean and non-Euclidean geometry doesn't mean that mathematics isn't linked together. Some have just assumed a long time ago that all numbers are rational or that Euclidean geometry is the "only geometry". Yet mathematics isn't an invented social construct that we can bend to whatever we want. All math is quite logical. The only thing is to be humble and understand that we can even now understand something wrong, just like the earlier Greeks who thought that all numbers had to be rational. It's even more easier to understand that we may have not discovered many new ways how math can be used, which will in the future opens new fields to us. To understand this doesn't in my view refute Platonism.

...the limited finite world is a part of the unconstrained world of math. — litewave

But there seems to be an obvious lack of a material principle in the formal realm of maths. Maths is spatial, or at best, spatiotemporal, and doesn't speak to energy or action in any basic way. It about the logical syntax of patterns and structures, and not about whatever breathes physical fire into those equations.

So the maths of the world would be the "maths" of constraining structure. The need to be constraining - constraining of material spontaneity or uncertainty or action - would itself be the big constraint on the maths that is physically relevant.

This is why statistical maths and symmetry maths does seem more real. It speaks to the naturally emergent structures of systems of constraint. There is a source of fluctuation or accident that is being limited by a global order. So that is where maths gets closest to the reality it might want to model. It incorporates the other thing of an action to be shaped.

Rovelli's realm of mathematical junk is then all the possible syntactical forms that can be generated when the forms serve no real organising purpose. Now you could say our finite material world is merely a part of that larger unconstrained universe. But I would say that Platonism - as a metaphysical position - is about the forms of nature that can do actual causal work. So there is the constraint that the forms do constrain. They must bring finitude to action, or the material aspect of nature. They must do the ontological job of stabilising accident and spontaneity and so allow a Cosmos to exist.

A lack of limits is pathological. It is the maths of constraining structure which promises to tell us the most about reality and the reasons for its existence.

mathematical objects do not exist apart from their concrete representations, but they are nevertheless real. — aletheist

My understanding exactly. And this same reasoning applies to all manner of transcendentals such as universals - they are also 'real but not existent'.

The objections to this understanding are usually based on the inability to make this distinction; hence the common objection to Platonic realism, 'where do numbers exist'? This is because we are by habit instinctively realist; we are oriented in respect of the domain of time and space, the objective realm, which for most of us defines the scope of what is real; everything that exists is 'out there somewhere' in the objective realm. Whereas logic, number, reason, and so on, transcend (or more accurately are prior to) the division of subject and object; they underlie and precede the ability to analyse the contents of the objective domain. It also includes the 'domain of possibility' which is likewise 'real but not existent' in that there are real possibilities - things that are likely to happen - but obviously this doesn't include what exists, otherwise it would be actual and not possible. (Methinks this has a lot to do with the probability wave.)

Rovelli's 'mathematical junk' is simply the all the possibilities that can be generated or conceived by a mathematically-literate mind once it exists. I don't think it says anything meaningful about the ontology of number per se. (A saying comes to mind - 'God created the integers - all else is the work of man.')

I believe that's exactly the point: M would be entirely trivial. This is the dilemma that the paper poses for Mathematical Platonism: either M is trivial and has no structure whatsoever (and thus largely says nothing at all about our world), or, if M is not trivial in this way, then it cannot be independent from our intellectual activity. In either case Platonism is undermined because if the former, then it has no explanatory power, and if the latter, then it simply isn't Platonism. — StreetlightX

But then M isn't a possible world, it's an impossible world. Under most analyses, impossible worlds have no ontology (because then you're accepting the existence of a contradictory object). Now I don't think this makes sense since impossible worlds ought to play the same theoretical role possible worlds do to the relevant modal statements, but put that aside.

All that's needed for math platonism is for the objects referred to and quantified over in maths to be real. I think recourse to possible worlds talk is at issue here. Consider we are in a possible world where intuitionistic logic/constructive mathematics obtains. Well, the results in standard maths (in our world) are still provable. That is, if we assume the ZFC set theory and classical logic, the formal derivations will be the same if I work them out in that world, with the same formalisms, as they are when we do them here (if that's confusing, what I'm saying is that the truths of the math hold in the formalism no matter the world I do them in).

So formal truths have a sort of... transcendality? Transcendency? Whatever. They go above and beyond possible worlds, basically. Their "truth" isn't quite the same as vanilla truth statements, and so too is their "necessity" not quite the same; they hold even if the logic of the world is different because formal truths don't involve any world at all. That's kinda what I was alluding to when I mentioned provability is what maths trades in.

part of this, in turn, has to do with the modal status of our math: contingent or necessary, and to what degree? Rovelli's answer is a kind of qualified contingency: our math is contingent ("Which tiny piece of M turns out to be interesting for us, which parts turns out to be \mathematics" is far from obvious and universal. It is largely contingent"), but this contingency in turn is premised upon the kind of beings we are, and the kind of things we encounter in the world, along with what we do with them - which lends our mathematics a kind of empirical necessity (Rovelli doesn't use that term, but I think it's appropriate in this context). — StreetlightX

Well, it's tricky. If we are talking about the truth (provability) of our mathematics then the answer is mathematics is necessary. But if we are talking about our mathematics's applicability to the world then that is contingent because so far as we know, there is no reason to think the structure our universe has is the only possible structure. As an example, our universe has a pseudo-Rimmenian manifold as its geometry. But it seems perfect possible that it could have had a Euclidean geometry or something else entirely. But irrespective of which one the universe does does have, the theorems about those systems are true about those systems. Maybe it's a sort of stratification of the modality. Neither geometry is made true or false based on what geometry our world happens to exhibit.

I think the "what is meant by interesting" is more about whatever math happens to hold at a world than about the platonism question. (Sorry if I'm taking this off track, I am trying to answer you, lol) I think I'll go read the paper because I'm probably mucking this up by not having done so.

As such, mathematical objects do not exist apart from their concrete representations, but they are nevertheless real. — aletheist

The problem I see with this is that if a mathematical "object", say the number five, has no existence apart from its concrete representations, then it cannot qualify as an object at all, except in the most abstract conceptual sense ( and it is already obvious that mathematical objects are real for us in this sense), since its representations are potentially infinite in number.

(By the way, by "concrete representations" do you mean the visual or verbal symbols that represent the number, or the actual instantiations of the number in groups of objects, or sounds and so on)?

I think it is better to think of a "mathematical object" as a way of thinking or speaking, so the sameness consists in the human action. It's like, for example, traveling by train from one station to another; the journey is both always the same and yet different every time, just as each instantiation or representation of fiveness is. There is no perfect form of fiveness, just as there is no perfect form of the train journey. The sameness in both cases is the result of the human process of abstraction.

Maths is spatial, or at best, spatiotemporal, and doesn't speak to energy or action in any basic way. It about the logical syntax of patterns and structures, and not about whatever breathes physical fire into those equations. — apokrisis

Spaces are indeed traditionally studied mathematical objects and time is treated in theory of relativity as a special kind of spatial dimension, completing a more complex mathematical object - spacetime. Now, this spacetime in which we live is not empty/uniform but it contains additional structure, like embroidery in a sheet of cloth, in the form of objects extended in space and time (which we perceive as extended in space and moving through space, changing and enduring in time), and there are also certain regularities in these extensions that we call laws of physics, laws that govern how spatially extended objects move and change in time and in relation to each other, how they interact with each other, and how these attractive and repulsive interactions define their boundaries. We may use concepts like "force" or "energy" to describe these movements, changes and interactions. Force is the product of an object's acceleration and mass, where mass is a quantity related to spacetime curvature in the place where the object is located. Energy is an object's ability to exert force over space, that is, a quantity that determines how the object accelerates another object in interaction.

So spacetime with its complex structure seems to be a specific mathematical object. One of all possible mathematical objects, and one whose structure allows the existence of what we call living conscious objects - like us.

I think it is better to think of a "mathematical object" as a way of thinking or speaking, so the sameness consists in the human action. — Janus

Rather than imagining counting train journeys, what about counting holes in a sphere. There is something perfect and absolute about the distinction between a sphere and a torus. Then you can keep on adding more holes.

So sure, some objects - like train journeys - seem pretty arbitrary. But then maths does arrive at cosmically general objects when every arbitrary geometric particular has been generalised away, leaving only the necessity of a pure topological constraint.

Yet mathematics isn't an invented social construct that we can bend to whatever we want. All math is quite logical. — ssu

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. 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?