So you talk about directly experiencing the substantial material concreteness of the actual real world. But that is still just a conception. — apokrisis

I think you're confusing yourself by over-thinking this. You're conflating talk about the experience with the actual experience. Try feeling some object in your vicinity right now. You can directly feel its material concreteness, its tangibility; it is from that basic experience that the idea of substantial material concreteness originates.

Nope. You want to apply a rigid if/then cause and effect logic to the situation. So for you, it has to be the case that one thing comes before the other thing. That is the habitual materialist Umwelt you are seeking to impose on your experiences of the world. — apokrisis

This seems totally wrongheaded. The experience comes before the idea about it. Animals experience the tangibility of the world as much as we do; the wind on their faces, the soil under their paws, the physicality of the struggle with the prey. This basic animal experience is what underpins all our notions about the nature of things, the distinctions between abstract and concrete. Animals cannot draw such distinctions since they lack the ability to grasp symbols.

Semiosis says what you "experience" is your Umwelt. — apokrisis

No, what you experience is the material world; your "Umwelt" is not experienced except upon reflection, it is your experience of the material world.

In fact to say that nature is constructed from number, or some such kind of metaphysical claim, as for example Tegmark makes, is a form of reductionism; reduction of the organic to the mechanistic. — Janus

I'm trying to say something more subtle. I am saying that number - the general thing of construction or composition - is emergent rather than basic. But having emerged as a habitual possibility of nature, it then does become basic to the greater complexity that ensues. Atomism and classicality are pretty much the truth of reality so far as we living forms are concerned, at the physical scale at which we arise.

A machine has a simple causality because it is just the sum of its parts. And nature isn't a machine - but it develops structural complexity by imposing an increasingly mechanistic order on itself.

The early universe was a relativistic gas, a hot featureless bath of radiation doing nothing but cooling and expanding. It was only because constraints emerged to cause mass particles to condense out of this spreading flow that more interesting stuff could happen. You had crunchy little electrons and protons bashing about and interacting at sub-light speed. There was now a localised form of time and action. A history of concrete or discrete events could get going against a backdrop of generalised continuity. It took a while, but a machine-like order clicked into place as the new normal. A bunch of identikit parts with constructive possibilities - individuated properties - could begin to produce a more complex world.

So physical reality as we know it is not founded in the mechanistic. But it wouldn't exist as we know it if it weren't also imbued with the propensity for a mechanistic and hierarchical form of organisation.

Numbers are then a pretty good representation of this developmental view as the very possibility of a number is emergent from the notion of an identity function. If you take a general action to the limit, like addition or multiplication, then a basic unit will emerge as the difference that doesn't make a difference, so terminating the symmetry breaking with a local symmetry. One times anything is still one. Anything plus zero is still zero. So - emergently - a basic limit, an identity function, will wind up grounding some space of functions. The units you need to justify a constructive or compositional (mathematical) reality just pops out as being eventually an atomistic regularity that can't be rotated or translated out of a freely dynamical existence.

So there is a reason why we do see a deep connection between our notions of mechanical construction and reality as it exists. The organicism of symmetry breaking or dichotomisation taken to the limit results in the emergence of fundamental units of action.

So the conception of nature as fundamentally mathematical - an atomistic construction - isn't wrong. It is a pretty good description of how reality is for us as complex creatures living in an era when the Cosmos is so extremely cold and large. But also we now know that this degree of classicality is an emergent fact. It isn't actually fundamental - unless we then go the next step that a complete holism would require, which is to realise that we are still seeking to impose a strict temporal order - a classical before and after - on a process of realisation.

So the mechanistic aspect of nature can be seen as now a finality - a cause acting from an organism's own future. Reality was being called towards the structuration that then did emerge. It was always inevitable that things would arrive there. The future becomes as real as the beginning - although now neither are real in the old privileged sense we want to give them. They have to share that foundational glory rather more equally in our conceptions of nature.

I think you're confusing yourself by over-thinking this. — Janus

Naive realists are always saying that. They like to under-think the metaphysical complexities.

Try feeling some object in your vicinity right now. You can directly feel its material concreteness, its tangibility; it is from that basic experience that the idea of substantial material concreteness originates. — Janus

Oh please. I press hard on the desk with my finger. I poke my finger with no sense of resistance through the surrounding air. I then pick up the physics textbook that tells me the solid matter is really a void of excitations, while the airy space is crammed with Newtonian particles exerting a collective pressure and resistance on my being.

Sensations are one level of semiosis - a biological level of meaning making. But an Umwelt is an Umwelt. Psychological science gives us all the evidence we need on that.

So sure, I make a perceptual distinction between what is substantial, what is void. Or what is actual, what is imagined. My psychological modelling of reality is organised by a bunch of useful conceptual dichotomies.

Thus yes, the dichotomies originate in the world in some sense. A characteristic of this era in Cosmic development is that you have solids, liquids, gases and plasmas. Concrete is a term that applies to one of the four "material state" in some everyday, not very philosophical or scientific, fashion.

But when you tell me to demonstrate the reality of "material concreteness" by poking something ... solid ... not liquid, or gas, or plasma ...

Machinery - mathematical machinery - can be "absolutely true" because it is based on deductive proof. — apokrisis

Machinery is a metaphor here. Surely the point about the axioms of arithmetic etc is that they're true for no further reason, they are apodictic, not dependent on some other truth. That's one of the things that differentiates them from the contingent domain of phenomena. But then the discovery was that maths and geometry reflected something real and constant in the constant flux of the phenomenal domain. Sure, one of the things that this enabled was machinery - but only one. But reason itself is strictly the relationship between ideas. It doesn't need to be validated with respect to any particular state of affairs - that is why it is associated with a priori truths.

If something counts as an object in the sense that Alethiest stipulates, then it certainly exists. If qualities exist, or subsist, only in their instantiations, then they are existent. My argument is only that there is no coherent sense in which we can say that they have an existence, or being or that they are real, whatever locution you prefer, beyond their instantiations and representations. — Janus

So do you think this paragraph in the SEP article on Platonism in the Philosophy of Mathematics means anything?

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences. — SEP

and also this:

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. — aletheist

That is the distinction that interests me.

I do agree that 'object' is a metaphorical description for number (etc). Likewise that 'realms' and 'domains' are metaphorical descriptions, in, for example, 'the domain of natural numbers'. Hence the equivocation between the concrete domain of sensory experience, and the metaphorical domain of meaning and reference. But nevertheless the point remains clear, I hope.

Oh please. I press hard on the desk with my finger. I poke my finger with no sense of resistance through the surrounding air. I then pick up the physics textbook that tells me the solid matter is really a void of excitations, while the airy space is crammed with Newtonian particles exerting a collective pressure and resistance on my being. — apokrisis

The textbook explanation is parasitic upon the experience of a world of tangible objects. Only a fool would deny that. I'm making no claim about the "ultimacy" of the material world, only about the foundational character of the experience of tangible things.

You love to resort to throwing around labels like "naive realist' when you cannot come up with cogent counterarguments. This is a fairly typical ploy around here. If you paid any attention at all to what I write you should know very well I am no naive realist, so don't insult me with that sub-standard shit.

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences. — SEP


and also this:

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. — aletheist — Wayfarer

That quote refers to "objects", but it does not explain how those purported objects qualify as objects beyond their conceptual, abstract dimension. So, it remains vaguely allusory, which is fine for poetry or mysticism, but not for philosophical analysis.

In reference to the second quote; I have already acknowledged that I think number is a quality or character that is real in its instantiations and representations; concretely in the former and abstractly in the latter. To say it is real is the same as to say that the quality or character exists, as far as i am concerned, since I can't see any significant distinction between saying they are real or saying that they exist. The point I have been making is that they are not real (they don't exist) apart from their instantiations and representations (including of course the conceptual), so I am not disagreeing with that passage @alethiest wrote, except perhaps in regard to what I think about the senses of terms.

The textbook explanation is parasitic upon the experience of a world of tangible objects. Only a fool would deny that. — Janus

What I questioned was your notion of "experience of a world". If you accept that experience is an Umwelt - conceptually structured from the get-go, then no problems. But if you can't see that speaking of tangibility is itself an abstraction, then there is a big problem.

You love to resort to throwing around labels like "naive realist' — Janus

Like you love to throw around labels like "over-thinking".

I'm making no claim about the "ultimacy" of the material world, only about the foundational character of the experience of tangible things. — Janus

So in claiming to escape naive realism, how are you managing to escape naive idealism here?

You are asking me to go out and feel the tangible quality of material things. And somehow that proves that materiality exists - exactly as conceived!?!

My response is that tangibility is a general conception I would apply as a contrast to some useful sense of its "other" - the intangible. And indeed, given the murkiness of what would be that dialectical other, tangibility already seems rather a confused term. Air is not tangible - until the wind blows or the plane cabin depressurises. Is the magnetic field between two magnets tangible? As a kid I spent a lot of time pushing two opposing magnets together and feeling the "bubble" or resistance that wanted to form in-between.

So you are now using "tangible" as another reified way of speaking about concrete reality - a mind-independent metaphysical framing. It sounds a little better because it also has its mind-dependent meanings. It can mean palpable or tactile - acts of mind - as well as physical, real, substantial, corporeal, or solid. Facts of the world.

I just want to draw attention to how language is being used here. And what the ontological commitments might actually be. Tangibility is a tad slippery and ambiguous about the very issue that needs to be made clear.

To again restate my own position, I am arguing that "ultimately" the material principle reduces to some kind of uncertainty or fluctuation. Naked instability. This is of course a highly abstract and not very tangible notion. And then what you are talking about is really substantiality. Formed matter. Constrained instability. Tangibility would become another word for individuation that persists due to some contextual strength of constraints.

That would be a mental picture of a concrete material reality that is the polar opposite of the usual atomistic one. So the two of us - to the degree we are immersed in these contrasting umwelts - would be taking home very different views about what we just learnt by reaching out and touching a "tangible object".

That was one of the points Rovelli argued in asking what kind of maths a Jovian would arrive at.

So again, experience turns out to be conception all the way down. Nothing enters phenomenology that hasn't already been shaped by some "fundamental" conceptual dichotomies. Sensory receptors are designed with the logic of switches that are in the business of saying yes or no to a question already posed.

This is an irreducible complexity that sits at the heart of all epistemology. It fundamentally screws up any simplicity about the relation between minds and worlds.

And I still believe this is an essential difference in our outlooks. I say that this irreducible complexity is what we must eventually embrace the best way we can. You reply by calling that over-thinking. And I still say no. That triadic irreducibility is the feature and not the bug here. It is how anything could even exist. It is what stops everything collapsing back into its own entangled confusion.

That quote refers to "objects", but it does not explain how those purported objects qualify as objects beyond their conceptual, abstract dimension. — Janus

There are physical objects like rocks, mental objects like whatever you happen to be sensing now, and abstract objects like mathematical entities. I think "object" here just sort of means thing.

The objects that appear to be dependent on the minds of individuals are mental objects. We know mathematical objects aren't in that category, and if intuition informs you as to why, then you know the basis of mathematical platonism.

Surely the point about the axioms of arithmetic etc is that they're true for no further reason, they are apodictic, not dependent on some other truth. — Wayfarer

They are formed apophatically. They arise as the opposed choices of some dichotomy. And then they are proved "true" retrospectively because the system of thought that follows turns out to be useful. Even "unreasonably effective".

So the discrete and the continuous arise as an obvious opposition. We can take one as axiomatic, the other as its emergent construction. In maths, a line could be built up as an infinite succession of points, so close together as to leave no gaps. Or we could just as well choose the opposite story that a line can be infinitely divided until it is just a bunch of discrete points - continuous intervals with no actual length.

So axioms might seem to be basic truths too self-evident to deny. But history usually shows that a dichotomising discussion first took place. And that resulted in a pair of contrasting limits. Which in turn offered two complementary descriptions of nature. We could have picked either one - as the "fundamental" from which its "other" was the mechanically emergent construction - and arrived at a useful description of the actual world.

But reason itself is strictly the relationship between ideas. It doesn't need to be validated with respect to any particular state of affairs - that is why it is associated with a priori truths. — Wayfarer

But what do you mean by reason?

Mathematical reasoning is pure logical deduction. It is mechanistic. And so it is both powerful, but also only exists as the limit case. It invents its own reality - one where constraints rule absolutely. So we know it is only a powerful fiction. To the extent that it locks reality down to a picture of efficient causation, it is "othering" the very finality that you so want to be part of your reasonable world.

So you ought to be much more comfortable with a full Peircean model of reason - one that is based on the irreducible triad of abduction, deduction and inductive confirmation. But that is then the scientific story on reason. You have to accept the validation by empirical particulars along with the "inspired leap" of the hypothesis/axiom forming, and the logical deduction of some formal description or mathematically-articulated theory.

So you put yourself in a weird position by celebrating the inspiration and the logic - seeing in them a transcendent step from the mind to the divine - and then rejecting the third leg of this triad, the empiricism that roots reasoning back in the world with which it engages.

You pick transcendence when reason actually functions - in a way we see all the time - as a pragmatic and immanent activity.

To connect back to the OP, Rovelli's paper illustrates the logical junk that can be generated if we imagine "reason" becoming just every possible output state of every possible bit of machinery or algorithm. It is just another sad multiverse tale.

But a Peircean understanding of reasoning - the scientific model you want to reject - book-ends that ultimately irrational fecundity with rational constraints. First, hypotheses are meant to be reasonable too. Inferentially, we can see why they would be a jump in the right direction. They feel like the right kind of grounding generalisation. And then second, the correctness of those inferential guesses is validated by their empirical outcomes. Judgement of truth is passed pragmatically. Certain mathematical models would be considered Platonically true because they worked. They wound up limiting unwanted surprises.

So we don't even start on generating maths unless we have a good reason to expect successful outcomes. And we don't take much notice of that maths unless it empirically delivers such an outcome.

Once Rovelli's infinite M has been shrunk in both fashions, then really the space of essential mathematical structures becomes very small. Some folk would claim it can all be shrunk down to set theory, or even category theory.

But anyway, modelling is triadic. Reasoning is a three-legged process. And your own defence of dualism is confused as it wants to conflate mathematical abduction with mathematical deduction - the creative spirit and its divine machinery - placing this unholy transcendental pairing in opposition to the dull inductive materiality of empirical measurement.

But an immanent metaphysics sees the circle being completed by these three elements working collectively. Nature is irreducibly hierarchical in its organisation. That view gives each element its proper place as aspect of a whole.

If you accept that experience is an Umwelt - conceptually structured from the get-go, then no problems. — apokrisis

This is a tricky point, it seems. Would you say the experience of animals conceptually structured?

So you put yourself in a weird position by celebrating the inspiration and the logic - seeing in them a transcendent step from the mind to the divine - and then rejecting the third leg of this triad, the empiricism that roots reasoning back in the world with which it engages. — apokrisis

It's not weird. I don't think the Western tradition of philosophy, and science, for that matter, is basically materialist in orientation. My view is that materialism kind of hijacked the Western tradition from within, although the times they are a'changing. It's great that Platonic and Aristotelian philosophy is being appreciated again, but its ultimate concern is not utilitarian in nature.

One thing I did notice:

Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic? The answer begins to clarify: because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space. In other words, this mathematics is of interest to us because it reflects very contingent interests of ours. — Rovelli

Everything is, after all, a function of biological adaption; we have the kinds of maths we have, because of the kinds of creatures we are. Remember our discussion a year or so back about Donald Hoffman? Don't you think that Rovelli's statement vindicates such an analysis?

Would you say the experience of animals conceptually structured? — Janus

Of course. Animals don't have a linguistically-structured conception of the world - so one that reflects a higher level social organisation. But they do have one that is structured by its ecologically useful generalisations, or habits of interpretance.

...but its ultimate concern is not utilitarian in nature. — Wayfarer

So who is defending a reductionist notion of utilitarianism? I was arguing in favour of the irreducible holism of pragmatism.

My view is that materialism kind of hijacked the Western tradition from within, although the times they are a'changing. — Wayfarer

Sure. Atomism reigns. But it can't be defeated by transcendent dualism. It must find its way to the immanence of holism. And that has a triadic logic at its heart.

Everything is, after all, a function of biological adaption; we have the kinds of maths we have, because of the kinds of creatures we are. — Wayfarer

Sure, things start there. All animals with large brains can count - at least they can count 1, 2, er many.

Linguistic humans then add on a level of social conceptualisation. And eventually a level of mathematical semiosis too. It became useful to count sheep so as not to be cheated at the market, or measure flat ground so buildings started off upright.

But I don't see the connection to Hoffman. Rovelli is not arguing idealism, only social constructionism surely.

In that phrase Rovelli is explicitly arguing on the basis of biological evolution - that mathematics is specific to 'creatures like us'. He is saying, the basis of the kinds of maths we have developed is contingent upon our evolutionary development, in almost exactly those words.

But they do have one that is structured by its ecologically useful generalisations, or habits of interpretance. — apokrisis

And you would count that as "conceptual"? Maybe 'proto-conceptual'; I would have no argument with that.

In any case there must still be a distinction between the concept we have of a thing and the shear appearance of a thing, and also the thing we have a concept of. 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.

If something counts as an object in the sense that Alethiest stipulates, then it certainly exists. If qualities exist, or subsist, only in their instantiations, then they are existent. My argument is only that there is no coherent sense in which we can say that they have an existence, or being or that they are real, whatever locution you prefer, beyond their instantiations and representations. — Janus

No, this completely ignores the accompanying definitions by which existence and reality are distinct. While qualities and habits only exist in their instantiations - as characters embodied in reacting things and laws governing such events - their reality does not depend on those instantiations; again, they are what they are regardless of what anyone thinks about them. Their mode of being is that of a conditional proposition; under certain circumstances, they would be instantiated.

Why wouldn't the ability to do mathematics be an evolved ability, underpinned of course by the evolution of animals we are descended from, and the ability they would probably have had to perform basic counting?

I still don't get what it could mean to say that they have a reality independent of their instantiations. What if nothing at all existed, would they still be real then? If you want to say that they are real over and above their instantiations, and then we imagine that there are no instantiations, then what could that reality consist in other than mere logical possibility? I mean obviously number is instantiated if anything at all exists, so the reality of number would seem to consist in that case in it's actual instantiations; it doesn't seem right to say that the reality of number consists in the infinite number of merely logically possible additional instantiations that could exist.

Before there was the word "round", there was obviously nothing which the word "round" signifies, because there was no word "round" to signify anything . Therefore it is absolutely impossible that there was "the real character of roundness" before there was the word "round". That there is something which the word "round" signifies is very clearly dependent on the existence of the word "round". — Metaphysician Undercover

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

That the world is round is a judgement. Whether any such judgement is true or false is irrelevant to the fact that such predications are judgements. — Metaphysician Undercover

That the world is round(ish) is a fact, whether anyone ever judged it to be so or not; i.e., the world is really round(ish), regardless of what anyone thinks about it.

I don't deny that the ability to do maths evolved, but does that mean you can justify mathematical proofs with reference to biology? I mean, how do you know anything is true, without appealing to reason? But if you then say, well the faculty you're appealing to is actually contingent upon biology - then you're questioning the sovereignty of reason by declaring that it's contingent on biology. (This argument is taken up in Plantinga's 'evolutionary argument against naturalism' and also the related 'argument from reason'. )

I still don't get what it could mean to say that they have a reality independent of their instantiations. — Janus

It means mathematical truths have the character of the discovered. One can be wrong about math.

There are options for explaining this, or you can leave it unexplained. It's the prevailing view among mathematicians. I don't think they bother much with trying to explain it.

But where is any connection to Hoffman's computational idealism?

Then as to algebra and geometry, I don't buy evolutionary contingency as the explanation. Rather it would be the necessity of such a structural conception of nature. The two ways of seeing the world are formally complementary. For instance, as Atiyah argues....

Algebra is concerned with manipulation in time and geometry is concerned with space. These are two orthogonal aspects of the world, and they represent two different points of view in mathematics.

So first up, evolutionary biology did not make us mathematical creatures. Although it did leave us - as the smartest animals, our brains already being reshaped by our language and tool-use - with highly lateralised neurobiology. Large brain animals already have that "algebra vs geometry" dichotomy wired in as a division between object recognition processing and spatial relations processing. Things vs their relations. And then humans continued on to have a strong left vs right brain dichotomy in terms of attentional style - the left specialised for focal differentiation and the right specialised for global integration (both always working together to produce the third thing of a task-appropriate balance).

So in a general way, biological evolution structured the animal brain with the kind of Gestalt figure~ground logic that would be needed to see the world "as it is". The world could be experienced as an intelligible whole because it was always being analysed in terms of its fundamental structuring dichotomies.

But then a mathematical turn of mind - one which actually see the world in terms of mechanistic construction, acts of counting and measuring - was a cultural evolutionary step. Ordinary language got it going. Then mathematical language crystalised the practice as an actual machinery, a teachable syntax.

So yes, a mechanistic conception of nature was of evolutionary value. But that couldn't become apparent until after language was developed.

And on the other hand, our neurobiological legacy was not absolutely devoid of logical structure. But it was the much more organic logic of dichotomisation and symmetry breaking. This is the logic that reflects the deepest structural reality of the Cosmos.

And then it is no surprise that both geometry and algebra developed hand in hand. They are simply the two complementary ways that structure itself breaks down. One is ground, the other is figure. One is global, the other local. We can choose either as the starting point from which to work back towards the other.

Except I found the author to be saying that the tree is not blue, and he did not tell us why. — Luke

Indeed, the paper stands or falls on whether the world M is a fair interpretation of mathematical realism. The process then becomes our selection fo the interesting bits of M. — Banno

Exactly. The structure of the argument of the paper is that of a reductio. Those who see the paper as begging the question seem to miss this entirely. The paper takes place on the grounds of mathematical Platonism, and attempts to dismantle it internally, and not from some position outside of it.

I still don't get what it could mean to say that they have a reality independent of their instantiations. — Janus

Again, something exists iff it reacts with other things; something is real iff it is what it is regardless of how anyone thinks about it. Numbers clearly do not exist, because they do not react with anything; yet they are clearly real, because they are what they are regardless of how anyone thinks about them.

If you want to say that they are real over and above their instantiations, and then we imagine that there are no instantiations, then what could that reality consist in other than mere logical possibility? — Janus

I am saying that the reality of numbers does not depend on their particular instantiations (existence), and hence that at least some "mere logical possibilities" are real - i.e., independent of how anyone thinks about them.

Mathematical proofs are justified by their consistency or lack of contradiction. They cannot be justified by anything outside of math as far as I can see. Mathematics is the elaboration of basic counting operations, an elaboration that is enabled by the ability to manipulate symbols and symbolic operations.

I can't see what counterpoint you are trying to make here.

It's true you can be wrong about math. But at the basic level this can be seen by manipulating physical objects. Say you want proof that 8 x 13 = 104. All you need to do is arrange 104 objects in groups of 8 and see if there are thirteen of them.

But "independent of our intellectual activity" is precisely what "real" means, assuming that "our" refers to any individual person or finite collection of people. — aletheist

No. That's what it is stipulated to mean. The idea that what is real cannot refer to things that are products of our activity is a malicious piece self-serving philosophical claptrap that Platonists have traded in since day one. I agree that it's the usual, most widely employed understanding of the term, but that only attests to the fact that people are not particularly bright.

Rovelli, who is happily a bulb above the rest, rightly avoids the whole semantic debate altogether.

I am saying that the reality of numbers does not depend on their particular instantiations (existence), and hence that at least some "mere logical possibilities" are real - i.e., independent of how anyone thinks about them. — aletheist

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.

The objects that appear to be dependent on the minds of individuals are mental objects. We know mathematical objects aren't in that category, and if intuition informs you as to why, then you know the basis of mathematical platonism. — frank

I agree that mathematical objects are not dependent on any individual mind; but it does not follow from that that they are real independently of their being instantiated in actual things or thought by actual minds.

It's true you can be wrong about math. But at the basic level this can be seen by manipulating physical objects. Say you want proof that 8 x 13 = 104. All you need to do is arrange 104 objects in groups of 8 and see if there are thirteen of them. — Janus

Proof isn't the issue, but I gather you're saying number is a property of physical objects and nothing more? Correct?

Rovelli, who is happily a bulb above the rest, rightly avoids the whole semantic debate altogether. — StreetlightX

Yes, I think the semantic debate is a red herring.