For one, I distinguish mathematics being objectively real, and mathematics being objectively true. The latter seems to hold, and the former I thought was what mathematical Platonism is about, but you say it's about being true. I am unsure if anybody posits that the truth of mathematics is a property of this universe and not necessarily of another one. — noAxioms

Well, platonism asserts that the mathematical truths are objectively real, so you aren't wrong. The problem is, however, that if mathematical truths are independent from both our minds and all the contingencies of the world, it would seem, indeed, that they are, in some sense, objectively real.

Mind you, not as 'things'. Plato himself for instance argued that they reside in a different level of 'reality', the reality of intelligible objects, accessible only from reason. I believe that many theists would say that mathematical truths are concepts in the mind of God and we are able to understand mathematics because our minds have a structure that is able to understand them. In recent times, Penrose popularized the idea of the 'three worlds', the physical world, the world of consciousness and the 'platonic realm'. All these worlds for him both transcend and relate to each other.

I believe that mathematical platonism is right because it seems to me that mathematical truths are objectively true and independent from both the world(s) and our minds. They can be known, so they are not 'nothing' (or figments of our imagination because they are independent from our minds) - they seem to have some kind of ontological reality.

Being objectively true (and not just true of at least this universe) does not imply inaccessibility. The question comes down to if a rational intelligence in any universe can discover the same mathematics, and that leads to circular reasoning. — noAxioms

Well, yeah, right. And also, this objection seems to miss the problem. The point would be "can a rational intelligence of any kind learn mathematics as we know it?". For instance, I read that some propose that a rational being that lives alone in an undifferentiated environment would not coinceive numbers. But the scenario presented here is made in terms of concepts that accessible to us and, even if that inteliggence could not conceive numbers, our arguments would be still correct.

Only a simulation of it. The things in themselves (all different seed states) are their own universes.
Funny thing is that our universe can be simulated in a GoL world, so it works both ways. — noAxioms

Well, you are assuming that our world can be simulated. That's a big assumption. Anyway, if our world were a simulation, I would not consider it a separated world from that which runs the simulation.

Totally agree here. — noAxioms

Good!

A perspective seems to be a sort of 5 dimensional thing, 4 to identify an event (point in spacetime), and one to identify a sort of point in Hilbert space, identifying that which has been measured from that event. All these seem to be quite 'real' (relative to our universe) — noAxioms

Interesting take, thanks. But maybe here the risk is to conflate the 'map' (the mathematical description) and the 'territory'.
But as far as descriptions go, probably one can describe a 'perspective' with a particular division of space time in 3d space, one dimension of time and a point in Hilbert space.
An interesting question would be what is the relation between spacetime and the Hilbert space.

I believe that mathematical platonism is right because it seems to me that mathematical truths are objectively true and independent from both the world(s) and our minds. They can be known, so they are not 'nothing' (or figments of our imagination because they are independent from our minds) - they seem to have some kind of ontological reality. — boundless

Well, there are infinitely many mathematical truths, so the realm they inhabit is going to be infinitely "large" (if that word even makes sense). Also, is some kind of interaction going on between our mental realm and the platonic realm? When you think 2+2=4, do you interact, in some way, with one of these mathematical truths, and that allows for the grounding of mathematical knowledge? If so, then the interaction between the specific mathematical truth and one of the infinite mathematical truths in this realm...how does that work, exactly? And if there is no interaction, why posit the existence of objective mathematical truths? To avoid contradiction?

Apologies, very long reply. Again, I don't expect any replies to these kinds of thoughts because when I come to these replies I am just ending up writing down going through my rambling thought process about how to produce a coherent view for these things, which goes way beyond being a self-contained reply. I am not restricting these thoughts as I would if expecting and requiring a reply to them. I am just going through my thought process.

I believe that realism is more like an epistemic position rather than an ontic one. — boundless

But realism is more a claim that we can have knowledge of that 'mind-independent reality' and it's where things get murkier. — boundless

Yes, very true. Its totally reasonable to have the thought process that: there is a reality out there independent of us even when we are not looking; when we do look, what we see is dependent entirely on out biology had how that biology relates to the world in a specific and non-unique perspectival way (based on the physical interactions mediating the relation between things in the outside world and our brains). It is then fair to say how we see reality and information we gain that can be put to use is dependent on a perspective of a mind.

For me, its acknowledging this fact but also arguing that this exact same situation can be also equally viewed as a brain receiving genuine information about the outside world which depends entirely on whats going on out in the world. I think there is wiggle room in deciding what constitutes mind-independent knowledge, or at the very least knowledge that is in some sense real.

Again, the motivation is that it seems paradoxical to say that all knowledge is false or not real, yet our mastery of the world is very good. And I think taking anti-realism to its logical conclusion, it makes sense to me to say that all knowledge, whether scientific or just your everyday knowledge, is not real in that anti-realist sense. That logical conclusion is why I tend toward a total deflationary attitude toward all knowledge and epistemic activities. However, apart from the fact that deflating knowledge and epistemic behavior actually requires a tentative story of what is objectively happening with regard to minds and brains and consciousness, I think just for the sake of a coherent story, there should be a kind of compromise position accounting for the fact that while knowledge and epistemic activities don't support the most extreme, almost ridiculously naive realist position, the aforementioned paradox makes the notion of anti-realism a bit misleading. The deflation of the anti-realism vs. realism dichotomy itself is part of a solution; we might even say that one can only view the construct of "real" perspectivally in a way that requires adding your own assumptions about what constitutes "real" that are not straightforwardly unambiguous. At the same time, the paradox can only be fully resolved imo by a story about some "real" engagement with the world. Often our engagement is completely erroneous; at the same time, such error is not dichotomous but a continuous, fuzzy gradation.

If I am not mistaken, ontic structural realism is the position that, while we can't know the intrinsic properties of mind-independent reality, we can, at least in principle, know some structural aspects of it. — boundless

That I believe is actually epistemic structural realism. Ontic structural realism is that there is nothing more than structural properties to reality. I think an ontic structural realist might say there is some objective, uniquely describable set of structures. My view would be considerably weaker than that.

I think part of my view is changing the standard of what constitutes "real" metaphysics and "real" knowledge.

Some say that what is "real" metaphysically is a world of objectively, uniquely defined stuff that kur words and sentences map to. For me, its sufficient just that our words consistently map to stuff in a self-consistent way. I don't need some notion of objective boundaries in the world beyond my senses, just that if I see stuff or say stuff, it always maps to the same part of the world with the same relations to other parts of the world.

Because what is knowledge but the ability to predict what happens next? In that sense, knowledge is entirely about structure. The idea of intelligible intrinsic properties separate from structure then doesn't make sense - there is simply nothing to know about that kind of thing and it doesn't even logically make sense unless it led to some perceptible structural distinguishability. Indistinguishable intrinsic properties are meaningless.

I would say it is this kind of argument which could be used to attack the idea that we cannot knoe about the intrinsic nature of the world mind-independently - it doean't really make any sense the idea that there is anything to know or anything intelligible about that. Everything that is meaningful and makes a difference in the world is about structure that makes a difference - a boulder is meaningful because it has structural relationships to everything else in the world and makes a difference to how things around it behave, whilst other things evince its existence by enacting change upon the boulder; when you push it it moves. If intrinsic properties cannot make a difference structurally then not only are they meaningless but they give no reason for us to even speculate on them. A Bayesian might say we should update out beliefs only as much as required by the evidence. If no difference is made, there is nothing to update.

There is then the possibility of knowledge being mediated by different structures that produce exactly the same predictions counterfactually. Like how Newtonian physics can be formulated dynamically or in terms of least action or complex Hilbert spaces. But then if there is no way of distinguishing what different models say about the world then how do they make a difference to our knowledge? How do they make a difference to our mastery? They don't. It then doesn't make much of a difference that we describe the same thing in different ways; we are still making a consistent mapping to the same world. Our conscious experiences are just that - informational structure about what is going on in the world, albeit form a specific persepctive limited by specific physical interactions with a small contained part of the world - nonetheless, when not going haywire they map to the world in a way that in principle would be vindicated by habitual engagement with the world. Sure we can be wrong or incorrect about how we see our mapping to the world but this is not so interesting if it is possible in principle for us to be errorless (albeit one could also be a radical skeptic about errors).

Ultimately, though we are often wrong and models often do make considerably different predictions. Even in something like quantum interpretations. Different interpretations make the same empirical predictions but clearly they do not make the same predictions, fullstop. Many worlds predicts a completely different universe to Bohmian mechanics or relational mechanica structurally; its just that physics so far has hidden the means from us to actually distinguish those different ontologies.

We are wrong all the time. We make idealizations that are often wrong in some parts, albeit vindicate the important predictions we are intetested in in other aspects.

I think many realists are not interested in the possibility that our theories are mistaken. For many realists, I think it is sufficient that it is in principle possible to have a model or maybe sets of many models that predict everything one could predict about the structure of the universe correctly. They would then say that many theories are wrong now, but the fact that they predict things correctly means some of the structure is correct and that those predictions will get better over time.

At the same time, I think most anti-realists would say this is nothing more than empirical adequacy, or empirical structure, especially if one ditched the idea that "real" requires unique, objective deacriptions.
I guess here it makes salient that my views about the issues of realism in regard to indeterminacy are no different to any anti-realist, and I embrace that anti-realism because I believe all knowledge and epistemic activities can be deflated, albeit deflated under a kind of scientific or scientifically-amenable description of how exactly we perform those activities.

But my concern I guess is that the upholding of predictions via empirical adequacy requires a form of real engagement with the world, albeit one that can be mistaken.
So in that sense many theories are just plain wrong on some level or some aspects; if they make acceptable predictions in some places, that needs elucidating about how it captures empirical structure or if it does so only by luck or too thinly to be interesting.
I guess it could be vacuous when one considers that structure must be scaffolded on other structures and that they could be plausibly scaffolded on many different incompatible ones.
I suspect many attempts at explanations are like this. Flat out wrong.
It is then valid to be truly skeptical of scientific theories that could be flat out wrong. But I think something like classical mechanics actually makes too little metaphysical assumptions to be vacuous in this way. In some ways, Newtonian mechanics is actually just a thin description of behavior we see in empirical structure. Its not like saying that the earth is flat and then finding out it is rouns - which invokes considerable extra metaphysical, structure depth beyond what we see in empirical behavior. Quantum thwory may be the same as Newtonian mechanica in that regard.... but quantum interpretations isn't as it goes metaphysically deeper.

I guess under my view is the idea that maybe there can be nothing more to say about reality than what could be perceived or distinguished in empirical structures counterfactually; albeit ones that can be mistaken and do scaffold on each other in some sense.

Again, I think because of the complicatedness it becomes difficult to unambiguously define real and not real in regard to descriptions and theories that themselves can be deflated in terns of physical activities. Nonetheless there is maybe a fuzzy gradation between consistent mappings, engagements with the world and ones that are erroneous, or at least our predictions of our own knowledge are erroneous

But then, if we accept that 'mind-independent reality' is intelligible, we might ask ourselves how is that possible. — boundless

And what is even more interesting is that if we do accept that we can know (part of) the mind-independent reality it is because it shares something with our own mental categories. So, it would imply that, say, mathematical platonists are in some sense right to say that mathematical truths are mind-independent, eternal and so on. — boundless

From my perspective, all it requires is a mapping so it is sufficient for a physical reality were things behave in consistent ways that structure of some parts of reality can be mapped to the behaviors of other parts (e.g. like say a mirror reflecting the image of the structure of a room). So I disagree about platonism.

The empirical knowledge that science gives us is undeniable. But, in a sense, we can't 'prove' in any way that this means that we do know the structure of 'reality as it is'. — boundless

On the other hand, basically everything seems to tell us that we can know something about a mind-independent reality. On the other hand, however, there is no logical compelling argument that we can. — boundless

Yes, I think skepticism is always real and healthy. Knowledge, or rather, beliefs can be and often are outright wrong. They cannot be compelled to be correct. I think maybe this though goes into a question of agnosticism about theories rather than anti-realism. If it were anti-realism then it would be: even if our predictions were correct would it be not real? Now, I have said that I believe all epistemic activities can be deflated as complicated, even instrumentalist, constructions, maybe they all are a bit erroneous too. But again, I have shifted my standard for realism - I think if one views realism in terms of unique mappings to reality then ofcourse there are always many possible descriptions of the world and none of them would be real. That is valid. But I think it is also valid to say that if our epistemic constructs are all deflations and all we have to go on is whether our predictions and mastery are correct, then there is in some sense a realism to it because it reflects some real engagement which - via the purview of the free energy principle - would reflect some genuine statistical coupling between us and reality; at least that is the story, and all we have are stories. But then again, those models can be wrong, they can only be valid in a small part of reality and turn out incorrect when we come across a novel context.

I think my main contention is the status of the connection between pluralism and realism or anti-realism; we may choose to construct different tools for describing reality, nonetheless they are describing or pointing at the same thing given that they are used appropriately. I think this is inherently ambiguous. Maybe this inclines to borderline paradoxical aspects about the relation of theories to reality. Maybe we can deflate all realism and truth but nonetheless still use those words meaningfully in a deflated context. But maybe we should all be scientific agnosticists though justified in choosing theories we believe are either the best of a bad bunch now or that we believe most likely to not become erroneous in the future.

The problem is, however, that if mathematical truths are independent from both our minds and all the contingencies of the world — boundless

How could they not be? I mean, OK, under idealism, mathematics is nothing but mental constructs. I get that, and there are even non-idealists that say something similar, but since they can be independently discovered, it seems more than just a human invention.

Contrast that with a god, who should very much be independently discoverable, and yet each isolated culture/tribe comes up with their own, none related to any of the others. That's huge evidence that it's all being invented, not discovered.

Plato himself for instance argued that they reside in a different level of 'reality', the reality of intelligible objects, accessible only from reason.

If a mathematical structure is going to supervene on mathematical truths, then those truths are going to need to be accessible by far more than just reason, which sounds like a mental act or some other construct that instantiates the mathematics (such as a calculator).

In recent times, Penrose popularized the idea of the 'three worlds', the physical world, the world of consciousness and the 'platonic realm'. All these worlds for him both transcend and relate to each other.

Great. 3 worlds on their own instead of each being stacked on the next.

I believe that mathematical platonism is right because it seems to me that mathematical truths are objectively true and independent from both the world(s) and our minds.

I agree with that bit, and perhaps the ontology can follow since such truth stands out from nonsense.

They can be known, so they are not 'nothing' (or figments of our imagination because they are independent from our minds) - they seem to have some kind of ontological reality.

I'm actually being moved by this reasoning, so yes.

The point would be "can a rational intelligence of any kind learn mathematics as we know it?". For instance, I read that some propose that a rational being that lives alone in an undifferentiated environment would not coinceive numbers.

I think not the point. Said intelligence would need to be presented with an environment where such tools would find utility. It need not be 'of any kind' for mathematics to be independently discoverable.

Well, you are assuming that our world can be simulated. — boundless

An approximation of it can be, yes. A classical simulation is capable of simulating this world in sufficent detail that the beings thus simulated cannot tell the difference. Another funny thing is that GoL is more capable of doing this than is our universe due to resource limitations that don't exist under GoL.

Anyway, if our world were a simulation, I would not consider it a separated world from that which runs the simulation.

A world is what it is, and a simulation of it is a different thing, sort of like the difference between X and the concept of X, something apparently many have trouble distinguishing..

An interesting question would be what is the relation between spacetime and the Hilbert space.

I need more of a mathematics background to give an intelligent answer to that.

Surely you can understand how unknown planets and unknown universes are on a different ontological plane? — Wayfarer

I sure do. A small fraction of the former are part of our causal past. None of the latter are, which makes a big ontological difference if ontology is based on us.

The universe being ‘the totality of what exists’.

If it's defined that way, then there's not such thing as other universes by definition, at least not existing ones, and that's presuming that we're part of the universe as thus defined. Per my prior topic, I find no empirical test for that sort of thing.

I’m open to Penrose’s idea of the cyclical universe

Do you understand how it works, what the duration of all the prior ones were (as measured by one of our clocks), and how long it will take for the next one to happen? It is a cool idea, I admit.

In the past 100 years our knowledge of the universe has expanded by orders of magnitude. I find the notion of a multiverse intriguing - but I'm just an armchair physicist. However, much smarter people than I think it's worth looking into.

https://www.thescienceblog.net/is-there-scientific-evidence-for-the-theory-of-the-multiverse/

https://organicallyhuman.com/googles-quantum-multiverse-exists/ — EricH

Those are interesting, but pop articles written by people no smarter than you or I, writing about view of people who are indeed smarter in their field.

The blog article speaks of 'the multiverse theory' like it's one theory. It goes on to admit that it's just a blanket term for several unrelated actual theories, but much of the discussion still treats it as one thing.
At least 9 types have been listed by Greene: Brane, Cyclic, Holographic, 2 Inflationary, Landscape, 3 Quantum, 1 Quilted, Simulated, 4 Ultimate.
The ones with numbers correspond to Tegmark's 4 types. Some of the others come from string theory.
The bolded ones are mentioned in the blog.
It has interesting assertions like: "MWI posits that every time a quantum event happens with multiple outcomes, such as a particle being in one state or another, the universe splits, creating a new branch for each outcome."
Well it posits no such thing. The sole premise is this: "All isolated systems evolve according to the
Schrodinger equation". That's it. It posits nothing else. All the rest follows from that one simple premise.

The blog is cute, but I spot several errors, meaning it's either not reviewed or it is written for entertainment rather than accuracy.

The google-chip article is attempting to sensationalize what is essentially: More progress has been made in the effort of realizing quantum computing. What it claims is that it will somehow prove the quantum multiverse (MWI), meaning that a successful quantum computer will somehow falsify all the other interpretations, which is by definition impossible since they all make the same empirical predictions.

Still, it will be interesting to see how proponents of each of the other interpretations choose to spin a functional quantum computer, similar to the wiki page showing how each interpretation spins the Schrodinger's cat scenario:
https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat#Interpretations

Interestingly, Bohmian mechanics isn't in there, and I thought it used to be. Was it removed? This is a significant hole in the page.

Well, there are infinitely many mathematical truths, so the realm they inhabit is going to be infinitely "large" (if that word even makes sense). Also, is some kind of interaction going on between our mental realm and the platonic realm? When you think 2+2=4, do you interact, in some way, with one of these mathematical truths, and that allows for the grounding of mathematical knowledge? If so, then the interaction between the specific mathematical truth and one of the infinite mathematical truths in this realm...how does that work, exactly? And if there is no interaction, why posit the existence of objective mathematical truths? To avoid contradiction? — RogueAI

Yes, that's the problem with platonism. If mathematical (and other types of) abastract concepts and truths abide in a separate realm from the physical world and the mental world (including our culture), how can we know them? How the realms 'interact'?

I don't think that there is a fully satisfying answer to this question. That's why I said that I think platonism is right, but I don't think that there are fully compelling arguments for it.
The strongest evidence is the apparent eternity and necessity of these truths. To me platonist positions are the best explanations. But I can't claim knowledge or certainty about this.

If mathematical (and other types of) abastract concepts and truths abide in a separate realm from the physical world and the mental world (including our culture), how can we know them? How the realms 'interact'? — boundless

The problem is that 'realm' is a metaphorical or allegorical description. It is not some place or ghostly ethereal realm. Here's a book called Thinking Being: Metaphysics in the Classical Tradition, by Eric Perl, which lays out the meaning of the Platonic 'ideas', with a chapter on 'the meaning of separation, from which this is excerpted:

Forms...are radically distinct, and in that sense ‘apart,’ in that they are not themselves sensible things. With our eyes we can see large things, but not largeness itself; healthy things, but not health itself. The latter, in each case, is an idea, an intelligible content, something to be apprehended by thought rather than sense, a ‘look’ not for the eyes but for the mind. This is precisely the point Plato is making when he characterizes forms as the reality of all things. “Have you ever seen any of these with your eyes?—In no way … Or by any other sense, through the body, have you grasped them? I am speaking about all things such as largeness, health, strength, and, in one word, the reality [οὐσίας, ouisia] of all other things, what each thing is” (Phd. 65d4–e1). Is there such a thing as health? Of course there is. Can you see it? Of course not. This does not mean that the forms are occult entities floating ‘somewhere else’ in ‘another world,’ a ‘Platonic heaven.’ It simply says that the intelligible identities which are the reality, the whatness, of things are not themselves physical things to be perceived by the senses, but must be grasped by reason. If, taking any of these examples—say, justice, health, or strength—we ask, “How big is it? What color is it? How much does it weigh?” we are obviously asking the wrong kind of question. Forms are ideas, not in the sense of concepts or abstractions, but in that they are realities apprehended by thought rather than by sense. They are thus‘separate’in that they are not additional members of the world of sensible things, but are known by a different mode of awareness. But this does not mean that they are ‘located elsewhere'... — Eric D Perl, Thinking Being, p28

The same general idea applies to all kinds of 'intelligible objects': they don't exist as objects in the phenomenal domain, but are more like principles.

Yes, that's a possible view and I sort of agree with it. But note that this raises the question: would those principles still 'exist' if they are not instantiated in the things they 'regulate'?

I believe that mathematical truths (and not only them BTW), would still 'exist' even if they were not instantiated. This of course would ask the question: how? What then would be their ontological support?

That's the debate between Aristotle and Plato in a nutshell: Plato has it that the ideas are real quite apart from any instance of them, Aristotle that they are only real as manifested in concrete particulars.

But such principles as the law of the excluded middle would presumably obtain in any world. That is what 'true in all possible worlds' means - although that is not highly regarded nowadays, because, as we've been seeing, we're prepared to entertain the idea of 'other universes' where such principles may not hold at all, But the question I have about that is, how could a world exist, if such principles didn't hold? In a sense, such principles are like constraints.

In any case, the specific point of the Eric Perl quote is to show that the idea of a 'separate realm' is not referring to a literal place. 'They are thus ‘separate’ in that they are not additional members of the world of sensible things, but are known by a different mode of awareness.’

How could they not be? I mean, OK, under idealism, mathematics is nothing but mental constructs. I get that, and there are even non-idealists that say something similar, but since they can be independently discovered, it seems more than just a human invention. — noAxioms

Well, it depends on the idealist, after all. Some idealists would contend that mathematical truths are concepts. But maybe there is an eternal and necessarily existing mind of some kind that always knows them.
A purely physicalist view, however, is difficult to reconcile with the existence of abstract objects. For instance, logical operations do not seem to be reducible to physical causality, which seems contingent.
Generally physicalists oppose platonism due to the fact that it posits an irreducible non-physical reality.

On the other hand, an idealist that doesn't posit any eternal mind shares the same difficulty.

If a mathematical structure is going to supervene on mathematical truths, then those truths are going to need to be accessible by far more than just reason, which sounds like a mental act or some other construct that instantiates the mathematics (such as a calculator). — noAxioms

It depends on what we call 'reason'. If by reason we mean the mental ability to make deductions, inductions, reasonings and so on, well, at least a good part of mathematical truths are accessible to our finite minds. Complex calculations do not but we do understand them. So, either mathematics transcends reason or reason at least potentially can understand everything in math.

I'm actually being moved by this reasoning, so yes. — noAxioms

:up:

I think not the point. Said intelligence would need to be presented with an environment where such tools would find utility. It need not be 'of any kind' for mathematics to be independently discoverable. — noAxioms

I take this as an agreement. I mean, the potentiality to understand 'our' mathematics would be there. So, at least in principle, that intelligence could understand our mathematics.

An approximation of it can be, yes. A classical simulation is capable of simulating this world in sufficent detail that the beings thus simulated cannot tell the difference. Another funny thing is that GoL is more capable of doing this than is our universe due to resource limitations that don't exist under GoL. — noAxioms

Well, to be honest, I don't think that conscious beings can be understood in purely computational terms. But, I still don't see how it can be considered a separate world from the one where the simulation is run (unless you mean from the 'perspective' of the simulated 'entities', assuming that such a concept makes sense).

A world is what it is, and a simulation of it is a different thing, sort of like the difference between X and the concept of X, something apparently many have trouble distinguishing.. — noAxioms

Ok! Yes.

I need more of a mathematics background to give an intelligent answer to that. — noAxioms

Don't worry, neither do I. It is an interesting idea nevertheless IMO.



I know that you do not expect a reply but thanks for the thoughts. Something you said is above my level. I'll think about your answer and maybe I'll write some thoughts about some parts of it.

↪boundless That's the debate between Aristotle and Plato in a nutshell: Plato has it that the ideas are real quite apart from any instance of them, Aristotle that they are only real as manifested in concrete particulars. — Wayfarer

Agreed! The problem with Aristotle's view is IMHO that at least some abstract concepts do seem completely independent from their particulars. Mathematical and logical truths are an excellent example of that. Incidentally, I believe that theistic philosophers mantained that God's mind was actually the 'receptacle' of those forms and we can understand them because we are also rational beings created by God (Christians would say 'created in image and likeness'). A middle way of sorts between Aristotelism and Platonism. So in this latter view ('conceptualism' I think it was called), these forms are neither ontological independent from anything else (as in Platonism*) nor dependent from the particulars (as in Aristotelism**).

*Of course there is the possibility that Platonism actually was closer to conceptualism that is often recognized. After all, there was a hiearchy of the Forms in Plato's thought, with the Form of the Good as the Highest. Neoplatonism, certainly, was close to 'conceptualism' and incorporated some of Aristotle's views.

**Similarly, one wonders how much Aristotle's thought was also far from conceptualism, given that Aristotle was also a theist.

But such principles as the law of the excluded middle would presumably obtain in any world. That is what 'true in all possible worlds' means - although that is not highly regarded nowadays, because, as we've been seeing, we're prepared to entertain the idea of 'other universes' where such principles may not hold at all, But the question I have about that is, how could a world exist, if such principles didn't hold? In a sense, such principles are like constraints. — Wayfarer

I think I agree with that. Mathematics, logic and so on seem 'transcendental' with respect of the world (at least if we assume that the worlds are at least partly intelligible).

In any case, the specific point of the Eric Perl quote is to show that the idea of a 'separate realm' is not referring to a literal place. 'They are thus ‘separate’ in that they are not additional members of the world of sensible things, but are known by a different mode of awareness.’ — Wayfarer

Agreed. It is useful to note that there are various forms of Platonism. Penrose's view seems to be indeed of a separate ontological realm, accessible to our reason, albeit certainly not a 'place'. Probably some platonists have a 'quasi-materialistic' view of the 'world of Forms', but generally do not.

Mathematics, logic and so on seem 'transcendental' with respect of the world (at least if we assume that the worlds are at least partly intelligible). — boundless

A safe assumption, seeing as how we've been able to successfully exploit those principles through the application of mathematics. (I'd say a bit more about 'conceptualism' but I don't think this is the thread for it.)

A purely physicalist view, however, is difficult to reconcile with the existence of abstract objects. — boundless

A problem with a materialist view perhaps, since only material things exist. A physicalist view only says that people are no more than arrangements of physical stuff. The view doesn't deny the potential existence of non-material things like forces and abstractions. At least that's how I distinguish materialism from physicalism.
Is mathematics abstract? That makes it sound like it's all mental concepts instead of anything objective, but I don't think you're using 'abstract' in that way here.

For instance, logical operations do not seem to be reducible to physical causality, which seems contingent.

Of course not. 'Physical' is a reference to our universe. If logical operations were physical, they'd be a property of this universe and not anything objective. Something in a non-physical universe (like GoL) could not discover mathematics.

Generally physicalists oppose platonism due to the fact that it posits an irreducible non-physical reality.

I'll let @wayfarer comment on that since I don't know Platonism enough to know what they assert.

If a mathematical structure is going to supervene on mathematical truths, then those truths are going to need to be accessible by far more than just reason, which sounds like a mental act or some other construct that instantiates the mathematics (such as a calculator). — noAxioms

It depends on what we call 'reason'. If by reason we mean the mental ability to make deductions, inductions, reasonings and so on, well, at least a good part of mathematical truths are accessible to our finite minds.

Being accessible to minds has nothing to do with the truth of them.

For one thing, the vast majority of real numbers are inaccessible to us. We only have access to countably many of them. Actual mathematics would not be thus restrained.

What else might you call 'reason'?

So, at least in principle, that intelligence could understand our mathematics. — boundless

Maybe it's us understanding some of theirs.

Well, to be honest, I don't think that conscious beings can be understood in purely computational terms.

Good to see that we don't agree on everything then.

But, I still don't see how it can be considered a separate world from the one where the simulation is run (unless you mean from the 'perspective' of the simulated 'entities', assuming that such a concept makes sense).

Difference of map and territory. There's the thing, and then there's a simulation of that thing. So while we can be simulated, by definition, we are not simulations.

Ok! Yes.

OK, you seem to grok that.

I need more of a mathematics background to give an intelligent answer to that. — noAxioms

Generally physicalists oppose platonism due to the fact that it posits an irreducible non-physical reality.
I'll let wayfarer comment on that since I don't know Platonism enough to know what they assert. — noAxioms

It's actually a very simple idea: that natural numbers (and other such intelligible objects) are real, but not materially existent. Which, of course, is anathema to materialism, which must insist that anything that exists is matter (or matter-energy). You can say it in a sentence or two, but it is the subject of thousands of volumes of argument. (Incidentally I'm not any kind of authority on Platonic scholarship but I regard this salient point as a matter of common knowledge.)

Is mathematics abstract? That makes it sound like it's all mental concepts instead of anything objective, but I don't think you're using 'abstract' in that way here. — noAxioms

The other point is that mathematics seems to be ‘true’ in a way that goes beyond the objective. We usually think of ‘objective’ as meaning something inherent in the object, or at least independent of our perception. But mathematics is often the means by which we define what’s objective in the first place—so in that sense, it seems to transcend the domain of the objective rather than just belong to it.

I’m not using ‘abstract’ to mean just 'mental' or 'subjective'—mathematical truths don’t seem to depend on individual minds. But it’s not clear that they’re part of the natural world either. That leaves a kind of philosophical gap: we trust mathematics to describe the real, but we’re not sure where mathematical truths themselves fit into our picture of reality.

These are by nature very hard arguments to adjuticate but I'm comfortable with the classical or Aristotelian understanding of them being transcendent truths - see this account of Aristotelian realism in mathematics, from which:

The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them.

I question Wayfarer's distinguishing between "existing" and "real". As a physicalist (more or less), I'd simply say that abstractions do not exist as independent entities in the world. We apply the "way of abstraction" - by considering several objects with some feature(s) in common, and mentally ignore all the other features. This process enables us to consider properties independently of the objects that possess these properties - even though those properties don't actually have independent existence; rather: they have immanent existence (they exist within objects). Example: we can consider several groups of objects, each of which has 3 members - and from this, we abstract "3". 3 is a property possessed by each of these groups.

This process is the basis of abstraction, but we can also conceptualize higher order abstractions by applying logic and extrapolating. That's the foundation of mathematics (from a physicalist perspective).

I question Wayfarer's distinguishing between "existing" and "real". As a physicalist (more or less), I'd simply say that abstractions do not exist as independent entities in the world. We apply the "way of abstraction" - by considering several objects with some feature(s) in common, and mentally ignore all the other features. This process enables us to consider properties independently of the objects that possess these properties - even though those properties don't actually have independent existence; rather: they have immanent existence (they exist within objects). Example: we can consider several groups of objects, each of which has 3 members - and from this, we abstract "3". 3 is a property possessed by each of these groups. — Relativist

Thanks for your comments. Needless to say, I will take issue.

First, I think this begs the question. You assume that to be real is to be an independent entity—or at least fo be some thing with “immanent existence” within particulars. But that’s the point at issue. For the physicalist, then of course abstractions like numbers can’t exist independently. But the philosophical question is whether that assumption is warranted and simply asserting it doesn’t settle it.

Second, the idea that numbers are just features abstracted from collections—say, that we form the concept of “3” by noticing trios in the world—follows a broadly nominalist and empiricist line (like J S Mill). But this has its own problems. For one, to even perform the act of abstraction, we already need the concept of number. We don’t derive the idea of “three” from objects; rather, we recognize objects as “three” because we already grasp the concept a priori. In that sense, the number is not a mere feature of things, but something we bring to experience through rational apprehension. (Try explaining 'the concept of prime' to a dog!)

Third, the truths of mathematics don’t seem to be empirical at all. The fact that 3 + 2 = 5 holds independently of any particular instance—it would be true even if there were no physical groups of five objects anywhere. This suggests that mathematical truths are not dependent on the world, but structure our ability to make sense of it. Mathematical physics sees the world through the prism of theory, hence is able to discern things about it which could never be seen by an eye not so trained. That’s a very different kind of reality than physical immanence, and it’s part of what motivates mathematical Platonism.

the philosophical question is whether that assumption is warranted and simply asserting it doesn’t settle it. — Wayfarer

Nothing's settled in metaphysics, but it does seem unparsimonious to consider them part of the furniture of the world.

We don’t derive the idea of “three” from objects; rather, we recognize objects as “three” because we already grasp the concept a priori. In that sense, the number is not a mere feature of things, but something we bring to experience through rational apprehension. (Try explaining 'the concept of prime' to a dog!) — Wayfarer

A priori? That's debatable, but I'll grant that we recognize more stuff vs less stuff, and could probably arrange collections into an order. Once we start counting, we're abstracting- but not until then.

To conceptualize a prime number, we first need to have learned some basics (abstractions).

The fact that 3 + 2 = 5 holds independently of any particular instance—it would be true even if there were no physical groups of five objects anywhere. This suggests that mathematical truths are not dependent on the world, but structure our ability to make sense of it. — Wayfarer

Twoness, threeness (etc) are certainly ontological properties of groups, and there are logical relations between these properties. Is this a truth? Not in my (deflationary) view, because a truth is a proposition. But we can formulate true propostions that correspond to the relations between twoness, threeness etc.

Mathematics is taught (and utilized) in a way that seems to imply platonism, but that doesn't make it so, and I don't think it justifies the belief that it is so. Why make the unparsimonious assumption that they exist?

They don't exist, but they're real. That's the point! In the classical vision the rational soul straddles this realm between the phenomenal and the noumenal. It's not an 'unparsimious assumption' but an insight into the nature of a rational mind.

More evidence of that, is the undeniable fact that man (sorry about the non PC terminology) has the ability to 'peer into the possible' and retrieve from it, many things previously thought impossible. The whole progress of modern science and technology is testimony to that - at the same time that the Armstrongs of this world deny the very basis on which this has been accomplished (as 'the possible' by very definition, does not comprise 'things that exist' but 'things that might exist'!)

In (a) new paper, three scientists argue that including “potential” things on the list of “real” things can avoid the counterintuitive conundrums that quantum physics poses. It is perhaps less of a full-blown interpretation than a new philosophical framework for contemplating those quantum mysteries. At its root, the new idea holds that the common conception of “reality” is too limited. By expanding the definition of reality, the quantum’s mysteries disappear. In particular, “real” should not be restricted to “actual” objects or events in spacetime. Reality ought also be assigned to certain possibilities, or “potential” realities, that have not yet become “actual.” These potential realities do not exist in spacetime, but nevertheless are “ontological” — that is, real components of existence.

“This new ontological picture requires that we expand our concept of ‘what is real’ to include an extraspatiotemporal domain of quantum possibility,” write Ruth Kastner, Stuart Kauffman and Michael Epperson.

Considering potential things to be real is not exactly a new idea, as it was a central aspect of the philosophy of Aristotle, 24 centuries ago. An acorn has the potential to become a tree; a tree has the potential to become a wooden table. Even applying this idea to quantum physics isn’t new. Werner Heisenberg, the quantum pioneer famous for his uncertainty principle, considered his quantum math to describe potential outcomes of measurements of which one would become the actual result. The quantum concept of a “probability wave,” describing the likelihood of different possible outcomes of a measurement, was a quantitative version of Aristotle’s potential, Heisenberg wrote in his well-known 1958 book Physics and Philosophy. — Quantum Mysteries Dissolved

They don't exist, but they're real. That's the point! In the classical vision the rational soul straddles this realm between the phenomenal and the noumenal. It's not an 'unparsimious assumption' but an insight into the nature of a rational mind. — Wayfarer

It sounds like equivocation, or cognitive dissonance.

More evidence of that, is the undeniable fact that man (sorry about the non PC terminology) has the ability to 'peer into the possible' and retrieve from it, many things previously thought impossible. — Wayfarer

The power of abstraction is present irrespective of the metaphysical interpretations we make of the process.

Reality ought also be assigned to certain possibilities, or “potential” realities, that have not yet become “actual.” These potential realities do not exist in spacetime, but nevertheless are “ontological” — Quantum Mysteries Dissolved

This sounds a bit like a presentist who considers as "existing" everything that exists, has existed, or will exist - i.e. a 4-dimensional landscape for identifying existents. We can make predictions about what will exist, but the act of prediction is just an intellectual exercise - epistemoligical. The same seems to apply to the possibilities you reference, but this seems epistemological (educated guesses about possible existents), not ontological.

I do see the utility of having a category for non-actual possibilities, but I don't see how this applies to mathematical abstractions in general. It only seems to apply to abstractions that describe non-actual possible existents- a small subset of all mathematical abstractions.

It sounds like equivocation — Relativist

I agree that this could sound like equivocation if you assume that existence and reality are synonymous. But again that begs the question of the reality of abstracta, which is the point at issue. To say something is “real” without existing in the spatiotemporal, empirical sense is precisely the point when discussing abstracta, mathematical truths, or modal possibilities. These are not “things” in the physical world, but they constrain what can be true of that world - hence their designation 'laws'/ The very framework of physics, for example, depends on mathematical structures that don't exist materially.

The power of abstraction is present irrespective of the metaphysical interpretations we make of the process. — Relativist

The capacity for abstraction is one thing, but the ontological status of what is abstracted - logical laws, symmetries etc - is the point at issue. If we’re to be strict materialists, then where do these structures reside? All in the mind? Just cognitive conveniences? or are they revealing something deeper about reality? That’s the live question, not the utility of abstraction per se.

It only seems to apply to abstractions that describe non-actual possible existents- a small subset of all mathematical abstractions. — Relativist

That is also not relevant to the fact that the ability to see via mathematical abstraction is so instrumental in the progress of science itself.

Bottom line here: the physicalist theory must be supported by some kind of 'brain-mind' identity. Why? Because it is necessary for them to argue that reason itself is somehow physical in nature. Whereas the non-materialist can simply say, look, reason comprises wholly and solely the relationship of ideas. These can be instantiated or realised in many different forms and many different media, so how can they be regarded a physical? The only fallback against that is to try and show that ideas are somehow identical with neural structures - as indeed D M Armstrong and other materialists insist.

The capacity for abstraction is one thing, but the ontological status of what is abstracted - logical laws, symmetries etc - is the point at issue — Wayfarer

The ontological status of a concept is that it is nothing more than a mental "object". You can apply whatever theory of mind you like to that (not just physicalism). I'm arguing that abstract objects are no more than mental objects- irrespective of what mental objects are. The mental objects that are abstractions are descriptions (e.g. detailing some or all the intrinsic properties that might be held by some objects in the world). Some such mental objects will correspond to something that exists - now, in the past, future, or perhaps in an independent universe (if such things exist). Others will correspond to nothing in the world (anywhere/anywhen).

We can also divide these mental objects into subsets: those that are physically possible (which may or may not exist) and those that are physically impossible.

It seems that you're defining as "real" : all the mental objects that are physically possible, irrespective of whether it exists, has existed, or will exist. If that's the extent of it, it's semantics. But I suspect you think it's something more than semantics.

the ability to see via mathematical abstraction is so instrumental in the progress of science itself. — Wayfarer

Sure, but this just suggests that scientists can extrapolate from what they know, to make good guesses as to what sorts of objects may exist. "Sorts of objects"= universals. Either a universal (or physically possible universal) is instantiated or it is not.

The only fallback against that is to try and show that ideas are somehow identical with neural structures — Wayfarer

That's not really necessary. Hebbian learning doesn't entail a structure being created, it entails patterns of neuron firings facilitated by changes to action potentials.

But that's a broader discussion. Let's focus on abstractions for now. I think they're nothing more than mental objects. If you think they are something more than that, then please describe.

These are not “things” in the physical world, but they constrain what can be true of that world - hence their designation 'laws'/ The very framework of physics, for example, depends on mathematical structures that don't exist materially. — Wayfarer

But mathematical structures are effectively tautologies so I don't see any reason for them to be meaningfully instantiated in some realm of their own or something like that where they magically affect the rest of reality.

The only fallback against that is to try and show that ideas are somehow identical with neural structures - as indeed D M Armstrong and other materialists insist. — Wayfarer

But there is overwhelming evidence that physical structures like brains are sufficient for all our reasoning, including mathematical. Why do you need to invoke anything else?

The ontological status of a concept is that it is nothing more than a mental "object". — Relativist

What is a 'mental object' in the first place? Consider a very basic one, namely equals ('='). Any child with a modicum of education will understand that symbol by age of 5 or 6. But there is no such physical object, is a pure concept, which can be grasped only by reason. You can form a mental image of the equals symbol, but neither the image nor the symbol is itself the concept 'equals'.

It seems that you're defining as "real" : all the mental objects that are physically possible, irrespective of whether it exists, has existed, or will exist. If that's the extent of it, it's semantics. But I suspect you think it's something more than semantics. — Relativist

The way I put it is that what Greek philosophy describes as universals are ubiquitous constituents of rational thought. I know that D M Armstrong, who you refer to, also defended the idea of universals, but on a materialist basis. Whereas I'm arguing that universals are reals that can only be grasped by reason ('equals' being one example.)

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'. ...In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness. The connected ambiguity in the word 'idea', which we noted at the same time, also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts. — Bertrand Russell, Problems of Philosophy - The World of Universals

Also:

Consider that when you think about triangularity, as you might when proving a geometrical theorem, it is necessarily perfect triangularity that you are contemplating, not some mere approximation of it. Triangularity as your intellect grasps it is entirely determinate or exact; for example, what you grasp is the notion of a closed plane figure with three perfectly straight sides, rather than that of something which may or may not have straight sides or which may or may not be closed. Of course, your mental image of a triangle might not be exact, but rather indeterminate and fuzzy. But to grasp something with the intellect is not the same as to form a mental image of it. For any mental image of a triangle is necessarily going to be of an isosceles triangle specifically, or of a scalene one, or an equilateral one; but the concept of triangularity that your intellect grasps applies to all triangles alike. Any mental image of a triangle is going to have certain features, such as a particular color, that are no part of the concept of triangularity in general. A mental image is something private and subjective, while the concept of triangularity is objective and grasped by many minds at once. — Edward Feser

Bolds added. So my argument is, that the coherence of reason depends on universal judgements which are not themselves found in the objective world - they're transcendental in nature. But that, due to the overwhelmingly nominalist and empiricist cast of modern thought, their reality cannot be admitted, as to do so undermines the materialism that it erroneously upholds. However, this also means that materialist arguments are inevitably circular and self-defeating, as they must rely on such non-material principles to even establish the meaning of 'material' and 'physical'. It is a hand that cannot grasp itself.

But mathematical structures are effectively tautologies so I don't see any reason for them to be meaningfully instantiated in some realm of their own or something like that where they magically affect the rest of reality. — Apustimelogist

But then, what's your account of the 'unreasonable effectiveness of mathematics in the natural sciences' (Eugene Wigner). If they were purely tautological, how could they be exploited to discover things that otherwise would never have been known? The example I often give is Dirac's discovery of anti-particles, which was predicted solely on the basis of mathematics, with no empirical evidence forthcoming till much later. How could tautological statements yield genuinely new observations? Not to forget the many predictions arising from Einstein's theories that took decades to empirically validate ('Einstein Proved Right Again').

But there is overwhelming evidence that physical structures like brains are sufficient for all our reasoning, including mathematical. Why do you need to invoke anything else? — Apustimelogist

What 'overwhelming evidence' is there that the brain is a 'physical structure'? A building is a 'physical structure', as is a machine. Both structures can be accounted for wholly and solely in terms of physical and chemical principles. But even very rudimentary organisms already instantiate order on a different level to that of the physical. Sure, the reductionist view is that living tissue is 'nothing but' physical matter, but that is highly contested and besides not in itself an empirical argument. But when we get to the human brain, which is the most complex naturally-occuring phenomenon known to science, I see no reason to believe that it can be described in terms of, or limited to, physical principles, nor to describe the brain as a physical object. It is an embodied organ, embedded in a body, culture and environment (subject of disciplines such as neuro-anthropology which by no means default to materialism as an explanatory paradigm.)

For the nature of mathematics, there is no reason to believe that this is grounded in or determined by any physical laws or relationships. You simply assume that on the basis of the inborn materialism of the culture we're sorrounded by, but there are plenty who will take issue (see What is Math? Smithsonian Magazine.)

But then, what's your account of the 'unreasonable effectiveness of mathematics in the natural sciences' (Eugene Wigner). If they were purely tautological, how could they be exploited to discover things that otherwise would never have been known? The example I often give is Dirac's discovery of anti-particles, which was predicted solely on the basis of mathematics, with no empirical evidence forthcoming till much later. How could tautological statements yield genuinely new observations? Not to forget the many predictions arising from Einstein's theories that took decades to empirically validate ('Einstein Proved Right Again'). — Wayfarer

Maths is like writing. Its a language that describes structure. The unreasonable effectiveness of writing is not magic, its that you can invent words to represent anything in the world, anything in imagination you want. Its the same with math. Math is a gigantic field with many many different topics where you can describe many different facets of structure and use math to invent structures that nicely fit things you observe. Its not miraculous at all.

I genuinely have never understood why people find it miraculous that people can invent a model that makes predictions, some of which havn't been observed yet, and they turn out to be the case. I don't understand why people find that miraculous or interesting. I don't need a special explanation of why that sometimes happens. All that maths does is describe structure in terms iof quantities. You observe stuff in the world with a structure and you fit math to it. Its very simple. What about maths that works well is it is flexible and diverse so you can invent math that describes a huge number of things completely disparate.

. Both structures can be accounted for wholly and solely in terms of physical and chemical principles. — Wayfarer

So is a brain.

But even very rudimentary organisms already instantiate order on a different level to that of the physical. — Wayfarer

Complexity doesn't make something not physical.

living tissue is 'nothing but' physical matter, but that is highly contested and besides not in itself an empirical argument. — Wayfarer

Outrageous statement.

I see no reason to believe that it can be described in terms of, or limited to, physical principles, nor to describe the brain as a physical object. — Wayfarer

Yes, we can describe it in terms of things like statistical inference and machine learning, neither of which assume anything other than the idea that learning is embodied by physicla stuff: i.e. cells, biochemistry, fundamental physics, all of which there is some substantial understanding.

It is an embodied organ, embedded in a body, culture and environment — Wayfarer

Yup, no conflict here.

For the nature of mathematics, there is no reason to believe that this is grounded in or determined by any physical laws or relationships. — Wayfarer

All I am assuming is that a physical structure called a brain or perhaps, another kind of machine, can learn to do math purely in virtue of its physical structure and the kind of learning or inference it can perform as describable via statistical inference and machine learning.

I don't understand why people find that miraculous or interesting. — Apustimelogist

In that case, there's nothing further to discuss. Philosophy begins in wondering about what is usually taken for granted.

It's actually a very simple idea: that natural numbers (and other such intelligible objects) are real, but not materially existent. — Wayfarer

Does this mean existent, but not in a material way? Because that implication is there, an equivocation of being real and existing. I point this out because there are those that very much distinguish the two, even if only by definition. Relativist detects the lack of the equivocation implied above.

I question Wayfarer's distinguishing between "existing" and "real" — Relativist

I have seen them used thus, as I have: Existing but not real or v-v.

If there are volumes of arguments about [reality of say mathematics], it must be a more important issue than simply one of the words being defined differently.

The other point is that mathematics seems to be ‘true’ in a way that goes beyond the objective. We usually think of ‘objective’ as meaning something inherent in the object, or at least independent of our perception. — Wayfarer

I've been kind of up front about my usage of 'objective' to mean 'not relative', but you're seeming to imply 'not subjective' here. A truth about an object (as if 'object' had any sort of objective meaning) seems to be relative to the object, which is fine for a predicate. Is 2+2 adding up to 4 an objective truth, or is it only relative to this mathematics we seem to have discovered? Maybe there's different mathematics where 2+2 is something else or is meaningless.

I remember going through a good tutorial on law of form, and it doesn't obviously get into numbers quite so fast, being somewhat more foundational than that.

But mathematics is often the means by which we define what’s objective in the first place—so in that sense, it seems to transcend the domain of the objective rather than just belong to it.

Point taken.

I’m not using ‘abstract’ to mean just 'mental' or 'subjective'—mathematical truths don’t seem to depend on individual minds. But it’s not clear that they’re part of the natural world either. That leaves a kind of philosophical gap: we trust mathematics to describe the real, but we’re not sure where mathematical truths themselves fit into our picture of reality.

Not sure indeed. The issue of descriptive vs. proscriptive comes to mind.

As a physicalist (more or less), I'd simply say that abstractions do not exist as independent entities in the world. — Relativist

I don't think you need to be a physicalist to agree with that statement.

Example: we can consider several groups of objects, each of which has 3 members - and from this, we abstract "3". 3 is a property possessed by each of these groups.

But what if numbers are more fundamental than the object. They certainly are in say GoL, where '3' definitely has causal powers, and 'objects' only exist if 3 does first. Of course, real numbers play far less of a role than do small integers.

This process is the basis of abstraction

The act of abstraction, sure, but abstract objects (like 3 itself, and not just the concept of 3) doesn't require an act abstracting.

For the physicalist, then of course abstractions like numbers can’t exist independently — Wayfarer

Why not? For the materialist, sure, but physicalist? Not sure exactly what defines a physicalist, but i thought it was something like 'mind supervenes on the physical'. It's a stance against mind not being fundamental.
My reply? Fundamental to what? Sure, I think mental processes are physical processes. That makes mind not fundamental, but there are still things that supervene on said mental processes.

We don’t derive the idea of “three” from objects; rather, we recognize objects as “three” because we already grasp the concept a priori. In that sense — Wayfarer

Be nice to know how it came about. If the concept is already grasped, then the roots of that concept go back further than Relativist's example. Perhaps tokens were grouped to match the count of something, but without knowing that there are 3 tokens. I don't think we'll ever know the early history of being able to count, but humans are not alone in the ability to do so.

The fact that 3 + 2 = 5 holds independently of any particular instance—it would be true even if there were no physical groups of five objects anywhere. — Wayfarer

The skeptic in me wants to doubt that, but how can it not be so? Does Platonism follow from it? It seems to come down to the issue of it being true implying its reality.

From your quoted bit:

By expanding the definition of reality, the quantum’s mysteries disappear. In particular, “real” should not be restricted to “actual” objects or events in spacetime. Reality ought also be assigned to certain possibilities, or “potential” realities, that have not yet become “actual.” These potential realities do not exist in spacetime, but nevertheless are “ontological” — Quantum Mysteries Dissolved

So the dead/live cat is real, but not actual. The measured dead cat is actual. Cute, but the Wigner's friend experiment seems to challenge this unitary notion of a wave function collapse into 'actual'. I'd like to see their take on that.

This sounds a bit like a presentist who considers as "existing" everything that exists, has existed, or will exist - i.e. a 4-dimensional landscape for identifying existents. — Relativist

There are those that assert this? Seems contradictory for some event to be 'existing' and also 'will exist', which seem to be two different contradictory tenses for the same event, relative to the same 'present' event.

About the posts of @Apustimelogist

But when we get to the human brain, which is the most complex naturally-occuring phenomenon known to science, I see no reason to believe that it can be described in terms of, or limited to, physical principles. — Wayfarer

Despite there no single tiny bit having been found that doesn't operate under said physical principles. Sure, the complexity might defy unwilling understanding, but that doesn't justify any claim that it does something dependent on more than just physical interactions.

Does this mean existent, but not in a material way? — noAxioms

That's exactly what it means, and I spelled it out in my response to him.

Despite there no single tiny bit having been found that doesn't operate under said physical principles. — noAxioms

Come on. When you study neuroscience, how much physics are you required to understand? Sure, the brain and other biological structures don't operate in defiance of physics but they instantiate principles which could never be predicted on the basis of physics alone. In biology itself, there is massive disagreement as to whether reductionism ('it's all physics plus chemistry') is adequate to account for the existence of even algae. One of the founders of the neo-darwinian synthesis, Ernst Mayr, certainly no starry-eyed idealist, said ' The discovery of the genetic code was a breakthrough of the first order. It showed why organisms are fundamentally different from any kind of nonliving material. There is nothing in the inanimate world that has a genetic program which stores information with a history of three thousand million years!’

Most of the materialism on this forum has a simple origin. It begins with Descartes' division of the world into res cogitans (thinking thing) and res extensa (matter). This becomes a major part of the 'new sciences' developed by Newton, Galileo, Boyle, et al in the beginning of the modern period. But 'res cogitans' is inherently problematical - what is it, where is it, and how does it affect or intervene with the physical order? Descartes himself couldn't answer these questions. So essentially it becomes shunted aside, in favour of exploration of the so-called purely physical, the objects of the hard sciences, definite, measurable, and with inummerable applications in technology. Why question that? How could it be considered that res cogitans was anything other than a ghost in the machine?

The modern mind-body problem arose out of the scientific revolution of the seventeenth century, as a direct result of the concept of objective physical reality that drove that revolution. Galileo and Descartes made the crucial conceptual division by proposing that physical science should provide a mathematically precise quantitative description of an external reality extended in space and time, a description limited to spatiotemporal primary qualities such as shape, size, and motion, and to laws governing the relations among them. Subjective appearances, on the other hand -- how this physical world appears to human perception -- were assigned to the mind, and the secondary qualities like color, sound, and smell were to be analyzed relationally, in terms of the power of physical things, acting on the senses, to produce those appearances in the minds of observers. It was essential to leave out or subtract subjective appearances and the human mind -- as well as human intentions and purposes -- from the physical world in order to permit this powerful but austere spatiotemporal conception of objective physical reality to develop. — Thomas Nagel, Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False, p33

Again, opinion, but the opposite opinion is to posit the existence of something (a preferred moment in time) for which there is no empirical evidence, only intuition, and I rank intuition extremely low on my list of viable references. — noAxioms

Well we agree there when it comes to questioning which of our resources is most likely to lead us to understanding the nature of things.

You also seem to agree that there are things independent of minds. In which case you would appear to be one the "anybodies" who support mind-independent reality.
Except for the 'reality' part, sure. Mind-independent, sure. Relation-independent, no. I think in terms of relations, but I don't necessarily assert it to be so. I proposed other models that are not relational and yet are entirely mind-independent. See OP.

Why must something be "relation-independent" in order to count as real? Is anything relation-independent? I would say probably not.

We have no relation to such worlds

Sure we do. It's just a different relation than 'part of the causal history of system state X', more like a cousin relation instead of a grandparent relation. The grandparent is an ancestor. The cousin is not. The cousin world is necessary to explain things like the fine tuning of this world, even if the cousin world has no direct causal impact on us.

We have no physical relation to such worlds. If we did they would be counted as of this world. The fine-tuning argument has never done it for me. I don't believe we can accurately calculate odds when the sample is but one. Even if we could the outcome is still not a zero possibility. That said I'm not against the 'Multiple Universes' idea. It does seem to be impossible to test. though.

How could we ever demonstrate that consciousness collapses the wave function

That interpretation can be shown to lead to solipsism, which isn't a falsification, but it was enough to have its author (Wigner) abandon support of the interpretation.

I'll take your word for it.

or that there really are hidden variables?

By definition, those can neither be demonstrated nor falsified.
They have proven that certain phenomena cannot be explained by any local hidden variable theory, but that just means that hidden variable proposals are necessarily non-local.

So it would seem.

There are those that assert this? Seems contradictory for some event to be 'existing' and also 'will exist', which seem to be two different contradictory tenses for the same event, relative to the same 'present' event. — noAxioms

There is a set of things that existed in the past, a set of things existing in the present, and a set of things that will exist in the future. The union of these three sets comprise the set of existents. This doesn't preclude tensed facts, but one must be careful with wording.

Contrast this with possible objects we might conceive of. The conception may or not correspond to a member of the set of existents.