I'm no expert on the subject, or even a (good) philosopher, but I tend to agree with Pneumenon here.

The author appears to argue that 'Mathematical Platonism...the view that mathematical reality exists by itself, independently from our own intellectual activities' is false, and it is false because mathematics is dependent on our own intellectual activities.

The point is, there are more than correlations between mathematics and nature; as Galileo said, and surely this is a Platonist sentiment, 'the book of nature is written in mathematics' (and as is well-known, Galileo was indebted to the revival of Platonism in the Italian Renaissance). — Wayfarer

No doubt we use mathematics to help (I don't see any reason to believe that mathematics is absolutely essential to the task) decipher the "book of nature"; but it does not follow from this that nature is somehow in itself reducible to mathematics. Obviously nature carries the possibility of math, otherwise there would be no math. For math to manifest,(although apparently some animals can count in rudimentary ways) the evolution of beings like us, who have developed the capacity for symbolic thought would seem to be necessary.

Nowadays there is an overwhelming urge to 'relativise' the whole matter, to say that number is something internal to or peculiar to humans - which is pretty well the impulse behind the Rovelli paper too. — Wayfarer

Honestly I think you should stick to trying to unpack the philosophical issues and refrain from indulging in tendentious and irrelevant social commentary.

'Concrete representations' are written symbols or representations in any material form. — Wayfarer

OK, so just to be clear you would consider five apples and the numeral '5' to be representations of an actual (something more than merely conceptual) object? For me the first is an instantiation and the second a representation of a characteristic or quality: namely fiveness. I don't believe the quality or characteristic of fiveness qualifies and an object except in the sense that it can be a conceptual object for us. I can't see any other coherent sense in which it could be said to be an object.

Abstract objects and their particular representations are inseparable. There cannot be one without the other. Representations cannot exist without that which they represent, and that which is represented cannot exist without its representations. If there is no number 5 then there are no 5 objects. — litewave

Perhaps, but do abstract objects exist independently of their being thought, and if so, how would that "existence" look?

The point was just that there is no single real (as opposed to conceptual or abstract) object, 'train journey' of which all train journey are representations or instantiations, — Janus

A train journey is a substantial act. So it is hylomorphically intantiated. My point is that we can unpack this in a general fashion by abstracting away both the formal and material principle involved. We can separate the formal necessities or constraints from the material accidents or fluctuations.

Topology seems to demonstrate this metaphysical principle in action. A compact surface is the constraint placed on the most generalised system we can imagine. We get a sphere. But then as a degree of freedom or accident that can’t be suppressed, the sphere could be punctured by holes. And so you get a primal model of countability that allows you to see what is really going on in something complicated, like the idea of countable train trips.

Train trips are a bad example because they start with a human mechanical imposition of a mathematical framework on a natural landscape. We are making it the case that there are some countable set of trips by some definition we all agree.

The argument here is then over the reality of mathematical structures themselves. And to follow what that argument would be - from my own hylomorphic and constraints based view - I would want to start with a clear mental picture of what the maths might actually be claiming.

So my claim was that the abstract object in question would be this kind of topological constraint that then still results in localised definite accidents. Limits turn out to be limited in this essential fashion. You can break a symmetry, but that symmetry breaking can in turn be broken by the arrival at some new terminating symmetry.

This is the physics we have discovered. It is why we wind up with the translational and rotational symmetries which all a cosmos of countable localised actions to exist. Constrain action to some spacetime point and it can still move or spin with inertial freedom.

So maths speaks deeply to the reality we observe. I just thought that counting train journeys was a misleading example because it is in no way a fundamental, or even natural, notion of a countable object. It is an action we mechanically impose on a landscape and so depends on our willingness to be indifferent about the acts which in the end do count.

If the train stopped only halfway into the station, we would all sit around debating if that not quite completed trip should still count. We would still be arguing about accidents vs necessities. But the answers wouldn’t carry much cosmological weight.

Bear in mind that I was originally talking about a train journey between two specific destinations. So, the train would need to pass through both those destinations, or begin at one and pass through the other, or begin at one and terminate at the other, to count as a train journey from one destination to the other. So, the question of whether we should think the journey completed doesn't seem relevant.

The analogy between the train journey and fiveness is not perfect, to be sure, since the train journey is an actual process or activity, whereas fiveness is a quality or characteristic. But both can coherently be understood to be abstract objects for us; and the question remains as to whether they could coherently be thought to be objects in any sense other than the abstract.

The argument here is then over the reality of mathematical structures themselves. And to follow what that argument would be - from my own hylomorphic and constraints based view - I would want to start with a clear mental picture of what the maths might actually be claiming. — apokrisis

I can't see how mathematics is, in itself and independently of any philosophical interpretation of its significance, claiming anything at all. And even if mathematics were claiming something in itself, I can't see how this would speak to "the reality of its structures". For me the very notion of "the reality of mathematical structures" beyond their being abstractions from concrete objects and processes, seems unintelligible. Mathematical "reality" seems to consist far more in possibility than it does in actuality, but then admittedly I am not much of a mathematician. I am open to explanations that I can make sense of, though.

Perhaps, but do abstract objects exist independently of their being thought, and if so, how would that "existence" look? — Janus

Yes, I think they exist independently of being thought because they are properties that different objects have in common; they are ways in which different objects are similar; they are conditions that different objects satisfy if they are of a particular kind. But I don't think we can visualize abstract objects, because they are not objects in space. (We may be able to visualize their particular instances or a typical instance.) But apparently we can inductively infer their existence by noticing in what ways different things are similar.

I agree that they are properties, and that properties as conceived can be understood to be conceptual or abstract objects. The problem I have is that I can't see any coherent way to think that properties are abstract objects in any sense other than their being thought as such.

If you are trying to imagine where objectively existing abstract objects would exist, one answer could be: nowhere. Just as the universe exists nowhere, as there is no underlying space for it.

Another answer could be: in the collection of all possible objects, which are related by the relations of similarity, composition and instantiation. Some of those objects form topological or metric spaces, including the one in which we live.

I'm not trying to imagine where they exist, but rather what kind of existence they could be said to have. For example fiveness is undoubtedly a characteristic or quality of many groups of things, and it is undoubtedly an abstract object of thought. So it has concrete existence as a quality and ideal existence as both a quality and an object. But it does not have concrete existence as an object. What more can we say about its existence? I can't see the point in saying that it has some further kind of concrete or objective existence if no account or explanation of that purported existence can be given.

The kind of existence of an abstract object (property) is such that the object has instances. And the kind of existence of a concrete object is such that the object has no instances.

Bear in mind that I was originally talking about a train journey between two specific destinations. — Janus

Sure. Two stakes were stuck in the ground. And so suddenly the landscape had your chosen metric imposed on it.

For me the very notion of "the reality of mathematical structures" beyond their being abstractions from concrete objects and processes, seems unintelligible. — Janus

Yeah. But what I was arguing is that your notion of material concreteness is itself just a matching abstraction.

So sure, there must be a material aspect to reality. And you are now insisting to me about the reality of that abstracted notion. If the fallacy of misplaced concreteness is a thing, it would have to apply to your claims about however you picture this idea of definite local particulars.

This is why Peirce would grant reality to both the formal and material aspects of nature. As generalisations, they are each "concrete" - or just as concrete as each other in terms of both being essential aspects of the whole.

In a holistic metaphysics, substance is emergent. It becomes localise individuation. But you sound as if you want to treat emergent individuation as the concrete baseline reality of existence. You begin with a world of objects, rather than arrive at that world.

Mathematical "reality" seems to consist far more in possibility than it does in actuality, — Janus

If we wind it back, I wasn't defending some kind of spooky seperate existence of Platonic structure. I was in fact arguing that forms are always instantiated - or would have to be intelligibly instantiable. So the kinds of structures that could exist are the kinds of structures that could dovetail with some kind of logically complementary material principle. They would have to be able to yield substantial being in interaction with that material principle.

This then leads to the question of how to conceive of that material principle in properly generic form. This would lead us towards Peirce's answer - vagueness of Firstness. Or more classically, Apeiron or Chora. Or in some modern physicalist sense, chaos or fluctuation or quantum foam. That is, a potential that is lacking in limits, but capable of being limited.

So essentially my point is that the maths that is powerful and useful when it comes to the metaphysics of possible cosmologies is the kind of maths which has this particular character. It can model the constraint of freedoms, the limitation of uncertainty, the emergence of stable habit or law.

And what is exciting is that maths could model both the formal constraints - by speaking to the necessity of certain such structures - and even the material accidents, the constants of nature that then ground that structure. These constants may turn out to be shapes - like the holes in a topological sphere. As I said, global symmetry-breaking is terminated by reaching local symmetries which it can't erase. That is why you have the particle zoo of the Standard Model. A quark or electron exists as fundamental - a fundamental excitation - because they can't be broken down any further. They put a stop to the symmetry-breaking cascade and now start to ground the construction of some kind of material content in the Universe.

That is what string theory is about. Topological irreducibility. If you curl up a higher dimensional space, you can't in the end get rid of all the kinks. You are left with some countable number of holes that then become the material character grounding the Universe. They are the knots that can't be undone.

So the material principle could be reducible to ontological structuralism - becoming the local kinks that can't in the end be rotated or translated out of existence. Matter would be part of Plato's realm, but exist in it apophatically, as the topological holes or features that can't be erased. The material part of being would be the inverse of the formal part of being.

So it was this organic conception of structure - the "co-arising holism" that physics is uncovering - that I'm contrasting to the mechanical conception of pattern generation which Rovelli is using to produce a landscape of mathematical junk.

Plato was speaking to that dawning metaphysical realisation that the intelligibility of reality is about a division of the substantial into the complementary things of the formal and material principle. Aristotle might have said it much more clearly, but the dim outlines of that emergent hierarchical view can be seen in Plato - as when he talks of The Good as a finality which acts to select certain forms, and the Chora as the need for some kind of material receptacle where structure could be instantiated.

Just consider the Platonic solids. In 2D, polygons can have any number of sides, as long as they have at least three. But in 3D, suddenly that adds a huge global constraint that limits local regularity to just 3x2 possibilities - the self-dual tetrahedron (4 triangular faces), the dual cube and octahedron (swapping faces for vertices), and the dual dodecahedron and icosahedron. So place a limit on dimensionality and only a limited number of perfectly exact resonances can fit that space.

The Platonic solids are examples of how local symmetry can become physically manifest if global symmetry is explicitly broken. And of course this mathematical realisation - this intelligible fact of any possible reality - was then used to give a Platonic account of material atomism. If material fluctuation was in fact bound by formal limitation, then these had to be the shapes that would emerge at the end of the trail. Atoms would be little triangles, and so be fiery, etc. (Of course, a sphere was the other emergent perfect shape - the one that then emerges at the infinite limit of "polygonicity".)

So yes, if maths abstracts and generalises, then of course it is stepping back towards the possible, and away from the actual or substantial.

But there are then two ways of stepping back towards generality. And hylomorphism would be about following both those paths - and being able to see the unity in the fact that they are a pair of reciprocally defined paths. Each is the other's inverse. And so the metaphysical formalisation of the description of the one can apophatically stand as the formalised description of its "other". Yin and Yang. Accident and necessity. Matter and spacetime.

It is that deep structural trick that would see Platonism - as understood charitably - being cashed out by a modern physical "theory of everything". If the material constants can be shown to be the irreducible holes produced at the limit of some process of constraint, some process of symmetry breaking, then reality would "pop out" of an intelligible mathematical description.

Again, this is the big prize that Rovelli himself is pursuing. So all his paper demonstrates is the paucity of a more conventional view of mathematics (and thence reality) as the infinite noodlings of mechanistic pattern generators.

If you want to call that "Platonism", I suppose you could. But Rovelli also wrote a book on Anaximander which showed him to be rather a lightweight on Ancient Greek metaphysics. I would rate him highly for his physical speculations, poorly for his history of philosophy.

Curvature is a geometric property of spacetime and is related by Einstein's field equation to energy. Spacetime curvature and energy determine each other through Einstein's field equation. — litewave

I'm baffled by your reply. What else did you think I said? And where yet did you say anything useful about the nature of this "energy density" which you have to go off and measure?

Sure you can quantify it as an act of measurement. But that still leaves "reality" as a number being read off a dial.

So we face a big choice at that point. Either we go with the usual naive realist view - reality is whatever we think it is that we are measuring. The phenomenological is mistaken for the noumenal. Or we instead make a virtue out the very fact that pragmatism and semiotics lies at the core of all this.

The noumenal becomes the fundamentally arbitrary or vague in our metaphysical picture. We conceive of the material principle as a state of radical indeterminism - like a quantum foam. And then structure is that formal principle which can constrain this indeterminism so that it forms an emergent state of order - like a classical realm of deterministic objects.

Again, you are talking about GR. And we know from QFT that spacetime would be material enough to be populated by an infinity of gravitational self-interaction. So nothing self-stable is specified at the level of GR modelling. Einstein's field equations had to include a cosmological constant just to prevent even a homogenous spacetime from immediately shrinking out of existence due to the smallest material fluctuation.

So you have to glue together some model of global spacetime symmetries, some model of an actual material content, plus a generic material fudge factor to keep the whole fabric expanding into a future, to get to a GR description that still needs to be fixed by constraints on is material self-interactions.

It is all kind of Heath Robinson. And yet, each of these components is well-motivated in terms the general principles they express - the need to satisfy that happy triad of constraints, the principles of locality, least action and cosmology. Through the glass darkly, the maths is expressing a holistic causal structure now. We are arriving at a Platonistic view of that kind - if not the other kind, the one that wants to conceive of nature as mere mechanistic construction.

The vast majority of it is simply useless, and of no interest to anyone whatsoever. — StreetlightX

So?

The argument straightforwardly conflates mathematical objects with mathematical practices developed using, or developed to describe, those objects. No one doubts that the mathematical practices of organisms are fluid, but that's not relevant to the Platonist's claim.

I'm baffled by your reply. What else did you think I said? And where yet did you say anything useful about the nature of this "energy density" which you have to go off and measure? — apokrisis

I said that energy is mathematically related to the acceleration that the space point imparts to another space point during an interaction. It means that energy can do work.

You said some posts back that energy and dynamics cannot be explained only by math; that we also need a "material principle". I don't understand what you mean by the material principle, and I just outlined how energy and dynamics (acceleration during interaction) can be explained by mathematical relations and how they are related to spacetime curvature, a geometric property of the mathematical object called spacetime.

The author appears to argue that 'Mathematical Platonism...the view that mathematical reality exists by itself, independently from our own intellectual activities' is false, and it is false because mathematics is dependent on our own intellectual activities — Luke

I'm not sure if you meant to phrase it how you did, but that... would be a perfectly valid argument ('it is false that the tree is blue because the tree is green - and here is why'). That said, that isn't the argument of the paper.

The argument straightforwardly conflates mathematical objects with mathematical practices developed using, or developed to describe, those objects. — Snakes Alive

It's only a 'conflation' if one assumes from the outset the Platonic position on mathematical objects. The point of the paper is to ask how tenable just such a distinction is, by setting out a disjunction ('dilemma'), the choices between which are claimed to put the Platonist in an untenable bind.

I'm not sure if you meant to phrase it how you did, but that... would be a perfectly valid argument ('it is false that the tree is blue because the tree is green - and here is why'). — StreetlightX

Except I found the author to be saying that the tree is not blue, and he did not tell us why. The author appears only to assert, or to assume the truth, that Mathematical Platonism is false. He could have written a shorter paper with the assertion that 'Mathematical Platonism is false'. But I'll take another look regarding the dilemma you mention above.

Precisely. Mathematics is mind-dependent because it's not useful to say that it isn't, and "utility" is a worthwhile measuring-stick here because mathematics is mind-dependent. It begs the question, and you can see this as soon as you ask why "utility" should matter here.

Mathematical Platonism is either true, or it is not. Why would utility help us answer that question?

Mathematical Platonism is either true, or it is not. Why would utility help us answer that question? — Pneumenon

It doesn't, anymore than utility helps us decide if mind-independent rocks and such exist.

The objections to this understanding are usually based on the inability to make this distinction; hence the common objection to Platonic realism, 'where do numbers exist'? This is because we are by habit instinctively realist; we are oriented in respect of the domain of time and space, the objective realm, which for most of us defines the scope of what is real; everything that exists is 'out there somewhere' in the objective realm. — Wayfarer

On the contrary, the problem with such a question is that most moderns are nominalists, rather than realists; they treat reality as coextensive with existence. The "objective realm" of reality is not limited to that which exists in time and space.

The problem I see with this is that if a mathematical "object", say the number five, has no existence apart from its concrete representations, then it cannot qualify as an object at all, except in the most abstract conceptual sense ... — Janus

In this context, an object is whatever is capable of being represented. Hence qualities are objects just as much as the things that embody them, and habits (including laws of nature) are objects just as much as the events that they govern. Some of these possibilities and (conditional) necessities are real - i.e., their characters are not dependent on what anyone thinks about them - even though they do not exist apart from their instantiations.

The question is explicitly about the independence of math from our intellectual activity. Rovelli - rightly, imo - does not say anything about what is or is not 'real', partly, I suspect, because the question of 'the 'real' causes more muddles than it solves. — StreetlightX

But "independent of our intellectual activity" is precisely what "real" means, assuming that "our" refers to any individual person or finite collection of people. The muddle comes from conflating reality with actuality/existence.

Within mathematics in general, there are numerous contradictions such as Euclidean vs. non-Euclidean geometry, imaginary numbers vs. traditional use of negative integers. — Metaphysician Undercover

Again, mathematics is the science of reasoning necessarily about hypothetical states of affairs. Euclidean and non-Euclidean geometry employ exactly the same (deductive) logic, but draw different conclusions because they begin with different premises; specifically, non-Euclidean geometry adopts one fewer postulate. Imaginary numbers are the perfectly logical result of defining "i" as the square root of -1, regardless of whether this corresponds to something actual.

Mathematics is a subject, so we cannot attribute to mathematics, opposing hypotheses, without contradiction. — Metaphysician Undercover

Mathematics in itself does not require the adoption of a particular set of hypotheses; it simply derives necessary conclusions from any set of hypotheses whatsoever - including, in some cases, the conclusion that those hypotheses are contradictory. Euclidean and non-Euclidean geometry are different subjects with different hypotheses. Algebra with imaginary numbers and algebra without imaginary numbers are different subjects with different hypotheses.

I conclude that you believe the word "round" existed before anyone existed, because this is what is required for the earth to have been determined as round, before anyone existed. Do you not recognize that whether or not an object has a specific property is a judgement, and nothing else? — Metaphysician Undercover

Nonsense. That which the word "round" signifies - the real character of roundness - existed in everything that possessed it before any human being existed, and would continue to exist in everything that possessed it after every human being ceased to exist. Do you not recognize that some judgments are true and others are false? This entails that there is a fact of the matter, which is independent of whatever anyone thinks about it. Any argument to the contrary is self-refuting.

Plato was speaking to that dawning metaphysical realisation that the intelligibility of reality is about a division of the substantial into the complementary things of the formal and material principle. — apokrisis

I think one thing that your analysis is missing is the understanding of what actually constitutes knowledge in Plato's philosophy. As is often said, Plato sets the bar for what constitutes ‘true knowledge’ very high. The gist is that empirical knowledge itself, sensory knowledge, can’t be considered knowledge of something real, because the senses lie, and because the real nature of ordinary objects of perception is such that they are a combination of being and non-being.

Whereas when logical and mathematical truths are known, they are known in a way that is not possible with respect to sensibles, almost in the sense that the mind unites with the object of knowledge. That is where Platonism verges dualism, although it is often implicit. It was the use that this understanding was put to, that gave the real power to Western science, but the original impetus behind the understanding was soteriological rather than utilitarian.

The Good as a finality which acts to select certain forms... — apokrisis

That thinking is Darwinian, I'm sure it has nothing to do with Plato. The Good doesn't do anything. The Good is not the demiurge.

maths speaks deeply to the reality we observe.. — apokrisis

Likewise, this is the modern understanding - not that it's wrong on that account. But one point that Aristotle makes, somewhere - I've never been able to find it again - is that metaphysics serves no useful purpose, it is not 'for' anything. Just to be able to get an understanding of it, is reward enough, the sole justification for understanding it. That is the intent of such sayings of his as 'thought thinking thought' (although I think 'thought' is a pretty lackluster word for what he is trying to convey here.)

the problem with such a question is that most moderns are nominalists, rather than realists; they treat reality as coextensive with existence. The "objective realm" of reality is not limited to that which exists in time and space. — aletheist

That's what I meant to say. Although I think 'transcendentals' can be distinguished from 'objective reality'.

Yeah. But what I was arguing is that your notion of material concreteness is itself just a matching abstraction. — apokrisis

Of course my notion of it is an abstraction, but material concreteness is experienced. If it weren't we would have no way of differentiating between the concrete and the abstract in the first place.

In this context, an object is whatever is capable of being represented. Hence qualities are objects just as much as the things that embody them, and habits (including laws of nature) are objects just as much as the events that they govern. Some of these possibilities and (conditional) necessities are real - i.e., their characters are not dependent on what anyone thinks about them - even though they do not exist apart from their instantiations. — aletheist

Well, I suppose it depends on your definition of object. I do agree that qualities have an objective (meaning mind-independent in the sense that they are not constructed by the mind) existence.

Perhaps, but do abstract objects exist independently of their being thought, and if so, how would that "existence" look? — Janus

‘Exist’ - that is the whole issue here. Again - does the number seven exist? If you point to a symbol, then, sure, that exists. But the number itself is something that can only be grasped by an intelligence capable of counting. But it’s real regardless - hence the distinction between ‘real’ and ‘existent’.

The point of the paper is to ask how tenable just such a distinction is, by setting out a disjunction ('dilemma'), the choices between which are claimed to put the Platonist in an untenable bind. — StreetlightX

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.

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. To say they have is to commit the dreaded "fallacy of misplaced concreteness" and to descend into obscurantism and incoherency.

Whereas when logical and mathematical truths are known, they are known in a way that is not possible with respect to sensibles, almost in the sense that the mind unites with the object of knowledge. — Wayfarer

Sure. That was the surprising new thing. Human thought could be organised by this new kind of logical structure. Maths cashed out a fully constrained, very mechanical, form of pattern generation. And that machine-like deductive and state-mapping approach to causality proved to be "unreasonably effective" at delivering technological control over nature.

So this was a big intellectual shock. The idea of the Machine was revolutionary. But then physical reality isn't in fact mechanical. So the "truth" of this mechanistic ontology is not the truth of the actual world.

Machinery - mathematical machinery - can be "absolutely true" because it is based on deductive proof. You assume some axiom. You derive some consequence. It seems perfectly water-tight. That is, it all degrees of freedom of rigidly suppressed. Nothing surprising can happen to derail the sequence of events. A perfect state of constraint is in effect. The only causality operating is that of linear cause and effect sequences - blind step by step deterministic construction.

So yes, this new machine mode of thinking seemed marvellous to the first logicists and mathematicians of Ancient Greece. For them, it seemed even "divine". It was thought and reason perfected. But it is ironic that you - given the way you rail against Scientism and other continuing cultural expressions of mechanical thinking - should still seem so in awe of mathematical forms. Reality ain't a computer, is it? The Cosmos is better understood as organic. Or better yet semiotic - because semiosis is the ontology which both accepts a mechanical twist to nature, but puts in its rightful place.

What I am saying is that you are still presenting an utterly confused history of the relevance of mathematics. You want it to be some kind of door to a transcendent divine aspect of existence. You respond to the reverential view taken by Pythagoreanism. And yet that exact path - that belief that maths is the royal road to Truth - is what winds up in modern technocratic reductionism. The belief that life, mind and physical reality in general can be accounted for fully and truthfully as mechanism.

...the original impetus behind the understanding was soteriological rather than utilitarian. — Wayfarer

Bullshit ontology is bullshit ontology, regardless of whether you are claiming god is a geometer or reality is a machine.

I agree with your analysis here. 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.

Again, mathematics is the science of reasoning necessarily about hypothetical states of affairs. Euclidean and non-Euclidean geometry employ exactly the same (deductive) logic, but draw different conclusions because they begin with different premises; specifically, non-Euclidean geometry adopts one fewer postulate. Imaginary numbers are the perfectly logical result of defining "i" as the square root of -1, regardless of whether this corresponds to something actual. — aletheist

Right, so if the "different premises" are contradictory, then it is impossible that they are each true. I think that this is a problem for those who claim Platonic realism concerning mathematics. If mathematical forms are so variable that they are contradictory, then what good are they? You use your forms for your purpose and I use contradictory ones for my purpose, and we each come up with contradictory understandings of the reality which we apply them to.

Euclidean and non-Euclidean geometry are different subjects with different hypotheses. Algebra with imaginary numbers and algebra without imaginary numbers are different subjects with different hypotheses. — aletheist

They are different subjects, as sub-classifications within the subject of mathematics, just like my example, biology and physics are different subjects as sub-classes within the subject of natural science. It is illogical to allow that sub-classes proceed with contradictory premises, as if each of the contradictory premises is true. Simply put, having different subjects which treat contradictory premises as if they are each true, is illogical.

Nonsense. That which the word "round" signifies - the real character of roundness - existed in everything that possessed it before any human being existed, and would continue to exist in everything that possessed it after every human being ceased to exist. — aletheist

This is very clearly false. 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".

Do you not recognize that some judgments are true and others are false? This entails that there is a fact of the matter, which is independent of whatever anyone thinks about it. Any argument to the contrary is self-refuting. — aletheist

I do not see how this is relevant. 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.

Of course my notion of it is an abstraction, but material concreteness is experienced. — Janus

Same old, same old. Semiosis says what you "experience" is your Umwelt. Sure, the "world" must stand in back of that as a noumenal constraint, some kind of recalcitrant actuality that limits the freedom of your interpretation. But then you need to pay closer attention to how conceptions are actually formed as logical dichotomies.

If you rely on a distinction like concrete~abstract, then it is the pairing that is itself the whole of the conception. Again, if you rely on experience~conception as the distinction, it is the whole of that differentiation which is the conception.

So you talk about directly experiencing the substantial material concreteness of the actual real world. But that is still just a conception. The more strongly you believe in that dualised "othering", the more deeply you are actually embedded in the Umwelt you are creating.

So the irony here is that you are strengthening your conceptualised distance from the noumenal thing-in-itself by insisting on the absoluteness of this concrete other. Your belief in the "material" - as you conceive it, conceiving it as absolutely "other" - is what makes it as abstract as it could be in terms of you "experiencing the world".

Now this is not to deny that the world is not out there in recalcitrant fashion. Nor that a conceptual division into the concrete and the abstract is not a pragmatically useful way to structure our understanding of this world. Umwelts have to achieve our lived purposes.

But to say the rock is hard, sometimes painfully hard, is no more "real" than to say a rose smells sweet or the sky is blue.

Science is at least honest in this regard. Material properties become the numbers we can read off dials. The formal aspect of existence becomes a mathematically-expressed theory. The material aspect of existence becomes an appropriately matched act of measurement. Reality is then whatever this pragmatic system of conception tells us it to be.

If it weren't we would have no way of differentiating between the concrete and the abstract in the first place. — Janus

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.

But I am arguing for an organicist or semiotic causality where the complementary aspects of any fundamental division must co-arise ... as each is the cause of its "other".

So before we could say our minds were divided by their abstract conceptions and their concrete perceptions, our state of experience would just have been an undifferentiated vagueness - the blooming, buzzing confusion of the newborn babe. A distinction between the concrete and the abstract is a structuring that grows. And no surprise really. As time passes, the generalities of the world will make themselves known as seperate from their specificities.

But in nature, the distinction is not some absolute or dualised difference. That is itself a further twist given to it by a modelling human mind. It becomes convenient to conceive of the world in the simplest fashion where the concrete and the abstract exists as actual absolutes, rather than merely as complementary limits to a useful metaphysical distinction.