So formal truths have a sort of... transcendality? Transcendency? Whatever. They go above and beyond possible worlds, basically. — MindForged

If you mean truths that hold in different possible worlds, then these truths constitute a more general/more abstract/higher-order possible world.

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

Pardon me for barging in.

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).

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. But thought has to conform to maths, not vice versa. I reckon the whole problem is, you can't fit this into the procrustean bed of neo-Darwinist epistemology. That's why it's such a subversive idea. It is supposed to be one of the things that died along with God. :smile:

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

There is no perfect form of fiveness, — Janus

The point is, it is precisely mathematics (etc) that is perfect in a way that empirical objects cannot be. X is always X (where X is a whole number); there's nothing else it can be. (Have a look at the paragraph on ontology in this Wiki article.)

BTW, Frege and Russell tried to derive mathematics from logic, and failed, for reasons later articulated by Godel (to my knowledge.)

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, since its representations are potentially infinite in number. — Janus

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.

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

The sameness is also a fact about the external world.

So spacetime with its complex structure seems to be a specific mathematical object. — litewave

Sure. Relativity falls out of the greater symmetry that results from switching from a distance preserving metric to an interval preserving metric. We are now talking about a world of objects with both a location and duration to be specified. Euclidean space was just too simple to stand as a model of physical reality. Lorentzian spacetime becomes the least number of symmetries we can get away with.

But even then, with general relativity, things are still too simple. We must tack on a tensor field to specify some energy density at every point in this spacetime. We have to tell Lorentzian spacetime how it should actually curve. A literally material constraint must be glued to the floppy Lorentzian fabric to give it a gravitational structure. And even then, the quantum of action - how G scales the interaction between the energy density and the spatiotemporal curvature - remains to be accounted for. This constant could have a purely mathematical explanation, but that is the big question for frontier physics. It might also be in some sense a pure accident of nature.

So the way that maths applies to physics is a complicated story. Mathematical symmetries do tell you about the zoo of possible constraints on any physical freedoms. But maths - being traditionally founded in spatial conceptions - may then tend to canonise the symmetries that are just to simple to be real. Constraints, in themselves, may be of irreducible complexity. And so the maths that really counts remains hidden from the conventional gaze as a result.

This is the case with Rovelli’s mathematical junkyard. Once you completely deconstruct maths so that it becomes just a flat and infinite syntactical machine, then it is going to spit out endless meaningless patterns. As Einstein said, the trick is to be as simple as possible, but not too simple.

And here is where Peirce, Aristotle, and other systems or hierarchical thinkers have got it sussed. They accept the irreduible triadicity of nature, where what exists is due to the fundamental reality of the possible, the actual and the necessary. Or the potential, substantial and final.

Rovelli’s relational interpretation of quantum theory and emergent approach to quantum gravity in fact have just this triadic character. So as a systems thinker, that is why he would latch on to the way that the overly simple conventional view winds up producing a world of unconstrained junk.

Should the mechanical view of reality be called Platonism? I think not. But Plato wasn’t a hierarchy theorist like Aristotle. So while he was groping in that direction with his positioning of the idea of the Good as the top of the pile constraint, and also with his talk of the chora as a complementary material principle, a fully triadic story was not cashed out. Platonism did get stuck in a dualism of opposed existences rather than united by a trichotomy of emergence.

Within mathematics in general, there are numerous contradictions such as Euclidean vs. non-Euclidean geometry, — Metaphysician Undercover

Not sure if you missed my reply:

https://thephilosophyforum.com/discussion/comment/219667

If you mean truths that hold in different possible worlds, then these truths constitute a more general/more abstract/higher-order possible world. — litewave

That's not what I mean (depending on what you mean by "hold"). Take this. Take any coherent math system and it's results (the theorems you derive) will not change no matter the possible world. I don't mean anything about those worlds remains fixed, I mean the formal system itself, no matter the system, doesn't have change depending on the world one is in or on the mathematical structure of the world one is in. E.g. the truths of Euclidean geometry are "true" even in a world that is non-Euclidean (true in the system, not about the world).

I think you can call this a sort of higher-order necessity, but recourse to possible worlds semantics is superfluous I think since these don't need those to explain their necessity the way other modal statements do.

We must tack on a tensor field to specify some energy density at every point in this spacetime. We have to tell Lorentzian spacetime how it should actually curve. A literally material constraint must be glued to the floppy Lorentzian fabric to give it a gravitational structure. — apokrisis

Energy density is a quantity (number) that is related via Einstein's mathematical equation to spacetime curvature. Pure mathematics.

And even then, the quantum of action - how G scales the interaction between the energy density and the spatiotemporal curvature - remains to be accounted for. — apokrisis

Maybe G can be derived from some general principles and maybe its value is specific just to the spacetime in which we happen to live and may have different values in other spacetimes.

In mathematics, existence is logical consistency, so everything that is consistent exists, and exists necessarily (because it cannot be inconsistent) and truths about it are necessary truths.

Energy density is a quantity (number) — litewave

Is it just that? The claim would be that it is some quantity of something. So the structuralism of the maths still leaves open the question of how to understand the material part of reality’s equation.

Maybe G can be derived from some general principles — litewave

I would expect it can. This is strongly suggested by the fact that the Planck scale is defined by a triadic system of constants. You have an irreducible triad of dimensionless constants in c, G and h. And the whole point of a theory of quantum gravity would be to unite all three in a single theory describing a single emergent geometry.

So if hierarchical organisation is the maths of existence - the Aristotelian metaphysical picture - then mathematical physics has arrived right at that very conclusion. That is the reality that a successful combination of quantum field theory and general relativity would reveal.

Again, Rovelli is right about mathematical junk. Much of maths is the result of mere syntactical complication - a mechanistic spewing that is just too simple to model a physical reality. But where maths models actual complexity - a hierarchical view of structure and development - then ontic structural realism, as the new metaphysics, is the right way to go.

But then M isn't a possible world, it's an impossible world. Under most analyses, impossible worlds have no ontology (because then you're accepting the existence of a contradictory object). — MindForged

No, Rovelli's 'M' explicitly excludes contradiction: "Then the platonic world M is the ensemble of all theorems that follow from all (non contradictory) choices of axioms": It contains everything that is true under any choice of non-contradictory axioms (so yes, read the paper!).

All that's needed for math platonism is for the objects referred to and quantified over in maths to be real. — MindForged

Again, this isn't what Rovelli's paper is about - nor do I think it ought to be about. 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. For my part, the metaphysical prejudice that equates the real with the Platonic is, I think horribly misguided, and simply bad philosophy through and through. I'm perfectly happy to accept the reality of math, with the caveat that what counts as 'real' needs to be rethought wholesale.

But irrespective of which one the universe does does have, the theorems about those systems are true about those systems — MindForged

But this is just tautological: theorems are by definition true (as distinct from hypotheses). You can't milk necessity out of analyticity. Or at least, you can't milk any non-trivial necessity out of it. The question of modality turns on something like: could math be otherwise? And again, the answer is a qualified yes - given what the world is, and how we utilize math for certain purposes, no, the math looks exactly as it 'should'. But without those constraints, in principle, math could well be - again, in principle - a whole bunch of junk: pure meaningless syntax unconstrained by the necessity provided by the world in which we live (which is to say: inseparable and thus not independent from it: this is the sense of contingency Rovelli is employing).

The question is explicitly about the independence of math from our intellectual activity. — StreetlightX

I think 'our intellectual activity' is another abstraction. It's along the lines of a set. I'm pointing to the difficulty in finding a vantage point on abstraction.

Not sure if you missed my reply: — litewave

I saw your reply. It looked too confused to be worthy of a comment. Here it is:

Only consistently defined objects can be part of the mathematical world.

Axioms are properties of an object (also called axiomatic system). Axioms like "The continuum hypothesis is true" and "The continuum hypothesis is not true" would be contradictory if they were properties of the same object but they are not contradictory if they are properties of different objects. — litewave

First, I see no definition of "object". Second, you say "axioms are properties of an object". Third, opposing axioms may describe different objects. Why this is totally confused is that you have no principle to differentiate one object from another object because you have no definition of "object". So, whenever opposing axioms are used, one might simply claim that they refer to properties of different objects. And mathematics might be composed of an endless number of inconsistent and opposing axioms each describing a different object, while each object is consistently defined by its one and only, and independent, axiom. In other words, we could make up an endless number of random axioms, each describing a different object, therefore mathematics would consist of an endless supply of random objects, each with its own axiom.

Now, let's get logical. Within logic we have subjects. You cannot attribute to the same subject, opposing predicates, without contradiction. Mathematics is a subject, so we cannot attribute to mathematics, opposing hypotheses, without contradiction.

Is it just that? The claim would be that it is some quantity of something. So the structuralism of the maths still leaves open the question of how to understand the material part of reality’s equation. — apokrisis

Maybe we could say that energy density is a mapping (defined by Einstein's field equation) from spacetime curvature to real numbers at a given point in spacetime? This mapping is in turn mathematically related to acceleration that this point can impart to another point during an interaction as the energy does work.

First, I see no definition of "object". Second, you say "axioms are properties of an object". Third, opposing axioms may describe different objects. Why this is totally confused is that you have no principle to differentiate one object from another object because you have no definition of "object" — Metaphysician Undercover

Object is something that has properties.

In other words, we could make up an endless number of random axioms, each describing a different object, therefore mathematics would consist of an endless supply of random objects, each with its own axiom. — Metaphysician Undercover

Yes. That's the most general idea of mathematics.

Now, let's get logical. Within logic we have subjects. You cannot attribute to the same subject, opposing predicates, without contradiction. — Metaphysician Undercover

Is there any difference between object and subject?

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

We don't attribute opposing axioms to the whole mathematical world, only to its parts (objects in the mathematical world). For example, zero curvature of space does not hold in the whole mathematical world but only in Euclidean spaces. And non-zero curvature of space does not hold in the whole mathematical world but only in non-Euclidean spaces.

I think 'our intellectual activity' is another abstraction. It's along the lines of a set. I'm pointing to the difficulty in finding a vantage point on abstraction. — frank

Something in your hand or room right now differs from your names for it, and ideations about them. Those are abstract, the thing in your perceptual field is not, and they are about it. Math differs in this though, and has deeper roots. Not as easy to hold in one's hands, to see in one's perceptual field, so far harder to imagine in a mind, or idea independent way, when you have it always at hand with the things in your vicinity.

I think that it is deeper level abstraction from the names themselves, and focus on the grammar or form of language itself, abstracting a step further, and removing the content entirely, and just hijacking the pure form from language and grammar and using it to talk far more precisely, recordably, trackably, and directly about the world.

So that, to speak of mathematical Platonic forms is to say that not only the names or ideas of how they relate, repeat, and can be talked about isn't wholly invented, but the form of language mirrors the form of the world so that language already is grounded in deep mathematical order that is then abstracted from language, and then reapplied in its pure form.

The problem of find with this though, is that it isn't wrong, but it isn't truer than the content of experience, which is where all of the quality is. It is the form of everything, but the dead form of everything, so that language involves the physical likeness, and qualitative likeness, and both have a universal abstractable nature. One into mathematics, and the other aesthetics. The Platonic view is then that these features, are real and independent and not projections by us. What kind of magical dimensions within which they reside, I cannot say, but I do think that language is abstracted in both parts from reality itself, and are real and mind independent.

But what would define the value of the curvature at every point? You are still left quantifying something beyond the metric that determines a number.

That doesn't really have anything to do with my response. Existence in the mathematical sense is, as I've already admitted, what's provable. My point is that the necessity of formal truths does not require (and in fact is incompatible with) possible worlds semantics.

No, Rovelli's 'M' explicitly excludes contradiction: "Then the platonic world M is the ensemble of all theorems that follow from all (non contradictory) choices of axioms": It contains everything that is true under any choice of non-contradictory axioms (so yes, read the paper!). — StreetlightX

Well then he's not talking about the trivial world, which is the world where everything is true. He is thus, in fact, making the exact baseline assumption that all mathematicians make: No mathematics that is interesting can entail triviality. The trivial world is the quintessential impossible world, because it's the world where all the contradictions are true. I will read the paper though!

Concerning the question of the sense in which numbers can be considered real. On a recent overseas trip, I picked up a copy of a new book on the recent history of the debates about the implications of quantum physics. The cover blurb:

The untold story of the heretical thinkers who dared to question the nature of our quantum universe. Every physicist agrees quantum mechanics is among humanity's finest scientific achievements. But ask what it means, and the result will be a brawl. For a century, most physicists have followed Niels Bohr's Copenhagen interpretation and dismissed questions about the reality underlying quantum physics as meaningless. A mishmash of solipsism and poor reasoning, Copenhagen endured, as Bohr's students vigorously protected his legacy, and the physics community favored practical experiments over philosophical arguments. As a result, questioning the status quo long meant professional ruin. And yet, from the 1920s to today, physicists like John Bell, David Bohm, and Hugh Everett persisted in seeking the true meaning of quantum mechanics.

The author comes down firmly on the side of Einstein on the grounds of his scientific realism, scorning the Copenhagen Interpretation as 'obscurantism'.

The title of this book: 'What is Real?', by Adam Becker.

So maybe it's correct to say that this is a problematical question. But it remains an open question, and one that has been exacerbated, not resolved, by the very physics of which Rovelli is a foremost expert. And as this is a philosophy forum, the question of ‘what is real’ and the sense in which numbers are real, remains a valid question, and an open one, in my view. (We probably wouldn't even be able to have the debate on Physics Forum, from my experience there.)

Furthermore, at the heart of the thirty-year debate between Bohr and Einstein was the argument about whether there are mind-independent objects - not numbers, but actual stuff. And, contrary to what Becker says, I think scientific realism lost that argument, although that is obviously one of the vexed questions of modern physics. But, be that as it may, one thing that everyone in the debate assumes, is that they're all dealing with the same measurements and observations; and all of those are measured and described in the common language of mathematics. It's precisely because of its independence that math is the language of physics - whether it is utilised by the inhabitants of Jupiter or Earth, I would hope.

So, I'm not convinced by Rovelli's argument; in any case, his conclusion that mathematical Platonism says that mathematics is 'fully independent' is not at all the case. Here, you're seeing the assumption that 'what is real' must, by definition, be mind-independent being smuggled into the argument. The whole question of what 'independence' means, and the relationship between the knower and the known, is a very deep one, and one that has been by no means resolved in either philosophy or science, as far as I'm concerned.

And besides - just what does ‘independent’ mean in this context? Frege, a mathematical Platonist, said that:

thought content [numbers and logical laws] exists independently of thinking "in the same way that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets."

’Frege on Knowing the Third Realm', Tyler Burge.

I personally believe that is actually a very modest claim. The only problem with it is, that the objects it is talking about, namely, numbers and logical rules, are not actually physical; so that poses an obvious problem for physicalism, which insists that only what is physical can be considered real; and that is the only point at issue in all of this.

Well then he's not talking about the trivial world, which is the world where everything is true. — MindForged

Well yes, but thats not was ever under consideration and is, if I may, an artifact of you not yet having read the paper.

So, I'm not convinced by Rovelli's argument; in any case, his conclusion that mathematical Platonism says that mathematics is 'fully independent' is not at all the case. Here, you're seeing the assumption that 'what is real' must, by definition, be mind-independent being smuggled into the argument. — Wayfarer

Either you haven't read the paper you actually qualify as a clinical imbecile. Rovelli's take on MP is not a conclusion he reaches - it is literally the first line of the paper where it is given as a premise against which the argument unfolds. A premise, moreover, supplied with two citations to Penrose and Connes. If emblazoning the first line of your paper with an explicit definition together with citations counts as 'smuggling' its no wonder your entire post is an exercise in obfuscation, invoking irrelevant debates about QM along with blurbs of books you show no evidence of even having read. Muddy the waters elsewhere you intellectual cretin.

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.

Any mathematical object can be regarded as a "possible world".

Object is something that has properties. — litewave

I can tell you that this is the problem right here. You have absolutely no restrictions on "object". This is what I referred to, any random thing may be an object, because properties are what we, as human beings determine and assign. So what exists as "an object" is completely arbitrary, and dependent only on the way that human beings assign properties. if someone assigns properties, there is an object there. There is no principle of unity here, nor is there a principle of identity, whereby "an object" might be an individual, particular thing.

Is there any difference between object and subject? — litewave

The difference between an object and a subject is found in the way that the law of identity is applied. An object is something we point to, and we identify it that way. It need not have any definite properties, so long as we can identify it as something we can point to, "what it is" may remain indefinite. Therefore an object may be identified, even named, without having any properties assigned to it. A subject is identified through a description, as having specific properties, it is identified by "what it is". As a tool of logic, this allows that numerous different objects may be identified as "the same" subject, when the differences between them are deemed as accidental. So an object is identified as something individual, particular, and unique, while a subject is identified as something specific. One is a particular, the other a universal.

We don't attribute opposing axioms to the whole mathematical world, only to its parts (objects in the mathematical world). For example, zero curvature of space does not hold in the whole mathematical world but only in Euclidean spaces. And non-zero curvature of space does not hold in the whole mathematical world but only in non-Euclidean spaces. — litewave

Yes, that's exactly the problem I referred to. Mathematics, as a subject, is allowed to have opposing and contradictory predications. You justify this by claiming that the contradictory descriptions describe properties of different objects. However, you cannot point to the objects, to say that this is the property of this object, and that is the property of that object, because your so-called "objects" exist only by specification; this axiom indicates the existence of this object, and that axiom indicates the existence of that object. So these so-called "objects" are really subjects. And, they exist as subdivisions of the original subject, mathematics. It is irrational and illogical to allow for contradiction within the subdivisions of one subject.

Here's an example. Suppose that natural science is specified as one subject, with subdivisions specified as biology and physics. We could say that biology and physics are distinct subjects within the subject of natural science, just like Euclidean space and non-Euclidean space are distinct subjects within the subject mathematics. However, we cannot allow that biology and physics proceed from contradictory axioms, because this would signify incoherency within the subject of natural science. Likewise, the use of both Euclidean and non-Euclidean geometry signifies incoherency within the subject of mathematics.

So what exists as "an object" is completely arbitrary, and dependent only on the way that human beings assign properties. if someone assigns properties, there is an object there. — Metaphysician Undercover

No, an object has properties even if no one assigns them to it. Planet Earth is round no matter whether someone assigns roundness to it. It was also round before anyone believed it was round, or before anyone even existed.

So an object is identified as something individual, particular, and unique, while a subject is identified as something specific. One is a particular, the other a universal. — Metaphysician Undercover

Huh? An object is a particular and a subject is a universal? Where did you get this terminology?

We could say that biology and physics are distinct subjects within the subject of natural science, just like Euclidean space and non-Euclidean space are distinct subjects within the subject mathematics. However, we cannot allow that biology and physics proceed from contradictory axioms, because this would signify incoherency within the subject of natural science. — Metaphysician Undercover

But biology is part of physics; properties of biological objects are physical properties. Curved space is not part of flat space and flat space is not part of curved space.

(...) Muddy the waters elsewhere you intellectual cretin. — StreetlightX

This is over the top and uncalled for. From where I stand, Wayfarer appears to have made some good and relevant points (which I was planning to comment on). But even is I'm wrong about that, your response still is uncalled for. Moderator, moderate yourself!

There's simply no possible world where Wayfarer's statement was made in good faith, or without wilful ignorance: the rudimentary confusion between the diametric opposites of a conclusion and a premise, one that ignores the very first sentence of the paper; the idea that Rovelli is trying to 'smuggle in' the notion of the mind-independence of math when the literal point of the paper is to argue against it; the baloney claim that the paper is making a claim for 'what is real' when it explicitly disavows precisely that vocabulary; referring to a doubly-cited claim as 'smuggling'. And this in two sentences of a multi-paragraph post.

This to say nothing of the usual modus operandi of citing blurbs (that is, second hand, twice-removed, assertions that themselves contain no argument) that have zip all to do with the topic at hand - Einstein and Bohr?? - all the while spending the bulk of the post talking about an entirely different topic altogether - 'scientific' claims about 'objects' - and referencing nothing, not a jot, of argument from the paper under discussion. And then to conclude, after this tangle of complete irrelevancy that "So, I'm not convinced by Rovelli's argument" - its a morass of obfuscatory sophistry, and simply the latest in a long line of it. By all means comment on it; it will dilute the muck.

This is over the top and uncalled for. — Pierre-Normand

I read the Rovelli paper, and also the book that I mentioned. Rovelli's concluding paragraph in the paper is as follows:

The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather.

I am question that the idea that mathematical Platonism insists on a 'fully independent Platonic world'. And I also think his motivation for the paper is because of the association of Platonism with Christian Platonism, which tends to oppose the current neo-Darwinian orthodoxy which dominates the secular academy.

It's impossible for me to discuss anything of any depth with StreetlightX without his flying into hysterical invective and insults. I am called an imbecile, cretin, imposter, and disseminator of intellectual poison. I will let others decide on why that might be motivating that, however I think it is driven by what Nagel describes in his essay Evolutionary Naturalism and the Fear of Religion.

No, an object has properties even if no one assigns them to it. Planet Earth is round no matter whether someone assigns roundness to it. It was also round before anyone believed it was round, or before anyone even existed. — litewave

Since it requires someone to determine what "round" means, and whether the object referred to as Planet Earth fulfills those conditions, it is impossible that what you say is true. 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?

Huh? An object is a particular and a subject is a universal? Where did you get this terminology? — litewave

OED: subject: 1a. A matter, theme, etc., to be discussed, described, represented, dealt with, etc.. object: 1. a material thing that can be seen or touched.

If what I said does not make sense to you, then you could perhaps explain why.

But biology is part of physics; properties of biological objects are physical properties. Curved space is not part of flat space and flat space is not part of curved space. — litewave

Have you ever been in a university before? Biology is not part of physics.

What you say is nonsense. If there is something referred to as "space", which has properties, then it is an object according to your own definition of object. It cannot be both curved and flat because this is contradictory.

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

I'm not clear how this relates to the point about train journeys. 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, and I am drawing an analogy between this and 'five'.

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

I agree, but then this generality is an abstraction not an ontologically robust object. To be sure, it is a conceptual object for us, but to imagine it has some existence independently of us is, to quote Whitehead to commit "a fallacy of misplaced concreteness".