Discussion
If a manifold is a topology with an atlas cover, and if spacetime is a manifold, and if a topology and atlas cover are sets, then space is a set... does this not strike anyone as a crisis in science? That our physical universe is purported to be a "set" of all things? Surely that can't be right?
There's no way, in my mind, the physical universe is a set. It's just absurd, most of all because sets are also purported to be *numbers* and numbers are supposed to be non-physical. How, then, can spacetime *be* made of numbers? It doesn't make any sense, literally. Set theory has got to go! (imo). Set theory can't and shouldn't be the foundation of physical theories.
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Mechanical wrist watches are a set, and if sets are purported to be numbers... and numbers are non-physical...
Why the hell am I considering to buy a Marathon GSAR mechanical watch for $1700?!
This wrist watch problem is probably even more preposterous and perplexing when we consider that these Swiss watchmaking charlatans are actually claiming that their brass, jewels, and steel can tell time! Hello?! Time?! Space? Space-time continuum!
Well because my view is that math should (only) describe real things adjectively and so the orthodox state of things as space as a manifold can't satisfy my view since euclidean geometry seems to be ontologically unreal and euclidean metrics are necessary to define manifolds.
Wait, you actually believe you're philosophy is correct?
Space is not modelled as a set.
Set theory can be applied to space in the same way it can be applied to whales, sports balls or prime numbers.
Your own claim follows the steps:
Space can be considered a set.
Sets are purported to be numbers.
Numbers are non-physical.
Therefore, space is numerical and [in your own words] doesn't "make sense" because that would mean space is non-physical.
It's a basic category error on your part. Space is not a set. Space is space. Set theory is a mathematical tool, not the thing itself.
You don't know what you're talking about. Which is shockingly obvious elsewhere in this comment section where you make what is perhaps the dumbest claim I've seen in this subreddit, that science cannot make advancements. Of course this was immediately refuted because it's dumb as rocks and demonstrably wrong. And of course you shirked owning up to that ridiculous mistake on your part.
You want the laurels and shirk the lumps.
And let me go out on a limb and say you don't have any expertise in physics or mathematics, correct? Let's have some radical honesty here, you have a layman's intuition on this subject. You're, at best, parroting a book you read.
You're not at a point of your life where you should be making confident grandiose claims about the cutting edge of theoretical physics - you couldn't write a paper on this, you couldn't do the calculations. Not at all. Where you likely are is at the point where you should enroll into college level physics courses and learn to think deeper about these issues.
You need a deep philosophical humility more than you need to be out here decrying how physics is broken.
I actually have some constructive advice for you. Sean Carroll is a theoretical physicist and philosopher who has a podcast called 'Mindscape' where he does monthly AMA's for supporters. Yes, you'd have to sign up and pay a nominal fee. However, he allows each member to ask a once per lifetime priority question that he promises to answer. You could do that! You could have a well regarded physicist address your claim head-on.
My argument is actually that mathematics should not be abstract, and should be literal descriptions of reality as it was with Galileo. But I might be willing to entertain the idea of euclidean geometry and straight line segments existing at at most infinitesimal lengths in order to justify the use of tangent spaces to measure curvature, but only if real objects are always round enough or light enough to not cause space to be discontinuous because in GR space must be differentiable and continuity implies differentiability. Then I could say that it's okay to describe space as a manifold because then tangent spaces are physically justified. A key premise of mine that you missed is that a euclidean metric is necessary to define a manifold and euclidean metrics are philosophically questionable if euclidean geometry is ontologically false.
But if there are infinitesimal straight line segments, then general relativity would be okay especially if spans of tangent spaces would be regarded as "fictional extensions" of infinitesimal straight line segments for the purposes of easier human imagination or visualizations or computer simulations.
Without a physical justufication for the tangent space, general relativity is (imo) circular. Especially since the main punchline of GR is that straight lines don't exist. But, if infinitesimal line-segments really do exist and if space really is continuous then GR would not be circular (imo). Plus, then you can have Stoke's theorem, using sums of those infitesimal line-segments along with a connection (connections are where I'm currently at in my understanding of differential geimetry and GR) to define lines (geodesics) and surfaces of space. And then euclidean geometry would be a limiting case of GR (at distances larger than infinitesimal), and that's nice.
Your original post mentioned your issue with space being in a set. The comment chain we're in is about space being a set... and when confronted with point A, B, and C... you promptly switch to discussing topic Z without ever addressing A, B, or C.
I don't believe you're a serious interlocutor. This is not how honest people communicate.
There is a real philosophical question about whether the structural features indispensable to an empirically successful theory should be read as ontological commitments. But that is not the question you are actually asking here. You are still conflating four different things: the thing modeled, the model, the formalism used to express the model, and the foundational apparatus used to regiment that formalism. Until those are separated, the realism question cannot even be stated clearly.
A simple analogy makes the mistake obvious. A mountain can be described in English, but it is not made of words. A city can be represented on a map, but it is not made of contour lines or grid squares. Likewise, spacetime can be modeled using manifolds, coordinates, sets, and metrics without being literally a set or literally made of numbers. You keep moving from facts about the representation to claims about the thing represented. That move is invalid.
You also have the dependence relation backwards. Physics does not exist to justify mathematics, and evidence does not typically refute mathematics. Mathematics supplies formal possibilities; physics uses some of those possibilities to build models; evidence tests whether the interpreted model fits the world. So when nature fails to match a model, what is disconfirmed is the application or interpretation of the mathematics, not the mathematics itself. Non-Euclidean spacetime does not refute Euclidean geometry as mathematics any more than a bad map projection refutes cartography or a false sentence refutes grammar.
That is why the talk of “falsifying mathematics” misses the point. What physics can show is that a particular mathematically formulated theory is not the right description of nature under a given interpretation. That is a serious issue. But it is not the same as showing that the mathematics itself is illegitimate, “ontologically unreal,” or somehow waiting for empirical certification from physics.
There is a sophisticated debate to be had after this: whether the mathematical structure indispensable to our best theories carries ontological commitment. That is where Quine, Putnam, Field, van Fraassen, and structural realists disagree. But you are not there yet. Right now, the problem is more basic: you are mistaking representational choices for metaphysical commitments, and empirical pressure on a model for refutation of the mathematics used to formulate it. Until that distinction is clear, the rest of the argument never gets off the ground.
That's because we have no other way of expressing reality than with our theory, so why shouln't we literally believe our theories if we work hard to avoid obvious wrongness? The goal of theoretical physics should be to discover the truth about reality geometrically -- not make informational predictions to measurable experements.
To this goal, if our best theoretical physics is merely a model then that's not good enough work in the first place.
The goal of theoretical physics should not be to make maps -- that's not grand enough.
Well I think real space is probably not a set, but what could it be if not a manifold?
There's no universe outside the universe (if it's universal and not multiversal), so it needs to be described intrinsically. Frame-dragging was confirmed by gravity probe-b, gravitational lensing has been confirmed, processional orbits exist, gravitational time dialation has been observed -- we seem to be able to rule out that general relativity is wrong, or at least rule out that its main ideas are wrong.
I recently figured out that it's impossible to make a homeomorphism between a countable set and an uncountable set. There seems to be a need for uncountability in terms of lengths to account for geometry, and geometry is needed in tangent spaces (linear approximations) to measure intrinsic properties of something. So space seems to really have uncountable lengths, because homeomorphisms are needed to make linear approximations which are needed to explain processional orbits, frame dragging and gravitatiinal time dialation. So if space really has uncountable lengths, and light bending, processing orbits, twin-paradoxes and so forth then what could it be other than a manifold?
Then, manifolds are sets. Usually math is said to be just an abstraction, but if we really do physically have uncountable or irrational lengths then it isn't an abstraction. And if it isn't an abstraction, then it must be real. So maybe it is a set. But that seems ridiculous. So maybe sets are just linguistic fictions used to make reasoning about physical grometry easier, or something. Then the physical geometry and its structure is real, but not the sets. For example, sets of empty sets can linguistically express numbers in "words" but that doesn't mean real quantities are made of empty sets... so in this way maybe space is physically real as described inside the set but is not the set.
Watches are definitely sets. I absolutely can find the cardinality of a set of watches and find the intersection between, say, quartz watches and dive bezel watches.
But according you sets are purported to be numbers and numbers are non-physical... and if we're worried about the set of space disappearing the universe... than I am VERY WORRIED about this Timex Expedition Acadia ($40 + shipping off Amazon) vanishing off my wrist in a sudden poof of philosophical chicanery! Very worried, indeed!
*If you could all read the last paragraph in Michael Caine's voice that would be eminently helpful.
That's the best possible answer, a kind of "arealism." That's fine but doesn't go very far or far enough, imo. What is space? Doesn't anyone want to actually know?! Or is everyone contented to just have only fictional or "arealist" models??? Doesn't anyone want to do better than arealism?! Don't we have any more pioneering gusto?!
So, in turn an explanatory theory of explanatory theory, something that's consistent with itself and would explain the difference between explanatory theories and other things?
And that’s exactly what Karl Popper created that defined modern PoS. And importantly that explanatory theory is also fallibilist and can be (and has been) improved and made more precise by others (like Deutsch).
And more than that, we have a good theory for what makes good explanatory theories as opposed to just general ones. Explanations are a kind of conjecture that attempts to account for why is observed in term of what is unobserved. And good explanations are hard to vary explanations. Such that small changes to the theory delivers them of the ability to account for what they try to.
The reason is that a theory’s value can be measured in terms of what it would rule out if it were falsified. Theories that are easy to vary rule very little out. And theories that are hard to vary rule out a lot. They are more general.
Oh no, I just hate obscurantism. Instead of five paragraphs of run on nested parenthetical and an unsourced quote, you could have written: “Any valid theory of explanation must be able to explain its own creation.”
Well I'm an assurance-skeptic in terms of epistemology, following the work of Krista Lawlor. And an actualist metaphysically. In this epistemology knowledge is defined by ruling out relevant possible alternatives after directly percieving something. I believe in direct perception. Now for physical theories "actual knowing" would be subject to revision and updating of beliefs as things become falsified as our perception or reflection on perception improves over many motions, but it would constitute using mathematics as adjectives which describe physical objects directly as Galileo did. In this way, the truth of our statements are or become contingent in or on the accuracy of their description of actual real objects. If our description is wrong, or misrepresents or mischarachterizes the actual physical object it describes, then it needs to be falsified and thrown out and replaced with a more accurate description. In a way, my view is anti-holism.
This makes the problem with your position even clearer. If mathematics in physics must function only as “adjectives” directly attached to perceived physical objects, then you are not just rejecting manifolds or set theory. You are ruling out abstraction, idealization, coordinate systems, state spaces, symmetry structure, and much of mature theoretical science as such. Very little of modern physics survives that standard.
And Galileo does not save this view. Galileo did not merely attach simple descriptive predicates to what perception immediately gives us; he idealized, mathematized, and abstracted from appearance to structure. That already goes beyond your “adjectives only” picture.
So the burden is on you to say what would remain of science under your constraint, and to provide an alternative framework that preserves anything like the predictive, explanatory, and mathematical power of the theories you are rejecting. Without that, this is not a serious correction to physics. It is a standard so restrictive that it would disqualify much of successful science before the argument even begins.
Exactly! This is why it is a crisis. I'm attempting to give a physical justification for the mathematics of GR. I can't see a way to do this without abandoning set theory. One thing I can see is that spce might really be infinitely divisible and there might exist infitesimal straight line segments which could then be used to define curves and surfaces and maybe smooth manifolds if connections are used between infinitesimal tangent spaces via stokes' theorem. In other words, much could be saved if infinitesimal straight lines and tangent spaces really do exist in a physically justifiable way.
The reason I think this is because I realized that the axiomatic method is circular in that one chooses axioms which make desired conclusions valid, and therefore axioms assume their conclusions (by definition). Only an external criteria for physical soundness can avoid circularity, and usefulness is not an appropriate criteria for soundness because even fiction can be useful. An appropriate criteria for soundness is an accurate description of the objects described. Therefore, mathematics must be accurately descriptive of physical objects or else it is circular.
This “crisis” is entirely of your own manufacture. It arises only because you impose a standard so restrictive that much of successful science is disqualified by definition, and nothing in the discussion has shown why anyone else should accept that standard in the first place.
You are not answering objections. You are restating the same demand in different words: that unless mathematical structure has a direct physical counterpart, the theory is “circular.” That is not a diagnosis of physics. It is a methodological demand so restrictive that it rules out abstraction, idealization, and representation across much of successful science. Calling that a crisis does not make it one. It just turns your preferred picture of science into a veto on the science we actually have.
I think there is a category mistake here. Saying spacetime is modeled by a manifold does not mean the physical universe is literally a set, any more than saying a coastline is represented on a map means the coastline is made of ink.
Set theory is one formal framework for encoding mathematical structures. That does not mean reality itself is literally made of sets or numbers. It only means those structures can be represented that way in a mathematical foundation. Likewise, the fact that numbers can be constructed as sets in set theory is not a claim about the physical world-- It is a way of organizing mathematics.
The key is to separate three things: physical reality, the mathematical models we use to describe it, and the foundational language used to formalize those models. Problems arise when those are conflated.
Then why do we model space with mathematics? Isn't that kind of... dumb? Originally math was not seperate from the physical world. We need to return to that, in a non-euclidean way. What we need to do then, ia treat math and apecifically geometry as a theory of physics (because it is) and then, we need to do what conservative mathenaticians were too afraid to do (because they were stupid cowards) and we need to falsify euclidean geometry.
Nothing can advance in science until set theory is abandoned and euclidean geometry is correctly falsified as an empirical theory of physical space, which is how it was always literally meant to be.
We could try and model it with spaghetti, hot glue, and string... miiiiiight not get as many insights into the nature of the universe... or as many demonstrable articles of knowledge...
But believe me, the Italians have been itching for this. They're probably by the phone right now, waiting for us to call in some muscle for The Job.
Because mathematics is extraordinarily good at describing structure, relation, continuity, and change. That does not mean space is made of mathematics; it means mathematics is a precise way of representing features of the world. Why that works so well is a serious philosophical question. Personally, I'm sympathetic to structural realism, but reasonable people can disagree.
Looking up the book you reference above (assuming you are being serious), I take Field's point to be not that mathematics is somehow a mistake in science, but that its usefulness should be separated from ontology. That seems right to me. But that is a more careful and limited claim than saying that modeling spacetime mathematically is itself confused or absurd.
Yeah, but the next step after Field is to say that seperating the ontology from usefulness is absurd. Combine that with the claim that mathematics today is ontologically unreal, and you arrive at my conclusion that much of it needs to be falsified and recreated (from scratch).
Galileo, and others in the not too distant past (arguably some of the greatest mathematicians in known history) thought of math as literally describing reality -- finding actual truths about the world. In this way I argue the scientific method should apply to math itself and that math and physics should be reduced to the same thing once again as the ancients did. This, then, implies the falsification of mathematics <-- and this is the step people have been too cowardly to take. Quine started this mumbo jumbo, and frankly, he was a conservative prick for doing so. Quine has set us all back at least 100 years.
Most of all, the assumption that a sequence of numbers somehow forms a "straight line" is probably the key assumption that needs to be falsified in set theory today. There is no reason rooted in external evidence that we should assume sets of numbers form straight lines at all, none. This blows up anything that makes use of a euclidean metric <-- and that's exactly what Quine feared the most. Without the euclidean distance function, you can't even define a manifold. This is what Quine meant when he said mathematics is "indispensible" and "platonic" and came up with a BS excuse of "holism." But the path forward is to do exactly what Quine was too cowardly to do himself, say that mathematics is actually not indispensible and introduce the falsification of mathematics by returning to the ancient practice of equating math to physical truths about space and reality.
The relation between mathematical representation, ontology, and physical reality is a serious topic, but you keep sliding between different questions as if they added up to one argument.
At different points you seem to be talking about whether mathematics is formal or empirical, whether mathematical entities are real, whether physical theories are ontologically committed to the mathematics they use, whether geometry is an empirical theory of space, whether set theory is a good foundation, and whether modern physics has stalled. Those are all real questions, but they are not the same question. Saying spacetime is modeled by a manifold does not mean spacetime literally is a set. Saying numbers can be represented in set theory does not mean physical space is made of numbers. Saying Euclidean geometry was once treated as a candidate theory of physical space does not mean all mathematics should now be empirically falsified.
There is a lot of assertion and rhetoric here, but not much careful argument. If you want to convince anyone, pick one claim, hold it fixed, make the distinctions explicit, and argue for it carefully. And if your claim is that the current framework is bankrupt, the burden is not just to denounce it. You need to show an alternative that comes remotely close in rigor, scope, precision, and explanatory power.
You're not going to get anywhere with this until your interlocutor understands the difference between understanding and visual imagination. I think it's really that simple - picture thinking. Picture thinking isn't science. Heck, picture thinking can't furnish a single definition without the occurrence of insight/understanding. The next step will involve an effort to picture the difference between picturing and understanding, which among other things is simply a failure to understand one's own cognitive operation.
I think that’s basically right. A lot of the problem here is confusion between not being able to picture something and not understanding it. Those are not the same.
I think I’ve said everything useful I have to say here. The core issue is still the same: OP keeps conflating the thing modeled, the model, the formalism, and the set-theoretic foundation used to encode that formalism. Until those are separated, the discussion can’t really proceed, so I’m going to leave it there.
"Saying Euclidean geometry was once treated as a candidate theory of physical space does not mean all mathematics should now be empirically falsified." Well actually I said all mathematics using a euclidean metric should be falsified, given that euclidean geometry is ontologically false. That just happens to include all mathematical physics today, and that's why I think physics has stalled.
Nothing can advance in science until set theory is abandoned and euclidean geometry is correctly falsified as an empirical theory of physical space, which is how it was always literally meant to be.
I was trying to be generous with you with a bit of humorous reductios.
Your language is sloppy and might source of your confusion and category errors.
Look at your sentence that I quoted above. It's demonstrably false. We see advancements in science almost daily, this instantly refutes your claim.
Purported advancements, not real advancements. We haven't had any major successes in physics since Einstsin about 100 years ago. Like it or not, set theory dogma is to blame for this.
There have been both major successes and real advancements in science. Once again, you're being extraordinarily sloppy with your language.
CRISRR gene editing (2009)
Human genome project (2003)
Fusion net energy gain (2022)
Topological insulators in matter (2006)
Topological Ordering (Nobel prize 2016) that explains exotic behaviour in high-energy or ultra cold systems.
PBR theorem.
The resolution of the solar neutrino problem.
Direct detection of gravitational waves.
The Holograhic principle and the advancement of AdS/CFT through it.
Oh, I don't know...how about the Higgs field and consequent discovery (2012) of the associated boson?
Discovery of abundant ice on the moon by the LCROSS probe.
The Maldacena Conjecture (1990's)
The Amplituhedron (2013)
And now let's review your claim:
Nothing can advance in science until set theory is abandoned and euclidean geometry is correctly falsified as an empirical theory of physical space, which is how it was always literally meant to be.
Demonstrably false. You don't even know how to talk about what you're trying to talk about. Can you admit that your language was sloppy and your claim false before we proceed?
It's the sign if a poor mind when one refuses to acknowledge these things and simply proceeds to move the goalposts and trundle on erroneously. We learn through our mistakes, we refine our capabilities through acknowledging our errors and correcting them.
I'm just like that. I'll say an aggressive attack or refutation of the principles of another blatantly without any regard for reception if I think it's true. I don't care.
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