> infinity is something only present in our math and not in the universe

This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.

Tell that to Plato.

Honestly I think that's a continuous claim, and comes down to differences in understanding. I can certainly have 7 of some object, does the 7-ness exist in the collection? Not really, but what about another phenomenon: colour? An object appears blue, and we say it is blue, and the blueness is due to physics, but it's a subjective delineation. A table is a delineation too, the leg is part of the table and the White House is not. In some sense, the table-ness category is just as real as the 7-ness category.

Of course you could just say that all that actually exists is some collection of particles/fields, but then you've abused all the words we're using until they stop being useful.

OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.

But the electron field has different values at different points in spacetime, and we have no evidence that either the number of points (locations) or the number of different possible values at those points is finite. Unless we posit that they are finite in number, infinity is quite present in the universe.

Would not an infinite electron field in a finite (observable) universe result in an infinite energy density and therefore the entire universe would collapse into a black hole?

Integrals over a finite interval can have (and often do have) a finite size even though the interval contains an infinite number of points, with an infinite number of different values at those point.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

Well, I was referring to the electron field, not to electrons. According to QFT, particles are excitations of an underlying quantum field. It’s the field that is fundamental, not the particle. See e.g. [0]. And those fields are continuous, not discrete, i.e. can only be described by an infinite number of points and values.

[0] https://www.quantamagazine.org/what-is-a-particle-20201112/

Great read, thanks. I think I see the point you're making now - any continuity necessarily requires an infinity by virtue of being continuous.

I'd argue you can't cram any electrons into any finite space without infinite energy. But we're not really talking quantum physics, here.

I'm not sure what you mean - any battery you have would seem to contradict you - would you elaborate?

The electrons are not wholly within the battery; they are mostly within the battery. (Batteries discharge over time for an almost completely unrelated reason.)

I think you're misinterpreting the meaning of the wave equation. Batteries self-discharge due to chemical reactions not quantum tunneling.

That sounds like a weird interpretation of "to be present in the universe" to me. Also I was under the impression that it's unknown whether the universe contains an infinite number of electrons or not.

It's certainly known that the observable universe does not contain an infinite number of electrons, as it has a finite size and finite mass. And it's rather moot to talk about the space beyond the observable universe that can never affect us or anything we can observe in any way whatsoever, so any other statements about it are inherently unfalsifiable, so all the science of physics is relevant only w.r.t. the (finite) observable universe.

Weirder than 'numbers are physical objects'? Can you suggest a more suitable interpretation of the OPs intent?

it's OK. electrons don't "exist" discretely, either.

At best, when you "hand me 7 electrons", you're directing me towards the fat part of 7 probability distributions, so we're back to math again...

Well that's a bold assertion about a theory with whose implications we are still grappling; electrons may not be point-like entities but they are nonetheless quantifiable 'packets' of energy, are they not?

But is that not the locations that are inexact, not the quantity of electrons?

I've always adhered to the idea that "infinity" encodes "allness". For instance, to say that the sum of 1/2^n for all natural numbers n converges to 1 is not to say that we're actually adding up infinitely many numbers, but rather that I can always win a certain game: you give me an arbitrarily small epsilon > 0, I can give you enough terms in the sequence such that their sum (a finite sum) is within epsilon of 1. No, I can't actually add up infinitely many numbers, but you can never win my game, so certainly "infinity" exists in that sense.

So, while I can't "point" to an infinite number of things like I can point to 9 things or 3.62 things, I still think it exists.

I'm not sure how well this generalizes to all infinite cardinals, ordinals, or to transfinite induction/construction. It is certainly strange that Cantor's theorem (the cardinality of a set is strictly smaller than that of its power set) implies there are different sizes of "all" implicit in my usage of the word.

Well, infinity is inherently unscientific, as there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time.

What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.

I don’t see how this makes infinity “unscientific.” Infinity is part of the language of mathematics. It’s no more scientific or unscientific than the definition of matrix multiplication.

Also wouldn’t your argument also apply to zero? You can never know if a quantity is zero as opposed to some enormously small epsilon that you haven’t detected yet. Is zero “unscientific?”

Well, I can say that there are precisely 0 african elephants in the room with me right now, so no, 0 and other integers don't have this problem. Similarly, the rationals are clearly realizable with perfect precision.

The reals however are a different problem, and it's not scientifically possible to prove that the ratio between the length and radius of any object is exactly pi (that it is a perfect circle). However, it's also impossible to prove scientifically that it is 3 or 3.14 or any other number.

Now my use of "unscientific" is more of a hyperbole or click-bait. I thought I explained my actual claim pretty well - that you can't measurably/scientifically distinguish between a universe that contains actual infinities and one that only contains some arbitrarily large numbers.

That's just because you use the word "exact", though. Exactitude doesn't exist in the universe as we understand it.

There's a difference between something not being instantiated in this universe and being unscientific, though.

If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?

> Exactitude doesn't exist in the universe as we understand it.

Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge.

> There's a difference between something not being instantiated in this universe and being unscientific, though.

Well, science is a particular way of studying what exists. Studying something that doesn't exist is unscientific (of course, you can use science to try to determine IF something exists).

But there are also things that are outside the reach of the methods of science, so they are unscientific in this sense. Questions such as "did some god create the universe" are unscientific because it is simply impossible to apply the methods of science to arrive at an answer to this question.

Similarly, asking "is the universe infinite in size" is unscientific, because it is impossible to apply the methods of science and arrive at a definite answer to this question.

> If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?

If it predicts actual infinities exist in certain conditions, than it is not going to be a testable theory in those conditions. It may still be a perfectly workable model, just as GR is perfectly workable despite predicting singularities at the center of black holes. That doesn't mean that the singularities exist, it means that GR breaks down at certain points.

But even if you had a physical theory that relied on something like a Banach-Tarski construction, you could never distinguish between an actual infinity of points, leading to two perfectly solid, perfectly identical spheres; and an arbitrarily large number of points, leading either to two perfectly solid but slightly different-sized spheres; or two identically-sized spheres with small holes.

Of course, without some need to specify the number of points, you would be well positioned to use the infinite variant. But if someone asked you if this means that the sphere really has an infinite number of points, the answer would have to be that you can't be sure.

> Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge.

Aren't claims like this unscientific according to your standard? You will never be able to measure that two electrons have the same charge to infinite decimal precision. You might have a theory that says they should have the same charge, but you won't be able to test that theory to infinite precision either.

Hmm, actually you may be right. I'm not entirely sure how powerful the 'requirement' in QM for these quantities to be quantized is though, but most likely you are right - the theory wouldn't be able to distinguish between identical charges and veeeeery slightly different charges.

You were talking about exactitude in space, rather than charge.

>Studying something that doesn't exist is unscientific

What about things that could exist, might exist, or even aren't expressly forbidden from existing? These have all been used as perfectly valid reasons for scientific inquiry, historically.

Asking "did some god create the universe" is unscientific by your reasoning so long as it is known that there is no in-universe trace or evidence that it was indeed created by a god. Proving that is proving a negative. I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something. It does certainly feel impossible that we will prove that the universe wasn't created by a god, though. The inquiry is deemed unscientific because we have no reason to go down that pathway, not because the question is fundamentally intractable.

Multiverse theory, on the other hand, would qualify as unscientific by your reasoning. If it were true, the different universes would be fundamentally inaccessible, according to our understanding. The model does not suggest that evidence could even possibly exist, as far as I understand.

A result being untestable doesn't, in my opinion, lead to it being unscientific. We cannot test whether black holes exist, except by looking for them. We cannot test whether wormholes exist, except by looking for them. These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result, just because maybe it is the case that our model is correct but there was never the appropriate state of the universe to prove our prediction.

Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct? Is that unscientific? Just like we assume that the charge on electrons is constant despite not actually measuring it always everywhere.

I was talking about exactitude in general, as a requirement for a physical interpretation of the integers.

> I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something.

That's actually a good point, there could be scientific proof of some intelligent creator in principle. The fact that there is no reason a priori to believe that we will find such a proof is a problem, but I don't think it would be enough to deem the theory unscientific. Otherwise, many actually used theories would be unscientific - for example, there is no scientific reason to expect supersimmetry to exist, but that doesn't make the search for supersimmetry unscientific.

> Multiverse theory, on the other hand, would qualify as unscientific by your reasoning.

Yes, multiverse theory is unscientific by my definition. I don't believe speculation about a multiverse can be considered science in any meaningful sense. Just like simulation theory, it is using science-sounding terminology for idle speculation (though the universe being a simulation could similarly be proven by the same kind of evidence as the intelligent creator idea, to be fair).

> These are predictions that we cannot "test" except by looking at the universe and seeing what we find, and even then we are not guaranteed a positive result

But this is exactly the definition of a test. It's true that you can't prove that something doesn't exist in this way, but saying that something is untestable goes beyond that. An untestable hypothesis is one that by definition doesn't make any predictions about the universe. Multiverse theory is in this bucket - whether you believe it to be true or not, you won't expect to see anything different in the world.

> Of course if something was actually infinite, you wouldn't be able to measure it to be so, but if the model (that you have shown to be correct in other case) predicts an actual infinity and you keep counting more and more orders of magnitude, does it not make sense to assume your model is correct?

Of course it's OK to assume your model is correct, and infinity will likely be the simplest assumption in this case. However, any model that predicts an infinity can be replaced with an equivalent model that makes all the same measurable predictions but replaces the infinity with some arbitrarily large but finite number (or arbitrarily small but not infinitesimal). This second model may well be harder to work with and will contain an extra assumption (an explicit upper bound for the infinite quantity), so I wouldn't advocate for its use. But it would have to be accepted that it is not empirically distinguishable from the infinity based model.

Exactitude in general is impossible, though. We assume that the charge on an electron is a constant, but there are limits to the precision.

An infinite value is theoretically testable. It simply implies that for however long you make your ruler, the value is larger. That is a prediction. You may not reach a conclusion, as you said, in finite time, but that is still a prediction.

The problem with replacing an infinity in a model with an arbitrarily large number is that, given enough time and a long enough ruler, you'll surpass that number, meaning your model is incorrect. In defence of "science", you're adding an arbitrary number into a model that you expect to be incorrect. That's not how it should work.

If the model says there's a singularity, we don't then say "okay but well that clearly doesn't make sense, so put a limit on the formula that clamps the values to uh 10^45". That is unscientific.

> Exactitude in general is impossible, though. We assume that the charge on an electron is a constant, but there are limits to the precision.

Yes, others have pointed that out and I am in fact conflicted right now.

> The problem with replacing an infinity in a model with an arbitrarily large number is that, given enough time and a long enough ruler, you'll surpass that number, meaning your model is incorrect. In defence of "science", you're adding an arbitrary number into a model that you expect to be incorrect. That's not how it should work.

I'm not against using infinities in scientific practice at all. I'm just pointing out that, when it comes down to it, that infinity is never necessary in the logical sense.

> If the model says there's a singularity, we don't then say "okay but well that clearly doesn't make sense, so put a limit on the formula that clamps the values to uh 10^45". That is unscientific.

Sure, picking some random big number would be unscientific. But saying "the model predicts a singularity or growing to infinity, so we're probably missing some piece of the picture that sets an upper bar" is not unscientific. It is in fact the common practice - just like no one believes that black holes or the early universe had an actual singularity at the center, we normally just believe the models break somewhere at those levels, and more powerful models (quantum gravity) will actually put a cap. Or how we keep saying that we know an upper bound for the possible mass of a photon, but don't actually know that it really is 0, and we keep trying to measure it.

I think we're not disagreeing, if we were before. Adding limits to functions without rationale other than avoiding infinities is arbitrary and unscientific. Given that we assume that the universe doesn't actually have the capacity for infinities (is this a scientifically grounded assumption, given the impossibility in measuring them?), finding them points us towards holes or limitations in the model, e.g. GR vs quantum gravity.

As the other comment implied, infinity and exactitude are two sides of the same coin. Exactitude is infinite precision. No finite amount of empirical evidence can afford infinite precision, so you’re back in math-land.

That's true, though I have a hard time squaring that with observations of classical objects. Would it make sense to say that I have approximately 2 legs, or can I actually be confident in saying I have exactly two legs? Even with things like MWI, in any particular world I would still have an exact integer amount of legs, as far as I understand.

Perhaps the problem here is one of mixing intuition (the idea of 'an object') with rigorous physics and mathematics, perhaps this is where I am going a bit wrong.

We have a very precise (although one might struggle to describe it) idea of what qualifies as a human leg. The number 2 is basically defined (at least in common usage) as the number of things you have when you have one thing and then another thing. I'd point out that, as you mentioned, this is a different level of abstraction to physics and mathematics.

I would also need to know more QM. I don't know if the theory would actually allow a small fraction of an electron or an electron and a bit to actually exist - common descriptions suggest that it wouldn't. If it doesn't, then electrons could be counted just as much as mathematical objects and legs.

> there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time

That depends on the model of computation you pick, doesn't it?

Only if you want to think about models of computation that allow performing infinite operations in finite amounts of time, which I don't think are that interesting.

I'd say if our universe makes such computations possible, then that would be very interesting.

Absolutely! Similarly, if our universe allowed instantaneous travel and free energy, that would also be very interesting.

Not holding my breath for either.

The fact that you can breath at all is quite a miracle. Not sure why you are so pessimistic about the rest.

What about negative numbers? Or complex numbers? They are only tools, which can be quite useful to build models of the world with predictive powers but shouldn't be confused for the underlying reality.

Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.

Whole numbers can be defined and proven to be necessary to describe the world pretty easily. From there, rational numbers are trivial to define. Negative numbers are somewhat more abstract, but they have very intuitive definitions in many domains, such as accounting. It may be possible to avoid them in a theory of physics, though.

The complex numbers (well, at least those with a rational imaginary part and a rational real part) have been recently proven to be necessary to describe the universe[0] (assuming quantum theory is correct).

The irrational numbers are then are the only numbers that are harder to pin down, and I'm not sure that there is a way to prove that any physical quantity has an irrational value, vs a rational value that is arbitrarily close to that irrational value.

[0] https://arxiv.org/abs/2101.10873

Infinity may be also "necessary to describe the world". But like every tool, you need to know its limits.

I'm not sure that it could be, actually. You can't use a finite amount of evidence to verify that something is infinite, so any infinity can always be replaced with a huge (or minuscule) number and the theory would make the same measurable predictions.

I'm not talking about proving that there exist infinite things.

I'm talking about using the abstract concepts of infinity as a useful mathematical tool to produce predictions. Notable example: calculus

Sure, but calculus makes infinity sufficient, but not necessary for describing the physical world. Integers, rationals, and apparently complex numbers (presumably those with rational components) are actually necessary for describing the physical world, given our current understanding. Irrational numbers and infinities are extremely useful, but not strictly necessary.

Sorry, I'm not getting it. A large use case for complex numbers is describing things that rotate, literally or not, like oscillations, waves etc. Trigonometry lies deeply in that math and the irrational number pi pops out left and right. An approximation of pi wouldn't cut it, would it?

Actually, for any practical use case or possible observation, there is an approximation of Pi that is good enough. The ancient Egyptians apparently did quite well in their architecture approximating Pi as 22/7 (3.(1428571)).

You only need the exact number Pi if you want to measure something like the ratio between the length of a perfect circle and its radius with infinite precision. But you can't be sure your measurement has infinite precision with a finite number of measurements, and so you can't observe the difference between a perfect circle and a many, many sided X-agon, even if perfect circles do exist in the geometry of the universe.

Just as a fun aside, even if perfectly circular shapes do exist, it's unlikely that perfect circles would exist in physical objects - at best, you would have ellipses, and there is no (known?) way to compute the ratio between the length of an ellipse and the properties of its foci.

We're clearly talking about different things.

You're talking about measuring something in the real world. Measuring is basically counting how many thing a given reference thing fits into the thing you're measuring. I have no problems with the assumption that for all intents and purposes we live in a finite (space and time) physical universe and there is a maximum precision that will ever be necessary.

What I am talking about is that in order for the math we use to describe that universe to work out we need irrational numbers; otherwise you couldn't be able to prove theorems and whatnot. I think this makes the irrational numbers (e and pi in particular) quite fundamental tools and I don't care if the real world doesn't allow objects (or positions) to be measured with irrational numbers.

> there is no (known?) way to compute the ratio between the length of an ellipse and the properties of its foci.

There is no closed-form expression for the circumference of an ellipse. There is an infinite series though. Same for a circle; there is no closed-form for computing pi either!

Mathematics doesn't really "exist" in the first place. It's more of a language that's rich enough and with enough logic to describe/approximate the laws of physics that actually do exist.

So quantity, structure, formal necessity don't exist?

Mathematics has nothing to do with laws of physics. Even if the laws of physics[0] were different, these mathematical[1] truths would remain the same.

[0] Laws of physics don't actually exist. They're shorthand generalizations about features of particulars. The notion of some kind of abstract disembodied "laws" that somehow "govern" everything is absurd.

[1] For clarify: mathematics is a field that studies such things.

I have never understood the opposition to the law of physics you seem to hold.

What is the alternative? That we merely observe rigid patterns that are baked into physical reality? Isn't whatever is 'baked in' more or less a 'law of physics'?

If these are just 'brute facts' are they not then 'laws'? Maybe governance is too strong an word for the correspondence but what is the alternative?

Laws in the physics sense don't govern, though, they describe. It's akin to saying moral laws don't decide what's immoral, they simply describe it.

Given that, "laws of physics" are certainly describable. We simply write the formulae that tell us what the next state of the dynamical system is. They are ways of delimiting what is physically possible, given the current state of the art.

What's your concept of time and its extension then?