Okay. I’d like to expand the question. As I understand it, modern physics (for present purposes I’m referring only to special and general relativity) does not deal with space and time in themselves. Special relativity deals with the measurement of time (or the measurement of bodies’ motion in time), and general relativity deals with the connections between the structure of space-time and gravitation. So in a certain sense, it seems to me correct to say that these two theories are not concerned with the separate existence of those two “entities.” For example, questions about the infinity of space and time, or about our knowledge of these “entities,” fall within the domain of philosophy (and therefore scientific theory has nothing to say about them).

But then it occurred to me—and your response strengthens my intuition—that there is still a hazier intermediate zone here. If I’m right, then claims like “the speed of light is relative to the observer” or “space is distorted because of gravity” walk the boundary of conceptual-philosophical discussion. After all, they seemingly express a principled and general claim, which—even if derived from observations and measurements—still carries more abstract information. In other words, they have philosophical implications touching on classical concepts from philosophy (for example: what is motion in general? what is a causal relation? what are space and time in themselves? etc.).

An alternative formulation of this last idea: with the appearance of special and general relativity, the rigid barrier between science and philosophy was partially breached (though I have a feeling that failed interpreters tend to say it was breached completely).

What do you think?

It was breached much earlier. There are quite a few philosophical assumptions in science, and quite a few scientific results have implications for philosophy. The sharp distinction between the fields is naive.
Even before relativity, the Newtonian conception of space and time reigned. That too is a philosophical assumption, and it too dictated physics for quite a few years—and vice versa.

I have no doubt that philosophical interpretation of modern science existed before that (and philosophical interpretation can be applied to almost anything).
My claim is that the turn represented by relativity is a step up. As I understand it, the innovation lies in relativity’s holism, which does not exist in the Newtonian conception. This holism arises because the new theory requires incorporating the observer’s actual point of view into the equation. That seems to me like an “expansion” of traditional scientificity, because the theory now comes to deal with a double system supposedly created by an interaction between nature and the human being observing it.
For example: if for Newton space is an entity independent of the observer (and therefore physics deals only with it itself), then Einstein opens the door to interpreting his theory as if space contracts and expands subject to the observer’s actual state of motion/velocity. Even the necessary union of space and time in relativity (Minkowski space, if I remember correctly) hints at a process of science “expanding” toward philosophy.

Agree?

I’m not talking about philosophical interpretation, but about philosophical assumptions and philosophical implications of science itself.
I don’t know how to define “a step up” (it reminds me of all the media interviews after every terror attack: is this an escalation?).

Okay, so it seems to me we’ve aligned, which leads me to the question that matters to me.
The claim of general relativity that space is distorted because of the gravitational force of the masses present in the universe is a convenient working assumption and advances science. But philosophically it is meaningless—there is no sense in saying that space is “curved” (just as you can’t say that it is “burgundy-colored” or that “it smells like strawberries”). The category of curvature is not relevant to space.
In the same way, it seems to me that we should relate to the data and predictions of special relativity regarding the flow of time. The rate at which time passes is indeed relative and measurement-dependent, but that says nothing about time itself, only about the way it appears to us (that is, appears to the observer who is actually measuring it). Therefore special relativity has nothing to say about the passage of time insofar as time is an “entity” (it has no ability to say anything about time as time).

Agree?

Again, this has nothing to do with relativity. Kant taught that all our claims concern phenomena and not noumena. Whatever you say about any other scientific variable, you can also say about space and time.
And such a statement about space and time definitely does have meaning (the categories that they are curved or flow at a different rate certainly do apply. Why not?).

I don’t understand, certainly not regarding the curvature of space.
I understand how one can say of a material object (which we can grasp with the senses or imagination) that it is “curved.” But space is abstract, so how can it be curved? What does that thought even mean—that it is curved?

Hi, my name is Ofek, and I have a PhD in the physics of gravitational waves (general relativity).
The “curvature” of space-time is measured mathematically using objects from differential geometry (specifically the Riemann tensor, Ricci tensor, Ricci scalar, etc.—all of which vanish in Minkowski space, which is therefore flat), so it definitely has a clear and defined meaning, just like the curvature of a material object. A non-mathematical way to understand this “curvature” is, for example, to aim two flashlights parallel to each other—in flat space-time the beams would never meet, but if there is some mass between them, that is, a curvature of space-time, the beams would “bend” (that is, each beam moves along a “straight line”—more precisely, a geodesic—of the curved space), and they can approach one another or even meet. This is curvature exactly like the curvature of the Earth, where two pilots starting in Tel Aviv and Jerusalem and both flying straight north in parallel will eventually meet at the North Pole.
I’m not much of an expert in philosophy, but these are definitely workable experiments that measure the curvature of space-time, and they are not theoretical—they were actually carried out in observations of starlight bending around the sun (Eddington 1919), or of galaxies seen in multiple copies in the sky because black holes along the way bend the rays from them (gravitational lensing).

If it were impossible to say this about space, then it would also be impossible to say it about our perception of space (because everything we say is about our perception of things).
As for the issue itself: in mathematics one defines the curvature of spaces, and that is a well-defined concept. Take the surface of a sphere as an example. It has a property different from that of some flat two-dimensional surface, regardless of how I perceive it. On a spherical surface there are situations where, if you keep moving in the same direction, you return to your point of departure, unlike on a flat surface. This is an objective distinction between those two two-dimensional spaces, and it means that one of them (the spherical one) is curved.
See here:

https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%9E%D7%95%D7%9E%D7%99%D7%95%D7%AA

And here (in section 3.1.2):

https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91-%D7%96%D7%9E%D7%9F

After I wrote this, I saw that Ofek had answered it in the previous message (which hadn’t yet been published when I wrote).

I thought the Rabbi held that one can grasp things directly and not only through their phenomenon.

Y.D., do you mean by way of immediate experience?

Not by immediate experience. But that a person’s intuition tells him something true about the world and does not merely impose its own laws on it, as Kant claimed.

Gentlemen, your humble servant cannot make head or tail of what you mean.

Ofek,
The example you gave about light rays and their trajectory does not seem relevant to my question. Light rays are entities bounded in space (and in time), and one can in principle measure their direction, shape, intensity, and other properties.
That is, light rays are actual to our senses (or at least to our measuring instruments, which are an extension of the senses).
By contrast, space as a whole (I do not mean limited areas or volumes within it) is an abstract infinite entity and is not actual for us. Space has no “dimensions” or form/shape that a person can represent in consciousness.
I have no doubt that science succeeded in determining that geodesics are curved, and perhaps even succeeded in proving that the reason for the curvature is the gravity of neighboring masses. But that determination applies to entities within space, not to space insofar as it is space. In other words: it is a scientific determination, not a philosophical one.

Michi,
I did not understand your first remark (“If it were impossible to say this about space, then it would also be impossible to say it about our perception of space [because everything we say is about our perception of things]”).

As for your second remark: I understand and accept that the distinction between a spherical surface and a flat surface is “objective.” But again, I do not see how such a claim goes beyond the domain of science.
Your example (moving along a spherical surface and eventually returning to the point of departure) also seems unrelated to me. In this thought experiment, you assume a finite and bounded “object”—which has a spherical geometric structure—and then you “walk” along it in a straight line, so to speak, and discover yourself, at the end of the motion, back at the point of departure. It is not clear to me how one can perform such a thought experiment (let alone an actual scientific experiment) regarding space as a whole (space as an infinite “entity”). The very comparison between the two bodies (the spherical and the non-spherical) does not exist when we speak of space.

If we use Kant’s terms, I would formulate it this way: the conceptualization of finite segments in space (theories about lines, surfaces, and volumes) belongs to the domain of phenomena. Clearly observations and measurements are also conducted in that domain.
By contrast, the conceptualization of space as a whole and infinite unit (that is, supposedly found “everywhere” in the universe) is done in the domain of noumena. Of course Kant himself did not mean to say this, and certainly would not have accepted my formulation. But I’m not dealing here with Kant; I’m dealing with the question of what is philosophically correct (in my view).

Doron,
Even in our universe there could be a situation in which you begin to move in a straight line and find yourself in the same place. Do you claim that is impossible? Who told you that our universe is not a spherical surface?
But it seems to me that here we’ve reached the point (the expected one 🙂 ) where I no longer see any point in continuing. What you’re really asking for is a lesson in differential geometry. This is not the place. I referred you to places where you can start reading.

1. I wasn’t talking about the universe but about space.
2. Nobody is talking to me anyway, and in any case no one told me that space (not the universe) is or is not a spherical surface.
3. Even if there were someone who wanted to talk with me and even went so far as to tell me that “space is a spherical surface,” I would present him with the same question I presented here (what is the meaning of the sentence: “space is curved”).
4. I think I already said that my troubles are many and my day is full of pain, and you, in an inconsiderate act, send me to study differential geometry? Have you not a drop of mercy in you? Are you even Jewish?
5. The beauty of rational philosophical discussion—and that is what we’re doing here—is that it requires no deep technical knowledge or details (even if that can help indirectly). Therefore the claim “the sentence ‘space is curved’ is a meaningless sentence” need not be discussed in specific disciplinary terms at all. Standard human logic, with which each of us is endowed to one degree or another, is enough. That, to my taste, is philosophy.
6. I repeat my claim (the philosophical one!): I find no more meaning in the claim that space is curved than in the claim that space likes to sleep between 2:00 and 4:00 PM. To my mind, this is a category mistake.
7. I think I’ll go nap a bit now.
Shabbat shalom

And indeed, I explained it to you very clearly even without any disciplinary knowledge, through the example of a spherical surface. Beyond that, mathematics is a branch of philosophy, and discussion in terms of curvature can and should take place within it. What can you do—that’s the field that deals with these concepts precisely.
For these two reasons I didn’t go into general relativity with you, as Ofek did. That is physics and not mathematics, and so the discussion really can and should be conducted without it as well. We are talking about logical and principled possibility and about the meaning of concepts, not about facts.

“We are talking about logical and principled possibility and about the meaning of concepts, not about facts.”

True and well said!

So given that you agree on that, please explain to me in your great kindness what the expression “curved space” means. Just try not to do it again through the example of the spherical surface, which as I already explained (and maybe also convinced you), does not have the same logical status as the concept of space.

And in general, I’m curious to know: is there among the readers one righteous person in Sodom who understands why I see a logical problem in this expression? Because even if I’m wrong (that is, even if the expression does after all have meaning), it is very easy to understand why it seems meaningless to me.

Convinced me? Of what? If we live on a spherical surface, that will be expressed in the fact that if we keep moving we will arrive at the same point. Exactly so it could happen in our space if that is its form. If such a simple explanation doesn’t seem acceptable to you—then this is the place to stop.

Doron, I volunteer to be the righteous man in Sodom. There is no meaning to the statement that space itself is curved, because you can’t measure it, only what is inside it. But on second thought, the example the Rabbi gave (if we keep moving we eventually arrive at the same place) does indeed define the curvature of space very well.

Dear B!
I’m glad about your response. In my opinion it only strengthens my claim.
As you say: “There is no meaning to the statement that space itself is curved, because you can’t measure it, only what is inside it,” (and I would add that one cannot even think about it in that way—curved space is a meaningless concept).
As for your second thought (the example of motion along the surface): in my opinion that too strengthens what I’m saying, for the simple reason that there is a sharp distinction between the actual path of a body/light ray—a path that can in principle be handled with the tools of science—and space, which is an abstract metaphysical entity about which science has nothing to say.

Using the image of a spherical surface only confuses things!

The logical problem is, in my opinion, easily solved. All one needs to assume is that the path itself curves (subject to relativity), and therefore ultimately returns to the point of departure. But that is only the path, not space itself. That is the meaning of the distinction I propose between these two “entities” (space and the path within it).

The concept of curved space does have meaning, and the example of an object moving in a *straight* line (in our eyes) and arriving at the same point is precisely the case. Space is what contains the expanse, and its curvature is relative to itself (I wrote that to prevent a few more objections I foresee you raising).

I find no meaning at all in what you wrote, sorry.
You write: “Space is what contains the expanse, and its curvature is relative to itself.”

Space contains the expanse? What does that even mean?

And as if it weren’t enough that the curvature of space is, in my opinion, a completely meaningless concept, you go further and claim that this is its curvature “relative to itself.”
Even regarding an ordinary material object, which certainly can be curved, I’m not sure I understand the meaning of the expression “relative to itself.” At most you can say of such an object (say an iron rod) that once it was straight and then it bent. I don’t think that’s what you want to claim about space…

For some reason I wrote that space contains the expanse; I meant that space is the expanse that contains everything else, and therefore its curvature cannot be measured (the curvature of everything else is measured relative to it). Since I nevertheless think that an object traveling in a straight line and arriving at the same point indicates that space is curved, I tried to describe how that is possible (it’s not clear to me either; at the moment the only possibility is that space is composed of some basic substance—in other words, that it is some kind of something and not nothing. Otherwise there is nothing that could bend).

Besides, if you are right and space is not curved (and cannot be at all), the meaning of that is that it is not curved in the sense you want to give that concept—but it certainly does have some meaning (people use it in physics every day), so what have we gained here?

It seems to me you’re getting tangled up unnecessarily.
Why assume that space is made of matter, or assume that it is “something” (and not “nothing”)?
If it were made of some substance that could bend, it would presumably have other properties that science could investigate.
Isn’t it simpler to accept the claim that space is an abstract metaphysical entity to which scientific discourse proper has no access?
You can accept that and still maintain the scientific truths of relativity (namely, as I argued before, by distinguishing between the actual curved trajectories of bodies and the spatial substrate in which they move).