taeto

Indeed. Coordinate independent means that it does not depend on the coordinates what is meant by measures of length and angle. I can say this with some confidence, since if we were to observe a geometric object all of whose points are represented by rational coordinates, then by shifting the coordinates by \(\sqrt{2}\) we obtain just another representation of the exact same object, but using all irrational coordinates. Shift by \(\pi\) and you get transcendent coordinates. Also, it is the measure of an angle which is a magnitude. The angle itself is a geometric figure. Numeric values may suffice, once you have the necessary additional notions in place. And this will need proof. But the definitions are not based on having numerical measures. I do not know what you mean by this. But I could get nasty (towards the constructivists) by suggesting that \(c,\) the speed of light, which features prominently in SR and GR, may just happen to not have any representation in terms of a quantity which is not rejected by constructivists. I always think of time in physics as one of the coordinates by which you describe the location of an event. And usually the range of this coordinate is real numbers, because that is just extremely convenient for talking about limits, such as when differentiating or integrating functions of time to get theoretical values of physical quantities. But it should not really matter which real numbers you use for any static description; you might use rational numbers for specifying positions of physical objects, or even whole numbers, if your choice of unit makes it small enough. And this is what I think of as the classical geometric view. As I mentioned, if you have any system given by coordinates in which all positions are represented by whole numbers, you may shift coordinates to represent the same system by non-constructible transcendental numbers. That being one reason why I cannot make any sense of the article that is referred to in the OP. I take up one view of time which is employed by constructivists in particular. They would say, I believe, that if you investigate a number with a decimal representation like \(0.4999999...\) then it will take you time to discover whether the number is actually equal to \(1/2.\) And to ascertain that it is actually equal to \(1/2\) will take you infinite time. It seems that the new explanation of time that gets offered is dependent on using numbers that require only finite time to represent. So that there is one view of time that is used to explain a different view of time.

Zulfi, can you figure out how much she earns every hour she works?

Yes, you can slide in Cartesian Geometry sideways if you want to. But the point of this thread was the role of the coordinate \(t\) for time in physics. And my point is then that it cannot really matter which kind of numbers we choose as the range of \(t\) geometrically, because that is not relevant for the kind of geometry that we posit as a model for physical space, which is Euclidean. Now you take an intuitionist view to say that we may define geometry any way we want. But if we talk about Euclidean Geometry, there are some ground rules. Actually, congruence in Euclidean Geometry does not involve "length". It only involves whether it is possible to move one object exactly on top of another object by using certain allowed operations. There is nothing about measurements and numbers.

So far as I can see, this whole thing is very hypothetical, and there is no proof that it ever happened anywhere, nor that it will happen. But of course, if it could happen, then it is reasonable to expect that it could happen anywhere with equal chance, including where we are. Of course that would seem extremely unlikely.

No, you do not do geometry with numbers, that is the point. You may in Euclidean Geometry have a picture of lines being composed of points that have rational coordinates, or of algebraic coordinates, or coordinates that are constructible numbers, it does not matter. The concept of a number being rational or not does not really enter into it. You have to think about geometry in terms of geometrical objects being parallel, congruent, similar, etc. all of which do not involve numbers. Actually, just think about what can be done if you do not have any \(x\)- or \(y\)-axes. Geometry is about the relationships that you can still identify and prove without those coordinates. Just go through Euclid's axioms for the Euclidean plane, and observe that it does not matter when you talk about points and lines whether the points are "rational" or not, in fact there is no way to even speak of this. Until you introduce added-on axioms like the Cantor-Dedekind axiom, which talks about limits. But that came in a bit later after the original axioms.

Classically an angle is made up of two rays (half-lines) that meet in a common endpoint. If you want to measure an angle, you can compare it to one of the arcs of a unit circle that gets cut by placing the meeting point of the two rays in the center of the circle. Then the length of this arc is some fraction of the length \(2\pi\) of the arc of the whole circle, and that determines the measure of the angle relative to \(2\pi.\) Again it is immaterial whether \(\pi\) is a whole number, a fraction of a whole number, or irrational.

You mean geometrically? For \(\pi\) I suppose you would be able to show that all circles of equal radius are congruent. Classically you do not have any immediate notion of measures or lengths, other than one length can be half of another one, or three times as long. When you later introduce length, you define the ratio between circumference and diameter to be \(\pi\). That does not require any further investigation into the nature of the number \(\pi;\) for what we know from this definition, it could be rational or not, supposing we have such a notion. Similarly with the length of the hypotenuse of a right triangle with two other congruent sides. The notion of "lengths" here are really quite separate from the specifics of the kinds of points that make up the line segments and arcs. Even line pieces that consist of only "rational points" are congruent to and have the same lengths as the equivalent line pieces in which all missing "real points" are added.

In Crichton's novel it seemed more of an arbitrary choice to name the invader after our neighbor galaxy. In Hoyle's case (if it was indeed Fred Hoyle), the Andromeda was its actual source, and it makes it scary on an additional level; that of the intrusion having been orchestrated by someone or -thing on purpose. Imagine receiving in our day a signal from some remote location, say originating from a billion light-years away, which clearly contains technical instructions for the construction of something. Would any laboratory agree to take a shot at it? The outcome could go in different directions, The exciting aspect is to not only have CETI, but you get to have an actual physical thing there to look at, even if you cannot communicate back and forth across the distance.

Judging from your avatar you might well have come across a few visionaries who got stoned 🙄.

We could easily describe every geometric aspect of spacetime by using only rational numbers, or if you prefer, all numbers that can be written as rational numbers with a power of 10 in the denominator (having so-called "finite decimal representation"). Geometry does not worry about continuum, at least not in the locally Euclidean case. The problem comes with the dynamics; a velocity is defined by differentiation, and derivatives of rational-valued functions are not rational-valued.

Wikipedia says: "Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics." Although it does seem to me personally, from my own philosophically intuitionist point of view, that these categories are purely philosophical and have little if nothing to do with the actual practice of mathematicians. By qualifying my stance as such, I believe that I can make stuff up of any kind I wish anyway.

The electron on the Moon versus on the Earth example at the end of the article seems suspect, as it clearly suggests that even positive numbers are zero, once they are small enough. That does not even agree with intuitionist mathematics. Anyway, it comes down to formalism what it all means. If limits are not defined, how is the momentum of a particle represented? Similarly how about forces and work? There is not much hint of that in the article, so I suppose it is necessary to look up the actual research papers. Oh, and before I forget again. "Constructivist mathematics" has a different view of reality than the scientific view. By saying that an object only "exists" if it can be constructed. A zoologist should be well able to define "a cat", even without demonstrating the ability to construct a cat. In practice, similarly as in ordinary non-constructivist mathematics, it is enough to have the ability to identify whether an object that gets presented is in fact a cat or not.

And Speed of the Sound of Loneliness https://www.youtube.com/watch?v=eFvenjll1Bk&list=RDSJPX2SpcCqw&index=3

Thanks studiot! I remember watching a TV series when I was a kid much along those same lines. Scary stuff for a kid 😨. I will try to find any episodes of The Big Pull that are still available, to find out if those are the ones I was watching.

That is getting into what QuantumT is saying. If we suppose that other universes keep bumping into our own universe, then it is practically impossible to tell when they might bump into one another at any particular location, such as ours. All that one can say then is that it will seem unlikely to happen.

I say "local timeframe", because what we can now see happening to them actually happened to them 3 billion years ago. The light did not arrive to us before now. Whether it is the only one, that is an interesting question. There are other cool spots, but it seems there is only this one particular very cold spot. What we might worry about is that a different universe collides with our universe at or around the spot of our location, if this is what happens. But seeing that this possibility is very hypothetical, I suggest that there is not real reason to worry.

The supervoid in itself is only hypothetical. But usually it refers to a unique very cold region determined by measuring the cosmic microwave background. It may be caused by not having a lot of galaxies present in a region of something like a diameter of 2 billion light years at a distance of about 3 billion light years from us, in our local time frame, and towards that particular direction. Or its detection can be caused by just local perturbance in the CMB itself called a cold spot. It seems not much to worry about.

The thing about the supervoid is that it is very cold. And very large. You can observe it from the southern hemisphere, and it takes up quite a substantial region of the sky as seen from there. It has not been observed to grow within our timeframe, but it is reasonable to assume that it has been growing since the time of the big bang. There are normal galaxies present in the direction of the supervoid.

I mean what happens with the research to figure out whether it is a supervoid, or just a normal cold region of the universe.

I think we simply have to wait and see what happens. We can of course be sure that it will not hit us during our lifetime.

Well, if there is indeed an observed cold spot originating from a collision with a different universe, then it stands to reason that quite many such collisions do occur, possibly even infinitely many of them. But as you say yourself, the idea is far fetched.

John Prine is in heaven now, due to corona. https://www.youtube.com/watch?v=JKPDFQRmG_M Life changed in many ways during the past four weeks. This however is one news that gives me most sadness.

Thanks Strange, Fred Hoyle definitely rings a bell. Apparently it was originally a TV series, which later became novelised and came out as a book from Macmillan. From what I have gleaned from the TV series itself, dating from 1961, it seems its character direction is even scarier than its scientific content.

I read it when I was a kid. A couple of times. It seems to be different from "The Andromeda Strain" by Michael Crichton, which comes up every time I try to Google search. It is an even more sinister story. About a signal that gets received from the general direction of the Andromeda Galaxy, and which contains the recipe for creating some kind of alien organism. It is when some laboratory follows this recipe and manages to create an exemplar of this organism that things begin to go very wrong. I remember that it was quite well written and scary, not some simple minded pulp story.

From usual definitions it would be \(\pi/\infty = 0\) and \(0 \cdot \infty = 0,\) so that means \((\pi/\infty)\cdot \infty = 0.\) But it seems you are trying to treat \(\infty\) as if it were a natural number, and that has only small chance or working out. I do not think that you need to introduce \(\infty\) in the first place anyway.