This is absurd. In what sense is "curved space" nonsensical, it means exactly that space is modeled by a (pseudo)-Riemannian manifold and that manifold has a nonzero curvature. This is perfectly well-defined and has easily explained and tested predictions (e.g. shapiro delay).
You are of course welcome to reject the continuum, then curvature *at a point* no longer makes sense (neither does instantaneous velocity ofc nor a whole host of very useful concepts that I think you'll quickly come to miss if you try to do physics without the continuum. It will quickly become apparent why physics prefers to model the clearly discrete (e.g. neutron flux) by continuus methods: it's a lot easier). However global curvature is not any less apparent. The Euler characteristic for example is a manifestation of global curvature. One can define it through say index theory (e.g. the Gauss-Bonet theorem for surfaces). Given a metric one can take ratios of surface areas to volumes of hyperspheres as a definition. This kind of curvature unfortunately depends on the chosen radius (in the continuous case the low radius limit gives the usual notion of curvature) making it less nice to work with but that's the price you pay for throwing away continuous methods.
Thanks for another great article. I believe that the concept of continuous space leads to far more absurdities than the concept of granular space. Therefore I tend to favour your opinion that the concept of continuous space, and by extension curved space, does not make much sense. By the way, I think you would enjoy my 'Karma Peny' videos on the halting problem proof.
OK. I mean, this is easy if you understand the implications of discrete space. "Circles" are actually pixelated. So, I made a circle whose area is 360,000 pixels, at least according to ChatGPT. Here's a screenshot. https://pasteboard.co/ZKJYD5GBC9r9.png . I trust you are able to make a square which is 600x600.
If the circle is a few pixels off, just add or subtract some from the edge as needed. It won't make a difference.
Euclidean and non-Euclidean are mutually exhaustive. You are describing a non-Euclidean and non-Riemannian geometry, and it is probably non-a-few-other-things. Clearer to describe it positively, maybe. A priori, finite, discontinuous, … more?
This is absurd. In what sense is "curved space" nonsensical, it means exactly that space is modeled by a (pseudo)-Riemannian manifold and that manifold has a nonzero curvature. This is perfectly well-defined and has easily explained and tested predictions (e.g. shapiro delay).
You are of course welcome to reject the continuum, then curvature *at a point* no longer makes sense (neither does instantaneous velocity ofc nor a whole host of very useful concepts that I think you'll quickly come to miss if you try to do physics without the continuum. It will quickly become apparent why physics prefers to model the clearly discrete (e.g. neutron flux) by continuus methods: it's a lot easier). However global curvature is not any less apparent. The Euler characteristic for example is a manifestation of global curvature. One can define it through say index theory (e.g. the Gauss-Bonet theorem for surfaces). Given a metric one can take ratios of surface areas to volumes of hyperspheres as a definition. This kind of curvature unfortunately depends on the chosen radius (in the continuous case the low radius limit gives the usual notion of curvature) making it less nice to work with but that's the price you pay for throwing away continuous methods.
Infinity isn't a coherent concept, let along "a thing", and our universe is discrete. Yes, and yes.
Space and time are both continuous in QFT and GR.
So your suppositions are wrong.
There is an approximately 0% chance that QFT and GR are final physics. So no, that's a terrible objection.
Thanks for another great article. I believe that the concept of continuous space leads to far more absurdities than the concept of granular space. Therefore I tend to favour your opinion that the concept of continuous space, and by extension curved space, does not make much sense. By the way, I think you would enjoy my 'Karma Peny' videos on the halting problem proof.
you dont get it; just make a square with area equal to that of a circle, simple.
I appreciate your look at the pixels comment, I think I found the curve
OK. I mean, this is easy if you understand the implications of discrete space. "Circles" are actually pixelated. So, I made a circle whose area is 360,000 pixels, at least according to ChatGPT. Here's a screenshot. https://pasteboard.co/ZKJYD5GBC9r9.png . I trust you are able to make a square which is 600x600.
If the circle is a few pixels off, just add or subtract some from the edge as needed. It won't make a difference.
What do I win?
what then makes a rectangle with 360,00 pixels any different from a circle?
I mean to say: a metric cannot be a measure of a shape, they must exist apart from any euclidean or hyperbolic(non-euclidean) space.
Its shape, duh. It's a set of relations--how the pixels stand in relation to one another.
Euclidean and non-Euclidean are mutually exhaustive. You are describing a non-Euclidean and non-Riemannian geometry, and it is probably non-a-few-other-things. Clearer to describe it positively, maybe. A priori, finite, discontinuous, … more?