Is the physical world proportional? What I mean is: is it possible, for instance, that we find a solar system exactly like ours except for the fact that every object (planets, stones, animals, trees, etc.) is one thousand times longer or less long? What if only twice longer? And what about a different universe where even atoms (and elementary particles, if they have any length at all) were one thousand times "longer"? Is this meaningless?

With regard to both questions, I understand you as imagining that objects are longer or shorter in all dimensions (not merely in one dimension). So spheres would still be spheres, except larger or smaller ones. Right?

On your first question, this is not possible if we hold fixed the laws of nature holding in this universe. To illustrate: In your Twin Solar System, scaled up by a factor of 2, Twin Earth would have eight times as much mass, and gravity near its surface would be roughly twice as great (surface gravity is proportional to the planet's mass divided by the square of it radius). Like the Earth, a scaled-up object would have eight times as much mass, so the gravitational force acting on it (its weight) would be 16 times greater. Now imagine this object suspended by a string. This string would be thicker in two dimensions, hence four times stronger. But the object's weight would be 16 times greater! So, on Earth, the string may be sufficiently strong to support the object even while on Twin Earth the counterpart string would not be strong enough to support the counterpart object. Examples could be multiplied. A Twin-Earth parachutist would also have 16 times as much weight as her counterpart on Earth while her parachute's surface would be only four times as large. Likewise for scaled-up planes, where the wing surface area (creating lift) would not keep up with the increased weight; scaled-up replicas of planes we use here would not be able to fly on Twin Earth. (The examples ignore more subtle differences: Because Twin Earth has higher surface gravity, it would have higher atmospheric pressure and density than our Earth, and it would also be a bit more compressed which would lead to a further increase in its surface gravity.) You see the general point: In many ways, other things would not be the same in a scaled-up (or scaled-down) solar system, because some of its parameters (e.g., forces acting) would vary with the scaling factor, others with the square of the scaling factor, and so on. Interestingly, however, one thing could be the same: Twin Earth could be circling Twin Sun in a stable orbit once a year. Gravitational and centrifugal forces acting on Twin Earth would balance out, both being 16 times what they are in our solar system.

On your second question, yes, I think this would be meaningless for lack of a common benchmark in terms of which lengths could be measured in both universes.

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