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Well, I don’t have the mathematical ability to argue against that, but I don’t think it’s true. The fundamental laws don’t distinguish time from the other dimensions except for the signature. You say that time is something measured by clocks, but clocks work the way do because of the laws of physics.PeterDonis said:I would not put it this way. I would say that there is no physical difference between a metric with a (3, 1) signature and a metric with a (1, 3) signature. That's because either way you can physically interpret the "1" dimension in the signature as the "time" dimension, which means, heuristically, that arc lengths along that dimension are measured with clocks instead of rulers, while arc lengths along the other 3 dimensions are measured with rulers. So there's no physical difference between the two, just a different choice of signature.
But saying that you have 3 time axes and 1 space axis is very different physically: it's saying that you have three orthogonal "directions" along which you measure arc lengths with clocks, and only one along which you measure arc lengths with rulers. That's a physical difference--and again, you could (if experiments supported it) adopt such an interpretation for either a (3, 1) or a (1, 3) signature.
If there were three time dimensions and one space dimension, then space would work like time, in that specifying conditions for all time at a particular location in space would determine conditions at all other locations in space. I think it’s the fact that there is only one time dimension that makes it work like time intuitively works.
The laws of physics written in terms of three time dimensions and one space dimension would look exactly like the laws of physics for 3 space dimensions and one time dimension, except with the opposite sign convention.