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At it most basic level, the answer is "We DON"T know that all galaxies are moving away from each other." However, IF (indeed) space is expanding as proposed by Georges LeMaitre, then we would see all distant galaxies moving away from our own, and the rate at which they are moving away from us would depend on the distance to those galaxies.

FACT 1) That's exactly what we see.

Note also that, if LeMaitre is correct, then all other observers in our Universe we also see exactly the same thing.

FACT 2) If the expansion we now see has been ongoing since the start of that expansion, then we would ALSO see microwave radiation coming to Earth with almost perfect isotrophy and with a black-body spectrum of about 4 degrees Kelvin. We also see exactly that.

Fact 1 COULD be explained by our Milky Way Galaxy just happening to be the ONE galaxy, out of the 170 billion we know about -- a large fraction of which are pretty much identical to ours -- that was at the point from which all matter started expanding from. That is an EXTRAORDINARY unlikely occurence. It is MUCH simpler to assume that we are NOT at that center, and that all the other tens of billions of galaxies that resemble ours, are no more (or less) special in our Universe.

In addition, Fact 2 can NOT be explained by the assumption of our galaxy being at the point from which matter began to expand.

The assumption that we are NOT at some special place in our Universe both (1) is a more reasonable assumption and (2) explains more of what we see in our Universe.

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Q: How do you know all galaxies are moving away from each other. From earth we see all galaxies are moving away but how do we know the same is seen from every galaxy?
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