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Heraclitus is famous for his argument that change is constant; that everything in the world is always changing. The saying that you quote shows this perfectly because it illustrates how even things that seem constant to us are actually undergoing change.
When we look at a river, we see a single entity that has always been there and always will, seemingly, be there. Therefore, a river seems to be almost the epitome of something that does not change. But, as Heraclitus says, it really is not possible to say that the river remains the same. The river is constantly changing and is, in effect, not the same river.
If you step in a river, you are stepping in a specific bit of water. If you withdraw your foot and step in it again, you are stepping in a different bit of water. Moreover, just by stepping in the river, you have changed it. You have altered the course of the water a little bit. You have eroded the river bed a little bit. The river is always changing and therefore it is never the same.
We can look at human beings and societies in the same way. We may seem to be the same person today that we were last week, but we have been changed in some way by our experiences. Society has been changed by all the things that happen. Human life is like the river that seems to stay constant yet is actually continually changing.
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