You can never step in the same river twice
People say this remark is from Heraclitus. The main idea is that the river keeps changing as the water flows. The other idea is that you
keep changing, too!
Jorge Luis Borges wrote:… each time I recall fragment 91 of Heraclitus, "You cannot step into the same river twice," I admire his dialectical skill, for the facility with which we accept the first meaning (“The river is another”) covertly imposes upon us the second meaning (“I am another”) and gives us the illusion of having invented it…
But actually it seems Heraclitus didn't exactly say "you cannot step into the same river twice".
He lived roughly from 535 to 475 BC. Only fragments of his writings remain. Most of what we know about him comes from Diogenes Laertius, a notoriously unreliable biographer who lived 600 years later.
For example: Diogenes said that Heraclitus became sick, tried to cure himself by smearing himself with cow manure and lying in the sun... and died, covered with poop.
But Diogenes also said that Pythagoras died while running away from an angry mob when he refused to cross a field of beans, because beans were sacred to the Pythagoreans. And Diogenes also said Pythagoras had a golden thigh - and was once seen in two places at the same time.
So we don't really know much about Heraclitus. And among later Greeks he was famous for his obscurity, nicknamed “the riddler” and “the dark one”.
Nonetheless a certain remark of his has always excited people interested in the concepts of sameness and change.
In one of Plato's dialogs the Socrates character says:Heraclitus is supposed to say that all things are in motion and nothing at rest; he compares them to the stream of a river, and says that you cannot go into the same water twice.
This is often read as saying that all is in flux; nothing stays the same. But a more reliable quote passed down through Cleanthes says:On those stepping into rivers staying the same other and other waters flow.
That's harder to understand - read it twice! It seems that while the river stays the same, the water does not.
No matter what the details are, to me Heraclitus was trying to pose the great mystery of time: we can only say an entity changes
if it is also the same
in some way — because if it were completely
different, we could not speak of "an entity" that was changing.
Of course we can mentally separate the aspect that stays the same and the aspect that changes. But we must also bind these aspects together, if we are to say that "the same thing is changing".
In category theory, we try to swim these deep waters using the concept of isomorphism. Very roughly, two things are isomorphic if they are "the same in a way". This lets us have our cake and eat it too: two things can be unequal yet isomorphic.
So when you step in the river the second time, it's a different but isomorphic
river, and a different but isomorphic you.
And the isomorphism itself? That's the passage of time.
So, isomorphisms exhibit a subtle interplay between sameness and difference that may begin to do justice to Heraclitus.
None of these thoughts are new. I'm thinking them again because I'm writing a chapter on "concepts of sameness" for Elaine Landry's book Category Theory for the Working Philosopher
. You can see a list of chapters and their authors here:https://golem.ph.utexas.edu/category/2015/02/concepts_of_sameness_part_1.html
Here and in future articles you can watch me write my paper, and help me out. It'll be more technical - and I hope more precise! - than my remarks here. But it's supposed to be sort of fun, too.
In Part 2, I talk about the Chinese paradox "when is a white horse not a horse?":https://golem.ph.utexas.edu/category/2015/02/concepts_of_sameness_part_2.html
In Part 3, I ask if you've ever used the equation x = x for anything. And I pose a precise conjecture which claims that this equation is useless. I would like someone to settle this conjecture!
But if x = x is a useless equation, why do mathematicians think it's fundamental to our concept of equality?
The picture here is taken from someone on G+ who is vastly more popular than me:https://plus.google.com/104293557269756681667/posts/KGGzk5nmQ4F