I came across this phrase recently and wanted to know what it meant. Here's what I found out.
Ancient astronomers believed that the evening star and the morning star were separate things, evidence of the perfect balance of nature. Greek astronomers named the evening star Hesperus and the morning star Phosphorus. When Pythagoras correctly understood that these two bodies were in fact the same entity, he was able to assert that "Hesperus is Phosphorus." This is the root of its first meaning.
A second meaning comes from philosophy, where the statement is problematic because once you accept its premise, it becomes a tautology, i.e. "(the thing that is Hesperus) is (the thing that is Hesperus)." To a philosopher, this illustrates that we do not discuss things directly, or extensively, even though we think we do. We discuss things referentially, or intensionally [sic]. Consider the following scenario.
Phil and Helen are planning their first dinner date. Phil asks, "Where would you like to go?"
Helen replies, "I really like the restaurants in Old Town." She has dined in the area before and knows there are only three restaurants there, each quiet, dim, and to her liking. In her mind, she has just communicated a list of three.
"Brilliant," Phil says, "I'll make reservations." After the call, he learns that there is a new hipster restaurant opening in Old Town, and, wanting to impress her, he makes reservations at the new place. In his mind he has satisfied her requirement. It's in Old Town. (You can see where this is going.)
The night of the date they head to this new hotspot where things are noisy and jumping. She is taken aback that the restaurant is not at all what she specified. She thinks he is incapable of listening, which predisposes her to disliking him. This first date is their last.
This sad little anecdote illustrates the problem of identity
in communications like, "I really like the restaurants in Old Town." Helen understood the expression to summarize a known list of restaurants. Phil understood the expression as a rule
to which any restaurant may be applied. Upon review of any given restaurant, he would need to apply the rule, "Is it in Old Town?" and, if so, then understand it as acceptable.
Per philosophic vocabulary, Helen was using extension to summarize a closed set that happened to share a common attribute, i.e. their location in Old Town. (One visit to the new restaurant and she will have revised the description of the set.) Phil was using intension, seeing Old Town as a qualifying attribute to test any restaurant for inclusion.
Fortunately, only philosophers and computers are so literal as to hear, "Hesperus is Hesperus." We understand Pyhtagorus' statement extensionally, i.e. "The thing with the attribute of (name = Hesperus) is the same thing that has the attribute (name = Phosphorus)." So in this sense its meta-meaning belongs in the realm of deep philosophy or possibly computational linguistics. It categorizes a problem being discussed as one of identity, or specifically identity over time in communication. For example, upon hearing the story of the mismatched diners, a philosopher may utter, "Hesperus is Phosophorus," implying that if Helen and Phil had understood the issues of identity over time, intension and extension, they might have gone on to a second date.
In everyday usage, we may use the phrase in its original Pythagorean sense as shorthand to imply that two things being discussed are in fact the same thing from different perspectives, instead of the longer "The Blind Men and the Elephant" parable.