This is most entertaining.
We have a report of this exchange from antiquity, involving the Stoic Epictetus:
“Come now, Epictetus, take off your beard.”
– If I am a philosopher, I answer, I will not take it off.
“Then I will take off your head.”
– If that will do you any good, take it.
And John Sellars tells this story in his book on The Art of Living (2003, p.15):
“In AD 176 the Roman Emperor and Stoic philosopher Marcus Aurelius created four chairs of philosophy in Athens, one for each of the major schools. When, a few years later, the holder of the Peripatetic Chair died, two equally well qualified candidates applied for the post. One of the candidates, Diocles, was already very old so it seemed that his rival, Bagoas, would be sure to get the job. However, one of the selection committee objected to Bagoas on the grounds that he did not have [a] beard saying that, above all else, a philosopher should always have a long beard in order to inspire confidence in his students. Bagoas responded by saying that if philosophers are to be judged only by the length of their beards then perhaps the chair of Peripatetic philosophy should be given to a billy-goat. The matter was considered to be of such grave importance that it was referred to the highest authorities in Rome, presumably to the Emperor himself…”
Over the page, Sellars suggests that it was the mission of the three philosophers to Rome in 155 BCE which created the popular link between philosophers and beards. That was the famous occasion (which haunts Grotius scholarship down to the present day) when the Sceptic Carneades made a speech in favour of justice one day, and a speech against it the next, very much annoying Cato the Censor in the process. But these were bearded Greeks in clean-shaven Rome, and the Romans remembered the beards.
Y'know, that's actually believable. And funny.
There's more; read the whole thing.