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I believe this ancient Greek philosopher is Epictetus--in all iconography, he has the longest beard and looks most like the button. The button is stamped copper embellished with Wedgwood blue paint. It measures 1 & 7/16" in excellent condition. It has been called Socrates, but it looks nothing like him :)

Notes the Dictionary of Greek and Roman Antiquities: Among the Greeks, longer beards had become a status symbol of philosophers, giving rise to the common saying “A long beard does not make a philosopher.” Later many Romans shaved their beards in order not to be taken for Greeks, Greek influence sometimes being politically suspect. The paranoid emperor Domitian once ordered philosophers to be banished from Rome, and many shaved their beards to disguise their profession. Stoic philosopher Epictetus wrestled lightly with the dilemma and, in a small act of defiance, refused to shave:
“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.