​ ​ Mathematical Quotes   ''Let no one ignorant of geometry enter here.'' (Plato, at the entrance of his school The Academy, where he trained future philosophers and politicians) ''Give him a coin, since he must make gain out of what he learns.'' (Euclid, to his teaching assistant, when a student asked Euclid, "What's the use of learning geometry?")  "There is no royal road to geometry." (Again, Euclid, to a Greek ruler who asked him if there is a quick way to learn geometry because, after all, he was king) ''Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language and forthwith it is something entirely different.'' (Goethe, a German poet) ''One must be able to say at all times (instead of points, lines, and planes) tables, chairs, and beer mugs.'' (David Hilbert, on the arbitrary nature of mathematical terms and their meanings) ''When I use a word, it means what I wish it to mean, neither more nor less.'' (Humpty Dumpty in Alice in Wonderland, again on the arbitrary nature of words we use.) ''Leave the science of parallels alone... No man can reach the bottom of the night.'' (Wolfgang Bolyai, to his son John, telling him not to spend his life trying to prove/disprove Euclid's Parallel Postulate) ''Out of nothing I have created a strange new universe.'' (John Bolyai, replying to his father, who told him NOT to study the Parallel Postulate. John discovered non-Euclidean geometry; this is one of the most significant moments in human history.) "Je le vois, mais je ne le crois pas (I see it, but I do not believe it)." (Georg Cantor, upon proving an amazing theorem on infinity) "Wir müssen wissen, wir warden wissen (We must know; we shall know)." (David Hilbert, 1930)"Common sense is nothing more than layers of preconceived notions stored in our memories and emotions….” (Albert Einstein)   **************************************************************************   And a few more philosophical ones...   "Gödel has taught us that not only is mathematics a religion but it is the only religion able to prove itself to be one." (John Barrow, on the Incompleteness Theorems of Gödel) ''As far as the mathematical theorems refer to reality, they are not sure, and as far as they are sure, they do not refer to reality.'' ''Mathematics is the subject in which we do not know what we are talking about nor whether what we say is true.'' ​