Dear Meg,

It’s not hard to see, in your question, a sense of – I don’t know – anticipated boredom, or perhaps some worry about what you’ve let yourself in for.  It’s all reasonably interesting now, but, as you say, “Is this all there is ?”  You’re reading Shakespeare, Dickens, and T.S. Eliot in your english class, and you can reasonably assume that while this is of course only a tiny sample of the world’s greatest writing, there is not some higher level of English literature whose existence has not been disclosed to you.  So you naturally wonder, by analogy, wether the math you’re learning in high school is what mathematic is. Does anything happen at higher levels besides biggers numbers and harder calculations ?

What you’ve seen so far is not really the main event.  Mathematicians do not spend the most of their time doing numerical calculations, even though calculations are sometimes essential to making progress.  They do not occupy themselves with grinding out symbolic formulas, but formulas can nontheless be indispensable.  The school math you are learning is mainly some basic tricks of the trade, and how to use them in very simple contexts.  If we were talking woodwork, it’s like learning to use a hammer to drive a nail, or a saw to cut wood to size.  You never see a lathe or an electric drill, you do not learn how to build a chair, and you absolutly do not learn how to design and build an item of furniture no one has thought of before.

Not that a hammer and saw arent’s useful.  You can’t make a chair if you don’t know how to cut the wood to the correct size.  But you should not assume that because that’s all you ever did at school, it’s all carpenters ever do.

– Ian Stewart dans Letters to a young mathematician

Les caractères gras sont de moi.  Je dis souvent à mes élèves : “vous pouvez tous aller à la bibliothèque et chercher les grands classiques de la littérature française : avec un peu de volonté, vous pouvez dévorer Notre-Dame de Paris ou Madame Bovary et apprécier l’oeuvre à juste titre.  Vous serez émerveillés, vous serez émus.  Personne parmi vous ne peut cependant aller à cette même bibliothèque et lire les grands classiques de la mathématique.   Personne ne sera ému par Introductio in analysin infinitorum d’Euler, personne ne sera émerveillé par Disquisitiones arithmeticae de Gauss et aucun d’entre-vous, certainement, ne sera en mesure de déchiffrer une seule phrase de Über die Hypothesen welche der Geometrie zu Grunde liegen de Riemann.  Pour y arriver, cela vous prendrait encore 10 ans d’études en mathématiques.  C’est une distinction importante.”