The Mathematical Experience
Mathematics is simultaneously a body of results, a practice of reasoning, and a historically situated human activity.
The book describes how mathematicians work, including intuition, conjecture, proof, collaboration, and disagreement. Finished demonstrations conceal much of the exploratory activity that produced them.
A finished proof hides the guesses, abandoned attempts, and conversations behind it. The book returns those human activities to mathematics without weakening the standards by which a result is checked.
Questions about mathematical objects lead through competing philosophical positions, including Platonism and formalism. The authors resist treating lived mathematical practice as fully explained by one abstract doctrine.
Certainty, computer-assisted work, applications, and the relationship between mathematics and society reveal different forms of justification and usefulness.
The central account presents mathematics as a communal practice with unusually disciplined standards, not merely an inventory of timeless facts or arbitrary symbols. Its human character does not eliminate the force of rigorous proof.
Discovery often begins with intuition that proof must subsequently discipline.
Certainty about an argument does not require agreement about what mathematical objects ultimately are. The authors make room for rigorous practice and unresolved philosophical interpretation at the same time.
A community decides how explanations are checked and communicated.
Formal rigour does not describe every step of creative practice.
Applications can emerge long after an abstract problem was posed.
Different philosophical accounts explain mathematical existence in competing ways.
The human history of mathematics does not make its conclusions arbitrary.