What is Mathematics, Really? was written by Reuben Hersh and published in 1997.
The book was written to challenge the common view that mathematics is either:
a collection of eternal, abstract truths that exist independently of humans (Platonism), or
merely a game of manipulating symbols according to rules (formalism).
Instead, Hersh argues that
mathematics is a human activity. Mathematical objects don't exist in a separate mystical realm, but neither are they arbitrary inventions. They are
socially created and maintained through the work of mathematicians, much like language, law, or music.
He wrote the book because he believed that:
- traditional philosophies of mathematics failed to describe how mathematics is actually practiced,
- mathematicians themselves often held inconsistent views about the nature of their subject, and
- students and the public deserved a more realistic account of what mathematics is.
The book explores questions such as:
- What are numbers?
- Why does mathematics work so well in science?
- Is mathematical truth discovered or invented?
- How do mathematicians know a proof is correct?
It's considered one of the most accessible introductions to the philosophy of mathematics, and it's often recommended alongside works by philosophers such as Imre Lakatos, especially if you're interested in the foundations of mathematics rather than learning mathematical techniques.