I’m in the middle of writing a long post about the tortured history of the book’s title. It’s taking much longer than I anticipated. In the meantime, let me get two possible interpretations out of the way.
The title Mathematics without Apologies expresses one of the book’s main themes, namely that the meaning and the appeal of mathematics can’t be understood unless mathematics is seen as an end rather than as a means to something else. Understanding the meaning of mathematics as an end rather than a means is not the same as defending the doctrine of “mathematics for mathematics’ sake!” Understanding is not the same thing as justifying! I obviously believe that mathematics can be justified, and maybe some day I’ll try to set out a systematic case for mathematics, using some of the material presented in this book. In the meantime, the title says that this is exactly NOT what the reader of the book will find.
The second thing the title doesn’t mean is that I am trying to set myself up as the 21st century gonzo postmodern answer to G. H. Hardy. That would be presumptuous in the extreme. Hardy’s A Mathematician’s Apology is irreplaceable as a great work of literature and as a document of its time, and it has undoubtedly inspired generations of pure mathematicians, perhaps number theorists more than others. Having heard the Apology cited reverentially as the sole reliable window into the mathematical mind, I did want to make an alternative available; and I wanted to rescue Hardy from the currently dominant misreading of his work, along the lines of “Hardy said number theory was useless; online encryption shows he was wrong.” But although my title inevitably echoes that of Hardy’s book, it was chosen at the end of a long process of accretion and elimination, and for much of the process I saw the allusion to Hardy as an argument against using the word “apologies” in the title. A future post will explain how it ended up there nevertheless.