The price of this book appalls. Would someone please see to it that it remains in print as an affordable paperback edition?
I begin by quoting from Lucas's Introduction: "To start from the absolutely incontestable is to start from nowhere... any actual starting point could be put into question by a skilful sceptic... a metaphor of Descartes metaphor is more appropriate: we should think of mathematics not as a building based on foundations, but as a tree grounded in the soil of our general conceptual structure, and growing both up and down." Thus the title of this book.
Lucas, a fine British academic philosopher whose interests have ranged over mathematics, physics, classics, and political philosophy, is the author of a startling and very controversial 1961 article on the interface among philosophy, computer science, and the philosophy of mind: "Minds, Machines, and Godel." This article much influenced "Godel Escher Bach," and Howard DeLong's "Profile of Mathematical Logic." Dale Jacquette agrees with Lucas's thesis; Judson Webb does not. I share Lucas's skepticism re artificial intelligence.
I think of Lucas's book as a recent and splendid addition to the saga begun by Russell's "Principles of Mathematics." Russell rightly believed that the philosophy of mathematics is fascinating, but his intellectual ethics left something to be desired. Frege and Brouwer fascinate as well, but are difficult. C S Peirce's contributions to logic and mathematics have only started to receive the detailed treatment they richly deserve, thanks to Geraldine Brady's 2000 book. Contemporary authors (e.g., Chihara, Hellman, Shapiro, Maddy, Crispin wright, Resnick, Detlefsen) are readable, but appreciating what they say requires knowing a lot about what their predecessors wrote. Such are the many pitfalls along the trail Russell blazed circa 100 years ago