Rigor and Structure

by John P. Burgess

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

Genres: Mathematics, Philosophy

224 pages, Hardcover

First published February 12, 2015

About the author

John P. Burgess (PhD, University of Chicago) is James Henry Snowden Professor of Systematic Theology at Pittsburgh Theological Seminary. He is the author of several books, including Holy Rus': The Rebirth of Orthodoxy in the New Russia, Encounters with Orthodoxy: How Protestant Churches Can Reform Themselves Again, and Why Scripture Matters: Reading the Bible in a Time of Church Conflict.

Reviews

[Zach Allison · Rating 4 out of 5]

Having had no background in philosophy of mathematics, this book was challenging. I was fortunate to have read it in a graduate seminar under the guidance of an established philosopher of science and mathematics. Had I not had that assistance, I might not have gained as much from reading this.
My favorite part of the book by far though is a section in chapter 3 that talks about the history of the relationship between philosophy and mathematics and gives some insight into how that relationship led to revolutions in not only those two fields, but in others as well. That section alone has me coming back to this book from time to time and is the primary reason I give this book 4 solid stars.

[Chip · Rating 4 out of 5]

I don't really like Burgess's writing, but for me he's the first one which is in analytic tradition, but still focuses on the historical account. His investigation of rigor in mathematics is a great clarification of the concept