In 1995, Andrew Wiles won worldwide acclaim when he finally laid to rest one of the oldest mathematical chestnuts of the modern era. After more than 350 years, Pierre de Fermat’s deceptively simple claim that he famously jotted down in a margin was now unequivocally true: the conjecture had at ...

James Robert Brown is Professor of Philosophy at the University of Toronto. He has written widely on a number of topics in the philosophy of science and philosophy of mathematics ranging from the sociology of science to seismology, but is most recognized for his pioneering work in thought exper...

The Laboratory of the Mind: Thought Experiments in the Natural Sciences
by James Robert Brown Thought experiments are performed in the laboratory of the mind. Beyond this metaphor, it is difficult to say just what these devices for investigating nature are or how they work. Though most scientists and philosophers would admit their great importance, there has been very little serious study of them. This volume is the first book-length investigation of thought experiments. |

Philosophy of Mathematics
by James Robert Brown Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value? In this book, Jim provides his unique answers to these questions as well as discussing the more traditional philosophical topics of formalism, Platonism, and constructivism. |

PI in the Sky: Counting, Thinking, and Being
by John D. Barrow Where does mathematics come from? What |

Platonism, Naturalism, and Mathematical Knowledge
by James Robert Brown This study addresses a central theme in current philosophy: Platonism vs Naturalism, and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. |

Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning
Lecture by James Robert Brown Do you have to see it to believe it? In this lecture, Jim discusses the highly interesting but controversial topic of the legitimate role of visual thinking in mathematics and science. Examples of picture proofs and thought experiments will be given. An explanation of how they work will be sketched. |

**The Deceptively Complicated Reality of Mathematics (Commentary Excerpt)**

If someone tells you that he’s just discovered that the fundamental laws of physics are wrong, you’ll likely be pretty skeptical. But depending on his reputation, you might be inclined to hear him out for a moment or two, curious to know what spectacular new evidence he’s claimed to have found.

But if he tells you that he’s stumbled upon a new proof that 7+5 no longer equals 12, I’m fairly confident that you won’t even take the time to listen to him for a moment longer, no matter who he might be.

Why the change in attitude?

Because mathematics is, well,* different *from science. It’s not terribly difficult to imagine nature being something else again from the way it happens to be: the proton much heavier, DNA made from three essential nucleotides instead of four, the Earth millions of years younger or older. Not only could things conceivably be quite different from the way that they actually are, we’re forever recognizing just how wrong we were. Indeed, the long march of scientific progress is little more than a series of discoveries forcing us to revise our old ways of thinking, from the causes of tides to why things burn.

Mathematics progresses too, of course. Its rapidly evolving sub-disciplines – from logic to fractals, network topology to optimization – have not only kept pace with physics, chemistry, biology, engineering and economics, but have fundamentally influenced the development of all of them. Meanwhile, the pristine world of “pure mathematics”, with its detached realms of number theory and multi-dimensional geometrical abstractions, plows methodically ahead, regularly opening up new domains of research while steadily clearing up niggling old mysteries.

In 1994, after several centuries of collective mathematical frustration, Princeton mathematician Andrew Wiles managed to prove Pierre de Fermat’s deceptively off-handed claim that no three positive integers a, b and c can satisfy the relation a^{n} + b^{n} = c^{n} for any number n greater than 2. In finally elevating this infamous 17th century conjecture to the status of a theorem, Wiles assured himself an intense 15 minutes of global fame together with a rather more lasting place in the mathematical Pantheon. His proof utilizes a combination of esoteric mathematical constructs such as elliptical curves, modular forms and deformations – none of which, it is worth emphasizing, existed in Fermat’s day.

So mathematics clearly evolves and makes progress. And yet, when a mathematician proves a new theorem or finds important connections between two areas previously thought to be independent of each other, what is she really doing? Well, making a discovery, it seems. But then, where is the “thing”, exactly, that is being discovered? Where was it before the discovery? And how did she “see” it, exactly?

In biology, when a new virus is found, or when we develop an understanding of the core principles as to how proteins are folded, we have uncovered something that nature had been hiding from us all along. Mathematicians, on the other hand, who spend their time in the abstract land of algebraic topology and infinite dimensional spaces, have no such recourse to the natural world to ground their discoveries. So what are they actually doing?

Opinions vary considerably. But most people, when pushed, will concede that mathematics must be some sort of man-made activity, an elaborate form of symbol manipulation according to a set of rules that we have all come to naturally agree upon: a sophisticated, rigorously consistent, abstract game.

University of Toronto philosophy professor James Robert Brown, however, will have none of it. For Brown, our age-old intuitions are quite correct; mathematical objects are every bit as real as their physical counterparts and exist completely independent of us. In other words, mathematicians are just like any other scientist when it comes to pushing the boundaries of their discipline: any new discovery they make simply corresponds to a recognition of what is already there....

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