How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

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A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics.

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Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics.

How Mathematicians Think. Using Ambiguity, Contradiction, and Paradox to Create Mathematics

Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions.

Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself. Ach, I really wanted to give this five stars. Byers does a great job of showing how ambiguity and paradox are at the core of what mathematics is about. Of course it is also a paradox that mathematics is paradoxical, because mathematics is the prime example of a discipline where paradox has been banished or at least securely caged.

Byers discusses briefly how this paradoxical nature of mathematics is important for science and culture at large. But in the end his conclusion falls a bit flat. He sees computers and software and algorithms as being stuck on one pole of the paradox and therefore essentially trivial. I must say, he triggered one of my pet peeves.

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On page he says: "The theory of chaos arises from the study of nonlinearity. Complexity is fundamentally nonlinear. If mathematics is non-linear, then its essence cannot be captured by algorithmic procedures or by the linear strings of reasoning that characterize both mathematical proofs and deductive systems.

How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

Through most of the book he has been quite careful to be clear and avoid confusing concepts. But clearly the term "linear" above is used in two very different senses. Deductive systems don't look like vector spaces very much at all! The crazy thing is, the whole business of complexity theory and chaos, this arose because of computers. It is just too much work to try to simulate those differential equations, to compute solutions for a variety of parameter values.

Here is a huge question that Byers just avoided. It is very nice to say that the human mind is not a deductive system. Sure, there is a school of cognitive science that would like to model minds as computers. I'm not sure whether very many folks in that field work from that premise any more. But, it seems pretty clear that the world of physics can be modeled quite nicely with differential equations.

It seems quite reasonable, in principle, to simulate a human being, i. OK, the computer would probably require cosmic-scale resources to pull this off. But there was recently some huge simulation of a decent sized chunk of a cat's brain, and it did simulate some interesting behavior. This is not modeling the human mind as a deductive system, this is modeling brain behavior as a biochemical system. This whole area is vast and deep. I think Byers is making a valuable contribution to the philosophy of mathematics.

But when he discusses computer science and cognitive science, he falls short. Both of these research areas are much more fertile than he seems to imagine.

How Mathematicians Think | Βιβλία Public

What would be much more fun is to extend his notions of the fundamental roles of ambiguity and paradox to those disciplines, to show how the internal conflicts in those disciplines are actually fertile, rather than flaws. For example, in computer science, contrast the view of Edsgar Dijkstra, that it is a mistake for students of computer science to run their programs on computers, with the common practice of agile development. Maybe computer science should be totally separated from software development?

That is really a beautiful paradox! Of course the whole mind-body distinction is an ancient paradox. Byers seems to be landing on one pole, mind is not body. Even life is not body. He seems to be proposing some kind of vital essence or soul.

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Wait a minute, Byers acknowledges, on p. Buddhism is practically founded on, hmmm, not exactly the non-existence of the soul, but the paradoxical nature of that issue. What if the paradoxical nature of the mind actually points to a paradoxical nature of the body? Does that mean that, after all, we can't really simulate the physical world? Playing with Byers's idea of objective subjectivity The idea that depth is associated with paradox, this is really nice.

I was just disappointed that he couldn't maintain that depth but ended up driven to resolution and hence landing in the shallows, just where the real fun could have begun. Finally after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of-and couldn't exist without-the many months of stumbling around in the dark that precede them.

This is the way it is! This is what it means to do mathematics at the highest level, yet when people talk about mathematics, the elements that make up Wiles's description are missing. Immediate Certainty Take the following famous quote from the great French mathematician Henri Poincare. Poincare had been working on proving the existence of a class of functions that he later named Fuchsian: Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines.

The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations that I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience's sake I verified the result at my leisure.

Poincare goes on to say: Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness, and immediate certainty, that the arithmetic transformations of inde- terminate ternary quadratic forms were identical with those of non-Euclidean geometry. Poincare's "immediate certainty" is an essential but often neglected component of mathematical truth.

Truth in mathematics and the certainty that arises when that truth is made manifest are not two separate phenomena; they are inseparable from one another-different aspects of the same underlying phenomenon. It all depends-on what you call a polyhedron Certainty arises in the mind, and one definition of "subjective" is "existing only in the mind and not independent of it.