A Mathematician Comes of Age by Steven G. Krantz

By Steven G. Krantz

This ebook is ready the idea that of mathematical adulthood. Mathematical adulthood is valuable to a arithmetic schooling. The aim of a arithmetic schooling is to rework the coed from anyone who treats mathematical rules empirically and intuitively to anyone who treats mathematical rules analytically and will keep watch over and control them effectively.

Put extra without delay, a mathematically mature individual is one that can learn, study, and evaluation proofs. And, most importantly, he/she is one that can create proofs. For this can be what glossy arithmetic is all approximately: bobbing up with new principles and validating them with proofs.

The ebook presents history, information, and research for figuring out the concept that of mathematical adulthood. It turns the belief of mathematical adulthood from a subject for coffee-room dialog to a subject for research and severe consideration.

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The tools of mathematics are abstraction, symbolic representation, and symbolic manipulation. However, being trained in the use of these tools no more means that one thinks mathematically than knowing how to use shop tools makes one a craftsman. Learning to think mathematically means (a) developing a mathematical point of view—valuing the processes of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure— mathematical sense-making.

This is intuition. Likewise, they can sense relationships between ideas without knowing exactly what the relationships are. It is important to learn to listen to one’s intuitive perceptions. This is how deep ideas are developed. (5) Stimulus-Response: This is learning to develop knee-jerk reactions to certain types of questions. If I ask you to calculate the eigenvalues of a matrix, you respond with a learned drill. If I ask you to find the tangent hyperplane to a surface in space, you trot out a standard procedure.

So Kepler was carrying out a very eclectic and specialized project. What is interesting here, from our point of view, is that Tycho Brahe’s work exhibits no mathematical maturity. He was a gatherer of data. He worked long hard hours, and gathered information that nobody had ever had before, but he exhibited no insight into the nature of mathematics. Kepler, by contrast, showed real sophistication in the way that he took the reams of raw data that Brahe’s work provided and turned them into concrete mathe9 An interesting historical note is that, instead of logarithms, Kapler used the theory of prosthaphairesis.

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