Tagged: mathematics

How to Solve It

how to solve it polya princeton science library 2004How to Solve It: A New Aspect of Mathematical Method by G. Polya, 3/5

This ambitious book tackles the fascinating topic of heuristics (practical problem-solving techniques) by focusing on a variety of naturally-occurring questions that can lead to solutions and discoveries in mathematics and other fields.  Using mathematical examples that I found challenging and somewhat inaccessible despite their stated simplicity, Polya demonstrates how questions like “What is the unknown?” “Do you know a related problem?” and “Did you use all the data?” can guide a potential problem-solver toward common-sense solutions even to problems that might seem dauntingly complicated at first.  Unfortunately, the book is both very dry and very confusingly organized–I never quite understood the layout and cross-references.  However, it is still a good resource on a surprisingly little-addressed topic.

Confession: I didn’t even attempt to complete the problems at the back of the book–even if I was smart enough to do them, I’ve forgotten most of the math I ever learned and my main reading time is right before falling asleep, which is not really conducive to mental acuity.

Why I read it: it was mentioned in The Organized Mind.

A picture quote I made:

A picture quote from How to Solve It by G. Polya. "No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all." Background image is of the "MegaZapper" Tesla Coil at the Spark Museum of Electrical Invention (Bellingham, WA).


Fermat’s Last Theorem

fermats last theorem aczel four walls eight windows 1996Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem by Amir D. Aczel, 2/5

In around 1637, Fermat hinted that he had “discovered a truly marvelous proof” that an + bn = cn cannot be true for n>2.  Proving this deceptively simple theory required the contributions of dozens of mathematicians over a span of some 350 years.  Unfortunately, while biographical aspects of the story are competently told, the author is unable or unwilling to explain important mathematical concepts in layman’s terms.  If the following paragraph makes sense to you, then you probably fit the target audience of this little book:

Here, a periodic function could be conceived as having a periodicity both along the real axis and along the imaginary axis.  Poincaré went even further and posited the existence of functions with a wider array of symmetries.  These were functions that remained unchanged when the complex variable z was changed according to f(z)——>f(az+b/cz+d)Here the elements a, b, c, d, arranged as a matrix, formed an algebraic group.  This means that there are infinitely many possible variations.  They all commute with each other and the function f is invariant under this group of transformations.  Poincaré called such weird functions automorphic forms (82).

If, like me, you were completely nonplussed by that description, then the only thing of value you will likely get from this book is a deeper understanding of the fact that concise ≠ comprehensible.

[Why I read it: I came across it while sorting through some of my Dad’s books.]