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Negative Math: How Mathematical Rules Can Be Positively Bent ReviewThis book was quite hard to review. There are parts I found extremely interesting, and other parts I thought were full of sloppy thinking and misleading analogies. But overall, I think it is a worthwhile book to read. I think it is appropriate to divide the book into four separate sections, each of which deserves to be reviewed separately.The first part is an attempt to show that some of the rules of algebra (particularly the rules for manipulating signs) are really counter-intuitive, and also an attempt to gain the perspective of an elementary algebra student who cannot understand why the rules are what they are. It is this part that I think is the worst part of the book, and in his attempts to show that the rules are counter-intuitive, all he manages to do is show that _his_ intuition works quite differently from _my_ intuition. This part is the section in which I found the sloppy thinking and resort to false analogy which I mentioned earlier. It seemed to me that there were things the author just didn't understand, but as I read further in the book, I found that he actually understood them even though he didn't seem to at first. This section would get three stars if I felt generous, or even two, if I were to review it alone.
The second part (actually intermixed with the first in its location in the book) describes the difficulties that mathematicians (even great ones) had in comprehending the concept of negative and imaginary numbers, and as such it provides some historical background for the rest of the book, which justifies its inclusion. If I were to review this part by itself, it would get three stars, meaning "it's OK," but it hardly justifies the book.
The best part of the book is the third. This is a very interesting attempt to come up with an algebra that differs from the usual, where he has to maintain consistency, and so he looks deeply into questions as to what further modifications to traditional algebra have to be made to go along with a postulated change. Much like the introduction of non-Euclidean geometry, the process leads to an odd-looking algebra, but one which fits together, and it is this part that I liked well enough to rate as five-star, bringing the overall rating for the book to four. This part made the book worthwhile for me.
Finally, the author ends with one very long chapter that probably summarizes what _he_ wants the book to be, though the previous section is what _I_ want of the book. He advocates a concept of a mathematics that would be suited to explaining problems of physics in a more natural manner, even if it might look different from traditional mathematics. I would have been happier if this part were shorter, though I think the author himself probably considered this part to be the major thesis of the book and this is why he devoted so much of the book to this part. This part is actually interesting enough that I'd rate it 4 stars, though as I said I'd prefer it more streamlined.
That is an overview of the book: very uneven, with both very good parts and bad, but if everything is all combined, the total package is a pretty good one.Negative Math: How Mathematical Rules Can Be Positively Bent Overview
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