Customer Reviews

Negative Math: How Mathematical Rules Can Be Positively Bent

5 star:		(4)
4 star:		(2)
3 star:		(0)
2 star:		(1)
1 star:		(1)

 

 

This 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.

The writing is strangely impersonal, though I'm not sure what's weird about it. At least it's easy to understand, no big equations, no calculus. The book has several historical parts, especially on negative numbers, which used to be considered fictitious and "evil" and I had not imagined that there used to be so many controversies and disagreements on them for centuries. The author essentially traces the birth of revolutionary algebras to controversies on the negative sign, just as non-Euclidean geometries came from controversies on the parallel postulate. Euler, D'Alembert, Carnot, De Morgan, Playfair, Hamilton, Frege and a bunch of others show up. There are good passages on Bishop Berkeley defending free-thinking in
mathematics.

Before reading this book, I always read and believed that minus times minus must be positive because of the distributive rule. But this book argues that the distributive rule is no more special than the commutative rule, and that accordingly it too can be restricted or rejected just like the universality of the commutative rule was rejected when the theories of quaternions and vectors were invented. Then the book presents an algebra in which minus times minus is minus, something which I haven't seen before. It's essentially a non-commutative algebra, and I expected that therefore the distributive rule would not hold, but surprisingly it does, albeit in a non-commutative form. A good part of this, I think, is where the author shows that this algebra can trace weird curves that are not given by analytic geometry. The last part of the book seems to criticize physics for restraining itself to mathematical methods that historically were not designed for its purposes. Apparently the author wants us to rush out to develop new "artificial" mathematics; but it's not clear to me what's so bad about the methods already in use. Still he wants to develop physical algebras just like mathematicians have developed physical geometries.

I give it a high rating because again and again I found myself agreeing with ideas that at first seemed ridiculous, such as that -1 is not necessarily less than zero. It has a lot of unusual reasonable ways of looking at the elements of math. Oh, but one more thing, I don't exactly understand what's the point of the spoon.

Consider the number line. It is very symmetrical. Zero is in the middle, with the positive numbers to the right, and the negative numbers to the left. But that symmetry rapidly breaks down under multiplication; if multiplication were symmetrical then multiplying two negative numbers should produce another negative number, but it doesn't. The symmetry breaks down even more when you take square roots. The square root of 4 is plus or minus 2. The square root of -4 isn't even a real number; it is plus or minus 2i. Here is another violation of the symmetry of the number line : 2^2 is 4, but -2^-2 is 1/4. That is weird.

Martinez develops an algebra that restores the symmetry of the number line under multiplication, while simultaneously dispensing with imaginary numbers. All you have to do is change the rule of multiplication so that a negative number times a negative number is still negative. Now the square root of -4 is -2. This also gets rid of the double roots for square numbers. It also makes -2^-2 = -4.

This algebra runs into some problems. For one thing, multiplication is not commutative. This seems odd, but we're already familiar with non commutative operations. Some examples include subtraction, division, and matrix multiplication. Martinez smoothes over this issue and a couple other potential pitfalls. He also shows that you can actually create simpler solutions to some problems in mathematics. Martinez's algebra also does a better job of corresponding to the real world. Thinking of negative numbers as "moving in the other direction" results in an arithmetic that does a better job of applying to the real world.

I would highly recommend this book because I'll never think of mathematics, or numbers, in the same way again. Martinez's experiment really forces you to think about what numbers mean, and what possible real world or geometric interpretation they may have. But having said that, I think his experiment is destined to be a failure.

Here is an example of how Martinez's algebra breaks down. 5 = (10 - 5), so 5 x 5 should be the same as (10 - 5) x (10 - 5). But according to Martinez's algebra, it is 75 (you can work this out - use the sign of the first number only for the inner terms). This experiment also provides insight into how negative numbers might work. We can think of -5 x -5 as ( 0 - 5) x (0 - 5) = 25. But according to Martinez's algebra, it is defined to be -25. [UPDATE: Martinez kindly explains in the comment that the artificial algebra uses a different distribution rule than traditional algebra, a nuance that I missed in this review)

Martinez's algebra also breaks down because you cannot use logarithms as a shorthand for division. Lets pick an easy example to demonstrate the point, dividing 4 by 8. The way you do this is to express them both to a common base and subtract the exponents. So you get log( 2^2 ) - log( 2^3 ) = log( 2^-1 ) = -1. Then you use re-exponentiate to get your answer. In traditional algebra, 2^-1 = 1/2 = 4/8. That is the correct answer. But with Martinez's algebra, 2^-1 = -2, which is the wrong answer.

I should point out at this point that Martinez successfully works out a few kinks that originally appear as though they would doom his system. So it is quite possible that someone with more mathematical maturity could figure out a way around these obstacles. But my instinct is that Martinez's system has run into a dead end. The real lesson I have taken from the experiment is that math isn't easy to bend, but I thoroughly enjoyed the attempt.

This should be compared to Paul Nahin's An Imaginary Tale (1998), and Barry Mazur's Imagining Numbers (2002)--Nahin's book is mentioned in it, but the latter isn't. All three of them deal with imaginary numbers and history, for different reasons, but Negative Math focuses actually on negative numbers. Nahin's book is packed heavy with equations, showing the many uses of complex numbers. Mazur's book is much simpler trying to charm humanists to like math and to explain why imaginary numbers make sense. If what you want is math popularization, get Mazur's book. Martinez's book is the more historical of the three insofar as it uses more primary sources etc. But actually this is not exactly a history book, and unlike a spoon, it's got a sharp edge.

Although both An Imaginary Tale and Negative Math were published by the same press (Princeton) Imaginary Tale claims that "there is nothing at all imaginary about imaginary numbers," while instead Negative Math argues that the historical lesson actually is that imaginary numbers were not discovered but were man-made and that we can even live without them.

Despite appearances this book is not what we call a popularization of math. Most of what it says does not follow other books (or none that I've read). And it's not a usual academic book either- it's well written in an interesting and matter-of-fact way that makes it very accessible. However the writing is also stiff, like the author is choosing Every Single Word, as if he wants to write not with pencil or pen but with a chisel. But okay- Martinez is right in treading with care because this is controversial stuff, math Platonists will HATE this book.

Negative Math begins from the history of mathematicians bickering on negative and impossible numbers to then later show that just as Euclid's geometry is not unique, so too the numerical algebra we learn in school is riddled by conventions. And to its credit fortunately it's not a hand-waving argument, Martinez actually formulates a new algebra in which minus times minus is minus, and comes up with some interesting results. For example, Mazur, a Harvard mathematician, argues in his book that minus times minus is plus because of the distributive rule (as is usual), but meanwhile Martinez does show that the distributive rule also holds in an algebra in which minus times minus makes minus. Still, my first reactions were: I refuse to be impressed. But that all changed in the section on analytic geometry: every curve that can be drawn with the geometry of Descartes can be drawn with the new one, but not vice versa, so traditional analytic geometry is therefore a subset of the Martinez algebra, and that's a big deal.

On the whole, I strongly recommend this book for using history to show how we can actually -make- mathematics, rather than just inherit, swallow or obey it.

I occasionally teach a course in the fundamentals of arithmetic and logic for elementary education majors. One of the concepts that the students find difficult is the rule of the negative sign in multiplication. Two negatives combining to make a positive is an idea that many find hard to comprehend. The main theme of this book is an explanation of the rules of how negative signs interact in multiplication and division.
At first glance that may seem to be far too little material for the creation of a book. While it is true that there are times when the discussion drags and grows repetitive, Martinez has managed to create an interesting book. He covers the historical development of the negative numbers, whose origin was in the areas of finance. As modern bookkeeping practices were being developed, a simple and efficient way to represent money owed was needed. Negative numbers provided a way to do that and in that context are easily understood.
The conceptual difficulties arose when people began developing multiplication problems involving negative numbers. While it is easy to understand negative numbers in the context of the transfers of money, when their use was extended to the general abstract case, people had difficulty grasping what the rules should be. As Martinez so very properly points out, some significant people in the history of mathematics had difficulty using them. Some called them a useful fiction, but as is always the case in mathematics, the useful dominates the fiction and we now easily teach them to children.
One of the strong points of the book is the sections where Martinez points out that the current rules for using negative numbers in multiplication and division are not the only viable set. If one is willing to open their mind to new ideas, then other consistent sets of rules can be created and applied. One of the wonderful things about mathematics is that the axioms and assumptions can be changed and sometimes the results are insignificant. However, other times the change is very insightful and leads to significant discoveries.
This is a very readable and entertaining book about how the rules of mathematics have evolved over time and how they can be modified to lead to interesting new results. While the explanations occasionally bog down in too much detail, it is not a significant weakness. I strongly recommend this book to teachers of the history of mathematics if their interest is in demonstrating to students how mathematical ideas have arisen and evolved over time.

I will leave out information that I feel other reviewers have already mentioned and just concentrate on a few points.

This writing really is wonderful, in my opinion. I think that, as a popular writing it is certainly accessible to the general public, but simplicity can be deceptive. The topics may not involve dense opaque notation and detailed treatment from abstract algebra, but if such approaches were resulted to, there would not have been any room to really deal with the historical, philosophical, and physical implications of new algebras. Often texts of that nature present the subject in its most compact form, which is rather out of step with any true learning process which gains an intuitive understanding first, and then builds up in layers upon the first insights, questioning and reworking as the process continues. I feel that the comment by one reviewer that the author is in error by not discussing the topic in abstract algebra form (that would, coincidentally make the book utterly incomprehensible to the common reader) is symptomatic of what happens when someone becomes very proficient at a subject and completely loses all touch with the reality of the people who don't have her/his expertise. It is not the job of this book to take that approach. That is your job. Take the ideas and fly with them, or reject them; either which way, I think you will come away from it with a deeper perspective. Even mathematicians dealing with hypercomplex algebras could deal with a short retreat to consider these basic ideas anew.

I am thankful to Mr. Martinez for writing this book. I hope it is well received.

I'll put it nicely. The author's project in this book is to motivate then introduce an alternative, non-commutative arithmetic which he feels may have use in some mathematical descriptions of the physical world. He does not, however, present this in a mathematically sophisticated way - for example, there is no mention of groups, rings, or fields, and there is no appendix containing a concise presentation of his ideas.

The book is written as a popularization and it remains at that level throughout. His attempt to motivate his alternative arithmetic involves trying to create cognitive dissonance in the reader: to convince the reader there is something unsatisfactory with the way we calculate with negative and imaginary numbers. He does this by obdurately insisting upon questionable physical interpretations of arithmetical equations, then complaining about their obscurity and pointing at the mathematics as the culprit behind the confusion.

His historical overview is essentially concerned with the persistent interpretation of numbers as representing quantity and the consequent discomfort with or disapproval of negative and imaginary numbers. This makes the book sound palatable, at least, and perhaps even interesting as a simplified look at one facet of the development and evolution of algebra. The historical material, we learn from the acknowledgements page, appeared in his dissertation.

The problem any reader will have is that to get to the historical material on a straight read through the book, there is a bog of muddleheaded obfuscation to slog through first. So little respect is left for the author upon reaching the historical section that any stylistic flaw, failure of clarity, or move into superficial expostition increases the exasperation and gives rise to distrust.

I checked this one out from the library on a whim, along with Kenneth Rosen's Elementary Number Theory (5th Edition) and John Watkin's Topics in Commutative Ring Theory, both of which I value and recommend. The book under present review has no value next to them.

the author claims that there's no device which measures an imaginary number; well, for EE's, an imaginary number (phasor) can be measured with an oscilloscope since oscilloscopes can measure amplitude and phase; this representation can then be converted to an imaginary number. the author also claims that you can't have -5 apples in a box and this seems sinister since -5, for example, would mean that there's an IOU in the box after the vender had 5 apples and bartered 10 apples for something he wanted. in this case, when the vender got a dozen more apples, 5 of them would be owed. perhaps I should read more than amazon's sample pages but the ones I read didn't seem that complete.