Ye Olde Natural Philosophy Discussion Group

Reviews and comments on
Joseph Mazur:
Euclid in the Rainforest:
Discovering Universal Truth in Logic and Math

      Our group really didn’t like this book, with one notorious exception (Scott). Our average rating (on a scale of 0 to 10) was only 2.3. In fact, the reaction was so negative that before we actually got together to discuss it we picked another book as our primary topic. We therefore didn’t really have a very thorough group discussion of why people didn’t like this book.

      John, however, sent around his personal summation of Mazur’s book via email:

My quick review of the book, Euclid in the Rainforest, was that I didn’t like it very much. It wasn’t so much the subject matter per se, rather it was how it was presented. Mazur’s method of presenting a little story and then talking about the actual point he was trying to present about math/logic just didn’t work for me. I didn’t feel that his cute little anecdotal stories related very well so I became frustrated with the book and actually stopped reading it about 4 chapters in. Also when he was actually talking about the subject matter, I started to feel like I was in the 7th grade or something, so it put me in an emotional funk too. So I am rating this book a one (1). I think this is a subject I should actually like, but I guess I just need it presented somewhat differently for me to enjoy it.

      Rosie also commented via email that “I, too, disliked the book. (Someone’s revenge for the grammar books we did before?) Tho it got wonderful reviews on Amazon. I give it a 2.” Kevin then added the remark that “Frankly I think the author is still wondering lost in the woods.”

      Scott, however, had a rather different take on Euclid in the Rainforest. In his view the arguments against this book were primarily that 1) too much of it was off topic and not about mathematics, and 2) too much of it was about mathematics and not about something else! Clearly Mazur has learned in his years of teaching math that for many people with math phobia it is useful to edge into the subject with stories and anecdotes; in other words, to sneak up on his audience! Scott didn’t really mind Mazur doing this and even thought many of the diversions were entertaining, though he was more interested in the math/logic itself—or rather, in the partly explicit, partly implicit, philosophy of mathematics and logic that lay behind it all. Scott comments further:

      I view the central theme of this book as being neither jungle travelogues nor even mathematics, but the philosophical question of what mathematical proof consists of, the essential nature of mathematical/logical truth itself and how people come to recognize it. I thought a lot about this sort of thing back in my university days (ancient times!) so this was right up my alley!

      I very much appreciated many of Mazur’s comments, such as that while mathematics enjoys a reputation for being logically perfect, in reality the truths of mathematics “are not communicated through airtight chains of logical arguments” (p. x), and his statement that “the whole notion of proof in mathematics has never been clearly defined” (p. 38). There are in fact a fair number of quite profound statements in Mazur’s book that a casual reader might not even notice, such as that “We become so accustomed to what we believe that we cannot believe otherwise” (p. 49) and “The mind has a big task: to interpret perceptions of the world so that it can live without contradiction” (p. 44). I like books that say things like this!

      On the other hand, there are quite a lot of things in Mazur’s book that I strongly disagreed with, some of which really made me cringe! On p. 61 he seems to suggest that mathematicians must either be Platonists or constructivists, but I reject both views. (One can accept the law of the excluded middle without being a Platonist!)

      Much of Mazur’s book is amenable to a materialist interpretation, but he is far from being consistent on this. On p. 83, while discussing non-Euclidean geometry, he remarks that instead of trying to prove Euclid’s fifth postulate “we should have been celebrating the liberation of mathematics from the physical world”. The fact remains, however, that mathematics has been mostly created by abstracting from relationships of things in the real world, and that even non-Euclidean geometries can be interpreted that way (as the surface geometry of spheres or saddles). Yes, once we have abstracted a world of mathematical objects we can sometimes manipulate them in ways divorced from actual reality. But the connection of mathematics to the real world is still primary and is of deep significance. We should perhaps be more inclined to lament “the liberation of mathematics from the physical world” than to “celebrate it”!

      Mazur has the irritating practice of referring to things which are merely internally consistent as “true” (cf. p. 85). This is a Platonist vice. Poincaré was indeed correct to say “One geometry cannot be more true than another; it can only be more convenient.” But this should be taken to mean that it is only correct to call any geometry (or any other branch of mathematics) “true” only insofar as it is applicable to describing some aspect of the real world. Thus even 1 + 1 = 2 is not always true when it comes to describing the physical world. (E.g., one liter of water plus 1 liter of ethanol makes slightly less than 2 liters of water-alcohol mixture—because the molecules somewhat fit between each other.)

      On pages 66-69, in the very interesting discussion of whether “Behold!” demonstrations in mathematics can actually constitute “proofs”, I thought Mazur went somewhat astray. My view is that what constitutes an adequate “proof” depends on both the evidence or arguments presented and on how well prepared the person is to understand that evidence and those arguments. For me the following illustration (from p. 66) does prove the theorem that the sum of any list of sequential odd numbers starting with 1 must be a square number, and it proves it at least as well as any verbal or algebraic argument could!:

A example “Behold!” proof.

      I grant, however, that there are reasons to favor an algebraic proof here, and that is the accepted standard for a proof of this kind of thing within the mathematical community. As Mazur appropriately remarks in other places, mathematicians have their own accepted standards of proof which are not, however, entirely precise and completely logical demonstrations, but rather merely enough in that direction so that most mathematicians in that particular field will recognize it as sufficient. Moreover, the typical procedure is for mathematicians to hide how they actually came to understand and prove the conclusion to their own satisfaction, by making the published “proof” more logical and complete than their real method of arriving at it! As another mathematician once wrote, “Our paper became a monograph. When we had completed the details, we rewrote everything so that no one could tell how we came upon our ideas or why. This is the standard in mathematics.”1

      I would concede that sometimes “Behold!” proofs are sufficient to convince someone, and sometimes they aren’t, and it may vary with the specific person. And in many cases they perhaps only suggest that something is true.

A “Behold!” proof that the sum of all fractions of the form
(1/2)n (where n is a positive integer) equals one.

      However Mazur goes on to argue (pp. 67-68) that “If anyone believes that the last picture proves that the sum of the fractions (1/2)n equals 1, then I have a picture that proves that 1 = 2.” But to my mind his sequence of pictures on p. 69 proves no such thing!2 A lot depends upon how one is prepared to interpret such “Behold!” illustrations. And a prepared mind is always the key to the correct understanding of anything. Moreover, it is not just the “Behold!” sort of “proofs” that can turn out to be fallacious; this can also happen with algebraic or any other form of mathematical proof.3

      There were many things of interest in the book, such as the young boy Evan’s own explanation of how he solved the bullseye puzzle (pp. 98-100). But I thought that there were weaknesses in the discussion of infinity and probability, such as too much acquiescence toward the false idea that infinity is inherently mystifying. The book often seemed to be on the right track philosophically, but then veered off in one way or another. This is why, even though I enjoyed the book, I only give it a 7 instead of a 10.

      The book could have benefitted from another round of proofreading. The worst error I spotted was on p. 296 where the formula for the sum of the first n integers is given as ((n+1)+1)/2 when in fact it is (n(n+1))/2.

1   David Berlinski, Black Mischief (1988).

2   In his series of diagrams on p. 68, at any specific iteration the combined lengths of the non-verticle two sides of all the triangles is always exactly equal to 2. The “limit” is therefore also = 2. Furthermore, the column of triangles—no matter how microscopic they get—never actually form a geometric line. But it is true that his series of illustrations might be misleading to some and might falsely suggest that the limit is 1 or even 0, or that they do “eventually” become a single straight line. But when Mazur says that his “students should have accepted my picture proof that 2 = 1” he is talking nonsense! For example it might depend in part on whether their intuition of such pictorial diagrams was already informed by the concept of limits.

3   For reference here is an old algebraic “proof” that 1 = 0, that you probably learned in high school:

1. Assume x = 1  

2. Then x2 = x [Multiply both sides by x]

3. Then x2 -1 = x - 1 [Subtract 1 from both sides]

4. Then (x + 1)(x - 1) = x - 1 [Factor the left side]

5. Then (x + 1) = 1 [Divide both sides by (x - 1)]

6. Then x = 0 [Subtract 1 from both sides]

7. But since we assumed that x = 1, then 1 must equal 0!

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