Joseph Mazur's book *Euclid in the Rainforest* is an earnest, rambling rehash of conventional topics in popular mathematics, spiced with disjointed tales of mathematical conversations in exotic places.

The book is divided into three parts. The first part, "Logic", treats the logic used in Euclid's *Elements* and the logic formalized by Aristotle (taken to be the same). According to Mazur, this kind of logic is unable to deal with infinity. The second part, "Infinity", discusses such things as the nature of irrational numbers and Cantor's distinction between countable and uncountable sets. The third part, "Reality", is supposed to be about "plausible reasoning", but in fact is mostly concerned with probability theory.

In the first part, the young Mazur travels to Venezuela and throws in with an Englishman named Roger. Together, they make their way to the Orinoco river, along the way meeting a young soldier named Jesus. As they go, they discuss the proof of the Pythagorean theorem and other results of Euclidean geometry. Then, suddenly, the author drops this story and begins to write about Camille Jordan and Lewis Carroll. We never find out what ultimately happened with Roger and Jesus.

In the second part, Mazur is travelling in the Aegean, where he meets a Norwegian named Carl and a Swede named Fredericka. Unlike the other characters in the book, Fredericka seems to have no interest in mathematics, but likes to swim in the nude. Mazur and Carl spend their time working on prime factorization. Later, Mazur slips into Turkey and discusses Zeno's paradoxes with the locals.

In the third part, Mazur visits a classroom at Columbia University, where he encounters a professor who looks like Lenin, or maybe Jacques Hadamard, and a young student from Harlem named Uriah. The professor, who mysteriously turns up everywhere Mazur and Uriah go, instructs them in the ideas of probability theory. For example (p. 192), the ideal, mathematical die is "entirely predictable" and "will fall with three dots facing up every six rolls."

It seems that Mazur's main objective is to ask in what way, if at all, logical arguments give us certain knowledge, and to point out — what no one has ever denied, as far as I know — that the conclusions of mathematics don't always hold in what he (along with many others, of course) refers to as the "real world". On the other hand, both Euclidean geometry and probability theory, if not Cantor's set theory, make predictions that are often strikingly accurate when applied to the world of our sense-experience. How can this be explained? Mazur raises this question (for example, on p. 223), but does not propose any answer to it.

"Mathematics enjoys a reputation for being an intellectual pursuit that generates universal *truths*. But contrary to what many of us think, those truths are not communicated through airtight chains of logical argument. The essence of proof contains something more than just pure logic, just as music is more than just musical notes" (p. x). Mixed metaphor aside, I'm not sure who it is that Mazur is arguing against here; the point he makes is hardly new. G. H. Hardy said, "If we were to push it to its extreme we should be led to a rather paradoxical conclusion; that there is, strictly, no such thing as mathematical proof; that we can, in the last analysis, do nothing but *point*; that proofs are what Littlewood and I call *gas*, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils … it is only the very unsophisticated outsider who imagines that mathematicians make discoveries by turning the handle of some miraculous machine" ("Mathematical Proof", p. 18).

Mazur goes on to claim (p. xii) that Cavalieri got correct results by using *illogical* arguments. In fact, we know today that Cavalieri's Principle is correct; we can prove it rigorously. That Cavalieri gave an inadequate justification of his principle does not make it *illogical*.

According to Mazur, "We develop powers of deduction through our experiences with cause and effect" (p. 250). I don't believe that "cause and effect" is something that we experience. Rather, we (or at least some people) *impose* the notion of "cause and effect" on our experience, as a way of structuring or clarifying it. The American mathematician Clifford Truesdell (1919–2000) has written, "Indeed, 'cause' is only a verbal crutch for beginners who do not grasp the concept of function in mathematics" (*Essays in the History of Mechanics*, p. 181).

Mazur repeatedly asserts that the axioms of mathematics, the starting points of our deductions, ought to be "self-evident". For example (p. 64), "Indeed, without some self-evident assumptions, we cannot have any notion of proof." This idea is apparently due to Aristotle. According to Aristotle (*Posterior Analytics*, I.2), scientific demonstrations must start from premisses which are *true*. But how can we know that they are true, if they are the starting point (rather than the conclusion) of our reasoning? Aristotle's answer (71^{b}21) is that they must be "immediate" (Greek *amesos*). Mazur (p. 159) points to Aristotle's doctrine that we get "immediate" truths by *induction* from sense-experience (*Prior Analytics*, II.23; *Posterior Analytics*, II.19). However this may work in Aristotle's philosophy, it is hard to see how it can apply to mathematics. How can we induce through sense-perception that all right angles are equal?

Mazur's stance here is odd, especially since he devotes a chapter to non-Euclidean geometry. The development of non-Euclidean geometry makes it plain that Euclid's fifth postulate, at least, cannot be "self-evident", since we have no idea whether it is even true. We still do not know whether the large-scale geometry of space is Euclidean or non-Euclidean, as Mazur remarks on p. 85. Thus, geometry, at least, cannot be based on "self-evident" truths. In fact, I do not believe that there is such a thing as a "self-evident truth"; certainly I have never encountered one.

In the section on logic, Mazur devotes a great deal of space to Aristotle's syllogisms, leaving the impression that Aristotle's logic is the logic used by Euclid. "If we accept them unconditionally [Euclid's Postulates 1 and 3], then, like the syllogism that claims Socrates mortal, these two postulates claim the truth of Euclid's first proposition. An equilateral triangle can be constructed on a finite straight line AB" (p. 64). Actually, even in the proof of this first proposition Euclid appeals to the principle that things equal to the same thing are equal to one another, a principle which does not fit into Aristotle's syllogistic.

Mazur states (p. xii) that "logical reasoning could not address the weirdness of infinity". Yet Euclid is able, in Book V of the *Elements*, to develop a theory of "magnitudes", which is basically equivalent to our modern theory of real numbers. His definition of magnitudes having the same ratio (Definition 5), a condition that has to be checked again and again in the proofs in Book V, involves universal quantifiers ranging over all the positive integers. What, then, is the "different sort of logic" (p. 86) that Mazur asserts is needed to deal with infinity? It turns out to be Cantor's set theory (p. xiii). Analogously, I suppose that geometry could be called the "logic of space", and calculus the "logic of motion".

According to Mazur, still another kind of logic is needed in order to deal with the "real world": "We accept a new form of logic when we presume that the practice of the real world behaves as the idealized theory of the mathematical world" (p. 192). Mazur seems to identify this new kind of logic with probability and statistical reasoning, even though it would seem that geometry, too, applies to the external world.

Mazur repeats the old saw that "deductive reasoning is not appropriate in investigations of the material world" (p. xiv). Mazur attributes this view to Francis Bacon; actually, it, too, goes back to Aristotle: "The minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Therefore its method is not that of natural science; for presumably all nature has matter" (*Metaphysics*, II.3, 995^{a}15–17). In a similar vein, the physicist Richard Feynman remarked, "One may be dissatisfied with the approximate view of nature that physics tries to obtain (the attempt is always to increase the accuracy of the approximation), and may prefer a mathematical definition; but mathematical definitions can never work in the real world. A mathematical definition will be good for mathematics, in which all the logic can be followed out completely, but the physical world is complex, as we have indicated in a number of examples, such as those of the ocean waves and a glass of wine" (*Lectures on Physics*, vol. I, p. 12-2). I have sometimes called this claim the "Aristotle-Feynman thesis".

On the contrary, Truesdell has written, "All too often is heard the plea that since the theory itself is only approximate, the mathematics need be no better. In truth the opposite follows … A result strictly derived serves as a *test of the model*; a false result proves nothing but the failure of the theorist … *In physical theory, mathematical rigor is of the essence*" (C. Truesdell and R. Toupin, "The Classical Field Theories", p. 231). In order to test a scientific theory by comparing its predictions with observation, we must know what its predictions are. The only way to obtain them is by means of rigorous mathematical proof. Otherwise, if we find a discrepancy between theory and observation, we will not be able to tell whether it is the theory itself which is at fault, or just an invalid derivation. (See also C. Truesdell and R. G. Muncaster, *Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas*, p. xviii.)

It seems to me that both Euclidean geometry and probability theory — along with other branches of mathematics, such as arithmetic and mechanics — have been developed in order to provide (as Truesdell suggests) *abstract models* for some part of our experience. I do not find it particularly surprising, though of course it is gratifying, that they agree so well with our experience, since the fundamental assumptions that are built into them were extracted from that experience in the first place — not "self-evident truths", but rather plausible hypotheses, selected to form the foundation of a theory giving an adequate description of some restricted class of phenomena. Indeed, if we try to extrapolate outside the range of experience on which the hypotheses are based, the theory may not be so successful. Thus, plane Euclidean geometry is not accurate if applied to the surface of the Earth in the large.

Mathematical logic, as developed since the middle of the 19th century, is also an abstract model, in this case a model for mathematical reasoning. Mathematical proofs are not justified by their adherence to the rules of formal logic. We consider them to be valid because they convince us that their conclusions follow from their assumptions. The rules of logic are just a description of what we already do. If mathematicians were to find a new kind of convincing argument, not covered by the rules of formal logic, those rules would simply be revised.

Statistical inference, of course, reaches conclusions that cannot be justified by deductive reasoning. In plain language, these "inferences" are nothing more than reasonable *guesses* based on a given set of data. (Statisticians call them "estimates".) There is a definite, known, chance that the data will lead us to make an incorrect guess. Possibly this is what Mazur is referring to when (p. 253) he writes, "Many questions in science are known only to 95 percent certainty." (Presumably, he means the *answers* to the questions, not the questions themselves.) I can only surmise that what he has in mind here is the common practice of using a 5% significance level for testing statistical hypotheses. The phrase "95 percent certainty" suggests that there is a 95% probability that the result is correct. But, as every student of elementary statistics is cautioned, 95% is *not* the probability that the "alternate hypothesis" is true. Indeed, in conventional statistics, no probability can be assigned to this condition, since it is not conceived as being *random*. (Although Mazur mentions Thomas Bayes on p. xv, he never takes up the subject of Bayesian inference.)

Mazur's book is written with enthusiasm but not with much care. Much of it reads like a first draft — some sections (such as pp. 224*f.*) like a mere outline, or a list of topics for discussion.

Mazur consistently misspells the name of the philosopher Carl Hempel (*e.g.*, p. 165). He also misspells the names of Girolamo Cardano (p. 217), Mark Kac (p. 240), and political commentator and mayoral candidate William F. Buckley (p. 196). We find "compliments" instead of "complements" (p. 96), "motes" instead of "moats" (p. 217), "decent" instead of "descent" (p. 242). On p. 78, we read that Euclid defined a line as "breathless" length. (This spelling perhaps reflects a New Yorker's pronunciation.)

When the Sumerians first entered the historical record, they lived in southern Mesopotamia, not in the "mountains of ancient Iran" (p. 91). Phoenicia, on the Mediterranean coast, was not "a huge area that included both Persia and the western part of India" (p. 131). The mathematician Pappus of Alexandria lived in the fourth century A.D., not "around 300 B.C." (p. 13). The Homeric passage quoted on p. 215 is from the *Iliad* (Book XV, lines 187–192), not the *Odyssey*.

There are surprisingly many mathematical misstatements. "[T]he factors of any number ending in 7 must end in either 3 or 7" (p. 42 — what about 77 = 7 × 11?) "For a conclusion to be true, it must come from a valid deduction with true premises" (p. 53). A geometry containing two non-similar triangles must be Euclidean (p. 84). If you take the product of the first k primes and add 1, the result must be prime (p. 111 — smallest counterexample, 2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509). "But if there is a natural number n for which the formula is invalid, it can never be discovered by a sequential search" (p. 166). "But there is a strange law called *The Law of Large Numbers* that tells us that if we flip a coin long enough, half the time it will come up heads and half the time it will come up tails" (p. 219). On p.176, Mazur states that the cardinalities of the iterated power sets of the set of integers are given by the sequence of alephs — even though he states on the same page that the continuum hypothesis is undecidable. Again on the same page, he seems to say that the continuum hypothesis is the hypothesis that there is no cardinal number between aleph-one [*sic*!] and the power of the continuum.

A footnote on p. 290 tells us, "This was how Archimedes calculated that π is between 2130/71 and 21/7" — that is, between 30 (!) and 3. This fact, though true mathematically, hardly requires the genius of Archimedes.

The "apocryphal story" on p. 43 is a garbled version of the true story of George Dantzig, inventor of the simplex algorithm for linear programming. During his first year as a graduate student at Berkeley, he arrived late to one of Jerzy Neyman's classes. Two problems were written on the blackboard. Supposing them to be homework problems, Dantzig took them home and solved them. They turned out to be two important unsolved problems in statistics. Dantzig's solutions became his Ph.D. thesis.

Good writing amounts to choosing in every case the one word that best expresses the writer's meaning. By this standard, Mazur is not a good writer. He frequently writes sentences which clearly do not say what he intends them to mean. For example (p. 220), "So how can we say The Law of Large Numbers is a law if it doesn't seem to obey any predictable behavior?" This is cock-eyed. Laws do not obey behavior; rather, behavior, in some circumstances, is supposed to obey laws.

Here are some further examples, which I leave as exercises for the reader to deconstruct: "Many theorems accepted and used in mainstream mathematics have proofs that hardly conform to any rigorous definition of proof" (p. x). "[R]igor would inevitably challenge any argument not based on strict rules of proof" (p. 14). "I felt that that was the answer, and yet, as I thought more about it, that answer seemed to devilishly slip away into that foggy, ungraspable place beyond conceptually finite margins" (p. 125). "In geometry, there is no sharp edge to the finite world" (p. 126). "Normally, existence is limited to objects that take up space" (p. 147). "Two schools of convention have opposing views on this last proof" (p. 162). "Of course, this set [*i.e.*, the set of real numbers] can also be represented by the decimal representation of a number" (p. 174). "In the spring of 1963, the answer was put to rest" (p. 178). "Though real numbers were used for centuries, it was the first time its illusive spirit was captured in a language that defined it" (p. 181). (Did Mazur mean "elusive"?) "Most convicted felons are guilty of the crimes they committed" (p. 234). "Every event in nature has to account for a vast number of indeterminate possibilities" (p. 242).

In mathematical writing, of course, lack of precision can be disastrous. "Mathematicians say that a set is countable if there is some correspondence that associates one and only one integer with one and only one member of the set" (p. 174). Well, then, the set of real numbers is countable: just associate one and only one integer, say 5, with one and only one real number, say π.

On p. 176, Mazur, without saying what the continuum hypothesis is, begins to discuss Cantor's attempts to prove it. "His attempts would alternate between believing that the continuum hypothesis was true and believing it false. Day by day, his opinions would change, but in the end, he hypothesized that it was true. This hypothesis became eminently known as the *continuum hypothesis*." In other words, the "continuum hypothesis" is the hypothesis that the continuum hypothesis is true. Also, it became known as the continuum hypothesis "eminently". I guess that means that it became known as the continuum hypothesis much more than any other hypothesis did.

On p. 181, Kronecker's remark that "God created the integers; all else is the work of man" is called "infamous" — like the attack on Pearl Harbor. But in the next sentence the same remark is called "distinguished".

Mazur has a tendency to launch into passages of irrelevant literary description: "The winter of 1654 was unusually cold for Paris. Even the Seine froze. Hundreds of people skated and slid on the river while fires burned on street corners and parish priests distributed bread to the poor. Those who had money suppressed their spending to provide food and warmth for the poor people who could not find work that difficult winter…" (p. 216). The temperature of the winter that year seems to have little to do with the 1654 correspondence between Fermat and Pascal on problems of probability theory, which this passage is leading up to. Mazur seems here to betray some embarrassment about his subject, as though he feared that, if he were to write only about mathematics, he would not be able to hold the attention of his reader.

To what kind of reader is the book addressed? There are footnotes that explain that a right angle is an angle of 90 degrees (p. 281) and that the square root of 23 is a number whose square is 23 (p. 291). On p. 149 (after several pages of discussion of the fact that the diagonal of a square is incommensurable with the side), Mazur explains that the diagonal is the line joining two opposite corners. Yet he takes only two pages to describe Poincaré's disk model for hyperbolic geometry (80–81) or the sequence of transfinite cardinals (175–176). I can't imagine that a reader who needs to be told what a right angle is would be able to understand these passages.

On p. 175, after remarking that the set of rational numbers is countable while the set of real numbers is uncountable, Mazur interjects, "Take a minute to think about how counterintuitive these arguments are." He is fishing for a "Gee whiz" response from the reader, and no doubt would be disappointed not to get one. But the statements about the cardinalities of the reals and the rationals are true. If, in fact, they are counter to our intuition, then it is our intuition that is off. Where does that intuition come from? It seems to me that, instead of emphasizing the allegedly counterintuitive nature of these results, we ought to be developing the reader's intuition in such a way that they would seem completely inevitable.

The book ends with a very good annotated reading list.

Pi Press, formed in 2003, is a subdivision of the publishing giant Pearson. According to its Executive Editor Stephen Morrow (in a statement on the company's web-site), "We are building books that will arrest the attention of the general science reader with great writing, authority and all the beauty of science." So evidently the twenty-first-century terminology is that books are "built", not written or edited. It also seems rather ironic to emphasize "authority" in a series ostensibly devoted to science.

**References:**

G. H. Hardy, "Mathematical Proof", *Mind*, **33** (1929), pp. 1–25.

Clifford Truesdell, *Essays in the History of Mechanics*, Springer-Verlag, 1968. C. Truesdell and R. A. Toupin, "The Classical Field Theories", in S. Flügge, ed., *Encyclopedia of Physics*, vol. III/1, Springer-Verlag, 1960, pp. 226–858. The quoted passage is reprinted in C. Truesdell, *An Idiot's Fugitive Essays in Science*, Springer-Verlag, 1984, p. 31.

C. Truesdell and R. G. Muncaster, *Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics*, Academic Press, 1980. The cited passage is reprinted in *An Idiot's Fugitive Essays in Science*, p. 75.

The quotation from Aristotle's *Metaphysics* is taken from Jonathan Barnes, ed., *The Complete Works of Aristotle: the revised Oxford translation*, Princeton University Press, 1984. For Aristotle's notion of induction (*epagōgē*), the process of going from particular to general statements, see W. D. Ross, *Aristotle's Prior and Posterior Analytics*, Oxford, 1949, pp. 47–51.

Richard P. Feynman, Robert B. Leighton and Matthew Sands, *The Feynman Lectures on Physics*, vol. I, Addison-Wesley, 1963.

Archimedes actually proved that π is between 223/71 and 22/7. See E. J. Dijksterhuis, *Archimedes*, Princeton University Press, 1987, pp. 222–238.

For George Dantzig's own account of the Neyman "homework" incident, see "An Interview with George B. Dantzig: The Father of Linear Programming", by Donald J. Albers and Constance Reid, *The College Mathematics Journal*, **17** (1986), pp. 292–314; see p. 301.

For Kronecker's view of transcendental numbers, see Harold M. Edwards, Essays in Constructive Mathematics, Springer-Verlag, 2005 (ISBN: 0-387-21978-1), pp. 201–204.

Stacy G. Langton ([email protected]) is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.