Ended up giving it away. My library Help Advanced Book Search. Euclid in the rainforest: Stay in Touch Sign up. See our Returns Policy. While some of the mathematics were beyond me, this book was well-written and enjoyable to read.

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The book is divided into three parts. According to Mazur, this kind of logic is unable to deal with infinity. 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. 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. How can this be explained? Mazur raises this question for example, on p. 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. 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.

Mazur goes on to claim p. 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. 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.

Mazur repeatedly asserts that the axioms of mathematics, the starting points of our deductions, ought to be "self-evident". According to Aristotle Posterior Analytics, I. But how can we know that they are true, if they are the starting point rather than the conclusion of our reasoning? Mazur p. How can we induce through sense-perception that all right angles are equal? We still do not know whether the large-scale geometry of space is Euclidean or non-Euclidean, as Mazur remarks on p. 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. An equilateral triangle can be constructed on a finite straight line AB" p.

Mazur states p. 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. 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. 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. 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.

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 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. 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. 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.

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.

Much of it reads like a first draft — some sections such as pp. Mazur consistently misspells the name of the philosopher Carl Hempel e.

He also misspells the names of Girolamo Cardano p. Buckley p. We find "compliments" instead of "complements" p. When the Sumerians first entered the historical record, they lived in southern Mesopotamia, not in the "mountains of ancient Iran" p.

Phoenicia, on the Mediterranean coast, was not "a huge area that included both Persia and the western part of India" p. The mathematician Pappus of Alexandria lived in the fourth century A. The Homeric passage quoted on p. There are surprisingly many mathematical misstatements. A geometry containing two non-similar triangles must be Euclidean p. If you take the product of the first k primes and add 1, the result must be prime p. 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!

A footnote on p. This fact, though true mathematically, hardly requires the genius of Archimedes. The "apocryphal story" on p. 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. By this standard, Mazur is not a good writer.

He frequently writes sentences which clearly do not say what he intends them to mean. 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.

Did Mazur mean "elusive"? In mathematical writing, of course, lack of precision can be disastrous. 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. 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.

But in the next sentence the same remark is called "distinguished".

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Euclid in the Rainforest



Euclid in the Rainforest: Discovering Universal Truth in Logic and Math



Joseph Mazur


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