Where Mathematics Come From

This is a book well worth reading if you can cut the authors a little slack. Having read it, I can now for the first time understand Euler's famous equation, e to the pie-eyed = -1. Wow! Euler wasn't just lubricating!A second or third edition could be very valuable as a textbook; what L&N have written is a great start, but there are a few rough spots.To paraphrase Ogden Nash,One thing that Cognitive Science would be greatly the better forWould be a more restricted (and more appropriate) employment of simile and metaphor.*Consider what is perhaps the metaphor most frequently quoted in WMCF, "Numbers Are Points on a Line."Some subsets of the numbers, including some infinite subsets (e.g. the 'real' numbers, or the 'pure imaginary' numbers {0 + xi}) can be mapped one-to-one onto points of a Euclidian straight line in such a manner that the directed distance from the point designated as zero [call it p(0)] to the point designated as x [call it p(x)] is proportional to the number x, and such a mapping can be very useful. Fine, use it. But why let yourself get so confused as to think that numbers are points. Points are points and numbers are numbers; points and numbers are two different concepts, and 'all you are entitled to say at the very most'* is that a particular subset of the numbers is, in some very useful respects, like the points on a line, which is a simile, not a metaphor. In general, I think similes would be more appropriate than metaphors in math. Sometimes, I think even a simile is a stretch.Near the top of p. 199: "The point of this is . . . to comprehend how we understand it."Is there only one way that anyone can understand something? Does every human mind work the same way?On p. 424: "-n and n are symmetrical points relative to the origin (zero)"This symmetry is conceptualized in terms of mental rotation . . . . Cognitively, we visualize the relationship between the positive and negative numbers using a rotational transformation . . . ." When dealing with the complex plane, this is clearly the better way to do it, but when dealing with 'real' numbers only, I have always thought of it as sliding a sort of mental tape measure along the number line until the point on the tape that was at n is at zero, and then the point that was at zero will be at -n. And lo and behold, for the 'real number line' this works just as well as the rotation, no better, no worse.On pp. 218-220 L&N discuss ordinal numbers. Another use besides cardinal and ordinal is nominal, using a number as simply a name for something, e.g. Joseph Heller's wonderful book  Catch-22 . There were no catches 1 thru 21; 22 was just part of the name for that particular catch.A few other items I noticed:Top of p. 86: "Moreover, we don't use binary notation, even though computers do, because our ten fingers make it easier for us to use base 10." Actually, the main reason we seldom use binary notation is because it is cumbersome. Our ten digits (8 fingers, 2 thumbs) led us rather naturally to base 10, but octal and hexadecimal are used quite a bit, and it is actually easier to keep from losing your place counting on your fingers in binary, in which 10 digits allow you to count from 0 to 31 on one hand, or to 1023 using both handsp. 140: The SIMILE Classes Are Like Containers sort of breaks down for me with the Venn diagram of two overlapping circles for two classes that have some members in common but neither is a subclass of the other.p. 331: "For example, we take the curvature at a point in a curve as being PART OF THE CURVE. (italics in original; I had to substitute CAPS because Amazon's text box doesn't permit italics.) I take the curvature to be a characteristic of part of the curve, not a part of the curve. And a tangent is most certainly not part of a curve.p. 332: "A Function Is a Point in a Space" ???p. 337: "What COMMONPLACE cognitive mechanisms do they use?" (emphasis added) You aren't even interested in any unusual cognitive functions they may use?p. 347: ". . . every bit of thinking we do must be carried out by neural mechanisms of EXACTLY the right structure to carry out that form of thought." Our brains haven't the flexibility to 'make do' to the slightest extent?p. 349: "The Pythagorean theorem hasn't changed in twenty-five hundred years and, we think, won't in the future." But what we know about it has changed considerably. I have discovered several facts that, so far as I have been able to find out, were not previously known, e.g. a set of 3 equations in r (row #) and k (column #) which define an infinity by infinity matrix of Pythagorean triangles, each row and each column of which is a family with a or b in arithmetic progression, and c - a or c - b constant:a(r,k) = 4rk + 2k(k-1)b(r,k) = 4r(r+k-1) - 2k + 1c(r,k) = 4r(r+k-1) + 2k(k-1) + 1p. 360: ". . . the form of arithmetic used in all computers." There is no such thing as THE form of arithmetic used in all computers, Many computers use two's complement binary arithmetic, but the Control Data 6600 and 7600 used one's complement binary arithmetic. The IBM 1401, 1440, and 1620 used binary coded decimal arithmetic. All computers I have ever programmed had integer arithmetic, and most, but not all, had floating point.p. 361: "All of that arithmetic used floating-point, not standard arithmetic." Simply not true. See just above.* Ogden Nash: Very Like a Whale.

I bought and read this book 4 years ago. The beginning of the book makes important statements about the "true nature" of mathematics. It is the perfect counterweight to the Platonic "universe of perfect forms" which some people think explains the source of mathematical meaning. Another philosophy of mathematics is the "mathematics in the universe" idea that the universe contains mathematics, and that's where we discover it from.In the Platonic view, mathematics is essentially completely disconnected from the physical world, but can be perceived by the mind. In the "math in the universe" view, it's "out there" outside the individual mind. The embodied mind viewpoint may be thought of as locating mathematics in the interface between individual minds and the universe around them. (Of course, there is no sharp boundary between individuals and their environments.)The essence of this book may be summarised by a single paragraph on page 9."Mathematics as we know it is human mathematics, a product of the human mind. Where does mathematics come from? It comes from us! We create it, but it is not arbitrary---not a mere historically contingent social construction. What makes mathematics nonarbitrary is that it uses the basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history."The rest of the book gives evidence for this claim and develops the consequences for some areas of mathematics. My favourite part of the book is the evidence for arithmetic in six-month-old human infants, using puppets, on pages 16-19. This shows clearly that some basic mathematics is not purely a matter of culture. It is innate.Chapters 2 to 11 present a series of "metaphors" and "schemas" for various mathematical concepts --- arithmetic, algebra, logic, sets, real numbers, infinity, transfinite numbers, infinitesimals. Initially it all seems quite plausible. Integers come from adding and removing objects from collections of objects. Real numbers come from measuring sticks. But the "metaphors" get less and less credible. My marginal notes pencilled in the margins changed from very positive to slightly positive, to lukewarm, to skeptical, to strong disagreement, to utter derision. My negative marginal notes started at page 91, concerning complex numbers interpreted as rotations. At page 108, my comments are "Huh?" and "infantile" and "Why not?". On pages 109 to 111, I commented "more nonsense", "unjustified conclusion", "nonsense", "total twaddle" and "nonsense".The metaphors are mostly like what are used to teach mathematics. In fact, a useful application of this book would be to mathematics teaching. That's what it seems like really.I do think that this is a very good book, but the authors could have said everything much better in 150 pages instead of 493 pages. After the first couple of chapters, the rest may be considered "metaphors for teaching mathematics". The first 4 chapters are well worth reading as an antidote to the Platonic "universe of forms" philosophy of mathematics. Modern mathematics is taught as if it were "analytic" truth, i.e. as absolute, universal truth which even life-forms in other galaxies would agree with.The fact that the basic concepts of mathematics are wired into the brain is very important. However, that is only the launch point for mathematics. The detailed content of modern mathematics cannot be mapped one-to-one with the "embodied mind".Maybe each individual component of mathematical thinking may be identified with innate capabilities of the human mind. But all cooking uses pretty much the same set of ingredients and the same set of techniques, and yet the art of great cooks cannot be reduced to mere ingredients and techniques. Painters all use the same colours, tools, concepts of perspective and composition etc., but that does not explain all of the art in the art galleries.The basic components of mathematical activity are indubitably in-born in the sensory/motor system of the human mind, but it is very doubtful that "cognitive science" (if it is indeed a science) can identify which mental process is being applied in each mathematical thinking-process. Probably each individual uses different innate brain functions to other individuals for the same mathematical concepts. For example, I use diagrams for almost all mathematics, whereas some mathematicians I have met say that they do all mathematics symbolically and algebraically without any visualisation.The "embodied mind" theory also does not explain how mathematicians at some points in history totally rejected concepts which at other times were accepted as self-evident. Examples are zero, negative numbers, irrational numbers, transcendental numbers, complex numbers, infinite sets, transfinite ordinal numbers and non-Euclidean geometry. The vicious debates between intuitionists and formalists in the late 19th and early 20th centuries show that even at one point in history, there can be strong disagreement on the most fundamental ideas of what constitutes mathematics. So it seems unlikely that the "embodied mind" theory can do explain anything more than the ingredients and utensils of mathematics. I cannot explain the recipes in the cookbooks.

最も有名な数学の関係式の一つに，e^πi＋1＝0があります．オイラーの式とよばれるこの式は，大学1年生レベルの数学（テイラー展開と複素関数論）で証明することができますが，「なぜこの式が成り立つか」というのは，不問に付されることが多いと思います．（「人間にはあずかり知らない神秘だとしかいいようがない」など？）このような考え方，つまり，人間の都合とは無関係に数学の世界が存在して，人間は証明を通じてのみ理解できるのだという考え方は一般的です．しかし本書は，そうした数学の見方をひっくり返します．認知科学者であるLakoffとNunezは，数学といえど人間の脳のなかで生まれるものであるからには，数学も認知プロセスの一つとして「経験科学」の立場から基礎付けることができるはずだ，という考えから出発し，数学を認知科学から基礎づけることを試みています．まずはS.Dehaene（『数覚とはなにか』早川書房，2010年）らの研究などを踏まえながら，人間や動物が生まれ持った数や算術の概念についての，認知実験からの知見を紹介します．ところが， Dehaeneらの研究がごく簡単な算術にとどまっていたのに対し，本書ではそこから敷衍して，一気に数学全体まで拡大し，自然数の算術から始まり，ブール代数，集合論，命題論理，群，「無限」の比喩から始まる実数，カントールの無限集合に関する議論，また，デデキントとワイエルシュトラスによる実数論，20世紀に展開された超準解析にいたるまでの数学を，「認知科学的に」解きほぐしてみせます．とにかく，著者らの知的体力には脱帽です．（ここまで認知科学的な議論を推し進めることができるとは驚きだった，と著者ら自身が書いています．）巻末では，彼らの「数学の認知科学」の"Case Study"として，冒頭のオイラーの式：e^πi＋1＝0が「成り立つ理由」を，彼らの方法で解説してくれます．他のレビューを見ていると，数学者や科学哲学の分野からはいろいろな批判がありうる（数学的厳密さに問題があるとか，哲学の近年の先行研究が全然踏まえられていない，など）ようです．しかし，学問的価値はともかく，インパクトの強い本であることは間違いないです．とくに数学を教える立場にある人には，読む価値があると思います．たとえば，　・負の数同士の掛け算が正の数になるのはなぜ？　・三角比を，θ＞90°に拡張するとはどういうこと？など，中高の数学で多くの生徒がつまずくだろう疑問は，この本を読めば，かなりの部分解消されるのではないしょうか．数学とは「本当は何であるか」ではなく，「人間は数学をどのように理解しているのか」という，知りたかったけれど誰も教えてくれなかった疑問に，真正面から答えている稀有な本だと思います．

I have never considered cognitive science to be a topic that could be studied and understood well enough to support the teaching, learning and understanding of mathematics. Oh how I wished that this text had been available when I was an undergraduate. Sadly the science was not well known in those days. It certainly would have made life a lot easier and answered the many questions that as graduates we bombarded our tutors with and never received a satisfactory explanation. It certainly is a topic that a lot of professional mathematicians shy away from and is certainly the case with a large number of teachers at all levels, from Primary through to post-grad. It is written in a style that is easily assimilated, and enjoyable to read. I would certainly recommend reading this work prior to any course on Philosophy, as it will place any such course into firm context and make it far easier to understand. It is an exciting text and introduces a topic that has far reaching implications for teachers and learners alike.

Some readers have dismissed "Where Mathematics Come From" as being "postmodern", i.e. advocating arbitrariness of mathematical reasoning. These reviewers don't seem to have made it to the end. The authors are actually very clear about their viewpoint: They disapprove of the romance of mathematics (that basically claims, mathematical truths have an objective existence independent of human cognition, and by the same token logic accounts for the (only) correct way of reasoning). But at the same time they acknowledge the universal nature of mathematical reasoning and its effectiveness in dealing with real world phenomena. There is a simple reason for the correspondence between human-made math and reality and it is given in chapter 15: Humans share a common brain structure, they live in fairly similar surroundings dealing with the same basic issues of everday life. Mathematics as well as language is modeled according to real life needs and conditions. Therefore mathematical reasoning is not arbitrary, albeit culturally shaped (c.f. the idea of "essence" leading to the need for axiomatization or the notion that all human reason is some kind of calculation).It is relatively easy to corroborate the author's thesis, that the development of mathematics can be accurately described in terms of application of metaphorical structures and conceptual blending mechanisms on mathematical concepts and thereby creating new concepts and so forth. Just take a contemporary mathematical advanced textbook on calculus or algebra and compare it to the writings of mathematicians before the invention of differential calculus (in Lakoffs/Nunez terms: the construction of infinitesimals and the mapping of numbers on the points on a line)or even Euler. The difference is striking: The idea that mathematical insights should rely on some essential axioms whence all mathematical truth can be derived must have seemed outlandish to mathematicians before the 19th century (although proved to be incorrect for quite some time now the notion of mathematics as being independent and self-sustaining seems to be quite widespread still).Of course, by exploiting the possibilies of metaphorical cross-mapping within mathematics itself mathematics has liberated itself from reality to a great extent and turned into an art. Why else would mathematicians claim that beauty, simplicty and truth are closely interrelated?The authors (and myself) obviously love mathematics and hold mathematicians in high esteem. And even more so by the fact that mathematics is "only" human.A great reader for anyone who loves mathematics and wonders how it connects to common sense!

Challenging....but enlightening.....why can't our kids do better at math...do they have embodiments of anything anymore or is everything virtual...and where is the body in such realities...

Una pieza extraordinaria de reunificación de lo que nunca debió disociarse: las ciencias analíticas y las disciplinas holistas, creativas, humanistas