George Lakoff. Rafael E. Núñez. Where Mathematics Comes From.

This is the book I was looking for since I obtained my degree in mathematics. In high school and later in college I was taught to perform all manipulations with mathematical symbols required from an average graduate; what I lacked was understanding of what I was doing. What's worse, since we moved to calculus studies, I started to feel uncomfortable with the whole endeavor. This concern is acutely addressed by the authors,

The study of infinitesimals teaches us something extremely deep and important about mathematics -- namely, that ignoring certain differences is absolutely important and vital to mathematics! This idea goes against the view of mathematics as supreme exact science, the science where precision is absolute and differences, no matter how small, should never be ignored.
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Generations of students have found these to be disturbing questions. Formal definitions, axioms, and proofs do not answer these questions. They just raise further questions, like "why these axioms and not the others?"

The mathematics itself can't provide answers to these questions; they have to be put into a broader context. The authors provide such context by means of cognitive science, looking at mathematical phenomena through the prism of human conceptual system, most notably conceptual metaphors and conceptual blends. ¹⁾

They start from subitizing -- the innate ability shared by humans and some animals (in particular, primates and birds) to instantly and accurately discern one, two, or three objects. This brings us to an understanding of addition and subtraction. Add here an understanding of grouping objects into sets, and we gain an understanding of multiplication and division. Then the authors take a look at various forms of infinity: points at infinity, infinite sets, mathematical inductions, infinite decimals, infinite sequences and sums, transfinite cardinals and ordinals, infinitesimals, points and continuity. In the discussion I met the names familiar from my college math, only this time I got a good idea of what these mathematicians really achieved. Here is what the authors have to say about Newton and Leibniz,

For Newton, the derivative of a function for a given number was (metaphorically) the tangent of the curve of the function at the point corresponding to that number. ... The tangent line was conceptualized metaphorically as the limit of a sequence of secant lines, with the secant lines becoming progressively smaller but always having a real length.

Leibniz solved the problem of arithmetizing "an infinitely small interval" in a very different way. He hypothesized infinitely small numbers -- infinitesimals -- to designate the size of infinitely small intervals.
...
As far as calculation was concerned, there was no difference between the two approaches. Both yielded the same results. For two hundred years after Leibniz, mathematicians thought in terms of infinitesimals, used them in calculations, and always got accurate results. But ontologically there is a world of difference between what Newton did and what Leibniz did. Newton did not need infinitesimals numbers as existing mathematical entities. His secants always had a real length, while Leibniz's had an infinitesimal length. Leibniz hypothesized a new kind of number. Newton did not.

And about Weierstrass,

Weierstrass, like Dedekind and Cauchy, sought to eliminate all geometry from the study of numbers and functions mapping numbers onto numbers, including derivatives and integrals in calculus.

Weierstrass's redefinition of continuity for a function over sets of discrete numbers was perhaps his greatest metaphorical magic trick. Think of the job he had to do. He had to eliminate one of our most basic concepts -- the natural continuity of a trajectory of motion -- and replace it with a concept that involved no motion (just logical conditions), no continuous space (just discrete entities), no points (just numbers), with functions that are not curves in a plane (but, rather, sets of ordered pairs of numbers). And in process he had to convince the mathematical community that this was the real concept of "continuity."

They thus characterize the discretization program,

An important dimension of the discretization program was the concept of "rigor." This meant the use of discrete symbols and precisely defined, systematic algorithmic methods, allowing calculations that were clearly right or wrong. They could be written down step by step and checked for correctness. The prototypical cases were the methods of calculation in arithmetic. Those methods of calculation provided for certainty and precision, which were taken as the hallmarks of mathematics in nineteenth-century Europe.

Geometry involved visualization and spatial intuition. But mathematics in nineteenth-century Europe increasingly became a field that valued certainty, order and precise methods over imagination and visualization. Indeed, visualization and spatial intuitions were eventually disparaged. ... Geometric methods came to be seen as vague, imprecise, not independently verifiable, and therefore unreliable.

Despite my high opinion about the book, I hesitate to recommend to others for two reasons. First, many reviewers, Amazon and not, found errors in authors understanding of some mathematical points. Next, because this is the first book about "What is mathematics, really" I read, I do not know where it stands compared to the other books available on the market.

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1) Metaphors and conceptual blends are examined in Lakoff's "Metaphors we live by", and "The Way We Think: Conceptual Blending and the Mind's Hidden Complexities" by Gilles Fauconnier and Mark Turner.