Where Does Mathemat- ics Come From?

How the Embodied Mind Brings Mathematics into Being
Some people believe in magic. Some believe in math. And some people believe that math is magical. Somehow, mathematics manages to describe the world with uncanny accuracy. Scientists use math to calculate the timing of eclipses, the strength of nuclear explosions, and the orbits of sending spaceships to the moon. Mathematics is essential for everything from jet design to bridge building. Math works better than anything else in the world, and scientists often wonder why. Many scientists suspect that the achieve- ment of mathematics expresses something profound and true about. the universe, revealing a natural mathematical structure that gov- erns the desmos or at least makes it intelligible. Somehow people were able to discover the laws that gov ern reality in the form of symbols
and the rules for combining them.

Or so it seems...

Where Does Mathematics Corne From? (WMCF)

Linguist George Lakoff and cognit- ive psychologist Rafael Núñez have long had interests in mathematics and have written a unique and fas- cinating book that attempts to "ap- ply the science of mind to human mathematical ideas" to discover where our mathematical ideas come from. This book introduces the discipline of "mathematical idea analysis": they seek to articulate how our understanding of human cognitive processes can explain the development of mathematical ideas. We can also say that some questions were left unanswered.

Metaphor is not just an ornament; It is the basic tool through which abstract think-
ing is possible. One of the main consequences in cognitive science is that abstract con- cepts are understood in terms of more concrete concepts, typically through metaphor,

This sentence in the book shows that the definition and meaning of mathematics is based on metaphor. It also gives the definition of mathematics as:

Precise, coherent, stable across time and human populations, symbolizable, calculable, generalizable, universally usable, consistent in each of its subjects, and a general tool for explaining, explaining and predicting in a multitude of daily activities, from sports to building, business, technology and science. as effective.
In fact, they emphasize that math- ematics is not a discovery about the outside world, but an invention based on metaphors associated with human thoughts, sensations and actions.

Lakoff and Núñez begin the book by reviewing the psychological lit- erature and conclude that humans have an innate ability to count to about 4 or 5, called addition and subtraction. They document this conclusion by reviewing the literature published in recent years describing experiments with infant subjects. For example, when babies are presented with "impossible" situations, they quickly become excited or intrigued, such as when three tovs appear when only two are initially present.

The authors argue that mathem-
atics goes far beyond this very basic level due to the large number of metaphorical structures. For example, the Pythagorean position where everything is numbers, and the associated crisis of confidence with the discovery of the irration- ality of the square root of two, arises solely from a metaphorical relationship between the length of a square's diagonal and the possible number of objects.

Most of the WMCF takes up the important concepts of infinity and boundary processes and tries to explain how finite humans living in a finite world can ultimately envision the true infinite. So much of the WMCF is, in fact, a study of the epistemological foundations of calculus. Lakoff and Núñez conclude that while the potential infinity is not metaphorical, the
actual infinity is not. Moreover, they see all manifestations of true infin- ity as examples of what they call the "Basic Metaphor of Infinity" as represented by the ever-increasing sequence 1, 2, 3,....

On the other hand, the WMCF absolutely rejects the Platonic philosophy of mathematics. They emphasize that the only thing we know and can know is human math- ematics, mathematics originating from human intelligence. In addi- tion, WMCF is primarily concerned with proposing and constructing an alternative view of mathematics that bases the field on the facts of human biology and experience,

And in terms of education, WMCF is still problematic. From the point of view of conceptual metaphor theory, metaphors reside in a different domain, the abstract, the
'real world', the concrete. In other words, despite the claims that math- ematics is human, the established knowledge of mathematics, which is what we learn in school, is assumed. and treated as abstract, completely disconnected from its physical ori- gin. It cannot explain how students can access such information.

Math may be not in the stars, but in

ourselves.

The book is introduced in the Dallas morning news as follows.

But not to George Lakoff and Rafael Nunez. As cognitive scientists, they see a world gov erned not by mathematics but by the human brain. Whatever the "truth," they write, human knowledge depends solely on the brain and its own ways of
finding things. Mathematics is not a discovery about the outside world, but an invention based on metaphors associ- ated with human thoughts, sensations and actions. "Where does math come from?" Dr. Lakoff and Nunez ask. "It comes from us," they answer in their new book Where Math Came From (Basic Books). "We create, but not arbitrarily," they write. "As it evolves in the real world, it uses the fundamental conceptual mechanisms of the embodied human mind. 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." Dr. Lakoff (University of California, Berkeley) and Nunez (of B Berkeley and the Univer- sity of Freiburg in Germany) argue that all mathematical ideas are elaborate metaphors. These metaphors are taken
then connected and mixed to guide the various realms of mathematical practice. The core principles of arith- metic, algebra, trigonometry, mathematical logic, and other mathematics subsections all rely on metaphorical reasoning to create rigorous deductive and computational systems. For example, arithmetic can be conceived as moving along a path of evenly spaced markers, which is the metaphorical basis of the concept of a number line. Other metaphors can be defined to illustrate the ideas underlying further mathematics. Therefore, Dr. Lakoff and Nunez conclude that mathematics is simply a human invention, a systematic way of capturing the way the brain sees the world. Dr. "The only math we know is math that our brain allows us to know," Lakoff said at a meeting of the
American Association for the Advancement of Science in San Francisco last month. As a result, he says, any question that mathematics is inherent in physical reality is moot be- cause there is no way to know if it does. Dr. "Mathematics may or may not be in the world, but there's no way we can tell scien- tifically," Lakoff says. Dr. Lakoff and Nunez argue in their book that math succeeds in science because scientists push for it. "All 'harmony' between math- ematics and the regularities of the physical world is done in the minds of physicists who understand both," they write. "Mathematics is in the mind of the mathematically trained observer, not in the regularities of the physical universe." All of this is very interesting and has earned his books a pretty high Amazon.com sales ranking. But their analysis leaves at least a few questions nonewared The authors for