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natural numbers. Dehaene argues that discoveries about the psychology of arithmetic confirm that “number belongs to the ‘natural objects of thought,’ the innate categories according to which we apprehend the world.” Following a separate study conducted with the Mundurukú—an isolated Amazonian indigenous group—Dehaene and his collaborators added in 2006 a similar judgment about geometry: “The spontaneous understanding of geometrical concepts and maps by this remote human community provides evidence that core geometrical knowledge, like basic arithmetic, is a universal constituent of the human mind.” Not all cognitive scientists agree with the latter conclusions. Some point out, for instance, that the success of the Mundurukú in the recent geometrical study, in which they had to identify a curve among straight lines, a rectangle among squares, an ellipse among circles, and so on, may have more to do with their visual ability to spot the odd one out, rather than with an innate geometrical knowledge.

The French neurobiologist Jean-Pierre Changeux, who engaged in a fascinating dialogue on the nature of mathematics with the mathematician (of Platonic “persuasion”) Alain Connes in Conservations on Mind, Matter, and Mathematics, provided the following observation:

The reason mathematical objects have nothing to do with the sensible world has to do…with their generative character, their capacity to give birth to other objects. The point that needs emphasizing here is that there exists in the brain what may be called a “conscious compartment,” a sort of physical space for simulation and creation of new objects…In certain respects these new mathematical objects are like living beings: like living beings they’re physical objects susceptible to very rapid evolution; unlike living beings, with the particular exception of viruses, they evolve in our brain.

Finally, the most categorical statement in the context of invention versus discovery was made by cognitive linguist George Lakoff and psychologist Rafael Núñez in their somewhat controversial book Where Mathematics Comes From. As I have noted already in chapter 1, they pronounced:

Mathematics is a natural part of being human. It arises from our bodies, our brains, and our everyday experiences in the world. [Lakoff and Núñez therefore speak of mathematics as arising from an “embodied mind”]…Mathematics is a system of human concepts that makes extraordinary use of the ordinary tools of human cognition…Human beings have been responsible for the creation of mathematics, and we remain responsible for maintaining and extending it. The portrait of mathematics has a human face.

The cognitive scientists base their conclusions on what they regard as a compelling body of evidence from the results of numerous experiments. Some of these tests involved functional imaging studies of the brain during the performance of mathematical tasks. Others examined the math competence of infants, of hunter-gatherer groups such as the Mundurukú, who were never exposed to schooling, and of people with various degrees of brain damage. Most of the researchers agree that certain mathematical capacities appear to be innate. For instance, all humans are able to tell at a glance whether they are looking at one, two, or three objects (an ability called subitizing). A very limited version of arithmetic, in the form of grouping, pairing, and very simple addition and subtraction, may also be innate, as is perhaps some very basic understanding of geometrical concepts (although this assertion is more controversial). Neuroscientists have also identified regions in the brain, such as the angular gyrus in the left hemisphere, that appear to be crucial for juggling numbers and mathematical computations, but which are not essential for language or the working memory.

According to Lakoff and Núñez, a major tool for advancement beyond these innate abilities is the construction of conceptual metaphors—thought processes that translate abstract concepts into more concrete ones. For example, the conception of arithmetic is grounded