The Biological Origin of Arithmetic: Why Is Arithmetic Universally True?

Friday - August 25 2023, 15:37 UTC - 1 year ago

Our research into the biological root of arithmetic has led us to believe mathematics is a realization in symbols of the fundamental nature and creativity of the mind. We explored the behavior of bees to gain insight into this, as they can take a meandering journey to find nectar and then return by the most direct route. We showed that the abilities of bees in returning to the hive suggested principles that could be used to explain simpler arithmetic operations. Our study brings together the insights from the psychologist Jean Piaget and the mathematician Jacques Hadamard.

Everyone knows that arithmetic is true: 2 + 2 = 4. But surprisingly, we don’t know why it’s true. By stepping outside the box of our usual way of thinking about numbers, my colleagues and I have recently shown that arithmetic has biological roots and is a natural consequence of how perception of the world around us is organized. Our results explain why arithmetic is true and suggest that mathematics is a realization in symbols of the fundamental nature and creativity of the mind .

Thus, the miraculous correspondence between mathematics and physical reality that has been a source of wonder from the ancient Greeks to the present—as explored in astrophysicist Mario Livio’s book Is God a Mathematician?—suggests the mind and world are part of a common unity. Humans have been making symbols for numbers for more than 5,500 years. More than 100 distinct notation systems are known to have been used by different civilizations, including Babylonian, Egyptian, Etruscan, Mayan, and Khmer .

The remarkable fact is that despite the great diversity of symbols and cultures, all are based on addition and multiplication. For example, in our familiar Hindu-Arabic numerals: 1,434 = (1 x 1000) + (4 x 100) + (3 x 10) + (4 x 1). Why have humans invented the same arithmetic, over and over again? Could arithmetic be a universal truth waiting to be discovered? To unravel the mystery, we need to ask why addition and multiplication are its fundamental operations .

We recently posed this question and found that no satisfactory answer—one that met standards of scientific rigor—was available from philosophy, mathematics, or the cognitive sciences. The fact that we don’t know why arithmetic is true is a critical gap in our knowledge. Arithmetic is the foundation for higher mathematics, which is indispensable for science. Consider a thought experiment. Physicists in the future have achieved the goal of a "theory of everything" or "God equation .

" Even if such a theory could correctly predict all physical phenomena in the universe, it would not be able to explain where arithmetic itself comes from or why it is universally true. Answering these questions is necessary for us to fully understand the role of mathematics in science. Bees Provide a ClueWe proposed a new approach based on the assumption that arithmetic has a biological origin. Many non-human species, including insects, show an ability for spatial navigation which seems to require the equivalent of algebraic computation .

For example, bees can take a meandering journey to find nectar but then return by the most direct route, as if they can calculate the direction and distance home. How their miniature brains (about 960,000 neurons) achieves this is unknown. These calculations might be the non-symbolic precursors of addition and multiplication, honed by natural selection as the optimal solution for navigation. Arithmetic may be based on biology and special in some way because of evolution’s fine-tuning .

Stepping Outside the Box To probe more deeply into arithmetic, we need to go beyond our habitual, concrete understanding and think in more general and abstract terms. Our work brings together insights from the psychologist Jean Piaget and the mathematician Jacques Hadamard. Hadamard showed that addition and multiplication expressed basic mental operations such as merging or splitting a “whole” into parts .

Counting, for instance, produces the same result, whether objects are merged into a “whole” to then be split, or vice versa. Our research studies this conception—known as Gestalt principles—as the basis for arithmetic mental operations. We explored the behavior of bees further to gain insight into this. We showed that the abilities of bees in returning to the hive suggested principles that could be used to explain simpler arithmetic operations .

For example, returning to the hive requires the bee to “join” the compass sense acquired while out searching for nectar, with the distance sense, to complete the navigation home.