Mathematics and the Embodied Mind

What is the mind and what is the body? Do they exist together on are they somehow separate? The mind/body problem and the idea of a duality of mind and body run through Western thinking especially. This goes deep into Greek philosophy and the Christian concept of the soul. Our minds feel outside of Nature, others, and our own bodies. Science fiction is full of tales where we can download ourselves into computers or back ourselves up and change worn out or damaged bodies like changes of clothes. Modern dualism is based on the idea that psychological states are irreducible to physical states and this allows for the mind to be separate from the body(1). This duality has been challenged by cognitive science and neural science by the idea of the embodied mind.. We inhabit a body and our mind is just a small part of our nervous system. However, the mind can be seen to be in a sort of partnership with the world. The body and the perceptions of the world through the body use the mind as a statistical prediction engine in which we learn how to manage the world, we learn about others, and we learn about ourselves. Psychological states are irreducible to physical states for exactly the reason that the mind is a statistical entity. We have a wild imagination. We can imagine anything but what the physical world tells us through our perceptions and the way we use tools to expand our perceptions, we begin to understand reality (2),

So what about mathematics? Math is confusing because it is clearly a mental endeavor yet it can tell us much about how the physical world works. Also, there is the ability of mathematics to describe some of the deep realities of our world. Many mathematicians today and throughout history, ascribe to mathematics a sort of super-reality, a place outside both the mind and the physical world. Mathematical objects are real objects and the terms 'pure mathematics' vs 'applied mathematics' come from a Neo-Platonist view. Again, cognitive science and neuroscience challenge this view. In a past post (4), I wrote about how experiments on babies have shown that we are born with innate expectations about limited arithmetic of numbers {1, 2, 3} and innate expectations about the physical world. These include expectations of the properties of solid objects in 3D space (continuity of motion, continuity of texture, in front and behind), container rules (within, without), and what the authors call 'substances' (smoke, water, sand) which have different rules than solid objects (3). I have wondered about all this. I have wondered whether this expectation could be somehow mapped to our knowledge of mathematics and whether human mathematics can be built from these basic innate assumptions of the world.

According to Where Mathematics Comes From (5) this is not only possible but the authors have attempted just this construction. I think they do a good job of creating the axioms of arithmetic from innate human arithmetic. Another section builds up the mathematics of Boole's algebra of classes to set theory. Their basic metaphor is the Venn diagram and the cognitive ideas of intersection and union. In this case, I don't believe they are correct. Babies expect objects in front of or behind not transparent like a Venn. They don't see the intersection in the center. They do have an innate knowledge of containers. This could be the start of the metaphor of intersection and union. An object can be both inside and outside a container object. Union is the container object plus the contained object while the intersection is the part of the contained object that is in the container. I really don't think that the current research is there yet but I'd like to see more of a study of the innate properties of containers. One of the most interesting parts of Where Mathematics Comes from is the idea that the notion of infinity is a human cognitive concept that arises out of the perception of continuity. Studies of babies so far support this. Babies expect an object entering their field of vision to cross and leave their field of vision. If it doesn't do this it gets their attention. They also expect substances to flow. The metaphor is that of iterative measurement. Take one of Zen's paradoxes. While going from place A to place B you go half the distance each time. Every time you do this you will get closer and closer to B but never actually reach it. Of course, we do reach place B. This reaching B is our metaphor for infinity. Research on babies supports innate knowledge of a basic concept that leads to mathematical infinity.

Lakoff and Núñez's work has taken many criticisms. (6) However their book has had over 4,800 citations since 2000, about 240 a year. This suggests that their work has not been ignored or rejected.

1. Robinson, Howard. “Dualism.” In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, Fall 2017. Metaphysics Research Lab, Stanford University, 2017. https://plato.stanford.edu/archives/fall2017/entries/dualism/.

2. Friston, Karl. Embodied Cognition Karl Friston. Accessed March 29, 2020. https://www.youtube.com/watch?v=HW0JnjgCO3o.“

3. Physics for Infants: Characterizing the Origins of Knowledge about Objects, Substances, and Number.” Accessed December 29, 2019. https://www.researchgate.net/publication/259853187_Physics_for_infants_Characterizing_the_origins_of_knowledge_about_objects_substances_and_number.

4. Beaver, Bill. “Babies Know Physics.” my site, February 2, 2020. https://wjbeaver.wixsite.com/mysite/post/babies-know-physics.

5. Lakoff, George, and Rafael E. Núñez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. 1st ed. New York, NY: Basic Books, 2000.

6. Sinclair, N., and M. Schiralli. “A Constructive Response to `Where Mathematics Comes from’.” Educational Studies in Mathematics 52, no. 1 (January 1, 2003): 79–91. https://doi.org/10.1023/A:1023673520853.