In a previous work (1), I went into some of the implications of singularities and paradoxes in mathematics, physics, and the natural world. This leads to a question. What is mathematics and why does it match our world so closely? There is a basic split in thinking. For many, mathematical objects exist outside of human thought. We discover mathematical truths in the same way we discover planets or new species. This is very Platonic. For others, mathematics is a cognitive, human endeavor that grew out of our evolutionary development. Mathematics is just another way we have to model and explore reality. Findings in neuroscience and psychology over the past forty years have shown that aspects of both are true. Yes, our cognitive processes include circuits that could allow for mathematics. These circuits are not built from scratch; we are born with innate expectations of the workings of 3-dimensional space and an innate basic number sense.
Number sense exists in bees. (2) Bees can add and subtract and the authors even suggest that bees have some conception of zero. If this is true, it is astonishing as the concept of zero originated in only a few human societies.
It used to be that educators thought that children were empty vessels that had learning poured into them by passive teaching. Then, with Piaget, ideas changed. Children are active learners who construct the world through activity and experiments. One idea did not change. Humans, according to Piaget, are born with a clean slate and develop all knowledge of the world from scratch. This has turned out not to be true (3). Babies as young as two months have a number sense of the numbers 1, 2, and 3; can compare basic quantities, and have an innate basic knowledge of 3D space. This knowledge forms a simple set of expectations about how things work. Challenges to these expectations lead to attention, curiosity, exploration, and learning.
Babies:
· Can tell the difference between solid objects and substances such as water, smoke or sand
· Have a sense of solidity, continuity, and cohesion in objects
· Substances are penetrable and lack cohesion
· Do not expect two objects to occupy the same space at the same time
· Expect an object (a container) to have an opening when something is put inside
· Expect an object inside a container to move with the container
· Do not expect an object suddenly to change a property (size, color, shape, surface, pattern, etc.)
· Can quantify larger and smaller objects
· Have similar but later forming (five months) expectations about substances
These abilities of number sense and spatial sense are hardwired at birth, and from this basic cognitive toolkit, humans learn.
A couple of questions arise:
· What is the current neurological knowledge of how we learn and how we think?
· Where in the brain do we do mathematics?
· Is human thought at base spatial rather than logical?
· Can this cognitive spatial and numeric toolkit be axiomatic in that they will generate mathematics?
These questions I hope to work on in future articles.
1. Beaver, William. “Black Holes, Singularity, and Human Thought.” Accessed February 2, 2020. https://wjbeaver.wixsite.com/mysite/post/black-holes-singularity-and-human-thought.
2. Howard, Scarlett R., Aurore Avarguès-Weber, Jair E. Garcia, Andrew D. Greentree, and Adrian G. Dyer. “Numerical Cognition in Honeybees Enables Addition and Subtraction.” Science Advances 5, no. 2 (February 1, 2019): eaav0961. https://doi.org/10.1126/sciadv.aav0961.
3. Hespos, Susan & Vanmarle, Kristy. (2012). Physics for infants: Characterizing the origins of knowledge about objects, substances, and number. Wiley Interdisciplinary Reviews-Cognitive Science. 3. 19-27. 10.1002/Wcs.157.
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