More than a feeling…
As animals with brains, we are challenged to make sense of a world full of rich sensory experience. There is a world created inside of our brains that we may see, hear, touch, taste, or smell. Built out of our senses, humans possess an innate ability to extract patterns and other meaningful features from the environment. 8-month old infants are capable of “segmenting” a stream of speech and determining when groups of syllables are likely to co-occur (Saffran 1996). Even immediately after humans are born, we can discriminate between a random and non-random sequence of shapes (Bluf 2011). From our brain’s ability to recognize patterns, we can make predictions based on previous experience, and we can also understand when small complex features of the environment are embedded within other, larger features (Fiser 2005). (To read this blog post, for example, you might accept that words fit within sentences, and that sentences fit within paragraphs.)
Like an electric current, the flow of environmental encoding (via our senses) and decoding (via the implicit learning structures in our brain) generates a mathematical field in the ethereal space. Humans have created language for describing conceivable formal structures, giving shape to the Math Universe– the world of numbers, geometry, and other abstract nonsense. But how did our brains get from the world around us — i.e. the world we sense — to the world of math? And given that we exist in the Math Universe, having created it, how might we further sense and navigate this new and interesting terrain?
Seeing is believing
How do our senses teach us about the world?
By focusing on one sensory modality at a time, neuroscientists can study the process of how animals order and contextualize information received from the environment. Ascribing order and precision to the world around us is an inherently mathematical undertaking, but math is not necessarily unique to humans. Other animals, including honeybees, display “implicit probabilistic learning”, a generalized term for describing how we interpret the environment via formal structures (Avargues-Weber 2020).
Studying the differences in how humans and bees process visual scenes can give us information about how human cognition differs from other animals. Abstractly, how might the Human Math Universe differ from the Bee Math Universe, or that of any other animal? Interestingly, Avargues-Weber and colleagues find that unlike humans, bees are incapable of learning the “predictability” of two elements given their conditional probability. In their experiment, the researchers applied the same behavioral tests to bees as those previously applied to human infants. Bees are sophisticated visual learners in a variety of other object recognition and association tasks. By observing and defining a concrete difference in how humans and bees process visual information, however, we can begin to draw conclusions about how our internal representations of the external world differ. Alternative “probabilistic machinery” which guides learning in human brains vs. bee brains, the researchers propose, lends to ultimately superior cognitive abilities (and perhaps a more nuanced Math Universe) in humans.
The flavor of numbers
How might we explore the Math Universe, using our senses?
“A mathematician is a blind man in a dark room looking for a black cat which isn’t there.”* For some humans, however, experiencing the “extrasensory” world of math takes on many of the familiar tones of the physical world we typically experience. Individuals with grapheme-color synesthesia deeply associate certain letters or numbers with an intrinsic color. A woman with this form of synesthesia described her color-coding intuition as often relating to prime divisors — whether a number is a multiple of 2 or 3, for instance. The color-number matching isn’t necessarily fully consistent with known rules of math and can be idiosyncratic or personal (Dixon 2005).
Of course, numbers have interesting properties which mathematicians often study. We still do not fully understand all the patterns and unique functionalities endemic to our numbering system. For example, we can define a prime number as a quantity greater than one which isn’t a product of 2 smaller numbers. We know that there are infinitely many of them. But there are many unsolved questions regarding prime numbers – can any number be written as a sum of two primes? Are there infinitely many twin primes — that is, prime numbers with just one even number between them?
Studying synesthesia may give us interesting clues regarding how humans visualize and learn about math in general. Importantly, the synesthetic “rules”, while they may vary by individual, are not arbitrary — they are shaped by the structure of the learned domain (Watson 2014, Mills 2009). Synesthetic experiences are characterized by a one-to-one mapping from the “inducer” domain (the number or letter that prompts an experience) and the “concurrent” domain (the experiences). This requires a consistent ability to parse, order, and understand the inducers. Additionally, growing evidence shows that adults can acquire synesthesia with training (Watson 2014, Colizoli 2012 ), suggesting that the “feelings” numbers give us may be inherent to how we think about them. What could this possibly tell us about the properties of prime numbers, or the various other abounding mathematical mysteries of our time? A neuroscientist’s unsolicited advice to a mathematician: trust your senses.
*quote (probably) misattributed to Charles Darwin https://en.wikipedia.org/wiki/Definitions_of_mathematics
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