Do Martians Enjoy Pi?
Published on 20230601
Innate Number Sense as an Argument Against Mathematical Platonism
In 1977, NASA launched Voyager 1 and Voyager 2 as part of the Voyager Program. These robotic probes were equipped with sophisticated instruments to study celestial bodies, but they also carried something unique: the Golden Record. This set of goldplated phonograph records contained mathematical definitions, pictures, and depictions of our culture, intended to communicate with potential intelligent extraterrestrial life.
This ambitious project presented a significant challenge: how to communicate complex information without relying on Earthspecific languages. The solution? Mathematics. This choice reflects a widely held belief that mathematics is a universal language, transcending cultural and linguistic barriers. But this raises a fundamental question: Is mathematics truly an independent entity of the universe, discoverable and understandable by any intelligent beings? Or is it a human construct, a system we've developed to describe our particular experience of the universe?
Mathematical platonism posits that mathematical objects (such as sets, prime numbers, and pi) exist independently of intelligent agents creating them (Linnebo, 2018). Platonists argue that humans have merely discovered these fundamental truths. However, I contend that mathematics is a creation of intelligent agents, developed to describe the universe we inhabit. Our innate number sense provides support for this antiplatonistic view of mathematical ontology: mathematical concepts are not an independent feature of the universe but a product of our cognition and evolution.
Empirical Evidence for Innate Number Sense
Number sense refers to the intuitive understanding of numbers and mathematical relations (Lipton & Spelke, 2003). This concept encompasses the ability to distinguish between "approximate numerical magnitudes" without relying on language or symbol manipulation, as well as a basic understanding of mathematical relationships like addition and subtraction. In essence, number sense allows us to have an intuition for distinct counts and to recognize that some numerical counts are larger than others, along with a rudimentary grasp of basic mathematical operations.
The question of whether humans possess an innate number sense or if it's learned has been the subject of extensive research. A notable experiment by Wynn (1992) investigated this by examining whether 5monthold infants, who had not yet developed linguistic skills or received any mathematical training, could recognize differences in counts of objects.
In this experiment, infants watched as Mickey Mouse dolls were placed behind an opaque screen. The screen was then lowered to reveal either the expected number of dolls (based on the dolls placed there) or an unexpected number. The results showed that infants who witnessed an unexpected number of dolls stared at the display case longer than those who saw the expected number. Wynn concluded that "infants know that an addition or subtraction results in a change in the number of items." This study supports the idea that a sense of numbers and basic arithmetic is innate, rather than learned.
Similar studies have been conducted with animals, notably rats. Mechner's (1958) experiment involved training rats to press a lever a specific number of times to receive a reward. The rats consistently pressed the lever the required number of times, suggesting an innate ability to track quantities. To address criticisms that the rats might be relying on timing rather than counting, Meck and Church (1984) designed a followup experiment using varyinglength audio tones. The rats' ability to distinguish between sequences of two and eight tones, regardless of the sequence duration, further supported the existence of an innate number sense across species.
The Evolutionary Basis of Number Sense
The evidence for innate number sense across species points to evolution as a key factor in its development. From an evolutionary perspective, the ability to distinguish quantities and perform basic arithmetic operations offers significant survival advantages.
Consider an early human foraging for food who sees two predators disappear behind a boulder. The ability to recognize that there are exactly two threats, rather than an indeterminate number, could be crucial for survival. Similarly, if a third predator joins the first two, the capacity to intuitively understand that the threat has increased could be lifesaving. This primitive mathematical competency extends to larger numbers as well, where approximating quantities (e.g., distinguishing between bushes with roughly 200 vs. 500 berries) could lead to more efficient resource gathering.
The Martian Thought Experiment
To further challenge the notion of mathematical platonism, let's consider a thought experiment involving a hypothetical Martian society. Imagine intelligent beings living deep within Mars' crust, in an environment where everything consists of tiny particulates and nothing can be distinctly counted. In such a world, there would be no evolutionary advantage to developing a number sense as we know it.
These Martians might instead evolve a "particulate sense" more suited to their environment. Their mathematical system, if they developed one, would be fundamentally different from ours and possibly incomprehensible to us. This thought experiment highlights a key point: mathematical systems are shaped by the specific environments and evolutionary pressures faced by the beings who develop them.
If mathematical objects like pi and prime numbers truly existed independently in the universe, as mathematical platonism suggests, shouldn't they be discoverable and meaningful to any sufficiently intelligent species, regardless of their evolutionary background? The fact that we can conceive of intelligent beings for whom our mathematical concepts would be meaningless challenges the platonist view.
Conclusion
Our innate number sense, shaped by evolution and our specific environment, forms the foundation of our mathematical understanding. The universality of mathematics, often taken for granted, may be more a reflection of our shared human experience than an inherent property of the universe.
This doesn't diminish the power or utility of mathematics. Just as color is our perception of light wavelengths rather than an intrinsic property of the universe, mathematical concepts like pi are our way of perceiving and describing universal patterns. They are incredibly useful tools, but they are tools we've developed to make sense of our particular experience of reality.
In conclusion, while mathematical platonism offers an appealing vision of a universe governed by immutable mathematical truths, the evidence from our innate number sense and its evolutionary origins suggests a different picture. Mathematics, in all its elegance and power, may be less a discovery and more a brilliant invention—a testament to human ingenuity in making sense of the world around us.
References

Dehaene, S. (2011). Our Numerical Heritage. In The Number Sense: How the Mind Creates Mathematics (pp. 3–78). New York, NY: Oxford University Press.

Hauser, M. (2000). What Do Animals Think About Numbers? American Scientist, 88(2), 144. doi:10.1511/2000.19.863

Lind, M. I., & Spagopoulou, F. (2018). Evolutionary consequences of epigenetic inheritance. Heredity, 121(3), 205–209. doi:10.1038/s414370180113y

Linnebo, Ø. (2018, January 18). Platonism in the Philosophy of Mathematics. Retrieved from https://plato.stanford.edu/entries/platonismmathematics/

Lipton, J. S., & Spelke, E. S. (2003). Origins of Number Sense. Psychological Science, 14(5), 396–401. doi:10.1111/14679280.01453

Mechner, F. (1958). Probability Relations within Response Sequences under Ratio Reinforcement. Journal of the Experimental Analysis of Behavior, 1(2), 109–121. doi:10.1901/jeab.1958.1109

Mechner, F., & Guevrekian, L. (1962). Effects of deprivation upon counting and timing in rats. Journal of the Experimental Analysis of Behavior, 5(4), 463–466. doi:10.1901/jeab.1962.5463

Meck, W. H., & Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9(3), 320–334. doi:10.1037//00977403.9.3.320

Pessoa, L., & Desimone, R. (2003). From Humble Neural Beginnings Comes Knowledge of Numbers. Neuron, 37(1), 4–6. doi:10.1016/s08966273(02)011790

Powledge, T. M. (2011). Behavioral Epigenetics: How Nurture Shapes Nature. BioScience, 61(8), 588–592. doi:10.1525/bio.2011.61.8.4

Schwartz, R. (1995). Is Mathematical Competence Innate? Philosophy of Science, 62(2), 227–240. doi:10.1086/289854

Starr, A., Libertus, M. E., & Brannon, E. M. (2013). Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences, 110(45), 18116–18120. doi:10.1073/pnas.1302751110

Suter, C. M., Boffelli, D., & Martin, D. I. (2013). A role for epigenetic inheritance in modern evolutionary theory? A comment in response to Dickins and Rahman. Proceedings of the Royal Society B: Biological Sciences, 280(1771). doi:10.1098/rspb.2013.0903

Wynn, K. (1992). Erratum: Addition and subtraction by human infants. Nature, 360(6406), 768–768. doi:10.1038/360768b0