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Philosophy of Mathematics: The Journey of Numbers from Tangible to Abstract

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        Numbers are universal. They’re present in every part of the world, and have been really important from the beginning (Boissoneault). Originally rooted in the tangible, numbers were once directly linked to physical objects: three oranges, ten trees, a hundred grains of rice. But as humanity progressed, so did our conception of numbers. Abstraction took over numbers and led us into a new era of understanding the physical world.

In ancient times, numbers were very tightly tied to the material world (Law). A farmer might use stones to keep track of his cattle, with each stone representing a cow. To trade, you had to engage in a bartering system, where physical goods were exchanged directly. However, as communities became more complicated, so did their trading and representation systems. Relying only on actual counting methods proved inconvenient. A more adaptable, scalable system was required. This prompted the transition from concrete counting systems to abstract number representation.


As abstraction took over numbers, they started to idealize concepts instead of physical objects. This abstraction wasn’t just a small development, however, it was a paradigm shift in thinking which led to deep mathematical and scientific findings. By detaching numbers from their physical counterparts, we were able to generalize a variety of problems, rather than thinking of them from a case-by-case basis.


Take the concept of currency, for example. As mentioned earlier, bartering systems were used to exchange goods and facilitate trade, but these systems also imposed a lot of restrictions regarding what a person could get with the resources they had (Anderson). If you had a couple of chickens, you might be unable to exchange them for cattle because of the trade being unavailable. However, with currency, an abstract form of value, there was freedom in purchase which led to the development of economic systems and the flourishing of civilization. With something as simple as a coin, value was no longer tied to specific objects but became a more fluid concept, adaptable to a variety of contexts (Law).


In the area of scientific investigation, the abstraction of numbers allowed us to analyze complex patterns (Boissoneault). These patterns led us to draw conclusions about the phenomena around us, and attain a greater understanding of the physical world. From working out the geometry of nature to using computer science through advanced machines, the abstract numbers and formulas have been influential to our modern world. They enable us to make generalizations about the world around us, analyze patterns about those generalizations, and use those patterns for an in-depth view of reality.


The transformation of numbers, from simple counting tools to abstract symbols, really underscores our human journey. Moving to a more abstract way of thinking about numbers didn't push us away from understanding our world. On the contrary, it handed us a richer toolkit to dive deeper, predict better, and truly grasp the world's complexities. It's a remarkable testament to our drive to learn and the inventive spirit we humans share.


Works Cited

Anderson, Somer. “Barter System vs. Currency System: What's the Difference?” Investopedia, 17 September 2023, https://www.investopedia.com/ask/answers/061615/what-difference-between-barter-and-currency-systems.asp. Accessed 10 October 2023.

Boissoneault, Lorraine. “How Humans Invented Numbers—And How Numbers Reshaped Our World.” Smithsonian Magazine, 13 March 2017, https://www.smithsonianmag.com/innovation/how-humans-invented-numbersand-how-numbers-reshaped-our-world-180962485/. Accessed 10 October 2023.

Law, Steven. “A brief history of numbers and counting, Part 1: Mathematics advanced with civilization.”
Deseret News, 5 August 2012, https://www.deseret.com/2012/8/5/20505112/a-brief-history-of-numbers-
and-counting-part-1-mathematics-advanced-with-civilization. Accessed 10 October 2023.

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