Abstract:
Throughout history, mathematical abstraction has been met with suspicion, and sometimes even murder. This post explores the historical resistance to replacing physical reality with symbols, from the abacus to algebra. I finally explore how my summer research project is connected to this mathematical “conspiracy”, outlining how the complicated physical phenomenon of water waves travelling in canals is transformed into a mathematical model.
The Power of Abstraction
Samuel Walsh, University of Newcastle
Around the 12th century CE, the Arabic numerals were gradually spreading to Europe. To the average accountant or merchant, they seemed unnecessarily complicated. The abacus had worked for generations, why replace something physical with an abstraction?
And yet, slowly, the system spread. Calculations moved from physical beads to symbols on paper.
Centuries later during the time of French occupation in Italy, Neapolitan mathematicians were untrusting of France’s “modern mathematics”. The Neapolitans studied what they saw as the purest form of the discipline and the true foundation of mathematics: geometry. Despite this, the French were making leaps and bounds solving countless problems with their algebraic techniques, and in doing so had abstracted the foundation of mathematics, removing its real-world ties.
And yet, despite the concern of the Neapolitans, today many branches of mathematics – such as calculus – are detached from geometry.
A millennium earlier, the Pythagoreans went to extreme lengths when one of their own proved the existence of irrational numbers – a number which cannot be expressed by ratios. It is said they drowned him. To them the world and numbers were deeply linked and built on ratios, and an abstraction would break this link.
And yet despite their efforts, irrational numbers exist.
Throughout history, it has seemed like abstraction has pushed mathematics forward, but at the same time has been met with resistance. Beads were replaced by pen and paper, quantities by letters, and shapes by symbols. Leaps forward always seem to come by severing a link to reality.
There is no easy way to say this:
There is a secret group of mathematicians that are plotting to remove the world, one abstraction at a time. An underground society that has existed for thousands of years.
First it was the decimals, a physical bead on a rod was replaced with symbols on a page.
Then they took our quantities and replaced them with symbols. The alphas, the betas. “x represents an unknown” they say, “solving for y” they say.
Have you ever seen an unknown?
They’ll look you dead in the eye and tell you so-called “imaginary numbers” are real. Tell you that the “Fourier transform” is useful. That waves – waves! – are “vectors” in a “space”.
Have you ever seen a vector?
When was the last time you went to the shops and saw something listed for an imaginary price?
Don’t believe their lies.
Or, at least, that’s how it feels sometimes.
The truth is that I am one of them. Over the summer I analysed how canals of water waves behave when they meet at a point. My research took a physical model grounded in the real-world behaviour of water waves and transformed it to a space where the problem was rephrased in terms of lines and circles on the plane. Seems a lot simpler right? Abstraction can seem scary, sinister even, like a magical way for mathematicians to make everything harder and full of jargon. But – perhaps counter intuitively – it is the ultimate tool for understanding the real world.
Samuel Walsh
The University of Newcastle
