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Double, Double Toil and Trouble: The Spiritual Danger of Neglecting Mathematical Study
  • Curriculum
  • History
  • Maths and Sciences
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Melissa Neacsu

A terrible plague ravaged the population of Athens around 430 BC. This plague made its way to the small island of Delos, and its inhabitants consulted the oracle at Delphi, believing the plague to have been sent by Apollo. The shape of Apollo’s altar at Delphi was a perfect cube, and through recounted writings of the Greek mathematician Eratosthenes, we hear the oracle’s reply:

“… when the god proclaimed to the Delians through the oracle that, in order to get rid of [the] plague, they should construct an altar double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid…”

-Eratosthenes quoted by Theon of Smyrna as recounted by T L Heath in A History of Greek Mathematics I (Oxford, 1931), and finally, copied from the article “Doubling the Cube,” by O’Connor and Robinson, University of Saint Andrews, Scotland.

Thus, if the Greeks took the sides of Apollo’s altar to be 1 unit long, making the volume 1 cubic unit, they were tasked by Apollo to construct a new cubic altar with twice the volume, 2 cubic units.

The Greeks were perplexed, writes Eratosthenes, and rightly so, for such a construction is provably impossible to complete with compass and straightedge alone, which are the tools that mimic the axioms of pure Euclidean geometry. This “Delian problem,” also known as the problem of “doubling the cube,” became a famous open mathematical question of antiquity. Historical evidence supports that its solution was sought by many great minds, including Plato (yes, the Plato), Archytas, Eudoxus, Eratosthenes, and Hippocrates (no, not that Hippocrates). Several solutions were proposed, but each relied on methods outside the realm of compass and straightedge construction, making them logically unsatisfactory to the ancient Greeks. Plutarch recounts Plato’s admonishment of those proposing such impure, mechanical solutions, saying:

“…proceeding in this way, [does] not one lose irredeemably the best of geometry, by a regression to a level of the senses, which prevents one from creating and even perceiving the eternal and incorporeal images among which God is eternally god[?]”

-J Delattre and R Bkouche, Why Ruler and Compass?, in History of Mathematics : History of Problems (Paris, 1997), 89-113, copied from O’Connor and Robinson’s article.

So how does the story end? The Greeks of course were never able to double the altar of Apollo using pure geometry. The plague eventually waned, though not without influencing the outcome of the Peloponnesian War and assisting the decline of classical Greece. Over 2000 years later, Pierre Wantzel (1837) used the methods of abstract algebra established by Evariste Galois to prove that the cube root of 2 is an impossible length to construct within the confines of Euclidean geometry, hence proving that the Delian problem is unsolvable. It is highly relevant to note that Galois was in and out of prison due to his political activism during the French revolution and died in a duel over a girl at age 20. Wantzel died young as well at age 34 but without ever having been shot or having served a single prison sentence.

The god Apollo had set the Delians upon an impossible task. To make sense of this, they called upon another sort of oracle: Plato, naturally. (Though Plato is not thought to have been born earlier than 428 BC, be so kind as to ignore this fact for the sake of the narrative.)

“…they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.”

-Eratosthenes quoted by Theon of Smyrna as recounted by T L Heath in A History of Greek Mathematics I (Oxford, 1931), copied from O’Connor and Robinson’s article.

Several different versions of this story are given in the writings of mathematical historians, but this version is my favorite. Perhaps Apollo has a valid point. If so, why is the study of mathematics so important, to the point of its neglect being worthy of severe rebuke? To the ancient Greeks, it isn’t so that math can be used for something practical, like constructing an altar. Plato makes this clear in the second quote above. Rather, mathematical studies move us beyond ourselves towards something eternal and transcendent. According to Saint Augustine, mathematics even aids in the pursuit of wisdom, which he sees as fundamentally bound together with number:

“…given the fact that both wisdom and number are contained in that most hidden and certain truth, and that Scripture bears witness that the two are joined together, I very much wonder why most people consider wisdom valuable but have little respect for number. They are of course one and the same thing.”

                                                 -Augustine of Hippo, On Free Choice of the Will, copied from James Bradley’s paper, “An Augustinian Perspective on the Philosophy of Mathematics.”

For Augustine, numbers are ideas in the mind of God which have been used in the creation of the physical universe. There is therefore no extricating mathematical studies from the study of reality. As is often quoted from Galileo, “Mathematics is the language with which God has written the universe,” and the theoretical physicist Eugene Wigner continues this thought:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

                                                 -Eugene Wigner, “On the Unreasonable Effectiveness in Mathematics in the Natural Sciences,” (1960).

So in neglecting to study mathematics, what are we missing? We miss a direct connection to some of the very thoughts of God - those He deemed so precious that they seem to govern all creation. I won’t go so far as to say that those who do not or cannot study mathematics (as mathematics is typically understood) are at a loss for their souls, but for those of us to whom it is offered, there may very well be a spiritual danger in intentionally spurning this great gift.

“…by Thy Providence, Thou orderest the world…”

                                                -From an Orthodox prayer for use at Theophany and baptism

 

Acknowledgements and Links

This post relied heavily on the article entitled “Doubling the Cube,” from the University of Saint Andrews, Scotland. The reader is highly encouraged to peruse the article and explore the various “impure” geometric solutions to the Delian problem found by the great minds of antiquity.

https://mathshistory.st-andrews.ac.uk/HistTopics/Doubling_the_cube/#:~:text=There%20are%20two%20different%20accounts,gave%20a%20somewhat%20different%20version.

 

Thank you to Geogebra for their (free!) geometric software, which was used to create all images.

https://www.geogebra.org/3d?lang=en

 

Eugene Wigner’s essay “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences” has become a classic in the philosophy of math and science communities for its clear formulation of the mystery of math’s ubiquitousness and unexpected usefulness in the formation of scientific theories. If you are interested in mathematical or natural philosophy, it may be intellectually dangerous to neglect reading it.

https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

 

Finally, I am most appreciative of James Bradley for posting his thoughtful paper, “An Augustinian Perspective on the Philosophy of Mathematics.”

https://acmsonline.org/home2/wp-content/uploads/2016/05/Bradley-Augustinian.pdf