1. Mathemagics

1.1.1 Numerologists seek insight in the patterned appearance of symbols.

1.1.2 Could this description not also describe a mathematician?

1.1.3 Mathematics and magic had little distinction in 1563, when the Council of Trent “took a hard line against [all] ‘mathemagic’” (referring predominantly to numerology).1

1.1.4 Nevertheless, while numerology has been thoroughly debunked, mathematics remains celebrated across popular culture and within the academy. To what can we attribute a fracture in which one discipline was repudiated and the other embraced—and was something perhaps lost in their disjunction?

1.1.5 Tracking symbols and values across various media, numerologists claim to draw meaning from their exceptional appearance. 666, the number of the beast, forebodes the presence of evil in Christian tradition, while for ancient Egyptions, 3 was said to represent plurality.2 And in China, the I Ching’s 64 hexagrams have played a central role in divination practices since the 9th century BC.

1.1.6 Likewise, mathematicians take interest in specific objects: π, the natural number e, the golden ratio, zero—but also the Mersenne numbers, the Woodall numbers, and the glorious primes.

1.1.7 These numbers hold significance, not due to qualities observed in isolation, but because they fit within deeper patterns and theories and patterns. π and e emerge widely across various areas of mathematics, primes are exactly that which is left from integer factorization.

1.2.1 For mathematicians, the more general a theory, the more powerful. A specific case is useful insofar as it can be extended. An abbreviation commonly found in mathematical proofs is WLOG, or, “without loss of generality.” WLOG is used in cases where the scope of the preceding discussion is narrowed, but the principle logic holds true for the entirety of the original domain.

1.2.2 In this case, the anecdote spans the theory.

1.3.1 Where numerologists focus on the significance of a particular incidence, mathematicians are interested in generality, structures which render patterns, rather than the patterns themselves.

1.3.2 This subtle—yet crucial—distinction separates math from mysticism.


2. Formulae For The Heavens

2.1.1 The Chinese had developed an effective base-ten system by the 2nd millennium BCE, and we find numerology in the historical record shortly thereafter:

there was a pervasive fascination with numbers and mathematical patterns in ancient China, and different numbers were believed to have cosmic significance. In particular, magic squares - squares of numbers where each row, column and diagonal added up to the same total - were regarded as having great spiritual and religious significance.”3

2.1.2 The three “predictive arts” appear in China after the third century BCE.a These divination techniques apply specific rules and cosmological patterns to numerical tables and a 64-part hexagram (1.1.5) to inform decision-making on topics ranging from geopolitical conquest to marital relations.

2.1.3 Reverence for the heavens and a desire to understand celestial mechanics were intertwined for much of history. In Dr. Gustav-Adolf Schoener’s essays on Astrology, he writes: “it is […] clear how closely the cultic reverence of the heavenly bodies is tied to observational science, how astrology wants to be religion and science at the same time.”4

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2.2.1 Of course interest in the physics of the cosmos was not limited to the East. Much of Sir Isaac Newton’s seminal Principia focused on codifying the laws which govern celestial bodies, building on Kepler’s influential work describing orbital motion. In essence, Newton’s formulation of calculus was a symbolic language for describing the heavens.

2.2.2 Though nonconformist in many respects, Newton remained committed to the Christian faith, maintaining that “gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done.” Studying the mechanics of the heavens was foremost a window into the work of God.

2.2.3 It wasn’t until the late Renaissance that astrology and astronomy became distinct fields. Newton’s mathematical physics grounded the discipline of astronomy, and subsequent work pushed out astrological teachings in favor of more descriptively powerful numerical techniques. A holistic “view of the world and man since scholastic times [had] been destroyed, depriving religion […] of a philosophical standpoint within a scientific consensus.”5

2.2.4 Application of the scientific method deepened the division between mechanistic and astrological understandings of the universe, meanwhile the underlying mathematical language became increasingly secular.

2.3.1 But symbolic language also carries a sense of runic magic. By manipulating symbols, mathematicians encode thought into forms that facilitate problem-solving and communication. Symbololgy is capable of capturing an abstract argument without reduction to anecdote.

2.3.2 However, access to information in symbolic language depends entirely on the possibility of decoding meaning. And just as some spoken or written languages resist legibility due to complex syntax, mathematics runs the risk of esotericism when the notation itself presents barriers to comprehension.

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2.3.3 The notation used in mathematics actively grows and changes, leaving inconsistencies among common symbols which often impede legibility, an issue made all the more difficult by mathematicians’ frequent abuse of notation.b Noted in a recent segment by Grant Sanderson on math notation, even elementary operations such as exponents have “counterintuitive and redundant standardised notation systems [which explain] why many students become overwhelmed by mathematics”.6

2.3.4 The problem gets worse, as the cryptic language of mathematics is also used to disguise the unsatisfactory analytical power of untested models in other fields as well.7 Alan Jay Levinovitz draws a parallel between today’s economic obscurantism and that of Chinese astrology circa 1st century BCE (2.1.3) in that a combination of institutional prestige, linguistic complexity, and appeals to “rationality” often serve to safeguard the field and its conclusions from criticism.

2.3.5 The rarified language of mathematics has slowly drifted from its roots in experimental reproducibility. With little remaining grounding in the observable world, and persistence of the notion that “math is hard,” public access to mathematical decision-making is greatly limited.

2.3.6 “Sufficiently advanced technology is indistinguishable from magic.”8 Indeed, in 2008, amidst a chaos of optimistic and contradictory economic models, attempts were made to avoid “paying debts by claiming that the economic meltdown was a “force majeure”—the legal equivalent, basically, of an act of God—and not a logical outcome of a set of observable circumstances.”9

2.3.7 Without a causal understanding, people are prone to magical thinking—trusting outcomes without understanding the underlying mechanics. To those outside limited technical circles, research conclusions can appear as if conjured.


3. Conjuring Platonics

3.1.1 Students taking algebra in the American Common Core curriculum are taught to start most of their answers with a let statement, as in “Let x equal 5.”

3.1.2 Similarly, math papers are typically written in the imperative tone: Let x be this, then that must follow.

3.1.3 Invocation is the magician’s trade. “Abracadabra” derives from the Aramaic expression, “I create as I speak,”c and has long been used to declare the act of pulling objects out of thin air.10

3.1.4 The ontological status of mathematical objects, i.e. whether the number “five” exists in the world, depends on one’s view of platonic forms.

3.1.5 Platonism posits that a number’s existence is independent of one’s conception: even without consciousness, numbers would intrinsically exist. An opposing view, fictionalism, holds that a mathematical objects arise in the way fictional characters do, as figments with conceptual presence.

3.1.6 Both platonic and fictionalist camps acknowledge (for different reasons) the existence of an abstract x, regardless its physical realization. Let x be.

3.1.7 Abracadabra.

3.2.1 Mathematical proofs follow a logical chain from statements taken as true (axioms and definitions) to a conclusion. Mathematicians trust that recording symbols effectively instantiate the mathematical object.

3.2.2 Nevertheless, proofs must ultimately be read and interpreted to enter the discourse.

3.2.3 Famously, Shinichi Mochizuki’s proof of the abc conjecture has resisted comprehension11 since its presentation in 2012. Only when a proof has been vetted by peers will it be deemed valid.

3.3.1 Many contemporary standards for proof writing grew out of the Bourbaki school, a collective of mathematicians who published under the pseudonym Nicolas Bourbaki throughout the 1900s. Their work developed set theory towards an airtight system of logic, with the aim of building a universal ruleset with perfect internal consistency.

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3.3.2 However, one shouldn’t presume mathematical symbology works purely for the sake of logic and argument. Nina Samuel in The Islands of Benoît Mandelbrot argues that a symbol invented by the Bourbaki school, the S-curve,12 marks a “dangerous turn” in the proof’s argument:

The Bourbakist S-curve became a kind of hex sign that called upon the magical power of the pictorial in a way reminiscent of the ancient stylized appropriation of snakes as a cultic means of defending against them.

Some readers may sympathize with the so-called turn, and others may simply ignore it. But where many symbols act to invoke objects, the S-curve produces a feeling—an nod to the reader, explicitly calling upon elements beyond the written proof.

3.3.3 With the S-curve, the Bourbaki, evangelists of formalism, introduced interpretive incantations into their symbolism. But even without the S-Curve, contextual interpretation is inescapable, the meaning of abstract mathematical objects is built up through shared understanding. Despite the utmost rigor, the mathematician must eventually make an appeal to subjectivity.

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4. Divine Abstraction

4.1.1 Kurt Gödel is principally remembered for his Incompleteness Theorems which demonstrated that a consistent logical system cannot prove itself true: that axioms must be taken on faith.

4.1.2 But axiomatic systems are leaky. Sloppy symbols are invoked (2.3.3), implicit information is presumed (3.3.2), and standards of proof are socially constructed (3.2.2). In fact, a whole field of Ethnomathematics arose to study these issues.13

4.1.3 The Bourbaki dream was defeated. Yet, research into formal systems proceeds uninterrupted (and remains useful), quietly perpetuating the Bourbaki’s absolutist creed.14

4.2.1 Alexander Grothendieck, who collaborated with the Bourbaki school early in his career, is widely credited with establishing a common set of tools for category theory, helping bring the field into prominence.

4.2.2 Contemporary category theory focuses on highly abstract structures among mathematical objects, formalizing patterns that appear across various distant areas of mathematics. And many category-theoretic results manifest as singular statements which express general behavior across multiple domains.

4.2.3 Since Grothendieck, categorical tools have been used to represent fundamental structures across physics, ecology, and computer science.d Applications of category theory at the intersections of observable reality and a platonic ideal lend support to belief that the universe is governed by unified mathematical structure.

4.3.1 To a category theorist, abstraction is sacred. Systems which progress further up the ladder of abstraction15 permit broader understanding, greater inclusivity, and the potential for deeper descriptive power—a God’s-eye view.

4.3.2 (I’m also told acid trips are an occasional topic of conversation at category theory conferences.)

4.3.3 Grothendieck’s extraordinary attention to the practice of mathematics had existential effects. “The extremity of Grothendieck’s focus on mathematics is one reason for the ‘spiritual stagnation’ he referred to in [his seminal works], which is one of the reasons behind his departure, in 1970, from the world of mathematics in which he had been a leading figure.”16

4.3.4 After numerous productive and heated decades, Grothendieck eventually “condemned [Bourbaki] formalism to hell as being inimical to life,”17 and withdrew from mathematical work, instead focusing his attention on political action outside the academy.

4.3.5 Despite its utility, Grothendieck saw formalism as too far removed from the vitality of human life, which he felt was increasingly threatened at the close of the 20th century.


5. QED

5.1.1 Mathematicians find beauty in studying the heavens, logics, and the limits of abstraction. Thus, the history of mathematics is also a history of mysticism, of symbolism, of gestures toward something more fundamental.

5.1.2 The same could be said of religious practice.

5.1.3 We have now positioned mathematics as a technology of symbolic incantation, whose implementation rests on faith.

5.1.4 No matter one’s metaphysical beliefs, mathematics is inseparable from the world of humans—its motivations, language, and means of abstraction all deeply culturally embedded.

5.1.5 Mathematical generalizations grant power but they must come with context, for human experience is fundamentally one of narrative and anecdote. Allowing for context also concedes that math is wrought with implicit faiths, biases, and subjective logics. This is not to devalue mathematical work, but rather to pull back the veil of mysticism.

5.2.1 It is a mystique of absolutism, built through culture, symbolism, and language—a practice of obscurantism that only serves to enshrine institutions by perpetuating stigma.

5.2.2 There is a vital magic to mathematics—its incantations and explanatory powers available if only we care to look. Let us then recognize the unreasonable effectiveness of abstraction, acknowledging the dangers and joys therein as implicit to the construction of imagination.

5.2.3 Henri Poincaré once said, ”if God speaks to man, it is in the language of mathematics.”



Lukas WinklerPrins is a mathematician and teacher living in Los Angeles.

Cover image: Leibniz’ copy of the I Ching hexagrams.


References

  1. These are the Tai Yi Shen Shu, Da Liu Ren, and Qi Men Dun Jia.

  2. When a notation is used in a way that is legible to the reader but formally incorrect.

  3. The origin of the term “abra­cadabra” is disputed, however—possibly Aramaic or Hebrew.

  4. I recom­mend looking at the work of Dr David Spivak at MIT or Dr John Baez at UC River­side for examples of applied category theory work.

  1. Davis, Philip J. A Brief Look at Mathematics and Theology. Humanistic Mathematics Network Journal, 14th ser., no. 27 (January 1, 2004).

  2. Semat, Aude. Art and Power in Ancient Egypt. Art and Power in Ancient Egypt. March 2016. Accessed July 20, 2017.

  3. Mastin, Luke. Chinese Mathematics. The Story of Mathematics. 2010. Accessed July 20, 2017.

  4. Schoener, Gustav-Adolf. Astrology: Between Religion and the Empirical. Esoterica. 2002. Accessed July 20, 2017.

  5. Harald, Walach Reconciling science and spirituality in the twenty-first century: Elements for a worldview incorporating spiritual and scientific insights. Studies in Spirituality. 2005. 13th ser., 283-308.

  6. Sanderson, Grant. Maths notation is needlessly complex. It can and should be better. Aeon. June 2016. Accessed July 20, 2017.

  7. Levinovitz, Alan Jay. The new astrology. Aeon. April 04, 2016. July 20, 2017.

  8. Clarke, Arthur C. Profiles of the Future: An Inquiry into the Limits of the Possible. London: Victor Gollancz, 1982.

  9. Paumgarten, Nick. The Death of Kings. New Yorker. May 18, 2009. July 20, 2017.

  10. Castelvecchi, Davide. The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof. Nature. October 17, 2015. July 20, 2017.

  11. Philologos. Are the Origins of “Abracadabra” Jewish?.Mosaic Magazine. July 2017. July 20, 2017.

  12. Bourbaki dangerous bend symbol. Wikipedia. July 20, 2017.

  13. Turkle, Sherry. Epistemological Pluralism and the Revaluation of the Concrete. Journal of Mathematical Behavior. March 1992. Vol. 11. No 1.

  14. Harald, Walach Reconciling science and spirituality in the twenty-first century: Elements for a worldview incorporating spiritual and scientific insights. Studies in Spirituality. 2005. 13th ser., 283-308.

  15. Victor, Bret. Up and Down the Ladder of Abstraction. worrydream.com. October, 2011. July 20, 2017.

  16. Jackson, Allyn. Comme Appelé du Néant—As If Summoned from the Void: The Life of Alexandre Grothendieck. Notices of the American Mathematical Society. March, 2016. Vol. 51. No. 10.

  17. Samuel, Nina. The Islands of Benoît Mandelbrot. New York (NY): Bard Graduate Center, 2012.


Thanks to

  • Sam Hart, for his emphatic and supportive editing.
  • NS, DV, and more friends for their thoughts and support.
  • Jordan Ellenberg and Acid Surfing for the inspiration.