Mathematics is often seen by outsiders as an austere discipline of pure reason. The goal of this post is to present an interpretation of mathematics as literal magic. More precisely, the practice of mathematics bears striking similarities (vibes) to the magical tradition known as goetia, or sorcery.

Goetia is a type of European sorcery, often referred to as witchcraft, that has been transmitted through grimoires—books containing instructions for performing magical practices. […] Grimoires, also known as “books of spells” or “spellbooks”, serve as instructional manuals for various magical endeavors. They cover crafting magical objects, casting spells, performing divination, and summoning supernatural entities, such as angels, spirits, deities, and demons.

To list the vibes I’m referring to, sorcery is

• Primarily performed and transmitted through written text, often in esoteric languages/scripts. More abstractly, sorcery is inherently associated with language and communication.
• Can be used to craft objects with magical power and predict the future.
• Often involves trading with supernatural entities for power. More abstractly, the powers of sorcery emanate from outside the empirical world.
• Is associated with scholarship, as the classical image of the sorcerer is of a scholar spending his time in books.
• Can have a collegial aspect, where small groups of practitioners gather and protect knowledge, as well as transmit it to apprentices.
• Can have a religious aspect, where founding figures are the subject of worship and the cult’s secrets are closely guarded.
• Has historical roots in Ancient Egypt, Greece and Mesopotamia, notably via the figure of Hermes Trismegistus.
• In popular culture, certain individuals have an innate affinity for magic.
• Can involve diagrams drawn in chalk.

In summary, a sorcerer is someone who studies magical texts and uses language to access another realm beyond the empirical world and obtain power that can be used back in the empirical world.

Now let’s look at mathematics:

• Primarily performed and transmitted through written text in an esoteric script
• Can be used to design magical objects and predict the future.
• For most mathematicians, it involves interacting with an abstract world beyond the empirical one.
• Is associated with scholarship, as professional mathematicians are found mostly in universities and academia.
• Has a collegial aspect, where groups of mathematicians work on gathering and preserving mathematical knowledge, as well as transmitting it to younger generations via education.
• Has historical roots in Ancient Egypt, Greece and Mesopotamia.
• Can have a religious aspect, with the most notable examples being Pythagoreanism and Numerology.
• In popular culture, certain individuals are thought to have an innate ability for mathematics.
• Can involve diagrams drawn in chalk.

In summary, a mathematician can be seen as someone who studies mathematical texts and uses language to access another realm beyond the empirical one and obtain power that can be used back in the empirical world.

The analogy between mathematics and sorcery is by no means an original idea. I’m simply explicitly articulating my personal take on it. My main reason for doing so is that taking this analogy seriously lets one apply an aesthetic layer over the modern practice of mathematics as literally performing magic by striking deals with demons, which I find whimsically funny.

As a disclaimer, since I’m primarily working off vibes here, don’t expect everything to be factual. This is a midpost I’m doing for Halloween.

Ancient roots

Let’s start with the easy stuff. Mathematics and magical traditions, as do many other things, have roots¹ in ancient history. The early developments of arithmetic and geometry occurred in Mesopotamia and Egypt, and met the practical needs of those developing civilizations to raise taxes and keeping accounting records. At the same time, astronomy and time keeping was developed both for the needs of agriculture and religious reasons. Those areas were also the birthplace of writing.

The Greeks would later continue developing mathematics, formalizing results like Pythagoras’ theorem (which was known to the Mesopotamians, but not proven formally as far as we know today). The most notable figure of that time is Pythagoras, who famously developed a cosmology and philosophical system centered on numbers and geometry, and indeed would be worshiped long after his death up to the 1st century AD, with Pythagorean communities² being simultaneously schools and mystery cults lasting into the 4th century BC. While the Pythagorean beliefs did not include magical elements, they were profoundly philosophical and religious, and would have a great influence on later thinkers like Plato, and from there, on Western thought as a whole.

On the magical side of things, the major mythical figure here is Hermes Trismegistus, who was a syncretism of Thoth, the Egyptian God of Knowledge, Wisdom, Writing Science and Magic, among other things, and Hermes, the Greek god of (verbal) communication, travelers, thieves and merchants. This figure is pretty much the Godfather of the western magical tradition, through the texts collectively known as the Hermetica, and the associated school of thought known as Hermeticism.

The point of this otherwise mid historical exposition is to highlight the common thread between the history of mathematics and sorcery. They originated in the same places at the same time period and were likely discussed among the same class of people (at least this was the case in Egypt), so they were certainly co-evolving. We also see the common thread of writing and scholarship emerging. Prime material for us to make up some head cannon.

The Process of Mathematics as Sorcery

Let us begin our aesthetic reinterpretation of mathematics as a kind of magic by examining what mathematicians actually do with their time. We can roughly describe mathematical research as the formalization of abstract problems and intuitions into the language of mathematics.

When presented with a problem, a mathematician will start by laying down some definitions to make more precise her intuition and carve out the shape of the problem into formal language. Once this is done she will manipulate those definitions according to the rules of the formal language to obtain the desired result, often in the form of a theorem. Those manipulations may involve calling upon existing theorems (which we may think of as casting spells).

The bounty of this process is widely known. We routinely use mathematical language to describe concrete problems from the empirical world and manipulate these descriptions to obtain new knowledge about that problem. This is how we are able to design any kind of technology (magical artifacts), or predict the future by modeling an empirical system (divination).

On top of that, the prevailing philosophy among mathematicians is Platonism, which posits that mathematical objects exist in a world separate from the empirical one, populated with abstract ideas. While few would go as far as literally claiming there is another world beyond ours, to most mathematicians, mathematics still feels like a reality of its own, which does not submit to the human will, and is not created, but rather discovered.

Sure sounds like consorting with another world to gain power over the empirical one to me.

Modern Mathematical Education as Ritual Initiation

The canonical image of the Sorcerer involves guarding their secret and powerful knowledge from the hands of the unworthy, while taking on apprentices and disciples and progressively initiating them to the mysteries after they have proved they are worthy of them.

While this is not literally true of modern mathematical education, which is in principle accessible to all, it is still preferable to be mentored into it, especially as one gets into the deeper math, and we can still spin some tale about it.

Consider the standard mathematical curriculum taught at school, which comprises euclidean geometry, elementary algebra, trigonometry, elementary calculus and complex numbers, and elementary combinatorics and probability. Some of these topics have been taught since Ancient Greece, and far too often they are taught in the most mind-numbing, creativity-erasing manner possible, leaving many students viewing mathematics as a dry subject consisting solely in applying algorithms by hand.

For those who have managed to find enjoyment and exert their creativity in mathematics in spite of this mistreatment, and have entered university studies in mathematics or physics, the typical 1st year undergrad math courses await. They are taught Calculus all over again, this time with much more rigor and proofs. They are also taught Linear Algebra, which is their first taste of true abstraction, and where many remain confused.

Once they have proved their ability to handle abstraction by passing their exams, their minds are opened to the greater mysteries. They are taught that Real Analysis is just a special case of General Topology. They are taught Differential geometry, which assumes they have well understood the contents of their linear algebra course. They are also taught the sublime beauty of Complex Analysis and the less sublime one of Differential Equations.

From there, they will usually be taught the mysteries of Functional Analysis and Measure Theory, alongside a diverse range of topics depending on the university they have chosen, and their own interests. By that point, most of them know whether they will use what they have learned in the mundane world or if they will seek yet more arcane knowledge.

The Realms of Madness

“Almost everywhere” is what you see on signs as you approach the realm of the gods

To conclude, let us elaborate on what exactly mathematicians are dealing with when dealing with the Platonic realm. The power of a sorcerer is said to come from trading with angels, demons and deities, but what might be the mathematical equivalent to those?

Well, remember that we said that theorems were kinda like spells, in that they are crafted by the mathematician from simpler building blocks and can be invoked later. If we’re looking for the source of these powers, we need to look for the source of the mathematical language, namely axioms.

Axioms are logical statements accepted as evidently true within some formal language. Some of them, like the axioms of Euclidean geometry or the Peano axioms seem so close to our common intuition that we have no qualms about them (at least at first), so we may think of them as angels or as neutral deities.

Some axioms on the other hand, are simultaneously necessary to prove some powerful and important theorems (major spells), or construct powerful objects (summon powerful entities), but also let us summon results that go wildly against our intuition. We may think of them as demons, and the greatest and most feared demon of modern mathematics is none other than the axiom of choice, which is why some mathematicians try to avoid its use in their work.

If you read my previous post, you know that to know a mathematical object is to have a name for it. As in demonology, names are power. The more names we have for a mathematical object, the more we understand it and can use it. The vast majority of mathematical objects have few names that are too complicated for us mere mortals, or are so beyond our comprehension that we could not even name them.

Luckily for us, mathematical entities suffer from the same weakness as magical ones. They ultimately derive their power from human belief in them. A mathematician can thus banish demons from his work by refusing to believe in the axioms that can bring them forth. Doing so limits his powers, however, so the temptation to make a pact with demonic axioms always remains.