God Created the Real Numbers

God created the real numbers 07-14-2025 7:37PM (ET) 08-28-2025 12:03AM (ET) (edited) W. Jherek Swanger writes in his introduction to Camillo Agrippa's "A Treatise on the Science of Arms, with a philosophical Dialogue (1553)" that: "Scienzia [...] was and is often held to relate only to the study of the eternal: that which exists in nature, or was created by God. Thus theology and astronomy/astrology are held to be sciences. Indeed, Ridolfo Capoferro held that strictly speaking fencing is not a science, but rather an art, because it was not divinely created." 16th Century Italy asks then if science is concerned only with divine (natural) knowledge and art is the realm of the secular and human-made, then was Leopold Kronecker right when he said: "God made the integers; all else is the work of man" The integers look little like the natural world and very much like the simple and powerful abstractions that humans find elegant. Much to the frustration of the human mind, that which created fig wasps, Leucochloridium paradoxum, quantum foam, and plasmids, does not create works of abstract simplicity. If such a divinity left fingerprints in mathematics, that inhuman complexity and irrational strangeness can be found in the reals but not the integers. One can imagine the God of the Book of Job creating the square root of two and pi. I have an imperfect rule of thumb for determining if something is of human construction or for the purposes of this exploration, divine. If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. If on the other hand it feels universal, simple, and ideal, it is the product of human effort. The Hierarchy of Weirdness This distinction science=divine/nature, art=made by human is more complex than just two separate categories. When humans create something, the created thing is contained, and is a function of, what already exists, so while art is the part of the universe that is created by humans, it also emerges from the part of the universe that isn't created by humans. The human idea of the triangle choke (sankaku-jime) only works because humans have arms, legs and necks. It wouldn't be meaningful to talk about it in a universe without chokeable necks, but necks existed before human ideas. So the idea of a triangle exists both because of the specific context of humanity and also because humans created it. More specifically, we can define a hierarchy of creation, or what a 16th Century Italian would call divine creation. At the top of the hierarchy we have eternal nature. Eternal nature gives rise to nature, which in turn gives rise to humans, which gives rise to human ideas. As shown below, the closer we get to eternal nature the greater the weirdness (from a human perspective). The real numbers are much closer to eternal nature, whereas the integers are much closer to human ideas. This is why to human mind the integers feel more perfect, feel more divine. A hammer feels like the perfect tool to hammer a nail not because a hammer is closer to eternal nature or more divine, but because it isn't divine. Hammers feel like the perfect tool because they were invented by humans specifically to solve the human problem of hammering nails. Kronecker, Cantor and God Kronecker's quote about the integers being divine was not made so much because Kronecker wanted to elevate the reputation of the integers but because he wanted to damn infinity. It was that reaction of horror at the creepy crawly bugs you see when you lift up the rock of Mathematics. The main target of Kronecker's campaign against infinity was the mathematician Cantor and his work on transfinites. Cantor viewed infinity in a much more positive light and he wasn't alone in this reaction. "From the paradise that Cantor created for us no-one shall be able to expel us." - David Hilbert (1926) Cantor's ideas also sparked a theological debate since if the mind can reason about infinity, perhaps the mind can also reason about God. Cantor himself was deeply invested in the theological implications of his mathematical work. He believed that God was beyond the mind and thus his mathematical work was not the result of his own effort but rather God was speaking through him. This is an inversion of God of the gaps where the gaps filled in by the God of the filling and the lack of gaps is evidence of God rather than the reverse. This echos Descartes' deeply unsatisfying Trademark Argument of the existence of God because finite beings like humans can perceive aspects of God like infinity, we must only be able to perceive them because God exists. "The idea of infinite substance, or God, must have “proceeded from some substance which really was infinite. [..] If a finite thing could produce the idea of an infinite thing, the Meditator reasons, this would violate the principle that there is more reality in a cause than in its effect, since the Idea of God is at the top of the Hierarchy of Ideas." - René Descartes, Meditations on First Philosophy, in which the existence of God and the immortality of the soul are demonstrated (1641) I much prefer the notion put forward by Chesterton that it is gaps themselves which are important. "The whole secret of mysticism is this: that man can understand everything by the help of what he does not understand. The morbid logician seeks to make everything lucid, and succeeds in making everything mysterious. The mystic allows one thing to be mysterious, and everything else becomes lucid" - Orthodoxy by G. K. Chesterton (1908) A Final Note Versions of this essay sat on my hard drive for the last few years and in private correspondence. I couldn't find a conclusion to draw these threads together so I'm leaving it as a vibes-based meandering through these ideas. This post originally came about because of a conversation with Madars Virza about my rejection of the Kronecker quote. I bounced it off of a number of other people including Andrew Poelstra, Russell O’Connor, Karina Poelstra, Peter Berard over the years. The observations of the real number line are taken from conversations with my dad, Ward Heilman. Hopefully I got it mostly correct, any errors are my own. Related Essays Ben Orlin - Why the Number Line Freaks Me Out (2016) - archived Joel David Hamkins - What are the real numbers, really? (2024) - archived Theology Of Georg Cantor (2014), Ochiai Hitoshi Russell O’Connor, How Dedekind Screwed Up a Hundred Years of Mathematics (2005)