Mathematics can be terrifying. After slogging through thorny symbols and thickets of corollaries, wading neck-deep in lemmas, creeping past proofs of non-existence, you find only more imposing maths, foreboding and insurmountable. Yet, almost as an afterthought, you might glance at your ink-stained hands and see a glimmering stone, untouched by the muck of mind-numbing equations — a meaning that rises like a bird, humming with an inner light. This! This is forever true.
How is it that we, imperfect, rooted in the rut of the material world, can grasp this wisp of truth? Where did it come from? Is there some portent in the fact that our minds can comprehend it? Are we meant to find this truth, here, and follow it to some undiscovered terrain? Mathematics tests our strength, and yields up inexplicable beauty, a truth that calls us further into the morass, or to some place far above it.
The greatest of these gems are symmetries. Those are the forces which change a thing in just such a way that it returns to itself. A mystical hokey-pokey, symmetries turn your sums around, back to where they began. Our hands are in mirror symmetry — each is the reflection of the other, and reflecting one hand twice brings it back into alignment. The cube has rotational symmetry — a quarter turn in any cardinal direction rights the cube to the same position, again. Symmetries are the folds of reality, where the seam bunches and turns back on itself. They are the truth girding truths.
Beyond our geometries, there are deeper symmetries, too, scattered like sapphires amidst the thick plumbum of functions and transformations. Each symmetry we uncover is like a philosopher’s stone, solving the insoluble, curing the ills of incomprehensibility and incomputability. Mathematicians mine for symmetry, pay toll of symmetry to pass through the gates of Dis, loft symmetries to shine through the darkest murk while they claw toward greater truths.
So, here is a symmetry you never knew, which transforms a function back into itself, whatever function it may be. You will find it in no weighty text, on no Babylonian tablet. Yet, it has always been true, hovering just out of sight, only waiting to be known. I give it a name: The Reciprocal Inverse Derivative Symmetry. Here:
What does it mean? First, the symbol of the integral, the elongated ‘S’ at the forefront: this represents the area beneath a curve. The curve is defined by the function following that ‘S’, which I address in parts. ‘1/’ expresses the reciprocal, which slips toward zero as the following term rises toward the infinite; g’ is the derivative, the slope of the curve g; and, g^-1(x) is the inverse of the curve g, which is obtained by reflecting that curve across the diagonal; ‘dx’ completes the integral, defining it in terms of the variable x.
It is a strange concoction of symbols, yet it is in gentle balance: the integral and derivative are two sides of the same coin. If you take a curve, and find the integral of the derivative, you return to that same curve. The area and the slope undo each other. Together, they are their own symmetry. Meanwhile, the reciprocal and the inverse express two forms of opposition — the inverse of a curve’s inverse is that curve again. That dual-inversion is a self-symmetry! Equally, the reciprocal of a curve’s reciprocal is that curve, as well. Another self-symmetry.
It is beguiling that, together, these symmetries unite to form a greater symmetry. The integral of the reciprocal of the derivative of a curve, taking the inverse of that curve as input, returns you to the inverse! Each operation is undone by the others, like a cube spun around four different ways, returning to its starting place.
I arrived at this truth through strange abstractions: an attempt to formalize the transformation of integrals of one function into integrals of another function, by changing those integrals’ bounds. A few simple diagrams illuminate the transformation between integrals, and an integration of their terms produces the above relation. I leave those components and proof for a later day. Truth is only more radiant when wrapped in mystery.