Is there a limit to the number of “true facts” contained in the discipline of mathematics? If (for the sake of argument) humans were around indefinitely, would the discipline ever end? Would we, one day, proclaim that we’d “reached the bottom”? Or does the field continue indefinitely into the depth, reaching arbitrary levels of complexity? If this is the case, where do these true facts come from, and why do they exist?
What will be the math of the future? What will math look like, a hundred years from now, or five thousand, or (again hypothesizing that we’ll be around indefinitely) even a hundred thousand?
These questions make math seem like a strange, magical and universal discipline. The pursuit of their answers has long eluded me. Here, I finally make some attempts.
Previously, we investigated how we make our way through the thick universe of mathematics. Is our path predestined, lighted by logic itself, waiting, only, to be discovered in the dark? Or do we ourselves light this wilderness, illuminating it with blazing color and creativity? We decided something intermediate: that though this light’s pathway may be predetermined, its color and feeling are determined alone by humanity.
Now, we ask a different question: where does this path lead? Does it have an end, and what could this end look like?
Proliferation of Axioms
We first investigate a basic question: whether arbitrary axioms indefinitely produce logical corollaries. In other words, given a single, particular set of starting principles, or axioms, is there an infinite supply of “true” facts which will logically follow from this set of axioms? This fundamental question will be important in answering our larger one.
I’m tempted to say that the answer is no. Each system of axioms produces a “space of consequences”, we might imagine, a universe of ideas which depend on these axioms. Upon reflection, however, it seems quite difficult to argue that this “space of ideas” could be infinite! Sure, this space may be massive, and very complex. The light has been shined into a “deep cave”, so to speak. But to argue endlessness would be to claim that these axioms trigger ideas, which then trigger more ideas, and so on, cascading indefinitely in an infinite chain reaction. From whence could the “material” for this continuation come? No cave – and nothing in our known universe, for that matter – is infinite. Ideas could in principle be infinite, probing endlessly through logic itself. But this would demand the existence of forces beyond those which I can explain.
Examples seem to support my hypothesis. Point-set topology depends on a miniscule, and highly abstract, list of axioms. The field has produced quite deep results, and remains extremely fundamental throughout all of mathematics. It’s largely considered, however, to be all-but solved. Group theory, too, rests on an elegant and extremely compact collection of axioms. Mathematicians, however, recently completed a hundred-year-old effort to classify every possible finite simple group: every possible direct ramification of these axioms (1). The cave was massive – the classification “theorem” spans tens of thousands of pages in hundreds of journal articles by hundreds of authors – but it was by no means infinite. The flashlight illuminates distant walls.
We’ve established that a particular set of axioms might be exhausted. What about math in general? Here, the picture seems much more subtle. Before, only logic itself extended outwards from the axiomatic core. Now, however, humans build on this very core itself, adjoining new structures, new properties, and new relations. These additions must logically cooperate with their predecessors. Other than this, however, the room for expansion seems unlimited.
Previously, we asked “to what extent must something logically follow.” Now, we ask “to what extent could something logically follow.” These questions are very different. Their answers seem different, too. Indeed, new structures can always be defined. New structures constantly indeed are defined, and these structures – in the utterly awe-inspiring world of “modern mathematics” – reach complexity unimaginably staggering.
Each new augmentation might produce only a finite extension. But the number of augmentations could be infinite.
Interests and Transitions
Will it be? Here, we must ask a very different question: will humans continue to find reason to accumulate augmentations? This is perhaps the hardest question of all. Our task has changed from philosophical epistemology to predicting the future!
The answer, thankfully, seems to be yes. Math, we must remember, has endless (and often mysterious) applications – in physics, computing, biology and more. Much more importantly, however, mathematicians explore this world for its own sake. They explore the mathematical universe because it’s interesting. The magnificence of this universe will only continue to explode.
In the truly long run? It’s hard to tell. We’re over our heads! Perhaps math will be phased out in favor of a new discipline altogether – super-ultra-logic, say. Perhaps things will change in a still-more-unpredictable ways. We can’t know.
Math’s warriors, however, push forward undeterred. I’ll continue to do the same. Onward I charge into the black depths. Onward! Onward into the impossible abyss!