“Mathematics is a game played according to certain simple rules with meaningless marks on paper.” – David Hilbert
Two intertwined musical careers – those of Pyotr Illych Tchaikovsky and Claude Debussy – came to a climactic head in 1885. The towering Russian Romanticist, Tchaikovsky, premiered his Orchestral Suite No. 3 in Saint Petersburg to overwhelming adulation. In the same year, Debussy, the revolutionary French Impressionist, won the prestigious Prix De Rome piano composition contest and began his work under a royal scholarship at the French Academy in Rome.
The mournful, melodic violin solos of Romanticism and the experimental tonalities of Impressionism contrasted drastically. “Not a single idea is expressed fully, the form is terribly shriveled, and it lacks unity,” Tchaikovsky once wrote of one of Debussy’s works. “Do you not remember the… music, able to express every shade of meaning,” Debussy himself reminisced, “which makes our tonic and dominant seem like ghosts?” 
Music, though, is not alone as a discipline of schools and schisms. Decades later, a similar division began to form in mathematics: the Platonists, led by Kurt Gödel, and the Intuitionists, led by L. E. J. Brouwer, began to stretch the very laws of logic themselves. Mathematics – just like music – became a house divided.
In this article, we take a tour of the fascinating and diverse branches of mathematical thought.
The first article in this series investigated the corollaries of a given set of mathematical axioms. Do these initial conditions fix the resulting theorems? We found that though the ensuing mathematical results might be predetermined, the methods and constructs of their discovery are certainly not. The second article asked about the scope of this process. Is math infinite? By adding and accumulating these axioms, we learned, our mathematical universe can continue to expand virtually indefinitely. Now, we ask a practical question. Where can these different axioms take us? We’ll investigate particular examples of axiomatic divergences and their mathematical ramifications.
I’ll first note that many possible axiomatic starting points are utterly meaningless. Beginning with an axiomatic system in which both some statement and its negation are true, in fact, we can quickly go on show that absolutely every statement becomes true!  These axiomatic systems are called “inconsistent”, and they’re the dread of set-theorists. Even within our well-accepted consistent (or so we hope) systems, though, much room for disagreement remains.
The Platonists, led by famed logician Kurt Gödel, argued that “mathematical objects and concepts are not constructed by humans,” but rather are “as objective as physical objects and properties.” Gödel even famously claimed to stand in a “quasi-perceptual relation” with these mathematical entities, through his incredible mathematical intuition. 
The Platonists’ faith in this abstract, pre-existing mathematical world led to fascinating mathematical consequences. The Platonists embraced the provocative Axiom of Choice , which – though initially formalized by Zermelo in 1904 as an “unobjectionable mathematical principle” – led, ultimately, to proofs of abstract Platonic phenomena generating, in mathematicians, everything from reverence to revulsion. These proofs typically featured assertions of the existence of objects which we simultaneously had no hope of ever finding – or, as they’re called in mathematics, non-constructive existence proofs. The apparently “unobjectionable” Axiom of Choice, for example, was used to prove that a ball in three-dimensional space can be cut into a finite number of pieces, taken apart, and re-assembled into two identical copies of the same ball!  (The “pieces” aren’t actually solid, but rather infinite scatterings of points.)
These procedures aren’t within our grasp, of course. They merely exist within the ephemeral Platonic universe. That’s why the Platonist insistence on their abstract mathematical world was so fascinating. That’s also why it was so contentious. In the Axiom’s early days, in fact, many mathematicians wholeheartedly rejected it, and proofs were often considered superior if they did not use the Axiom of Choice. (Now, it’s widely accepted.)
The Intuitionists, led by L. E. J. Brouwer in the early twentieth century, took an opposite tack. The Intuitionists claimed that “mathematical meaning is a mental construction,” arising from its human creators alone.  Math didn’t extend past the humans creating it.
Under the Intuitionists’ purview, math began to assume its own distinct form. The Intuitionists rejected not only the Axiom of Choice, but non-constructive existence proofs altogether. Mathematical proof, to the Intuitionists, only made sense if it not only argued that a mathematical object exists but also provided a procedure to construct it in a finite number of steps. If a human can’t produce it, they argued, – well – it’s nonsensical to talk about its existence!
The Intuitionists actually began a project to rebuild math from its beginnings using only constructivist methods. They made considerable progress. A comparison of, for example, classical mathematical analysis with the Intuitionists’ constructive analysis reveals fascinating differences. A fundamental – and very intuitively plausible – theorem in analysis asserts, roughly, that if a function crosses the x-axis, it must at some point touch the x-axis. The Intuitionists, though, – despite the theorem’s apparent obviousness – refused to accept it! (Notice the non-constructive existence statement.) The Intuitionists, ever scrupulous, instead suggested a weaker version: that given any desired closeness to zero, we can, in a finite number of steps, produce a point which achieves that closeness. Intuitionist mathematics is unfamiliar territory indeed.
The Intuitionists, by rejecting a handful of key axioms, created a mathematics which – though careful and modest – existed entirely within human control.
These dogmatic divisions might offer us profound insight into how humans perceive and understand mathematical truth.
Have the Intuitionists given up on mathematical truth altogether? Certainly not! The Intuitionists surely believed, in fact, that their local, human-constructed results were still true – quite as true, perhaps, as the Platonists considered their more grandiose non-constructive existence statements. Both camps felt their statements to be true. The difference lay in the scope and breadth of their conclusions. How can we make sense of these competing conceptions of mathematical truth?
Humans must collectively agree upon a system of starting points, axioms, as well as a set of logical principles by which we might move from these initial axioms to their eventual mathematical corollaries. We might call these foundations a mathematical system. The statements ultimately regarded true by the mathematicians, then, are precisely those statements which are produced within and sanctioned by the mathematical system.
Which system better captures mathematical truth is perhaps a question for the philosophers. In a way, though, the choice doesn’t matter. The statements or theorems produced from within a mathematical system are – no matter the system – nothing more than the results of formal manipulations of symbols on paper. Are these results merely local? Or do these they accurately represent a larger mathematical universe?
Well, do you enjoy Tchaikovsky or Debussy?
- Tchaikovsky: The Man and his Music
- Music: An Appreciation
- An excellent and amusing explication of the fact that all statements are true in an inconsistent system.
- The Stanford Encyclopedia of Philosophy’s “Philosophy of Mathematics”
- The Axiom of Choice
- Creating two spheres from one: The Banach-Tarski Paradox