The Mathematics of Wishful Thinking: On Category Theory and Its Mythology

Yesterday, a conversation with a friend turned to category theory—not its technical machinery, but the cultural aura surrounding it. We acknowledged its genuine value: as a precise language for describing structural similarities that appear across disparate mathematical domains, it allows mathematicians to recognize when a proof in one context might translate to another with minimal adjustment. It streamlines arguments by isolating the essential relational structure, stripping away irrelevant particulars. This is straightforwardly useful work, the kind that accumulates in the scrapbook of mathematical practice.

Yet we also noted how frequently category theory is invoked with expectations that outstrip its actual capabilities. The belief surfaces that mastering its concepts will somehow unlock solutions without the corresponding investment in domain-specific difficulty—that it offers a free lunch of insight. This is where the conversation shifted from useful tool to cultural phenomenon.

Later, encountering a tweet from Qiaochu Yuan crystallized the distinction: 'category theory is just some straightforwardly useful stuff for some purposes in some fields! you can elegantly simplify and streamline some proofs. then there is the mythology of category theory, which is some other thing entirely, mostly wishful thinking and projection afaict.' His phrasing gave name to what we had been circling—the separation between category theory as a collection of proven techniques and category theory as a screen onto which hopes for effortless understanding are projected.

This mythology manifests in several recognizable ways. Sometimes it appears as the assertion that category theory reveals 'the true nature' of mathematical objects, as if the categorical perspective were more real than the concrete constructions it describes. Other times, it takes the form of believing that categorical language alone can solve hard problems in algebraic geometry or quantum physics, ignoring that the hard work lies in verifying that the categorical framework actually applies to the specific situation at hand. The mythology treats the abstraction as a source of power rather than as a lens whose value depends entirely on what you bring to it.

Why does this persist? Partly, the abstraction itself feels profound. When you first see how diverse constructions—products in set theory, greatest common divisors in number theory, conjunctions in logic—all manifest as instances of the same categorical pattern (the product), it genuinely feels like uncovering a deep unity. This genuine insight can easily tip into overestimation of the framework's reach. Additionally, category theory's reputation for difficulty creates a halo effect; mastering something perceived as abstruse lends an aura of exclusivity that makes its pronouncements seem inherently weightier.

The implications extend beyond this single field. Any sufficiently powerful abstract framework risks accumulating similar mythology. Consider how 'blockchain' or 'neural networks' sometimes function in tech discourse—not as specific tools with defined trade-offs, but as incantations promising transformation regardless of context. The pattern is familiar: a genuine technical advance captures imagination, then accumulates projections of what it should be able to do rather than what it demonstrably can do under specified constraints.

Counter-perspectives exist, of course. Some argue that the mythology serves a generative function, motivating exploration that might not occur otherwise. Others contend that dismissing the mystical aspects ignores how mathematical progress often relies on intuition that precedes rigorous justification—where the 'wishful thinking' is actually a form of guided speculation. Yet even if we grant these points, the danger remains when the mythology displaces the actual work of verification and application. Category theory's strength lies not in revealing hidden truths independent of effort, but in organizing effort more effectively once the domain-specific groundwork has been laid.

The useful stuff remains useful: a well-honed pattern language for those who need to compare structures across fields. The mythology, meanwhile, persists as a reminder that our relationship to abstract tools is never purely instrumental. We do not merely use them; we live inside the stories we tell about what they mean. Recognizing the difference between the scrapbook and the sorcery is not to diminish either, but to use each where it belongs—applying the patterns where they fit, and examining the myths as cultural artifacts rather than mathematical techniques.