Mathematicians, list your ingredients!

A wise friend of mine once compared being a constructivist to being a vegetarian: even if you are convinced it is the right choice, it is hard to fully commit to it in practice. As a vegetarian and a (developing) constructivist myself, I have come to appreciate just how fitting that comparison is. Both philosophies involve conscious decisions not to use certain things, coming from a deeper commitment to a way of thinking. Just as vegetarians avoid meat, constructivists avoid non-constructive principles like the Law of the Excluded Middle or the Axiom of Choice.¹ Over the years, it has become easier and easier to follow a vegetarian diet and eating meat is (in certain circles) no longer the norm. Perhaps this analogy can be used to our advantage: what we can learn from this development? How could we make it easier to be a constructivist in a world where one often assumes everyone eats “meat” by default?

One small but transformative change that made the life of a vegetarian easier was the rise of labeling. Suddenly, you no longer had to interrogate the waiter or read through ingredient lists with a magnifying glass — a small green “V” quietly told you what was safe to eat. The system did not require anyone to stop eating meat; it simply acknowledged that not everyone does. In mathematics, we are not quite there yet. Most results are served without a list of logical ingredients. Some assumptions, like the Axiom of Choice, tend to come with a warning label, but others, like the Law of the Excluded Middle, are folded so deeply into the dough of standard reasoning that they are rarely mentioned at all.

For the omnivores in this world, this is unproblematic, since they do not have to watch what they eat. However, for those who follow a constructive diet, this makes the landscape hard to navigate. One may only realize a proof contains meat when it is already on their plate — provided it has not been so thoroughly processed that you can no longer tell. So, maybe we can take inspiration from the transparency movement in the food industry and start labeling our results; not to restrict, but to inform. Not only would it be helpful to constructivists, it would give any mathematician more insight into what depends on what, and remind all of us that logical assumptions are not neutral background noise: they shape what can be said, and how.

Of course, not everyone is aware of the assumptions underlying their proofs — largely because we are not trained to recognize them. But that very invisibility is part of the problem: how can we be precise in our reasoning if we are unaware of the tools we rely on? As mathematicians, we value precision, so it only makes sense to be exact about the very foundations we build upon.

Once we start listing our ingredients, a natural next step follows: we begin to wonder which recipes succeed without certain controversial components. We try reworking them, testing whether the dish remains just as satisfying when prepared with alternative methods. This does not only benefit constructivists — it enriches our collective mathematical cuisine. Sometimes, we discover that the meat was never essential. Other times, it proves so central that removing it transforms the result entirely. But even then, the exercise teaches us something important: apparently, this dish is unavailable to vegetarians. On the bright side, when we do manage to find a constructive proof of a formerly classical result, the dish becomes one that everyone can enjoy. Classical mathematicians can still add the Law of the Excluded Middle to taste — maybe to enrich the flavor of the result — but the base recipe is now accessible to all.

If we care about foundations — and as logicians, I certainly hope that we do — then we should care about transparency. So perhaps it is time we start treating our proofs a little more like recipes. Let us label them as being constructive or not, adapt them, and occasionally rethink them from a position that steps away from the norms - after all, history shows us that what is common practice is not always the most desirable or enlightened way forward. If we are transparent about our ingredients, everyone, regardless of their diet, knows exactly what they are being served. And who knows? Maybe one day, using excluded middle in a proof will raise as many eyebrows as pineapple on pizza — an unexpected ingredient that sparks a heated debate.