From now on an axiomatic accompanies the scientific process. We have written the accompaniment after the melody, yet the mathematician plays with both hands. And it’s a completely new way of playing; it requires multiple plans of consciousness, a subconscious affected yet acting. It is far too simple to constantly repeat that the mathematician doesn’t know what he manipulates; actually, he pretends not to know; he must manipulate the objects as if he didn’t know them; he represses his intuition; he sublimates his experience.
Le nouvel esprit scientifique p 52, 1934, Gaston Bachelard
Here Bachelard is analyzing the development of non-euclidean geometries. His point is that the biggest hurdle to discover these new geometries was psychological: euclidean geometry is such a natural fit with our immediate experience that we intuitively give it the essence of geometry. It’s no more a tool or a concept engineered for practical applications, but a real ontological property of the physical world.
Faced with such a psychologically entrenched concept, what recourse do we have? Formalization, answers Bachelard.
For formalization explicitly refuses to acknowledge our intuitions of things, the rich experience we always integrate into our concepts. Formalization and axiomatization play a key role here not because we don’t know what our concepts mean, but because we know it too well.
It’s formalization as suspension of intuition.
What this suspension gives us is a place to explore the underlying relationships and properties without the tyranny of immediate experience. Thus delivered from the “obvious”, we can unearth new patterns and structures that in turn alter our intuitions themselves!
A bit earlier in the book, Bachelard presents this process more concretely, highlighting the difference between Lobatchevsky’s exploration of non-euclidean geometries and proofs by contradictions:
Indeed, we not only realize that no contradiction emerges, but we even quickly feel in front of an open deduction. Whereas a problem attacked through proof by contradiction moves quickly towards a contradiction where the absurd comes about, the deductive creation of the Lobatchevskian dialectic anchors itself more and more concretely in the reader’s mind.
Le nouvel esprit scientifique p 46-47, 1934, Gaston Bachelard
Of course, formalization has many other uses. But I still find this Bachelardian function enlightening. It not only points at the constructivist nature of our models of the world; it also gives us a concrete tool to realize the perpetual rectification which Bachelard sees as the core process of scientific discovery and progress.
This post is part of the work done at Conjecture.
Note that I translated all quotes myself — for better or worse — because I’m reading the book in French and Bachelard’s philosophy of science is criminally untranslated anyway.