Uncertainty: A Mathematician's Perspective

by Sean Crowell, Dept. of Mathematics, Univ. of Oklahoma
One thing we enjoy in mathematics is certainty of our knowledge. This is because the things we know are logically certain (neglecting Gödel), and logic is all there is for mathematicians. In fact, unjustified certainty can be very disturbing to professors attempting to teach you to prove things rigorously in foundation courses. Never mind all of the debate in recent years about the philosophy behind these things. Most of us have some inner threshold beyond which we say “the definitions and axioms are satisfied, and so the theorem is proven!”
For a long time I had heard applied mathematics described as “messy.” Really this meant that the calculations done were unpleasant, or the proofs were complex and tedious and inelegant. But I’ve discovered a far more terrifying aspect about this work, one which nonmathematicians take in stride.
There is no certainty in science. There is merely evidence. Again, we have an inner threshold, only once it’s crossed, we say “I now believe the conjecture supported by the evidence.” How is this different from logical proof? The biggest difference is that we have no perfect measuring stick to go by. All data has errors in it. All models are flawed. And yet, through a mysterious bootstrapping process, data is used to improve models, which is used to improve observational techniques and to tell researchers what they should be observing.
The casual and especially mathematically trained reader will shake their head and say “Errors upon errors! How can we say we know anything?” At this point we can trot out statistics that point to the advance of science. Warning times for severe weather have gotten much better due to this process. New particles have been discovered by this process. Diseases cured and sheep cloned. All using noisy data and imperfect models. It works! For whatever reason, the universe that we can interact with, though mysterious, is far more regular than irregular. If we watch long enough, we can uncover its secrets.
(Editors’ note: This post was first published on Sean’s blog, A Mathematician in a Meteorologist’s World, on Wednesday 15 September. We liked it too much to merely excerpt it. Sean is a doctoral candidate in mathematics studying tornadic flow.)