Sue Archer at the Doorway Between Worlds blog recently completed an A to Z challenge. I decided to follow her model and challenge and intend to do a little tour of some mathematical terms through the alphabet. My intent is to focus on some that are interesting terms of art that I feel non-mathematicians never hear. Or that they never hear clearly. Indeed, my first example is one I’m not sure I ever heard clearly described.
I first encountered this term in grad school. I can’t tell you when. I just realized that every couple sessions in differential equations the professor mentioned the ansatz for this problem. By then it felt too late to ask what it was I’d missed. In hindsight I’m not sure the professor ever made it clear. My research suggests the word is still a dialect rather than part of the universal language of mathematicians, and that it isn’t quite precisely defined.
What a mathematician means by the “ansatz” is the collection of ideas that go into solving a problem. This may be an assumption of what the solution should look like. This might be the assumptions of physical or mathematical properties a solution has to have. This might be a listing of properties that a valid solution would have to have. It could be the set of things you judge should be included, or ignored, in constructing a mathematical model of something. In short the ansatz is the set of possibly ad hoc assumptions you have to bring to a topic to make it something answerable. It’s different from the axioms of the field or the postulates for a problem. An axiom or postulate is assumed to be true by definition. The ansatz is a bunch of ideas we suppose are true because they seem likely to bring us to a solution.
An ansatz is good for getting an answer. It doesn’t do anything to verify that the answer means anything, though. The ansatz contains assumptions you the mathematician brought to the problem. You need to argue that the assumptions are reasonable, and reflect the actual problem you’re studying. You also should prove that the answer ultimately derived matches the actual behavior of whatever you were studying. Validating a solution can be the hardest part of mathematics, other than all the other parts of mathematics.