A Summer 2015 Mathematics A To Z: ansatz


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.

Ansatz.

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.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

16 thoughts on “A Summer 2015 Mathematics A To Z: ansatz”

    1. I’d thought it had a meaning like that. There are a good number of mathematical terms that are German in origin — well, “eigenvalue”, along with related words like “eigenvector” and “eigenfunction” are evidence of that — though I’m surprised to find one that’s in the process of becoming part of the English mathematician’s vocabulary.

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        1. If the eigenvalue, or other eigen-thing, has to be rendered in English only it’s usually turned into “characteristic value” or “characteristic vector” or so on. And that’s fair enough.

          The eigenvalues (or other things) can be seen as kind of the spectroscopic analysis of a mathematical object. (This is a very loose metaphor.) If you’ve got a mathematical object describing a system, then the eigenvalues (eigenvectors, eigenfunctions, et cetera) can be simpler ways to describe how the system behaves.

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    1. It’s, sadly, the boring part of learning something. You know the part of mathematics class where you get, say, the root of a polynomial and then you’re supposed to go back and put it in to the polynomial and see if it really is zero? That’s validation. But for a simple problem it’s dull because if you did the work right there’s nothing being revealed.

      Where it’s important is when you try modeling something new and interesting because you don’t know that you made all the right choices in your model. (Also you might not be sure you calculated things based on the model right.) But that’s not something most mathematics classes reach, not below the upper levels of undergraduate life anyway.

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