This post was inspired by a recent discussion about “mathematics vs cryptography” over on Bruce’s blog, which I found particularly interesting given the discussions going on regarding the nature of scientific theories in my current graduate school class.

There is a fundamental difference between an axiomatic system (like, say, Geometry) and a Science. In an axiomatic system, you have a number of set definitions and a number of statements that are *given to be true* (these are the axioms). The practitioner then formulates a theorem, and proves the theorem to be true using the tool of logic, on the foundation of the axioms. If one of the logical consequences of the theorem contradicts one of my axioms, then the theorem is not true.

Compare this to Science, where instead you have a theory, which you attempt to prove or disprove by running experiments, gathering data, and analyzing the data in the context of what is considered to be the body of knowledge in your particular field. Science is the iterative process of trying to explain observable data using propositions that enable you to expand your capabilities to predict the outcomes of future events.

Mathematics is a *constructive* process – you are attempting to **build** a logically consistent system. In a real sense, it doesn’t matter at all if your system has any current practical use (the example I gave on Bruce’s blog is that there are no currently known applications for N-space topology). Of course, people will probably be more interested in your theory if you can show it has some real world application, but if your system is consistent, that’s all that is really required.

Science is a *de-constructive* process -> you are attempting to ***derive*** a logically consistent system given a bunch of experimental data and a bunch of additional theories that are supported in turn by experimental data.

This is one of the fundamental disconnects between a large percentage of non-scientists and people who practice science for a living; people who don’t understand how science actually works think that science is axiomatic; something is either true, or it is not true. Science isn’t like that, kids. As my high school AP Physics instructor said in class one day, “If you’re looking for Truth, go take a Philosophy class”.

Misunderstanding this is where the oft-repeated statement, “[Some particular scientific theory] is JUST A THEORY, you can’t PROVE that it’s true!” comes from. ** In an axiomatic sense**, this is correct,

**I can’t PROVE that the Theory of Gravity is true in an axiomatic sense. I can, however, prove that the theory of gravity is useful and therefore ought to be accepted as part of our our default understanding of the universe (well, to be precise, I have no desire to do this, but you can read plenty of Newtonian mechanics and physics and if you don’t get it, you’re just not a scientist).**

*but in a scientific sense*it is totally irrelevant, because Science is not an axiomatic system.Someday, someone will come along that has a better Theory of Gravity (Einstein did this – ed note), one which explains some things that the original Theory of Gravity did not explain. If those are useful things (they are), and there is no evidence that contradicts the newer Theory of Gravity, then scientists will adapt the newer, improved Theory of Gravity. Yay! This is how Science advances.

[Edited to add *08-22-2008*] – another version of this post over at Cosmic Variance, with a follow up here. Both are good reading.