The other day, I had my students respond to a question about how mathematics relates to science.

In class, I had pointed out that mathematical definitions are very precise while scientific measurements can be rather messy. A mathematician has a very precise meaning when they say that two variables are proportional or have a linear relation. But when we get real data, even if they do not satisfy these precise meanings, we still gain significant information about the relation and might even say that the measured variables are proportional or linear. Unfortunately, many students seemed to think I was looking for a repeated discussion of this point.

Science can be thought of as the study of the physical world through the scientific method. Essentially, we make observations on what happens in the world (whether that be physical, chemical or biological interactions) and want to understand why that is happening as well as to predict what will happen in the future. In order to do this, scientists propose various hypotheses based on their observations (and past accumulated scientific experience) and then test those hypotheses. Experiments quite often include quantitative measurements, and part of the prediction is often to propose relationships between independent variables (the variables in treatments and control) and the dependent variables (outcomes). Experience may support a hypothesis or falsify the hypothesis, but it never can prove a hypothesis.

Knowledge based on patterns that we predict will continue, but which we can support but never prove, is called inductive knowledge. Science is an example of inductive knowledge.

Mathematics can be thought of as the study of structures that satisfy very specific rules. We have properties of arithmetic, algebra, and calculus. We establish specific axioms that describe our basic assumptions about the structures and then use logical argument to deduce the behavior of more complicated constructions. We might look at examples to see what ideas might be true or false, and in this sense mathematics can also take advantage of inductive knowledge. However, the objective in mathematics is not just to suppose that a pattern will continue; the objective is to determine conclusively if it must continue. We seek for proofs (that it is true) or counterexamples (break the pattern).

Knowledge based on basic assumptions (axioms) and logical argument that determines conclusively what must follow from these assumptions is called deductive knowledge. mathematics is an example of deductive knowledge.

Models form a connection between mathematics and science. Data often appear to follow a general trend, even in the presence of the noise of messy observation. A mathematical model takes that messiness and forms an abstract clean relationship that mathematics can work with. Based on the deductive approach of mathematics, we can often establish consequences of the assumed model form. We then apply those consequences as hypotheses in our scientific framework. The predictions from the deductive approach provide the predictions that can be used to falsify these hypotheses.

## Thursday, September 3, 2009

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One way to think about this question is to identify the difference between mathematical truths and scientific “truths”.

Mathematical truths are contextual truths. Contextual truths are the only kind of truths for which one can attain 100% certainty. Mathematics is, by design, an internally self-consistent system by which we can prove certain statements true within the context of the fundamental axioms of math.

Scientific truths can only asymptotically approach 100% certainty. This is partially due to the limits of experimentation and uncertainty, but there’s more to it. Mathematical scientists use mathematics as a tool to analyze reality, but there is no guarantee that reality is mathematical. That is to say, we use the rules of our field of numbers to build models, but reality isn’t a field of numbers and may not always behave as our models predict.

It’s conceivable that, like a world map that very accurately portrays the region around the equator but distorts Greenland, reality may just be approximately mathematical in our local region of space and time. For example, maybe there are no singularities in space. Just because mathematical functions have asymptotes doesn’t guarantee that space-time is asymptotic too. Our knowledge about black holes and big bangs is inferred from our mathematical models.

Every time we do an experiment and our observations match the predictions of our models, we become a little more certain. But the next prediction could fail. Since we can’t do infinite experiments we can never gain 100% certainty on any question about reality, but we can approach probable certainty to a very high number of decimal places if we keep experimenting (hence conventions like the 5-sigma requirement in particle physics).

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