Hi! Welcome to my fifth week of BeyondResearch Insights! This time coming from the Differential Geometry side of the programme.
Previously, we have been asked to develop ideas on how to define the length of a curve in a mathematically rigorous way. We have struggled greatly to answer this, so we ended up proposing the familiar arclength equation: the integral of the absolute value of the derivative of the parametric function giving the curve, . However, this failed on multiple fronts, which we briefly discussed at the beginning of the session and arrived at some deep insights.
It appears that our confusion stemmed from the fact that we didn’t fully understand what it means for a definition to be mathematically rigorous. Well, firstly, it needs to be intuitive. In our case, it should have been clear from the definition how it relates to length. This is the first point where our attempt failed since we couldn’t explain why this strange integral, , should produce the length of a curve in the first place.
Secondly, a rigorous definition should also be built with respect to a set of axioms. Although this might be beyond our abilities right now, we should have at least based the definition on fundamental principles. In other words, we should stay away from such constructions like the integral from our previous attempt, .
In summary, we can create a rigorous definition by modelling our intuition in a formal mathematical system. Then, we can develop our ideas and work with the definition following the rules of the theory, to test whether our formalization still reflects the intuition we started with. If it doesn’t, which can often be the case, we have two options: either we alter our definition, or we accept the result and learn something deeper about our model.
Here we can draw an interesting link to the other side of the course: abstract analysis, specifically logic. If we are using a sound logical deduction system to reason about the formalization we developed, then through derivations we should only obtain results that are compatible with our model. But is this true the other way around? As in, if something is true in the model, can we derive it from our formal system?
Well, Gödel’s (first) incompleteness theorem tells us that if we are using second-order logic, then this is not necessarily the case. It showed that you can’t discover all the truths in mathematics by the means of proof! In other words, in second-order logic there can always be true but unprovable statements. Thankfully, most logical systems we use are complete, i.e., we can in fact derive everything that is true from the axioms.
The way Kurt Gödel proved his theorem is fascinating: by encoding logical statements in arithmetic! But this will have to wait for some other time!