Arstechnica (hey, they’ve had some back-to-back articles I want to comment on!) has posted an article discussing a panel talking about the Limits of Understanding. Particularly, the panel seemed to focus on apparent deficiencies in math – notably, the practical intractability of biology to mathematical modeling and Godel’s Incompleteness Theorem.

Now, the proper modeling of biology may simple be a problem of time…sometime, a mathematical model may be devised of sufficient complexity and rigor that it is considered an adequate analysis of the underlying systems leading to observable reality. What exercises the mathematicians more is whether an “adequate model” is a true model, that is.

The statement which interests me the most, though, is the final two sentences, uttered by the author of the article himself:

If math turns out to be just a tool (and a tool with some substantial limits), that may disappoint mathematicians, but it won’t necessarily slow down our ability to understand and model the natural world. This may be my background as a scientist talking, but that seems like the most important consideration, and I’m willing to live with a community of disappointed mathematicians in order to get there.

This demonstrates an infuriating philosophical stance. A model does two things: it relates observations to one another, expressing a causal relationship chain between them, and elucidates the underlying real factors which produce these observations. There exist two parts to the natural world: the observable, particular parts and the invisible relations underlying them. Two rocks are observable, the force holding them to the ground, or the force value we say they exhibit as a property is not. We infer force, based on changes over time. Force isn’t so much real as a very successful description of an inferred relationship. Mass is similar: it’s a relationship one objects exhibits vis a vis all other possible objects. Importantly, mass was very successfully describes as a constant property of a given set of material. However, mass can equally well be described as a property of curved space surrounding an object (if we’re talking about the source of the “attractive” force) or as a measure of the type of curvature produced by a massive object.

The statement by the author above exhibits a certain lack of philosophical follow through. A model which successfully relates observed experience to one another implicitly makes statements about the invisible relationships observations have. We cannot “understand and model” the natural world correctly unless we also correctly model those invisible relationships. Unfortunately, those relationships are invisible; the best we can do is infer their nature. If we have inferred incorrectly, then we are wrong, and our understanding is flawed. We DON’T understand. Since these relationships are invisible, this is somewhat untestable, so we won’t know. The hope is that, by using rigorously consistent systems, systems which exhibit the same properties as the fundamental relational properties of existence, we can leave behind some of those inferential worries.

What is irritating the mathematicians is that Godel seems to have annihilated that possibility. What really frustrates them is that if Godel truly killed the possibility of a formal system being provable, or that it is composed of two distinct subsets of theorems: those which are provable and those which are not, then do the relationships underpinning the universe dividable into a set of provable relations? That is, do all the relationships of the universe derive from a rational order. Importantly, Godel showed that certain theorems of a given system cannot follow from an entire set of other theorems. It’s like saying a relationship could be true, or false, and it bears no relation to other theorems. That’s fine, you might say, we can just look and see, right?

That’s part of the mathematicians problem, though: you can’t. We’re speaking here of the fundamental relations of the universe, not the observable particulars. Physics and biology aren’t much use if they can’t predict what’s going to happen, based on other things it knows. Godel showed that, if you have a formal system of rules that is good enough to describes math, then there are going to be statements in that system YOU CAN’T PROVE. Which means you can’t predict. If someone asks, you have to say “I don’t know”. Hofstadter shows (in *Godel, Escher, and Bach*) that there exists an infinite set of such statements.

If we take the positivist approach of simply using math as models where appropriate, we’ve thrown up our hands and admitted: we can’t actually describe the universe. We can describe bits of it, but we can’t relate it together. Further, because we cannot assert that the universe is describable by fundamental, consistent laws – because there cannot exist any such systems, including the universe – we can’t actually say the models we use are true, correct descriptions of how the universe works. That’s not decidable. All we can say is they work. There demonstrably exists at least one other formal system which can accurately describe the observable phenomena a model describes, and it is impossible to decide between them. In fact, it’s not even possible to say the universe is rational, because that, too, is undecidable.

If the purpose of the University is the search for truth, then it becomes problematic for the University if it can be rationally demonstrated that Truth is unreachable. That means its purpose is finished: it has found the one Truth: there is no Truth. Everything else is mental masturbation.

## Leave a Reply