# A simple mathematical example for pseudo-teleology

The Lagrangian formulation of mechanics seems to be an interesting example for 'pseudo-teleology': a particle at starting position (a, b) with final position (x, y) will take the path of smallest action (the path integral of T - V, with T kinetic and V potential energy) between these positions.

At first, it seems there is foresight of the particle involved, teleology. But on a closer look this is wrong, it is an illusion to think that the particle calculates paths ahead of time. The Lagrangian formulation can be mathematically derived from Newton's laws, in which normal, instantaneous causation determines everything.

But I'm not so happy with this example. It's mathematically relatively complicated (in the context of philosophy), i.e. it cannot be understood with high-school math. Also, contrary to what I said before, I actually believe that there is some "cheating" involved, because it is not so clear that the Lagrangian formulation can really be deduced from Newton's laws without some extra assumptions.

Do you know a similar but mathematically simpler example? It doesn't have to come from a physical theory, the model can be completely fictional.

• Euclid an geometry? I'm not sure math does teleology. you could argue that triangles "want" their inner angles to sum to 180 degrees, but that's not a mathematical proposition. I guess I'm not sure what you're looking for, can you clarify?
– user20153
Commented Dec 3, 2016 at 20:37
• This is entirely unclear. Mathematics is descriptive. How do you define teleology on that, and then what is 'pseudo'-teleological? You can put a teleological interpretation on addition, that the two addends 'want' to become the sum, but that is just fuzzy headedness, apophenia, projection, ascribing emotional states to abstract patterns. Commented Dec 8, 2016 at 16:42

An ancient example comes from optics, on the path taken by a ray of light reflecting from a mirror.

The analogy to Newton's Laws, the "instantaneous" law, is the statement that the angle of incidence equals the angle of reflection.

The analogy to the Lagrangian formulation is that light takes the shortest path. The equal-angles formulation can be derived from this with high school calculus, or even with a geometrical diagram that includes the ray image -- if the path from eye to image deviates from a straight line it is clearly no longer the shortest path, and the equal-angles property follows immediately from the path being a straight line.

Since the math formulations are equivalent, then on the basis of the math alone there seems to be no basis for deciding that one or the other of the instantaneous interpretation or the planning-ahead interpretation is an illusion. Rather, they are two different ways of saying the same thing.

One reading of evolution by natural selection follows a similar pattern. (I think I'm cribbing this from Dennett's Darwin's Dangerous Idea, but that could be wrong.) Here's the idea:

Pre-Darwin, the standard explanations for the functional organization of organisms was teleological. Specifically, organisms were designed by God, who designed hearts to circulate blood and lungs to respirate and eyes to see and so on. Organisms themselves have designed functions — the function of deer is to eat plants and be eaten by large predators — and all of creation fits together into God's master plan. Without an alternative, non-teleological way to explain the functional organization of organisms, the argument from design was a powerful argument for the existence of God.

Evolution by natural selection gives exactly this alternative. Natural selection doesn't work to realize an intentional design or other predetermined end (telos). Competition and reproduction tend to drive clades up fitness landscapes, which gradually produces greater complexity of organization and functional specialization. The functional organization of a circulatory system is the product of millions of generations of minute trial and error.

Arguably this narrative is oversimplified. "Theistic evolutionists," like Asa Gray have argued that evolution by natural selection is compatible with the existence of God and a divine plan. (Roughly, the idea is that God sets a plan, then uses evolution to achieve it.) Natural selection is only one form of evolution. It's not clear how compelling the argument from design was pre-Darwin. And there's a lot of contemporary work in philosophy of biology on how we should understand biological function-talk (the purpose of the heart is to pump blood, gene XYZ is for this phenotypical trait).

• evolution is not a mathematical example for pseudo-teleology Commented Dec 7, 2016 at 11:18
• Oh, I must have missed that aspect of the question. If the questioner likes the example but wants a mathematical model for it, take a look at this question: <biology.stackexchange.com/questions/6939/…> Commented Dec 7, 2016 at 11:37
• The problem is that this is not easy mathematics. Certainly not easier than the example from Lagrangian mechanics, which can at least be understood after taking a class on differential equations and multivariable calculus. Commented Dec 7, 2016 at 11:42