What Naturalism sounds like to natural theologians

August 11, 2014 at 8:07 pm (Uncategorized)

Assume you’re an intellectual living in Alexandria around 350 AD. Euclideanism is a runaway favorite for the most effective scientific system of all time. It’s successfully determined the size of the earth, it’s been used to make a system of astronomy so precise that no one has improved on its accuracy for centuries, it’s built devices that can measure the distance of ships or allow for perspective in architecture, it’s showed the governing ratios for art, painting, sculpture, and the proportions of natural bodies. Plato has insisted that everyone learn it before they attend his ancient academy, some neo-Platonists mimic Euclid’s framework, and Aristotle’s whole theory of science takes it as a point of departure. The science has lead to many conclusions that seem shocking to common sense (like asymptotes, irrational numbers, angles less than any given angle…), while at the same time being based on principles that seem irrefutable and self evident.

You, however, work in a comparatively small backwater of inquiry: the nature of mathematicals. You spend your time engaged in the various disputes about whether they are Platonic substances, Aristotelian abstractions, Pythagorean monad or dyads, etc. You get in disputes with collegues, go to conferences, get alternately enthused and bewildered by all the various arguments, and gradually appreciate more and more how difficult the problem is.

One day, however, you hear a new theory about the nature of mathematicals, namely that the whole discourse is pointless and probably meaningless. You’re intrigued and a bit worried about the idea and so you ask some of those who advance the theory to explain themselves. Here are some responses you get:

1.) There’s no need to explain where mathematical things come from, and it’s probably impossible to do so. There is no explanatory gap within Euclid’s system for either Platonism or Aristotelianism to fill. What is your theory except a Platonism in the gaps?

You respond that you’re not trying to give a Euclidean theory, but you want to explain why there are Euclidean entities at all. Your interlocutor brushes off all that mystical sounding woo-woo. Why can’t the mathematicals just be there? What do Platonic heavens add to the rigor of Euclid?

You respond that the opinion he is giving just now is not a Euclidean one, but he laughs at that suggestion. All he’s making is a denial of supra-Euclidean reality. It’s not a belief, but the absence of a belief.

2.) Another guy, who is far more rigorous and analytic, brushes off the coarseness of the above opinion and tries to give a more subtle interpretation of it. To what extent, he asks, does any theory of mathematicals possess the theoretical virtues that we usually require of explanatory systems? Both Aristotelianism and Platonism and all the other theories are completely undefined, since the acceptance of one or the other makes absolutely no difference on the ground for how Euclid’s geometry advances. Does any theory yield a mathematically testable theorem? Can any of your theorems be used to falsify a theorem or give us new insights into the ones we have? Don’t all of these consequences reflect negatively on your whole domain of inquiry?

Here again, you try to explain that you’re not trying to give a theory in mathematics, but about mathematics. He insists you completely missed his point, namely that your theory is totally undefined and can make no mathematical difference.

3.) Others object that the whole idea of an extra-Euclidean system is nonsensical. How does this extra-Euclidean domain interact with the Euclidean one? If it has real effects in the Euclidean world, then it has to exist within it; and if it doesn’t, then how can it explain anything about it?

4.) Another guy is far more pragmatic, but agrees with the general thrust of what was said. The discussion of mathematicals has been going on for centuries and has gotten nowhere while Euclideanism has proved an unrivaled tool for explaining practically everything. Shouldn’t you try to stop solving all these problems by logic and try to use a method that’s actually gotten results? After all, Euclideanism has showed us the problem of just trusting logic to solve problems – there are all sorts  of counter intuitive conclusions that are nevertheless completely certain. He doesn’t say exactly what he means by logic, and it seems very odd to you to hear your arguments described as “solving things by logic”, but you figure that the debate won’t get anywhere.