Empirical mathematics

Suppose I want to figure out a good approximation to the eigenvalues of a certain Hamiltonian involving a moderately large number of Coulomb potentials. It could well be the case that the best way to do so is to synthesize a molecule with that Hamiltonian and then measure its spectrum. In other words, there are mathematical problems where our best solution to the problem uses scientific methods rather than mathematical proof.

Would it have been crazy to think everything is made of water?

According to Aristotle, Thales held that the whole physical world is made out of water.  I don't know if Aristotle was an accurate interpreter of Thales, but let's suppose he was.  The claim that the whole physical world is made out of water seems really wacky.

But I think it was quite defensible before we got the successes of modern chemistry.  And I want to sketch one line of thought why.  I am not claiming that this line of thought was in fact Thales'.  But it could have been: it won't rely on any science inaccessible to Thales.  The line of thought has three steps.

Step 1. All liquids are the same substance.

Here is a line of thought towards this.  Granted, obviously samples of liquids differ widely in shape, color, opacity, taste, wholesomeness, viscosity, miscibility, temperature and inebriativeness.  But such variability does not challenge the claim that all liquids are in fact forms of the same substance.  After all, apples differ widely in size, color, taste, wholesomeness and hardness, but they are all fundamentally apples.  Moreover, samples of liquids apparently of the same sort can differ in most if not all of the above properties.  The most obvious are shape and temperature: simply by varying the environment, the shape and temperature of a sample of a liquid can change.  More interestingly, the opacity, taste and inebriativeness of grape juice changes over time.  The viscosity and at least apparent miscibility of honey changes over time.  The taste and wholesomeness of milk change significantly very quickly.  It is a very reasonable hypothesis, then, that mere differences in these observable qualities do not correspond to a fundamental difference in kind, that the grape juice and the wine, or the milk and the yogurt, are one and the same liquid, despite significant differences in causal powers.  But the differences between water and, say, milk or oil seem to be precisely differences in respect of qualities that do not make for different kind of thing.

The alternate hypothesis of explanation of the differences in properties between liquids, and that is that they have different ingredients rather than different properties.  But there was good reason to doubt this alternate explanation.  The significant changes happening in juice, milk and honey apparently do not require the introduction of any additional ingredients, nor the removal of any ingredients.

Step 2. All liquids are water.

If all the liquids are forms of the same kind of substance, we may want to figure out what the basic, generic or fundamental form of that substance is.  And here water seems a very plausible choice.  It is colorless, transparent, tasteless, devoid of medicinal effects except relief of thirst (which it shares with many other common liquids), non-viscous, quite miscible and non-inebriative.  It is reasonable to suppose, for instance, that when water acquires the properties of whiteness, milky taste and a bit of viscosity, it becomes milk.  One might wonder: how does one get water to acquire these properties?  Well, a reasonable thing to say is that female goats make rainwater acquire milky properties.  Certainly, when you stop the access to water, the goats stop making milk (and die).

Water, on this hypothesis, is the fundamental liquid, having the minimal set of properties needed for being a liquid, and when it acquires different properties, we call it by different names.

Step 3. Everything physical is water.

Step 3a. Everything solid is water.

Water can turn into a solid without anything being added to or substracted from it.  One might think that "cold" is being added to it, but supposing cold as a substance may be questionable (though, less so to the Greeks than to us), and besides if cold had to be added to water in order to turn it into ice, it would follow that the formation of ice should reduce the amount of cold in the air.  But the freezing of water does not appear to warm the surrounding air.

Furthermore, metallurgy shows that certain kinds of rocks can be turned into liquid, and then solidified into metals.  Some varieties of mud, which appears to be a particularly viscous liquid, can be solidified into brick or ceramic.  Sand can be melted into a clear liquid, and then solidified into glass.  Since Step 2 hypothesized that all liquids are water, it becomes very plausible to generalize to the hypothesis that all solids are water, too, albeit with the property of solidity.

Step 3b. All gases are water.

When water is heated, it turns into steam, which looks like a cloudy form of air, and hence one can suppose that air is a kind of steam.  Smoke comes from solid objects when these are on fire.  Since solid objects are water by Step 3a, smoke is also water.  It is reasonable to hypothesize that all gases are water.

Step 3c. Fire is water.

This is perhaps the toughest step, intuitively.  Fire and water seem to be opposites.  But one might hypothesize that when water is added to fire, the fiery qualities of the fire simply become diluted by the water. Steam is hot and burns, though it does not glow.  Fire, thus, could be reasonably thought of as a particularly vivid kind of steam.

We can generalize from Steps 2-3c to conclude that everything physical is water.

Final remarks.  On this highly speculative interpretation, Thales' thesis stands in sharp contrast to ingredient-based theories of the natural world, such as we have in modern Mendeleevian and ancient four-element chemistries.  We do not need to posit differences in ingredients to explain differences between things.  We generally don't posit differences in ingredients to explain differences in shape.  So why should we posit them to explain differences in, say, color or taste?

Of course, if a particular ingredient-based theory comes to have significant predictive and explanatory power, Thales' thesis needs to be abandoned.  I do not think the four-elements theory that some other ancients preferred was all that superior in predictive or explanatory power.

Mendeleevian chemistry, on the other hand, was far superior in predictive and explanatory power over either Thalesian or four-elements chemistry.

More on science and theism

One of the puzzles in the scientific explanation literature is how it is that one can genuinely have explanation in the special sciences—chemistry, biology, geology, etc.—given that the facts in the purview of the special science in question appear to reduce to facts of physics, and hence it seems that only physics-based explanations are appropriate. One strategy is to resist the reduction move. I am happy to resist the reduction move for biology, but to me reduction seems exactly right for geology and probably for chemistry.

The theist, however, has a neat story, like the one in my previous two posts. The patterns that the special science identifies are valuable. They are valuable intrinsically—they exhibit an aesthetic good (and scientists talk of the beauty of theories, though admittedly they do so less in the special sciences)—and they are valuable instrumentally as they make it possible for us to make predictions and organize our knowledge. Because these patterns are valuable, God intends them and their presence is causally explicable. This holds whether or not the given generalization in the special sciences rises to the level of laws or not.

What happens, though, when one pattern is subsumed into a wider pattern? As long as the narrower pattern is still there, it can be correctly used for explanation. But explanation in terms of the wider pattern is better, because the wider pattern is more valuable, and hence more explanatory of God's creative action. Thus, early on we may have learned that all mammals have hearts, and later on we learned that this is true of all vertebrates. It is still correct to explain the presence of the heart in Socrates by his mammality, but better to do so by reference to his being a vertebrate.

A similar move explains why it is that when there are two formulae that equally fit all the data, the simpler is the one to be preferred in explanation. This is relevant to both the curve-fitting problem and the problem of which of two mathematically equivalent formulations is the more explanatory.

Furthermore, that some simple theory approximately fits a body of phenomena is also of value. Hence, theories that are mere approximations can yield genuine explanations. And that is how it should be. In particular, Newtonian mechanics continues to be explanatory, and not simply because of classical-limit stuff in quantum mechanics.

The theistic story also explains why it is that Lagrangian mechanics was genuinely explanatory, despite not fitting well in the mechanistic model of explanation. This is, of course, an application of Leibniz's discussion of teleological explanation.