Monday, April 22, 2013

Why Math is Useful in Natural Sciences and not in Social Sciences

I'll touch on some fundamental questions here:
  1. Why use Mathematics in Science?
  2. Why does it work so poorly in Social Sciences (e.g. Economics, Psychology)?
  3. Why does it work so well in Natural Sciences (e.g. Physics and Molecular Biology)?

Why use Mathematics in Science?

You might think that mathematics is just about being more precise than words can convey. For examples, instead of conveying nuances such as "it's very warm today" and "it's a little warm today", we can be more precise and say "it's 32 degrees today" and "it's 23 degrees today". So it seems like numbers are useful to remove the subjective interpretation of "very warm" and "a little warm".

But this is not the main strength of using numbers. The main strength is not that you can quantify things more precisely, but that you can reach conclusions that are otherwise completely inaccessible. Consider this:

An artillery piece fires a shell right up into the sky. Will the shell return back to earth? You might think it's obvious that it will, in which case you'd be wrong. There's two opposing things here: gravity, which pulls the shell back to earth, and the velocity at which the shell was fired.

And here's where the true power of science comes in: one can calculate that the velocity needed to escape earth's gravity is 40 000 km/h. That's a lot, but not impossibly much: in 1957, during Operation Plumbbob, the Pascal-B test detonated an extremely weak nuke (300 tons) which propelled a metal plate into the sky at  240 000 km/h, about 6 times escape velocity. That piece of metal is believed to have disintegrated in the atmosphere, but one things for sure: it did not return to earth.

This is the strength of using quantities: we can tell that a pistol bullet will come back to earth, whereas a nuke propelled piece of metal will not. There's two opposing "forces": gravity and velocity, up and down. And because we have a formula to balance them, we can actually get an answer.

Now compare this to the following:
Alice is offered $20 to eat a fresh cockroach. Does she say yes or no? Two opposing forces: money (good) and eating fresh cockroach (bad). Naturally, everybody's different, so the answer depends on the person. There's certainly validity to that argument, and I'll return to this issue when answering the next question.

For now, I want to point out why quantitative science is useful: not only because we can quantity results, but because we would not even be able to reach some conclusions without using quantitative science. In particular, we can't balance two or more opposing things without using quantities.

Sometimes numbers are not necessary. Here's an example from evolutionary psychology: on average, who is a person closest to among these four people: his/her maternal grandmother, maternal grandfather, paternal grandmother, or paternal grandfather? Which is a person the least close to? Evolutionary psychology is based on a set of basic assumptions (much like Newton's laws, except they're not quantitative but rather purely logical), from which one can derived that people tend to be closest to their maternal grandmothers, and least close to their paternal grandfathers? And indeed the predictions are correct. What is the logic behind this? That people know for sure that their mothers are theirs, but not so for their fathers. Thus there is a certainty gap that arises every time a parent is a male. Because people tend to favor their own kin (also a pillar of evolutionary psychology, follows from the modern insight that the unit of survival and replication is the gene), the conclusion follows.

The argument is statistical in nature, but no quantities are involved (it is qualitative). You might argue that, due to the genetic nature of the assumptions, the argument is essentially one of probabilities over shared genes. It would be so if the assumption of evolutionary psychology where that narrow, but they are not: evolutionary psychology studies the interaction of genes in environments (and thus the assumptions are weaker too, but presumably more accurate).

So in this example it works great, but we kindly run into dead ends again if we start balancing different aspects that are in conflict. The problem is that here we could conclude that a person should be close to their mother than father, and that the parent should be closer to their mother than father. So:

closeness(mother(A)) > closeness(father(A)) => closeness(mother(mother(A))) > closeness(father(father(A)))
In short, there is no conflict: the inequalities go in the same direction both times. In form, it is similar to when arguing that if Alice is taller than Bob and Bob taller than Eve, then Alice is taller than Eve:

height(A) > height(B) and height(B) > height(E) => height(A) > height(E).

This holds true of numbers too, but numbers are not necessary for such arguments to hold, such as in this case, where we don't know the actual heights of anyone, yet can still draw a conclusion based on the fact that heights are ordered. Logically, a partial order is sufficient for this reasoning to go through (by definition, any transitive relation).

In general, the more we assume, the more we can conclude. And quantitative sciences (those based on mathematics) make far more assumptions about their domain of study. Consequently, the theories are richer in predictions, and, as a result, research becomes more fruitful as more experiments can be performed.

This is the reason why social sciences do not live up to the impressive achievements of physics: they are not quantitative. This, of course, is something to blame social scientists for. Or is it?

Why does math work so poorly in Social Sciences?

We know that social sciences cannot make strong predictions when many opposing features need to be considered together, due to their lack of quantitative precision. But why can't we get a quantitative precision?

Here's a couple of explanations:
  1. Social scientists are less smart than natural scientists, so they can't find a mathematical foundation.
  2. Social sciences are more complex than natural sciences.
  3. Social sciences are by nature non-mathematical, no mathematical foundation exists.
The main reason why explanation (1) doesn't work is that a lot of mathematicians have worked in the fields of economic theory and behavioral sciences. In particular, John von Neumann, widely regarded as one of the greatest mathematicians to ever live, attempted to create a solid mathematical foundation for social sciences: Game Theory. He even showed that randomly bluffing in poker is not a non-mathematical aspect of the game, but actually falls out from the calculations as the optimal strategy.

Von Neumann is not the only person who has made signifcant theoretical contributions to (theoretical studies of) social sciences, so we can safely rule out (1).

This laid a foundation upon which countless economists have built their academic careers, but there's a reason why almost none have made it in the industry: it's is useless for practical purposes. It may well be that game theory can capture all of social behavior, the problem lies in the fact that it is not much easier to formalize the "game" correctly than it is to simply write down mathematical equations for human behaviors. We have created a mathematical framework that is no simpler than mathematics itself in terms of describing human behavior.

So is there a mathematical foundation? Everybody knows how well math works in physics. The laws of nature seem to be mathematical. At the very least they are very well approximated by mathematics. Here's the key insight: social sciences have a mathematical foundation if, and only if, the laws of nature are truly mathematical (i.e. there are laws that not only approximate reality well but actually describe reality exactly). This follows from straightforward reduction of everything (brain states, body movements) to the laws of nature. This is of course a horrible theory for social sciences due to the microscopic view of things (humans are seen as a pile of atoms), and would be completely useless. But it gives a partial answer to (3).

In fact, (3) isn't even the right question to ask. It may be that the laws of nature are not mathematical, but we still have good approximations today. Perhaps there are no exact laws of social science, but we should nevertheless be able to find good approximations.

An important comparison can be made to physics: the laws of physics are not necessarily "real". Temperature is not a real concept of nature: it is the speed at which atoms bump into each other. Viewed from a microscopic perspective (imagine we seen every atom individually) there is no "heat", just movement of atoms. From a macroscopic perspective, it is the amount of such movements that causes heat in a gas. But "macroscopic perspective" is certainly a human centric view. It may even be a natural point of view for all intelligent biological lifeforms, but it is certainly a creation of our perception, a way to simplify down reality.

The same goes for many other quantities: pressure is the amount of "push" atoms have towards the enclosing surface (e.g. the bottle containing the gas). Again, it's all about the atoms and their movements. 

Why can't we define concepts like "temperature" and "pressure" in social sciences? We could, it's just that they tend not to make any non-trivial predictions. So why is that? Because human brains are not like atoms. In classical physics, atoms only have three useful properties: location, mass, and velocity. Brains need significantly more information than a couple of numbers to describe it.

In mathematics, there is a notion of complexity as "the smallest possible computer program that describes it". This captures the intuitive notion that the string "1212121212" has lower complexity than "7387904583".
The reason is that the first string is described by a computer program that says "output 12 5 times", whereas the second says "output 7387904583". Intuitively, we need a very short program to describe the state of an atom ("location = (x,y,z), mass = m, velocity = v") but a very long and complicated description for the state of a brain.

Complexity in this sense is not the reason. Atoms are memoryless: what an atom did yesterday does not affect its current state. But brains have memory: they remember the past, and their behavior today may significantly have been altered by the past.

A third factor is that atoms influence each other on a very basic level: they knock each other around (they can also be entangled in quantum mechanics, in which they have a relatively simple relation, e.g. opposite spin, which is only one number that is correlated). Brains can have enormously complex influence on each other.

Once you mix plenty of atoms, you get a predictable behavior. The idea that, once you mix plenty of brains, you get predictable behavior, well, is simply not true, as is made obvious to anyone who has invested on the stock market.

We don't need to move all the way from atoms to brains to see the effects of complex interactions. An example from statistics will illuminate the problem: if you want to know the behavior of throwing 100 dices, that's pretty straightforward: the sum will be approximated by a normal distribution with average 350 and standard deviation 0.292. But if you start mixing different kind of dices (some land on 6 more often than 1, or vice versa) as well as interactions between the dices (if one gets a 4, then the others are more likely to land on the same number), then you can't say anything about even what the average is going to be. The reason you can't is because you would need to specify the distribution of probabilities of each dice and how they interact.

The easiness of solving the first case lies in one hidden assumption that is extremely useful: the outcomes of each individual dice does not affect the others (the dices are independent). Once you have dependency, you have two choices: if you know all the correlations between different outcomes, there's still a chance you might get a mathematical solution. If not, you're out of luck. For the sum of 100 brains (whatever that means, say voting), you're going to have B1+B2+...+B100 and that's where the math begins and ends: no useful interpretation, no useful conclusion.

 It is in this sense that I think explanation (2) is the correct one: social sciences are more complex than natural sciences. It's not that it is in impossible to analyze such complex situations: it may be possible with a mind that can handle extremely complicated equations and long, counter-intuitive lines of reasoning by combining a wide variety of techniques (numerical optimization, computer algebra systems, automated theorem provers, simulations, and lots of methods that don't even exist today). It's just that our brains are incapable of such herculean tasks, or we would have done it already - it's not for lack of trying we've failed.

Why does it work so well in Natural Sciences?

Nobody knows the answer to this. We live in a world filled with science and technology (it is tempting to call it the age of science and tech, except that we'll have even more in the future). We've gotten so used to the fact that science can accurately predict the motion of planets, when a comet is going to pass by and whether it will be visible to the naked eye, that thousands of commercial airliners can take off and land everyday (they also achieve this without a pilot), satellites orbit our planet pinpointing our exact location, and robots crawl on the surface of Mars.

All of this is possible because of one thing: the laws of nature are possible to approximate with mathematics. Put differently: we are capable of creating an approximate representation of the real world in our minds. We can compress reality.

We could turn the question and instead ask: what could go wrong when trying to represent reality in our minds?

For one thing, the laws of nature could be hopelessly complex ("incompressible"), just like social sciences are. All of the laws of nature as we know them are simple, in the sense that they admit a very short description: Newton's laws, Maxwell's equations, Schrödinger's wave equation, the equations of general relativity. Now, the laws of nature are at the fundamental level of reality: there is by definition nothing "underneath" them. But why would they have short descriptions? Apparently the description is not as short as we once thought, since Newton's laws had to be replaced by general relativity and quantum mechanics. The latter two contain more equations, and the equations have longer descriptions.

It is in principle conceivable that there is are no final laws of nature: the universe is not mathematical, we only approximate it by inventing (not discovering) laws that are useful for our purposes. Science is essentially an engineering task from that point of view: all that counts is what works. That's not so bad, since it apparently works really well, but more interesting question now is: How would it be otherwise?

It would be that the universe is actually mathematical. The laws of physics as we currently are not nature itself; they only approximate nature. But perhaps that's only because we haven't found them yet, and we're on the right track (that's two different assumptions: mathematical correct laws of nature exist, and we will find them in the future). This would be the case if the universe worked like a computer program (perhaps because we truly live in a Turing-complete simulation). Another explanation, provided by Tegmark, is that all logically consistent theories are actualized as a separate universe. One problem with this explanation, however, is that the vast majority of universes would have extremely complex logical/mathematical laws, making it impossibly unlikely for us to end up in our current universe with simple laws (assuming they ultimately turn out to be simple, perhaps they're not).

This is essentially the concern Wigner had when we wrote a paper titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Why does math work so well in physics?

Clearly, even if the universe is non-mathematical, there's the mystery of why mathematics approximates reality so well. An interesting thought is that perhaps it doesn't: perhaps all mathematics does is describe our understanding of the universe, as we perceive it with our limited minds. As mentioned before, temperature and pressure are not properties of the universe, they are properties of the world as we perceive it macroscopically. A hypothetical divine being that perceives quantum states directly may not have any notion of temperature, since that would not be needed: it is after all about compressing lots of information into just one number, so we can make sense of it. We keep repeating this pattern over and over: in statistics, "all information" is all the available samples. We then compress this data into mean, median, standard deviation, etc. We even replace real world data with known statistical distributions because they approximate them well. But that is something we do to comprehend reality, the fact that it throws away information makes it perfectly obvious that it is man made.

But those notions are not the most fundamental. At the most basic level, we have something like quantum mechanics (general relativity is not the correct fundamental theory). The rest of science (even physics) is built on top of that. The wave equation describing reality does not throw away information. The laws of physics seem reversible, so that we can freely "move" between past and future in the mathematical formulation. That makes it clear that we're no longer throwing away information, so perhaps this is something like reality itself. So even though most of science is "engineering" in the sense that it defines concepts useful to humans, it may also be the case that reality is mathematical: if we ever find unified mathematical laws that describe everything  in the universe, so there is no need for further refinements, we will know that the answer is "yes, the universe is mathematical". The question of why, however, may still remain a mystery.

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