Math and Nature*…

I am looking for a cool site full of book reviews and recommendation of books that I would love to read, but that it would never occur to me to look for myself. You know what I mean?

Here’s a sight that is clearly not what I’m looking for. Checking them out, I immediately went to their hard science section, and clicked here for this book on General Relativity.  The publisher’s website associates this quote with the book:

” Mathematics is not the language of Nature. It is the language of mathematicians. “

This is so true! Although, honestly, the rest of the book seems like complete bunk. But really, there is the enduring mystery of why nature† is so well described by math. Well, I guess it sort of is well described by math. Simple arithmetic isn’t so good for describing quantum mechanics, but the theory of partial diffential equations does a pretty great job of it, if you interpret it correctly–which is a straightforward, if not intuitive, process.

Some people think it’s quite profound that math is so good at describing nature. It’s really really good for it, almost as if nature was written in math by a cosmic mathematician. Math is really everywhere, not just in physics, but in biology…and I suppose that exhausts all of nature. And the same math comes up so often, in seemingly unrelated problems; for example things like Laplace’s equation, which comes up in heat transfer, diffusion, wave mechanics, quantum mechanics, and lots of other things that don’t have anything to do with that stuff, like probably economics.

Well, here’s why math is so successful at describing nature. It’s because that’s what people have been creating math to do. Math is a tool, created by people, to do things with**. And we have expanded our ideas of what math is until we could use it to describe nature with high precision. Now, maybe it’s remarkable that nature is so predictable that it can be described at all, but given that the universe has been stable enough over the past 4 billion years for us to evolve, it only stands to reason that the universe we see should be predictable.

As fer the strange coincidence of some equations and constants (Euler’s constant comes to mind) popping up everywhere, that has more to do with those equations and constants being very easy to make approximations than that systems really behave that way. I mean, they do behave that way, but only if you aren’t looking closely enough.

*and video games

†I really do mean to get  on with the fascinating project of describing video games mathematically, really!

**it’s been somewhat frustrating to me that math professors seem oblivious to the fact that a lot of the math that they teach was developed with specific problems in mind. I think math education would be much easier on the student if students were introduced to the types of practical problems that math is good for before they are introduced to the proofs and theorems of the theory.

Thermodynamics for Climate Change Denialists

Let’s take a look at the latest ideas in global warming denialism (with a screencap, in case the original source ever gets embarrassed by this). It goes something like this:

Greenhouse gasses can’t increase the temperature of the Earth: they don’t add heat to the climate (causing global warming), they just trap what heat is already there, slowing down any cooling. Those are two different things.

It’s a nonsense argument, so trying to really understand it is impossible, but I think that’s a decent approximation. My first reaction was similar to PZ’s, that it’s true as far as it goes, but omits the effects of any external source of heat.* But really, it’s not even true if you forget about the sun (but remember anything else about the climate). If you slow the cooling of the Earth at evenings and during the transition from summer into winter, you will increase the average temperature of the Earth without increasing the maximum temperature of the Earth. Therefore, the globe will warm.

But let’s see what happens when we don’t forget that big hot yellow Sun that’s heating us up. Generally, as you pour more energy (sunlight, and to a much lesser extent these days geothermal energy) into a system, the system heats up, temperature rising. As the temperature rises, the body (the planet Earth) emits radiation, shedding heat into its surroundings (in the case of the Earth, the radiation is primarily infrared, although atmosphere evaporating into space would count, too). The rate that heat is radiated off is proportional to the temperature of the Earth. The rate that heat is trapped is proportional to the reflectance r (albedo) or absorbance A = 1- r of the object,^† as well as surface area, and maybe some other things that will be constant here. The temperature T equation looks like this:

CdT/dt = AP₀ – kBT

with C the heat capacity, heat per unit temperature, of the Earth, P₀ = the (constant) rate of energy flow into the system, from the sun (I_sun•σ), geothermal sources, cosmic rays, whatever, and kB the constant describing the rate of heat loss per unit temperature due to radiation (and other sources, such as evaporating atmosphere or maybe chemical reactions^‡, which are mostly constant-”conduction” and “convection” wouldn’t really apply to the planet Earth as a whole). Roughly speaking, to determine the temperature of the Earth, you solve this equation for dT/dt = 0, the thermal equilibrium condition where heat flow in is equal to heat flow out. Increasing the absorbance of the Earth by increasing greenhouse gas concentrations increases the equilibrium temperature, which we’ll interpret as an increase in extreme temperatures of the climate, as well as increased average temperature.

Of course, the fine details of weather and climate are MUCH MUCH MUCH more complicated than this–but they’re much more complicated that you’d expect to see on a children’s climate website, too, and they’re more complicated than Kate and Hans Schreuder seem to realize, too.

*It’s amazing to me that they have made fundamentally the same mistake that some people make when they claim that evolution violates the 2^nd law of thermodynamics, i. e. they forget that there is a Sun.

^†Alright, here goes: the atmosphere looks like a thin film, with light shining down, with no transmission through the surface of the earth, some absorption, some reflection; some of the reflected sunlight is reflected by greenhouse gases back to the Earth’s surface, and either absorbed or re-reflected back to the atmosphere to be re-re-reflected, and so on. With each reflection back to the surface, a bit more gets absorbed; each reflection back to the atmosphere a little more gets back out to space; and also, there is a bit perpetually trapped reflecting back and forth between surface and atmosphere (which helps keep us from dropping to near 0 kelvins at nights). Turn the Sun off, and that trapped bit will decay exponentially, but the Sun isn’t going anywhere anytime soon.

^‡Chemical reactions and some other things are technically additional degrees of freedom, and change the heat capacity C of the system. More chemical reactions, for example, would slow down temperature rise, but not necessarily the rate of heat gain or loss.

Book Review: Kluge by Gary Marcus

Natural selection can only* work in small steps, and without foresight. So evolutionary progress is constrained to small improvements on what we’ve already got in place. Now, those small steps can accumulate to some pretty wonderful and complicated organs and organisms, such as the human brain, but the evolved design is strongly constrained by its evolutionary history. We wouldn’t expect those organs to be as nifty as they might have been if they’d been designed from scratch for a specific purpose. In Kluge, Gary Marcus details the mental errors we make all the time, and suggests that we are prone to making them because of the peculiar history of the evolution of our brains.

Pet peeve: in the first chapter, Marcus describes natural selection as a “satisficing” process, that doesn’t achieve optimal results, but good enough results. That always rubs me the wrong way–natural selection is a local optimizing process, with the caveat that what it’s optimizing is reproductive fitness, not, well, whatever we think a particular interesting organ is supposed to be doing. For example, we look at a liver, think, “Hey, it’s for filtering out poisons!” But no, a liver is meant to increase reproductive fitness, by a) filtering out toxins while b) not wasting too many resources on keeping blood clean as opposed to other vital functions. And as a local process, it suffers from the same failing of any local optimizing process, such as Newton’s Method: it tends to get stuck at the top of whatever local maximum it reaches first.

Anyway, we don’t expect evolution to achieve the best of all possible solution to a problem, but we can often expect that it’s doing as well as it can. The distinction is important, because any sloppiness or sub-optimality we see in an organism can tell us something about how it got to it’s present form, or about how costly a process would be to improve. And Marcus agrees with me about that, so no complaint there, I just don’t like his word choice.

Throughout the book, Marcus does a fine job cataloging our mental mistakes–reasoning error; faulty, fuzzy memories; our clumsy and illogical languages; our miscalibrated pleasure system that we’re eager to cheat. He refers to plenty of cute psychology experiments that explore the holes in our minds that we usually manage to overlook.

For each of our faults, he presents his imaginings of how our brain would work if it were built by a good engineer. To demonstrate how sub-optimal our brains actually are, he tries to tell us how much better they could be. These are often not very convincing–he proposes that we could have “postal code” memories that keep perfect, computer like track of where every datum is in our memories. But…how would we keep track of which code goes with each datum? Postal codes on postal codes? Sure, computers do it, but not really. All a computer does is run a set of arithmetical instructions that someone programmed into it. Interpretation of a computer’s output has always been done by clumsy human brains. When AI researchers try to build software that has contextual/interpretive abilities, that software always makes plenty of mistakes. And it doesn’t come with the ability to introspect it’s own software guts. Anyway, I will concede that our brains could have been built with a relatively small hard-memory module, for example; but expecting all of our memory to be perfect by working on a “postal code” scheme seems pretty far fetched.

I’ll counter-argue with Gary Marcus that our subconscious brain could be more optimal in some of those ways than he realizes, and that some of the mistakes and limitations of our conscious minds will be shared by the most sophisticated AI running on the best computers. I’ll of course concede, again, that we could have, in principle, been built with an integrated 16-bit calculator to help us with arithmetic, and that evolutionary history kept that from happening.

Similarly with language–he discusses logical languages that we could have evolved to understand. These languages would use words with phonic structures (for example) that give clues to their meanings, i.e. similar sounding words could mean similar things. A logical language would also result in a clear, distinct meaning for each statement it’s allowed to make. But–imagine a language that has no ambiguities. How would you (or evolution) know it has no ambiguities? If you only have a few simple concepts to express, such as a language that describes just arithmetic, it can’t be done. Gödel’s theorem. Simple arithmetic contains statements that can’t be proven true. I don’t know what they are. I’m not sure you can know that a particular statement can’t be proven. Now, in English and other human languages, there are plenty of obvious ambiguities, and we could probably do better if we put our minds to it. But how much effort would it take, and would it make evolutionary sense to invest it? Maybe evolution has built our language systems exactly as well as it needed to.

Another thing about language to illustrate the point–we are good at inferring meaning from partial information. At least, better than a computer is when it misses a little bit of information, which can result in a fatal program error. It may well be that the same system that allows us to infer meaning from partial information means we can’t tell so well when we’re saying exactly what we mean to say.

My other main gripe about the book is that Marcus never really discusses what these mental limitations tell us about our evolutionary history. It says right inside the front flap: “How the accidents of evolution created our quirky, imperfect minds.” But if we’re lucky, he’ll tell us that he can easily imagine that there’s a just-so story explaining the mental fault. Without even telling us his story! Great! I’d much prefer examples from other creatures minds, and what they tell us about how our minds work, or don’t work, as the case may be. Not a just-so story, and definitely not a hint that just-so stories are easy to imagine.

The penultimate chapter of the book is almost the most interesting. He discusses mental disorders, and suggests that the particular mental disorders that people suffer can tell us about how our brains are assembled. Great stuff, except that after he makes a persuasive and satisfying argument that it’s possible, he fails to do it. He offers some interesting guesses about what some common disorders might tell us about our brains, though.

Overall, I thought the book was somewhat interesting, and pretty fun and quick to read, but I already knew most of this information from other places that discussed them in more detail, with better context. And since it failed to live up to it’s purported unique, evolutionary history perspective, I was mostly disappointed:/

* “only” in practice; in principle evolution can take very large steps, it’s just that large steps are vastly more likely to break things more than they improve things, while I’ve heard that infinitesimally small steps are about as likely to improve things as they are to break things, so that half of those will be worth keeping, if they can bubble up above the noise

Abiogenisis and Evolution

I’m reading the tail of a very long comment thread at Pharyngula, and I come across one of my pet peeves. A creationist claims that evolutionary theory can’t explain the origin of life, and someone claims in response that evolution doesn’t have anything to do with that, it’s a separate problem called “abiogenesis.”

Uh, well, half right. Certainly, the theory of evolution doesn’t depend on a theory of abiogenesis: however life first arose, once it came about natural selection started working on it, driving evolution. Abiogenesis was also certainly an evolutionary (by natural selection) process, and we should expect any theory of abiogenesis to look like a smooth, evolutionary transition from non-life to life, without a dramatic boundary between inanimate chemicals and primitive organisms. So the creationist isn’t exactly wrong to conflate the two theories, if he’s making a theological point or something; but what we already know about evolution doesn’t depend on how life started in the first place.

Because once there is some sort of life going on, it’d take something like a god culling the fittest out of the gene pool to keep natural selection from taking it’s course.