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taw comments on Mathematical simplicity bias and exponential functions - Less Wrong
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I have this vague idea that sometime in our past, people thought that knowledge was like an almanac; a repository of zillions of tiny true facts that summed up to being able to predict stuff about stuff, but without a general understanding of how things work. There was no general understanding because any heuristic that would begin to explain how things work would immediately be discounted by the single tiny fact, easily found, that contradicted it. Details and concern with minutia and complexity is actually anti-science for this reason. It’s not that details and complexity aren’t important, but you make no progress if you consider them from the beginning.
And then I wondered: is this knee-jerk reaction to dismiss any challenge of the keep-it-simple conventional wisdom the reason why we’re not making more progress in complex fields like biology?
For classical physics it has been the case that the simpler the hypothetical model you verify, the more you cash out in terms of understanding physics. The simpler the hypothesis you test, the easier it is to establish if the hypothesis is true and the more you learn about physics if it is true. However, what considering and verifying simpler and simpler hypotheses actually does is transfer the difficulty of understanding the real-world problem to the experimental set-up. To verify your super-simple hypothesis, you need to eliminate confounding factors from the experiment. Success in classical physics has occurred because when experiments were done, confounding factors could be eliminated through a well-designed set-up or were small enough to neglect. (Consider Galileo’s purported experiment of dropping two objects from a height – in real life that particular experiment doesn’t work because the lighter object may fall more slowly.)
In complex fields this type of modeling via simplification doesn’t seem to cash out as well, because it’s more difficult to control the experimental set-up and the confounding effects aren't negligible. So while I've always believed that models need to be simple, I would consider a different paradigm if it could work. How could understanding the world work any other way than through simple models?
Some method trends in biology: high through-put, random searches, brute force, etc.
I must disagree with premise that biology is not making progress while physics is. As far as I can tell biology is making progress many orders of magnitude larger and more practically significant than physics at the moment.
And it requires this messy complex paradigm of accumulating plenty of data and mining it for complicated regularities - even the closest things biology has to "physical laws" like the Central Dogma or how DNA sequences translate to protein sequences, each have enough exceptions and footnotes to fill a small book.
The world isn't simple. Simple models are usually very wrong. Exceptions to this pattern like basic physics are extremely unusual, and shouldn't be taken as a paradigm for all science.
The catch is that complex models are also usually very wrong. Most possible models of reality are wrong, because there are an infinite legion of models and only one reality. And if you try too hard to create a perfectly nuanced and detailed model, because you fear your bias in favor of simple mathematical models, there's a risk. You can fall prey to the opposing bias: the temptation to add an epicycle to your model instead of rethinking your premises. As one of the wiser teachers of one of my wiser teachers said, you can always come up with a function that fits 100 data points perfectly... if you use a 99th-order polynomial.
Naturally, this does not mean that the data are accurately described by a 99th-order polynomial, or that the polynomial has any predictive power worth giving a second glance. Tacking on more complexity and free parameters doesn't guarantee a good theory any more than abstracting them out does.
I actually entirely agree with you. Biology is making terrific progress, and shouldn't be overly compared with physics. Two supporting comments:
First, when biology is judged as nascent, this may be because it is being overly compared with physics. Success in physics meant finding and describing the most fundamental relationship between variables analytically, but this doesn't seem to be what the answers look like in biology. (As Simon Jester wrote here, describing the low-level rules is just the beginning, not the end.) And the relatively simple big ideas, like the theory of evolution and the genetic code, are still often judged as inferior in some way as scientific principles. Perhaps because they're not so closely identified with mathematical equations.
Further, and secondly, the scientific culture that measures progress in biology using the physics paradigm may still be slowing down our progress. While we are making good progress, I also feel a resistance: the reality of biology doesn't seem to be responding well to the scientific epistemology we are throwing at it. But I'm still open-minded, maybe our epistemology needs to be updated or maybe our epistemology is fine and we just need to keep forging on.