As computational neuroscientists, the members of this podcast take for granted the fact that math can be used to describe the brain. But is this ability of math trivial or deeply surprising? On this episode we tackle the large philosophical issues behind computational sciences, with the help of a set of articles entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (1960) and "The Unreasonable Effectiveness of Mathematics" (1980). These articles, written by mathematicians, claim that the ability of mathematics to explain the world is unexpectedly impressive. This leads us to a set of questions: What is math? What is the world? And how do we know what to expect from either? Is this ability of math truly unreasonable or did it just appear so at a certain time? As we explore these ideas we discuss thought experiments involving artificial intelligence and real experiments involving the Cold War. We also hypothesize on the role of elegance in mathematics and what "understanding" really means to us as humans.

We read:

The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The Unreasonable Effectiveness of Mathematics
And mentioned:

The Unreasonable Effectiveness of Data
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As always, our jazzy theme music "Quirky Dog" is courtesy of Kevin MacLeod (incompetech.com)

Listening to the newest podcast "Unreasonable Effectivenes..." and feel like you guys are missing something fairly large and important. Though I must say I am firing this off without having read the essays. But at least some of you keep pushing this idea that "Well, mathmatetics exist because we wanted to talk about the world so why it surprising that it does a good job of talking about the world?"

ReplyDeleteBut I feel that this vastly underestimates how little we of the world we had seen when laying down the fodunation of modern math, not to mention how far mathematics has carried us since then.

Let me make the point with a simple example. For sure,the idea of the natural numbers seems likely to have been the direct result of observing the world around us. We found ourselves wanting ways to talk about joining and dividing and so on.

The part where I think you're supposed to be impressed is that what we have found out is that that these things... natural numbers... do so much more than help us join and divide. They are filled with surprising properties that describe aspects of the world that had nothing to do with their creation. Like, how can anyone be jaded at the idea that if you make a sequence of numbers that are each the usm of the previous two you have a model that describes things in the natural world from snail shells to galaxy spirals. No matter how you slice that is not something that I think is obvious. Certainly, there's no reason to suspect that the tools we invented to help us make money in the agora would generalize in such grand ways.

Obviously, a super simplified toy example but we could keep increasing the sophistication and coming up with all kinds of these examples -- everything from Black-Scholes to DNA to Deep MInd. It's fairly shocking how much of the natural world can be modeled with the a couple logical axioms and the concept of the number one.

I thin your point would resonate more if we had invented mathematics recently - after all this observation -- or if we were constantly tweaking the foundations.. .then I would agree: what do we expect to happen? But as it is, I'm impressed. By the unreasonableness. Of the effectiveness. Of that paper's title.

Thanks for your comment. I think you make clear an important property of mathematics that is part of what is supposed to awe us. And I see the way in which it is impressive, but I just want to highlight two points that I think we were trying to make in response to that:

Delete1.) One of the arguments that Hamming brings up--and I think that we were agreeing with--is that a lot of the world is still not well-explained by mathematics. So in the example you give, the fibonacci sequence can capture something about the pattern of seashells, but there is still a lot of detail and variety in that patterning that it doesn't capture. So our simple understanding of the world when creating the natural numbers gave us a system that (surprisingly?) could explain the basic pattern that underlies the generation of things like seashells and galaxies but was not enough to capture more complex observations.

We didn't discuss this very explicitly in the episode, but in order to formalize more complex facts about the world, math has indeed required many rounds of revision as our understanding of the world increased. Things like zero and negative numbers and decimals and irrational numbers were added and even whole new fields of math were born. You could look at this from the vantage point of: you build a whole bunch of machinery on top of notions of number and number doesn't break! And that's impressive. But there is also the perspective that a lot of iterative tweaking of the foundations actually was there and might be hidden by the present vantage and revisionism (some good books on this history: "Number" by Tobias Dantzig and "Mathematics and Its History" by John Stillwell). In this way, some parts of math are actually somewhat recent inventions. And some of the more recent fields, like Probability Theory for example, are crucial for some of the more sophisticated abilities of math that you listed (like AI).

And even after all of this there are still huge parts of the world that aren't easily mathematized. Our entire field of computational neuroscience is based on how hard it is to quantitatively describe just one organ.

2.) If math is just a language to describe relationships in the world, and the world is somewhat regular and conserved in terms of the types of relationships that can exist in it, then math that we developed to describe something simple should apply to some more complex things. This is a slightly murkier and more philosophical point. In a way it is saying that everything that seems surprising and impressive about math (its regularity and generalizability etc) should really just be attributed to the world and the underlying rules that govern it. Math then is just a reflection of those rules, insofar as we are able to describe them. Now perhaps our ability to identify and describe them is truly impressive and surprising. But that is slightly different than being impressed by the math itself.

I think ultimately though this all comes down to a question of magnitude, and as such it's a rather subjective one. There are certainly examples where we've gotten more out of math than we've explicitly put in. Is it enough to be flabbergasted by? That's in the eye of the beholder. I certainly hope, however, that we don't come across as jaded (!) but just on the not-entirely-overwhelmed end of the spectrum.