December 31, 2020

Alien Mathematics

Is human mathematics universal? What if it isn’t?

Alien Mathematics

In "The Beginning of Infinity"* David Deutsch argues that the human brain is what he calls a universal explainer. There is a lot to unpack in that statement, but the main idea is that there exists an objective physical reality outside subjective human experience, that this reality is governed by natural law, and that the human brain has evolved to be able to discover any and all such natural laws (hence, universality) through the creation physical theories expressed in the language of mathematics and falsifiable through empirical measurement.

Deutsch's statement lies at the intersection of many very old philosophical questions, and much of what he says is assertion rather than synthesis (or explanation, as he calls it). Deutsch takes great pains to defend the universality of human eplanatory power, including an attempt to refute Dawkins' assertion that the selection pressures the brain underwent actually makes the human brain limited in its explanatory capacity.

Core to Deutsche's idea of universality is that the brain's explanatory power goes beyond direct human experience. No human being and in fact no instrument we have has ever observed the core of a neutron star directly, but we have explanatory theories of thow they work which we can nevertheless test and refute. One of the reasons for this, according to Deutsch, is that the laws of logic (which ultimately govern the language of mathematics at its very foundation) are direct consequences of the (natural) laws of physics.

This is a neat intellectual judo move. It provides an explanation for the unreasonable effectiveness of mathematics in the natural sciences - i.e; the observation made by Eugene Wigner (and others) that the abstract process of mathematical theorem and proof, seemingly detached from physical reality, nevertheless is extremely powerful in explaining the physical world.

The thing is, while mathematics is often unreasonably effective, it is not always so. It is possible to imagine all manner of fantastical phenomena, e.g. Dark Matter, which fit this description, but one need not go so far to find a barrier in our ability to explain things.

Some very simple physical systems exhibit inexplicable behavior, and have remained so for centuries.


Universality Breaks Down

Our physical models account very well for linear phenomena. Loosely put, linear phenomena are those which can be expressed in systems of equations in which the terms involve only addition, and multiplication by constants**. The equations themselves might be diferential equations describing dynamical processes, stochastic equations expressing probabilities, or some other form, but the underlying pattern of linearity is the same.

Many physical phenomena like two-body gravitational mechanics, heat transfer, free-body forces, ideal gas models, and others, can either be expressed exactly or else can be very well approximated as linear systems. Some nonlinear phenomena can be accurately modeled by assuming they are locally linear, and making tiny linear-steps, correcting as we go.

Unfortunately many (in fact, most) actual physical phenomena are not only non-linear in character, they cannot even be well approximated as linear systems. Three famous examples are the n-body gravitational dynamics problem, the problem of fluidic turbulence, and my personal favorite, the dynamics of the double pendulum.

Double pendulum in action. 

In each of these examples, the problem is not that we don't have a mathematical model which describes these physical phenomena. We do, and the models are in fact very simple. The dynamics of the double pendulum, that is the equations which precisely specify its position in space at any instant in time, are expressed exactly in the following four equations;

{\displaystyle {\begin{aligned}x_{1}&={\frac {l}{2}}\sin \theta _{1}\\y_{1}&=-{\frac {l}{2}}\cos \theta _{1}\end{aligned}}}
Position of the center of the first pendulum with respect to angle of the first pendulum
{\displaystyle {\begin{aligned}x_{2}&=l\left(\sin \theta _{1}+{\tfrac {1}{2}}\sin \theta _{2}\right)\\y_{2}&=-l\left(\cos \theta _{1}+{\tfrac {1}{2}}\cos \theta _{2}\right)\end{aligned}}}
Position of the center of the second pendulum with respect the angles of the first an second pendulums. 

The motion of the pendulum through space is thus also governed by these equations. But despite being able to describe the dynamics of the pendulum exactly, we are unable to exactly compute the time evolution of this dynamical system. Put another way, there is no equation (that we know of) that would allow us to write down where the pendulum would be at some point in the future, given its current position.

But actually the problem is even worse, we cannot even approximate it well. The double pendulum has the special property that very small changes in initial conditions result in very large changes in eventual outcome. And that means small approximation errors compound much faster than we can deal with them - the system diverges***. One way to think about it is, to get an additional digit of precision about where the pendulum will be one second from now requires an order of magnitude increase in the number of approximation steps, i.e. computations.

It must be reiterated that the equations are exact. We don't require any outside physical explanation for why we can't predict the pendulum's dynamics, it's a property not of randomness, quantum physics, or anything else - it's a property of the governing equations themselves.

To the extent that physical models are what allow us to explain (and thus make predictions about) reality, and that these models are expressed in the language of mathematics, our failure in predicting the state of such a simple system at arbitrary times in the future with unbounded accuracy seems to indicate that our mathematics is far from universal.

This strange duality has fascinated me for as long as I have been learning mathematics.

How is it that we can exactly describe these systems, in very simple ways, but we cannot use our descriptions to make predictions about them? This is a very different problem from not having enough information to describe some physical phenomena - all the information is there, there just seems to be an unseen barrier between it and our ability to grasp it.

Is there a hole at the foundation of our mathematics? A problem with logic itself which raises barriers to human explanatory power? Moreover, is that hole there because of some limitation of human perception? And are there other physical phenomena which are even further beyond the barrier?


Alien Mathematics From Beyond The Barrier

In Fire in the Sky, Mike Solana makes the case that we should be taking UFO phenomena seriously, because the whatever the explanation (government coverup, aliens, time travel...) it would be of incredible importance. He identifies this as a belief barrier; people seem unable to take these phenomena seriously.

Solana makes another interesting point, that these phenomena might be something which exposes the limits of human perception - we are observing physical phenomena so far outside human experience that our cognitive-sensory apparatus has no chance to make sense of them. They represent a sheer wall, which our human explanatory power has no hand-holds to climb. A perceptual or cognitive barrier.  

As we have demonstrated with the double pendulum example, this is not a far-fetched idea. We run into these barriers inside our own mathematics, when we care to notice them. Do these things imply fundamental limitations to what we can ever hope to explain?

Let's assume for a moment that Deutsch is right about an objective physical reality, and Solana is right about limits to human perception and cognition. Suppose those limitations in our ability to even think about these phenomena - our inability to predict the evolution of the double pendulum - are a result of the environment we evolved in; that most things of interest to early humans as they obtained the ability for abstract thought were linear phenomena.

What about an intelligent species which evolved under different conditions?****

Faced with a nonlinear physical reality, in order to reach any explanatory power their alien mathematics must be able to deal with nonlinearity as a first class property, just as linearity and infinitessimals are in ours. They would be able to explain and hence harness and predict an entirely different, perhaps wider set of physical phenomena than us (or perhaps to them it would appear that most phenomena in the universe were linear, since those are the phenomena they'd have trouble explaining). The resulting effects of this species harnessing those phenomena may well be incomprehensible to our own explanatory apparatus.

The alien mathematics would have different explanatory power. It would have a different part of objective physical reality within its barrier, and might have a great deal of trouble conceptualizing the part of reality within ours. Our interactions would be confusing at best.

They might look like Fire in the Sky.


These ideas contradict neither Deutsch nor Solana. They preserve the idea of an objective physical reality that both we humans and the aliens experience through physical phenomena, and that physical reality can be explained, while showing the possibility for phenomena outside human ability to comprehend.

Deutsch may be right that logic is a consequence of physical law; but humanity was not handed all of physical law when we became conscious for the first time. We should not assume we have been handed all of logic as well. And perhaps someone else has arrived at a different logic, through the need to develop different explanations.

Whatever the case, in looking for where the barrier lies and finding things that lie beyond it, we at least admit the possibility to expand what it's possible for humans to think, and eventually, the foundations of our explanatory power. Perhaps if we meet a species with their alien, nonlinear mathematics, we could find a way to exchange our knowledge, and synthesize a more universal logic in our drive toward becoming universal explainers.

But if they would not share, or we never meet them, well, through our determined observation and fundamental human need to understand, maybe we can climb.


Postscript:
After a conversation with Mike, I became aware of another possibility. If we take the assumption that human cognition - the process of generating explanations - is emulatable***** by a Turing machine (which Deutsch and other authors often do), then one need only find in objective reality a physical phenomenon which is not only itself computationally intractable by a classical Turing machine, but whose explanation is incomputable.

In this scenario, we really are in a great deal of trouble in terms of what we'll ever be able to explain. An airtight intellectual box it's physically impossible to breach.

That's a cosmic horror level thought and I'll probably think about it further.


*I have a difficult relationship with The Beginning of Infinity. On the one hand, Deutsch constantly makes assertions that I know to be untrue (for example, about computability) which makes the book overall quite suspect, and on the other, reading it pushes me to think about things I didn't realize one could think about. Maddeningly, Deutsch makes true assertions in his scientific papers on the same subjects (notbaly, physics and computability theory).

**Another short-hand is "if you can express it as the product of matrices and vectors, it's linear".  

*** A formal quantification for how fast the trajectories of a dynamical system diverge over time is called Lyapunov Time. Russian/Soviet mathematicians, notably Lyapunov, Kolmogorov, and many others, did foundational work in this area.

**** In Cixin Liu's aptly named Three Body Problem, an alien species evolves in an environment with properties very much like the double pendulum; they live on a planet orbiting a three-star system. They are forced to develop an alien, nonlinear mathematics much sooner, resulting in great technological advantages. But their abstract thought is missing the depth of communication ability humans have - they are unable to lie, or even conceive of lying.

***** Emulation has a strtict mathematical definition; one state machine emulates another if it can achieve the same output sequence given the same input sequence.