Mathematics is usually thought of as a tool that quantifies things in our lives and there is good reason for this. Early in our experience, it is presented to us as a counting and measuring device, not as a way to see something. But this characterization of mathematics is misleading. Quantification alone would not get us very far. The true value of numbers is that they give us a way to perceive order and relationship, and these produce the images and forms in mathematics that have become so powerful. Despite the ubiquitous presence of these forms in living things and social phenomena, we still tend to associate mathematical ideas with physics, or the forces that structure material. Yet mathematics itself emerges, is brought to life, from our own biology. How or why we find it is as mysterious today as it ever was.
It is for this reason that every indication of its living presence is interesting to me, like the instinctual vector analysis that ants seem to manage to find their way home. Or what I found today – the living phase transitions of starling flocks. The video posted on Wired Science is definitely worth a look. The text of the article describes the unexpected character of the patterns displayed by the flock.
What makes possible the uncanny coordination of these murmurations, as starling flocks are so beautifully known? Until recently, it was hard to say. Scientists had to wait for the tools of high-powered video analysis and computational modeling. And when these were finally applied to starlings, they revealed patterns known less from biology than cutting-edge physics.
Starling flocks, it turns out, are best described with equations of “critical transitions” — systems that are poised to tip, to be almost instantly and completely transformed, like metals becoming magnetized or liquid turning to gas. Each starling in a flock is connected to every other. When a flock turns in unison, it’s a phase transition.
Another great video is featured here.
In the abstract of a paper on biological criticality the authors explain:
Many of life’s most fascinating phenomena emerge from interactions among many elements–many amino acids determine the structure of a single protein, many genes determine the fate of a cell, many neurons are involved in shaping our thoughts and memories. Physicists have long hoped that these collective behaviors could be described using the ideas and methods of statistical mechanics. In the past few years, new, larger scale experiments have made it possible to construct statistical mechanics models of biological systems directly from real data. We review the surprising successes of this “inverse” approach, using examples form families of proteins, networks of neurons, and flocks of birds. Remarkably, in all these cases the models that emerge from the data are poised at a very special point in their parameter space–a critical point. This suggests there may be some deeper theoretical principle behind the behavior of these diverse systems.
It’s not just interesting that these events can be modeled using mathematics. What’s noteworthy is that the living actions themselves seem to contain mathematics. They manifest the mathematical forms we investigate as plainly as do the organic structures in this very pretty film about the Fibanocci numbers.
What this suggests to me is that mathematics is opening our awareness to something truly fundamental about our reality by giving it conceptual shape. And this window we’ve created should ultimately tell us something about ourselves.
A quote I like from Blaise Pascal (that I found on MAA Mathematical Sciences digital library) is this one:
Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.
Thank you for your thoughtful reply. I do appreciate it.
You make a fine point, but you misunderstand me. And perhaps I should make myself more clear. I’m not expecting science to tell us more about ourselves. What I hope is that a fresh look at how we make science, and how we make mathematics (as well as art and music) will reveal a side of life (even larger than our own) that we may be missing. That’s why I work on this blog. In this post it’s the fact that we capture a living event in a concept that I find noteworthy. Or that the concept has life outside of our making it. I haven’t read Spaemann, but I will look into his work. I don’t believe that our experience with science conflicts with our experience of God. And I probably chose to quote Pascal for reasons you would appreciate.
“And this window we’ve created should ultimately tell us something about ourselves.” You assume (no scientific evidence here)that ‘ourself’ is countable, therefore I will find out what I am made of with science. In other words, I will pull myself up with my own boot straps.
Reading ALL of what Blaise Pascal wrote might help.
I would like to introduce to you somebody who could really help, Robert SPAEMANN . Here is a little taste. (Please do not let the word God throw you off “Science is the investigation into causal conditions. To be a self means to be liberated from the conditions determining a thing’s existence. And that which is unconditional cannot appear within the search for conditions per definitionem. But God is the Unconditional per definitionem. When I said that the progress of science has driven the rumors away from the Unconditional, it is not because science has produced some argument against this rumor. The reason is instead a psychological one: scientific progress has dazzled us. It exerts a fascination. It creates the impression that science will ultimately end up explaining everything, even though this notion is as absurd as it would be for a person watching a film to expect the film ultimately to illuminate itself so that the projector would become superfluous.