Can we see where math begins and science ends?
Galileo is often called the father of modern science because of an insight he had about the relationship between mathematics, and what we are able to see in our world. Two of John Horgan’s recent blog posts (and the writing to which they refer) nicely demonstrate what I think is a remarkable oversight in discussions about the prospects for the future of science as we know it. Neither John, nor any of the writers to whom he refers, consider the significance of the role that mathematics is playing in the development of scientific ideas and analyses. None of them wonder about how mathematics shapes our views of reality. If we want to consider that we’ve reached some limit to the progress that science can make, perhaps we should revisit Galileo’s original insight about what science is, and think again about the role mathematics plays.
Galileo understood that science could not be done without mathematics. And it’s this science that so many seem to be worried about. From his book Il Saggiatore (The Assayer) published in 1623:
Philosophy [i.e. physics] is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.
The words that strike me are “continually open to our gaze, but not understood.” Galileo’s insight was that mathematics could bridge the rift between the gaze and the understanding. But how or even why does it happen? Trying to get at the how might shed new light on what we call physical law. Getting at why might lead to fresh ways to consider questions about consciousness and objectivity. These questions are at the bottom of any other questions we might have about the value or future of science. I’m not meaning to suggest that there are any easy or definitive answers to these questions, but they are certainly relevant to the questions being asked about what science has or may yet accomplish, and consistently overlooked. The Euclidean geometry that lit Galileo’s way was not originally motivated by pragmatic concerns or by the use Galileo made of them. There is something independent about the spirit of mathematics that we barely understand. And the mathematical landscape has exploded with ideas since Euclid’s survey of geometry. It is the wealth of conceptual forms, provided by mathematics, that has shaped the often bewildering, counter-intuitive ideas in modern physics. It is mathematics that resolves the flood of experimental observations into the space-time continuum of Relativity, or into the quirky laws of quantum mechanics. It is mathematics that leads some physicists to consider multiverse ideas, or leads others to the dispute over which is primary to the universe – material or information. It should not be possible to leave mathematics out of a discussion of science, and of physics in particular.
Questions that address the emergence of mathematics, and the cognitive structures it may mirror, could give us a new way to tackle the enigmatic relationship between mind and matter (as we now imagine them). Mathematics is, after all, an almost purely introspective science, yet it builds the science of material, the structure of modern physics.
I remember reading John Horgan’s 1996 book The End of Science and really enjoying it. The intimacy of his interviews brought real vitality to the ideas. His latest post is called The End-of-Science Bandwagon is Getting Crowded. In it he quotes from some of the responses to Edge.org question, What should we be worried about? He’s chosen scientists concerned with the future of science. Horgan also references a Nature essay by Dean Keith Simonton who argues:
Our theories and instruments now probe the earliest seconds and farthest reaches of the Universe, and we can investigate the tiniest of life forms and the shortest-lived of subatomic particles. It is difficult to imagine that scientists have overlooked some phenomenon worthy of its own discipline alongside astronomy, physics, chemistry and biology…Future advances are likely to build on what is already known rather than alter the foundations of knowledge.
But it seems to me that if we can manage to get our attention on some useful questions about the more specific nature of this knowledge – about what it’s made from and how we built it – then we might begin to see a revolutionary way to extend its limits. We might replace our current notion of objectivity with something more appropriate to the interplay between ourselves and the world that we see in pure sensation as well as mathematics. Mathematics is not just the tool. It is the strategy.