Sunday, February 8, 2009

Kant, Math and Perennial Questions of Philosophy

Philosophy addresses questions that flow from the fundamental human condition, and so are relevant to all times and places. These are the questions that lie behind all other questions; a great philosopher is one who identifies and addresses these fundamental questions. The great philosopher is validated in his intuitions by the fact that, over time, men keep returning to the very same questions originally asked by the great philosopher. 

An example of this is found in the Ideas section of this Sunday's Boston Globe. The article starts this way:

Mario Livio is an astrophysicist, a man whose work and worldview are inextricably intertwined with mathematics. Like most scientists, he depends on math and an underlying faith in its incredible power to explain the universe. But over the years, he has been nagged by a bewildering thought. Scientific progress, in everything from economics to neurobiology to physics, depends on math's ability. But what is math? Why should its abstract concepts be so uncannily good at explaining reality?

The question may seem irrelevant. As long as math works, why not just go with it? But Livio felt himself pulled into a deep question that reaches into the very foundation of science - and of reality itself. The language of the universe appears to be mathematics: Formulas describe how our planet revolves around the sun, how a boat floats, how light glints off the water. But is mathematics a human tool, or is reality, in some fundamental way, mathematics?

What's funny about the contemporary world is that, after dismissing philosophy as an anachronistic waste of time, when we finally discover truly philosophical questions, we believe in all innocence that we are the first to ask them. Then we breathlessly announce our bold quest to answer these novel questions, in which we think we are blazing a new trail of human knowledge, but in reality are only retracing the footsteps of the philosophers of old, usually doing a much poorer job than they did. Here is Immanuel Kant writing in the eighteenth century in his Critique of Pure Reason:

In the solution of the above problem there is at the same time contained the possibility of the pure use of reason in the grounding and execution of all sciences that contain a theoretical a priori cognition of objects, i.e., the answer to the questions:

How is pure mathematics possible?
How is pure natural science possible?

About these sciences, since they are actually given, it can appropriately be asked how they are possible; for that they must be possible is proved through their actuality.

The Globe interview goes on to say the following:

Livio, an astrophysicist at the Space Telescope Science Institute in Baltimore, concludes that math has to be thought of, at least in part, as a human invention. That's a profoundly weird notion in a world where math has always had a special status, untainted by people's opinions and biases. Religion, politics, and picking a great work of art can all incite vigorous debate, while 2+2=4 has always seemed like a cold, hard fact. Math isn't a figment of our imagination, but perhaps it isn't quite as far from great art as we thought.

It's only a "profoundly weird notion", or a novel notion, for someone who has never read Kant. Kant proposed that math is a human invention insofar as it reflects the structural nature of human cognition. "Time" and "Space" are the forms under which man experiences reality according to his nature; geometry has the certainty it does because its conclusions are a necessary consequence of the nature of man.  Now we may agree or disagree with Kant, but there is nothing new in these ideas.

It follows, of course, that other intelligent creatures who do not share the cognitive apparatus of man may not develop math and science the way we do. Our "time" and "space" may mean nothing to them, let alone the geometry and physics we develop from them. Kant made these points in his Critique. Livio uses the example of a jellyfish. Would jellyfish develop the natural numbers - 1,2,3,4, etc. - when perhaps their entire experience is purely analog, e.g. the pressure temperature, and motion of water? Maybe not, but again, this is not a new point, only an old point - stated clearly and profoundly by Kant - presented as new.

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