I noticed an irrepressible smile on my face as I listened to Peter Sarnak’s lecture in a complex analysis class I took in graduate school. I tried to account for the pleasure I felt about ideas I could just barely comprehend. It seemed I understood the significance of the exploration, that these abstractions were discerning realities not accessible to the senses.
This weekend I picked up The Lightness of Being by physicist Frank Wilczek. About the title Wilczek says:
A central theme of this book is that the ancient contrast between celestial light and earthy matter has been transcended. In modern Physics, there’s only one thing, and that thing is more like the traditional idea of light than the traditional idea of matter.
This statement is grounded in work done in particle physics, which the book explores. In the first few pages he makes the following remark:
But the ultimate sense-enhancing device is a thinking mind. Thinking minds allow us to realize that the world contains much more, and is in many ways a different thing, than meets the eye.
Seasonal repetitions, the motion of celestial objects, hidden regularities in harmonies, or in material in general (both organic and inorganic), are not captured by the fundamental action of the senses, but only by the thoughtful observation of pattern. And mathematics is the way we do this.
I’m convinced that my smiling in that complex analysis class was a consequence of my recognizing how far-reaching and liberating the observation of pattern and relationship can be. We do it in other things. In painting, music, poetry and fiction, we produce maps to what we cannot see directly by finding relationships in our experiences. We create what one might call emotionally defined aggregates, of the things we see and feel, and organize them with the words or images that best capture the sensations . And the precision of mathematically defined classes and equivalence relationships is the creativity that builds what we call science.
It is our nature to find our reality in relationship and not in ‘the thing itself.’ Our bodies are built for it. Mathematics helps us get behind misleading appearances by relying exclusively on discerned relationship. It admits no other kind of judgment. As a thorough investigation of relationship, it ultimately breaks through stubborn prejudices (like the work of Riemann described in my previous post). Richard Courant points to the necessary dissubstantiation of mathematically defined objects in What is Mathematics? when he says:
What points, lines, numbers “actually” are cannot and need not be discussed in mathematical science. What matters and what corresponds to “verifiable” fact is structure and relationship….
This dissubstantiation of mathematical objects is the prerequisite for finding our way. On this rests The Lightness of Being…..
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