Weyl’s take on some things
Unfortunately for us, philosophies of science and mathematics are rarely brought to the attention of individuals who are not engaged in these efforts. Yet, while difficult to access, the views of the world provided by mathematics and science are pregnant with meaningful implications for all of us. I have always been struck by the depth of reflection in Hermann Weyl’s writing. These are thoughts that come clearly from a person, an individual working to reconcile all of the images of himself and the world that have taken shape in his mind’s eye.
At the end of his essay The Unity of Knowledge he summarizes that at the basis of knowledge is (1) intuition “the mind’s originary act of seeing what is given to him,” (2) expression, “the active counterpart of passive understanding” and (3) Thinking the possible, or imagination (he then apologizes for some of his indecision of mind) For me, these three things are mathematics. And seeing mathematics in this way tells us something about ourselves and the extent of our potential.
I’ve pulled out other excerpts that I found particularly provocative from a collection of selected Weyl pieces, edited by Peter Pesic, entitled Mind and Nature.
The first is from The Open World published in 1932. It is a reflection on the non-causal aspect of quantum theory, which Weyl sees as bridging organic and inorganic nature.
According to vitalism the living organism reacts as a whole; its functions are not additive. The manner in which its structure is preserved throughout growth, in spite of a outside influences and perturbations, is not to be explained by small scale causal reactions between the elementary parts of the organism. Now we see that according to quantum physics the same applied even to inorganic nature and is not peculiar to organic processes. It is out of the question to derive the state of the whole from the state of its parts. This leads to conditions which may most plainly if not most correctly be interpreted as a peculiar non-causal “understanding” between the elementary particles, that is prior to and independent of the control exercised by differential laws which regulate probabilities…It seems therefore that the quantum theory is called upon to bridge the gap between inorganic and organic nature; to join them in the sense of placing the origin of those phenomena which confront us in the fully developed organism as Life, Soul and Will back in the same original order of nature to which atoms and electrons also are subject. So today less than ever do we need to doubt the objective unity of the whole of nature…..
Within a lengthy discussion of the concept of infinity in mathematics he makes the following remarks:
Mathematics is not the rigid and uninspiring schematism which the layman is so apt to see in it; on the contrary, we stand in mathematics precisely at the point of intersection of limitation and freedom which is the essence of man himself.
Something entirely new takes place when I embed the actually occurring number symbols in the sequence of all possible numbers…The given is embedded in the ordered manifold of the possible, not on the basis of descriptive characteristics, but on the basis of certain mental or physical operations and reactions to be performed on it – as, for example, counting.
Consider the definition “n is an even or an odd number according as there exists or does not exist a number x for which n = 2x. For one who accepts this with its appeal to the infinite totality of numbers x as having a meaning, the sequence of numbers open into infinity has transformed itself into a closed aggregate of objects existing in themselves, a realm of absolute existence which “is not of this world,” and of which the eye of our consciousness perceives but reflected gleams.
We reject the categorical finiteness of man…..On the contrary, mind is freedom within the limitations of existence; it is open toward the infinite…The completed infinite we can only represent in symbols. From this relationship every creative act of man receives its deep consecration and dignity. But only in mathematics and physics, as far as I can see, has symbolical-theoretical construction acquired sufficient solidity to be convincing for everyone whose mind is open to these sciences.
The last one is from the piece called Mind and Nature (1934).
The impossibility of designing a picture of reality other than on the background of possibility appears to be founded on the circumstance that existence is a penetration of the what and the how, and consequently arises from a contact of object and subject, of pure factuality and freedom.
I was particularly taken by the words, “existence is a penetration of the what and the how, and consequently arises from a contact of object and subject” Penetration is not a word one might expect to use to characterize our existence, but I think it leads to the right images – like probe, permeate, immerse, sink into, as well as fathom and understand.
In his introduction Pesic makes the point that Weyl’s praise of the symbol “includes the mathematical no less than the literary, artistic, and poetic.” He tells us that
Weyl gained perspective, insight, and altitude by thinking back along the ever-unfolding past and studying its great thinkers, whom he used to help him soar, like a bird feeling the air under its wings.