I’ve become a bit preoccupied recently with the world of early 20th century mathematicians, partly because of a book I’m working on, but also because of how late 19th and early 20th century thinking largely defines the mathematics students learn today. In this light I found a book of selected writings of Hermann Weyl, a prominent 20th century mathematician. The book, Mind and Nature, was edited by Peter Pesic. One of the Weyl selections from 1932 is called The Open World: Three Lectures on the Metaphysical Implications of Science.
Weyl’s preface to the lectures begins like this:
One common thought holds together the following three lectures: Modern science, insofar as I am familiar with it through my own scientific work, mathematics and physics make the world appear more and more as an open one, as a world not closed but pointing beyond itself……It remains to be added that science can do no more than show us this open horizon….
He follows this with an analysis of perspectives that touches on the thoughts of individuals from Plato to W. Pauli and includes references to the work of some of his contemporaries. Weyl’s lectures and papers are never easy reading, but they are an opportunity. Rarely do mathematicians write about what they do.
I would like to point today to just a few of Weyl’s thoughts as they correspond to ideas I’m trying to build.
About the very nature of reality, Weyl says that it is “an error of idealism” to assume that the mind or the ego’s mental images guarantee a reality to the ego which is more certain than the reality of the external world. What he says next reminded me of the kinds of ideas in the notion of embodied cognition.
….in the transition from consciousness to reality the ego, the thou and the world rise into existence indissolubly connected and, as it were, at one stroke.
Weyl distinguishes two domains in human life, creation (or construction) and reflection (or cognition). He warns that constructive activity that is not aided by reflection may depart from meaning. Passive reflection, however, “may lead to incomprehensible talking about things which paralyzes the creative power of man.” Truth, he says, must be attained by action, “..we do not perceive it if we merely open our eyes wide…” And science cannot be done with intuitive cognition alone. He uses this idea to conclude that Bouwer’s restriction of mathematics to “intuitively cognizable truths” would not help us in the sciences. The action, to which he refers, is the mathematics that begins with axioms followed by what Weyl calls “the practical rules of conclusion.”
With respect to the infinite, the subject of much disagreement among late 19th century mathematicians, Weyl tries to put down “the experiences which mathematics has gained in the course of its history by an investigation of the infinite.” His words again:
The infinite is accessible to the mind intuitively in the form of the field of possibilities open into infinity, analogous to the sequence of numbers which can be continued indefinitely; but the completed, the actual infinite as a closed realm of absolute existence is not within its reach. Yet the demand for totality and the metaphysical belief in reality inevitably compel the mind to represent the infinite as closed being by symbolic construction.
Mathematicians who would not allow the existence of an actual infinity rejected the set theoretic work of Georg Cantor which distinguished different kinds of infinities, established equivalence relations among them, and now defines the the structure of much of modern mathematics.
Weyl makes a final point when he rejects “the categorical finiteness of man.” in both its atheistic and religious forms. He concludes that mind is “freedom within the limitations of existence” and is “open toward the infinite.” Further, “God as the completed infinite cannot and will not be comprehended by it.”
Recent Comments