Journeying in and out

When ecently reminded of the images in Catholic texts and prayers, I considered, again, my hunch that mathematics could somehow help connect unrelated aspects our experience, in particular counterintuitive religious images and familiar sensory experience. I am not suggesting that mathematics would explain these images, but more that it could be used to encourage, perhaps even contribute to, their exploration. This possibility would rely on a refreshed understanding of what mathematics is.  There are numerous mathematical ideas or objects – necessary, productive and useful ideas – that do not correspond to anything familiar to the senses.  Some of the most accessible are things like the infinite divisibility of the line, the point at infinity, bounded infinities, or the simple fact that the open interval from 0 to 1 is equivalent to the whole number line.  The infinite divisibility of the line relies, to a large extent, on the fact that there are no spaces between ‘individual points.’  A similar construction happens in complex analysis, where one can consider layered complex planes with no space between them.  My hope is that these mathematical possibilities become more widely known and considered.

Today I looked back at a post that I wrote in 2010 with the title, Archetypes, Image Schemas, Numbers and the Season.  The subject of the post is a chapter from the book, Recasting Reality: Wolfgang Pauli’s Philosophical Ideas and Contemporary Science.  The chapter was written by cognitive scientist Raphael Nuñez who uses Pauli’s collaboration with Jung to address Pauli’s philosophy of mathematics. Jung understood ‘number’ in terms of archetypes, primitive mental images that are part of our collective unconscious.  But Nuñez seems most interested in addressing Platonism.  Pauli’s interest in Jung doesn’t address Platonism directly but it is nonetheless implied in many of the things he says.  As a cognitive scientist, Nuñez rejects Platonism.  Despite the complexity of mathematical abstractions, he argues that the discipline is heavily driven by human experience. While his observations of Pauli’s interest in Jung’s psychology are nicely laid out, and some parallels to his own theory are highlighted, Pauli’s ideas don’t really contribute to the non-mystical position that Nuñez has staked out.

But today I looked at the entire text to which Nuñez contributed, and could see that there are a number of things in Pauli’s view that address my own preoccupation with the nature of mathematics, and more deeply than does the question of whether mathematics exists independent of human experience or not. Pauli was preoccupied with reconciling opposites, finding unity, making things whole, and was strongly motivated to think about the problem of how scientific knowledge, and what he called redemptive knowledge, are related to each other. I find it fairly plausible that mathematics could help with this since it exists in the world of ideal images as well as the world of physical measurement and logical reasoning.

Today I found a really nice essay by physicist Hans von Baeyer with the title Wolfgang Pauli’s Journey Inward.  It tells a more intimate story of Pauli’s ardent search for what’s true, and is well worth the read.

In time Pauli came to feel that the irrational component of his personality, represented by the black, female yin, was every bit as significant as its rational counterpart. Pauli called it his shadow and struggled to come to terms with it. What he yearned for was a harmonious balance of yin and yang, of female and male elements, of the irrational and the rational, of soul and body, of religion and science

During his lifetime, Pauli’s fervent quest for spiritual wholeness was unknown to the public and ignored by his colleagues. Today, with the debate between science and religion once more at high tide, Pauli’s visionary pursuit speaks to us with renewed relevance.

I particularly enjoyed von Baeyer’s description of the famous Exclusion Principle for which Pauli received a Nobel Prize in 1945. It went like this:

The fundamental question had been why the six electrons in the carbon atom, say, don’t all carry the same amount of energy — “why their quantum numbers don’t have identical values… it should be expected that the electrons would all seek the same lowest possible energy configuration, the way water seeks the lowest level, and crowd into it.” If this rule applied to electrons in atoms, there would be very little difference between, say, carbon with its six and nitrogen with its seven electrons. There would be no chemistry.

Pauli answered the question by decree: the electrons in an atom, he claimed, don’t have the same quantum numbers because they can’t. If one electron is labeled with, say, the four quantum numbers (5, 2, 3, 0) the next electron you add must carry a different label, say (5, 2, 3, 1) or perhaps (6, 2, 3, 0). He proposed no new force between electrons, no mechanism, not even logic to support this injunction. It was simply a rule, imperious in its peremptoriness, and unlike anything else in the entire sweep of modern physics. Electrons avoid each other’s private quantum numbers for no reason other than, as one physicist put it, “for fear of Pauli.” …With the invention of the fourth quantum number and the exclusion principle Pauli opened the way for the systematic construction of Mendeleev’s entire periodic table.

What struck me from reading von Baeyer’s account was the depth of Pauli’s concern. And the boldness of his Exclusion Principle somehow makes him seem particularly trustworthy. The reconciliation he sought was not one that just allowed for the accepted coexistence of different concerns, but rather one that changed both of them to accommodate something new. As von Baeyer points out, physics has become more and more dominated by “the manipulation of symbols that facilitate thinking but bear only an indirect relationship to observable facts.” Pauli seemed to expect that symbol was the link between the rational and the irrational.  This would easily support my hunch.  He seemed to expect that science be able to deal with the soul, where the soul in turn informs science.

Eventually, he hoped, science and religion, which he believed with Einstein to have common roots, will again be one single endeavor, with a common language, common symbols, and a common purpose.

This is what I expect. And, at the moment, mathematics seems to be my most trustworthy guide. It lives on the boundary that we think we see between pure thought and material, between mind and matter. Pauli’s conviction is particularly reassuring.

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