How do Words & Numbers Actually Intersect?

Actually, quite a bit, as explaining math requires words, and explaining words requires math.

More interestingly, it is how questioning words leads to great math & physics, and how it is questioning math and physics which leads to great words. Both are meaningful code for what we actually mean to say, write and imagine. Before we can write the words or the math we must first imagine a shape, a direction and perform some pretty spectacular thermodynamics.

Thinking is expensive, for both biological neural networks and for electronic ones. We are involved currently in a worldwide master class indicating how very expensive thinking is — both economically and thermodynamically.

What we may not recognize is that how we derive our explanation of the universe, scientifically and mathematically, depends upon the thermodynamic flexibility of that expensive process. Thinking matters, and in this respect we can recognize that the universe possesses an uncharacteristic profligacy in this respect. We might ask: what insane universe would possess such a profligate accounting system? One which seems to prioritize the dramatically expensive process of either biological or electronic thinking systems?

This encompasses the direction, the shape and the thermodynamics upon which this entire book — and all the books I have written previously — depends. And depending means the whole teetering edifice of my thinking and ideas actually depends upon these questions.

This book — and all the previous ones — have sought to answer, in a decidedly non-linear and at times in a circuitous, spiraling out of control, fashion — these precise questions.

For an answer I give you a sentence. A sentence written by legendary mathematician and physicist, Paul Dirac. A sentence questioned by taking one word in that sentence, and then subjecting that one word to the ultimate Socratic methodology.

The Socratic questioner was legendary Nobel laureate — Richard Feynman — able to leap tall physics problems through questioning mathematics with words, and the ability to question words with mathematics.

We could engineer a very complex chapter or several books on this one questioning of a word, but our intentions here are not to write a book on physics precisely, but to write a book about habits of thinking and how these habits of thinking determine the lives of individuals and whole societies.

The word Dirac wrote: analogous

Dirac's 1933 paper The Lagrangian in Quantum Mechanics opens: "Quantum mechanics was built up on a foundation of analogy with the Hamiltonian theory of classical mechanics. This is because the classical notion of canonical coordinates and momenta was found to be one with a very simple quantum analogue…" Later, on the transformation connecting one representation to another across a slice of time: "It is now easy to see what the quantum analogue of all this must be…" And again: "…it is satisfactory to find that they have their classical analogues, expressible simply in terms of the Lagrangian."

Analogy. Analogue. Corresponds to. These are Dirac's words for the relation between the classical quantity and the quantum one — a likeness held loosely, not an equation.

The word Feynman wrote: proportional

Feynman's word was proportional. Working through Dirac's paper with the physicist Herbert Jehle at Princeton in the early 1940s, he saw that across an infinitesimal slice of time the two quantities were not vaguely analogous. They were proportional — the same relation, differing only by a normalizing constant.

At Princeton's bicentennial celebration in the fall of 1946 he found Dirac lying on the grass outside the conference, and asked him:

Dirac had written the word and walked right past it. Replacing a qualitative metaphor with the strict quantitative reality of a proportionality opened the path integral formulation: particles effectively traverse every possible path through space-time, and the classical path emerges out of the sum of them.

One word, questioned. That is the whole of the demonstration, and it is enough.

Citation note

Dirac's paper. Paul A. M. Dirac, "The Lagrangian in Quantum Mechanics", Mathematical Proceedings of the Cambridge Philosophical Society 29(3), 1933, pp. 292–299. The opening sentence is quoted verbatim from the reprint in J. R. Brown (ed.), Feynman's Thesis — A New Approach to Quantum Theory (World Scientific, 2005), p. 111.

The grass dialogue. Reported by Feynman and repeated in Krauss's Quantum Man and Farmelo's The Strangest Man.

Jehle. Feynman first heard of Dirac's paper from Herbert Jehle at Princeton, about 1940–41.