[2]A short while ago, a narrator on PBS radio was presenting a theory by a fellow named Rye-man (as in rye bread). Riemann (rhymes with tea or the T in t-shirt) would have readily recognize the theory as his own, because of the clear, concise and authoritative presentation of the piece. But he would not have recognized his name from its pronunciation.
[3]See Riemann, 1929b, pp. 411-425.
[4]If Riemann's words sound like the fixed-point theorem we used in the last chapter, there is very good reason. In a later investigation, he virtually laid the groundwork for modern topology (and I believe, but cannot prove, contemporary mathematical logic). Riemann's lecture to the distinguished Göttingeners contains the germ of much of his later work (the part I'm familiar with, that is).
[5]In a right triangle, the tangent of an acute angle (A) equals the opposite side (a) divided by the adjacent side (b); in shorthand, tan A = a/b. With r the hypotenuse of the right triangle, sin A = a/r; cos A = b/r.
[6]Once again, let me remind those who are terrorized by mathematics of the existence of Salvanius Thompson's Calculus Made Easy.
[7]The exceptional derivative is the base of natural or Naperian logarithms.
[7a] What do we say to those who insist that what we can't see doesn't exist? Few of us have experienced a million dollars. How many among us would tear up a check in that amount? Or does the National Debt of a nation disappear because no one can see a trillion dollars, yen, pounds....?
[8]In our own day, Benoit Mandelbrot has lifted the restrictions even further with entities he calls "fractals"; his work is cited in the bibliography.
[9]See Cohen and Hersh, 1967.
[10]If you think the notion of negative space is nothing but pure theoretical baloney, consider this fact. On the very morning I am keying this very shtick into my computer, I heard on the radio that physicists had finally succeeded in constructing anti-atoms. As atoms occupy space, anti-atoms must occupy anti- (or negative) space.
[10a]Out at the fringe of things, there may be only a single infinity, and thus a confluence of our positive and negative infinities; we don't know this for sure, one way or another; but the uncertainty is why one instead of two uncountable sets for the minimum.
[11]The reader who enjoys problems of this sort may also enjoy chapter 3 of Martin Gardner's book (see bibliography).
[12]Brillouin, 1964, p.3.
[13]Brillouin, 1964, p.3.
[13b]The serious student of neuroscience who doubts that tensors can be rigorously and directly applied to brain function should read Pellionisz, A. and Llinas, R., "Brain modelling by tensor network theory and computer simulation. The cerebellum: distributed processor for predictive coordination." Neurosciences vol. 4, pages 323-348, 1979.
Unfortunately (for me), I only became aware of the remarkable research of these two investigators after the first edition of Shufflebrain had been published and Pellionisz sent me a nice letter and some reprints.
[14]Clinical literature abounds with reports of this general sort, some related to obvious brain damage, others not. N. J. David (1964, p. 150), presents an illustrative case history of a young man two policemen arrested for public drunkenness when he asked them why they were only two feet tall. The young man was not drunk. He was suffering from an attack of psychomotor epilepsy.
[15]Existenz refers to an entity that exists by force of implication. The existential philosophers (see summary in Flew, 1979) use the term more comprehensively then I do here. But their usage and mine connote more than existence in the sense of being present (Dasein, in German). Infinity is a good example of Existenz .