From Osher Doctorow
The habit of not frequently examining one's Foundations is sometimes
justified by claiming that it is more valuable to "go in new
directions," but in the case of tensors it has resulted in some rather
curious "blocking of comprehension".
Consider for example a matrix, which represents a tensor in a
particular coordinate system. With the emphasis on "invariants", we
certainly regard determinants and traces for example as key aspects of
matrices, but we regard the matrix itself as something "deep" beyond
its invariants. Just how deep is it? The question is usually left
unanswered if ever asked.
Yet let's write a 2 x 2 matrix M this way:
1) | a1 b1 |
| a2 b2 |
The elements of the first column, call it a (or vector a, or transpose
of a), are "matched" with the elements of the second column b. We
then have:
2) det(M) = a1b2 - a2b1
3) Trace(M) = a1 + b2
The corresponding 3 x 3 matrix, call in N, will be labelled with
columns a, b, c, from left to right, with a = (a1, a2, a3)^T (T =
transpose), etc. We have then:
4) det(N) = a1(b2c3 - c2b3) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2)
5) Trace(N) = a1 + b2 + c3
This is very reminiscent of curl in its matrix formulation with first
row vectors i, j, k, and even of Lie brackets and Poisson brackets in
regard to (2) and (4).
What is going on here?
Well, all the invariants of matrices involve PROXIMITY of columns (or
rows) or even of elements of diagonals or subdiagonals. In (2) and
(4), the subscript 2 is not only proximal to 3, but so is the element
respectively labelled 2 and 3 (of the corresponding columns). It is
true that a "non-proximal" element enters in (4), for example the
factor a1 outside the parentheses in the first term, which is not
proximal to c3 in the sense that column b intervenes, but a1 is
"safely" isolated from the parentheses.
Similarly, the rows or columns 1, 2, 3 of (5) are proximal (differ by
one).
Is it possible that when writing a matrix itself, there isn't anything
to it but its elements like a1, a2, a3, b1, b2, b3, etc., and the
proximities of its elements to each other?
This is not only a plausible conjecture, but is arguably very
plausible, and now for the "topper" or "clincher" to this story,
namely, that Probable Influence/Causation (PI) is the mathematics of
proximity at least in the probabilistic formulation, since as easily
verified (and as I've proven fairly often in previous threads here),
PI is a one-sided partial inverse of Euclidean type distance-functions
or metrics which increases as distances decrease, that is to say, PI
measures "nearness" or "proximity", while Euclidean type distance-
functions measure "farness"! For readers who don't recall how PI
generalizes from 1 + y - x to n dimensions, scalars x and y are
replaced by means of vectors x = (x1, x2, ..., xn) and y = (y1,
y2, ..., yn), although other generalizations are also possible as I
discussed earlier in this thread.
Osher Doctorow
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