So, after some months in which I have been thinking about Wigner's famous remark on the strange almost magical power of mathematics in the sciences, I have some thoughts, a sketch if you will. Just for fun and stimulation:
Now, for argument consider an abstract, empty world. (If such be possible.)
To see how numbers are real in even such, consider the well-known empty set, which collects nothing: { }. Then, consider -- all of this is a mental, abstract exercise -- the following steps:
i: Assign { } the symbol, 0: { } --> 0
ii: collect 0 as the sole member of the set 1: { 0 } --> 1
iii: Similarly, collect to get 2: { 0, 1 } --> 2. The number 2 thus exists without a beginning or cause, nor can it cease from being -- it is a necessary being. We cannot create a possible world in which 2 would not exist, given the abstract steps so far!
iv: This recognition of the reality of numbers can continue indefinitely to yield the Natural Numbers, N. So far, some odd math but nothing too weird. But let us go on:
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v: For the more mathematically inclined (fair warning . . . !), this can be extended by defining fractions and decimals to express Real and Complex numbers, by setting any real number as being a composite, WHOLE + FRACTION, where:
Fraction = 0 + b1/10 + b2/100 + b3/1,000 + . . .
. . . so that we get say 19.79 etc with the usual meanings, and where also we define a complex number c = p + i*q, i being the square root of minus 1 (very useful in Math) and p and q are real numbers. So we can have the complex number 1.978 + i*19.79. And, as we can see that for any two "neighbouring" points u and v defined on such whole + fraction terms (which differ by some tiny amount, e), we can always extend to a number between them, by adding in more terms -- or, simply, by taking the average (u + v)/2, i.e. we have here defined a continuum, the Real Numbers, where we are taking in the negatives as simply the reversal of the positives such that w + [-w] = 0.
A complex number, (SOURCE: Wiki)
vi: Now i*q is often assigned to a Y-axis and p the X-axis, so that p and i*q can be plotted on the Argand plane. We can then draw a vector r from the origin, to the point defined by the co-ordinates. Then, angles made by such a vector can then be defined relative to the X-axis from the usual trig ratios, and rotations can be defined, introducing time, t. By Pythagoras' theorem, of course r^2 = p^2 + q^2, defining the magnitude (length) of r. BTW, a rotating vector is called a phasor.
vii: Similarly, we can extend to three dimensions [using the i, j, k unit vectors along x, y and z axes], and allow a virtual particle p -- notice, we are now in the world of contingent possibilities -- to traverse on set coherent laws of motion, including introducing mass, force, momentum, energy etc. in what is now a virtual model world.
viii: Bodies in such a world would be collections of linked particles, even as geometrical figures are clusters of linked points.
ix: We now have a three dimensional virtual reality with a physics! (Computer graphics uses techniques related to this outline sketch.)
x: This can then move to the or a real world by instantiation. This would of course require a creator with the skill, knowledge, intent and power to move from virtual worlds to actual ones.
xi: As a corollary, it is worth noting on the parallel lines postulate of Euclidean Geometry. It is often said that the fact of non-Euclidean Geometry renders moot the idea that parallel lines never meet, which is equivalent to the angle sun triangle assertion that the sum of the three angles is 180 degrees of arc. Just look on how triangles on Earth's surface can sum to a different value, and how parallels of longitude converge at the poles:
xii: But, a subtlety lurks. Parallel lines lie in a flat plane, as is specified by the vertices of a triangle, say ABC. (Such a plane can be set up algebraically using the Argand plane, with origin at say vertex A and 0X axis along the line AB, so we can define a vector of arbitrary length r pivoting from A within the plane and rotate it to sweep the plane, guaranteed to be flat by the mathematics involved. And with a spot of thought, this can be extended to the three-dimensional case.)
xiii: Within the plane, straight lines can be specified by the usual expression y = m*x + c, and for a given m, the slopes will be the same for a family of lines with a range of values of c, say c0, c1, c2, . . . cn. These lines will be parallel, separated by the fact that 0Y-intercept is c and oX-intercept will be -c/m. The triangle between the origin, the 0Y and 0X intercepts will specify that the triangles for c0, c1 etc are similar, and the lines for two different values ci and cj will have a guaranteed separation linked to the value of cj - ci at any given point along the lines.
xiv: That is, within a Euclidean, planar space, parallel lines indeed will never meet. Spaces where this fails are not planar. Hence the following from Wolfram Math World:
In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid’s postulates, but each uses its own version of the parallel postulate. The “flat” geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry). Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry.xv: So, equidistant straight lines in the same flat plane will be at the same separation anywhere. That is locked into what parallel means in this context -- and to shift from such a world without notice is a case of equivocation, i.e. there has been a subtle shift in the meaning of triangle and parallel.
xvi: In other words, the dismissive assertion that this fifth Euclidean postulate does not hold (as was used to create non-Euclidean Geometries) is equivalent to leaving such a space, e.g. cf. a “triangle” on the curved surface of the earth. The problem was that evidently such spaces had not been thought through as possible. Where also of course, post Relativity, whether the world we live in in the large scale is Euclidean is doubtful, but in the small scale it is sufficiently close that it suggested the idea of such a space.
xvii: This little excursus shows us how the astonishing relevance and power of Mathematics in analysing the physical world can be easily explained on the use of such mathematics as the means of contemplating and creating the world, by God. Again,
xviii: God is the best candidate explanation of a world in which mathematics (necessarily including logic) shows such astonishing power.An eternal mind that is all-knowing and capable of such contemplation, reasoning and creation etc, is of course one way of describing God.
So, let us pause and let us ponder . . . and note this is not a proffered proof or even a claimed warrant: could God be the ultimate mathematician? And, could the mathematical order we seem to discern be a signature of the intelligent design of our world? (Good enough to at least be sand inside the Oyster's shell, methinks. ) END
