By Lax M.

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**Sample text**

This makes them a convenient codification of mathematics, but not a simplified starting-point. The number 1 is not the most obvious initial step from 0 because it contains, for example, the notion of discreteness, as well as ordinality (or ordering). Further, although there are many ways in which mathematics can be generated from foundational elements, physics seems to indicate that discreteness at the foundational level is always associated with dimensionality (see chapter 2), and hence is not a truly primary concept.

The integers are loaded with a mass of assumptions about mathematics. They are not fundamentally simple but already contain packaged information about things beyond the integer series itself. This makes them a convenient codification of mathematics, but not a simplified starting-point. The number 1 is not the most obvious initial step from 0 because it contains, for example, the notion of discreteness, as well as ordinality (or ordering). Further, although there are many ways in which mathematics can be generated from foundational elements, physics seems to indicate that discreteness at the foundational level is always associated with dimensionality (see chapter 2), and hence is not a truly primary concept.

The commutativity of im and in is equivalent to defining (imin)2 as 1, while the anticommutativity of in and jn defines (injn)2 as the conjugate, or –1. If i1 forms a quaternion system with j1, and the complex product i1j1, then no product of i1 with any other complex number of the form i1, i2, i3, i4, … or j2, j3, j4, … will itself be complex. If we take i1, j1, and i1j1 as a quaternion system (i, j, k), then any further complexification, to produce, say, i2i1, i2j1, and i2i1j1, will produce a system equivalent to the multivariate vectors, complexified quaternions, or Pauli matrices (i, j, k), which square to positive scalar units.