So as not to hijack this thread, I'm responding here. First, an obligatory opening potshot. Isn't that difficult, since you're not a pure mathematician? :tongue: Rather than going over the other stuff, I'm going to hone in on this one point: You are apparently talking about one of the symmetries of an ordered group: if you reverse the order, you're left with the "same thing". IOW, you have to "choose" a direction before using such a thing. Now, I would agree with some of the points you're making. Some applications of real numbers do require a "choice" of direction, such as a common freshman quandary of whether acceleration due to gravity is a positive or a negative number. However, you neglect to account for many applications of multiplication. The fact that 1*1=1 forces a unique choice of order, because the product of two negative numbers must be positive. As a physical example, recall that, in Euclidean space, the dot product of two vectors is an invariant quantity -- it doesn't depend on any choices. The dot product of oppositely pointing vectors will be negative, period. Another is that of scale factors. There is a "physical" difference between scaling by 1 and by -1 that, again, cannot be explained away by any sort of argument that it's all "relative".