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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?
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".
First, an obligatory opening potshot.
I too fully understand where the pure mathematicians are coming from.
Isn't that difficult, since you're not a pure mathematician?
Rather than going over the other stuff, I'm going to hone in on this one point:
I fully accept the notion of negative numbers and imaginary numbers. ... Like as if that concept has merit without any relation to anything else.
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".