Hi Roger,

Sorry for the delay, I should have said thank you a long time ago for putting the Book onto the Internet.

I'm slowly going through and reading it; it is excellent.

I have spotted one typo error (sorry) on chapter 1 p 8 line 26 you have:

y=n; dy/dx = undefined

I don't think that is right, if n is a constant, then dy/dx = 0

Dear Roger,

Glad you are enjoying OP.

Thanks for the feedback on OP, 1.8. By “undefined” I mean that y = n is undefined in terms of x. If y = f(x) = n, we know that f(x) can be any value of n. If n is constant, then the derivative of f(x) certainly becomes 0. What I say at OP 1.8 regarding y = n is "no slope at all", which amounts to saying that dy/dx gives zero slope if we interpret n as a constant. Such a case is usually written as f(x) = c. My point is that there is a subtle difference between saying y = x^0, where y, that is f(x) is always equal exactly to 1 for any real value of x and therefore is a constant (c), and saying y = n, where n can be ANY real number, and thus y = f(x) can have any value and is completely independent of any value we give to x. For example, if x = 5, what is y? Don’t know. If x = 27, what is y? Don’t know. There is no connection defined between x and “f(x)”, that is, y. Technically y = n is NOT a function (f) defined on a set of real numbers such that it assigns to each number x in the set of real numbers exactly one real number that we refer to as y = f(x) because we can assign any number (n) from the set of real numbers to y for each possible x, as opposed to the case where n becomes a constant and has a fixed value. Therefore we technically can not differentiate y = n.

On the other hand, if we say x = n, then we have any value of x considered independent of the y axis, and thus independent of f(x). Again, this is not a function since there is not a one-to-one correspondence between each value of x and each value of f(x). We just say that we can pick any value for x on the x axis. Any x we choose can map to any value of y. That is not a function. If we say x = x, then we have an identity function where we define f(x) = x, or y = x. Identity is one of the fundamental properties of an equivalence relation, and it is a function because for every value we assign to “x”, “f(x)” (that is, y) will have that one unique value we assigned as “x”. That gives a constant slope of 1 (45 degrees), a standard light line of a light cone.

If we call dy/dx the "slope" of the function y = f(x), then in the case of (y = x^0) the function is a flat horizontal line at y = 1 that obviously has no slope and runs parallel to the x-axis. It is a flat and level "plain". The case of (y = n) is totally indeterminate with regard to "slope" since there is no reference to the x axis and "slope" is defined as the ratio of the rise (value of y) to the run (corresponding value of x). If we set an arbitrary constant value for x, say x = 1, then y = n becomes a vertical line with "infinite" slope situated at the x = 1 value and parallel to the y axis. We can also write such a condition as an inverse function: x = y^0. In this case every possible value of y produces the value 1 for x. If we write y^2 = x^1, we can take the square root of both sides and rewrite this as the function y = x^1/2. On this analogy the "function" (y^0 = x^1) can be formally rewritten as y = x^1/0. According to the "rules" of mathematics, you can not divide by zero. What you get is that y^0 = x = 1 for every value of x, which is nonsense since x is supposed to range over the whole set of real numbers for us to have a function, but 5 is not equal to 1, nor is 6, or 7. Division by zero produces an undefined state of affairs, even in the case of an exponent, which can have any real or complex value.

If you have a pie and divide it by three, you get three equal portions of the pie; and if you divide it by two, you get two equal portions of the pie. But if you divide it by one, you get the whole pie intact. If you divide it by nothing, what do you get? Well let's see. If we divide it by "1/2", the pie doubles in size. If we divide it by "1/4", the pie quadruples in size. Proceeding in this fashion, we discover that if we divide the pie by an infinitely small portion (viewpoint), we get an infinitely huge pie. Now that is really "pie in the sky"!

Perhaps I need to rephrase my statements in the text a little bit, but what I am doing is shifting people's attention from the formal manipulation of symbols (which is what generally happens to students of calculus) and bringing them back to a visual, geometric model that connects them to the "real" world. You can do the pie routine as an actual experiment by setting a pie, or just a disk of paper, in front of you. By adjusting the distance of your viewpoint as an observer relative to the pie you can experience the pie apparently changing its shape to get larger or smaller. If you shift yourself to a viewpoint infinitely far from the pie (you can do this one in your imagination or physically, if you know how) , the pie becomes a mathematical point. If you shift yourself to a viewpoint that is infinitely close to the pie, you BECOME the pie, and it becomes your infinite universe and not just a pie in the sky. You can also do this exercise with imagination or for “real” depending on how well you can manage your attention.

In terms of actual physics, taking a “mathematical” viewpoint into a pie causes the pie to appear to expand while the observer observes smaller and smaller details of the pie. Eventually he hits the quantum level and then the Planck level. At this point the pie explodes into a Big Bang that appears to generate an entire universe. The act of observing involves an interaction between the observer and what he observes. Observing the vacuum state from the Planck scale is like becoming the Big Bang. Done thoroughly there is no longer distinction between the observer and what she observes. Observing the core of a single nucleon takes you into what we unfold in Observer Physics as the Unity Boson. This also is the same as observing the vacuum state from the Planck scale.

DAW