The Simon Lectures. Series II, Part 1.

Originally published on greatawakening.win, 2023 January 27.

Introduction.

Series II pertains to the foundations of knowledge and its inherent limitations. I am certain that, to many, this sounds a lot like saying that it pertains to nothing of practical value, and has little to do with what this site is all about. I promise this is all connected. But you won’t see it at first.

I confess an idiosyncratic affection for this topic. But that’s not why I’m writing about it. This topic is necessary, friends. We are in the midst of a war for minds. Everything turns on this topic, as you will soon learn.

I suspect some or many of you may anticipate that a topic such as this will dry and tedious. Rest assured, I am constitutionally opposed to arid tedium. So I’ll make this interesting or die trying. Moreover, I will write no more of this Series than is necessary to support Series III. These Parts 1-3 of Series II are absolutely necessary for Part 1 of Series III. Trust me, you’ll see.

As always, take this or leave it as you please.

-simon_says

Series II, Part 1.

I said that the purpose of this Series is to explore the foundations and limitations of knowledge. Which suggests that – in my view – knowledge may encounter barriers. Or else it wouldn’t have limitations. It also suggests that – in my view – knowledge rests on something. Or else it wouldn’t have foundations.

So let’s get started. Let’s look at a topic that forces us to encounter a barrier.

Let’s consider 0.999…. . Which you may articulate as “point nine repeating.” I am referring a number described by zero, followed by a decimal point, followed by an infinite succession of nines. I’ll present that herein as 0.999… . As far as I know, this site does not permit me to include “bar notation” – which is the term for that line you were taught to write over a decimal to indicate repetition. So 0.999… will have to do.

Now let’s ask ourselves a question: is it the case that 0.999… = 1?

I know that during your school years many of you may have heard that these two quantities do, in fact, equal one another. And it is the prevailing viewpoint. But set all that aside for now. Let’s address the question for ourselves.

Let’s begin by simplifying the question, shall we? Let’s begin with a simpler one. Is it the case that 0.9 = 1?

We all know the answer to that one. No. They’re not the same. And we all arrived at that answer in largely the same way. We subtracted 0.9 from 1, and arrived at a difference of 0.1.

1 – 0.9 = 0.1

And it stands to reason that if there is a difference between 1 and 0.9, then they are not equal to one another. We found a “difference” between these quantities. Which is to say they are literally different. As in not the same. So they don’t equal one another. And the matter is settled.

Now let’s redirect that method to the original problem.

1 – 0.999… = __?

That’s a tough one. We can’t exactly write it out, but the “answer” seems to be something akin to a zero, followed by a decimal point, followed by an infinite succession of zeros, followed by a one: 0.000…1. What is this oddity? Is that a non-zero number? Because if it is, then we have identified a difference between the two quantities. And if there’s a difference, they’re not equal. But if it’s really just “0” written in another format, then there’s no difference between the two. And if there’s no difference, that means they’re the same. As in equal.

Let’s reframe the question and see if we can’t reason our way through this, shall we?

At this stage, we can see that the original question is, in reality, an inquiry into whether 0.000…1 is equal to 0. Let’s direct our powers of reason to this question. How hard could that be? Right?

Let us assume we had access to a “decimating machine” that we could operate in cycles. It would receive an input, decimate it (i.e., divide it by ten), and yield an output. So if we initially input a 1, then on the first cycle, the output would be 0.1 (1/10 = 0.1). And if we fed the output of the decimating machine back into its input, to operate the machine for another cycle, then upon the conclusion of the second cycle, the output would be 0.01 (0.1/10 = 0.01). And if we operated the machine for a third cycle, the output would be 0.001 (0.01/10 = 0.001). And so on. So, as you can see, the question becomes an inquiry into the nature of the output of the machine after an infinite quantity of cycles.

The first thing to notice about this question is that upon the first cycle, the input to the machine is a non-zero number (i.e., 1). And the output at the conclusion of the first cycle is a non-zero number (i.e., 0.1). So the next cycle begins anew with a non-zero number (0.1) and concludes with another a non-zero number (0.01). Moreover, decimation, when performed upon a non-zero number, always yields a non-zero number. So the third cycle will begin with a non-zero number derived from the output of the second cycle, and conclude with another non-zero number. So will the fourth cycle. And the fifth. And the sixth. And so on. Non-zeros all the way through.

The process appears air-tight. You will start with a non-zero number and conclude with a non-zero number – every time. No matter how many cycles you operate the machine. A non-zero number goes in, and a non-zero number comes out. Forever. So there’s no room for the output of this machine to be anything other than a non-zero number. This means that 0.000…1 is a non-zero number. Which means that 1 and 0.999… have a non-zero difference. Which means that they’re not equal. Whew. Problem solved. Thank God for reason.

Not so fast.

Let’s look at this another way. Let us refer to the output of the machine as Q. And let us refer to the quantity of cycles for which the machine has operated as C. Then, after a quantity of C cycles, the output of the machine is: 1/10^C. You can intuitively see that is true. If operated for a single cycle (i.e., C = 1), the output us 1/10^1 = 1/10 = 0.1. If operated over a period of two cycles (i.e., C = 2), the output is 1/10^2 = 1/(10*10) = 1/100 = .01. And so on.

Q = 1/10^C.

Of course, instead of expressing the output (Q) as a function of the quantity of cycles (C), we could reverse that. We could express the quantity of cycles (C) as a function of the output (Q). Thank you, eighth-grade Algebra II.

C = log(1/Q).

Again, you can intuitively see that this is true. Consider the output (Q) of the machine after one cycle, i.e., Q = 0.1. C = log(1/0.1) = log(10) = 1. Try it with the known output after two cycles, i.e., Q = 0.01. C = log(1/0.01) = log(100) = 2. And so on.

Now what is the implication of this? Here’s what. For so long as the output of the decimator (Q) does not equal 0, then it is the case that 1/Q has a defined finite value. And for so long as 1/Q has a defined finite value, so does log(1/Q). Which means so does C, because C = log(1/Q).

If Q is non-zero, then C is finite. C is 100,000. Or 1,000,000. Or 1,000,000,000. Or some other very large but finite number. And C is the quantity of cycles for which the machine has been operated. And if C is finite, then quantity of cycles for which the machine has been operated is finite, meaning that the machine was not run infinitely. It was, instead, run for a finite quantity of cycles. Any non-zero value of Q (the output of the machine) means that the machine has not been running infinitely.

So, upon having been operated for an infinite quantity of cycles, Q must be 0 – because a non-zero value of Q indicates a finite quantity of cycles. And if Q is 0, that means that 1 and 0.999… have no difference. Which means they are equal. Which means that reason has brought us to a contradiction.

We have a line of reasoning that supports the notion that 0.999… = 1. And we have a line of reasoning that supports the very opposite notion. So now what? Now it’s time to make an observation.

The outcome of any line of reasoning depends on assumptions. Assume the same things, and you will get the same outcome. But if you have a choice of independent assumptions – assumptions that are more than just different ways of expressing the same thing – then there is no guarantee that reason carries you to the same conclusion. This means that in order to use reason as a tool, you must first decide what to assume. And to decide what to assume, you must answer a question: what do I believe is true?

Reason cannot guide an unanchored quest for truth. It can take you from one truth to another. But if you cannot locate even a single truth, reason can do nothing to help you. I’ll close here. Stay tuned for Part 2. Or don’t. It’s your decision.

Ever yours, simon_says

p.s. Series I is not concluded with Part 8. Part 9 is coming. If you have not read Parts 1-8 of Series I, you may find Part 8b here: https://greatawakening.win/p/16ZqiECU58/the-simon-lectures--series-i-par/

Part 8b contains a link to Part 8a, which, in turn, contains links to Parts 1-7.

I know this Series II is taxing. The first three Parts of this Series II set up Series III, which you will find in line with the topicality of this board, and, I hope, find intriguing and enjoyable.