The Simon Lectures. Series II, Part 2.

Originally published on greatawakening.win, 2023 February 10.

Series II, Part 1 can be found here: https://greatawakening.win/p/16a9lhEbrc/the-simon-lectures--series-ii-pa/

Series II, Part 2.

Permit me to highlight the salient points of Part 1. I don’t care for repetition but this is a mental journey we’re on together and context is important. The major points of Part 1 are crucial for this Part 2.

1. We began by posing a question: Is 0.999… = 1?

2. We determined that the original question could be reframed: Does 0.000…1 = 0?

3. If 0.000…1 = 0, then 0.999… = 1; otherwise, if 0.000…1 is non-zero, then 0.999… does not.

4. We traversed a first line of reasoning that showed that 0.000…1 must not be zero. Which means it is non-zero.

5. We traversed a second line of reasoning that showed that 0.000…1 must not be non-zero. Which means it is zero.

6. We concluded that the operation of reason had led us to a contradiction.

7. Then I said something somewhat cryptic: “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?”

That cryptic conclusion? That was my attempt at a philosophical cliff-hanger. I know. I shouldn’t quit my day job to pen Hollywood scripts. Don’t worry. I can’t imagine Hollywood will come knocking.

If my cliff-hanger worked at all, it should have caused you to wonder: “What assumptions? I don’t see any assumptions here. Just reason. And reason failed me.”

Fair enough. Let’s dig those assumptions out.

When we think of a quantity such as one or two or one-half or one-tenth or so on, we should consider that it is constituted of two parts: (1) a numeric representation or “number” – i.e., “1” or “2” or “1/2” or “0.5” or “1/10” or “0.1”; and (2) a related meaning or “amount.”

“1” is a numeric representation relating to the idea of unity or oneness and the amount that such idea represents. “2” is a numeric representation relating to the idea of duality or twoness and the amount that such idea represents. And so on.

Number <--> Amount

The first thing to notice is that every number corresponds to an amount. If you can conceive of a number and express it as a number, then it has a corresponding amount. Three? You can have three apples. One-tenth? You can have one-tenth of an apple. And so on. Every number refers to an amount.

But is it the case that every amount corresponds to a number? Can you have an amount that is inexpressible as a number, for one reason or another? This is the critical question.

Could it be the case that an amount can be so small that it slips right out of the realm of the numerical and descends into the realm of the ineffable – and is therefore inexpressible as a number? Could it be the case that an amount could be so small that it could be simultaneously different than zero – truly distinct from zero – while operating within the arithmetic realm in a manner no different at all than zero? A non-zero amount that functions arithmetically identically to zero. That’s what we’re considering here.

Let’s call this fantastically small quantity “µ.”

5 + µ = ___? We are saying that 5 + µ = 5.

3 - µ = ___? We are saying that 3 - µ = 3.

And so on.

µ is not 0. It refers to a non-zero amount. But when treated as a number inside of a system of reason designed to operate upon numbers – arithmetic – it behaves no differently. That’s the idea.

Let’s consider the ramifications of assuming that µ exists.

Turn your attention our first line of reasoning that demonstrated that 0.000…1 must not be zero. Would that not leave open the possibility that 0.000…1 = µ? After all, µ is not zero.

Now turn your attention to our second line of reasoning that demonstrated that 0.000…1 must not be non-zero. Armed with the understanding that µ exists, we now see that the second line of reasoning actually demonstrated that 0.000…1 must not be a non-zero quantity expressible as a number. And does this not leave open the possibility that 0.000…1 is equal to µ? For µ is not expressible as a number.

If we were to assume µ existed, we would eliminate logical contradiction. Both lines of reason would lead us to conclude that 0.999… does not equal 1.

Without belaboring matters, if we were to start with the opposite assumption – i.e., that µ did not exist – we also could resolve the contradiction. Albeit in the opposite direction. We would end up concluding that 0.999… = 1.

(I note that to resolve the contradiction, we would have to make a further clarifying assumption about the nature of infinity. (1) assume that µ does not exist, and (2) make a further clarifying assumption about infinity, and – voila! –both lines of reason takes you to the conclusion that 0.999… = 1. For the sake of brevity, I will not get into the further clarifying assumption, in this Part 2. I could be talked into it, though, in a separate post, if anyone is interested.)

So, in order to pursue the first and second lines of reasoning to a consistent outcome, we must either: (1) assume that µ exists; or (2) assume it does not exist, and augment such assumption with an additional clarifying assumption. In Part 1, we did neither of these things. So our lines of reasoning resulted in contradiction.

The outcomes of our lines of reasoning actually turn on what we assume. Moreover, it matters not whether our assumptions are explicit. In Part 1, we “assumed” that µ did not exist by virtue of neglect: we simply hadn’t considered the matter, meaning that – in practical effect – we assumed it didn’t exist. And we failed to augment that “assumption” with the aforementioned additional clarifying assumption – again, out of neglect – meaning that we were doomed to follow our lines of reasoning to contradiction.

Ok. So outcomes of reason are determined by assumptions. And in the case of 0.999… and 1, we must decide whether we believe µ exists (set aside the “other” assumption for now).

But how do we decide whether µ exists?

Suppose I were to scoff at the notion of µ. Suppose I were to say: “Let me get this straight. Whereas every other quantity can be expressed as a number, you propose that perhaps there exists one oddball quantity – this “µ” as you call it – that eludes such expression? It’s not zero, you say. It just functions identically to zero. In every way. Is it not the case that the reason you cannot express µ as a number is because you are too stubborn to call it by its proper name: zero? And is it not the case that µ functions identically to zero within the realm of arithmetic because µ’s true identity is zero? You have taken zero and renamed it µ! There’s no other way to see it!”

And suppose you were to reply: “Tell me, when you contemplate the present – that span of time interposed between the past and the future - how much time is that? When you refer to ‘now,’ how much time are you referring to? One second? One half of a second? One-millionth? No. Even in the span of one microsecond, some portion of that will either reside in the past or in the future. But how much time resides in the ‘now?’ I challenge you: present me with a number to express such quantity of time. It cannot be done. And does this therefore mean that ‘now’ refers to nothing at all? Is that even remotely sensible? Of course not! You live in the now! In fact, it is the only time in which you live. So you know it exists. With absolute certainty, you know it exists. And I’ll tell you how much time ‘now’ is constituted of: µ! A non-zero quantity of time smaller than any expressible number. µ is what permits ‘now’ to exist. It’s what your life is constituted of! An infinite succession of µ-after-µ-after-µ-after-µ.”

Well, shit. It doesn’t look like you and I will get to the bottom of it yet. But that’s not the important part. The important part is to see the nature of what where our conversation has gone.

We began with reason. And we came to contradiction.

Then we realized that contradiction was the consequence of assumptions we had failed to explicitly make. To eliminate contradiction, we either need to decide that µ exists or that it does not exist.

And then – in our hypothetical conversation – we tested the idea of µ against itself. We examined the idea that is the subject of the assumption under consideration. We examined it against itself. Which is to say that we examined it to see whether it was self-consistent in its nature. To see whether any contradiction lurked inside of the very essence of the idea. When we test an idea against itself, we are examining the coherence of an idea. We are asking: is this idea coherent? And, in some circumstances, examining the coherence of an idea is sufficient to bring about agreement concerning whether we ought to make one particular assumption or another. Regrettably, this is not such a circumstance. We’re still stuck.

Reason --> Contradiction --> Unearth Assumptions --> Test Coherence of Assumptions.

That’s the meta-structure of our journey so far. And it typifies many journeys. So it’s important to recognize its structure.

We’re not done with this journey yet. Part 3 is coming, and all of this continues. I know that in writing this Series II, I am asking much of you, my readers. Hang in there. One more Part. Then I’ll have laid sufficient groundwork to move on to Series III, and you’re going to love that. But we need to get through Part 3 of this Series II, first.

Stick around for Part 3.

Or don’t. It’s your decision.

Ever yours, simon_says