The Simon Lectures. Series II, Part 3.

Originally published on greatawakening.win, 2023 March 19.

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

Series II, Part 2 can be found here: https://greatawakening.win/p/16aA97Pst1/the-simon-lectures--series-ii-pa/.

Series II, Part 3.

This Series has been a test of faith. You must have faith that this goes somewhere and is not an exercise in self-indulgence. I promise it goes somewhere. After this Part 3, we will have developed a sufficient foundation to launch Series III, and, with that, much will be made clear. Plus I suspect many of you will find it intriguing. When I “tease” some of these ideas in my day-to-day life, they are received with surprising levels of interest. And that’s among “normal” folks. We’re something like birds-of-a-feather here, so I suspect most of you will be at least as interested. So, hang in there. And thank you for your faith.

I usually recap previous Parts as a courtesy: I don’t expect people to remember what I’ve written previously. I know we all read much, and I write these Parts in irregular intervals, so that makes instant recall of my past material an unreasonable expectation. I’ve tried recapping Parts 1 and 2, and I just don’t think they can be summarized in a way that is both succinct and complete. So I’ll skip the recap this time. The bottom line is this: we’ve advanced along a journey that began with asking whether 0.999… = 1, and has resulted in asking whether a peculiar quantity I have called µ exists.

µ is a speculative idea. µ refers to a quantity so small it evades proper expression as a number. If we tried to express it as such, it would be something along the lines of 0.000…1. In other words: a zero, followed by a decimal point, followed by an infinitude of zeros, followed by a one. When we speak of µ, we are referring to a speculative miniscule quantity that is distinct from zero – truly different than zero – but, by virtue of its vanishing minuteness, behaves no different than zero within the confines of arithmetic.

3 + µ = 3.

5 - µ = 5.

It behaves just like zero. But is, nevertheless, a non-zero quantity. That’s the idea.

In Part 2, we tested the idea of µ against itself, which is to say that we examined the idea to see whether it contradicted itself. If an idea contradicts itself, then the idea is incoherent. Suppose I were to say to you: “I ask you to call to mind a triangle with internal angles of 45°, 90° and 50°.” Could your mind’s eye behold such a figure? Of course not. For the internal angles of a triangle sum to 180°, and you cannot even imagine a triangle structured differently. The idea I asked you to hold in your mind is, itself, broken. And, as a consequence, it cannot be held in your mind or anyone else’s mind – not out of a defect arising in your imaginative capacity, but out of a defect located in the idea, itself. The idea is incoherent. And, without belaboring the point, it is the case that an incoherent idea cannot be true. Every incoherent idea is false, but not every coherent idea is true. So if the idea of µ were to be incoherent, we’d know it was untrue. But, unfortunately, in Part 2, we discovered that the idea of µ was coherent. So we didn’t get off the hook that easily. So where do we go from here?

Suppose I were to say: “This µ you suggest, you tell me it has an essence that is non-zero. But nowhere does it make that essence known. At all times and in all circumstances, it behaves as zero. An essence is known by its behavior. Nothing else. Only its behavior. If you disagree, then tell me: in what other capacity could an essence make itself known? If something behaves as zero – at all times and in all circumstances – then it is, in essence, zero.”

And suppose you were to retort: “At all times, and in all circumstances? Wherever did you get that idea? Consider, my dear Simon, if we regard µ as 0.000…1, i.e., a ‘1’ separated from a decimal point by an infinitude of zeros, then what would be the result of multiplying µ by 10? Would that not “remove” a singular zero from the infinite body of zeros interposed between ‘1’ and the decimal point situated far to its left? And would not multiplying µ by 102 remove two such zeros? And would not multiplying µ by 103 remove three of them? And, finally, would not multiplying µ by 10∞ remove an infinitude of them? Which is to say all of them. And, therefore, would you not be left with the quantity ‘1’? I am saying that µ * 10∞ = 1. But what about zero? I’ll tell you what: 0 * 10∞ = 0. There’s your behavioral difference, Simon. There’s the behavior that reveals µ’s essence. µ can harness the baffling qualities of infinity to escape its low-altitude orbit of zero. But zero can do no such thing, for zero does not merely orbit Planet Nothingness; it is nothingness, itself.”

In view of your retort, I might respond: “You have imagined a realm, and then supplied a behavioral difference found exclusively therein. Infinitudes don’t exist. They are ideas. Notions. Concepts. Abstractions. And they may be manipulated via thoughts to arrive at other abstractions. If I were to chop off a unicorn’s horn, how many horns would it have left? As a concept? Zero. For I would have removed its only horn. As a reality? Still zero. But not as a result of horn mutilation. But because unicorns don’t exist. Not in reality. They are only an idea. Our world is finite. You can locate nothing infinite within it. There is nothing incoherent about an infinitude of unicorns – they are both perfectly well formed ideas. But neither are real. You say µ and zero behave differently in the realm of infinity. But I say infinity does not exist as a reality. Which means your professed behavioral distinction does not exist.”

To which you might respond: “Simon, I’m shocked. Of course reality includes infinitudes. Because reality includes ideas. There are material realities. And there are immaterial realities. Some ideas find correspondence in material reality. And some do not. But those correspondences – when found – do not inject their counterpart ideas into the fabric of reality. Circularity is real, even if not a single truly precisely perfect circle is to be found in material reality. Infinitudes are but examples of immaterial realities. And it goes without saying: realities – material or immaterial – are real.”

What’s going on here, in this exchange? We could not settle our differences when testing the idea of µ against itself. For the idea of µ is well formed – it is coherent. So we have shifted gears. We are testing the idea against the particular philosophical frameworks each of us has adopted. And were it the case that we shared philosophical outlooks, we could settle the matter this way. But we do not. For in my example, I am a materialist (believing only in material constituents of reality) and you are not.

What now?

Well, suppose I said to you: “You know, I’ve been thinking about the matter for some time now. I took an entire week. Meditated. Concentrated. Visualized. The whole shootin’ match. And you know what? A quantity subjected to infinite decimation – µ – must be zero. It simply must be. Not because reason compels it. Not because the alternative is incoherent. Not because no sensible philosophical framework could incorporate a non-zero infinitesimal. But because I have visualized the matter clearly in my mind’s eye. With stark, perfect clarity. And do you know what I saw? I saw nothingness. Zero. I know µ is zero because I saw it directly and with perfect clarity.”

And suppose you should respond: “This is spooky, Simon. I’ve been doing the same thing – all week long, same as you. But you know what? In my mind’s eye, infinite decimation resolved to a point. Not to nothingness, as it did in your mind’s eye, but to a point. I saw it clearly. With perfect resolution and nothing left to doubt. It’s not nothingness. It’s not zero. Not because reason compels it. Not because the alternative is incoherent. Not because no sensible philosophical framework could exist without non-zero infinitesimals. But because I saw it. I saw µ clearly. As a point. With existence as opposed to non-existence. And it’s non-zero because that’s what I saw.”

This is a critical juncture in our journey. Each of us has formed the idea of µ in our minds. We have done so with excruciating care, constructing it with cognitive precision and with detail. And, having constructed our respective notional figments, each of us beheld the product of our own intellection. And we saw what we saw. Our “vision” was not preceded by reason or expectation or philosophy or any other matter. It was preceded by nothing at all. Each of us saw what we saw. And the reason each of us saw what we saw was because that was what we saw. Which is another way of saying our respective visions cannot be explained on any other terms. Each of us saw what we saw. Period. This is a descent into the realm of the subjective. When the same matter – having been clearly and precisely apprehended in our respective minds – appears one way to you but another way to me, we are having a subjective difference of opinion.

Subjective impressions are brute facts. My mind’s eye saw what it saw, and there is no way to explain it, nor anything to be done about it. And the same can be said of your mind’s eye. So it might appear that nothing remains to be said or done. This is oftentimes the occasion for an appeal to authority. You might say to me that Newton and Leibniz both believed in non-zero infinitesimals, and made use of them in developing calculus. And if infinitesimals made sense to Newton and Leibniz, perhaps, Simon, there may be something defective with your mind’s eye? But I could reply: Cauchy (and many others) explicitly reformulated the entire basis of calculus for the purpose of excluding infinitesimals. And Bertrand Russel referred to infinitesimals as pseudoconcepts because such figments, while coherent, lacked empirical counterparts – they simply lack correspondence in reality and are therefore meaningless. So perhaps it is your mind’s eye that needs glasses!

Appealing to authority may be useful to test the reasonableness of a particular individual’s claims about the vision beheld by his mind’s eye. But, some matters simply possess the capacity to present two mental visages – one to you and a different one to me. These matters divide humanity. You are the variety of person that sees a matter one way, and I am a different variety of person that see the same matter in a different way. So, if we are to get along, it may seem that one segment of humanity will have to bow to the other. But which segment bows and which segment stands? I hasten to point out that this may suggest that either there is no truth pertaining to such matters or that there may be plural truths – perhaps one for you and one for me. But this is not the case. It means only that a limitation has been encountered (recall: this Series pertains to the limitations and foundations of knowledge). It means that, today, we see as through a glass, dimly. There is much to be said about the nature of the limitation we have encountered, but I will omit this discussion for now. I suspect most of you just want to get on with things and reach Series III.

We might be tempted to appeal to the majority. The prevailing viewpoint among mathematicians is that non-zero infinitesimals do not exist. The calculus text you used in high school undoubtedly deliberately omitted the topic, skirting it with careful use of epsilon-delta proofs. Yet you may say: “It is true that the prevailing majority view is that non-zero infinitesimals do not exist – this is why the real number system excludes them as a matter of axiomatic choice. But there are those who see them in their mind’s eye and know them to be real. We are a minority but not a fringe minority. And we have developed our own number system that makes use of them: the hyperreal number system. And we have formed an alternative foundation for calculus that assumes their existence. This alternative foundation is called non-standard calculus, and while it is not the traditional entry point into the subject of calculus – it will not be taught at your local high school – it is universally regarded as logically sound. So I am not alone, and those of us who believe in the nonequivalence of 0.999… and 1 are not alone. And I am not crazy and my community is not crazy!” And you’d be right. An appeal to the majority is not fruitful when there exists a non-trivial minority that fruitfully employs an opposing viewpoint. And who knows? Tomorrow, the majority view may change – for the wind bloweth where it will, and thou hearest the voice thereof, but knowest not whence it cometh, and whither it goeth.

What now?

At this point I ask you to suspend disbelief. What follows is a hypothetical experiment by which we could in principle ask the universe to answer the question for us. I ask you to consider a scenario in which we had access to a perfect balance. On one arm of the balance we place a perfect one-pound weight. On the other arm we place a weight that is perfectly nine-tenths of a pound. Then, to this other arm we add another weight, this one weighing perfectly nine-one-hundredths of a pound. Then we add another weight, this one weighing perfectly nine-one-thousandths of a pound. And so on. We carry on this exercise of adding additional successively tenfold smaller weights infinitely. Recall: you are to suspend disbelief – it is, of course, impossible to perform an infinitude of steps. But if somehow we did complete the infinitude of steps, what would the actions of the universe tell us? Would the arms of the scale come to equipoise? Or would they remain imbalanced in favor of 1?

Here's the point. Whatever the action of the arms, we must defer to it. If the balance comes to equipoise, then 0.999… = 1. Period. And if it remains imbalanced in favor of 1, then we have our answer, too. We must defer. Our reason must defer. Our presuppositions on which our reason turns must defer. Our judgments of coherence must defer. Our philosophical frameworks must defer. Our subjective impressions must defer. Our authorities must defer. And the majority must defer. Objective experience – experience grounded in observables that you and I can both encounter – forms the foundation of all knowledge. It is not the only source. We will discuss other sources later. But it is the foundation on which the entire edifice of knowledge is built.

I will conclude here, as I am out of space!

Ever yours, simon_says