“What are the odds?” I said upon discovering an old classmate in the lobby of the very same hotel in St. Thomas where I had just checked in. “And look, you’re reading Leibniz too!” My old classmate was carrying a transcript of Leibniz’s letters to Clarke, and I had a copy of his letters to Arnauld in my bag at that very minute. Just as we were marveling at the incredible coincidence, in walked the professor under whom we had both studied Continental Rationalism — including a good deal of Leibniz — at university! In his bag was a biography of Wilhelm Gottfried Leibniz.
“This is too much,” I said. “The odds of the three of us arriving here, in the US Virgin Islands, on St. Thomas, in this hotel lobby, on April 4th at 3:00 pm, all carrying books on Leibniz…the odds are astronomical! This can’t be a coincidence. The probability of this event happening is nearly zero. There is some meaning, some purpose, in this. Of that we can be sure.”
The other two slapped me on the mouth for being an idiot, because the odds of the three of us all arriving in the US Virgin Islands, specifically St. Thomas, at 3:00 pm on April 4th, all carrying Leibniz books, are exactly the same as the odds of me arriving on April 3rd, my former classmate arriving on April 4th, and the professor arriving on April 5th, with me arriving at 2:00 pm and my former classmate and professor arriving at 1:00 pm and noon respectively. Those two events have the same probability of occurring; the two men slapped me because I was ascribing meaning and significance inappropriately — by which I mean to a situation due to its low probability, and at the same time ignoring another thousand situations that were just as equally likely to occur. “Ah ha,” you may say, “but those other thousand events didn’t occur. This one did, which is why it has significance!” But I wouldn’t recommend saying that because then you’d get slapped too. If any of those other events had happened you wouldn’t have seen any significance in them at all, despite them having just as astronomically low odds of occurring.
I TOLD YOU THAT STORY SO I COULD TELL YOU THIS ONE
I’ve been hearing about Sacred Geometry in one form or another for a long time now, but it feels that it’s been growing in popularity in recent years (although that may very well be the result of a heuristic bias, one whose name I can’t presently recall – [Baader-Meinhof Phenomenon – ed.]). Regardless, something about it had always bothered me and I couldn’t put my finger on it until I recalled something one of my old metaphysics professors had said about Wittgenstein long, long ago: “He was a whacko who beat kids too hard. Or not hard enough, depending on where you stand on that sort of thing”.
[Ludwig] Wittgenstein had a pretty wild philosophy of mathematics, one I didn’t appreciate until I learned more about math and logic. He’s got genuine (contingent) propositions, used to assert a state of affairs in the real world, which he contrasts with mathematical propositions, which “have no subject matter” and “say nothing”.
From his Tractatus: “If an elementary proposition is true, the state of affairs exists; if an elementary proposition is false, the state of affairs does not exist.” Easy, right? While the linguistic components of the proposition aren’t inherently meaningful, we endow it with a conventional meaning and use it to do stuff (compliment someone, alert someone, hurt someone’s feelings). “A proposition is true if we use it to say that things stand a certain way, and they do” (also from Tractatus). The notion isn’t terribly complex.
These genuine propositions stand in contrast to mathematical “propositions” — mathematical equations, essentially — which even while we describe them as “true” or “false” we’re using those terms in a radically different sense than when we use it to describe genuine propositions. Mathematical propositions are “pseudo-propositions” that, when true (“correct”, rather), simply “mark the equivalence of meaning of two expressions.” Even better (or worse, depending on where you stand), since tautologies and contradictions are simply cases where the combination of signs destroys itself, and that “the conditions of agreement with the world—the representational relations—cancel one another, so that [they] do not stand in any representational relation to reality,” tautologies and contradictions do not reflect reality and cannot refer to any possible facts or any possible state of affairs, and so cannot be either true or false. They don’t have meaning the way genuine propositions do and so cannot be used to assert anything. From the Stanford Encyclopedia of Philosophy:
Given linguistic and symbolic conventions, the truth-value of a contingent proposition is entirely a function of how the world is, whereas the “truth-value” of a mathematical proposition is entirely a function of its constituent symbols and the formal system of which it is a part. Thus, a second, closely related way of stating this demarcation is to say that mathematical propositions are decidable by purely formal means (e.g., calculations), while contingent propositions, being about the ‘external’ world, can only be decided, if at all, by determining whether or not a particular fact obtains (i.e., something external to the proposition and the language in which it resides).
Basically, the signs and propositions of formal mathematics don’t refer to anything. It is purely syntactical and completely without meaning. The numerals are the numbers — “Arithmetic doesn’t talk about numbers, it works with numbers”. For instance, Wittgenstein says “What arithmetic is concerned with is the schema | | | |.—But does arithmetic talk about the lines I draw with pencil on paper?—Arithmetic doesn’t talk about the lines, it operates with them.” Mathematical symbols aren’t a proxy for things that would otherwise give them meaning, they’re just part of a calculator.
WITTGENSTEIN: (sarcastically) “Chess only had to be discovered; it was always there! Ha ha ha no but seriously that is a silly thing to say”
From the SEP again:
This is the core of Wittgenstein’s life-long formalism. When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover pre-existing truths that were “already there without one knowing” (PG 481)—we invent mathematics, bit-by-little-bit. “If you want to know what 2 + 2 = 4 means,” says Wittgenstein, “you have to ask how we work it out,” because “we consider the process of calculation as the essential thing” (PG 333). Hence, the only meaning (i.e., sense) that a mathematical proposition has is intra-systemic meaning, which is wholly determined by its syntactical relations to other propositions of the calculus.
This covers only a very small portion of his entire philosophy of mathematics, of course. I don’t agree with all of it, nor do I claim to understand all of it — but this, arguably the first step to understanding the rest, struck a chord with me when I began researching Wittgenstein. I had studied plenty of Plato at that point, but I’ve always secretly agreed with the afore-mentioned metaphysics professor when he told me “Plato was a guy that asked all the right questions but got all the wrong answers”. As such, I never had a problem with Wittgenstein contradicting Platonism (which he does. Pretty hard) and its theory of geometric forms.
When looking into sacred geometry, I felt I was being shown syntactical expression after syntactical expression with no inherent meaning to it all. “So what?” was my response to each proposition. I didn’t — and still don’t — buy that this sacred geometry is just there, sitting around and waiting for us to recognize it. It was people placing a geometric image over the universe and wherever some points or lines matched up, they’d point and say “There! Look! Surely all of this must mean something!” while completely ignoring all the places in which it didn’t match up at all, also ignoring the fact that the latter exists in far greater quantities than the former unless they use every geometric shape they can get their hands on and do some serious rationalization.
When I started reading about Wittgenstein I felt like someone had finally pulled an irritating thorn from my side. I mean, if humanity had developed on a significantly smaller planet, we’d learn Elliptical geometry before Euclidean and Pythagoras wouldn’t even be a thing since on a curved surface the sum of a triangle’s angles — any and every triangle — is greater than 180. What would the “sacred geometry” look like to those poor bastards? What if humanity had developed on a constantly fluctuating 2-dimensional surface and Hyperbolic geometry was what one learned in school? Would they be missing out on everything? Isn’t sacred geometry unwittingly trying to make the calculator sacred?