On Wittgenstein’s Skepticism of Sacred Geometry

PIC: PD

Pic: PD

“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?

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  • Simon Valentine

    who be putt’n the lambda clamp down on captain marvel?!?

    the odds are mein!

    • Simon Valentine

      measure? weight? source code escrow?

  • Tchoutoye

    This reeks a bit of Dawkins implying that we should discard our emotions and psychology as being irrelevant, and behave as genetic machines. But just because we assign meaning to patterns doesn’t make such meaning less important to us. Recognising patterns and manipulating symbols is what makes us human after all. Science may strive to be objective, but every scientist works from a subjective mindset.

    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

    If correlation occurred more often than non-correlation it would lose its uniqueness and become boringly trivial. Then non-correlation would become meaningful instead.

    • mannyfurious

      There’s probably some truth to this. Sacred geometry isn’t pure mathematics. There is meaning ascribed to it in a similar way meaning is ascribed to a poem.

      • sonicbphuct

        I feel it could best be summed up via analogy – sacred geometry is to geometry what astrology is to astronomy. That they both use numbers and derive completely arbitrary answers, whereas astronomy and geometry result in meaningful answers.

        Or another one – Ken Ham’s Creation *museum* and sacred *geometry*: adding the rational word at the end does not make the whole thing rational. Or, to apply geometry, there is no transitive property here.

        • mannyfurious

          Ehhh… I want to agree with you, because I like what you wrote. But I’m not sure the analogies fit all that well. Certainly astrology and sacred geometry are madness–but there is a method to the madness, and therefore the answers aren’t strictly “arbitrary.” It may be nonsense, but not arbitrary.

          • sonicbphuct

            Fair enough; not exactly arbitrary results, but perhaps arbitrary interpretation, or, subjective interpretation. But nonsense is also valid.

    • Simon Valentine

      i’m afraid, as a figure of speech, that your idea, noun, genetic machine, symbol, “trivial” is neither scientific nor what you might purport :/

      walk the doubt. graph it out.

    • Tuna Ghost

      I’m not sure how you’re using “correlation”, but similarity does occur more often than non-similarity. This is partly why nature, by means of evolution, strives for diversity.

  • Patrick Ryan

    I agree that The word “sacred” in sacred geometry is just that, a word. But that doesn’t make the incredibly complex and symmetric patterns produced by natural systems any less beautiful or mathematically complicated.

  • BuzzCoastin

    once I met a girl who said she was from Cuba
    and I said
    I’ve lived in Cuba
    my youngest brother was born in Cuba
    I have a t-shirt that says property of Cuba on it
    I’m contemplating a trip to Cuba
    she found no sacred geometry in it

    • Simon Valentine

      did the ghost of Ben Franklin appear in the form of an owl
      perched atop a plantation notated “Jefferson Ranch”
      did all the sense made make a black hole
      or do people still treat it like a block hole
      like they were the knower of block holes
      whilst the owls are not what they seem

  • https://twitter.com/anti_euclidean ÿ

    Interesting presentation, thank you for sharing.

    However, you could have just said: “the map is not the territory”

    Mathematics™, like Language™, is a medium.

    2 + 2 = 5

  • emperorreagan

    HP discontinued the only line of sacred calculators in 2003 due to low profit margins:

    http://en.wikipedia.org/wiki/HP-48_series

    • Simon Valentine

      “compiler” they said
      “trade secret” they said
      “genetic programming doesn’t exist yet”, they didn’t exactly say

  • heinrich6666

    Ugh. There isn’t much I hate more than the sophomoric attempts to use philosophy to support this or that banal prejudice. In this case, it’s a prejudice against sacred geometry, which is mainly attributable to this writer’s fairly superficial take on it.

    To wit: The writer seems to hold that the whole of sacred geometry is akin to the whole of mathematical expressions in that it’s a closed system that’s ultimately self-referential. He seems to further hold that the ‘meaning’ that is supposedly discernible in sacred geometry is mostly wishful thinking, foisted upon it while ignoring all the ways sacred geometry doesn’t work.

    For one, using Wittgenstein’s perspective as a basis for this criticism is pretty tenuous. But worse, there’s the obvious objection: why do our brains find certain relationships like the golden ratio more ‘pleasing’? Anyone who has ever done graphic design can tell you how some designs feel intuitively correct (and end up obeying some inherent formal principles) and others totally out-of-whack. Even if you take this property of ‘rightness’ as a pure artifact of our brains, that at least would still serve as a basis for the ongoing attempt to find ‘meaning’ in geometrical relationships, and save sacred geometry from this writer’s faux-Wittgensteinian criticism.

    Basically, the writer ignores the whole dimension of sacred geometry that concerns ratio and proportion, and instead focuses on the attempt to find meaning in discrete syntactical expressions. This not only conflates geometry with arithmetic, but spatial reasoning with discursive or symbolic reasoning.

    • Tuna Ghost

      For one, using Wittgenstein’s perspective as a basis for this criticism is pretty tenuous.

      In what way?

      Basically, the writer ignores the whole dimension of sacred geometry that concerns ratio and proportion, and instead focuses on the attempt to find meaning in discrete syntactical expressions. This not only conflates geometry with arithmetic,…

      “You could say arithmetic is a kind of geometry; i.e. what in geometry are constructions on paper, in arithmetic are calculations (on paper).—You could say it is a more general kind of geometry.” — Wittgenstein (PR §109; PR §111)

      As the above quote shows, Wittgenstein considered both arithmetic and geometry to have the same meaning — none at all. How exactly is his philosophy of mathematics only tenuously applicable here?

      …but spatial reasoning with discursive or symbolic reasoning.

      Your use of “spatial reasoning” here does not reference your argument. It seems to be a synonym you’re using for “geometry”, which makes it the same thing as conflating geometry with arithmetic. In your view, in what way is geometry different from mathematics, and how does that exempt it from Wittgenstein’s criticisms?

      • heinrich6666

        Importing the whole apparatus of Wittgenstein’s critique of mathematical expressions to criticize sacred geometry is like borrowing your neighbor’s Rube Goldberg device to swat a fly. Not only does that critique only make sense and fully come to life in a certain philosophical context, you also import all its shortcomings. For example, the distinction between so-called genuine propositions and pseudo-propositions requires a theory of reference. Whatever Wittgenstein’s differences with the logical positivists, any theory that has recourse to a ‘world’, an outside ‘state of affairs’, etc., involves a referentialism. This was a major weakness IMHO in the early Wittgenstein. Regardless, invoking a fairly vast argument for the purpose of saying sacred geometry is false and disingenuous seems disproportionate. Especially when the only link is that arithmetic statements engender other arithmetic statements, and the wishful findings of sacred geometry only seem to produce more such findings.

        As for arithmetic and geometry — that there’s overlap between arithmetic and geometry cannot be denied — but the question is whether something essential is not lost by treating geometry as a set of discrete propositions only. To begin with, if your target is sacred geometry, it’s worth saying that geometry and sacred geometry are not necessarily the same. For confining itself to pure proportions, etc. (e.g. insisting on the ‘harmony of the spheres’) the latter may be so primitive as to be immune to the criticisms of geometry. But if we’re talking about what may be essentially different about geometry compared to arithmetic, you can consider the logical difficulties associated with, for example, reducing the act of tying your shoes to a propositional argument. Also topology: spatial reasoning, as it might be called, is sustained by a kind of logic that resists being broken down into discrete propositions. Something like the concept of symmetry is not explicable in terms of logical statements — yet it seems to sustain our symbolic expressions.

        • mannyfurious

          I don’t think he was “importing the whole apparatus of Wittgenstein’s critique of mathematical expressions….” That’s almost like saying, “You don’t have to import the whole apparatus of Newtonian physics to explain why the apple falls from the tree.” I think it’s safe to say he was simply tackling the topic from a “Wittgensteinian” point of view, much in the same way one can tackle a similar topic from an “Existentialist” point of view. You don’t have to have read all of Kierkegaard and Nietzsche and Schopenhauer and Heidigger and Sartre to take an existentialist view on something.

          Wittgenstein was well aware of his ideas requiring a “theory of reference.” That’s why he advised his readers to disregard the reference after it had served its purpose. However, it’s pretty difficult to talk about anything without some kind of point of reference. It’s pretty difficult to talk about the limitations of symbols without using symbols to do so. A lot of people get lost in the minutia of Wittgenstein, but that’s a quick road to nowhere, admittedly. Wittgenstein is best read from a “big picture” perspective.

          • heinrich6666

            The piece invokes Wittgenstein to make a far simpler and less interesting point that could have been expressed without invoking Wittgenstein at all: that sacred geometry is sustained by wishful thinking and by ignoring the numerous instances in ‘reality’ where it doesn’t work. If you claimed that after dropping a ball from your outstretched hand that it would shoot upward into the sky, I would be guilty of the same disproportion if I invoked some arcane distinction of Newton’s to ‘prove’ it would not. Now at issue is whether having recourse to Wittgenstein to make a much more rudimentary point is a bit of thinking out loud, or whether it is basically an appeal to authority functioning to shut the sacred geometers up because they can’t talk Wittgenstein.

            My view is that it is both: a harmless piece of thinking-out-loud *and* an unnecessary attack. It’s a stretch to import Wittgenstein’s argument for use against sacred geometry because to do so you have to disregard 90% of Wittgenstein and 90% of sacred geometry. As I pointed out above, sacred geometry is distinguished by – if anything – its preoccupation with intuitively satisfying (‘harmonious’) relationships. The supposed harmony created by an instance of symmetry or by the golden ratio is *not* the product of a certain proposition or set of discrete propositions but can be put down to spatial perception, etc. While no doubt there are those who find ‘proof’ for this or that metaphysical worldview in a variety of discrete mathematical expressions, in general sacred geometry is more primitive even than that: it’s a wishful sort of Platonism that holds a higher reality of harmonious formal relationships exists and bleeds over here and there into this one.

            Now, in an effort to make his point, the writer treats sacred geometry as if it were *nothing more* than looking at a set of arithmetical statements and forcing a meaning on to them. This ignores the better part of sacred geometry devoted to what was mentioned above: ‘harmony’, formally perfect relationships and shapes, natural numbers, etc. It’s precisely on the basis of this latter that sacred geometry-types feel empowered to look elsewhere for these ‘proofs’.

        • Tuna Ghost

          the question is whether something essential is not lost by treating geometry as a set of discrete propositions only.

          Given that the very core of Euclidean geometry is a set of discrete propositions — five of them, to be precise, from which all of Euclidean geometry follows — I can’t imagine how something essential, or anything at all for that matter, would be lost. That’s what geometry is. We get Elliptical and Hyperbolic geometry with only four of those propositions.

          But if we’re talking about what may be essentially different about geometry compared to arithmetic, you can consider the logical difficulties associated with, for example, reducing the act of tying your shoes to a propositional argument.

          Why would one do that reduction? I don’t see how that applies here.

          Also topology: spatial reasoning, as it might be called, is sustained by a kind of logic that resists being broken down into discrete propositions.

          Topology is a branch of geometry and set theory, which absolutely can be broken down into propositions. Why do you say it “resists”?

          • heinrich6666

            You’re doing a lot of dancing. Either you are immune to the basic insight or avoiding it purposely. I’ve already stated that there is overlap between geometry and the language of mathematics (i.e. discrete mathematical expressions). That much is clear, and bringing up Euclid after this much is clear is begging the question. The real question is: is what we call geometry simply a set of discrete expressions and nothing besides? I’ve given several examples of where this point of view fails — from concepts in spatial perception like symmetry/dissymmetry to graphic design where certain proportions look ‘right’ to the so-called Golden Ratio, which incidentally is a mainstay of sacred geometry. The problem of capturing a basically spatial operation like tying your shoes in propositional language is apropos because, though you may not be aware, it has been *tried*. There are real problems in reducing the logic of space to a propositional language. This is clear in knot theory, where different knots constitute different logical systems. As for topology, what makes topology what it is — the most basic concept — is that by pretending an object is made out of rubber, you can stretch and deform that object revealing unusual equivalences. A coffee mug transforms into a donut (to use that old example). This makes ‘sense’ before any surface is described in terms of algebraic points, etc.

            Now how is it possible that I might perceive a logic to such a rubber-sheet transformation before it has ever been established using a propositional language? How am I able to tie my shoes without it being taught to me first through a series of airtight propositional steps? The fact is, there are plenty of mathematical issues today from the different forms of the traveling salesman problem to the general problem of parallel computing where the classical algorithm fails. These problems all have in common some issue having to do with space. Geometry is non-identical with arithmetic to the extent that it grapples with space in ways that pure arithmetic does not. Sacred geometry, for focusing on meaningful spatial relationships only, could as easily be defended as attacked for this reason. Your piece treats sacred geometry as bibliomancy and nothing more, and geometry as arithmetic. This misses the whole obsession of the sacred geometry-types with intuitively clear relationships (‘harmony’) as demonstrated by the golden ratio, the Platonic solids, etc.

          • Tuna Ghost

            If I appear to be dancing, it’s because I have no idea what your basic insight is.

            Geometry and arithmetic don’t overlap, they are the same kind of thing. They are both mathematical systems. As such, it is fair to say that, per Wittgenstein’s philosophy of mathematics, their meaning is purely syntactical and not semantic and definitely not sacred.

            It is not a “point of view” that geometry is at its core a set of propositions, that is a fact. That is its historical origin. Geometry, all geometry, was built over the centuries by the study of those propositions. The fact that some graphic artists apparently think the golden ratio “looks right” — whatever that means — doesn’t change that.

            Geometry is non-identical with arithmetic to the extent that it grapples with space in ways that pure arithmetic does not.

            But that’s just it — it doesn’t grapple with space. That’s Wittgenstein’s whole point. It doesn’t “grapple” with anything; it has no connection to the real world. Neither does arithmetic, or any mathematical system. The fact that it’s difficult to use propositional language to describe how to tie a shoe has no bearing on this.

            I’m not missing the obsession with intuitively clear relationships as demonstrated by Platonic solids or the golden ratio. I’m agreeing with Wittgenstein that they don’t “demonstrate” them at all, that they can’t, because they have no meaning.

          • heinrich6666

            I congratulate you. You wrote a piece using an already shaky distinction (genuine vs. pseudo-propositions) originally aimed at the ‘meaning’ of arithmetical statements to critique sacred geometry, which you did by way of a double conflation: arithmetic = geometry and sacred geometry = geometry. Now you continue to defend the first conflation to preserve the second.

            Let’s address what you call ‘Wittgenstein’s whole point’. Of course, what you call his whole point *isn’t* his whole point. His point has to do with the meaning, but more precisely the truth of mathematical statements. If I declare ‘X’ to equal 2, neither term need be ‘true'; indeed, the statement as a whole cannot contain truth or be inherently true. Being a mathematical statement, it’s simply a statement of relationship. In the same way that if I’m presented with a passport and photocopies of the passport, I can affirm that the copies match the passport, but cannot be sure as an everyday citizen that the passport itself isn’t a forgery.

            But all this does is simply move the locus of truth away from an outside world into the system itself, or what Wittgenstein would call ‘use’. Properly understood, this does not mean all mathematical statements are false — as you mistakenly end up suggesting — but that the criteria for judging the truth of mathematical statements is simply different.

            Now, you claim that geometry ‘doesn’t grapple with space’ and that it has no ‘connection to the real world’. This is nonsense. It is taking what might be called Wittgenstein’s agnostic view (mathematical expressions are neither inherently true nor false but simply relational) and turning into an atheistic view (mathematical expressions can have zero relationship to the world).

            This is a misreading of Wittgenstein — at the least, a failure to recognize the substance of his early philosophy and how it functions for the rhetorical gestures he must make to differentiate himself from the logical positivists. Be that as it may, if you’re willing to insist that there is absolutely no difference between geometry and arithmetic; that geometry is simply arithmetic and has no proper object of its own despite the examples I’ve given, then you’re simply defending your piece and not thinking. What else can be said?

          • Tuna Ghost

            …which you did by way of a double conflation: arithmetic = geometry and sacred geometry = geometry.

            I never said arithmetic is geometry, which is what “arithmetic = geometry” means. I said they were the same kind of thing (a mathematical system), which they very obviously are. I don’t know why you’re pretending otherwise. Also, the first doesn’t preserve the second in any way, so why anyone would do what you’re suggesting I can’t imagine. And if you’re trying to suggest that sacred geometry isn’t derived from actual geometry I’m going to start doubting your commitment to the truth, sir, because you’re obviously not simply a fool.

            In the same way that if I’m presented with a passport and photocopies of the passport, I can affirm that the copies match the passport, but cannot be sure as an everyday citizen that the passport itself isn’t a forgery.

            In regard to Wittgenstein and our conversation, you’d be declaring that because the copy appears to match the passport, it functions in the same way as a passport.

            Properly understood, this does not mean all mathematical statements are false — as you mistakenly end up suggesting —

            Where did I suggest that?

            but that the criteria for judging the truth of mathematical statements is simply different.

            Incorrect! What Wittgenstein suggests — in fact, what he outright states — is that we’re using the word “truth” to mean something else when we talk about mathematical propositions. He says repeatedly that mathematical propositions cannot be either true or false in the sense that we use those words regarding genuine propositions.

            Now, you claim that geometry ‘doesn’t grapple with space’ and that it has no ‘connection to the real world’. This is nonsense.

            You say that, but a pretty good case has been made for it.

            It is taking what might be called Wittgenstein’s agnostic view (mathematical expressions are neither inherently true nor false but simply relational) and turning into an atheistic view (mathematical expressions can have zero relationship to the world).

            Can you show me where Wittgenstein admits that they can have a meaningful relationship to the real world?

            Be that as it may, if you’re willing to insist that there is absolutely no difference between geometry and arithmetic;

            I’m not, and never have. Show me where I’ve stated that.

            …that geometry is simply arithmetic and has no proper object of its own despite the examples I’ve given,

            What examples? Graphic artists thinking the golden ratio “looks right”? That it “grapples with space”? It’s still a mathematical system, and still subject to Wittgenstein’s criticism’s regardless of what graphic artists think of it. Will you now argue that it’s not a mathematical system? Because there’s pretty convincing proof that it is.

            (I’ve enjoyed our talk, by the way, despite my occasional snarky tone. But don’t tell anyone; I have a reputation to protect)

          • heinrich6666

            You’re now repeating my points, but assuming that I mean the opposite.

            1. I’m glad that you admit that arithmetic and geometry are not identical. The question is what the difference is and whether the difference is significant when it comes to your argument. For your piece to stand up, sacred geometry has to be as susceptible as geometry to Wittgenstein’s distinction between genuine and pseudo-propositions, and for that to be the case, geometry has to be a set of mathematical propositions and nothing besides. But you might ask yourself these questions:

            – Could you describe a triangle to someone who has never seen one before using propositional language?

            – Could you explain the difference between left and right using only propositional language?

            The many everyday examples I’ve given in previous comments should not be sneered at — they point to the essential difference between arithmetic and geometry without defining that difference completely. Here I’d invite you to do what Warren McCulloch used to admonish his students with: ‘Don’t bite my finger; look where I’m pointing at’.

            2. “Incorrect! What Wittgenstein suggests — in fact, what he outright states — is that we’re using the word “truth” to mean something else when we talk about mathematical propositions.” But this is simply what I said. When I say the locus of truth moves, it is simply another way of saying this. The meaning of ‘truth’ changes. In the same way that in formal logic today, the distinction between valid and sound arguments is taught, with the latter being ‘true’ and the former merely being internally ‘true’ or correct according to the rules of argument, mathematical propositions, in Wittgenstein’s view, do not require an outside world to assess their ‘truth’. The criteria change. The ‘truth’ of the statement X + 2 = 5 is in doubt if I also declare X = 0, and vice versa. So *absolutely* the truth of mathematical propositions is not the same as the truth in the ordinary sense (according to Wittgenstein).

            3. It’s beyond dispute that mathematical expressions (pseudo-propositions in this Wittgensteinian context) can describe geometrical objects. The area of a circle can be described as pi * R squared. This is a statement of relationship — but just one of many possible. You ask for proof where W. says mathematical statements have a meaningful relationship to the world. I could equally ask: could you show where he says the relationship is *meaningless*? The point is, they are neither meaningful or meaningless. They fall outside of truth in the ordinary sense. The fact is, I *can* calculate the area of a circle using the statement of relationship given. The proposition is ‘meaningless’ when viewed alongside the other type of proposition W. favors, but not meaningless in the absolute sense of being random, unintelligible, indecipherable. I don’t think W. would ever say that mathematical propositions are meaningless in the same way a scratch on the pavement is meaningless, and this is my point about his rhetorical strategy: he may declare mathematical propositions ‘meaningless’, but this is mainly to sharpen what he is trying to say by ‘meaning’. He would never declare them meaningless in the most radical sense.

            4. In my view, geometry is not simply a set of mathematical propositions, though naturally the better part of geometry is spent using the logic of propositions to prove existing notions or to make new discoveries. However, geometry has as its proper object a spatial reasoning available to the brain for navigating the human body through a 3D(+) world. One can draw a circle without computing it first. One can judge an equilateral triangle different from others without knowing the right mathematical proposition (i.e. definition) that establishes that formally. So geometry is concerned with a mental object, a kind of spatial ‘intuition’ that, in fact, sustains mathematics and logic more fundamentally. You could say as a matter of fact that our whole two-valued logic system used since Aristotle (true/false) is based on nothing but a mental prejudice in favor of symmetry.

            How does this relate to sacred geometry? While there are certainly those in sacred geometry who point to certain mathematical propositions and the values derived to make their case that the moon was artificially created, etc., in my experience the real animating spirit behind sacred geometry is not this computational aspect at all, but rather the preference for intuitively satisfying spatial aspects or relationships and the expectation that intuitively correct (usually primitive) proportions and relationships are proof of an underlying or transcendental reality more correct than our own.

            Cheers.

          • heinrich6666

            Though this conversation appears to be dead, it’s worth noting for whomever may read it that the distinction you highlight (genuine propositions vs. pseudo-propositions) is the perfect lens through which to see Derridean deconstruction. Far from being obscurantist bullshit, deconstruction can be seen as just an extension of Wittgenstein’s distinction where the category of pseudo-proposition is expanded to include all natural languages and symbolic systems and ‘genuine’ propositions are in turn considered a myth or fantasy. The claim is that just as with mathematical propositions, any formulation in natural language has a meaning (and a truth-criteria) based on its relationship to the rest of the system and, ultimately, nothing besides. Though you might try to invoke this or that extra-symbolic reference point (as I did above with geometric mental objects), the fact is you can only do so *again* through language. This effectively means according to deconstruction that there is no magical moment when the operational context of the symbolic system falls away and the word becomes united with the thing; the only thing that is ‘sure’ is that words lead to other words (‘there is no outside of/to the text’) and meaning never really appears or becomes present. It’s always amazed me — the Anglo-American helplessness when it comes to Derrida, when so much of deconstruction is already prefigured in Wittgenstein.

          • Tuna Ghost

            As an aside: the fact that a copy of a passport can’t function as a passport is somewhat mitigated by the fact that a copy of a passport is, I’m willing to admit, far more helpful than not having a passport at all, in that a copy of a passport will convince the authorities (to a degree, anyway) that the passport does or did in fact exist somewhere. There’s an interesting ontological discussion there, or an epistemological discussion, but either way I believe it’d lead far afield from where we’re at now.

          • heinrich6666

            Wasn’t the point. In some countries a lawyer in a foreign country may be hired to attest that photocopies of official identity papers are legitimate. This, when a foreign national is asked to present his passport, etc., for the purposes of employment but does not wish to send them by post. The lawyer does so by comparing the copies later to be sent by post with the originals with which he has been presented. Of course, he can’t verify that the original is itself genuine. He can only say that the copies match the passport he’s been given. He signs the copies; the copies are sent.

  • Tuna Ghost

    I don’t know if they do, but I sure don’t

    • https://twitter.com/anti_euclidean ÿ

      I think it’s one of these…

      • Tuna Ghost

        ah, one of those. Why can’t people just tell me they want to make out, I’m very open-minded

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