Semiotic Investigations: Towards an Effective Semiotics
Alec McHoul
Part Two
From Formalism and Ethnomethodology to Ethics
Chapter 12
Gatekeeping Logic
In his important ethnographic study of science in action, Bruno Latour has shown how scientists often have an interest in severely narrowing the indexical potential of signs within their own (scientific) domains.1 That is, they can be seen to act in ways which prevent "aberrant" uses of key pieces of terminology, formulae, constants, technical devices, forms of argument and so on. If Latour is right, then we could expect such "gatekeeping" tactics to be at their most vigilant at those points where scientific communities come into contact with other communities. A particularly "high risk" area would be popular science texts where writers cannot control who it might be that comes to read, interpret and, potentially, appropriate scientific signs. In this investigation I want to look at one such strategy of control as it operates in a popular account of mathematics and physics.
The mathematical physicist Roger Penrose has had great success with his popular work, The Emperor's New Mind. There, he makes strong claims against the idea that machines can be intelligent in the sense that humans are. By way of argument, he presents a very wide-ranging discussion of events and discoveries in contemporary mathematics and physics and goes on to show that mind-machine analogies arise out of mistakes within and abuses of scientific ideas. It's quite clear from his writing that physics (and in particular the manipulation of mathematical-physical symbols) is not for just anyone. But this provides him with a problem on the other side: for he is precisely, in (and as) the very book which sustains his argument for scientific exclusivity, trying to make technical and professional details accessible to any reader. How can he be sure that bizarre varieties of physics do not ensue? How can he be sure that his part-simplifications won't lead to "ethno-physicses" springing up among the "laity"?
One way in which he does this is to invoke a variety of logico-mathematical realism: he argues that physical-mathematical utterances are not valid unless the one who utters them can show that they somehow map on to really-existing equivalents in nature. Expressions which fall short of this criterion (which is, presumably, what non-professional instances would do) ought to be rejected. However, there is no precise way in which this argument can be sustained without a certain kind of tautology arising. How could one show that there exists a "real" pi to which a very very long and complicated number corresponds? Somehow the number itself (or the best approximation of it yet calculated) has always to be, itself, the proxy for this real pi. Somehow, it is the number's elegance which guarantees its attachment to what is presumed to be "real." But, at the same time, only "insiders" (mathematicians) can make judgments on the question of what is elegant.
Hence, what starts off as a heuristic for getting mathematics done (a heuristic of the form "let's imagine that mathematical expressions map directly on to nature") is transferred into an object of an entirely different epistemic status. It becomes a philosophical principle. And in fact, when it comes - textually - to argue for mathematical Platonism, all Penrose can do is to make a stipulation (where, in an ideal world, he would be able to offer a proof). For example, writing about the real number system, Penrose asks:
Why is there so much confidence in these numbers for the accurate description of physics, when our initial experience of the relevance of such numbers lies in a comparatively limited range? This confidence - perhaps misplaced - must rest (although this fact is not often recognized) on the logical elegance, consistency, and mathematical power of the real number system, together with a belief in the profound mathematical harmony of Nature.2
Having raised the possibility of an empiricist approach to number systems (that they should go so far and only so far as our experience - such that numbers like 10100 would be "too big" to use), Penrose dumps it straightaway as a principle of faith. Naturally, if you believe that "Nature" has a "profound mathematical harmony" (a phrase which is merely added to the idea of "logical elegance" but is made, linguistically, to look as though it were entailed by it), then there is nothing else for it. There is no other argument to be entertained.
This is precisely the battle of faith which Penrose raises with Brouwer and other so-called mathematical "intuitionists" who believe that mathematics is no more than a social practice involving community-based rules for the manipulation of symbols. But note that how he achieves this is via stipulation rather than argument:
Can [mathematical objects] be other than mere arbitrary constructions of the human mind? At the same time there often appears to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician.3
Note the slippage between concepts and objects, as though statements about the one could be identical with those about the other - thus accomplishing their identity by the attribution of identical predicates to each. Note too that the main weight of Penrose's claim is taken by the phrase "there often appears to be" - for he is in the predicament of not being able to prove that "there is." Note further that the naturalness of mathematical objects is contrasted, not with their particularity to a community (which is Brouwer's actual argument), but with their situation in a particular person's mind.4 This last slippage is between a semio-sociological and a psychological account of mathematics as synthetic. Quite simply: the one cannot simply substitute for the other; and a rejection of one is not a rejection of the other - if only because a semio-sociological version of intuitionism would not need to consider "the mental deliberations of any particular mathematician." Penrose persists in this form of "disproof":
The very system of complex numbers has a profound and timeless reality which goes quite beyond the mental construction of any particular mathematician.... Later we find many other magical properties that these complex numbers possess, properties that we had no inkling about at first. These properties are just there. They were not put there by Cardano, nor by Bombelli, nor Wallis, nor Coates, nor Euler, nor Wessel, nor Gauss ... such magic was inherent in the very structure that they gradually uncovered.5
Strangely (or aptly) enough, the Platonic properties of numbers are referred to here as their "magic." Mathematical symbols map on to mathematical objects by magic! And this is crucial to Penrose's argument throughout the book. For his view of specifically human intelligence is that it is able to intuit this magic, to see its beauty, to be driven by it, since it pre-exists all human action and thought. Human intelligence, to take this somewhat further, is aesthetic:
the true mathematical discoveries would, in a general way, be regarded as greater achievements or aspirations than would the "mere" inventions. Such categorizations are not entirely dissimilar from those that one might use in the arts.... Great works of art are indeed "closer to God" than are lesser ones. It is a feeling not uncommon among artists, that in their greatest works they are revealing eternal truths which have a kind of prior etherial existence, while their lesser works might be more arbitrary, of the nature of mere mortal constructions.6
And this is precisely why machines cannot model human intelligence. Machines can only perform functions which are calculable (in Turing's very specific sense of this term): they cannot feel the sublimity of what they produce by virtue of such calculations. Human beings can do this, Penrose argues, by virtue of the "anthropic principle": the belief (be it noted, belief) that "Nature" itself is specially constructed so as to provide for beings like ourselves, the proper interpreters of natural (hence mathematical) principles.
Penrose goes on to argue that the main bastions of insight into the mapping of mathematical expressions on to (hypothetical) natural objects are the two principles of logic: the law of the excluded middle and the law of contradiction. What he omits is an equally important methodological principle of logic: that nothing should be defined in terms of itself. In fact he has completely ignored this principle in his argument for (or statement of faith in) Platonism. If we now take this principle (call it, the principle of external definability), we can see that it throws a spanner in the works of Penrose's argument.
The proposition that Penrose wants to show to be true, let us call it proposition Q, is that mathematical truth is absolute. More formally, proposition Q would state: "For all mathematically true statements, we have absolutely true statements." As I mentioned above, he cites two axioms which support this proposition - the law of the excluded middle and the law of contradiction. Let us call these E and C respectively. E states that "a proposition is either true or false," while C states that "a proposition cannot be both true and false." So it is now propositional truth which will go proxy for "naturalness" or "reality." By satisfying E and C, that is, a consequently true mathematical statement will be considered Platonically real.
So, to put Penrose's position simply, Q (the absoluteness of mathematical truth) depends on E and C. If they are true, then Q is true. But there is a circularity to this since mathematical truth is, itself, defined by this very same argument. Or, to put this another way: mathematical truth exists only in a world where E and C hold, and where they entail Q (the absoluteness of mathematical truth). So we can see that this definition of mathematical truth contains itself: it contravenes the principle of external definability ("anything to be defined, say x, must be defined by a proposition which does not contain it, x").
Hence the absoluteness of mathematical truth can only be stated tautologically. If we believe it, it must be true: if we don't, it can't be. And a further complication is this: as a logical argument, Penrose's position on mathematical truth would seem to require the principle of external definability; yet it also seems to contradict it.7 That principle seems to be both true and false at the same time. And if we remember back to the very beginning of Penrose's argument, we will see that this contradicts his own absolutely crucial axiom, C: "a proposition cannot be both true and false." And if that proposition doesn't hold, then the whole house of cards called "absolute mathematical truth" looks very shaky indeed. It's clear then: you either believe Platonism or you don't; but you can't ground it on the standard logical canons. The social rather than logical paradox is that it's perfectly all right for anti-Platonists to live with contradictions of this kind, but not for Platonists of Penrose's ilk.
These peculiar grounds are those on which Penrose tries to deny a Brouwerian version of mathematics (the view that mathematics is cultural invention). But why does he want to deny it?: perhaps because that view would mean that whoever wanted mathematics, for whatever purpose, could have it and use it freely without the restrictions of professional control. The control comes, precisely, from the enlistment of Aristotelian principles for symbolic manipulation by a Platonic faith in absolutes. A curious philosophical mix. And these principles are, in essence, ones which guarantee a closely binaristic version of the world. In this version of logic, propositions are either true or false: there is no available space between the binary alternatives, and there is no possibility of combining truth and falsity in the same proposition. Commonsense logics of various kinds cannot operate: for example the commonly held view that statements may contain "a grain of truth," or that views and opinions can be true on some occasions but not on others. Mathematics is not allowed to slip into any domain of this kind. It is at once a strict binaristic formalism and a key to the "magic" of "Nature"'s own structure. The world is Platonically real, for Penrose, and the essence of its reality is a binary logic. We are invited to share this - and thus to do mathematics on what appear to be narrowly professional terms - or to relinquish any desire to manipulate mathematical symbols. The signs are kept highly narrow and strictly within a specific community.
As with mathematics, so with formal semantics, where a similar problem exists. Formal semantics treats natural language as though it were a calculus: a set of symbols to be manipulated according to a fixed set of rules. But, at the same time, it believes those rules to be the key to the truth-functionality of sentences or propositions. Sheer calculability (a formal procedure) and truth (an onto-epistemological judgment) are conflated. Formal semantics makes the mistake of looking for the truth of descriptions as a formal property of their relation to the world.
If the argument from calculability to truth can be sustained in mathematics only by statements of faith, then this must be the case a fortiori in natural language studies. Accordingly, effective semiotics would want to take a more Brouwerian approach. This would mean that, instead of analyzing descriptions in terms of their (natural) truthfulness, effective semiotics would be more interested in their (social, historical) plausibility. Let us take an example - another instance of graffiti:
The Beatles were: a great man, his side-kick and two props!
The truth of this description is eternally contestable - no amount of formal analysis (logical or semantic) will take us to a point where the linguistic matter maps on to a true (or indeed false) state of affairs. But at the same time, for any community with even the most meager knowledge of the Beatles, the description is a plausible one.
How do we know this? In terms of the semiotic model developed so far, we can say that the description sets up a problem: which Beatle is which? And anyone who knows the Beatles quite simply can see, at a glance, which is which. They find the solution right away - even if they disagree with the truth of what it asserts. The most ardent fan of Paul McCartney will still know that it is not he who is intended by the description, "a great man."
Finding a puzzle-solution, in this way, can establish the plausibility of an utterance for its reader. As with the sexual graffiti analyzed in chapter 7 (above), displaying the solution displays one's competence. That is, it's just as competent to respond with "That's a bit unfair to Paul," as it is to respond with "Yeh, John was always the real leader." The first response takes the initial utterance to be plausible but false; the second takes it to be both plausible and true. In this way, the T/F criteria can be inverted, while the plausibility criteria must remain constant - at least for any competent reading.
In everyday life, it's not agreement but understanding that is most basic to communication (that is, to community membership). If you don't understand, you don't get past first base and into the further space of agreement or disagreement. But this is not to say - in fact on the contrary - that understanding, plausibility and competence are always and necessarily part of any communicative situation. Sometimes, "first base" itself is not reached and - as I hope to show in the next investigation - this does not mean that we have a case of "failure to communicate," "distorted communication" and so on. In fact, such situations may make it much more clear and obvious to all concerned just what it is to be social: to be a specific community member, a specific kind of social subject.
=> chapter 13
This page incorporates material from Garry Gillard's Freotopia website, that he started in 2014 and the contents of which he donated to Wikimedia Australia in 2024. The content was originally hosted at freotopia.org/people/alecmchoul/seminv/12.html, and has been edited since it was imported here (see page history). The donated data is also preserved in the Internet Archive's collection.