Logicism and Anti-Logicism alike are both bankrupt and unnecessary

 

            Consider the following:

0) The concept horse is a concept easily attained.

1) The concept horse is a concept.

2) The class of horses is a class.

3) The class of horses is not a horse.

4) The class of horses is not a member of itself.

5) The class of things which are not horses is a member of itself.

6) The class of classes which are not members of themselves is [not] a member of itself.

 

            Items (1) through (5) on this list seem perfectly innocuous. Once one has granted (1) and (2), there seems no reason not to move right down the list to (5). But notoriously, if that is allowed, then we reach a paradox ((6)). We reach the paradoxical situation wherein the class of classes which are members of themselves is a member of itself if and only if it isn’t. This paradox, due to Russell, apparently very much required some kind of resolution. Russell’s ‘Theory of Types’ was at length born. And so the programme of Logicism remained a hope, for a while longer. That is to say, it was possible to continue to hope that maths (arithmetic) could be founded on logic, ‘logic’ including set theory, set theory centred for instance on the notion of ‘class’,[1] a notion allegedly rather clearer and ‘purer’ -- freer of certain logico-philosophical obscurities -- than the notion of ‘concept’.

            The above is a (thumbnail) historical sketch then of the situation up until the time when Gödel apparently showed, roughly, that a special, subtle version of the Liar Paradox -- his ‘Incompleteness Theorem’ -- ended any hope of basing arithmetic upon ‘logic’. Gödel is thought by most philosophers and logicians to have decisively shown the incompletability of Logicism.

 

            Now I hold no brief for Logicism. But I am unhappy, at a level of fundamentals, with the above paragraphs as a sketch of key developments in the history of logic in the twentieth century. Before assenting to the claim that the twentieth century has seen the increasingly general and correct recognition of the decisive triumph of Anti-Logicism over Logicism, let us cast our minds back for a moment to the start of this story: to Frege, and to my opening list.

           

            (0) of course was the subject of Frege’s famous philosophic triumph over Benno Kerry. Kerry argued that proposition (0) was perfectly fine. This appeared to problematize Frege’s ‘context principle’; for this principle, Frege’s dictum never to look for the meaning of a word in isolation, but only in the context of a proposition, has as its concomitant that one ought always strictly to separate the logical and the psychological, but this, Kerry thinks he has shown us (with (0)), we do not actually need to do. Frege countered that (0) is not just alright as it stands. A certain concept is ‘easily attained’ only in a person-relative psychological sense, whereas the notion of being “easily attainable” has no relevance to the logical sense of the concept horse.

            Frege recognised that his claim was somewhat ‘counter-intuitive’, and he held that in fact the seas of language run very high here, and that it is almost impossible to find a way of expressing oneself that does not mislead. He argued that all that philosophical logicians could hope to do hereabouts was to provide elucidations, elucidations of what we (hopefully) already ‘know’ and are willing to acknowledge. For example, that there is a fundamental difference in use between the concept concept in the proposition “The concept horse is, logically, closely related to the concept quadruped” on the one hand, and in propositions such as “The concept horse is a concept easily attained” or “The concept concept is not a concept easily attained” on the other. Frege held that the surface appearance of natural language is such that in all three of these propositions, and actually in pretty much the whole list of sentences with which we began this paper, there is an ever-present and serious risk that we will mistake the use and nature of (for example) the word, ‘concept’. For this word, which Frege thought it best to use in a strictly logical sense, almost inevitably and invariably appears to identify itself as (in Frege’s terms) an object-word. What Frege hoped was that he would help his readers find ways of not being bemused by the non-obvious logical category-distinctions which the surface appearance of language could mask.

            And so Frege held that, strange as it might sound, the least misleading thing to say is that “The concept horse is a concept easily attained” is not an ordinary, sensical, truth-evaluable proposition. For there is an important sense in which the word ‘concept’ is being used inappropriately, almost-inevitably misleadingly, in it.[2]

            We may usefully phrase the elucidation that Frege was trying to make for us here as follows: That there is no such thing as the defining of the logical categories and distinctions which constitute the ‘basis’ of any efficacious begriffsschrift. Rather, these categories, these ‘concepts’, can only be elucidated; they can in fact only be understood by someone who already implicitly understands them. In short, there is no such thing as taking a ‘metaperspective’ on logic.

 

            After having endeavoured to become clearer about the nonsensicality of the project of stepping outside logic, of giving logic foundations, if we turn back now to our series of ‘propositions’, (0) through (6), they may start to look rather different. Frege’s discussion of (0), which I have endeavoured to recapitulate the gist of, leads naturally into the following, Fregean, thought about (1): That “The concept horse is a concept” (or similarly, “Concepts are not objects”; etc.) is least-misleadingly construed not as a true statement, say as an analytic truth, but rather as an inevitably-misfiring [3] attempt to say something which can only be shown, which can only be understood in linguistic practice. At best, such ‘propositions’ are themselves elucidations.

            Now, if “The concept horse is a concept” ((1)), a seemingly innocuous (and seemingly true) statement, is itself best-construed, if one is to avoid falling into deep error through failing to respect the ‘context principle’ (and its concomitant strict separations between the logical and the psychological, between concepts and objects)... if (1) is itself best construed simply as nonsense, nonsense which can perform an elucidatory function for us, then it follows that “The concept horse is not a concept” is not false, but also nonsensical; that “The concept horse is not a horse” is also nonsensical (and at best elucidatory); and so on. And let us note that “The concept horse is not a concept” too may be elucidatory nonsense -- Frege himself of course used this example, to draw our attention to the ‘objecthood’ of concepts, when they are predicated of.[4] As Cora Diamond puts it, “Nonsense-sentences are as it were internally all the same; and are einfach Unsinn, plain nonsense. Externally, however, they may differ... For a sentence that is nonsense to be an elucidatory sentence is entirely a matter of features external to it.” [5] Nonsense-sentences do not stand in logical relations to each other, not even if they ‘appear’ to blatantly contradict one another![6]

            Now review (2) through (5), with which we began; which led to Russell’s Paradox. If we apply Frege’s own rigorous thinking about concepts (and elucidation, and nonsense) rigorously to thinking about classes then we quickly reach the following conclusion: That neither (2), nor (3), nor (4), nor (5), (nor indeed any of their ‘contraries’) are sayable at all; except (at best, in a very attenuated sense) as elucidations  (We could imagine (3) being uttered perhaps as a grammatical joke, by a teacher, for example). But elucidations are not truth-evaluable. Thus they do not provide us with truths that can stated; but nor can they be counter-exampled or refuted.

            My conclusion is, then, that the reasoning which appeared to take us to (6), to Russell’s Paradox, to an apparent counter-example to Frege, is flawed. There is no decisive reason for us to see Russell’s Paradox as a flaw in Frege’s symbolism; but no reason either to see either Russell or Frege as actually providing (or failing to provide) foundations for mathematics. Rather, what Frege was actually doing, when read ‘charitably’, was giving us elucidations to help (us) avoid misunderstanding the logic of our language and of arithmetic. The ‘propositions’ about classes given here are themselves already nonsense, and at best elucidatory nonsense. They yield no contradictions, no surprising ‘results’, no ‘statements’ with which mathematical logicians have to reckon.

 

            Now it will be objected that my account does not distinguish, as one should, between Frege’s elucidatory sentences, which are given in ordinary language, and statements made within Frege’s Begriffsschrift, which, at least as Frege understood them, are straightforward assertions. “Concepts” and “objects” are excluded from the Begriffsschrift, it will be said, but “classes” and so on are not. The statements which give rise to Russell’s paradox can all be said to occur within the Begriffsschrift itself (at least in the system of the Grundgestetze). Thus Russell’s Paradox can be constructed within Frege’s symbolism, and does not merely occur in sentences which elucidate it. As a result, Frege cannot reject the paradox in the same way that he rejects Kerry’s statements about the concept horse. Russell’s paradox appears as an inconsistency in the system itself, and employs only legitimate concepts, legitimate moves in Frege’s game.

            But I have already suggested that no good reason -- or at least, no decisive reason -- is given us by Frege not to treat e.g. (4) through (6), above, in the same way as (0) and (1). We can understand why Frege would have found this proceedure dissatisfying, but I’m suggesting reasons -- and resources from within his own set of ideas -- for him to have actually taken the route (away from apparent defeat at the hands of Russell’s ‘Paradox’) that I am suggesting. Some statements which can arguably be developed in the Begriffsschrift have just as little right to be seen as non-nonsensical as (e.g.) the ‘statement’, “The concept horse is a concept” (or its ‘opposite’, “The concept horse is not a concept”). We ought not to hold on to the usual view that every ‘statement’ in one of Frege’s symbolisms must be a proper, truth-evaluable statement. Some of Frege’s would-be elucidations, and some other nonsenses, frame (e.g.) the Begriffsschrift, or are even to be found within it. So there can be nonsenses within the Begriffsschrift!?!  ....Polemically: So what? We might compare here Wittgenstein:

 

            “Let us suppose that people originally practised the four kinds of [arithmetic] calculation in the usual way. Then they began to calculate with bracketed expressions, including ones of the form (a minus a). Then they noticed that multiplications, for example, were becoming ambiguous. Would this have to throw them into confusion? Would they have to say [as Frege did on learning from Russell of the Paradox]: “Now the solid ground of arithmetic seems to wobble”?[7]

           

This remark is crucial for my argument. For we see here that Wittgenstein did not think it would be compulsory for them -- i.e. for us -- to say such a thing. We just don’t talk about -- we ‘systematically’ leave out, ignore -- division by zero, etc. Likewise, Wittgenstein thought that Frege’s logical excavations and elucidations, even some of those accomplished via the Begriffsschrift, did not simply collapse in the face of Russell’s Paradox. Frege took himself to be giving arithmetic a foundation in logic, but the very idea of providing such a foundation is an absurdity. Frege misunderstood what he was about in the Begriffsschrift -- we need to re-read what he was about, ‘charitably’, as I have put it; and then we can hold on to what is useful in Frege, to his real achievements of insight. (Wittgenstein put this point as follows: “ “But didn’t the contradiction make Frege’s logic useless for giving a foundation to arithmetic?” Yes it did. But then, who said that it had to be useful for [that] purpose?” (1978, p378) That was Wittgenstein’s way of understanding how Frege’s work on logic could be intelligibly thought of and still used once the idea of Logicism were given up as a chimera.)

            What of the role of (6), the Paradox, in Frege’s symbolism? Doesn’t it undermine the symbolism as a whole? We can just ignore it. So this ‘statement’

-- the purported Paradox -- can be generated in the Begriffsschrift...  So what? Once we note firmly that ‘statements’ (1) through (5), wherever they occur, are at best elucidations, then we should realize that nothing can be meaningfully concluded from -- generated from -- them. They are not truth-evaluable statements from which other statements can be derived. Again, they have no logical relations with (other) statements. So (6), Russell’s dread Paradox, cannot be generated from them. If one insists that it occurs, if one chooses to state it, it just stands there in the Begriffsschrift, alone, uselessly.

            ‘But what use can a concept-script be, after it is no longer a sufficient condition of something being sensical that it can be written in the concept-script?’ Well, indeed, we may want to give up the name ‘concept-script’, after we see that nonsensical expressions can appear in it. But we may not. Here is one reason why we may not: We may still have reason to think that it is a necessary condition of something’s being sensical that it can be written in our concept-script. Admittedly, this will now need some further reasoning beyond the lines of argument exploited by Frege himself -- and I have no space to try to give a full argument here. But the thought that there can be no sensical sentences which are not concept-script-able seems at least a not-unreasonable and somewhat attractive one. (In fact, it sounds quite like a central thought of Wittgenstein’s in the Tractatus.)  If we cannot find a way to render for ourselves or others how a sensical thought means in a way which is perspicuous after the fashion of Frege (and early Wittgenstein), is that not at least a good prima facie reason for worrying about whether we have succeeded in thinking something actually worth calling a thought at all?

            ‘But look, Frege wants his Begriffsschrift for two reasons. Firstly, to provide foundations for logic, foundations excluding all intuition. You have dismissed this first aim. Secondly, to see clearly the structure of our thought. This, you want to say, remains a pretty sound project. But once nonsenses are ‘allowed into’ the concept-script, then the primary reason Frege had for thinking that his concept-script ‘limned’ thought-proper is gone. What are your grounds for proposing that being ‘concept-script-able’ is a necessary condition for being a thought?’

            My response to this formulation of the objection to my argument is implicit in what I have already argued. For I suspect that the reasonable thing to say, at least for someone at all impressed by Frege, is that the boot is on the other foot. Once we have ‘admitted’ nonsenses into the concept-script, then it looks pretty unlikely that the concept-script is insufficiently generous and open-textured! If one wants to argue that something that cannot even be gotten into concept-script is not nonsense, the onus seems to be on one to say why.

            So, one might back away from the term ‘concept script’, and instead call what Frege produced (say) a ‘[useful and perspicuous] logical notation’. A change in appellation does not remove all use from the notation, even uses including claims as to sense.

 

            Frege unfortunately did not take the route I have suggested. Due to his insistence on an extreme kind of purity in the system he was constructing, he responded to Russell’s Paradox rather as a potentially fatal counter-example to his own system; “unfortunately”, because Frege thus did not realize ... that the paradox is fatal only on the basis of an incoherent goal for one’s symbolism. Frege realized rather more than Russell, for sure; Frege realized clearly, at his best, that Philosophy is self-deceived if it takes itself to be able to enunciate the form of our language, and even that all that we can actually do -- and all that is necessary -- is to apply or enact or attempt an elucidation or two, on those occasions when someone falls into the grip of illusion concerning the functioning of words. Moreover, Frege again and again stated, in the advices to his readers on how to read his works, that they were not to be taken as issuing in  ... statements. (Advice which Frege’s ‘Analytic’ followers have almost entirely ignored.) Frege is travestied, whenever his cautionary words on what it means to give an ‘elucidation’ are ignored (and the same is true of Wittgenstein).

            It took Wittgenstein -- in the Tractatus [8] and in his later remarks on maths etc.[9] -- to see entirely clearly what the matter was:

   to see how Russell’s Paradox could tenably be seen as uncompelling, as posing a problem only for an incoherent ambition;

   to see how Russell’s ‘Theory of Types’ was philosophically unsatisfactory, and thus quite orthogonal to the supposed paradoxical ‘problem’ with Frege’s logic ... and moreover that the ‘Theory of Types’ was a fortiori unnecessary to a proper (understanding of) logic;[10]

   to see, in sum, how Logicism itself is an absurd project, and an unnecessary one;

   and  to see therefore that the Anti-Logicism of Gödel quite failed to undermine Russell’s and Frege’s logics, when those were thought of outside the deforming ambition of Logicism! That Gödel only played a new game, with a new calculus; and that the application of that calculus to carry out substantive work in the philosophy of maths was an incoherent aspiration, a nonsensical effort to directly combat and ‘refute’ a nonsense -- the nonsense, that is, of a fantasised Logicist foundation for arithmetic.[11]

 

            In conclusion then, Frege/Russell Logicism and Godelian Anti-Logicism are both bankrupt and unnecessary -- for reasons not only Wittgensteinian, but also Fregean. In saying this, I am of course being revisionary especially in respect of Frege’s own conception of what he was about. We need to think not only of Frege’s prose introductions and prefaces, and his attempts at producing mutual understanding with other logicians and philosophers, but also of some of the statements within the Begriffsschrift itself as being at best elucidations -- and there is no overwhelming reason for us not to do so. Such an attitude toward the Begriffsschrift,  while not consistent with Frege’s wishes to be producing a science of logic, does  fit naturally with an idea which is, again, at heart Fregean -- namely, as cited above, the idea that, strictly, there cannot be such a thing as a meta-perspective on logic. The Begriffsschrift cannot give us such a meta-perspective ‘mechanically’, or by the back door. We should not expect it to achieve a fantasized ‘absolute purity’ which ordinary language cannot. (Again, this is what Wittgenstein realized clearly -- arguably, in the Tractatus itself. It is a serious mistake, though an extremely widespread one, to see Tractatus as itself a Logicist work.)

            We can, if we wish, treat Frege’s symbolism simply as an uninterpreted ‘symbolism’. In which case (e.g.) his Grundgesetze etc. yields simply a perhaps-amusing (or perhaps arcanely mathematically-interesting) system of ‘symbols’. If we rather have a charitable view of Frege’s Grundgesetze symbolism, which he himself did not -- if we import into it his own ‘context principle’ and the understanding of elucidation which goes with that principle -- then Frege’s symbolism is again harmless, and potentially-elucidatory, and again there cannot be any undermining of it. Understood aright, then, Frege’s symbolism is not refuted or even problematized by Russell’s Paradox: because ‘all’ that Frege’s symbolism does is provide a (potentially-misleading) schema of elucidations. Such elucidations just do not allow the supposed problems of self-inclusion etc.

-- ‘problems’ which Russell ‘delineated’ -- to arise. Our language is alright as it is, arithmetic is alright as it is, and logic must take care of itself; all these were held by Wittgenstein, on the basis of a comprehension of and extension of fundamental insights of Frege’s, and in the Tractatus. So, as Wittgenstein elucidated for us in the Tractatus, there is in turn no need whatsoever for the Theory of Types, a ‘Theory’ which would eff ‘the ineffable’. All that we can do, all that we need to do, as Frege began to do, and Wittgenstein from the Tractatus onward into his later work continued to do, is to offer elucidations etc. when anyone is confused into thinking anything other than that our everyday language is in order as it is, or when they are tempted to conflate the logical and the psychological, etc. .

 

            And it remains only to add, in clarification, that when in this paper I have used terms like ‘Fregean’, I have not been meaning to be speaking of what were Frege’s fixed and unassailable views. This paper has of necessity been too brief to constitute a serious intervention in the exegesis of Frege, let alone in the history of Early Analytic Philosophy. Rather, I have attempted to partially reconstruct an aspect of Frege’s thought (and development), in particular, of his thought at the height of his powers (at around the time of the controversy with Kerry). I have also done some substantive philosophy on that thought -- i.e. I have worked out some philosophical consequences of the notion of ‘elucidation’ etc. for the materiel exegeted from Frege. And I have fed both the ‘reconstruction’ of Frege and the substantive philosophy by means of which I extended Frege’s thinking into a speculative outline ‘alternative history’ of the last hundred years of philosophy of maths. An alternative history, both in the sense of suggesting how that history as it actually was should be re-read, and in the sense of hinting at how the historical process actually would probably have developed differently, making my (revisionist, ‘Wittgensteinian’) re-reading still more plausible, had Frege himself stuck by the aspect of his thought which I centrally highlighted above.

            So, my paper has been about what Frege (and Wittgenstein) actually said and thought, and also about what Frege could (and should) have done and said, beyond that. If he had done so, the title of my paper might have been far more obvious to most readers than, in 2002, it actually (I suspect) is.

 

 

 

 

 

References

 

  Beaney, M. (1997), The Frege Reader, Oxford: Blackwell.

  Conant, J. (2000) “Elucidation and Nonsense in Frege and early Wittgenstein”, in Crary & Read, 2000.

  Crary, A. & Read, R. (eds.) (2000), The New Wittgenstein, London: Routledge.

  Diamond, C. (1991a), The Realistic Spirit, Cambridge, MA: MIT.

  Diamond, C. (1991b), “Ethics, Imagination and the method of the Tractatus”, in Heinrich and Vetter, 1991; reprinted in Crary and Read, 2000.

  Heinrich, R. and Vetter, H. (eds.) (1991), Bilder der Philosophie (Wiener Reihe: Themen der Philosophie), Vienna: Oldenbourg, 1991.

  Wittgenstein, L. (1922), Tractatus Logico-Philosophicus, London: Routledge.

  Wittgenstein, L. (1978), Remarks on the Foundations of Mathematics, Oxford: Blackwell.

   Wittgenstein, L. (1975), Wittgenstein’s Lectures on the Foundations of Mathematics (Cambridge 1939); London: Univ. of Chicago Press.

 



[1] In the present context, I believe that we can leave aside the ‘no class theory’ option introduced by Russll -- it makes no difference to the central philosophical issues here.

[2] This is perhaps an appropriate point at which to head off parenthetically a general objection perhaps growing in the reader’s mind by now: that my ‘reconstruction’ of Frege and of the history of early Analytic philosophy here may seem to be turning Frege into a ‘Philosopher of Language’. NO: I aim rather to be ‘elucidating’ a tension in Frege’s project. I try in what follows to bring out an oft-underplayed (and ‘Wittgensteinian’) aspect of his early and mature thought (and an aspect of the development of his thought), and suggest that this aspect of his thought (which I explicate in greater detail in “What does ‘signify’ signify?”, in Philosophical Psychology 14:4 (Dec. 2001), pp.499-514) casts a different light both upon Logicism and upon the history of twentieth century philosophy of maths and logic, and indeed upon the whole ‘development’ of Analytic philosophy. If Anglo-American philosophers had ever taken on board Frege’s arguments in “On concept and object”, the course of twentieth century philosophy could have been fundamentally altered (and improved). (I expand on these remarks at the close of my paper, below.)

[3] This use of the word “misfire”  -- in which the inevitability of the misfiring, and thus the nonsensicality of the result, is crucial -- I draw from Jim Conant’s (2000).

[4] For detail, see Cora Diamond’s (1991), pp.130-1 & p.143; and Wittgenstein’s (1922), section 4.1272. I mean in this paper to be using the word ‘nonsense’ in a manner roughly consistent both with Frege and Wittgenstein, but there are of course differences between Frege, early Wittgenstein, and later Wittgenstein here -- see again Conant’s (2000) for details.

[5] Diamond (2000), p.70. What I have done here, applying a Diamondian spin to Fregean insights, is to cast serious doubt on the interpretation of Russell’s ‘Paradox’ which Russell himself unfortunately managed to convince Frege of, in his famous letter of 1903 (see Beaney (1997)).

[6] The same applies to elucidatory nonsense, wherever we may find it -- even in Wittgenstein’s later work. Elucidatory nonsense -- exemplifications of nonsense at particular moments -- does not show us any fact or thing. This is why ‘grammatical remarks’ or ‘reminders’ -- the terms that later Wittgenstein prefers to ‘elucidations’ -- do not contradict. One can even make ‘opposite’ grammatical remarks in different circumstances, remarks which would if ‘eternalised’ be in both cases simply plain nonsense. One isn’t reminded of any fact by Wittgenstein’s reminders.

[7] P.204 of his (1978) (and see also p.205, p.212, pp.395-6).

[8] An objection might be raised that the crucial element in Wittgenstein’s progressing beyond Frege in the Tractatus was his relatively principled giving up of Frege’s Basic Law 5. One can read the Tractatus that way. My suggestion in this paper has rather been the following: that in Wittgenstein’s work on the philosophy of maths, we see pretty explicitly that it is not compulsory to give up Basic Law 5. Rather, one can keep it, except where it actually causes problems, where one just suspends it, or ignores the results. Those made unhappy with this, as a seemingly ‘unrigorousproceedure, have yet to come to terms in particular with Wittgenstein’s later philosophy of maths, a philosophy which, I have suggested, most clearly renders Logicism and its negation absurd, while building on and preserving many of the ‘insights’ of Frege concerning language and concepts.

[9] For instance, an attentive reader of Wittgenstein’s Lectures on the Foundations of Mathematics cannot fail to be struck by the serious value accorded by Wittgenstein to the philosophical advances made by Frege and Russell, including quite specifically those things made clearer by their Logicistic moves (see e.g. p.267f.). A fuller task for another occasion would be: to bring out the philosophy of Logicism’s rejection -- and great value -- as seen throughout Wittgenstein’s career. Throughout his career, Wittgenstein holds that reduction of maths to logic is the mistake. Thus he does not uphold Logicism in the Tractatus -- yet nor does he in his later work condemn the impulses that led to Logicism and some of the elucidatory impulses and proceedures which it involved in Frege’s work especially.

[10] On this, see Kellly D. Jolley’s excellent (unpublished) paper, “Logic’s Caretaker”.

[11] Thanks to the audience which heard and very helpfully responded to this paper in an earlier (and briefer) form at the annual Wittgenstein Symposium, Kirchberg-am-Wechsel, Austria, 12-18 August 2001; particularly Phil Hutchinson, Dan Hutto, Laurence Goldstein and Jim Conant. Thanks also to Cora Diamond, Michael Kremer and Wes Sharrock.