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,
Conant, J. (2000) “Elucidation and Nonsense in Frege and early Wittgenstein”, in Crary
& Read, 2000.
Crary, A. & Read, R. (eds.) (2000), The New Wittgenstein,
Diamond,
C. (1991a), The Realistic Spirit,
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),
Wittgenstein, L. (1922), Tractatus Logico-Philosophicus,
Wittgenstein, L. (1978), Remarks
on the Foundations of Mathematics,
Wittgenstein, L. (1975), Wittgenstein’s
Lectures on the Foundations of Mathematics (
[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 ‘unrigorous’ proceedure,
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”.