Frege
against logicism [1]
Consider
the following propositions:
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. Or at least, once one
has granted (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. 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 is
‘represented’ above by (6).) This
paradox, due to Russell, apparently very much required some kind of resolution.
Russell’s ‘Theory of Types’ seemed to do the trick. And so the programme of
Logicism remained a hope, for a while longer. That is to say, it as possible to
continue to hope that maths (arithmetic) could be founded on logic, ‘logic’
including set theory, set theory centred on the notion of ‘class’,[2] a notion allegedly rather clearer and ‘purer’ --
freer of certain logico-philosophical obscurities or difficulties -- than the
notion of ‘concept’.
Until
Gödel apparently showed, roughly, that a special, subtle version of the Liar
Paradox [3] -- his
‘Incompleteness Theorem’ -- ended any hope of basing arithmetic upon ‘logic’.
Gödel is thought by most -- by nearly all ‘mainstream’ -- philosophers and
logicians to have decisively shown the incompletability of Logicism (and also
to have refuted Hilbert’s Programme[4] ).
Thus, from Frege’s hopes of a foundation
for arithmetic in logic, we go to Russell’s Paradox of self-inclusion of
different ‘levels’ of language; from Russell’s Paradox to Russell’s solution,
thus saving Logicism, through explicit separation of ‘hierarchical levels’, of
Types; from Russell’s solution to Gödel’s decisive demonstration of the
inadequacy of all Logicism(s), via a
particularly subtle self-referential device, a device ingeniously bridging
those ‘levels’: a sentence which is true but unprovable, which cannot be true
if it is provable.
Now
I hold no brief for Logicism. None whatsoever. But I am unhappy, unhappy at a
level of fundamentals, with the above
one-sentence paragraph as a sketched ‘history-in-brief’ of arguably the key
developments in the history of logic in the twentieth century. And I suspect
that even those historians of logic who would find the above sketch of course
horribly crude and over-simplified will nevertheless not find it unsatisfactory
at a level of fundamentals; or at least as a ‘rational reconstruction’ of what
happened. That worries me. And so: Before concluding that the twentieth century
has seen (roughly speaking) 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 supposed start of this story: to Frege.
Now
you will probably have noticed that, before we got to propositions (2) through
(6), I listed two other propositions:
0) The concept horse
is a concept easily attained.
1) The concept horse
is a concept.
(0)
of course was the subject of Frege’s difficult and 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 subjective and the objective, 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, strange as it might seem, (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/interpersonal sense of the
concept horse. (Similarly, Frege of
course distinguished rigorously between the psychological and logical senses of
the word “thought”.)
Frege
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
oneself and others. He argued (both against Kerry, and elsewhere [5] ) that all philosophical logicians could hope to
do hereabouts was to provide elucidations,
elucidations of what we already ‘know’. 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 -- indeed,
in this last, we almost see the problem, and Frege’s point, quite directly and
immediately. More fundamentally still, he 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. His hope was that he could provide this help by means of
providing elucidations.
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,[6] almost-inevitably misleadingly, in it.
We
may usefully phrase the elucidation that Frege was trying to make for us
hereabouts, then, as follows: that the form of our language cannot be fully
enunciated. Or, better still, a little more finely-grained: That there is no
such thing as -- no coherent understanding available of what it would be to
effect -- the defining of the logical
categories and distinctions which effectively constitute the basis of any
efficacious begriffsschrift (‘concept-script’).
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: logic cannot be taught to someone who doesn’t
already ‘know’ it.
After
having endeavoured to become a little 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”; or for sure any other ‘proposition’
involving the terms which, while not in
his Begriffsschrift, were or could
have been used to frame it) is
least-misleadingly construed not as a true statement, say as an analytic truth,
nor even as a tautology, but rather as an inevitably-misfiring [7] attempt to say something which can only be shown,
which can only be understood in linguistic practice. At best, such ‘propositions’ are themselves
elucidations.
It
might be objected that “The concept horse
is a concept” is not nonsense, on the grounds that it could be quite
meaningfully employed: e.g. when explaining the meaning of the word “concept”.
To explicate this analogically -- it might be said -- we should note that one
could intelligibly use the sentence
(1^)
“The animal horse is an animal”
when explaining what an animal is... And it is true, we might employ the latter
sentence in that fashion (though I think that the first occurence of “animal”
in the sentence gives it really quite an odd -- and
potentially-very-misleading? -- sound). But notice a crucial difference between
that sentence ((1^)) and (1): “The
animal horse is an animal” is a sentence that could only be intelligibly used
to define “animal” in conjunction with other similar utterances. One might say
to a child, “And so is the dog, and the lizard; but not the seaweed, nor the
Venus fly trap.” But it’s different with concepts. One can’t give other
examples: just because everything
that one could name would, in a way, be a concept. “Object” is a concept, as is
“concept”, and “horse”, and “everything”, and so on... (And -- and here is
Frege’s point again -- just as we could say that everything that one could name is a concept, so one could say that nothing is! (This is the
inevitable-misfiring of attempts to refer to concepts.) Which of these two
things one says -- whether one says that in a way “x is a concept” is true for any value of x, or that in a way “x is
an object” is true for any value of x
-- will depend on one’s elucidatory purpose; Frege himself emphasised the
latter.) So: “The concept horse is a
concept” is not relevantly analogous
to “The animal horse is an animal”, even if -- or rather, especially if -- we
allow the latter to stand as ordinary and sensical. As explicated above, one
should not fall into the illusion of thinking that one can explain the word
“concept” ... to someone who doesn’t already understand what it means. For no
‘contrast-class’ to “concept” can in any ordinary sense be exemplified.
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), as either plain nonsense or
at best as plain nonsense which can function for us as an elucidation,[8] 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 carefully that “The concept horse
is not a concept” (or similarly, even, “Concepts are not concepts”!) too may be elucidatory nonsense -- Frege himself used this example, to draw
our attention to the ‘objecthood’ of concepts, when they are predicated of.[9] 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.” [10] Nonsense-sentences
do not stand in logical relations to each other, not even if they ‘appear’
to blatantly contradict one another![11]
Let
us now review (2) through (5), with which we began, and which led -- which
apparently lead -- to Russell’s Paradox:
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.
I
hope it is now obvious what my thought is. If we apply Frege’s own rigorous
thinking about concepts (and elucidation, and nonsense) rigorously to thinking
about classes -- and surely to do so is to do nothing more than ensure that we
are not falling into philosophical error(s) in our thinking about classes,
either -- 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, and in a very attenuated sense) as elucidations (We could perhaps imagine (3) being uttered
as a potentially-illuminating grammatical joke, by a teacher, for example). But
elucidations are not truth-evaluable (and are not in -- are not parts of --
Frege’s symbolisms, unless we give up
the usual view that every statement in one of Frege’s symbolisms must be a
proper, truth-evaluable statement...on which possibility, see below.).[12] Thus they do not provide us with truths that can
stated;[13] 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 (we might say) charitably, was giving
us elucidations of how to avoid misunderstanding the logic of our language and
the logic 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.[14] “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 (or at least, surely, in the slightly-‘extended’ 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.
It
is perhaps already evident what my response to this objection will be. I have
already suggested, that no good reason is given us by Frege not to treat (4)
through (6), above, in the same way as (0) and (1). We can understand why Frege
would have found this 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 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 sensical as (e.g.) the ‘statement’, “The concept horse is a concept” (or its ‘opposite’,
“The concept horse is not a
concept”). We ought not (I suggest)
to hold on to the usual view that every ‘statement’ in one of Frege’s
symbolisms must be a proper, truth-evaluable statement. To say it again: what
Frege was actually doing, when read charitably (he, unfortunately, not being
very clear at this moment in his texts about what was entailed by his own
methods), was giving elucidations on
how to avoid misunderstanding the logic of our language and of arithmetic. Some
of these would-be elucidations, and some other nonsenses, frame (e.g.) the Begriffsschrift, some are even to be
found within it. So there can be nonsenses within the Begriffsschrift! So what?[15] We might here compare -- and this is very
important to my own view of the situation -- some words of Wittgenstein’s:
“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”?” [16]
Wittgenstein did not think it would be compulsory
for them -- and of course, ‘they’ are us
-- to do so. 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 in fact the very idea of providing such a foundation
is an absurdity. Frege misunderstood what he was (necessarily, willy-nilly)
about in the production (and consideration) of the Begriffsschrift -- we need to re-read what he was about,
‘charitably’, as I have put it; and, providing we do so, we can hold on to what
is useful in Frege, to his real logical achievements of insight.
Wittgenstein
put this crucial 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 this purpose?”
[17] 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. (It is notable that Wittgenstein uses here the expression “Frege’s
logic”. If he had written, say, “Frege’s philosophy”, we might think that he
was simply making the (trivial) point that Frege’s entire lifework is not
rendered philosophically vacuous by the implosion of Logicism. His actual
wording suggests that he thought something stronger, something which I am
recommending we think: that a Begriffsschrift
may turn out to be useful in one’s conceptual thinking (and pedagogically),
even after Frege’s own motivation for it has disintegrated.)
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 generated from them. They are not truth-evaluable statements from which other
statements can be derived. Again, they have no logical -- nor even any
self-evident analogical -- relations with other statements. Or better still:
they have no logical relations with statements. Full stop. 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, an irrelevant isolated object. Unless and until it actually causes
problems in the application of the Begriffsschrift,
it can simply be ignored.[18]
‘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 may be 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 (nor am I sure I actually would want to, for more
or less later Wittgensteinian reasons). 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?
[Expand and reintegrate:]
‘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
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 the
above. 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 may 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.
Now
unfortunately for us all, perhaps out of his desire for a kind of argumentative
and foundational purity (of the kind that Russell and Whitehead also sought to
offer, in Principia Mathematica),
Frege himself did not see or at any rate would not accept what I have been
arguing above, and indeed arguably moved somewhat in the opposite direction in
some of his later work.[19] Frege’s periodic or partial realization that
there can be no such thing as speaking -- enunciating -- the form of our
language, and that elucidation must suffice, thus did not carry over to a
realization that the very idea of grounding our concepts -- the very idea of
providing a foundation for mathematics, say -- is itself an absurdity, a nonsense.
(And if one sees clearly that this very idea is nonsensical -- is not in fact
even well-described as an idea -- then one will not be tempted to enunciate the
‘opposite’ of it. The opposite of nonsense is nonsense. Anti-Logicism is as
utterly absurd as Logicism. There was no great task there for Gödel to carry
out; and so, as I elaborate a little below, the most famous ‘task’ he in fact
carried out -- the proof of the Incompleteness result -- was in the end, from the point of view of assessing its philosophical
interest/implications, merely a
misleading production of further verbiage.)
Frege thus unfortunately responded to Russell’s Paradox as a potentially
fatal counter-example to his own system; “unfortunately”, because Frege thus
did not realize, did not see clearly, what was available to see clearly: 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; he realized
clearly, at his best, that Philosophy is in the grip of a terrible
self-deception 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.[20] Thus 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.) But it took Wittgenstein to see entirely
clearly what the matter was, and to begin in earnest the difficult process of
persuading and enabling others to see so too:
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, an
arbitrary saying, and thus quite orthogonal to the supposed paradoxical
‘problem’ with Frege’s logic ... and that it (the ‘Theory of Types’) was a fortiori unnecessary to a proper
(understanding of) logic;[21]
to see
(more widely) how Logicism itself is
in any case an absurd project, and an unnecessary
one;
and
to see (in his notorious and unjustly much-maligned Remarks on the Foundations of Mathematics) that Gödel quite failed
to undermine Russell’s and Frege’s logics, when those were thought of outside
the deforming ambition of Logicism! [22] 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 supposed
Logicist foundation for arithmetic concepts.
So,
I am 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 [23] -- 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
of course 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 complete mistake, though an extremely widespread one,[24] 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 it -- 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.[25] 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’.[26] 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. .
To
sum up, and conclude. No calculus could hope to undermine anything, or to
provide a foundation for anything, in the sense in which Russell (and Gödel)
had more or less just these hopes. Russell’s system can be seen as a calculus,
a harmless mathematical ‘game’. In it, there is no undermining of Frege (if
Frege is understood aright, in the sense just described; i.e., if Frege is
understood as applying consistently his own insights, which unfortunately he
did not always do [27] ). Russell does not undermine Frege via the Paradox; nor does he later
effect any undergirding of Frege’s hope, of Logicism, via the Theory of Types. Gödel’s Incompleteness Proof, provided
there are no technical flaws in it, can of course be seen as a strictly
mathematical proof, just proving (arcanely or interestingly, for
mathematicians) whatever it proves. But there can’t be any such thing as its
combatting Frege’s concept-script, or proving that Russell’s system is flawed,
or even that Logicism is false or even incompletable. Logicism is not false or
incompletable, but nonsensical, and unnecessary. (No sense can be made of the
idea of something’s being proven to be incompletable, if no sense could
actually be made in the first place of the idea of its being completable.) And so Gödel’s Proof, at least in the
normal substantive understanding of it which Gödel himself initiated,[28] is equally nonsensical and unnecessary.
Logicism
is almost universally thought these days to be defunct. But all ‘refutations’
and ‘disprovings’ of Logicism, and all alternative ‘foundations’ for
mathematics -- e.g. Intuitionism, Formalism, any form of Platonism,
Conventionalism -- are as absurd as Logicism itself. No more, but also no less!
For to defeat Logicism mathematico-logically, to show its falsity or
incompletability, you have to imagine it as making
sense.as a project. But that is simply absurd, nonsensical! We may also put
this point the other way around: If Logicism can be disproved, then it makes
sense. But if it made sense, then it would be able to resist Gödel etc.: if we could make any sense out of
the idea of founding Arithmetic on Logic, then no mere formal mathematical
proof could defeat the idea.[29] But, as I showed earlier -- as was plain to
Wittgenstein, and could have been plain to Frege had he chosen to heed and
develop his own key insights -- the (‘only’) problem is that no such sense can be made of that ‘idea’. (Any
more than sense can be made of the project of trisecting an angle with ruler
and compass.[30] )
Set
Theory is of course in some ways a wonderful and extremely impressive
intellectual edifice. And Logicism and its opposing systems are, similarly, big
and impressive projects in their own ways, real responses to deep human
disquietudes and wishes: disquietudes such as are induced by the possible
presence of contradiction, wishes such as the quest for certainty.[31] But --
for those who understand what Wittgenstein, building on Frege, understood --
Logicism and Anti-Logicism are nevertheless ultimately equally absurd efforts to ‘formalize’ or systematize the so-called
‘foundations’ of mathematics.[32] Will the twenty-first century understand and
realize in practice the bankruptcy of
Russell and Gödel alike in the philosophy of maths, which the
second half of the twentieth has been so unwilling to acknowledge? If it does,
it may be due in part to the recognition I have attempted to foster in this
essay: of the elements of Frege’s own
thought which point in a different direction to that of Frege’s own
overwhelming Logicist ambition. The most valuable parts of Frege’s thought, I
have suggested, run up against (and overcome) Logicism.
Re-write/integrate:
And it remains only to add, in clarification, that when I say ‘Fregean’ here, I
do not of course mean 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, or 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 ‘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 was should be re-read, and in
the sense of hinting at how the historical process actually would have
developed differently, making that (revisionist, ‘Wittgensteinian’) re-reading
still more plausible, if Frege himself had stuck by the aspect of his thought
which I centrally highlighted.
[1]
I have substantial intellectual
debts, vis-a-vis the writing of this
paper, to J. Guetti, W.Coleman, W.Sharrock, and (especially) to M.Kremer, C.
Diamond and J. Conant (thought not all of these people would endorse even most
of what I am saying). My ideas having taken the particular form they have here
is due to my reading of L. Goldstein’s Clear Thinking and Queer Thinking
(London: Duckworth, 1999); though I suspect that Goldstein will certainly
prefer not to associate himself with mostof my conclusions! Thanks also, in
perhaps a similar vein, to B. Worthington, S. Ferguson, an anonymous referee,
and especially to the audience which heard this paper in a slightly earlier
(and much condensed) form at the annual Wittgenstein Symposium,
Kirchberg-am-Wechsel, Austria, 12-18 August 2001, particularly Phil Hutchinson,
Dan Hutto, Laurence Goldstein and Jim Conant. An earlier, much briefer version
of this paper, entitled “Logicism and Anti-Logicism are both equally bankrupt
and unnecessary”) is forthcoming in the Conference Proceedings from Kirchberg
(to be published by Holder Pichler Tempsky).
[2]
We need not, I think, be concerned here with the complications consequent upon
taking seriously Russell’s ‘no-class theory’ (which regarded classes as logical
fictions) -- because this metaphysical/ontological move does not, I believe,
make a substantive difference vis-a-vis
the logical points I shall be making
concerning concepts, classes, etc. .
[3]
This claim, of the importance of something taking itself as an argument, of the
central importance for instance of self-reference to (our understanding of)
Gödel’s Proof, is implicit in the work for example of both Quine (see especially p.17 of The
Ways of Paradox (Cambridge, MA: Harvard, 1976) and Laurence Goldstein (see
n.1, above), and is explicitly defended in my “There is no such thing as de re self-reference” (forthcoming);
but, excepting rhetorically, it is
actually quite peripheral to the main concerns of the present paper... and so
those who find it unconducive may without hazard ignore it.
[4]
I think a structurally similar argument to mine in this paper could be made
concerning the absurdity of Hilbert’s aims and (thus) the absurdity also of any
effort to disprove him/them. It
might be entitled, “Hilbertism and Anti-Hilbertism alike are both bankrupt and
unnecessary”.
[5]
See e.g. Posthumous Writings (Hermes, Kambartel and Kaulbach (eds.),
Chicago IL: Univ. of Chicago Press, 1979), pp. 119-120, pp.177-8, p.207, and Collected
Papers on Mathematics, Logic and Philosophy (McGuinness (ed.), Oxford,
London: Blackwell, 1984), p.182, p.189.
[6]
The word is Frege’s: see e.g. pp.119-120 of his Posthumous Writings (ibid.). (And so 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 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). (See also n.21, below.))
[7]
This use of the word “misfire” -- in
which the inevitability of the misfiring, and thus the nonsensicality of the
result, is crucial -- I draw directly from Conant, from his “Elucidation and
Nonsense in Frege and early Wittgenstein” (in Read & Crary (eds.), The
New Wittgenstein (London: Routledge, 2000)). (Frege wavered slightly -- as
I do, ‘in sympathy’, in this paper -- as to whether the misfiring was
absolutely inevitable or not -- see again Conant for detail, and David
Cerbone’s “How to do things with wood”, also in Read & Crary. Conant and
Cerbone argue that Wittgenstein very
largely managed throughout his career to overcome such wavering.)
[8]
If further detail be needed concerning how to understand the concepts of
“(plain) nonsense” and “elucidation” hereabouts, and on the circumstances in
which it is tenable to regard nonsense-sentences as elucidatory, it is
available in Cora Diamond’s “Ethics, Imagination and the Method of the Tractatus”
(reprinted in Read and Crary (op.cit.)), especially on p.70. (See also
n.11, below; and Remarks on the Foundations of Mathematics (Cambridge,
MA: MIT, 1978 (1956), revised ed. (‘RFM’)), p.402, in which Wittgenstein
is guardedly willing to consider allowing that there can be something worth calling a ‘language game’ centred upon
elucidations.)
[9]
For detail, see Cora Diamond’s The Realistic Spirit (Cambridge, MA: MIT,
1991), pp.130-1 & p.143; and Tractatus Logico-Philosophicus (London:
Routledge, 1922; henceforth ‘T L-P’) 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 (and developments) between
(and from) Frege, (to) early Wittgenstein, and (to) later Wittgenstein here --
see again Conant’s (2000) for details. The key question when faced with a
potentially-nonsensical sentence, the key criterion for sense, is, What could
this sentence be used to do? I suggest that the ‘germs’ of this criterion can
already be found not only in the Tractatus
but also in Frege.
[10]
“Ethics, Imagination...” (op.cit.), 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. To see the parallelism in
more detail, consult p.89 of Julian Roberts’s The Logic of Reflection
(New Haven, CT: Yale, 1993).
[11]
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; and why the apparent ease of catching the later Wittgenstein
himself in a contradiction is usually of little philosophical interest. One can
make ‘opposite’ grammatical remarks in different circumstances, remarks which would if ‘eternalised’ be in both cases simply plain nonsense. (For detail and examples, see my “Beyond
pluralism, relativism, realism, etc.: Reassessing Peter Winch” (paper given at
the BSA ‘Peter Winch’ Conference, Bristol U., Sept. 8-10 2000), and n.11,
below.) One isn’t reminded of any thing
by Wittgenstein’s reminders; this suggests a strong sense in which they are at
best senseless, and (‘better’!) themselves (akin to) latent nonsense. (In a fuller
presentation, we should go into how this point relates to Wittgenstein’s
marvellously exact, hesitant and tortured style in his later work.)
[12]
Are elucidations not themselves nonsense? Yes. So how am I, and how are
Wittgenstein and my Frege -- the part of Frege I like, especially, a key strand
in the early and mature (not so much the late) Frege -- any better off than (on
my account) Russell, or Gödel, or indeed the Positivists? A question too large
for the present paper, beyond saying that self-concsiousness about one’s
nonsenses is far preferable to lack of same; but a fuller ‘answer’ is
available, in (e.g.) Diamond’s “Throwing away the ladder” (in her ibid.;
cf. also “What does a concept-script do?”, in the same volume, which finds some
real philosophical utility in both the frame and the substance of Frege’s Begriffsschrift); and also in the papers
by Cerbone, Conant and Diamond in Read and Crary (op.cit). After Cerbone and Conant, I am drawing
attention to there being 2 different strands
in Frege, one of which leads in a direction very different to what is usually
supposed to be Frege’s inheritance. (See additionally n.19 & n.14, below.)
[13]
If elucidations are kept radically apart from truth-evaluable propositions, is
one not committing some version of the analytic vs. synthetic distinction? Well, it is true that my writing is
largely out of sympathy with Quine’s, and more in sympathy with those (e.g.
Hacker, Dilman) who question the hegemony of Quine in English-speaking
philosophy of language; but technically I do not need in the present paper to
set out a stall radically opposed to that of Quine, for such Quinean issues are
largely orthogonal to mine. Why? Because ‘elucidations’ in Frege and
Wittgenstein are not, properly, candidates for truth or falsity at all; whereas the analytic vs. synthetic distinction is a
distinction between truths supposedly
arrived at by meaning alone and truths
supposedly arrived at with the aid of the world.
[14]
Though on this, see n.28, below: ultiimately, though this is strictly beyond
the arguments given in the body of this paper, I would wish to raise some
questions concerning the very idea, presupposed by Frege, Russell, Gödel, etc., of ‘mathematical propositions’.
(See also F. Waismann in Wittgenstein and the Vienna Circle (henceforth
‘WWK’; Oxford: Blackwell, 1979), p.240f.)
[15]
As Diamond shows in her “Frege and Nonsense” (in The Realistic Spirit),
Frege’s symbolism is meant to exclude only (some?) misleading appearances, and
thus to get us to see some nonsenses plainly. But there is no nonsense
‘expressible’ in ordinary language which is excluded by a concept-script. For
there is no nonsense literally formed by ‘category-mistakes’; all there is (see
above) when there is nonsense is ... plain nonsense, words in combination to
which we do not succeed in giving any sense. What a concept-script sometimes
enables us to do (see again Diamond’s “What Does a Concept-Script Do?”,
especially p.143 & p.132) is to see that certain alleged ‘philosophically
interesting claims’ cannot be translated into a concept-script-based language at all. The attempt at such translation
helps us to see such ‘statements’ in their full nonsensicality. As explained
further below, I am suggesting that this point holds also for certain
‘statements’ that can apparently be
made in a concept-script. Arguably, a
concept-script helps us see more plainly for instance that ‘statements’ (2)
through (6) (and (6) most obviously of all) are nonsense. It helps us to see
them in their plain nonsensicality -- that’s precisely what it’s good for, and
not a refutation of it! Not something to make it fail (and arithmetic totter)!
Some things may appear in a good Begriffsschrift
itself which are plain nonsense (but it is not so obvious that anything which
is not nonsense can fail to appear in our concept-script, as I suggest in the
main text, below). Some of the same
kinds of nonsenses which are to be found in misleading or systematically
ambiguous sentences of ordinary language get reproduced there -- they are to be
noted, and ignored if they cause
trouble; or, if you like, thrown away.
If
this all seems simply too scandalous, perhaps the following rendition of what I
am up to might be preferable: We could choose to distinguish between two senses
of being ‘in’ the Begriffsschrift.
In one (‘narrow’) sense, something can
only be in the Begriffsschrift if it
is not nonsense. In another (‘wide’) sense, some
of the nonsenses to be found in ordinary language, or logico-mathematical
‘versions’ of them, would be constructible (not ‘expressible’) in the Begriffsschrift. If we adopted this
proposal, we would then speak of some things in the ‘wide’ concept-script
coming to be seen as needing to be excluded from the ‘narrow’ (‘true’?)
concept-script. However, we would then exclude from the Begriffsschrift narrowly construed much of Set Theory. (Though that might not be such a bad idea...) Again, a concept-script yields no special
quasi-metaphysical vantage-point whatsoever; it simply helps to make
perspicuous certain features of our talk. There is in fact no reason why,
viewed aright, our ordinary language
itself cannot be seen as a concept-script. Whereas Logicists wanted to
found maths on logic, and ‘Ideal Language’ theorists wanted to found language
on logic, Wittgenstein drew out the strand in Frege (see also n.19 & n.20,
below) according to which all a concept-script is is, roughly, a useful means
for institutionalising grammatical reminders. As for instance in the following
words he wrote to Ogden, concerning how to translate, how to understand, the Tractatus:
“[T]he propositions of our ordinary language are not in any way logically less correct or less exact or more confused than propositions written
down, say, in Russell’s symbolism or any other “Begriffsschrift”.” (P.50 of Letters
from Ludwig Wittgenstein to C.K. Ogden (London: Routledge, 1973).
[16]
P.204 of RFM (and see also p.205, p.212, pp.395-6). It is not, it should
be noted, quite clear that Frege ever did say quite this. Wittgenstein may (though I myself think he was not,
given the letter of Frege’s texts) have been interpreting Frege a little
uncharitably, a risk I run similarly. I leave the reader to judge -- the quote
which is, to my knowledge, probably closest to the phrasing Wittgenstein gives
here is to be found in the famous letter to Russell, reprinted on p.254 of The
Frege Reader (ed. M. Beaney; Oxford: Blackwell, 1997).
[17]
P.378 of RFM (emphasis mine). This remark, and the ‘affirmative’ reading
of Frege I suggest is implicit in it, leads into one of my main themes here:
that an attentive reader of Wittgenstein’’s later work (e.g. see p
p.267f.
of his 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 very specifically
those things made clearer by their Logicistic moves. A fuller task for another
occasion would be: to bring out in detail 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 (see TLP 6.2f.). Thus he
does not uphold Logicism in the Tractatus
-- and nor does he in his later work condemn the impulses that led to
Logicism and some of the elucidatory impulses which it involved in Frege
especially.
An objection might be raised that, even
if it be concede to me that Wittgenstein has already overcome Logicism in TLP,
nevertheless the crucial element in Wittgenstein’s progressing beyond Frege in
the Tractatus was his giving up of
Frege’s Basic Law 5, whereas I am focussing rather on controversial
applications of Frege’s thought involving ‘elucidations’ and nonsense, and thus
not strictly following either Frege or Wittgenstien. To anticipate my reesponse
(below) to this objection: in
Wittgenstein’s later 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 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 the insights of Frege concerning language and
concepts.
[18] Much as we ignore too the (useless) supposed
self-referential sense of the statement “I am lying”; see p.120 and p.255 of RFM.
(It is worth noting that a serious emphasis on use in one’s philosophy of language avoids the impression, possibly
given by some of my formulations early in this paper, that sentences can be
inspected, in isolation, for their sensicality. No; sentences only mean in a
context, in use. ‘Indexicality’ is, if you like, a vital feature of all (meaningful) sentences.)
[19]
See for instance p.23 of P. Carruthers’s Tractarian Semantics (Oxford:
Blackwell, 1989). (Frege eventually moved away from ‘classical’ Logicism
altogether; but that late part of his
work need not concern us here.)
[20]
For more on this version of Frege as proto-Wittgensteinian, see J. Conant’s
work; and Kelly D. Jolley’s “Frege at Therapy” (paper presented to the ‘Mind
and Society Seminar’, Manchester Metropolitan University, June 6-7 2000). As
hinted above, a reasonable suggestion as to why Frege did not make the further
move here which Wittgenstein did make is that Frege regrettably came to place
less weight on the Context Principle etc.
in his later work (See n.18, above). His Begriffsschrift etc. work is that which, when ‘applied’ and extrapolated in the
manner which I am undertaking in this paper, best yields the complete
deflation of the Logicism vs.
Anti-Logicism debate.
[21]
Unless, of course -- again, a charitable thought -- we try to see the Paradox
as an elucidatory reductio ad absurdum
of the very idea of something’s being a member of itself, and thus of the whole
tendency of classical set theory (see RFM p.330); and try then to see
the ‘Theory of Types’, as Russell quite plainly did not see it, as in turn a
(rather crude) attempt at elucidation, at reminding us of what we must do with
signs if we are not to come up with something useless. (For Wittgenstein’s
severe critique of the very idea of a Theory of Types, see T L-P 3.326 -
3.333 (and Kelly Dean Jolley’s powerful (unpublished) paper on the same topic,
“Logic’s Caretaker”). In these sections of TLP, Wittgenstein’s
fundamental aim, again following a basically
Fregean line of thought, is to make plain that a Theory of Types is
unnecessary for any language (i.e.
for any language which, as any language
does, consists of “legitimately constructed” propositions (cf. T L-P
5.4733, & 5.5563, remarks which not incidentally make obvious how strongly
the Tractatus anticipates Wittgenstein’s later work), and which, as any
language does, -- and this comes to the same thing -- stands in ‘logical
relations’ to, roughly, a concept-script). For his dismissal of the alleged
foundational role of ‘classes’, see e.g. T L-P 4.1272 and 6.031, and
also RFM pp.401-3. For Wittgenstein’s suggestions as to how to react to
the contradiction in a manner other than that of constructing a ‘Theory of
Types’, see RFM pp.217-8, p.376, & p.410. A full investigation of the grammar(s) of
‘contradiction’ is a task for another paper; but it is worth noting the
fairly-extensive investigation undertaken by Laurence Goldstein, in his Clear
thinking and queer thinking (op.cit.), on pp.147-160. Goldstein emphasizes
that Wittgenstein in his work on maths emphasized that contradictions are not
best-construed as statements of any kind, and that they can in some
circumstances be quite harmless. ‘Superstitious’ fear of contradiction may
largely result from thinking of contradictions as a kind of statement, and from
thinking of statements’s meanings as literally being formed compositionally or
additively: Wittgenstein, after Frege, rejects the latter notion, also. (This,
of course, is they key fault-line between Frege and Wittgenstein on the one
hand and Russell and Moore on the other. Russell and Moore seem to have won the
battle over the unity of the proposition in Analytic philosophy -- part of the
thrust of my work here is to try to ensure that they lose the war. To his
credit, Russell was quite often relatively honest about some of the deep
difficulties facing both the Theory of Types and anti-‘propositional-wholism’
-- see, e.g., pp.162 and 166-7 of Ray Monk’s Bertrand Russell: the Spirit of
Solitude (London: Jonathan Cape, 1996).)
[22]
I trust that this makes clear what mystifies some of Wittgenstein’s critics on
Gödel: why didn’t he welcome Gödel? Why not welcome the ‘proof’ that Logicism
was wrong, as he (Wittgenstein) himself had in a sense long maintained? But to
have done so would have been little better than welcoming Chomsky’s ‘disproof’
of Skinner, or (closer to home) Logical Positivism’s ‘refutation’ of
metaphysics. The straight opposite of nonsense is nonsense; Wittgenstein had
(less than) no need of Gödel.
(Unless, again, we were to attempt (over-!)charitably to re-read Gödel: as simply spelling out -- drawing out --
how the attempt to found mathematics
(whether logicistically or formalistically or what-have-you) leads to paradox.)
As for the objection that Wittgenstein
simply failed to understand Gödel’s maths; this claim is to my mind effectively
rebutted in Juliet Floyd’s (and also in Stuart Shanker’s) work.
[23]
If the reader is losing a grip on the force of the term “elucidation”, please
consult n.7 & n.11, above.
[24]
See for example the “Introduction” to Putnam and Benacerraf’s influential
collection, Philosophy of Mathematics: Selected readings (2nd ed.,
Cambridge: C.U.P., 1983), p.16. (A careful reading of T L-P 6.2f. indicates that in fact the
similarities between Logicism and Wittgenstein’s ‘early view’ are mostly only superficial: One may think of maths
as in certain respects analogous to logic, but one can hardly think of it as
the same as it nor as reducible to or foundable on it -- one can hardly be a
Logicist -- if, like Wittgenstein in T L-P, one thinks that there are no
logical constants, etc. . (See also WWK,
pp.218-9.))
[25]
And indeed before the Tractatus:
see e.g. n.15 of M.McGinn’s Between metaphysics and nonsense: Elucidation in
Wittgenstein’s Tractatus (Philosophical
Quarterly 49:197 (Oct. ‘99), pp.491-513). (As will be evident, my
Wittgenstein is both strictly therapeutic (like Diamond’s and Conant’s, unlike
McGinn’s) and focussed on elucidation
(like both Conant and McGinn; though I fear that McGinn has not understood that
elucidations are not in any way assertions nor (even tautologous) truths (See
again n.11 & n.7, above).))
It might be objected that I have got
Wittgenstein’s move beyond Frege in the Tractatus
wrong; ‘Isn’t what Wittgenstein did in the Tractatus essentially to take Frege
on board, but abandon Axiom 5 (which
caused all the trouble) and Identity, and add the truth-tables? Isn’t that a radical -- and wholly necessary --
revision of Frege’s logic?’ This objection raises some issues concerning the
interpretation of Wittgenstein much too large to be settled here; but a brief
response would be to say that, as I have suggested in the main text, above,
with quotations from RFM, that, while one could say what this objection say, one could equally say, while
still following Wittgenstein, what I have tried to say: namely, that one can
just keep ‘Frege’s system’ of logic as is, and then apply its results with
sensitivity. I.e. When it generates nonsenses, don’t get too worried by them.
It can arguably fulfill a certain logico-conceptual purpose perfectly well as
is.
There is no interesting or useful system of logic (or maths) which is
invulnerable to its rules being applied so as to generate falsities or
absurdities. We simply do sidestep ‘the contradictions’, ‘the paradoxes’,
except when there is a special reason not
to. (One ought if at all possible to understand every bit of logic -- and every
bit of concept-script -- which one chooses todevelop, but that is all.) So one shouldn’t get too fussed by (e.g.)
paradoxes, and try to expurgate them once and for all. To do so is just
pointless. (Again, I am implying that one doesn’t need Gödel to see any of
this.)
[26]
There is a risk here of appearing to court what Jim Conant (in his (op.cit.)) has called the (very popular)
‘ineffabilist’ reading of (the early) Wittgenstein. I show how to avoid this
risk in my “Meaningful Consequences” (jt. with J. Guetti, Philosophical Forum, Winter 1999). Provided this risk, a risk which
Frege is continually in severe danger of, of explicitly stating what one has
oneself ruled out as unstatable ... provided
that this risk is avoided, then it is safe to say that the avoidance of
‘effing’ the ineffable is invariably to be preferred to the (e.g. Russellian)
theoreticist option of quasi-positivistically trying to state the unstatable.
(For a similar case, see my “The Unstatability of Kripkean Scepticisms” (Philosophical Papers XXIV: 1 (1995)).) I am suggesting that Frege is right to
emphasize elucidation and ‘unstatability’ over the (fantasized) theorization,
e.g. á
la Kerry, of ‘the foundations of logic’; and that he should have
extended this compunction full-bloodedly to ‘the foundations of arithmetic’.
[27]
See n.18, n.19, & n.13, above.
[28]
See e.g. p.19 of and p.42 of "On formally undecidable propositions of
'Principia Mathematica' and related systems I" (1931; reprinted in [& page numbers from] Gödel's
Theorem in focus (ed. Shanker; Beckenham: Croom Helm, 1988)). Again, it is
crucial to be clear, with Floyd, that Wittgenstein had NO objection to Gödel, given a construal of his work simply as
maths. So, we can say quite happily that Wittgenstein had NO objections to
Gödel’s proofs -- only to their mode of presentation and reception.
[29]
To go beyond what I have argued in the body of this paper, one might suggest
that no mathematico-logical tricks will be felt to be needed hereabouts --
either to ‘found’ maths on logic, or to ‘disprove’ the legitimacy of such
founding -- once one tries looking at maths, for mental-cramp-reducing
purposes, roughly as grammar, ‘rather
than’ as a body of statements/ truths/propositions. One will then
see how very different mathematical ‘statements’ are from (other)
statements. (For explication, see e.g.
p.90 & pp.162-4 of RFM, p.138ff of Goldstein (op.cit.), and
Baker & Hacker’s Wittgenstein: Rules, grammar and necessity (Oxford:
Blackwell, 1985, passim, especially
p.288 & p.6. Note that Baker and
Hacker’s account -- of which I am endorsing the broad sweep, not necessarily
the ‘details’ (such as their rather excessive liking for a thesis of ‘the autonomy of grammar’, and their questionable
‘meta-philosophical’ presuppositions and practices) -- carefully distinguishes
Wittgenstein’s own view from any conventional form of Conventionalism (see
p.338ff.), as well as from Logicism itself. (What Baker and Hacker do not bring
out so well is yet a further move in the therapeutic dialectic: Wittgenstein
emphasises, for example on pp.40-43 of Lectures on the Foundations of
Mathematics (‘LFM’; Hassocks, Sussex: Harvester, ‘76), the sense in
which mathematics is a set of techniques,
or a set of actions of calculation,
rather than a set of linguistic items, no
matter of what kind. But we have no room here to explore this further).).
Baker and Hacker pointout further that, if one thinks of arithmetic etc. basically as (akin to) grammar, then
one will no longer be inclined to place the question of the ‘ontological
status’ of numbers centrally. And then, unlike Frege, Russell, etc., one will not be nearly so
interested in the potential question of logicising arithmetic -- via
‘logicising’ numbers into sets -- in
the first place. But to go more fully into this point would require that we
take seriously also the huge question-mark which the von Neumann vs. Zermelo etc. ‘debate’ over what sets the numbers supposedly are (see e.g.
Paul Benacerraf’s “What numbers could not be” , reprinted in Benacerraf and
Putnam, op.cit.) places -- more or less independently of the present
discussion -- over the very intelligibility of Logicism... and that is clearly
a task for another occasion.) To take
this thought about ‘maths as grammar’ seriously -- this thought whose
consequence is that mathematical
‘propositions/statements’ are in fact only
quite misleadingly described as such
-- involves a further step away from the ‘mainstream’ than does Wittgenstein’s specific
point (in ‘defence’ of Russell) that one must strictly distinguish statements
within a mathematical system from those outside the system, and his related
point (see RFM pp.119-122) that an option that Gödelians arbitrarily
ignore is that of giving up the interpretation
of the Gödel sentence as “I am unprovable”.
[30]
For detail, see J. Floyd’s paper on Gödel, the concept of proof, and the
trisection of the angle, “Wittgenstein, Mathematics and Philosophy”, in Read and Crary, (op.cit.) .
[31]
These disquietudes and wishes are arguably all of a piece. Those many who
follow Wittgenstein on epistemological questions should ask themselves why they
do not join those few who follow him in the philosophy of maths: “Now, what is it for us not to know our way
about in a calculus? // We went sleepwalking along the road between
abysses.___But even if we now say: “now we are awake”, ___can we be certain
that we shall not wake up one day? (And
then say:___so we were asleep again.) // Can we be certain that there are not
abysses now that we do not see?... // [I]s it wrong to say: “Well, I shall
go on. If I see a contradiction, then will be the time to do something about
it.”?___Is that: not really doing mathematics? Why should that not be
calculating? I travel this road untroubled; if I should come to a precipice I
shall try to turn round. Is that not ‘travelling’?” [RFM, pp.205, 212]
The
fear of contradiction, common to Logicists and Anti-Logicists, is very much
like the fear of uncertainty which characterizes so much Modern Epistemology.
(See also n.20, above.)