FOM: Response to adverse comment on the Arche project
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Wed Mar 14 15:35:06 EST 2001
On Sat, 10 Mar 2001, Roger Bishop Jones wrote:
> I have two critical remarks to make in relation to the Archie project
> recently mentioned on fom.
>
> Firstly, in its statement of "the research problem"
> http://www.st-and.ac.uk/~arche/ahrbresearch.shtml
> the Archie website grossly distorts the history of its subject by suggesting
> that logicism received absolutely no further support from philosophers after
> its abandonment by Frege.
>
> In fact, the word logicism does not seem to have been applied to the
> philosophy of mathematics until decades later (1931)
Would it not be irrelevant whether the term "logicism" was in use only at
a later date? The question is rather whether any doctrine recognizably
logicist in nature was held by any philosophers in the immediate aftermath
of the discovery that Frege's class theory was inconsistent.
> Secondly I observe that it seems to me a sad comment on contemporary
> Philosophy of Mathematics that philosophers should consider that the status
> of the axiom of infinity is materially changed when it is disguised as
> "Hume's Principle".
> I doubt very much that Frege would have taken refuge in such equivocation.
For a disguise to be effective, it must be possible to attribute to the
one devising the disguise the intention to mislead his audience as to the
true nature of what is (supposedly) being disguised. But there can be no
doubt that Wright and other thinkers interested in neo-logicism have no
such intention. For them, it is obvious---and obvious that it will be
obvious to any other competent and interested party---that Hume's
Principle packs even more ontological punch than a standard axiom of
infinity. (An axiom of infinity usually states only that a countable
infinity exists, and it is left to the powerset axiom to "blow it up" to
get even higher infinities, via Cantor's Theorem.)
Hume's Principle immediately implies---very ambitiously---that for every
property (or open sentence) F(x) there exists *the number of Fs*, denoted
(say) by #xF(x). The axiom of infinity is hardly being *disguised*
here---rather, it is being shouted from the rooftops, for all to hear.
What, then, would be interesting about the neo-logicist program?
The answer lies precisely in the web-page to which Roger Jones refers.
Note how carefully the philosophical claims therein are formulated; no one
in this program would be claiming that one can miraculously get something
for nothing. It is simply a proposed study of logical and conceptual
*dependencies*, involving a set of premisses (including Hume's Principle)
which *of course* embody strong existence claims. (For the reader's
convenience I shall attach a relevant exerpt from the web-page in question
at the end of this posting.)
While I would not agree, therefore, with Roger Jones's rather sarcastic
assessment of the prospects of this kind of neo-logicism, I do have my own
---and different---criticisms to level against it. These are criticisms
from the standpoint of the "constructive logicist" (for which, see my book
"Anti-Realism and Logic", Clarendon Press, Oxford, 1987).
Whereas the Wrightian neo-logicist works from the starting point of Hume's
Principle:
#xF(x)=#yG(y) iff there is a 1-1 mapping of the Fs onto the Gs,
the constructive logicist proceeds in a much more cautious fashion,
ontologically speaking. Since the professed concern is just to recover the
laws of *arithmetic* (the theory of the natural numbers) from the
proposed logicist basis, why not do so---"logically"---from a basis that
affords one the natural numbers and not necessarily any infinite numbers?
Thus the logicist takes as an analytic constraint not Hume's Principle,
but the following Schema N:
Schema N
#xF(x)=[numeral n] iff there are exactly n Fs.
Here, [numeral n] is the term of Peano arithmetic built up from the symbol
0 and n occurrences of the successor symbol: ssss...s0. Note that the
righthand side of the biconditional uses "n" only adjectivally. Thus, for
example, "There are exactly 2 Fs" is the sentence
ExEy(-x=y & Fx & Fy & (z)(Fz -> (z=x v z=y)),
which involves no reference to the number 2. Such reference would occur
only on the lefthand side, and indeed in two ways: once via the
abstractive term #xF(x), in which the variable-binding term-forming
operator #x is at work; and once via the numeral ss0, which is the
mathematically canonical presentation of the number 2. Schema N serves
thus as a conceptual control on number-talk, linking the progression of
natural numbers 0, 1, 2, ... with predicates whose extensions have the
corresponding (finite) numerosities.
Unlike Hume's Principle, which serves the Wrightian neo-logicist as an
axiomatic *starting point*, Schema N serves the constructive logicist as a
conceptual constraint, or adequacy conditions, on his theory of numbers.
In this regard, it is like Tarski's Schema T for a theory of truth.
The axiomatic starting points for the constructive logicist are principles
that secure the existence of the number 0 (outright) and then
(conditionally) the existence of the successor of any number that exists.
But this is not just a matter of simply postulating that 0 exists and that
given any number m its successor s(m) exists. Rather, a link is made to
the operation of numerical abstraction. First there is a rule that can be
stated in natural deduction format as follows:
___(i)
Fa
:
*
________(i)
#xF(x)=0
This says that if one can derive absurdity (*) from the parametric
assumption Fa ("arbitrary object a has property F) then one can discharge
that assumption and infer that that number of Fs is 0. This immediately
yields the theorem that #x(-x=x)=0.
Secondly, there is a rule that says that if one has shown that t is the
number of Fs, and can exhibit a 1-1 mapping of the Fs onto all but one of
the Gs, then the number of Gs is s(t).
Thirdly, there is an inductive scheme of definition for the concept N(x),
"x is a natural number". The Peano-Dedekind axioms (which are to be
derived on this approach) will all be formulated with N-restricted
quantifiers in the obvious way.
So the constructive logicist's axiomatic principles take the form of
introduction rules (balanced by corresponding elimination rules) governing
the symbols 0, s and N. See "Anti-Realism and Logic" for details. Those
rules are devised both with an eye to considerations of harmony within a
theory of logico-mathematical operators, and with an eye to satisfaction
of the adequacy condition involving Schema N. That condition is that all
instances of Schema N should be theorems of the theory; as indeed they
are.
The upshot is that the constructive logicist gets what is wanted by the
Wrightian neo-logicist, but in an ontologically much more parsimonious
way. Only the natural numbers get into the picture. Omega does not.
Moreover, the derivations of the Peano-Dedekind axioms can be carried out
in a weakly second-order intuitionistic relevant logic.
The constructive logicist makes no claims concerning the logical
derivability of axiomatic principles concerning the real numbers; this is
where the extra ontological commitments of Wrightian neo-logicism might
make a difference.
To return to my earlier defence of Wrightian neo-logicism against the
rather different---and in my view, misguided---criticism by Roger Jones, I
now append the St.Andrews statement, and would draw the reader's attention
to its rather careful and hedged claims about the significance of what is
being undertaken.
Neil Tennant
__________________________________
EXTRACT FROM THE ST. ANDREWS WEB-PAGE FOR THE ARCHE PROJECT ON NEO-LOGICISM
The neo-Fregean thesis about arithmetic is that knowledge of the
basic arithmetical laws (essentially, the Dedekind-Peano axioms)---and hence
of the existence of a range of objects which satisfy them---may be based
a priori on "Hume's principle", that (informally)
The number of F's is the same as the number of G's just if the F's and
G's can be put into one-to-one correspondence.
More specifically, the thesis involves four ingredient claims:
(i) That the language of higher-order logic plus the
cardinality-operator, "the number of...", as introduced by Hume's
principle, provides a sufficient definitional basis for an expression
of the axioms of arithmetic;
(ii) That those axioms, so expressed, may be derived within a
consistent system comprising Hume's principle and a standard
higher-order logic;
(iii) That someone who understood a higher-order language to which the
cardinality operator was to be added would learn, on receiving Hume's
principle as an implicit definition of that operator, everything
needed to understand any of the new statements that are then
expressible.
(iii) Finally and crucially, that Hume's principle may be laid down
without significant epistemological obligation beyond that of any
implicit definition: that the principle may simply be stipulated as
explanatory of the meaning of statements of numerical identity, and
that---beyond the issue of the satisfaction of the truth-conditions
thereby laid down for such statements---no competent demand arises for
an independent assurance that there are objects whose conditions of
identity are as it stipulates.
Claims (i) and (iii) concern the epistemology of the meaning of
arithmetical statements, while (ii) and (iv) concern the recognition
of their truth. Claims (i) and (ii) are proven: Frege himself
established (i) and the first explicit demonstration of (ii) was given
in [Wright's "Frege's Conception of Numbers as Logical Objects"]. So
the neo-Fregean thesis about arithmetic turns on the informal
philosophical theses, (iii) and (iv). If all four theses can be
sustained, then arithmetical knowledge emerges (not indeed as of a
species with knowledge of pure logic, but) as derivable from logic and
implicit definition --- a sufficient basis to sustain Frege's claim of
the analyticity of arithmetic.
END OF EXTRACT
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