FOM: And Another Thing

[Dean Buckner]

The history of the predicate calculus is closely bound up with early work on
the foundations of mathematics.  It was a notation and a method, explicitly
designed to provide an adequate basis for the theory of number.

In this posting I want to question whether it really does provide a basis of
this sort.  I want to show that there are simple numerical propositions,
easily expressed in statements of natural language, whose semantics cannot
be captured by any formal system with predication as its basis.

This is a long (ish) one, please get a cup of tea or coffee now.  Note all
cc's are blind cc's.

1.  I start (as usual) with fiction.  There are three ways of introducing
characters in a story.  (i) A character may be introduced without specifying
whether s/he is the same or different from a character previously introduced
to the story.  The usual way of doing this, in English, is to use the
indefinite article.  So "A man was standing at the fountain" .  (ii) A
character may be explicitly introduced as someone different from any of the
other characters  in the story.  The usual way of doing this is to use
"another" or "some other".  So "Another man was standing at the fountain"
(iii)  The character may be specified as one of those already introduced.
So "the man was standing at the fountain".

2.  Hence we can speak of type, (i) (ii) or (iii) relations between
propositions expressed by different sentences in the story.

    (i)  That there was an F ... that an F was G.
    (ii)  That there was an F ... that another F was G
    (iii) That there was an F ... that the F was G.

Type (i) propositions combine without specifying whether there are one or
two individuals.  Type (ii) propositions combine to specify that there are
two individuals.  Type (iii) propositions combine to specify that there is
only one individual in question.

By "combine" I mean we can take the individual propositional specifications
made by separate "that" clauses and combine them into a single
specification, in a single "that" clause.  Thus

    (i')  That there was an F and an F was G
    (ii')  That there was an F and another F was G
    (iii') That there was an F and the F was G.

3.  Note (important) that propositions expressed by grammatically indefinite
sentences can only combine with propositions expressed by sentences in some
previous part of the story, to form type (i) relations.  For example

In 1816, a young commercial traveller, a regular client at the Café David,
got drunk between eleven o'clock and midnight (Balzac).

There can be no character in any previous part of the story, such as we must
necessary conclude an identity between this character and the young
commercial travellerone and the other.  Such an identity may be intimated,
but is not part of the sense.  Concerning any already-introduced character
X, we can always deny that the commercial traveller was X.  By contrast, we
cannot say that there was a man standing at a fountain, that _the man_ had a
drink, and that the man who had a drink was different from the man who was
standing at the fountain.

By "previous part of the story" I mean a part which, in the conventional
order
of reading, should be read before the part to which it is previous.

3.  Consider

    (S1) A unicorn was standing by a pool.
    (S2) Another unicorn was by the pool.

The propositions expressed by the two sentences have a type (ii) relation.
To understand what the two sentences say, as read in the conventional order
(first sentence the second), you have to understand that there are two
unicorns in question.  (I don't mean there have to be unicorns or
"intensional objects" or whatever, I mean you have to understand that the
story _says_ this).

But no proposition expressed before the first sentence can bear a type (ii)
or type (iii) relation to the proposition expressed by the first sentence.

4.  Hence (simple conclusion), the proposition expressed by the second
sentence cannot be expressed at any part of the story before this.  That is,
a proposition (a meaning) that cannot in principle be expressed, by any
sentence of any language, until after the first of the sentences above.

This has proved a little difficult for many of my correspondents, who buy
this Fregean idea that "With a few syllables [language] can express an
incalculable number of thoughts, so that even if a thought has been grasped
by an inhabitant of
Earth for the very first time, a form of words can be found which in which
it will be understood by someone else to whom it is entirely new."

5.  Type (ii) relations are the way we build number series.
For example, we can continue the story above by saying that there was a
third unicorn at the pool, i.e. one different from the one said to exist by
the first and from the one said to exist by the second.  I.e. we can
introduce a third proposition that has type (ii) relations with the first
and second, then a fourth that has the same relation to the first second and
third, "and so on".   In general, n such propositions will assert that n
unicorns were by the pool.

Note also that at any point, say n+1, we can assert the proposition  "there
were no more unicorns by the pool".  This is of course the negation of
"there was another unicorn by the pool", which would have asserted the
existence of the n+1th unicorn.  In other words, for any n such
propositions, there is an n+1 th which we are free to assert or deny.  I.e
"there is another thing" must always have a meaning, _even if it is false_.
We are thus guaranteed that, however far we continue the series, it has a
meaning.  The proposition that there are an infinite number of unicorns
would of course have to consist of an infinite number of propositions of the
series, and it's not entirely certain what the meaning of these, fitted
together, would be!  (A note for those who corresponded offline on the
subject of infinity).

6.  This raises questions about how mathematical thinking is "founded",
since it shows there are certain arithmetical propositions that cannot be
expressed using the existing stock of predicates within a language, or
indeed anything which is genuinely a predicate.  Despite the incalculable
number of thoughts we can express using the store of proper names and
predicates with which our language is furnished, given even the capacity for
defining new ones, there are thoughts that we cannot express in this way.
This is not a matter of the deficiency of "stock", as with like English in
the fifteenth century, when they had to import Latin words to make up for
the crudeness of Anglo-Saxon.  I am arguing that are thoughts which we
cannot _in principle_ express in language, however we define the meaning of
the terms we use.

Any theory of number that bases it on predicates or concepts is therefore
bound to be flawed

7.  Take for example the Fregean idea that is standard to all discussions of
the subject, that number is something attached to predicates.  Frege
discusses this in S. 45-54 of the Foundations of Arithmetic, culminating in
S. 55 where he says that the number (n+1) belongs to the concept F if there
is an object *a* falling under F such that the number n belongs to the
concept "falling under F, but not [identical with] a".

This cannot be correct.  Translate it to the unicorn example.  The first
difficulty (for Fregeans) is that there are no unicorns, the story merely
says there are.  But let's pretend for the sake of argument that there are
such things.  Then we have to suppose

(a) that there is a concept F under which both unicorns fall.  Perhaps
"being one of the the two unicorns mentioned by (S1) and (S2)", since there
is no other property that uniquely characterises the two creatures.
(b) that *a* is the second unicorn
(c) There is a concept F' "falling under F but not a".  Perhaps "one of the
2 unicorns, but not the second"

This makes sense, sort of, and of course "one of the 2 unicorns, but not the
second", fits only one object, whereas F fits two.  However, there is the
difficulty, central to my whole argument, that if being non-F is asserted of
any unicorn at any
previous point in the story, the resulting proposition would bear a type
(ii) relation with either of the propositions expressed by (S1), (S2), when
of course there can be no such proposition.  The sentence (S1), "A unicorn
was standing by a pool" does not say which unicorn was in question, it's not
part of its sense that the unicorn is the same or different from any
creature talked about previously in the story.

8.  In consequence, it cannot be that "a statement of number contains an
assertion about a concept".  For it can be proved that any such statement,
if it has the form "there are n F's", can be decomposed into separate
propsoitions which are also statements of number, and which bear internal or
essential relations to each other, which they cannot bear to any statement
which has a merely general or predicative meaning.

9.  But the idea that statements of number contain assertions about
concepts is implicit to every foundational theory of number that I have so
far seen.

10.  With the possible exception of the formalist-looking theory briefly
sketched by Wittgenstein in Tractatus 6.24 - 6.241.  "The method by which
mathematics arrives at its equations is the method of substitution".

This example, and line of argument is adapted from my earlier work on
fiction, where I have been exploring how stories define numbers of
characters.


Dean Buckner
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London, SW15 1PL
ENGLAND

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