FOM: On the linguistic frame of mathematical reasoning
Walter Felscher
walter.felscher at uni-tuebingen.de
Sun Jan 11 17:14:38 EST 1998
There have, in some contributions to FOM, been occasional
complaints about the mathematical public's disinterest,
particularly in formal languages as used for the analysis of
logical arguments. It is the purpose of the following
considerations to (1) point out some historical background
encouraging such disinterest, and to (2) draw attention to
some situations where basic mathematical arguments are
intimately connected with the linguistic environment in
which they are performed.
1. The elimination of the linguistic frame as a tendency in
20th century mathematics
Reviewing the development of mathematics during our century,
there can be noticed, far below the important discoveries, a
certain tendency to liberate the explanations of mathematical
concepts from the burdens of notation, and to eliminate from
their definitions such references which do not belong to the
mathematical objects themselves, but to the language employed
to speak about them. For instance,
Example 1 : Polynomials, explained as written (or printed)
expressions, yet with coefficients from an arbitrarily
large ring R, may easily appear as bastards, bred from
concrete graphics and abstract concepts (e.g. real numbers).
It seems that it first was L.E.Dickson in 1923 ("Algebras and
their Arithmetics", para. 111) who avoided the reference
to written (or "formal") expressions by speaking of finite
sequences of elements of R instead.
Example 2 : In textbooks on linear algebra, only 50 years
ago, vectors v written as columns [for coordinates of
points] were distinguished from vectors w' written as rows
[for coordinates of hyperplanes], and a witty, ingenous
process of transposition was required to write down a
quadratic form such as vAw' . Finally, a small red book
from the Ann.of Math.Studies, destined to illustrate the
finite dimensional case of Hilbert space, taught how to
think coordinate-free, and within a few years it made a
career from a research monograph to an undergraduate text.
Example 3 : Also 50 years ago, tensors still appeared as
objects with multiple indices, transformed under change of
coordinates "in the manner of coordinates of a tensor".
Similarly, for the "formal" use of differential forms etc.
The first book to give a conceptual account of these basic
notions of differential geometry seems to have been
Chevalley's Theory of Lie Groups I of 1946 .
The tendency illustrated here then may be said to amount to
the abandonment of explicit linguistic descriptions of what
mathematicians work with, and to their replacement by
conceptual descriptions of mathematical objects as mental
entities.
Not that there then was a universal agreement about what
these objects were, but in practice conceptual descriptions
meant descriptions in the terminology of sets, as first made
popular in algebra by Dedekind, Weber and Hilbert (and with
additional invigoration drawn from the breathtaking discoveries
of Cantor's transfinite set theory). In so far, as the
mathematician's education led him away from linguistic
descriptions, it particularly did not invite him to the
study of formal languages; and the success of the use of set
terminology was an essential contributing factor in that
many constructions that logicians might describe with
formulae are replaced by set manipulations. They seem
"more concrete"
as Mr.Scott formulated it here only a week ago.
2. The seams between language and mathematics
Every mathematician, of course, believes that his proofs
proceed by the laws of logic. To know more about these
laws, and about their relationship with the foundations of
mathematics, may appear as a challenging mathematical
occupation - such as knowing more about Hill's theory of the
Moon - but it is a different matter to convince a
mathematician that such knowledge may actually be
indispensable. It is, again, proof of the power of set
terminology that the places are rare where the stitches
become visible of the seams along which language and
mathematics are sewn together - but it are they that should
be exhibited.
Example 4 : An amusing new axiom ?
There are, fortunately, a few of our mathematical colleagues
who, without being trained in logic, are quite serious
about the claim that the logical principles employed in
proofs should be made explicit. For instance the
principle [often used within indirect proofs] that, if a
set E [defined by a property e(x)] should not be empty,
then we can choose an element y of E and use it to prove
certain facts C about E . The late professor Erich Kamke,
reknowned not only for his work on differential equations,
but also for his courageous stand against NS racism, in
1928 wrote a book on set theory, which experienced several
editions and in which he stated the above principle as the
true 'axiom of choice'. And as recently as 1975, the even
more reknowned topologist professor Peter Hilton, for many
years a leading member of the committee's of the IMU,
wrote together with H.B.Griffiths "A Comprehensive
Textbook of Classical Mathematics" , in which he again
stated the above principle as the 'small' axiom of choice,
preliminary to the (usual) 'strong' one.
The logician, of course, will smile, but he should not
undervalue the intellectual honesty which moved our
uninformed colleagues. And he will not help them, or
others in their position, when dryly pointing out that
their principle can be derived as a consequence of the
axioms of first order logic [and particularly not when
taking these from a Hilbert type calculus ! ] Rather, he
might draw attention to the one of the two defining rules
for the existential quantifier (Ex) : from
e(y) => C
proceed to
(Ex) e(x) => C ,
in a calculus of deductive situations (Gentzen's sequents),
and emphasize that y here is a letter which serves as an
eigenvariable, i.e. is not free in the conclusion (hence
not subjected to additional restrictions which might be
expressed there). To this end, of course, language must
be taken seriously: the variable y here is a letter
(Frege in his "Begriffsschrift" of 1879: ein Buchstabe)
and nothing but that.
Example 5 : Can mathematical induction be understood without
referring to language ?
Another place where language and mathematics are sewn
together - again through the use of letters as free
variables - is the principle (I) of mathematical
induction: from
a(0), a(y) => a(s(y))
proceed to
=> (Ax) a(x)
where y again is a letter which serves as an eigenvariable
(and where 0 and s are term symbols for zero and for the
successor function). Poincare/ (Sur la nature de
raisonnement mathe/matique. Revue Me/taphys.Morale 2
(1894) 371-384 , reprinted with slight changes in La
Science et l'Hypothe\se, Paris 1902) wrote "le characte\re
essentiel du raisonnement par re/currence c'est qu'il
contient, condense/e pour ainsi dire en une formule
unique, une infinite/ des syllogismes" . Nothing could be
farther off the mark ! Because a derivation of the
inductive hypothesis
(Hy) a(0), a(y) => a(s(y)) ,
with a free variable y , NOT only gives, for every
numerical constant n , a derivation of
(Hn) a(0), a(n) => a(s(n)) ,
RATHER it gives all these derivations in a single, uniform
manner, independent from the constants n . That (Hy) be
derivable with a free variable is, clearly, a much stronger
assumption than to say
(HP) for every numerical constant n : there is a derivation of (Hn) ;
hence the induction principle (I) is much weaker than
would be one based on (HP) . But of course, and again, in
order to formulate (I) I need to be able to seriously
speak about a language with variables that are letters.
Compare this with the philosophical jabberwock written by
Poincare/ l.c. about mathematical induction:
Cette re\gle, inaccessible a\ la de/monstration analytique
et a\ l'expe/rience, est le ve/ritable type du jugement
synthe/tique a priori. On ne saurait d'autre part songer
a\ y voir une convention, comme pour quelques-uns des
postulats de la ge/ometrie. Pourqui donc ce jugement
s'impose-t-il a\ nous avec une irre/sistible e/vidence ?
C'est qu'il n'est que l'affirmation de la puissance de
l'esprit qui se sait capable de concevoir la re/pe/tition
inde/finie d'un me^me acte de\s que cet acte est une
fois possible. L'esprit a de cette puissance une intuition
directe et l'expe/rience ne peut e^tre pour lui qu'une
occasion de s'en servir et par la\ d'en prendre conscience.
Example 6 : For all and every ?
The rule (IA) introducing the universal quantifier (Ax) was
stated 1879 on p.21 of Frege's "Begriffsschrift" (in the
paragraph beginning with "Auch ist einleuchtend ..."): from
C => a(y)
proceed to
C => (Ax) a(x)
where y is a letter which serves as an eigenvariable. [Frege
did not list (IA) among his nine basic laws, or "Kernsaetze",
though it is implicit in his formula (58. on p.51 .]
Clearly, this rule is governed by the same idea which
underlies the induction rule from the previous example: a
derivation of the premiss
(Ay) C => a(y) ,
with a free variable y , NOT only gives, for every term t ,
a derivation of
(At) C => a(t) ,
RATHER it gives all these derivations in a single, uniform
manner, independent from the terms t . That (Ay) be
derivable with a free variable is, therefore, a much
stronger assumption than to say
(HQ) for every term t : there is a derivation of (At) ;
hence (IA) is a much weaker rule than would be a rule (QA)
based on the assumption (HQ) . But of course, and again,
in order to formulate (IA) I need to be able to seriously
speak about a language with variables that are letters.
It is, of course, amusing to conceive of beings, viz. angels
or demons, able to acquire insights such as (HQ) [or such as
(HP) above] about infinite totalities WITHOUT the rigid
linguistic crutches of (Ay) [or of (Hy) above]. It would be
such beings, then, who also would have the ability to verify
satisfiabilities in infinite domains and would be able to
recognize truth and falseness even of sentences undecidable
by our more limited, only linguistically enabled reasoning.
The amused reader then is also invited to ponder about the
aspects of the seeming paradoxon implied by the so-called
completeness of elementary logic: a sentence true is also
provable. Because a sentence C -> (Ax) a(x) , derived say
by an infinitary rule such as (QA), hence true, then also
should have a proof employing only finitary rules such as
(IA), applied to uniformly established premisses (Ay).
W.F.
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