FOM: On the linguistic frame of mathematical reasoning

[Walter Felscher]

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.