Both Peano's books from 1891 , our "I Principii di Geometria
Logicamente Esposti" as well as "Arithmetices Pricipia Novo Methodo
Exposita" [the latter in Latin and conveniently accessible in North
America in van Heijenort's "Source Book in Mathematical Logic"] --
they both have the main part of their text, theorems and proofs,
written in the symbolic notation invented by Peano; only the
introduction and a few comments (such as the one I quoted in my
last note) appear in ordinary language. To read them, therefore,
amounts to first decode long strings of symbols - only to find
out that their mathematical content does not go much beyond what
was said earlier, say by Pasch.
Footnote 3 : During the last decades of the 19th century,
Peano was one of the more remarkable mathematicians. When
writing down his theorem on the existence of solutions of
(systems of) OEDs, he noticed that one of his arguments
consisted in making undeterminate choices from infinitely
many intervals on the line (i.e. used the as yet unformulated
axiom of choice), and he then succeeded to replace this
argument by another one, avoiding choices with help of a
unique algorithmic construction.
This sharp minded analysis of arguments, as they are used
habitually by mathematicians, led him to the conviction that
involved mathematical statements should be formulated with
utmost perspicuity, and to this end he invented his formal
notation which he then used in his publications. Not too
surprisingly, that notation then took on a life of its own, was
not appreciated by his contemporaries and was not used except
by his immediate pupils. Just as we today look with disdain
at certain modern philosophers who use one or two lines of
logical symbols to give the appearance of rigour to the
presentation of their disorderly thoughts, so the one or other
of Peano's contemporaries may have viewed his formal notation
as a ridiculous mystification of trivialities.
Peano invented a good notation, but what he lacked was the
conceptual apparatus to put it to use with: the formal
analysis of logical arguments by their syntactical form, the
setup of logical rules, expressing by purely syntactical
descriptions that one notational string be a consequence of
one or several other notational strings. Such apparatus,
however, was present already in form of Frege's "Begriffsschrift"
of 1879/1884 which used a much less convenient, two-dimensional
form of notation. Still, it took until Russell's article in
the American Journal of Mathematics of 1908 and Whithead-Russell's
monolithic treatise of 1910 , that Frege's machine was married
with Peano's notation, giving rise to mathematical logic as
we know it. Curiously enough, it was Hilbert, not having even
mentioned Peano in 1899 , who in his articles after 1920
introduced the simplifications of the Peano-Russell notation
which mathematical logic uses today. [End of Footnote 3 ]
Footnote 4 : In my note from Friday last I wrote
the first author to admit implicit definitions with
non-unique solutions seems to have been Peano with
his axiomatization of geometry in "I Principii ...
where it should have read
the first author to admit implicit definitions with
non-unique solutions in geometry seems to have been
Peano with his axiomatization in "I Principii ... .
As quoted in Footnote 2 , already Boole considered the possibility
of different interpretations of the basic notions and relations
appearing in systems of axioms. But already the inventors of
Symbolical Algebra, George Peacock and D.F.Gregory, had expressed
the similar thoughts, e.g. the former in his "A Treatise on
Algebra, vol.2 : On Symbolical Algebra and its Applications
to the Geometry of Position", London 1845 , where he defined
algebra as a formal discipline whose truth criteria do not
come from observation or measurement, but alone by deduction
from given axioms governing their operations. And Gregory, in
"On the real nature of Symbolic Algebra", Trans.Roy.Soc.Edinburgh
14 (1840) 208-216 , had written
The light then in which I would consider Symbolical
Algebra is that it is the science which treats of the
combination of operations defined not by their nature,
that is, by what they are or by what they do, but by the
laws of combination to which they are subject. ... The
step which taken from the arithmetical to the symbolical
algebra is that, leaving out the view of the nature of the
operations which the symbols we use represent, we suppose
the existence of classes of unknown operations subject to
the same laws. We are thus able to prove certain relations
between the different classes of operations, which, when
expressed between the symbols, are called algebraical
theorems.
Here we have a program to study unspecified structures with
operations subjected to laws - apparently conceived as general
equations. It does not seem that it was carried out beyond
trivialities, possibly because there still lacked non-trivial
examples (such as those which thirty years later inspired
Dedekind when he wrote the supplements to Dirichlet's lectures).
[End of Footnote 4 ]
Biliographical notes
Freudenthal's essay, of which M. Cabillon located a
version in the 1960 Stanford Congress Volume, appeared
also as
Zur Geschichte der Grundlagen der Geometrie. Nieuw Arch.
Wiskunde (4) 5 (1957) 105-142 .
A still remember a most impressive conference by
Professor Freudenthal on this theme, given at the
DMV-Tagung in Bonn in October 1959 .
W.F.
Concerning the question why it was the case that Hilbert did not
mention Peano's booklet of 1891 , we are left with surmises only.
We do not know whether, when writing the "Grundlagen", he knew of
the book's existence; we do not know whether, provided Yes to the
above, he ever held it in his hands; we do not know ... . In
para 21 of the "Grundlagen", Hilbert cites publications by de
Zolt written in Italian - which makes it unlikely that it was the
use of this language which prevented him from reading Peano.
But provided Hilbert looked at the book at all, or received
colleagues' comments about it, it is conceivable that it was
another 'difficulty of language' which made him abstain from taking
further notice.
Both Peano's books from 1891 , our "I Principii di Geometria
Logicamente Esposti" as well as "Arithmetices Pricipia Novo Methodo
Exposita" [the latter in Latin and conveniently accessible in North
America in van Heijenort's "Source Book in Mathematical Logic"] --
they both have the main part of their text, theorems and proofs,
written in the symbolic notation invented by Peano; only the
introduction and a few comments (such as the one I quoted in my
last note) appear in ordinary language. To read them, therefore,
amounts to first decode long strings of symbols - only to find
out that their mathematical content does not go much beyond what
was said earlier, say by Pasch.
Footnote 3 : During the last decades of the 19th century,
Peano was one of the more remarkable mathematicians. When
writing down his theorem on the existence of solutions of
(systems of) OEDs, he noticed that one of his arguments
consisted in making undeterminate choices from infinitely
many intervals on the line (i.e. used the as yet unformulated
axiom of choice), and he then succeeded to replace this
argument by another one, avoiding choices with help of a
unique algorithmic construction.
This sharp minded analysis of arguments, as they are used
habitually by mathematicians, led him to the conviction that
involved mathematical statements should be formulated with
utmost perspicuity, and to this end he invented his formal
notation which he then used in his publications. Not too
surprisingly, that notation then took on a life of its own, was
not appreciated by his contemporaries and was not used except
by his immediate pupils. Just as we today look with disdain
at certain modern philosophers who use one or two lines of
logical symbols to give the appearance of rigour to the
presentation of their disorderly thoughts, so the one or other
of Peano's contemporaries may have viewed his formal notation
as a ridiculous mystification of trivialities.
Peano invented a good notation, but what he lacked was the
conceptual apparatus to put it to use with: the formal
analysis of logical arguments by their syntactical form, the
setup of logical rules, expressing by purely syntactical
descriptions that one notational string be a consequence of
one or several other notational strings. Such apparatus,
however, was present already in form of Frege's "Begriffsschrift"
of 1879/1884 which used a much less convenient, two-dimensional
form of notation. Still, it took until Russell's article in
the American Journal of Mathematics of 1908 and Whithead-Russell's
monolithic treatise of 1910 , that Frege's machine was married
with Peano's notation, giving rise to mathematical logic as
we know it. Curiously enough, it was Hilbert, not having even
mentioned Peano in 1899 , who in his articles after 1920
introduced the simplifications of the Peano-Russell notation
which mathematical logic uses today. [End of Footnote 3 ]
Footnote 4 : In my note from Friday last I wrote
the first author to admit implicit definitions with
non-unique solutions seems to have been Peano with
his axiomatization of geometry in "I Principii ...
where it should have read
the first author to admit implicit definitions with
non-unique solutions in geometry seems to have been
Peano with his axiomatization in "I Principii ... .
As quoted in Footnote 2 , already Boole considered the possibility
of different interpretations of the basic notions and relations
appearing in systems of axioms. But already the inventors of
Symbolical Algebra, George Peacock and D.F.Gregory, had expressed
the similar thoughts, e.g. the former in his "A Treatise on
Algebra, vol.2 : On Symbolical Algebra and its Applications
to the Geometry of Position", London 1845 , where he defined
algebra as a formal discipline whose truth criteria do not
come from observation or measurement, but alone by deduction
from given axioms governing their operations. And Gregory, in
"On the real nature of Symbolic Algebra", Trans.Roy.Soc.Edinburgh
14 (1840) 208-216 , had written
The light then in which I would consider Symbolical
Algebra is that it is the science which treats of the
combination of operations defined not by their nature,
that is, by what they are or by what they do, but by the
laws of combination to which they are subject. ... The
step which taken from the arithmetical to the symbolical
algebra is that, leaving out the view of the nature of the
operations which the symbols we use represent, we suppose
the existence of classes of unknown operations subject to
the same laws. We are thus able to prove certain relations
between the different classes of operations, which, when
expressed between the symbols, are called algebraical
theorems.
Here we have a program to study unspecified structures with
operations subjected to laws - apparently conceived as general
equations. It does not seem that it was carried out beyond
trivialities, possibly because there still lacked non-trivial
examples (such as those which thirty years later inspired
Dedekind when he wrote the supplements to Dirichlet's lectures).
[End of Footnote 4 ]
Bibliographical notes
Freudenthal's essay, of which M. Cabillon located a
version in the 1960 Stanford Congress Volume, appeared
also as
Zur Geschichte der Grundlagen der Geometrie. Nieuw Arch.
Wiskunde (4) 5 (1957) 105-142 .
A still remember a most impressive conference by
Professor Freudenthal on this theme, given at the
DMV-Tagung in Bonn in October 1959 .
W.F.