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Artur Schoenflies and Felix Hausdorff were particularly hostile to logic, targeting the
famous 20th-century logician Bertrand Russell. (Even the extensive dispute over the axiom
of choice focused mostly on its legitimacy as an assumption in set theory and its use of
higher-order quantification:^2 its ability to state an infinitude of independent choices within
finitary logic constituted a special difficulty for ‘logicists’ such as Russell.) Russell, George
Boole, and other creators of symbolic logics were exceptional among mathematicians in
attending to logic, but they made little impact on their colleagues. The algebraic logical
tradition, pursued by Boole, Charles Sanders Peirce, Ernst Schröder, and others from the
mid-19th century, was nothing more than a curiosity to most of their contemporaries.
When mathematical logic developed from the late 1870s, especially with Giuseppe Peano’s
‘logistic’ programme at Turin University from around 1890, Peano gained many followers
there but few elsewhere.^3
Followers of Peano in the 1900s included Russell and Alfred North Whitehead, who adopted
Peano’s logistic (including Cantor’s set theory) and converted it into their logicist thesis that all
the ‘objects’ of mathematics could be obtained from logic. However, apart from the eminent
Cambridge mathematician G. H. Hardy, few mathematicians took any interest in Russell–
Whitehead logicism.^4 From 1903 onwards, Russell publicized the form of logicism put forward
from the late 1870s by Gottlob Frege—which had gained little attention hitherto, even from stu-
dents of the foundations of mathematics, and did not gain much more in the following decades.
In the late 1910s David Hilbert started the definitive phase of his programme of ‘metamath-
ematics’, which studied general properties of axiom systems such as consistency and complete-
ness, and ‘decision problems’ concerning provability. Hilbert’s programme attracted several
followers at Göttingen University and a few elsewhere; however, its impact among mathemati-
cians was limited, even in Germany.^5
The next generations of mathematicians included a few distinguished students of the foun-
dations of mathematics. From around 1900, in the United States, E. H. Moore studied Peano
and Hilbert, and passed on an interest in logic and metamathematical ‘model’ theory to his
student Oswald Veblen, and so to Veblen’s student Alonzo Church, and from Church in turn
to his students Stephen Kleene and Barkley Rosser.^6 At Harvard, Peirce showed his ‘multiset’
theory to the philosopher Josiah Royce, who was led on to study logic and (around 1910) to
supervise budding logicians Clarence Irving Lewis, Henry Sheffer, Norbert Wiener, Morris
Cohen, and also Curt John Ducasse—the main founder of the Association of Symbolic Logic
in the mid-1930s.^7 In central Europe, John von Neumann included metamathematics and
axiomatic set theory among his concerns,^8 but in Poland a distinguished group of logicians did
not mesh with a distinguished group of mathematicians, even though both groups made much
use of set theory (even their joint journal Fundamenta Mathematicae, published from 1920,
rarely carried articles on logic).
The normal attitude of mathematicians to logic was indifference. For example, around 1930
the great logician Alfred Tarski and others proved the fundamental ‘deduction theorem’;^9 this
work met with apathy from the mathematical community, although it came to be noted by the
French Bourbaki group, who were normally hostile to logic. (Maybe the reason was that French
logician Jacques Herbrand had proved versions of the theorem—if so, this was his sole impact
on French mathematics.) Also, while logicians fairly quickly appreciated Kurt Gödel’s theorems
of 1931 on the incompletability of (first-order) arithmetic, the mathematical community did
not become widely aware of Gödel’s results until the mid-1950s.^10