[FOM] The Search for Mathematical Roots

[Schaefer, Marcus]

I recently purchased Ivor Grattan-Guinness' "The Search for Mathematical
Roots 1870-1940". The book is a history of the modern foundations of
mathematics, taking into account the developments both in mathematics
and logic, and how they were related. This made the book rather
attractive to me: the table of contents promised a detailed history of
algebraic logic and mathematical analysis and their relationship to the
foundations of mathematics. Furthermore, some figures that do not always
get treated thoroughly in foundational histories, including Grassmann,
Schroeder, Bolzano, and Husserl were each dealt with in a separate
section. A first glance at the text also gave the impression of a
carefully researched book with numerous references, in particular to
rare original sources. However, when I began reading the book, I came
across many rather doubtful statements and claims (some of which I'll
describe in some more detail below). Most of these are related to
mathematics and logic. I don't feel comfortable (or competent) to judge
the philosophical and historical claims made in the book, and I was
wondering whether anybody else had attempted to read the book and formed
an opinion on those aspects of it?

The first sign that the book was going to be unusual was the author's
use in Section 1.2.1 of the term "incompletability theorm" for Goedel's
result. He later explains (in Section 9.2.3) his use of
"incompletability" over the standard "incompleteness"; his point seems
rather too trivial to warrant this change in standard usage. In a
similar vein he uses the phrase "bicimal expansion" instead of "binary
expansion" (Section 4.5.8), which goes against standard usage and, I
believe, good taste.

These, of course, are quibbles, but there is a similar laissez-faire
attitude in the author's treatment of translations. For example, in
Section 4.5.3 he quotes Frege speaking about mathematics (in the
"Grundlagen") as saying

"It would be wonderful, if the most exact of all the sciences had to be
supported by psychology, which is still groping uncertainly."

This sounded strange to my ears, and, indeed, going back to Frege's
Grundlagen, it becomes obvious that Frege frequently uses the word
"wunderbar" in the sense of "strange" or "hard to believe". The author's
English rendition gives the text an entirely different meaning. Another
mistranslation, this time of Schroeder, can be found in Section 4.4.8.
Here is the German; the text, as it is, is not quite grammatically
correct, most likely due to faulty transcription:

"Herr G. Cantor, mit dessen Genialitaet ich weit entfernt bin; meine
bescheidnen Anlage im Vergleich stellen zu wollen, hat sich mit seiner
Forschungen beschaeftigt, obwohl einer Vertiefung in diese mir stets als
Desideratum vorgeschwebt."

Grattan-Guinness translates this to:

"Mr. G. Cantor, from whose geniality I am far distant; to want to place
my modest talent in comparison, he has occupied himself with his own
researches, although a deepening of them always hovers for me as a
desideratum."

which looks remarkably like the result of a google translation. Maybe
Raymond Roussel could have made sense of it, but little of the original
meaning is left. The German is a bit tricky (Schroeder was an awful
writer), and the semicolon misleading. A rough, but more correct
translation would be:

"Mr. G. Cantor, with whose genius I am far from comparing my own modest
talent, has occupied himself with his own researches; a deeper
understanding of these has always been a desire of mine."

There is, unfortunately, also evidence to suggest that the author does
not actually understand all of the mathematics or logic involved. Here
are some examples; most of them speak for themselves. In Section 4.7.3,
talking about Huntington, he writes

"His main mathematical interest was finding axiom systems for various
mathematical theories and studying their consistency, independence,
completeness, and `equivalence' (his word for categoricity)."

The author does not seem to understand the difference between
equivalence of axiom systems and their categoricity. Just before that,
in the same section, he discusses Hilbert's axiomatization of the
natural numbers (from "Ueber den Zahlbegriff"), and misinterprets one of
the axioms:

"The first axiom of this last group was Archimedes's, as usual; it
guaranteed the existence of real numbers and thereby the real line,
hence underpinning geometry."

Shortly afterwards (Section 4.7.5), he writes, about Hilbert's
axiomatization of logic:

"The axioms were
	`1. x = x. 	2. {x=y u. w(x)} | w(y)'
for some (unexplained) prepositional function w."

apparently misunderstanding Hilbert's use of an axiom schema (`u' is
Hilbert's symbol for conjunction, and `|' his symbol for implication).
Also, here is his explanation of Cantor's diagonal argument (in Section
3.4.6):

"In the second part of the paper Cantor took as M the set of
characteristic functions {f(x)} (to use the modern name) of all subsets
of the closed interval L = [0,1]. ... ; to show that it was definitely
greater he took the function phi(x,z) of two independent variables,
where z was the member of L with which f(x) was associated by the
relation
	f(x) = phi(x,z) for 0 <= x <= 1.
He then considered the function
	g(x) /:= phi(x,z) for 0 <= x <= 1:
while an element of M, it took no value for z, thus establishing the
greater cardinality of M."

(I can't typeset mathematics in ASCII, so /:= means := with a bar
through it, i.e. not equal by definition). For an author that regularly
complains about inaccuracies in other authors' translations and
mathematics this does not read well. He also complains about waffling.
Here is one of his own contributions:

"Apparently Goedel found his theorem when he represented each real
number by an arithmetical prepositional function phi(x) and found that,
while `phi(x) is provable' could also be so treated, `phi(x) is true'
landed him in liar and naming paradoxes (Wang 1996a, 81-85). Maybe
because of Viennese empiricist doubts over truth, he recast the paradox
in terms of unprovability and `correct' (`richtig') propositions;"

The most surprising claim I have found so far is in the following
paragraph (from Section 3.2.4), in which he discusses Dedekinds's
"Stetigkeit und die irrationalen Zahlen":

"He also proved here the existence of irrational numbers by a lovely
reductio argument that has never gained the attention that it deserves:
assume that the equation t^2 = Du^2 in integers (D not a square) has a
solution, and let u be the smallest integer involved; then exhibit a
smaller integer also to satisfy the equation, a contradiction which
establishes sqrt(D) as irrational."

After reading that paragraph, I had to look at the back cover of the
book again. There it says, in black and white: the author is the
President of the British Society for the History of Mathematics. He is
also on the advisory board of important projects such as the Peirce
project and the Russell project. 

Marcus Schaefer