How was Mathematics
modernized?
Apr.7, 2018, JACAP 2018@Nagoya university
Susumu Hayashi
Professor, Department of Media and Culture Studies
Division of Contemporary Culture
Graduate School of Letters
Kyoto University
ver.2021.07.11

Modernization of Mathematics• The previous talk is about modernism in mathematics.
• This talk is about the modernization of mathematics.
• They are different although they are closely related.
• My research on modernization of math. profited much by themodernism idea by Prof. Gray as they are related.

• The modernization of mathematics, as I call it, is thetransformation of the attitudes of mathematicians
which was completed after the second world war.
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What is Modernization of Mathematics? 1/3• The modernization theory was originated by the Germansociologist Max Weber, and influenced many sociologists
such as T. Parsons and A. Giddens, etc..
• My modernization theory of mathematics is also based onWeber-sociology, and can be seen as a particular case ofWeber’s modernization theory of societies.
• Weber’s modernization idea has many elements such assecularization, formal rationality, warring deities, etc..
• Each specific domain of society has its own characteristic ofmodernization, as Weber defined the concept of formalrationality in different ways for justice and economy.
3What is Modernization of Mathematics? 2/3• The main characteristics of the modernization ofmathematics are secularization and formal rationality.
• Since I do not have enough time to explain the whole picture,in this talk, I will concentrate on the first issue“secularization of mathematics”.
• Secularization of mathematics?
• I am not talking about religious values in mathematics.
• Secularization of mathematics in my sense means thatmathematicians distance themselves more and more fromphilosophy.
4What is Modernization of Mathematics? 3/3• Thus modernized mathematicians would say:
No Philosophy Please, We’re Mathematicians.
• We will see a typical example of the modernization ofmathematics in an episode between a secularized great
mathematician and a non-secularized great mathematician,and will see the meaning of the sentence “No Philosophy
Please, We’re Mathematicians” in the episode.
• The non-secularized great mathematician is Hermann Weyl,on whom the previous talk was.

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Weyl and Kodaira• Weyl, on the right, was the hero for Kunihiko Kodaira, on theleft, Japan’s first Fields medalist, who was awarded theFields medal for his genelrization of Weyl’s theory ofRiemann surfaces.
K. Kodaira
小平邦彦

1915-1997

H. Weyl

1885-1955

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Kodaira’s “Memories of Princeton”• Kodaira wrote in his essay “Memories of Princeton (プリンストンの思い出)”, 1981 August issue, Sugaku Seminar (数学セミナー), a
popular magazine of mathematics as follows:• As a student, Kodaira was fascinated by Weyl’s famous booklet“Die Idee der Riemannschen Fläche” (The Concept of a RiemannSurface) 1913.
• Kodaira extend Weyl’s theory of one-dimensional Riemannsurfaces of the booklet to n-dimensional surfaces.
• Weyl appreciated this work, and invited Kodaira to the Institutefor Advanced Study in 1949. When Kodaira visited Weyl for the
first time in his office, his hero said to him like this:7

Although I may be old-fashioned…
• From Kodaira’s essay: 先生曰く『私は古い(old-fashioned)かも知れないが、直交射影の方法はよくないと思う。君の論文も直交射影を用いない形に書き直した方がよい』と。これはショックであった。…ワイルの直観主義の立場ではL2可積分な可測微分形式全体のヒルベルト空間というような大きな集合はその存在があやしい、ということであったらしい。ワイル先生
は、ゲーデルの不完全性定理によって数学の無矛盾性を証明する希望が失われた以上は、数学をあまり一般化すると矛盾に撞着するのではないか、と本気で心配しておられたようである。
…それにしてもヒルベルト空間がすでに危険であるという感覚は私には理解できなかった。
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Although I may be old-fashioned… translated• From Kodaira’s essay: Prof. Weyl said , "I may be old-fashioned,but I think the orthogonal projection method is not good. It would
be better to rewrite your paper without orthogonal projection. "This shocked me. ... It seems that the existence of the large set
such as the Hilbert space of the whole of the measurable
differential forms, which are L2 measurable, is doubtful from thestandpoint of Weyl 's intuitionism. I guess that Prof. Weyl wasseriously worried that mathematics may confront contradictions, if
it is too much generalized, as the hope of proving the consistencyof mathematics had gone by Gödel’s incompleteness theorem. ...
At any rate I could not understand the feeling that Hilbert space
was already dangerous.
9The feeling of secularized mathematicians• The following two highlighted parts represent the feeling ofsecularized mathematicians after the second world war.
• From Kodaira’s essay: Prof. Weyl said , "I may be old-fashioned,but I think the orthogonal projection method is not good. It would
be better to rewrite your paper without orthogonal projection. "This shocked me. ... It seems that the existence of the large set
such as the Hilbert space of the whole of the measurable
differential forms, which are L2 measurable, is doubtful from thestandpoint of Weyl 's intuitionism. I guess that Prof. Weyl wasseriously worried that mathematics may confront contradictions, if
it is too much generalized, as the hope of proving the consistencyof mathematics had gone by Gödel’s incompleteness theorem. ...
At any rate I could not understand the feeling that Hilbert
space was already dangerous.
10Not only Kodaira. Bourbaki said:• Certainly, the vigorous attacks from the intuitionist camp have fromtime to time obliged not only the avant-garde mathematical schools,
but even the partisans of traditional mathematics, to be on thedefensive. A well-known mathematician has admitted to being
impressed by these attacks to the point that he voluntarily
restricted his work to those branches of mathematics considered tobe “certain’’. But such cases must have been rather uncommon.
The intuitionist school, whose memory will undoubtedly survive only asa historical curiosity, has at least rendered the service of having obliged
its opponents, that is to say the vast majority of mathematicians, toclarify their own positions and to become more consciously awareof the reasons (whether logical or sentimental) for their confidence
in mathematics.
(2004 English translation “Theory of Sets” of Bourbaki’s Théorie des ensembles, 1970, p.336)
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Not only Kodaira. Bourbaki said:• Certainly, the vigorous attacks from the intuitionist camp have fromtime to time obliged not only the avant-garde mathematical schools,
but even the partisans of traditional mathematics, to be on thedefensive. A well-known mathematician has admitted to being
impressed by these attacks to the point that he voluntarily
restricted his work to those branches of mathematics considered tobe “certain’’. But such cases must have been rather uncommon.
The intuitionist school, whose memory will undoubtedly survive only asa historical curiosity, has at least rendered the service of having obliged
its opponents, that is to say the vast majority of mathematicians, toclarify their own positions and to become more consciously awareof the reasons (whether logical or sentimental) for their confidence
in mathematics.

Weyl(2004 English translation “Theory of Sets” of Bourbaki’s Théorie des ensembles, 1970, p.336)
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Around 1960, secularization completed• Around 1960, most of pre-secularized mathematicians as
Weyl passed away.
• After that, no great mathematicians at the rank of Weyl caredabout philosophical issues on mathematics.
• They were like Kodaira and Bourbaki.
• It is interesting that secularization of physics took placealmost the same time.

13Secularization of Physics 1/2• Physics experienced a similar transformation. A notable historianof science Helge Kragh claimed in his book “Quantum Generations:
A History of Physics in the Twentieth Century”, 1999, p.441, (a very
good Japanese translation available「20世紀物理学史（上下巻）」
by Okamoto, Ariga, Inaba et al.) as follows:• One result of the postwar generational shift and that general turntoward instrumentalist and pragmatic modes of thought was thatphilosophy lost its place in physics. Many members of the earlier
generation of physicists had a deep interest in philosophicalquestions and were sometimes inspired by philosophers in theirinnovative work, or they discussed competently the philosophicalimplications of the new physics. (continued)
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Secularization of Physics 2/2• (continued) For Plank, Bohr, Schrödinger, Weyl, Heisenberg,
Einstein, Eddington, and many of their colleagues, philosophy
was an important aspect of physics. Few of the leaders of
postwar physics cared about philosophy or had more than asuperficial knowledge of the field. In the 1960’s, with thedeath of Bohr and Schrödinger, the once proud and vital
tradition of physicist-philosophers came to an end.

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Modernized Physicists and Mathematicians• In short, Kragh maintained that the attitude of thegenerations of physicists after the second world war was“No Philosophy, Please, We’re Physicists”.
• Any physicist, who takes this attitude to philosophy, is a“modernized physicist” in my sense, just as Kodaira andBourbaki are “modernized mathematicians”.

16Difference between Physics and Math.
• Modernization was practically achieved almost at the same timein physics and mathematics.
• However, the processes were very different.
• In the case of mathematics, there was a stage of modernization atwhich a considerable number of great mathematicians believedthat philosophical problems on mathematics would be fixed bymathematics itself.
• They believed that secular mathematics would solve the mostnon-secular (philosophical) problems of mathematics.
• There was not such a stage in the history of physics.
• What is the stage then?
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The Stage: the Period of Hilbert’s Program 1/2• The stage is Hilbert’s Program. Hilbert tried to solve many foundational
problems on mathematics such as consistency and completeness by
his proof theory.
• It is widely believed that this program was initiated by the discovery ofantinomies of set theory.
• It’s only half true. Solving the problem of set-theoretic antinomies byconsistency proof is definitely an important motivation.
• However, the real origin of Hilbert’s program was a plan to prove thesolvability of mathematics mathematically.
• The solvability of mathematics means that there is a method by which
all mathematical problems can be decided true or false in finite steps.
In the modern words, decision method of mathematics.
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The Stage: the Period of Hilbert’s Program 2/2• In his 1905 lecture, after proving his version of the completness ofpropositonal logic, Hilbert claimed that it is the simplest case of his
grand plan to prove that there is no Ignorabimus, “limitation of humanwisdom”, in mathematics. He further mentioned that this plan was thestarting point of all of his research on the foundations of mathematics.
• The starting point took place around 1888-9. Then he wrote about theplan of the solvability proof and other related problems, relating themto Kant’s philosophy in the first of his three mathematical notebooks.
• I do not have time to explain them in detail, but there is a Web page on
the findings of Hilbert’s notes: David Hilbert's Mathematical Notebooks,
at which we will takes a glance, and we have written a paper on it longtime ago, but not yet published.
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Solving philosophical problems by math. 1/2• Hilbert called Brouwer and Weyl “philosophers disguised asmathematicians” in his 1931 lecture in Hamburg.
• Apparently, he was disgusted with them and philosophy. DidHilbert change his mind from his youthful days when hementioned Kant?
• No, he didn’t!
• In 19th century, some scientists tried to draw philosophicalconclusions by help of science. The greatest example is theprogram of Hermann von Helmholtz to rescue Kant’s
philosophy by physiology (pp.196-205, “The Genesis of NeoKantianism 1796-1880”, 2014, by F.C. Beiser).

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Solving philosophical problems by math. 2/2• The young practically unknown mathematician in 1880’s was doing
similar thing in mathematics, and three decades later, then he was oneof the greatest mathematicians in the world, was pursuing the dream of
his youth.
• It was Hilbert’s Program.
• However, the dream was shuttered by Gödel’s incompletenesstheorems.
• The best strategy left for mathematicians was just to forget about
philosophy and to strengthen their confidence in mathematics.
• Anyway, there are formal systems of sets (ZF, NBG) seemingly
consistent and very enough to describe their concepts and proofs.
• Then, the step of modernization was forwarded.
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Gödel and Modernization• In a sense, Gödel made a step forward in the process of themodernization.
• However, he had a different view on it.
• I claimed that after 1960’s no great mathematicians at therank of Weyl cared about philosophical issues.
• Gödel was an exception.
• He viewed the step of modernization brought by his great
theorems as a step backward!

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Gödel’s historical view 1/3• In Kurt Gödel Collected Works, Volume III, Unpublished Essays
and Lectures, there is an interesting and enigmatic article titled“The modern development of the foundations of mathematics inthe light of philosophy” (1961/?).
• Gödel presented a classification of philosophical world-views(Weltanschauungen) according to the degree and manner of how
they are away from metaphysics or religion.
• Some world-views from the left side to the right side:
Left
Right
skepticism
theology
materialism
idealism
positivism
spiritualism
more metaphysical or religious ➡
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Gödel’s historical view 2/3• Then Gödel wrote (verbatim quotations): Now it is a familiar
fact, even a platitude, that the development of philosophy
since the Renaissance has by and large gone from right toleft-…. Particularly in physics, this development has reacheda peak in our own time, …, the possibility of knowledge ofthe objectivizable states of affairs is denied, and it isasserted that we must be content to predict results ofobservations. … (although this predicting can be completely
sufficient for practical purposes such as making television
sets or atom bombs).
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Gödel’s historical view 3/3• We may guess that Gödel wrote about the same historical process
which I quoted from Kragh’s book. Good!
• However, then, he continued as follows: Mathematics is an apriori science, and so, it went in the opposite direction!!
• He even claimed that Hilbert’s program was a strange hybrid(merkwürdige Zwitterding) attempt to rescue the rightward
aspects of mathematics by the leftward proof theory.
• The Hilbertian combination of materialism and aspects ofclassical mathematics thus proves to be impossible. (verbatim)• It was as if he was saying the failure of Hilbert’s program by hisincompleteness theorems was a destiny.
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The Origin of My Research• I happened to read the above article 19 years ago.
• Then I was an amateur historian, who was a professor of
applied mathematics and software engineering.
• Just by chance, I was about to study Weber sociology forunderstanding of my problems in software engineering.
• Then, it became my fascination to understand the process ofhistory that Gödel described by Weber’s modernization
theory, and it made me even a professional historian.
• It took nearly 19 years to reach a satisfactory view.
• It is my idea of Modernization of Mathematics.
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Realities for Mathematicians 1/3• I claimed, after 1960’s, no great mathematicians at the rank of
Weyl cared about philosophical issues on mathematics.
• However, there was an exception at least: Gödel!
• He was known as a philosopher of mathematics, as he hadclaimed “Gödel’s Platonism”. He believed that transfinite setsexist just as in the same sense as physical bodies exist.
• It is doubtful that Bourbaki members thought in the same way.
• For most mathematicians, axiomatic set theory is no more than alingua franca (common language) to describe their own “realities”.

27Realities for Mathematicians 2/3• They have their own realities, subjects of research, in theirminds, e.g. complex manifolds, Hilbert spaces, spectrums ofrings, etc.
• They do not care about “materials” of their realities, as
modern mathematicians care about only structures, “formalcauses” in a sense.
• Thus, some of them prefer category theory to set theory, as
they can do mathematics without talking about materials of
their subjects of research.

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Realities for Mathematicians 3/3• We may consider axiomatic set theories or the like, e.g. NBG, ZF,
Tarski-Grothendieck set theory, elementary topos, etc., asexamples of the formal rationality in mathematics.
• Thinking so, mathematicians’ realities become “re-enchantmentthrough disenchantment”, a sociological issue developed after
Weber sociology.
• Anyway, it is impossible to do research of mathematics without
thinking that subjects of research really exist so that you can eventouch them. As an ex-mathematician, I deeply believe in this frommy own experiences.
• However, as a historical sociologist or historian, I see this as a reenchantment through disenchantment.
29Modernization of Mathematics 1/3• From this standpoint, the whole development of the foundationsof mathematics, modernization of mathematics, looks like thefollowing:• Since Riemann, there was a metaphysical trend in mathematics.
• It was questioned by 19th century rationalists such as Kummer,
Kronecker and Weierstrass, but the history proved they werewrong and Riemann was right.
• Metaphysical ideas such as Riemannian surface and infinite set
made mathematical researches much more productive.
• It was contradictory, as Gödel illustrated in the 1961 article that Iquoted.
• Thus, as Gödel pointed out, it was natural that contradictions
were found in the metaphysical foundation, i.e. set theory.
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Modernization of Mathematics 2/3• What early 20th century rationalists did was to control too
strong a power of the metaphysics in mathematics by theformal rationality in mathematics, i.e. formal systems.
• Although Hilbert’s program, which was an attempt to validate
this disenchanted metaphysics by the left most standpoint,failed, mathematicians did not mind using the disenchanted
metaphysics.
• Reason: they have their own metaphysical realities in theirminds and the disenchanted metaphysics is no more than alingua franca for mathematics for most of mathematicians.
31Modernization of Mathematics 3/3• Spanish historian José Ferreirós pointed out that the early set
theories were considered as improved Aristotelian logic.
• Dedekind called his version of set theory in 1888 booklet “Was sind undwas sollen die Zahlen?” Logik. Thus he identified the singleton of an entitywith the entity itself in the booklet.

• Namely, axiomatic set theory is a rationalized metaphysics,
Aristotelian logic improved by the algebraic and formal methods
initiated by George Boole.
• It is a land which was once a part of the kingdom of philosophy
but is now a part of the republic of mathematics.
• Since they can find enough metaphysics for their research in theirown land, contemporary mathematicians ignore philosophy.
• The modernization of mathematics was achieved in this way.
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