The architectural metaphor of foundations in mathematics is dead among mathematicians for several reasons: 1) Mathematics has become more specialized with the rise of abstract algebra, moving away from thinking of operations on elements to relations between subsets and homomorphisms. 2) Developments in logic and set theory brought "roots" and "branches" of mathematics together rather than viewing them as separate, undermining the tree analogy. 3) Mathematicians by the 1920s took for granted the use of set theory for basic definitions and reasoning rather than viewing it as providing foundations in an architectural sense. The metaphor died sometime between the wars as mathematics ceased requiring foundations in the architectural sense due to its changing nature and methods becoming

1. Foundational Research as Mathematical Practice
Brendan Larvor
www.herts.ac.uk/philosophy

2. Plan
1. Philosophy of mathematical practice: what and why?
2. Why are foundational questions interesting for PMP?
3. The architectural metaphor is dead among mathematicians
(so why does it live on among philosophers?)
4. When and why did it die?
5. Some concluding thoughts on PMP and foundations
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3. Philosophy of Mathematical Practice
Motivation
Foundational and metaphysical attention:
• Is too often motivated and/or conditioned by extraneous concerns
• Washes out differences of practices
• Makes a mystery of human mathematical success
• Obscures (or leaves subjective) questions of value and development
• May be parasitic on experience of mathematical practice that it cannot theorise
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4. Philosophy of Mathematical Practice
Context
• Mature, professional history of mathematics and science
• Emerging ethno-mathematics
• Cognitive science/psychology/evolutionary accounts of natural mathematical abilities
• Broad shift to practice (e.g. in philosophy of experimentation, ethics, philosophy of
language).
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5. Philosophy of Mathematical Practice
Some problems and questions
• “Studying what mathematicians actually do.” (Compare: history “as it actually happened”)
• “Mathematics is a human activity.” (And therefore…?)
• Philosophy of MP a broad church with conservative and radical wings
• How to turn all this observation into philosophy
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6. Philosophy of Mathematical Practice
Some consensus on proof
• Real proofs are not abbreviations of fully formal proofs. They are (recipes for) rigorous
conceptual arguments.
(Quick way to see this: Gentzen's call for conceptually conservative (i.e. cut-free) proofs
impossible to satisfy with real proofs)
• In other words, Traduttore, Traditore!
• So PMP tends to oppose foundationalisms that depend on formal accounts of proof
• But! Historically, these are not unrelated to mathematical practice
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7. Why are foundational questions interesting for PMP?
Two disquiets
• Because PMP has a delicate relationship with foundationalism
(opposes it in philosophers, seeks to understand it in mathematicians)
• Because the best story on the nature of proof entails localism, while foundationalism
aspires to be universal.
Domain-specific moves 
Domain-specific practices 
Diverse communities of practice
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8. The architectural metaphor is dead among mathematicians
Michael Harris
Two kinds of foundations are at issue in mathematics: the foundations which represent the
starting point for building what one wants to build and the foundations without which the
entire structure will crumble.
Grothendieck had strong feelings about the former, referring in his letter to Quillen to
"people such as yourself, Larry Breen, Illusie and others, who … may not think themselves
too good for indulging [sic] in occasional reflection on foundational matters and in the
process help others in the work which should be done." (p. 13 of Pursuing Stacks)
…
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9. The architectural metaphor is dead among mathematicians
Michael Harris
…
But it's only a matter of habit to call these "foundations" as if they were necessarily below
our feet; given that the only stipulation is that the work we are actually doing needs to be at
eye-level, there's no reason not to situate foundations of this sort up above our heads.
Indeed, what Grothendieck is calling foundations in Pursuing Stacks, like Weil before him
(and Bourbaki more generally) is really a common language sufficiently rich as well as
precise to address the problems that may arise in the development of the theory one was
going to develop in any case.
Such a language has more in common with the satellites and transmitters that carry the
signals that permit electronic communication than with the 120 meters of concrete attached
to the base of the 452 meter high Petronas towers, to protect it from destabilization by the
forces of nature.
(book in preparation, private communication)
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10. The architectural metaphor is dead among mathematicians
Yuri I. Manin
Foundations of mathematics [are] a historically variable set of principles appealing to
various modes of human intuition and devoid of any prescriptive/prohibitive power.
At each turn of history, foundations crystallize the accepted norms of interpersonal and
intergenerational transfer and justification of mathematical knowledge.
Existential anxiety can be alleviated if one strips “Foundations” [of] their rigid
prescriptive/prohibitive, or normative functions and considers various foundational matters
simply from the viewpoint of their mathematical content and on the background of whatever
historical period.
Talk at the international interdisciplinary conference “Philosophy, Mathematics, Linguistics:
Aspects of Interaction” , Euler Mathematical Institute, May 22–25, 2012
(Thanks to Michael Harris for pointer)
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11. The architectural metaphor is dead among mathematicians
Raymond L. Wilder : the tree of mathematics
As long ago as the 17th century when Fermat and Descartes introduced their consolidation
of algebra and geometry to form analytic geometry, the ‘tree’ representation began to break
down.
And later, work in the foundations of mathematical logic not only brought about
consolidation between roots and branches — e.g. Algebra à la Boole — but brought the
roots up to aid in the solution of problems belonging in the branches (e.g. the continuum
problem in the theory of sets, Souslin's problem, etc.). Such events not only made the tree
analogy quite unrepresentative of the way mathematics grows, but emphasised the
interrelatedness of all parts of mathematics…
(Mathematics as a Cultural System 1981 p. 15)
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12. The architectural metaphor is dead among mathematicians
Emmy Noether
…[Noether’s] idea of set theoretic foundations for algebra. This was not what we now mean
by set theoretic foundations. It was not the idea of using set theory for basic definitions and
reasoning. She took that more or less for granted, as did other Göttingers by the 1920s.
Rather her project was to get abstract algebra away from thinking about operations on
elements, such as addition or multiplication of elements in groups or rings. Her algebra
would describe structures in terms of the relations between selected subsets (such as
normal subgroups of groups) and homomorphisms.
Colin McLarty “Emmy Noether's `Set Theoretic' Topology: From Dedekind to the rise of
functors'', In Jeremy Gray and José Ferreirós eds The Architecture of Modern Mathematics:
Essays in history and philosophy, Oxford, 2006, 211--35.
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13. When and why did the architectural metaphor die?
(Among Mathematicians)
It was alive in the late 19th century:
• Dedekind, Cantor, Frege, Peano, Russell, Hilbert
• General axiomatic movement of later 19th century mathematics
An attempt to provide mathematics with mathematical foundations
(Ferreiros & Gray)
• This in response to the anxiety associated with galloping abstraction
(Gray Science in Context 17 (1/2) 23-47 (2004)
• But by 1948, Bourbaki had switched to the picture of mathematics as a city (op. cit.)
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14. When and why did the architectural metaphor die?
(Among Mathematicians)
So it died sometime between the wars. Candidate explanations:
• Professional specialisation—mathematicians stopped doing philosophy
No. Disciplines reach for philosophy when they feel the need—sociology, etc.
• Zeitgeist shock—everybody stopped doing Philosophy (Hersh)
No. What explains all explains nothing. Besides, International Encyclopedia of
Unified Science
• Mathematics ceased to be, in the eyes of mathematicians, the sort of thing that requires
foundations in the architectural sense.
What sort of change was this?
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15. When and why did the architectural metaphor die?
Albert Lautman following Hermann Weyl
All this new mathematics, that of the theory of groups and abstract algebras, is animated by
a spirit that is clearly different from that of ‘classical mathematics’, that found its highest
flowering in the theory of functions of complex variables.
Weyl Gruppentheorie und Quantenmechanik preface 1928, quoted in Lautman
In contrast to the analysis of the continuous and the infinite, algebraic structures clearly
have a finite and discontinuous aspect. Though the elements of a group, field or algebra…
may be infinite, the methods of modern algebra usually consist in dividing these elements
into equivalence classes, the number of which is, in most applications, finite.
Lautman 2006 pp. 86-87
Lautman also contrasts local with global
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16. When and why did the architectural metaphor die?
Methods independent of operations
[Noether] described ‘purely set-theoretic’ methods as ‘independent of any operations’
(Noether 1927, 47). These methods do not look at addition or multiplication of the
elements of a ring (or a group, etc.).
They look at selected subsets and the corresponding homomorphisms. The
homomorphism theorem for commutative rings correlates homomorphisms to all ideals. In
groups in general, homomorphisms are correlated only to normal subgroups. So she
looked at groups in terms of their normal subgroups and homomorphisms. She looked at a
ring in terms of its ideals and ring homomorphisms, since those correspond in the
homomorphism theorem for rings.
Colin McLarty “Emmy Noether's `Set Theoretic' Topology: From Dedekind to the rise of functors''
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17. When and why did the architectural metaphor die?
Did Noether do it in?
History is never so simple
Van der Waerden’s Modern Algebra is marked by foundational scruples:
“…I have tried to avoid as much as possible any questionable set-theoretical reasoning in
algebra. Unfortunately, a completely finite presentation of algebra, avoiding all non-
constructive existence proofs, is not possible… On the other hand, it was possible to
compile the building stones for a constructive foundation of algebra… “
Preface to the second edition, p.v (1953)
But we can distinguish:
– Hilbert/Gray-modernism
– Noether-modernism
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18. Final thoughts on PMP and foundations
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19. Mathematical Cultures
Research Network
A series of three conferences with associated publications on mathematics as culture and
mathematics in culture.
A research network funded by the Arts and Humanities Research Council under the
'science in culture' highlight notice, with additional support from the London Mathematical
Society.
The first conference will gather research that explores and maps the variety of and
connections among contemporary mathematical cultures.
It will take place at De Morgan House (London) 10-12 September 2012
https://sites.google.com/site/mathematicalcultures/
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