Palimpsest and Bricolage.pptx

Palimpsest and
Bricolage
CLPS seminar 24 June 2021
Brendan Larvor - Onbezoldigd (emeritus) 1

2
Conceptsdevelopanddifferentiate. Sometimes,earlyversionsofaconceptwillbecompletelysuperseded,butphilosophically
importanttermsoftenretainmultiplesensesthatplayintoeachother(sotheyarenotsimplyhomonyms). Nietzschemakes
thispointwiththeword'punishment'andtellsastoryofsemioticaccretionthroughtheexerciseofpower. Nietzscheismore
orlessrightformoralandreligiousterms,exceptthatheneglectsthegreatstructuringtendencyofmodernlife:rationalisation
inthesenseofWeber.
Turningtomathematics:someconceptsaretechnicallydefinedandmaytherebyfalloutofthisdiscussion(suchas
'group'). Othersareasrichandinvolvedas'punishment'--forexample'mathematics'. Butwhataboutintermediatecases,
suchas'number'and'proof'? Inthesecases,I'llarguethatFerreiróspointsintherightdirectionbutneedsabitofhelpfrom
NietzscheandWeber.
How do versions of the same concept relate to
each other?

My long-term problem: proof
What is the relation between the proofs
that mathematicians actually write and
the proofs studied in formal logic (and
invoked in the ‘standard view’)?
Yehuda Rav, JodyAzzouni, Reuben
Hersh, JeremyAvigad, Ken Manders,
Yacin Hamami… etc..
The Standard View:
(approximation sufficient for
motivation):
Any rigorous informal proof can be
routinely or mechanically translated
into a formal proof.
Me: Traduttore, traditore!

‘Proof’is a word with a long history with
related meanings (including the formal
ideal and various mechanical
realisations). How should we think
about such words?

‘Proof’is a word with a long history with
related meanings (including the formal
ideal and various mechanical
realisations). How should we think
about such words?
We could just insist that philosophy
gives accounts of terms with stable
essences:
“[My] account is not actually of proof in
general, but only of rigorous proof...
…questions like “what is mathematical
proof?”, asked in full generality, are
unlikely to receive a satisfying answer:
there is no univocal notion of proof in
mathematics, or at least not one we can
expect to obtain a substantial philosophical
analysis of.”
—Oliver Tatton-Brown,
Review of Symbolic Logic
This assumes that tidy concepts
can be fully understood with no
residual arbitrariness without
reference to their scruffy relatives.

Thinking diversity in relation
“That the Historyof the World,with all the
changingsceneswhich itsannalspresent,is this
process of developmentandthe realizationof
Spirit– thisis the true Theodicaea,the
justificationof God inHistory.Only this insight
canreconcileSpiritwith the Historyofthe
World – viz.,thatwhat has happened,and is
happeningevery day, is notonly not“without
God,”butis essentiallyHisWork.”

The Idea of History
The Idea of Nature
…
The Idea of Proof
The Idea of Numbers
The Idea of …
RG Collingwood

Mathematics is a “MOTLEY/COLOURFULmix of
techniques of proof” (RFM III, §46)
(“ein BUNTES Gemisch von Beweistechniken”)
—LWnotices the diversity but does not think about the
relations
‘Monadology of language games’
(Also: family resemblance concepts are not helpful.)

Ein BUNTES Gemisch von Beweistechniken

Ein BUNTES Gemisch von Beweistechniken
“Philosophers are now faced with the conceptual agility of 20th-
century mathematics: hopping a functor to another category any
time superfluous details are sensed (homology groups in topology);
bringing in details seemingly extraneous to one’s question by a
representing functor (group representations in the theory of
abstract groups); explaining families of simple number-theoretic
facts anyone can see one-by-one, by some hard to master
“underlying” abstract structure (Fermat’s problem).”
Ken Manders ‘Euclid or Descartes? Representation and
Responsiveness’

“Today it is impossible to say
precisely why people are actually
punished: all concepts in which an
entire process is semiotically
concentrated defy definition; only
something which has no history can
be defined.”
(GM II 13)

Punishment as a means of rendering harmless, of
preventing further harm.
Punishment as payment of a debt to the creditor in any form
(even one of emotional compensation).
Punishment as a means of isolating a disturbance of
balance, to prevent further spread of the disturbance.
Punishment as a means of inspiring the fear of those who
determine and execute punishment.

Punishment as a sort of counter-balance to the privileges
which the criminal has enjoyed up till now (for example, by
using him as a slave in the mines).
Punishment as a rooting-out of degenerate elements
Punishment as a festival, in the form of violating and
mocking an enemy, once he is finally conquered.
Punishment as an aide memoire, either for the person
suffering the punishment – so called ‘reform’, or for those
who see it carried out.

Punishment as payment of a fee stipulated by the power
which protects the wrongdoer from the excesses of
revenge.
Punishment as a compromise with the natural state of
revenge, in so far as the latter is still nurtured and claimed as
a privilege by more powerful clans.

Punishment as a declaration of war and a war measure
against an enemy of peace, law, order, authority, who is
fought as dangerous to the life of the community, in breach
of the contract on which the community is founded, as a
rebel, a traitor and breaker of the peace, with all the means
war can provide.
Etc..
–(GM II 13)

“…the concept ‘punishment’
presents,… not just one meaning
but a whole synthesis of ‘meanings’
[Sinnen]: the history of punishment
up to now in general, the history of
its use for a variety of purposes,
finally crystallizes in a kind of unity
which is difficult to dissolve back into
its elements, difficult to analyse…”
(GM II 13)

Rationalisation, including
intellectualisation and analysis.
Nietzsche can make that list of the
meanings of punishment because
he is a late product of Weberian
processes.
Max Weber

Thinking diversity in relation:
the interplay of practices

These examples run the risk of making Ferreirós look
like Collingwood!
This is not essential to the logic of his scheme.
Later examples and remarks suggest it’s not his view.

E.g: we might think that the natural development of reckoning
arithmetic is the Peano axioms.
BUT
Reckoning practice makes no use of the inductive properties of
the natural numbers. It connects much more easily with
commutative algebra.
e.g. Algebraic integers and square matrices are like the regular
integers because they are commutative rings—but they are not
inductive sets.
If you want something that feels number-like, go for a
commutative ring.

E.g: we might think that the natural development of reckoning
arithmetic is the Peano axioms.
BUT
Reckoning practice makes no use of the inductive properties of
the natural numbers. It connects much more easily with
commutative algebra.
e.g. Algebraic integers and square matrices are like the regular
integers because they are rings—but they are not inductive sets.
If you want something that feels number-like, go for a
commutative ring.
Maths Nietzsche might say:
…the concept ‘number’ presents,… not
just one meaning but a whole synthesis
of ‘meanings’ [Sinnen]: the history of
numbers up to now in general, the
history of its use for a variety of
purposes, finally crystallizes in a kind of
unity which is difficult to dissolve back
into its elements, difficult to analyse…
IOW the history of the concept of
number is not the uncovering of an
essence. It’s more like Weberian
rationalization acting on a complex
unity arising from practices.

Aquestion for native speakers of
German:
I’m leaning hard on GM II 13. Is the
Diethe translation reliable?

What about proof?
Formalisation programmes and programs are guided in the details of their
design by the whole BUNTES Gemisch, and we’ll find out as they develop
how that goes.
E.g. students trained on Lean write informal proofs differently, and their
professors’experience with the whole BUNTES Gemisch guides their
judgments about whether this is a Good Thing.

What about philosophy?
Nietzsche is right, the interesting practices are always a BUNTES Gemisch.
Resist unnatural tidiness!
Study the scruffy cousins of respectable concepts.

Thank you

Palimpsest and Bricolage.pptx

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Palimpsest and Bricolage.pptx