[FOM] Syllogistization thesis / Formalization thesis

BEZIAU Jean-Yves jean-yves.beziau at unine.ch
Sat Dec 29 07:37:30 EST 2007

I think it is worth comparing the Formalization thesis according to which
all mathematics can be reduced to set theory to another thesis:
(SY) Syllogistization Thesis: all mathematics can be expressed by chains of syllogisms
There was a time when famous mathematicians believed that this thesis was true.
Nowadays such kind of thesis seems absurd.
I think that the Formalization thesis will appear in the future as absurd as SY,
in fact it looks already quite absurd.
As we know syllogisms catch only very superficial aspects of mathematics,
and we can say the same about set theory,
one may think we are getting closer,
following a scientist *line* of thought,
according to which science is improving
getting closer to reality
but science maybe is not a linear order.
Mathematical proofs can be expressed for example by pictures
and it makes no sense to say that these pictures
can be expressed by set theory
or by chain of syllogisms
maybe there is something  "corresponding",
but it is not the same.
Wittgenstein in his second period was supporting proofs as pictures
inspired  by Schopenhauer.
See also the more recent book by Nelsen, "Proofs without words".
Here is an interesting quotation by Jean-Yves Girard: 
"the XXth century has been the century of totalitarisms of every possible kind,
in particular the linguistic totalitarism (rebaptised "turn").
This extreme form of scientism, which consisted in the reduction of
any mathematical question to a question of formal, linguistic, bureaucratic, protocols:
Kafka was waiting behind the door." 
(Lectures on essentialism and incompleteness, 2006)
JYG was one of my teachers
a guy with great ideas.
It is interesting to examine and understand 
the Formalization thesis and to see its limitation.
Formalization, what a nasty world !

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