[FOM] query on the history of the philosophy of mathematics: mathematics as formal

jaN. bov234 at gmail.com
Wed Mar 5 05:35:06 EST 2008

dear catarina;

i have recently read a book that might be of interest for your query. it 
deals with the history of formalization both in mathematics and logic. 
therein the history of mathematics and logic is reconsidered with 
respect to the emergence of  formalization. in this view formalization 
is bound to three conditions:
(1) written symbol manipulation
(2) schematic use of symbols
(3) interpretation-free use of symbols
formalization is thus bound to (1) a medium that permits to fix 
unambiguous, graphic symbols in a decisive order (the book places 
special emphasis on this point). the (un-)correctness of a formal 
description results (2) from its compliance with a scheme. it is this 
infinitively repeatable compliance that constitutes the meaning of the 
formal description. formal descriptions presuppose (3) the distinction 
of a formal language and a metalanguage. the punchline of this 
distinction is that one can decide whether an expression inside of a 
formal language is right or wrong without referring to the 
interpretation of that expression.
the book tries to catch the step-by-step development of the conditions 
(1)-(3). it suggests that in logic the conditions were fully achieved 
for the first time in leibniz' logical calculus, in mathematics in 
francois vieta's algebra.
the title of the book is:

sybille krämer: 'symbolische maschinen. die idee der formalisierung in 
historischem abriß.' darmstadt, wissenschaftliche buchgesellschaft, 1988

roughly  in english: 'symbolic engines. a historical survey of the idea 
of formalization.' unfortunately i do not know if there is an english 


catarina dutilh schrieb:
> Dear all,
> A while ago, we had a very interesting discussion within this list concerning the formalizable character of mathematics (which was meant to be captured by Chow's Formalization Thesis). Among other things, one interesting fact that emerged from the discussion was that some took being formal -- and thus formalizable -- as an essential feature of mathematics, while others insisted that actual mathematical practice goes much beyond what can be formalized into mechanically checkable proofs.
> In this context, I now have a historical query: since when is it widely (even if not unanimously) held that what is distinctive about mathematics is its *formal* character? Who were the first people to hold such a thesis, and what were their underlying motivations? What was the 'pedigree' of the notion of formal in question: was it the Aristotelian form vs. matter distinction or the Platonic idea of Forms (or perhaps neither)?
> Just to give an idea of what I'm after: in his excellent PhD dissertation, 'What does it mean to say that logic is formal?', John MacFarlane argues that the source of the idea that what is distinctive about *logic* is its formal character is to be found in Kant. I'm looking for a similar analysis concerning mathematics, so basically this is a query concerning the history of the philosophy of mathematics.
> At the moment, I am working on the history of the notion of 'formal', in particular but not exclusively with respect to logic. So even though mathematics is not my main concern, I feel that my story would be missing an important piece if I did not at least mention the history of the attribution of formality to mathematics. So your help on this matter would be much appreciated!
> Many thanks in advance,
> Catarina
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