[FOM] mathematics as formal

Michael J Barany mjb245 at cornell.edu
Wed Mar 5 09:24:58 EST 2008

Hi everyone,

This seemed like a good opportunity to make a first post to this list.

It looks like you're distinguishing between "formal mathematics" (mechanically
verifiable) and "mathematical formalism" (abstract manipulation of signs and
symbols), though they are certainly quite connected in history and in
practice.  For formalism in modern Western mathematics, the iconic figure is
Euler, who was quite nearly the only person in millenia before him and a
century afterward to produce great mathematical results without bothering to
explain them in terms of Euclidean geometric constructions.

My work is on Cauchy's Course in Analysis, which I argue formed a sort of
conceptual bridge between Euler and the founders of "formal mathematics" a
century later in Germany (Weierstrass, Dedekind, Kronecker, Cantor, Heine,
Hilbert,... and of course Peano and others elsewhere).  Cauchy's innovation
was making formal mathematics, *when subject to additional rules and
methodological constraints*, viewable as rigorous.  In his case, these
additional rules were made to make algebraic manipulations work more like
geometric manipulations, which is why Cauchy was so insistent on convergence

The second half of the nineteenth century was dominated by attempts to try out
different axiom systems and to experiment with different formal structures. 
Mechanical formalism reached a height of absurdity in Russell and Whitehead's
Principia Mathematica from 1910, which is 3 volumes of formal derivations,
half of which is a lead-up to 1+1=2.  Then, there was the interwar period,
which was full of debates between positivists and anti-positivists, and
included Godel's theorem.  Positivism and logical formalism share many similar
philosophical premises and ambitions.  Formalism in mathematics underwent its
next big transition with the rise of electronic computers.  See Donald
MacKenzie's Mechanizing Proof for an excellent discussion of those

There's another piece which might be more closely related to your question. 
Early modern mathematics (mostly 17th century) was marked by a huge
deification of Euclid.  Hobbes was famous for arguing that Euclid should be
the model not only of math but also of philosophy: start with areas of
agreement and proceed by rational and logical arguments, thereby compelling
assent.  It has been argued that the popular view of Euclid as one of the
first *formalizers* originated around or just before this period.  Indeed,
Euclid's axioms have been mechanized and formalized in many different ways by
many different people over the last century and a half (see as bookends
Hilbert and computer programs like Geometer's Sketchbook).



> 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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