[FOM] Sazonov on intuitive and formal mathematics

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Tue Jul 31 15:47:24 EDT 2007

Quoting Gabriel Stolzenberg <gstolzen at math.bu.edu> Sun, 29 Jul 2007:

>>   Without intuition formalization is not interesting. Without
>>  formalizability intuition is too poor and vague to be considered
>>  as mathematical.
>   I'm sympathetic to this view.

- the view of a convinced formalist.

>> >   [Hilbert] succeeded in saving classical mathematics by a
>> >   radical reinterpretation of its meaning without reducing
>> >   its inventory, namely, by formalizing it, thus transforming
>> >   it in principle from a system of intuitive results into a
>> >   game with formulas that proceeds according to fixed rules.
>      (Weyl)
>> "Game with formulas" assumes "meaningless". But that is wrong.
>> Interplay of intuition with formalism is not a meaningless game.
>> Who said that formalization excludes or removes the intuition?
>   I think you did because formalization excludes the part of the
> intuition with which there is the interplay about which you speak.

There are possible two reading of "formalizing":

1st reading: Stripping out skin, hair, heart, brain, etc. till the skeleton.

But in mathematics, before formalizing, we had rather cartilages 
instead of rigid skeleton.

2nd reading: Strengthen the cartilages in the body to solid bones.

Which one do you prefer?

The above statement of Weyl means that by formalizing mathematics 
something happened quite a bad (some unfortunate "reinterpretation of 
its meaning"); the meaning in mathematics was replaced by a game with 
formulas. But it was not. What, in fact, happened is that formalisms 
were introduced TO INTERPLAY with our (already existing and possibly 
renewed due to formalization) intuitions and imaginations and TO 

>   Finally, perhaps you could also explain how Hilbert's remark
> that "[the logical laws] that Aristotle taught do not hold" fits
> with your point of view.

Literally, it looks to me as a contradiction: which logical laws of 
Aristotle do not hold in ZFC, for example? Also I do not know the exact 
context of this phrase. I think it is fair enough to interpret it as 

When trying to formalize some new imaginary world (say, of infinite 
objects - sets) we should realize that it is only imaginary one and 
nothing is true or false there in the same sense as in the ordinary 
physical world or for not so big finite objects, and no logical laws 
hold there just because they are "objectively" true - "do not hold" in 
this sense. Thus, when we DECIDE to impose the ordinary (or any other 
preferable) logical laws onto this world, it is not because they are 
true there. It is because this is OUR decision, assuming it is 
sufficiently coherent with our imagination and sufficiently robust. 
(The coherence is typically incomplete; also various surprises - 
counterexamples - are possible which would rather "correct" our 
intuition. And we so much respect these formalisms that we usually are 
quite happy with these corrections.) We create our own worlds and 
"play" there by the laws of some logic we choose, let Aristotelian.

Vladimir Sazonov

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