[FOM] Mathematics ***is*** formalising of our thought and intuition

Vladimir Sazonov vladimir.sazonov at yahoo.com
Mon Jun 7 19:45:39 EDT 2010

----- Original Message ----
> From: Marc Alcobé <malcobe at gmail.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Sent: Mon, June 7, 2010 8:48:47 AM

Dear Mark, 

> As I told in some post before, I guess one should think about how
> mathematics come to existence. 

For me the brightest examples are 

1. Multiplication of natural numbers in decimal notation 
by using known school formal rules. How much this economised 
thought (the goal of mathematical formalisms). 

2. Euclid's Geometry giving templates of practically contemporary 
mathematical rigour, the templates used by mathematicians 
again and again and so making our thought powerful. 

3. Analysis/Calculus. Recall how a planet was discovered by 
calculations based on the formal rules of the Calculus 
(and, of course, the lows of mechanics which without the Calculus 
would be able to give only qualitative picture of moving planets) -- 
another strongest confirmation how formalisms make our thought 

That is why formalising is the goal of mathematics and why 
mathematical rigour is in the blood of all mathematicians 
even when it is not directly seen that it will lead 
to so strong formal tools for our thought. 

Take the problem of Königsberg bridges
> as an example. If you pay particular attention to the means by which
> it is solved you'll realize that one departs from a more or less
> idealized situation with riversides, islands and bridges, and ends up
> dealing with entities that not only do have nothing to do with all
> that, but also serve equally well to deal with other kind of different
> situations.
> Mathematics exist because problems exist, and because some of these
> problems can be solved by certain means: mainly idealization,
> abstraction, generalization, particularization, etc... the kind of
> heuristic means Polya used to stress (and others possibly to be
> discovered), we call mathematical (proof is only a small, though
> important, part of the whole process, I believe).

Not small (and simply important) but crucial. 

All you mentioned above (idealization, abstraction, etc.) is 
covered by the phrase 

        formalising our ***thought and intuition***

but not vice versa! How abstraction, etc. without assuming 
formalisation would lead to, e.g. the Calculus? To get 
a calculus we should anticipate in advance that we will be 
looking for some kind of formal rules. Otherwise the whole 
enterprise would give something vague and hardly would lead  
to the ability to predict that some unknown planet should 
appear in this part of the sky at this particular time. 

All these abstractions and idealisations are comfortable for 
philosophers to discuss on mathematics. Working mathematicians 
think in terms of (besides of abstractions, etc.) something much 
more concrete -- in terms of some formal "instruments" 
(formal rules like the rule for derivative of multiplication: 
(uv)' = u'v + uv'; this is their working apparatus). 

Stressing on the ***form*** of thought does not reject or 
diminish the thought itself and all its fundamental attributes. 
Also devising a formal proof is not an algorithmic or 
mechanical process. It involves our thought and intuition related 
with a formal system which formalises some our specific thought 
and intuition. 

> Physics deals with idealizations of physical situations (with things
> that stand for other things, like the painter who paints a landscape,
> or a portrait). Similarly for any other sciences: they model realities
> different from the objects they use to model them. Mathematics deals
> with abstractions, i. e. with objects that are seen as standing for
> themselves, and nothing else, no matter where they have come from
> (maybe this is not the usual sense of the term abstraction, it could
> be thought of as a kind of extreme idealization).
> So, the task of the physicist is to test his models against the
> reality he pretends to model. But so is the task of the mathematician,
> unless he or she is an extreme formalist, because his/her results must
> conform to the "self-reality" he pretends to model, be it either
> numbers, sets, cats or mathematical statements.

There are no (and there were never) working mathematicians who are 
extreme formalists. The term "extreme formalists" (or the like) 
was especially created to win in the ideological (religious?) "war" 
by other "...ists" to ridicule the formal view of mathematics instead 
trying to realise what is the rational point of this view, that this 
view actually does not reject intuition, abstraction, idealisation, 
imagination, etc., but rather assumes all of this if to formulate 
this view appropriately. 

If formal character of mathematics and mathematical rigour are 
considered only as some additional feature of mathematical thought 
then it becomes unclear why it should be ever compulsory and why all 
mathematical proofs are actually formalisable, and it is also unclear 
what is mathematics about since then it has no clear "pivot". 
Then either Platonism with its supernatural absolute truth and 
pretension on a kind of Realism (having nothing to do with the 
real world) becomes the last refuge or concurring Intuitionism 
(although quite a reasonable approach, initially presented as 
a strong constraint on mathematics; later represented by various 
formal systems of a specific kind and the related intuition), 
or whatever else. 

Formalist view on mathematics covers everything (essentially 
by one simple phrase) and does not reject (but rather assumes 
implicitly) anything whatever we could want to reasonably 
include in mathematics. Abstractions, idealisations, etc. 
do not explain the nature of mathematics because they belong 
to ANY kind of human thought about any subject matter and so 
cannot clearly distinguish mathematics from anything else. 
On the other hand, "formalising our thought and intuition" 
automatically assumes abstraction because pure form (without 
the content) of thought automatically implies that this thought 
is abstract. (The smile without the cat as you wrote in other 
posting.) Moreover, formalisation implies a kind of automation 
of our thought (see above examples) and thereby making our 
thought so incredibly powerful. And it is exactly this what 
is the goal of mathematics. 

If a human activity with exactly this goal would not exist 
it should have been invented by people earlier or later. 

Best wishes, 



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