[FOM] Mathematics and precision

[Vladimir Sazonov]

Quoting Henrik Nordmark <henriknordmark at mac.com> Sun, 04 Mar 2007:

> And how high should the threshold be?
> High enough that we feel confident that in principle what we are
> doing can be formalized even though we may not want to bother doing so.

Of course (I hope) nobody intends to force mathematicians to write 
absolutely formal proofs. (However, e.g. programmers do write 
absolutely formal programs, and there are also some projects of writing 
absolutely formal mathematical proofs.) In the case of proofs or 
algorithms we deal, indeed, with formalizability. But, nevertheless,

the SUBJECT MATTER of mathematics is, I believe, just FORMAL SYSTEMS 
(provided they serve to support, organise and strengthen our inevitably 
vague, amoeba-like intuition).

Proving mathematical theorems (with interesting meaning) in these 
systems, relating (interpreting) systems one with another and also with 
the real world or with other sciences leads to making them 
usable/applicable in various senses. Unlike software engineering, 
formal systems considered by mathematicians are quite simply 
describable (for example, ZFC, or the rules for manipulating with 
derivatives or integrals, etc.). In a sense, mathematics is also a kind 
of engineering of formal tools to support our abstract thought ability 
(abstract = applicable virtually to anything whatever we could 
imagine). On the other hand, software engineering is concerned mainly 
with the routine part of our thought and informational activity. This 
makes the difference between these two "relatives" sufficiently strong. 
And, indeed the style of thought of programmers and mathematicians is 
usually rather different. It might look strange (although quite 
explainable), but however dealing with absolutely formal objects 
programmers are usually not inclined to mathematical standards of 
rigour when they are reasoning about their programs or formal systems. 
They are too huge and complex to be considered rigorously in the truely 
mathematical manner. This is difficult, but there is the place for the 
cautious optimism. This is the subject of Theoretical Computer Science 
which I would consider as a branch of mathematics also related with 
f.o.m.


Vladimir Sazonov


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