[FOM] Formalization Thesis vs Formal nature of mathematics
S. S. Kutateladze
sskut at math.nsc.ru
Wed Jan 2 20:58:18 EST 2008
Sazonov wrote :
Mathematical proofs are always sufficiently formal (just
to such degree to which it is necessary/essential).
Otherwise they are not mathematical.
___________________________________________
I appreciate that you use the words "sufficiently"
and "necessary" in regard to proofs so informally.
Sazonov wrote:
It was mathematics (not just speculations) to that degree to
which it was formalized (in the wide sense of this word),
in fact, "fragmentary" ...
Mathematicians of that time realized that something
was non-well. This was the lack of rigour (the lack of
mathematics in these considerations).
_________________________________
I disagree that there was any lack of mathematics in
the mathematical discoveries of Newton, Leibniz, and
other forefathers of the calculus.
Sazonov wrote:
formalist paradigm (or view) on mathematics...
I see nothing in mathematics what is not covered by this paradigm.
________________________________________________
You see all of mathematics from a formalists
point of view. I do not doubt this, but this
does not exclude other views of the whole of mathematics.
Sazonov wrote:
This probably was so trivial to him [Euclid]... We, nowadays, also
allow various such omissions which are usually easily
recovered by any qualified reader.
_____________________________________________
This claim is far from formal and, well, disputable as regards "easily."
It took two thousands years to propose a formal modern definition
of a triangle. However, I'll add an argument in favor of formalism at this point.
We often end a proof with claiming "obvious." However,
we abstain from refer to the Indian Sutras pictures of the
Pythagorean theorem as proofs. The reason is that "obvious"
is personal or subjective. Mathematics starts with a quest
for more "impersonal" or objective proofs.
Sazonov wrote:
We have reached a kind of absolute rigour,
at least a "stable" point.
_______________________________________________
I doubt this. We are not smarter that our ancestors.
Please note that the drastic twists and turns of the
mathematical views of foundation in the twentieth century.
I do not think that this will terminate ever.
Sazonov wrote:
> The quest for mathematical knowledge is mathematical but may be
> informal.
There is always the place for speculations on formal devices
for thought created and studied in mathematics. This does not
change its formal nature in the sense that it is an activity
by creating such formal "devices".
___________________________________________
Mathematics is human rather than formal. "Human" relates to the mentality of man
as a human being. We remain the same biologically. "Formal" relates to the form of mathematics.
The form of mathematics changes, whereas its human nature remains the same.
Sazonov wrote:
> The nature of math is human whereas the objects of math are formal.
Should I conclude that you actually agree with me?
_______________________________________________________
Yes, as regards the objects of mathematics which are indeed the forms of human reasoning.
Sazonov wrote;
Do you believe (or may be know???) that mathematical objects
and truths exist objectively and that they are not results
of interplay between our intuition/imagination and formalisms?
______________________________________________
My answers to the two questions differ.
In my opinion, the mathematical objects and truths exist objectively, in a sense,
provided the existence of mathematicians.
However, I think that the mathematical objects and truths emanate from playing humans.
---------------------------------------------
Sobolev Institute of Mathematics
Novosibirsk State University
mailto: sskut at math.nsc.ru
copyto: sskut at academ.org
http://www.math.nsc.ru/LBRT/g2/ruswin/ssk/index.html
Sazonov wrote :
Mathematical proofs are always sufficiently formal (just
to such degree to which it is necessary/essential).
Otherwise they are not mathematical.
___________________________________________
I appreciate that you use the words "sufficently"
and "necessary" in regard to proofs so informally.
Sazonov wrote:
It was mathematics (not just speculations) to that degree to
which it was formalized (in the wide sense of this word),
in fact, "fragmentary" ...
Mathematicians of that time realized that something
was non-well. This was the lack of rigour (the lack of
mathematics in these considerations).
_________________________________
I disagree that there was any lack of mathematics in
the mathematical discoveries of Newton, Leibniz, and
other forefathers of the calculus.
Sazonov wrote:
l formalist paradigm (or view) on mathematics...
I see nothing in mathematics what is not covered by this paradigm.
________________________________________________
You see all of mathematics from a formalists
point of view. I do not doubt this, but this
does not exclude other views of the whole of mathematics.
Sazonov wrote:
This probably was so trivial to him [Euclid]... We, nowadays, also
allow various such omissions which are usually easily
recovered by any qualified reader.
_____________________________________________
This claim is far from formal and, well, disputable as regards "easily."
It took two thousands years to propose a formal modern definition
of a triangle. However, I'll add an argument in favor of formalism at this point.
We often end a proof with claiming "obvious." However,
we abstain from refer to the Indian pictures of the
Pythagorean theorem as proofs. The reason is that "obvious"
is personal or subjective. Mathematics starts with a quest
for more "impersonal" or objective proofs.
Sazonov wrote:
We have reached a kind of absolute rigour,
at least a "stable" point.
_______________________________________________
I doubt this. We are not smarter that our ancestors.
Please note that the drastic twists and turns of the
mathematical views of foundation in the twentieth century.
I do not think that this will terminate ever.
Sazonov wrote:
> The quest for mathematical knowledge is mathematical but may be
> informal.
There is always the place for speculations on formal devices
for thought created and studied in mathematics. This does not
change its formal nature in the sense that it is an activity
by creating such formal "devices".
___________________________________________
Mathematics is human rather than formal. "Human" relates to the mentality of man
as a human being. We remain the same biologically. "Formal" relates to the form of mathematics.
The form of mathematics changes, whereas its human nature remains the same.
Sazonov wrote:
> The nature of math is human whereas the objects of math are formal.
Should I conclude that you actually agree with me?
_______________________________________________________
Yes, as regards the objects of mathematics which are indeed the forms of human reasoning.
Sazonov wrote;
Do you believe (or may be know???) that mathematical objects
and truths exist objectively and that they are not results
of interplay between our intuition/imagination and formalisms?
______________________________________________
My answers to the two questions differ.
In my opinion, the mathematical objects and truths exist objectively, in a sense,
provided the existence of mathematicians.
However, I think that the mathematical objects and truths emanate from playing humans.
---------------------------------------------
Sobolev Institute of Mathematics
Novosibirsk State University
mailto: sskut at math.nsc.ru
copyto: sskut at academ.org
http://www.math.nsc.ru/LBRT/g2/ruswin/ssk/index.html
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