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

[Vladimir Sazonov]

----- Original Message ----
> > From: Robert Lindauer <rlindauer at gmail.com>
> > To: Foundations of Mathematics <fom at cs.nyu.edu>
> > Cc: vladimir.sazonov at yahoo.com
> > Sent: Fri, May 28, 2010 7:48:53 PM
> > Subject: Re: [FOM] Mathematics ***is*** formalising of our thought and intuition
> > 
> > Consider: "Political science is the art and practice of devising and
> using 
> > conceptual spaces where long sequences of reasoning steps can be
> carried out 
> > without error"
> 
> The problem is that your definition 
 
 
Please note that you are reformulating a definition of hendrik at topoi.pooq.com. 
I only confirmed that it is good enough version of my version which 
is briefly stated in the Subject of this posting. Although it is not my 
formulation (which is more preferable to me), I would try to defend both  
(with more stress on my definition). Of course, I cannot be absolutely 
sure that my opinion coincides with that of Hendrik. 
 
 
would make mathematics 
> -the only-
> science, and that's not reasonable.  

 
Mathematics is indeed ***the only*** science (or a kind of engineering 
creating formal tools for making our thought and intuition powerful; 
see also a comment on Computer Science below) as I wrote in a recent 
posting where, also as Hendrik suggests, "long sequences of reasoning 
steps can be carried out without error" with a guarantee. In all other 
science, even in Physics (and least of all in Political Sciences and 
Philosophy), long (and even not so long) sequences or reasoning steps 
are much less reliable. In mathematics these steps are formal (with 
"formal" understood either as "semi-formal" or as "absolutely formal" 
like in computers; see also my other recent postings). Correctness 
in mathematics is determined exclusively by the form of reasoning 
rather by its content. This does not hold even in Physics which 
stresses on the truth in the real world but not on the (mathematical) 
rigour. 
 
 
Surely mathematics 
> is foundational in that every science must make some use of 
> mathematical concepts and procedures, however, it does not follow that the 
> whole content of those other sciences (e.g. chemistry) is 
> mathematical!
 

Here I completely agree. No other science creates formal tools 
for thought. Other sciences study some kind of reality. 
Only Computer Science (Programming, Software Engineering and AI) 
is like mathematics in this respect (creating also formal tools for 
thought), but it deals with the ***routine*** part of human thought, 
unlike mathematics. 
 

> 
> Hegel's definition is classic and correct:
> 
> "One could still, however, conceive the thought of a philosophical
> mathematics, namely, as a science which would recognise those concepts
> which constitute what the conventional mathematical science of the
> understanding derives from its presupposed determinations, and
> according to the method of the understanding, without concepts. "

 
I could guess that (1) "derives", (2) "according to the method of the 
understanding" and (3) "without concepts" above could mean (1) derivations 
(2) based on the form of reasoning (3) without taking into account 
the content. 

 
Your reformulation below seems somewhat similar, but I believe 
that my (based on the explicit opposition of the form of the reasoning 
vs its content) is more clear and relates more directly to the contemporary 
mathematical formalisms. Hegel refers to "method" which I explicate 
as "form of reasoning". You seemingly omit this at all: 
 
 
> 
> Or, think more plainly, pure mathematics is the systematic
> consideration of proto-mathematical concepts (space/time, infinite,
> finitude, recursion, grouping, shape, etc.) without regard to the
> conceptual content (that is, without regard to the concrete concept
> itself or the things to which they are to be applied).

You write "without regard to the concrete concept itself or the things 
to which they are to be applied". I think this should be explained 
in the way as I do --- in terms of "formal" (in a general philosophic 
sense of this word or in the limit, contemporary sense of "absolutely formal". 
I also explain in my postings why "formal" is so important in itself 
(as providing a mechanisation and automation tool of thought). 
"Without regard to the concrete concept itself or the things 
to which they are to be applied" means a kind of universality 
--- applicability potentially anywhere. But how this is achieved? 
If we abstract from the content of our reasoning, then only the 
form of the reasoning remains, and the form could be filled in 
by various content, so it implies universal applicability. 
Thus, "formal" seems to me explaining better the nature and 
the power of mathematics. 

 
Best wishes, 
 
Vladimir
 
> 
> Best,
> 
> Robbie Lindauer