[FOM] mathematics as formal

[Vladimir Sazonov]

> From: Vaughan Pratt pratt at cs.stanford.edu

> Vladimir Sazonov wrote:
> > My definition: 
> > 
> > Mathematics is nothing else as taking and exploring the form of 
> > human thought seriously and consistently

in mathematical practice 

[- seemingly important addition; 
also exploring should seemingly be better replaced by exploiting, 
developing]

> 
> I can't accept this as I don't see how it distinguishes an Archimedes, 
> Euler, Gauss, or Erdos from those who debate logical positivism, 
> freewill, formalism, etc. I don't consider the latter mathematicians 
> unless they bring mathematical tools to bear effectively on their subject.


In my understanding, 

"bring mathematical tools to bear effectively on their subject" 
= 
"taking and *exploiting* the form of human thought seriously 
and consistently (in mathematical practice)".

Does not this make the distinguishing you want? Note that 
"seriously and consistently" above does not mean taking 
the form of thought by working mathematicians philosophically. 
They can well have quite queer and probably wrong philosophical 
views on math. The point is HOW they do mathematics objectively, 
irrespectively on what they THINK about HOW they do it. 
That is why I used the word "instinctively" in my previous posting. 
HOW they do this can be observed and explained by an objective 
observer, may be psychologist. 


> 
> I also have trouble with "human". Aliens might not be DNA-based, 

I do not insist on "human", but do not want to generate my own 
fantasies on aliens (and even on artificial intelligence) either, 
at least for a while. 

By saying "human" I rather alluded to possible participation 
of psychologists (as soon as they understand mathematics 
sufficiently well) in explaining what is mathematics 
by making an objective research on mathematical thought. 

I guess, you want to say that there is some objective subject of math. 
independent on human beings (or even on aliens) or on the "thought" 
which is ascribed to some human or non-human beings. (A way to 
Platonism and so called "Realism" - in my opinion just subjectivism.) 
I consider that the goal of mathematics is to generate tools of 
thought making it stronger. (Examples - again the Calculus of 
differentials and integrals, the rules of multiplication of decimal 
numbers, etc, etc.) These tools are applicable both inside and 
outside of mathematics, say, by physicists (thereby making their 
thought on the movement of physical bodies or the like miraculously 
stronger). The source of strength is exactly in the formal character 
of mathematics (the Calculus, etc.) That is why mathematicians take 
the form of thought seriously and consistently (recall also Algebra 
- another kind of Calculus or formal system to mechanize/automate 
thought) - does not matter whether they do this consciously or 
instinctively. All of this is just an external observation of 
their behaviour. 


but it 
> seems to me that they'd more likely to arrive at the concept of prime 
> number than of logical positivism once they get past mere street smarts. 
> Mathematics is surely the least specifically human of all thought 
> processes, with physics, chemistry, etc. close behind.


I agree, but in some further step they could also come to their own 
views on the nature of math. (My fantasy which I do not want to 
develop further.)

> I do however appreciate our shared goal of defining mathematics. 
> Applied to myself as much as to anyone, "better to have defined and lost 
> than never to have defined at all."


Yes. Without having a good definition, discussions on the nature 
of mathematics often reminds me the known parable on blind men and 
the elephant. Formality/rigour (taken seriously and consistently 
in mathematical practice) is the only feature of mathematics I know 
which distinguishes it of anything else. Evidently, it should be 
put at the center from which all other essential features can be 
naturally "derived" or understood. For example, the abstract style 
of mathematical thought evidently follows from the formal/rigorous 
style (but not vice versa). 


> Indeed. You have correctly discerned that I am not a formalist.


It remains unclear why. Your posting I replied to shows that you 
are not a formalist in the extremely narrow sense of the word formal 
(just a caricature sense as a view on math! so, I am also not a 
formalist in that sense). What about the wider and flexible 
sense I suggest? Have I clarified it above (jointly with my previous 
posting explaining the wider sense of "formal")? 


Vladimir Sazonov


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