[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

Allen Hazen <allenph at unimelb.edu.au> wrote:

> ONE version of "logicism" would be a kind of "if-then-ism": mathematical
> truths are propositions that follow LOGICALLY from (perhaps arbitrarily
> chosen) "axioms," or, more precisely put, the REAL content of a
> mthematical statement is simply that the statement follows logically
> from certain axioms.  So, if you are willing to settle for this (weak?)
> understanding of "logicism" and willing to take "logic" to be "standard"
> Second (or higher) Order... logicism is true for "ordinary" mathematics.

One should add that "if-then-ism", or "deductivism", has many 
problems as a general philosophy of mathematics. 

For eample, it cannot explain why we are more interested in certain axioms 
(such as the axioms of arithmetic and analysis) rather than their 
negations.  And what were the mathematicians studying before the 
axiomatization of the fundamental theories? The view  makes the choice of 
new axioms incomprehensible. Throughout the history the mathematicians 
have also used considerations, e.g. based on analogies and probabilities, 
which are not strictly deductive. Further, what are the axioms which are 
parts of such conditionals, themselves talking about? "If-then-ism" also 
entails that a large part of traditional mathematics is not in fact 
mathematics. Moreover, it has difficulties in explaining why mathematics 
is so succesful when applied in physical reality. 

(I don't intend to suggest that Allen commits himself to "if-then-ism".) 


Best, Panu



Panu Raatikainen

Academy Research Fellow, The Academy of Finland
Docent in Theoretical Philosophy, University of Helsinki

Department of Philosophy
P.O.Box 9
FIN-00014 University of Helsinki
Finland

e-mail: panu.raatikainen at helsinki.fi