[FOM] MathOverflow question on f.o.m.

Antonino Drago drago at unina.it
Tue Dec 13 17:59:04 EST 2016

 Tuesday, December 13, 2016 6:55 PM Timothy Y. Chow" <tchow at alum.mit.edu>
There are currently two answers [about what are the FoM], one by Andrej 
Bauer and one by myself,
 but additional answers would be welcome.

Since more than a decade I suggest a solution which relies on (a variant of) 
Leibniz' two main suggestions about the human mind:
1) two dichotomies (instead of "two labyrinths") on the infinity (either 
potential infinity - constructive mathematics or actual infinity - classical 
mathematics) and on the organization of a theory (either the deductive one 
from principles/axioms - managed by classical logic or solving a problem 
through the invention of a new method  - managed by intuitionist logic).
2) The last dichotomy in logic was anticipated by Leibniz' suggestion of 
"the two great principle of human mind", the principle of non.contardiction 
and the principle of sufficient reason, being the latter one the more 
general and characteristic predicate of intuitionist logic.
My main publications on the subject are the following ones:
Pluralism in Logic: The Square of Opposition, Leibniz' Principle of 
Sufficient Reason and Markov's principle, in J.-Y. Béziau and D. Jacquette 
(eds): Around and Beyond the Square of Opposition, Birkhaueser, Basel 2012, 

The foundations of mathematics from a historical viewpoint, Epistemologia, 
2015, 133-151

They take in account my historical analyses on Cavalieri & Torricelli's 
calculus, Lazare Carnot's geometry, Lobachevsky's non-Euclidean geometry, 
Poincaré's four geometries, Poincaré's principle of mathematical induction, 
Weyl's elementary mathematics, Kolmogorov's foundation of minimal logic, 
Markov's theory of computable numbers, application of constructive 
mathematics to theoretical physics.

All in all, my solution is not a basic notion (e.g. a set) or a theory (e.g. 
categories) but dichotmies, i.e. incompatible alternatives. This novelty 
(actually an old Leibniz' suggestion) may explain the historical difficulty 
in solving the problem of FoM notwithstanding the great debate of the first 
half of the 20th Century and the efforts elicited by great mathematicans 
along at least a century.

Bets regards

Antonino Drago, f

retired associate professor of Hisiory of Physics at Naples University.

----- Original Message ----- 
From: "Timothy Y. Chow" <tchow at alum.mit.edu>
To: <fom at cs.nyu.edu>
Sent: Tuesday, December 13, 2016 6:55 PM
Subject: [FOM] MathOverflow question on f.o.m.

> On MathOverflow last month, someone asked the following question:
> "What is some current research going on in the foundations of mathematics 
> about?  Are the foundations of mathematics still a research area, or is 
> everything solved? When I think about foundations I'm thinking of reducing 
> everything to ZFC set theory + first order logic. Is this still 
> contemporary?"
> http://mathoverflow.net/questions/254669/what-is-some-current-research-going-on-in-foundations-about
> The question was closed as being "too broad" but then was re-opened. There 
> are currently two answers, one by Andrej Bauer and one by myself, but 
> additional answers would be welcome.
> Tim
> _______________________________________________
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> FOM at cs.nyu.edu
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