[FOM] R: Multiple types of foundation for mathematics (cognitive, biological, mathematical...)

[Antonino Drago]

Aaron Sloman' presentation of the foundations of Mathematics suggests to me
the following answers: 
	a) Why he do not take into account that William Beth (Foundations of
Mathematics, North-Holland, 1959 declares to be declive by his PhD about
Kant's philosophy of science as applied to the foundations of mathematics? 
	b) Why he do not mention Leibniz, the last philosopher-mathematician
at the highet levels of the history of Western philosophy? . 
	c) On the subject I wrote two papers: (2015) Foundations of
Mathematics from an historical Viewpoint, Epistemologia (prenstly
Axiomathes) 38(1):133-151 and  (2017) A Pluralist Foundation of the
Mathematics of the First Half of the Twentieth Century, J. Of Indian Council
of Philosophical Research, DOI 10.1007/s40961-016-0089. I suggest a
foundation which can be traced back to the two Leibniz's labyrinths; i.e. a
dichotomy on the kinds of mathematics (classical or constructive) and a
dichotmoy on the kind of logic (classical or non-classical; the latter one
correpsonding to the philosophical notion of the deductive organization of a
theory or a problem-based organization of a theory. These dichotomies are at
the same time formal and philosophical in nature; therefore they overcome
the traditional gap between philosophy and science.  

Best 
Antonino Drago

P.S. Your words 

I would particularly welcome pointers to work on how biological evolution
was able to produce brains able to make ancient mathematical discoveries in
geometry and topology, e.g. leading to Euclid's Elements. 

recall to me that there is a neural basis to geometry. Mosquito's eye sees
by means of each ommatidium what is in a little solid angle. Frog's eye sees
borderlines. Octopus sees the projetions of a figure on the two Cartesian
axes of a plane. Sapphirinide Copilia female sees through a scanning of the
figure.  

-----Messaggio originale-----
Da: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] Per conto di
Aaron Sloman
Inviato: venerdì 17 marzo 2017 00:59
A: Foundations of Mathematics
Oggetto: [FOM] Multiple types of foundation for mathematics (cognitive,
biological, mathematical...)

Some notes comparing and contrasting investigations into several different
kinds of foundation for mathematics, including cognitive,
biological/evolutionary, mathematical and metaphysical foundations:

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/maths-multiple-founda
tions.html
(Also pdf)

Comments, criticisms, suggestions, and pointers welcome.

I would particularly welcome pointers to work on how biological evolution
was able to produce brains able to make ancient mathematical discoveries in
geometry and topology, e.g. leading to Euclid's Elements. Can those
processes be replicated in AI theorem provers?

The answers should refer to explanatory mechanisms not the competitive
advantages of having mathematical abilities.

Thanks.

Aaron Sloman
a.sloman at cs.bham.ac.uk
http://www.cs.bham.ac.uk/~axs
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