[FOM] Predicative foundations

José Félix Costa fgc at math.ist.utl.pt
Fri Feb 17 08:09:06 EST 2006

Eray Ozkural: «Thus, if we would carefully review what exists when a person
is performing
arithmetic, we would find that there are marks in his brain (as
electrochemical events, etc.) and/or on the paper. There is nothing else
that is physically happening.»

In fact we know a little about what happens when a person is performing
arithmetic. Some experiments have been made to teach arithmetics to a
«neural net».

E.g., James Anderson at Brown University has a book on neural nets (An
Introduction to Neural Nets, MIT), somewhat driven by biology,
neurophysiology, learning theory, «Turing tests» on subjects, confronted
with neural nets representation of the brain. In his chapter 17, Anderson
discusses the learning of arithmetics and «concept formation» in the brain
about numbers and mathematical operations -- indeed quite different in
several aspects from Turing machines or even finite automata. But since
finite automata (performing some bounded arithmetics) can be simulated by
neurons and action potencials flowing through the net, we can look at
«marks» as abstract electrochemical phenomena.

The problem is with the perception and conception of «infinite».

P:S.: in my mail of yesterday, about Zeno's four paradoxes, where each
paradox is put in relation with {continuous, discrete} times {continuous,
discrete}, I forgot to mention that in Koyré's analysis each paradox can not
be understood (or taught) in isolation: the four paradoxes make the PARADOX
within the Eleatic school, where the concept of infinite is implied.

J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm

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