Abstract

If our mind is just an algorithm running on a flesh hardware, then it seems that there is no place for the free-will. An algorithm decides everything based on deterministic computations, or on random inputs, but neither inevitability nor pure hazard is free choice. Hopefully, some day, Science will be able to understand, monitor and simulate all the mind processes. Even then, it will still be a possibility for the free-will to exist, based on the convergence of the initial data. I propose a crucial experiment to test this hypothesis.

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Preprint

Creators:

Creators	Email	ORCID
Stoica, Ovidiu Cristinel

Keywords:

Mind, mind-body problem, free will, reductionism, artificial intelligence

Subjects:

General Issues > Thought Experiments
Specific Sciences > Psychology
Specific Sciences > Artificial Intelligence
General Issues > Reductionism/Holism
Specific Sciences > Cognitive Science
General Issues > Science and Religion
General Issues > Determinism/Indeterminism
Specific Sciences > Physics > Quantum Mechanics

Depositing User:

Ovidiu Cristinel Stoica

Date Deposited:

23 Nov 2008

Last Modified:

07 Oct 2010 15:17

Item ID:

4356

Subjects:

Date:

November 2008

URI:

http://philsci-archive.pitt.edu/id/eprint/4356

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JOI’RNAI.
OF APPROXIMATION
THEORY
30, 139-148
On the Convergence ANDRAS Mathematical
(1980)
of Pdya’s KRO~
Institute of the Hungarian Budapest. Hungay Communicaled
Algorithm
Academy of Sciences.
bj, E. W. Chene!’
Received Mav 4. 1979
One of the central questions of Tchebycheff approximation is computing the polynomial of best approximation. The underlying idea of the algorithms of computation is usually approximation of Tchebycheff norm by other norms. Consider for example the Polya algorithm. Let f E C[O, 11, let p,(f)c be the algebraic polynomial of degree n of best Tchebycheff approximation toJ and p,(f), (q > 1) the algebraic polynomial of degree n of best L, approximation to 1: Then as was shown by Polya [ I] p,(f), converges uniformly to p,(f), as q + + co. The analogue of this theorem for the de la Vallee-Poussin (or discrete) algorithm was proved by Motzkin and Walsh 12. 3 1. Moreover Cheney 141 proved that
where p,(f), is the best Tchebycheff approximation tofon Y c [0, 11, uA6) is the modulus of continuity off and 1YI = supXpro, inf,=, 1.y- ~1. Some theorems on uniform convergence of de la Vallee-Poussin algorithm for classes of continuous functions were proved in 151. In the present paper we shall investigate the rate of convergence of Polya algorithm. As it was shown by Peetre 161, if f E C[O, l] is continuously differentiable then for q > qO
IIP,(f),. - P,(f),llc < CC% f) 7. Our aim is to prove a theorem on convergence of Polya algorithm for arbitrary f E C[O, I]. Moreover we shall verify the sharpness of our estimations. At last we give a theorem on uniform convergence of Polya algorithm. 139 0021
Y045/80~lOO13Y-
lOSO2.OOiO
Copyright C 1980 by Academx All rights
of reproductton
Press. Inc. ,n any form reserved
In what follows Ci(.. .) and si(. . ) denote positive constants depending only on quantities specified in the brackets; while Ci and 9; denote positive absolute constants. MAIN THEOREMS
Let f E C[O, 11. We shall use the following notation
p,(J), and p,(f), are algebraic polynomials of order at most tt of best approximation in C and L, norm respectively (n E Z . ). Further define E, = E,(q) as the unique solution of the equation
It can be easily verified that E,(q) monotonously 9 + + 03 and 9/E,(q) > C In 9 for 9 > 9”. THEOREM
1.
Letf
tends to infinity
as
E ClO, 11. Then for any 9 > 1 and n E I + , (2)
Let us consider some concrete cases. If wxd) < 6” (0 c: LI Q 1; 0 ( b‘ 6 I ). then E,(q) > nq/ln q (q > en). If u,(S) < exp[-a lnb(1/6)1 (0 < b < 1, a > 0), then E,(q) > a'~bq/lnl~hq (q > e”). For w,{S) < l/ln”(1/6) (a > 0) we have E,(q)Z9”‘““’ (42 1). It turned out that estimation (2) is in general the best possible. We shall need some additional definitions. Let W be the set of all moduli of continuity of continuous functions. cc), w2 E W are said to be equivalent, written u, - WI iff C,w,(6) < w,(6) < Czw,(6) (0 < 6 < 1). Let n E B +. Then for f E C[O, 1) such that wf- Q and THEOREM 2.
any to E W there exists a finctiorl
‘ml E,(q) IIP,u-),- ~- P”u-),llc > 0. w-x where E,(q)
is the unique solution of ( I ).
(3)
PbLYA
141
ALGOKITHM
By this theorem estimation (2) is sharp in general for functions with arbitrary moduli of continuity. From Theorems 1 and 2 we obtain following COROLLARY. Let f E C(0, 11, nE “7 +. Then jbr an??q > en
0x8) < 8’
(0 < U < 1;
0 < 6 < t ).
IIPn(f)c - Pn(f),llc G C.3hf) y and for artjs 0 < u < 1 this order of convergence is in general the best possible. Finally, we give a theorem on the uniform algorithm for Lip a.
convergence of Polya’s
THEOREM 3.
O< />
For any n E L + , f E C(0, 11 with w,(S) < 6” (0 < 6 < 1; l)andq>q,(n,a) IIp,(f),.
- p,(f)&
< C,(n, a) [y]

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