[FOM] 204:Finite Extrapolation -- two suggestions
Matthew Frank
mfrank at math.uchicago.edu
Mon Jan 19 13:33:39 EST 2004
1) Harvey Friedman asks: given a finite set of numbers, what is the
simplest definition of an infinite set which begins with those numbers?
Stephen Wolfram would say instead: look at the simplest possible
definitions of sets, and see what sets they lead to. Something
interesting might turn up.
E.g. {x: (exists y)(exists z)(x + y.y = z.z.z)} already leads to
interesting number theory.
2) Harvey Friedman also asks: given a finite sequence of numbers, what is
the simplest definition of an infinite set which begins with those
numbers?
There's an issue here with recursive definitions of sequences. The
standard treatment of sequences in the suggested language is so awkward
that intuitively simple initial sequences lead to odd definitions.
(1,1), (2,2), (3,3), (4,5), (5,8), (6,13) can be defined by
2y = (x + |x|) + (x-3 + |x-3|) + (x-4 + |x-4|) + 2(x-5 + |x-5|);
(1,1), (2,2), (3,6), (4,24), (5,120) can be defined by
2y = (x + |x|) + 3(x-2 + |x-2|) + 14(x-3 + |x-3|) + 78(x-4 + |x-4|);
and the simplest expressions of these formulas (writing 78 as 3.3.3.3-3,
etc.) are simpler than the simplest (obvious) definitions of the Fibonacci
sequence or the factorial sequence in the suggested language.
So the question about sequences would become more interesting by adding
some capacity for recursive definitions to the suggested language.
--Matt
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