robin2

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May 20, 2007

Misc

1

For future reference, should I ever need to name a demon: Anabibazon.

2

Also for future reference, should I ever find myself in the past trying to earn a living by anachronistically inventing logarithms: repeated multiplication by a base only fairly close to 1, together with linear interpolation, gives better results than I would have thought.

This:

#!/usr/bin/perl

my $BASE = 1.05;

sub logarithm
{
    my ($x) = @_;
    my ($x1, $x2) = (1, $BASE);
    my $y1 = 0;

    while ($x2 < $x)
    {
        ($x1, $x2) = ($x2, $x2 * $BASE);
        $y1 += 1;
    }

    return $y1 + (($x - $x1) / ($x2 - $x1));
}


print  "   x    |           log(x)           | %-age error\n";
print  "--------------------------------------------------\n";
foreach my $x (2 .. 10)
{
    my $logarithm = logarithm($x);
    my $log = log($x) / log($BASE);
    printf "%4.1f    |%10.3f    %10.3f    | %4.2f\n",
            $x, $logarithm, $log, 100 * abs($logarithm - $log) / $log;
}

Give this:

   x    |           log(x)           | %-age error
--------------------------------------------------
 2.0    |    14.203        14.207    | 0.03
 3.0    |    22.511        22.517    | 0.03
 4.0    |    28.407        28.413    | 0.02
 5.0    |    32.987        32.987    | 0.00
 6.0    |    36.719        36.724    | 0.01
 7.0    |    39.881        39.883    | 0.01
 8.0    |    42.614        42.620    | 0.01
 9.0    |    45.033        45.034    | 0.00
10.0    |    47.190        47.194    | 0.01

3

Regarding what I said about physics, it may be objected: “The bad end implausibly, the good end untidily. This is what theoretical physics means.”

4

I had in mind to write something about model theory, which I haven't done. But I did come across a footnote in A Thousand Plateaus that I'd not noticed before:

Historically, these have been the major problems of axiomatics: “undecidable” propositions (contradictory statements are also nondemonstrable); the powers of infinite sets, which by nature elude axiomatic treatment (“the continuum, for example, cannot be conceived axiomatically in its structural specificity since every axiomatization one can give it will rely on a denumerable model”). See Blanché, L'axiomatique, p. 80.

I assume the latter of these refers to Skolem's paradox: that although set theory covers uncountable sets, its axiomatization will admit countable models. This isn't strictly a paradox, in the sense of being a formal contradiction. It isn't necessarily even a problem. (I'm not sure what to make of the word ‘rely’ in the extract.) Hilbert's programme aimed at a more thorough paradox: grounding these infinities on finite objects (the formal proofs themselves).

But it is odd. Like the story of the man who went fishing with a picture of a worm, and caught a picture of a fish.

Posted by robin2 at May 20, 2007 07:01 PM

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