Das Kalenderblatt 090818

17/08/2009 - 15:36 von WM | Report spam
I have seen some ultrafinitists go so far as to challenge the
existence of 2^100 as a natural number, in the sense of there being a
series of "points" of that length. There is the obvious "draw the
line" objection, asking where in 2^1, 2^2, 2^3, ... , 2^100 do we stop
having "Platonistic reality"? Here this ... is totally innocent, in
that it can be easily be replaced by 100 items (names) separated by
commas. I raised just this objection with the (extreme) ultrafinitist
Yessenin-Volpin during a lecture of his. He asked me to be more
specific. I then proceeded to start with 2^1 and asked him whether
this is "real" or something to that effect. He virtually immediately
said yes. Then I asked about 2^2, and he again said yes, but with a
perceptible delay. Then 2^3, and yes, but with more delay. This
continued for a couple of more times, till it was obvious how he was
handling this objection. Sure, he was prepared to always answer yes,
but he was going to take 2^100 times as long to answer yes to 2^100
then he would to answering 2^1. There is no way that I could get very
far with this.

[Harvey M. Friedman: "Philosophical Problems in Logic"]

http://www.math.ohio-state.edu/~friedman/manuscripts.html
http://www.math.ohio-state.edu/~friedman/pdf/Princeton532.pdf

Gruß, WM
 

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#1 Herbert Newman
17/08/2009 - 16:47 | Warnen spam
Am Mon, 17 Aug 2009 06:36:51 -0700 (PDT) schrieb WM:

I have seen some ultrafinitists go so far as to challenge the
existence of 2^100 as a natural number, in the sense of there being a
series of "points" of that length. There is the obvious "draw the
line" objection, asking where in 2^1, 2^2, 2^3, ... , 2^100 do we stop
having "Platonistic reality"? Here this ... is totally innocent, in
that it can be easily be replaced by 100 items (names) separated by
commas. I raised just this objection with the (extreme) ultrafinitist
Yessenin-Volpin during a lecture of his. He asked me to be more
specific. I then proceeded to start with 2^1 and asked him whether
this is "real" or something to that effect. He virtually immediately
said yes. Then I asked about 2^2, and he again said yes, but with a
perceptible delay. Then 2^3, and yes, but with more delay. This
continued for a couple of more times, till it was obvious how he was
handling this objection. Sure, he was prepared to always answer yes,
but he was going to take 2^100 times as long to answer yes to 2^100
then he would to answering 2^1. There is no way that I could get very
far with this.

[Harvey M. Friedman: "Philosophical Problems in Logic"]



Was sehr schön die logisch-mathematisch-philosophischen Probleme des
Ultrafinitismus belegt.

"The logical foundation of ultrafinitism is unclear; in his comprehensive
survey /Constructivism in Mathematics/ (1988), the constructive logician
A. S. Troelstra dismissed it as "no satisfactory development exists at
present". This was not so much a philosophical objection as it was an
admission that, in a rigorous work of mathematical logic, there was simply
nothing precise enough to include."

Source:
http://en.wikipedia.org/wiki/Ultrafinitism

Aber natürlich ist das KEIN Grund für WM, hier irgendwelche Schwierigkeiten
zu sehen: im Gegenteil! :-)


Herbert

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