Das Kalenderblatt 090719

18/07/2009 - 23:11 von WM | Report spam
Potential vs. Completed Infinity

Let us distinguish between the genetic, in the dictionary sense of
pertaining to origins, and the formal. Numerals (terms containing only
the unary function symbol S and the constant 0) are genetic; they are
formed by human activity. All of mathematical activity is genetic,
though the subject matter is formal.

Numerals constitute a potential infinity. Given any numeral, we can
construct a new numeral by prefixing it with S.

Now imagine this potential infinity to be completed. Imagine the
inexhaustible process of constructing numerals somehow to have been
finished, and call the result the set of all numbers, denoted by |N.

Thus |N is thought to be an actual infinity or a completed infinity.
This is curious terminology, since the etymology of “infinite” is “not
finished”.

We were warned.

Aristotle: Infinity is always potential, never actual.
Gauss: I protest against the use of infinite magnitude as something
completed, which is never permissible in mathematics.

We ignored the warnings.

With the work of Dedekind, Peano, and Cantor above all, completed
infinity was accepted into mainstream mathematics. Mathematics became
a faith-based initiative.

Try to imagine |N as if it were real.

A friend of mine came across the following on the Web:
www.completedinfinity.com
Buy a copy of |N!
Contains zero—contains the successor of everything it contains—
contains only these.
Just $100.
Do the math! What is the price per number?
Satisfaction guaranteed!

Use our secure form to enter your credit card number and its security
number, zip code, social security number, bank’s routing number,
checking account number, date of birth, and mother’s maiden name. The
product will be shipped to you within two business days in a plain
wrapper.

My friend answered this ad and proudly showed his copy of |N to me. I
noticed that zero was green, and that the successor of every green
number was green, but that his model contained a red number. I told my
friend that he had been cheated, and had bought a nonstandard model,
but he is color blind and could not see my point.

I bought a model from another dealer and am quite pleased with it. My
friend maintains that it contains an ineffable number, although zero
is effable and the successor of every effable number is effable, but I
don’t know what he is talking about.
I think he is just jealous.

The point of this conceit is that it is impossible to characterize |N
unambiguously, as we shall argue in detail.
[...]
Over two and a half millennia after Pythagoras, most mathematicians
continue to hold a religious belief in |N as an object existing
independently of formal human construction.

[Edward Nelson (Department of Mathematics Princeton University):
"Hilbert’s Mistake",
Talk given at the Second New York Graduate Student Logic Conference,
March 18, 2007.]

http://www.math.princeton.edu/~nelson/papers/hm.pdf

Gruß, WM
 

Lesen sie die antworten

#1 Ralf Bader
19/07/2009 - 00:11 | Warnen spam
WM wrote:

Potential vs. Completed Infinity

Let us distinguish between the genetic, in the dictionary sense of
pertaining to origins, and the formal. Numerals (terms containing only
the unary function symbol S and the constant 0) are genetic; they are
formed by human activity. All of mathematical activity is genetic,
though the subject matter is formal.

Numerals constitute a potential infinity. Given any numeral, we can
construct a new numeral by prefixing it with S.

Now imagine this potential infinity to be completed. Imagine the
inexhaustible process of constructing numerals somehow to have been
finished, and call the result the set of all numbers, denoted by |N.

Thus |N is thought to be an actual infinity or a completed infinity.
This is curious terminology, since the etymology of “infinite” is “not
finished”.



Das ist natürlich ein Nullargument (daß für einen Sachverhalt vor dessen
Entdeckung noch keine Terminologie bereitsteht, sagt über die "Existenz"
des Sachverhalts exakt überhaupt nichts aus).

We were warned.



Und das ist reine Rhetorik.

Aristotle: Infinity is always potential, never actual.



Piffle. Von Nelson, nicht von Aristoteles. Aristoteles' Satz steht
mutmaßlich in einem gewissen Kontext. Ich habe dazu vor ca. 2 Wochen hier
etwas zitiert.

Gauss: I protest against the use of infinite magnitude as something
completed, which is never permissible in mathematics.



Piffle. Von Nelson, nicht von Gauss. Ich habe dazu wiederholt etwas
geschrieben, was ich nicht nochmal wiederkàuen mag.

We ignored the warnings.



Und das ist reine Rhetorik.

With the work of Dedekind, Peano, and Cantor above all, completed
infinity was accepted into mainstream mathematics. Mathematics became
a faith-based initiative.



Nein. Mathematik wird dadurch nicht zu einer "faith-based initiative", im
Sinne etwa von Nelsons persönlichem katholischen faith.

Try to imagine |N as if it were real.



Ich weiß nicht, was das heißen soll. Die "Aktualitàt" von IN wird
angenommen, nicht imaginiert.

A friend of mine came across the following on the Web:
www.completedinfinity.com
Buy a copy of |N!
Contains zero—contains the successor of everything it contains—
contains only these.
Just $100.
Do the math! What is the price per number?
Satisfaction guaranteed!

Use our secure form to enter your credit card number and its security
number, zip code, social security number, bank’s routing number,
checking account number, date of birth, and mother’s maiden name. The
product will be shipped to you within two business days in a plain
wrapper.



Wahrscheinlich gibt es auch Idioten, die darauf hereinfallen.

My friend answered this ad



Ah ja. Ich habe diesen Satz vorher nicht gelesen, Ehrenwort.

and proudly showed his copy of |N to me. I
noticed that zero was green, and that the successor of every green
number was green, but that his model contained a red number. I told my
friend that he had been cheated, and had bought a nonstandard model,
but he is color blind and could not see my point.

I bought a model from another dealer and am quite pleased with it. My
friend maintains that it contains an ineffable number, although zero
is effable and the successor of every effable number is effable, but I
don’t know what he is talking about.
I think he is just jealous.

The point of this conceit is that it is impossible to characterize |N
unambiguously, as we shall argue in detail.
[...]



Genau, Mückenheim. Die relativ ausführlich zitierte Nelsonsche Philosophie
ist leider wenig überzeugend und daher entbehrlich. Dort, wo er aber
tatsàchlich etwas zu sagen hat -denn seine pràdikative Arithmetik enthàlt
mathematische Substanz- werden 3 Pünktchen gemacht.

Over two and a half millennia after Pythagoras, most mathematicians
continue to hold a religious belief in |N as an object existing
independently of formal human construction.



Abgesehen von der m.E. in die Irre gehenden Charakterisierung
dieses "Glaubens" hat sich eben auch seit "two and a half millennia" kein
Grund ergeben, von diesem "Glauben" abzufallen. Am wenigstens übrigens für
Mückenheims Quackelbruder Albrecht Storz. Der "glaubt" zwar nicht an
das "aktual Unendliche", aber an eine Art Naturwüchsigkeit der natürlichen
Zahlen; und findet damit, wie gelegentlich zu lesen war, durchaus
Zustimmung seitens Mückenheim. Ungeachtet ihrer (fehlenden)
Überzeugungskraft schlàgt also Nelsons "Philosophie" nicht zugunsten des
bescheuerten Mückenheimschen "Matherealismus" aus.

[Edward Nelson (Department of Mathematics Princeton University):
"Hilbert’s Mistake",
Talk given at the Second New York Graduate Student Logic Conference,
March 18, 2007.]

http://www.math.princeton.edu/~nelson/papers/hm.pdf

Gruß, WM



Ach ja, wenn Nelson fragt ob man ihm Glück wünsche bei dem Versuch, die
Inkonsistenz der Peano-Arithmetik aufzuzeigen: Warum nicht? Diese
Inkonsistenz wàre jedenfalls ein spannendes, mithin unterhaltsames,
Resultat. Allerdings glaube ich nicht, daß Nelson bei diesem Unterfangen
große Chancen hat. Es ist aber seine Sache, ob es die Zeit wert ist, womit
er sich beschàftigt.

W. Hughes, in sci.math.: "No set of natural numbers without a last element
[is finite]"
Prof. Dr. W. Mückenheim, mathematical mastermind of "Augsburg University of
Applied Science": "There is no natural number called "out a last element".

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