Re: Food for thought... [Way OT] (Message 1 to 8 of 8)
From: Alexander Enzmann
Subject: Re: Food for thought... [Way OT]
Date: 3 Sep 1999 07:14:56
Message: <37CFAED1.174B9ED0@mitre.org>
Larry Fontaine wrote:
> > Nothing can be proven without an assumption.> For example, numbers can only be defined using numbers. Numbers were> made up by humans out of thin air...
No - the Peano axioms don't require any numbers to start with. Only a
thing and the concept of a successor of that thing. Standard formal
logic stuff. Of course it takes Bertrand Russel a couple of hundred
pages in his Principles of Mathematics to get to the point that you can
show that 1 + 1 = 2.
Xander
From: Alberto
Subject: Re: Food for thought... [Way OT]
Date: 3 Sep 1999 10:59:02
Message: <37CFE16B.31FB2747@usb.ve>
Alexander Enzmann wrote:
> No - the Peano axioms don't require any numbers to start with. Only a> thing and the concept of a successor of that thing. Standard formal> logic stuff. Of course it takes Bertrand Russel a couple of hundred> pages in his Principles of Mathematics to get to the point that you can> show that 1 + 1 = 2.>> Xander
I doubt it. 2 is the successor of 1 and the equation
1 + 1 = 2
is a symbolic way to state this fact.
Alberto
From: Margus Ramst
Subject: Re: Food for thought... [Way OT]
Date: 3 Sep 1999 11:17:33
Message: <37CFE680.E38B89E7@peak.edu.ee>
Alexander Enzmann wrote:
> > Of course it takes Bertrand Russel a couple of hundred> pages in his Principles of Mathematics to get to the point that you can> show that 1 + 1 = 2.> > Xander
Does it serve any practical purpose or is it of purely philosophical value?
Margus
From: Alexander Enzmann
Subject: Re: Food for thought... [Way OT]
Date: 3 Sep 1999 16:48:30
Message: <37D0353C.AD4B3B82@mitre.org>
Alberto wrote:
> > Alexander Enzmann wrote:> > > No - the Peano axioms don't require any numbers to start with. Only a> > thing and the concept of a successor of that thing. Standard formal> > logic stuff. Of course it takes Bertrand Russel a couple of hundred> > pages in his Principles of Mathematics to get to the point that you can> > show that 1 + 1 = 2.> >> > Xander> > I doubt it. 2 is the successor of 1 and the equation> > 1 + 1 = 2> > is a symbolic way to state this fact.
You doubt that the Peano axioms don't require numbers or that Russel
takes a long bit of analysis before getting to ordinal numbers? In
either case you'd be off the mark.
Xander
From: Alexander Enzmann
Subject: Re: Food for thought... [Way OT and back again]
Date: 3 Sep 1999 16:52:25
Message: <37D03627.E58F5A7@mitre.org>
Margus Ramst wrote:
> > Alexander Enzmann wrote:> >> > Of course it takes Bertrand Russel a couple of hundred> > pages in his Principles of Mathematics to get to the point that you can> > show that 1 + 1 = 2.> >> > Xander> > Does it serve any practical purpose or is it of purely philosophical value?
While Russell hoped to prove the consistency of math through logic, he
wound up with one of the more famous paradoxes in math (stated at
another point in this thread). His principles of math are for people
who want to understand the foundations rather than memorizing things.
[Desparately thinking of a way to get this back to povray...... Ok I
have it but it's a stretch.]
Reading Russell (and contemporaries) lead to my reading a bunch of
history of Math books, which led to reading about the first techniques
found for solving cubics and quartics through radicals, which led to the
code I wrote for polynomials in POV-Ray. [There, nearly back OT.]
Xander
From: Alberto
Subject: Re: Food for thought... [Way OT]
Date: 3 Sep 1999 20:34:18
Message: <37D0681D.E2A708CE@usb.ve>
I think I misunderstood your message. Now I read it as
It took Russel hundred of pages to go through the logic stuff before he could
talk about "the thing" and "the successor of the thing". And naturally, I
agree with that.
Alberto.
From: Mark Wagner
Subject: Re: Food for thought... [Way OT]
Date: 5 Sep 1999 03:05:15
Message: <37d2162b@news.povray.org>
Alberto wrote in message <37CFE16B.31FB2747@usb.ve>...
>>>Alexander Enzmann wrote:>>> No - the Peano axioms don't require any numbers to start with. Only a>> thing and the concept of a successor of that thing. Standard formal>> logic stuff. Of course it takes Bertrand Russel a couple of hundred>> pages in his Principles of Mathematics to get to the point that you can>> show that 1 + 1 = 2.>>>> Xander>>I doubt it. 2 is the successor of 1 and the equation>> 1 + 1 = 2>>is a symbolic way to state this fact.
Prove that 2 is the successor to 1.
Mark
From: Ralf Muschall
Subject: Re: Food for thought... [Way OT and back again]
Date: 7 Sep 1999 19:13:46
Message: <37D59B71.6DEDD5EA@t-online.de>
Alexander Enzmann wrote:
> > > Of course it takes Bertrand Russel a couple of hundred
How did he enumerate the pages before having numbers?
> > > pages in his Principles of Mathematics to get to the point that you can> > > show that 1 + 1 = 2.
He did it the hard way :-)
With Church numerals, it gets quite short.
> > Does it serve any practical purpose or is it of purely philosophical value?
E.g. the proofs of commutativity, associativity and distributivity
of addition and multiplication of natural numbers become trivial.
> [Desparately thinking of a way to get this back to povray...... Ok I
Rewriting it (or at least it's macro language) in Haskell ?:-)
Ralf
