From: Virgil on
In article <1166166464.741143.287220(a)f1g2000cwa.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Dik T. Winter schrieb:
>
>
> > > You have not understood, deplorably. The shares of every edge I use add
> > > to 1 edge. And every share is mapped on one path only. And every path
> > > gets as many shares to restate two full edges on its own.
> >
> > I have no idea of the meaning of the last sentence.
> >
> > > According to your proposal, every decimal number gets only some shares
> > > of the digits 1 to 9, but not as many shares as one digit is divided
> > > into. So not one full digit is mapped on a real number.
> >
> > Why not here? And why is that the case in your mapping?
>
> You map 10 digits on many more real numbers. Therefore not every real
> numbers can get the shares of a full digit. An example:
>
> If you map the digit 1 in the numbers 0.1153 and 0.3421, then you must
> divide the digit 1 into 2 + 1 = 3 shares, of which the number 0.1153
> gets 2 shares and the number 0.3421 gets one share. But if you consider
> some more real numbers with, say, together having 100 digits, then it
> is obvious that not every real number can get a full digit (uot of
> these 10 digits).
>
> In the tree we have enough edges so that every path gets the shares of
> two full edges by
> 1 + 1/2 + 1/4 + ... = 2
> > >
> > > Look at the edges. Do you deny that every edge is the beginning of that
> > > part of a path where it is separated from other paths?
> >
> > Right.
>
> Then you see that not more than countably many separated parts of paths
> can beginn in the tree. Where do you believe beginn the others which
> you claim to be there? Or do you claim that there are more paths than
> parts of paths?
> >
> > > Do you believe that paths beginn to run separated wihout any edge being
> > > inolved?
> >
> > No. And so what? But only looking at the paths that way you never get
> > the full infinite paths.
>
> If we look at last at one part of each path, then we look at all paths.

Not so. To separate any infinite path from /all/ other infinite paths
you must look at ALL its edges, not just some of them.


> If we look at all separated parts of paths, then we can be sure that we
> look at more objects than by looking on the whole paths. Or do you
> claim that there are more separated paths than sperated parts of paths?

I am not sure what you mean by "separated paths" or "separated parts of
paths". If those parts are restricted to finite lengths then there are
more paths than parts.
>
> Regards, WM
From: Han de Bruijn on
Jonathan Hoyle wrote:

> Bob Kolker wrote:
>
>>Virgil wrote:
>>
>>>And it is plain that no sound mathematics can be developed unless based
>>>on some axiom system as its solid foundation.
>>
>>Arithmetic was around long before it was axiomatized and people were
>>proving theorems about integers. For example Gauss and Euler.
>
> True, but you are ignoring Virgi's adjective "sound". Calculus existed
> way back during the time of Newton and Leibniz, but you could hardly
> call their use of the infinite and infinitessimals at all "sound" by
> today's standards. It wasn't until Bolzano and Weierstrass made things
> truly rigorous in the 19th century was Calculus anywhere near sound.

Allright. And they should have _stopped_ at this point in time.

> It is in fact their essential treatments that we are taught Real
> Analysis today, not Newton's. (Newton's work would be barely
> recognizable today with its "fluxions" and "fluents".)

> Bolzano and Weierstrass gave way to more rigor in numbers by Cantor,
> and then rigor in Set Theory by Zermelo and Fraenkel.

There was a pre-emptive war. Set Theory invaded Calculus.

> Then with the
> wonderful contributions of Hilbert, Lebesgue, Godel, and others,
> mathemaatics today is far more rigorous than it was over a century ago.

Thought we were talking about Calculus ...

> Even infinitessimals were consistently defined by Robinson.

I've never seen such useless things as Robinson's infinitesimals.

> With the
> exception of Aristotle's Logic and Euclid's Geometry, much of
> mathematics would not be considered acceptable by today's standards.

Who's "standards"? The problem with standards is that you have so many
to choose from. Are you imposing _your_ standards upon the rest of us?

> Even Guass and Euler played a bit fast and loose (although they were
> considered impeccably precise in their day.)

Huh? Would you say that absolute nitwits like Zermelo and Fraenkel, who
have contributed nothing to actual mathematics, are greater individuals
than Gauss and Euler? Did I really read this? Can't believe my eyes ...

> In ancient times, arithmetic was discovered in much the same way
> physical laws were. "Hey, notice that when we do this, that always
> happens..." As centuries of very hard work, mathematicians have boiled
> arithmetic assumptions down to some basic axioms, and all of the
> remaining theorems flow forth.

According to the Holy Gospel of Modern Mathematics, I suppose ...

http://web.maths.unsw.edu.au/~norman/views2.htm

Han de Bruijn

From: Han de Bruijn on
Bob Kolker wrote:

> Jonathan Hoyle wrote:
>
>> It is in fact their essential treatments that we are taught Real
>> Analysis today, not Newton's. (Newton's work would be barely
>> recognizable today with its "fluxions" and "fluents".)
>
> By the way the dot notation for differentiation with respect to time is
> frequently used in physics text books, so to that extent fluxions are
> alive. Fortunately the Leibniz notation predominates so algebraic
> manipulations can be readily used.

Snipped that out of an earlier comment, but it's true. Physicists have
no trouble with Newton's fluxions.

Han de Bruijn

From: Han de Bruijn on
Virgil wrote:

> Actually, we today have a number of improvements on Aristoteles logic,
> and Euclid's axiom system had to be revamped by Hilbert to bring it up
> to modern standards. But they have both stood the tests of time
> remarkably well.

Brings me to the question: who is doing Geometry these days?

Han de Bruijn

From: mueckenh on

William Hughes schrieb:

> mueckenh(a)rz.fh-augsburg.de wrote:
> > William Hughes schrieb:
> >
> > > mueckenh(a)rz.fh-augsburg.de wrote:
> > > > Virgil schrieb:
> > > >
> > > > > > > (It is contained in the union of all lines, but the
> > > > > > > union of all lines is not a line)
> > > > > >
> > > > > > That is a void assertion unless you can prove it by showing that
> > > > > > element by which the union differes from all the lines.
> > > > >
> > > > > Not quite. In order to achieve that the diagoal is not in any linem all
> > > > > that is required is:
> > > > > Given any line there is an element of the diagonal not in THAT line.
> > > > > It is not requires that:
> > > > > There is an element of the diagonal that is not in any line.
> > > >
> > > >
> > > > For linear sets you cannot help yourself by stating that the diagonal
> > > > differs form line A by element b and from line B by element a, but a is
> > > > in A and b is in B. This outcome is wrong.
> > > >
> > > > Therefore your reasoning "there is an element of the diagonal not in
> > > > THAT line. It is not required that: There is an element of the diagonal
> > > > that is not in any line." is inapplicable for linear sets. You see it
> > > > best if you try to give an example using a finite element a or b.
> > >
> > >
> > > In every finite example the line that contains
> > > the diagonal is the last line.
> >
> > Every example with natural numbers (finite lines) is a finite example.
> >
> > > Your claim is that there is a line which contains the diagonal.
> >
> > Because a diagonal longer than any line is not a diagonal.
> >
> > > Call it L_D. Question: "Is L_D the last line?"
> >
> > There is no last line
>
> Then, there is a line that comes after L_D.
>
> Therefore :L_D does not contain every element
> that can be shown to exist in the diagonal.

All elements that can be shown to exist in the diagonal can be shown to
exist in one single line.

If the diagonal contains infinitely many elements, then there is a line
containing infinitely many elements.

Therefore, there is no diagonal contaning infinitely many elements.
Every element that can be shown to belong to the diagonal can be shown
to belong to a finite set (intial segment, line):

Regards, WM