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Topic: John Gabriel's Thread on Mathematics.
Replies: 231   Last Post: Mar 22, 2014 9:23 PM

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ross.finlayson@gmail.com

Posts: 2,714
Registered: 2/15/09
Re: New Calculus for Dummies.
Posted: Mar 3, 2014 9:30 PM
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On 3/3/2014 12:13 AM, John Gabriel wrote:
> On Monday, 3 March 2014 09:22:46 UTC+2, Roland Franzius wrote:
>> Am 02.03.2014 20:46, schrieb dullrich@sprynet.com:
>
> <snip>
>
> Agree with previous comments.
>

>> For their proofs most of the corpus of formulas of elementary functions
>> and higher transcendental functions need only one or two essential limit
>> theorems about absolutely, equicontinuous convergence for derivatives.

>
> In fact, we don't need limits at all, as Newton showed.
>
> If f(x)=sin(x)/x, then f(0)=1. There are no limits here. The so-called "limit" is a fallacy, because f is exactly computable at x=0.
>


Eh, that doesn't help you that in linearization as numerical method,
just like Cotes', there is an error term (here as in examples where
the error term vanishes).

No, these various methods, routines, or tricks a) don't contradict
anything already in the calculus, b) use a subset of analysis and
the calculus, also with that the results via Cotes are proved with
standard real analysis, c) work on a subclass of continuous curves,
and a quite general one, but not a general one, d) besides
connecting to the line integral, as is built of the continuous
function then to shade the space, which could have various functions
in that, it is otherwise the bit disconnected from transformation of
coordinates. Built as it is generously read to be, there are a
variety of new sources of error terms and bias, in construction,
then there are as from the usual orthonormal basis and the usual
fundamental theorems of the integral calculus.

Besides, Gabriel's nettlesome or offensive enough, I don't see
people in real life being such cretins without much expecting to eat
their words or teeth. This, even along lines of simple truisms
dashed with invective, it would be as easy to explain the
fundamentals to another who would go about perhaps at least being
conscientious in definition insofar as mathematical claim.

Or, "congratulations, that's 300 years old", and the reason it's not
primary in mathematics is because it's not primary in mathematics.

That's firm and not unharsh, if you were arduous but instead
forthright it would be easier to count you. Besides, if you expect
it to develop greater connections to greater mathematics, as it has,
unless you plan on setting up an answer shop for the perpetuity,
it's forgotten.

Similarly I maintain that the sweep N <-> R[0,1] with f constant
monotone, as is modeled by finite functions. There are undoubtedly
others on that, for example any who ever saw [0,1] divided as points
in order: the finite continuum.

So, while you claim priority on these, I am to the point where I
point to Leibniz and say "this is what I say", Leibniz is among
those who've defined this function, as for example Newton did with
fluxions (not as nilpotent, but as not their own fluxions). Again
in various regular definitions and modes of reason since antiquity,
there is a working definition of this function since Man had
infinite integers in one integer. Then, I have gone about
establishing and exploring certain of its properties.

I find it ironic no-one else had. This based on imperfect
information, I look forward to the day of finding great true
extra-dogmatic features in mathematics, here as: they are. For,
there are, and, in the foundations and in simplest developments and
in first principle(s), there are: as primary.




Date Subject Author
2/22/14
Read John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Dirk Van de moortel
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/25/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/25/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Art Hopkins
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
Dirk Van de moortel
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/23/14
Read Space time and the universe (STATU).
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
Giddy Armstrong
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/24/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
YBM
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Roland Franzius
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Inverse 18 Mathematics
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Inverse 18 Mathematics
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Karl-Olav Nyberg
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Karl-Olav Nyberg
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read the story of number (what is the canonical representation of
base-one, not "hashmarks [Roman numeralS
Brian Q. Hutchings
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/26/14
Read New Calculus for Dummies.
thenewcalculus@gmail.com
2/27/14
Read Re: New Calculus for Dummies.
Wizard-Of-Oz
2/27/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/27/14
Read Re: New Calculus for Dummies.
David C. Ullrich
2/27/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/28/14
Read Re: New Calculus for Dummies.
David C. Ullrich
2/28/14
Read 'rithmetical (trivial
Brian Q. Hutchings
2/28/14
Read Re: 'rithmetical (trivial
Peter Percival
2/28/14
Read Re: 'rithmetical (trivial
fom
2/28/14
Read 'rithmetical (trivial
Brian Q. Hutchings
2/28/14
Read Re: New Calculus for Dummies.
John Gabriel
3/1/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/1/14
Read Re: New Calculus for Dummies.
ross.finlayson@gmail.com
3/1/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/1/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/1/14
Read Re: New Calculus for Dummies.
Karl-Olav Nyberg
3/1/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/2/14
Read Re: New Calculus for Dummies.
Wizard-Of-Oz
3/2/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/2/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/2/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/2/14
Read Re: New Calculus for Dummies.
John Gabriel
3/3/14
Read Re: New Calculus for Dummies.
John Gabriel
3/3/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/3/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/4/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/4/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/4/14
Read Re: New Calculus for Dummies.
Karl-Olav Nyberg
3/4/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/4/14
Read Re: New Calculus for Dummies.
Karl-Olav Nyberg
3/5/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/4/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/5/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/5/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/5/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/6/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/6/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/6/14
Read "skwaring the cyrcle
Brian Q. Hutchings
3/3/14
Read Re: New Calculus for Dummies.
Roland Franzius
3/3/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/3/14
Read Re: New Calculus for Dummies.
ross.finlayson@gmail.com
3/4/14
Read Re: New Calculus for Dummies.
ross.finlayson@gmail.com
3/3/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/3/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/3/14
Read Re: New Calculus for Dummies.
Roland Franzius
3/4/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/4/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/8/14
Read Re: New Calculus for Dummies.
Peter Percival
3/8/14
Read Re: New Calculus for Dummies.
Wizard-Of-Oz
3/8/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/8/14
Read base-zero for fun (but not infinite series
Brian Q. Hutchings
3/8/14
Read Re: New Calculus for Dummies.
Peter Percival
3/8/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/9/14
Read Re: New Calculus for Dummies.
Peter Percival
3/9/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/10/14
Read calculi
Brian Q. Hutchings
3/12/14
Read Re: New Calculus for Dummies.
Peter Percival
3/12/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/12/14
Read Re: New Calculus for Dummies.
Peter Percival
3/13/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/12/14
Read Re: New Calculus for Dummies.
Mutt Buncher
3/8/14
Read Re: New Calculus for Dummies.
Virgil
3/8/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/9/14
Read Re: New Calculus for Dummies.
Peter Percival
3/9/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/12/14
Read Re: New Calculus for Dummies.
Peter Percival
3/12/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/12/14
Read geometry for ... folks (ratios in constructive [synthetic geometry
Brian Q. Hutchings
3/13/14
Read Re: New Calculus for Dummies.
David C. Ullrich
3/13/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/13/14
Read calculi
Brian Q. Hutchings
3/22/14
Read Re: New Calculus for Dummies.
Al Bundy
3/9/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
3/1/14
Read Re: New Calculus for Dummies.
Tom Traubert
3/1/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/27/14
Read Re: New Calculus for Dummies.
fom
2/27/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/27/14
Read Re: New Calculus for Dummies.
Ross Donglemeier
2/27/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/27/14
Read Re: New Calculus for Dummies.
PianoMan
2/27/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/28/14
Read Re: New Calculus for Dummies.
Harman Kardan
2/27/14
Read Re: New Calculus for Dummies.
Ross Donglemeier
2/27/14
Read Re: New Calculus for Dummies.
thenewcalculus@gmail.com
2/26/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/27/14
Read What you had to know in mathematics but your educators could not tell you.
thenewcalculus@gmail.com
2/27/14
Read Re: What you had to know in mathematics but your educators could not tell you.
Wizard-Of-Oz
2/27/14
Read Re: What you had to know in mathematics but your educators could not
tell you.
thenewcalculus@gmail.com
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/28/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/28/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/28/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
2/27/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
3/1/14
Read Construction of rational numbers and arithmetic axioms.
thenewcalculus@gmail.com
3/1/14
Read Re: Construction of rational numbers and arithmetic axioms.
sceptic
3/1/14
Read Re: Construction of rational numbers and arithmetic axioms.
thenewcalculus@gmail.com
3/1/14
Read Re: Construction of rational numbers and arithmetic axioms.
Wizard-Of-Oz
3/1/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/1/14
Read Re: John Gabriel's Thread on Mathematics.
Sal Honda
3/1/14
Read Bourbaki did iT
Brian Q. Hutchings
3/1/14
Read Re: Bourbaki did iT
thenewcalculus@gmail.com
3/1/14
Read The Gabriel Polynomial
thenewcalculus@gmail.com
3/1/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/2/14
Read Re: John Gabriel's Thread on Mathematics.
Karl-Olav Nyberg
3/2/14
Read Re: John Gabriel's Thread on Mathematics.
Karl-Olav Nyberg
3/2/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/2/14
Read Re: John Gabriel's Thread on Mathematics.
John Gabriel
3/2/14
Read Re: John Gabriel's Thread on Mathematics.
John Gabriel
3/2/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
3/2/14
Read Gabriel's Polynomial
thenewcalculus@gmail.com
3/2/14
Read Re: Gabriel's Polynomial - absolutely useless
Wizard-Of-Oz
3/2/14
Read Re: Gabriel's Polynomial - absolutely useless
thenewcalculus@gmail.com
3/2/14
Read The Gabriel Polynomial - One of the many great features in the New Calculus.
thenewcalculus@gmail.com
3/2/14
Read How New Ideas and Knowledge are often treated.
thenewcalculus@gmail.com
3/2/14
Read Re: How New Ideas and Knowledge are often treated.
Wizard-Of-Oz
3/2/14
Read Re: The Gabriel Polynomial - One of the many great features in the New Calculus.
Wizard-Of-Oz
3/2/14
Read Re: Gabriel's Polynomial - absolutely useless
Wizard-Of-Oz
3/3/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/4/14
Read Construction of Rational Numbers.
thenewcalculus@gmail.com
3/4/14
Read Re: Construction of Rational Numbers. - still useless
Wizard-Of-Oz
3/4/14
Read Re: Construction of Rational Numbers.
Peter Percival
3/4/14
Read Re: Construction of Rational Numbers.
thenewcalculus@gmail.com
3/4/14
Read Re: Construction of Rational Numbers.
Major Doctor
3/5/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/5/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/6/14
Read True knowledge or Cantor's Fools Paradise, which?
thenewcalculus@gmail.com
3/6/14
Read Re: True knowledge or Cantor's Fools Paradise, which?
Wizard-Of-Oz
3/6/14
Read The first and only rigorous formulation of calculus in human history.
thenewcalculus@gmail.com
3/6/14
Read Re: The first and only rigorous formulation of calculus in human history.
Dirk Van de moortel
3/6/14
Read Re: The first and only rigorous formulation of calculus in human history.
thenewcalculus@gmail.com
3/6/14
Read Re: The first and only rigorous formulation of calculus in human history.
Dirk Van de moortel
3/6/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/7/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
3/8/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/9/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/11/14
Read Re: John Gabriel's Thread on Mathematics.
Hoofington P. McSnort
3/11/14
Read Re: John Gabriel's Thread on Mathematics.
thenewcalculus@gmail.com
3/11/14
Read Re: John Gabriel's Thread on Mathematics.
Wizard-Of-Oz
3/11/14
Read Re: John Gabriel's Thread on Mathematics.
Brian Q. Hutchings
3/11/14
Read hrea
Brian Q. Hutchings
3/12/14
Read Re: hrea
thenewcalculus@gmail.com

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