Dan asks if G. Polya is qualified to teach Calculus
I don't usually engage the trolls directly. I compile lists of their typical rantings and post them as a warning to others who may be taken in, wasting their valuable time or worse. I don't usually post more than one such warning per thread.
#26#
G. Polya
--- quoting HOW TO SOLVE IT, G. Polya, Princeton Univ. Press,
pages 166-167 ---
3. Indirect proof. The prime numbers, or primes, are the numbers
2,3,5,7,11,13,17,19,23,29,31,37,... which cannot be resolved into
smaller factors, although they are greater than 1. (The last clause
excludes the number 1 which, obviously, cannot be resolved into
smaller factors, but has a different nature and should not be
counted as a prime.) The primes are the "ultimate elements" into
which all integers (greater than 1) can be decomposed. For instance,
630 = 2x3x3x5x7 is decomposed into a product of five primes.
Is the series of primes infinite or does it end somewhere? It is
natural to suspect that the series of primes never ends. If it ended
somewhere, all integers could be decomposed into a finite number of
ultimate elements and the world would appear "too poor" in a manner
of speaking. Thus arises the problem of proving the existence of an
infinity of prime numbers.
This problem is very different from elementary mathematical
problems of the usual kind and appears at first inaccessible. Yet,
as we said, it is extremely unlikely that there would be a last
prime, say P. Why is it so unlikely?
Let us face squarely the unlikely situation in which,
hypothetically, supposedly, allegedly, there is a last prime P. Then
we could write down the complete series of primes 2,3,5,7,11, . . .P.
Why is this so unlikely? What is wrong with it? Can we point out
anything that is definitely wrong? Indeed, we can. We can construct
the number Q = (2x3x5x7x11x...xP) +1. This number Q is greater than
P and therefore, allegedly, Q cannot be a prime. Consequently, Q
must be divisible by a prime. Now, all primes at our disposal are,
supposedly, the numbers 2,3,5,...P but Q, divided by any of these
numbers, leaves the rest 1; and so Q is not divisible by any of the
primes mentioned which are hypothetically, all the primes. Now, there
is something that is definitely wrong; Q must be either a prime or it
must be divisible by some prime. Starting from the supposition that
there is a last prime P we have been led to a manifest absurdity.
How can we explain this? Our original supposition must be wrong;
there cannot be a last prime P. And so we have succeeded in proving
that the series of prime numbers never ends.
Our proof is a typical indirect proof. (It is a famous proof too,
due to Euclid; see Proposition 20 of Book IX of the Elements.)
We have established our theorem (that the series of primes never
ends) by disproving its contradictory opposite (that the series of
primes ends somewhere) which we have disproved by deducing from it
a manifest absurdity. Thus we have combined indirect proof with
"reductio ad absurdum"; this combination is also very typical."
--- end quoting HOW TO SOLVE IT, G. Polya, Princeton Univ. Press ---
Apparently in the 20th century there was no clear precision definition
of series versus sequence, for we see both Polya and G.H.Hardy
and some others using the term "series" when it at best is described
as a sequence of primes. And later on in this book of Correcting Math, a precision
definition
of Series versus Sequence is given which is very important for the
Riemann Hypothesis
in that it is a pseudo hypothesis since it never precision defined
Series as it relates
to the Euler encoding and the Riemann zeta function.
Precision mathematics is not a hallmark or cornerstone of 20th century
mathematics
and just as the world experienced hippie-ism and laissez faire
attitudes and
marijuana smoking, one can say that much if not most of 20th century
mathematics
itself was "marijuana-mathematics" with all its ill-defined notions
passing off as math, worst being infinity.
Polya makes the usual 3 mistakes, although Polya, half-heartedly defines prime first off, but seems to have forgotten his definition of prime by the time he reaches his "multiply the lot and add 1".
First, Polya thought Euclid gave a
indirect proof but in actuality, Euclid gave a direct proof of
IP by increasing set cardinality. Secondly, Polya did not realize the
pure clarity and absoluteness of Logic that once "multiply the lot and
add 1" is formed it
must be necessarily the new prime that discharges the initial
assumption
and leads to the contradiction. Polya sort of like Paulos and Peterson
juggle with
Euclid's number when formed, and juggle because they never had their
first step
as the definition of prime which would have pinpointed Euclid's number
as necessarily
a new prime, and second mistake, instead of understanding this "multiply the lot and add 1" as necessarily a new prime, Polya goes on to make a prime factor search.
And thirdly, if Polya had been required by the new
standard that you must provide both a Direct alongside an Indirect
Method proof of Euclid IP, it is likely that Polya would have
delivered a mistake-free proof. But such was the ill-defining of
mathematics
of the 20th century, the sort of hippie mathematics that was so
prevalent, but
can anyone blame them considering they were under the awful threat of
nuclear
war in the Cold War, or the malaise of the Vietnam war, so you may as well have much ill-defining,
even
mathematics.
Let me discuss something that Polya brilliantly mentions in his above.
I want to complain about something which I have not complained
enough about. Why is it that so many authors are so irritatingly
worried about the number 1 in Euclid's IP, as if some obsessive
compulsion disorder. Probably because Euclid was worried about
1 in his proof for he mentions 1 often. If Euclid mentions "1" in his
proof, it is probably because his proof is Ancient, and not because
it is any key element of the proving
technique. But in modern day times
whenever a person attempting Euclid IP Indirect that mentions
1 unit as having any significance in the proof is a logic-misfit. And
this
is added reason that Symbolic Logic should be a prerequisite course
for
anyone wanting to major in mathematics, so that they fully understand
what is needed and not needed in a proof of mathematics.
Polya does mention 1 unit, but to his credit points out why
it is logically-wacko to worry about 1 unit in Euclid's IP indirect.
When Polya mentions the prime series in this sentence:
"primes, are the numbers
2,3,5,7,11,13,17,19,23,29,31,37,... which cannot be resolved into
smaller factors, although they are greater than 1. (The last clause
excludes the number 1 which, obviously, cannot be resolved into
smaller factors"
Or when Hardy mentions the prime sequence (not series) 2,3,5,7,..
Then no-one cares anymore about 1 or 1 unit as being involved in
the proof itself that a mentioning of the sequence eliminates the 1
unit.
But try telling that to those in mathematics that never took a
Symbolic
Logic course, who get fixated on some
irrelevant thing and then making themselves a perpetual pest over the
issue.
Proofs in mathematics do not require us listing all the facts and
information of things gone before. And novices of mathematics
often commit this error. That a proof of
mathematics should be deplete of irrelevancies. One irrelevancy of
Euclid IP is the fact of "1". Another irrelevancy in Euclid IP is the
fact of what is composite. Novices of Euclid IP often dwell on "1" and
on
"composite". When doing a mathematics proof we do not need to
throw in all of mathematical knowledge that has come before. We
only have to throw in the bare essentials of the proof. And a strict
course
in Symbolic Logic would teach many want-to-be mathematician what a
proof
needs and does not need.
Direct method---
Definition of prime. Given any finite set of primes we can augment
that set with one more new prime and thus since
we augment *any finite set* means it is infinite. Given a
set of finite primes we
multiply the lot and add 1, call it W+1. Either W+1 is prime or W+1 has a prime factor not
in the original finite set because W+1 is not
divisible by any of those finite listed primes. QED
Indirect method---
Definition of prime. Suppose set of all primes is finite with P_k its
last and largest prime. Multiply the lot and add 1, call it W+1. This
new number W+1 is necessarily prime
because of the definition-of-prime.
Contradiction since W+1 is larger than P_k. Hence set of
all primes is infinite. QED
AP1990s Survey
AP1990s Survey
Many of AP's postings can be safely ignored as well. But once he starts peddling his mathematical snake oil promising an "easier way" for students to do calculus, for example, I think they really ought to be warned about him.
Yeah, I probably have too much time on my hand and should get out more. Sigh...
Dan
Dan seems to have lost sight that sci.math is about math, not about hatred of people and plastering hate-sheets all over the place, but when a person is insane, insanity is where "no reasoning can enter" and only repetitive hate sheets. Even if a poster like John Gabriel is wrong in everything he says about calculus, at least the reader deserves to judge for himself/herself, and not have a lunatic predator like Dan Christensen stigmatizing and branding other people.
AP