Favorite Quotations, Poems and Ideas
One of the most poetic and thrilling tributes to Mathematics is the following:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty
--a beauty cold and austere, like that of sculpture, without appeal to any part
of our weaker nature, without the gorgeous trappings of painting or music, yet
sublimely pure, and capable of a stern perfection such as only the greatest art
can show. The true spirit of delight, the exaltation, the sense of being more
than man which is the touchstone of the highest excellence, is to be found in
mathematics as surely as in poetry. What is best in mathematics deserves not
merely to be learnt as a task, but to be assimilated as a part of daily thought, and
brought again and again before the mind with ever-renewed encouragement.
Real life is to most men, a long second-best, a perpetual compromise between
the ideal and the possible; but the world of pure reason knows no barrier to the
creative activity embodying in splendid edifices the passionate aspiration after
the perfect from which all great work springs. Remote from human passions,
remote even from the pitiful facts of nature, the generations have gradually
created an ordered cosmos, where pure thought can dwell in its natural home,
and where one, at least, of our nobler impulses can escape from the dreary exile
of the actual world."
Bertrand Russell, in his book Mysticism and Logic.
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"Before you enter on the study of law a sufficient groundwork must
be laid . . . Mathematics and natural philosophy are so useful in the most
familiar occurrences of life and are so engaging and delightful as would
induce everyone to wish an acquaintance with them. Besides this, the
faculties of the mind, like the members of a body, are strengthened and
improved by exercise. Mathematical reasonings and deductions are, therefore
a fine preparation for investigating the abstruse speculations of the law."
- - - Thomas Jefferson, in a letter responding to an
enquiry as to what to study in preparation for
the study of law. (Natural Philosopy = Physics)
When students of mine who majored in mathematics have switched to law,
and wondered if I would write letters of reference for them, I give them
this Jefferson quote. . . . A former WVU Law School Dean used to tell me
he was always glad to accept good math. students for Law School.
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"Doing good Calculus requires the careful use of your accumulated algebraic,
geometric and trigonometric wisdom." H. W. Gould, March 2006
"Grading papers for 50 years is sending people to Heaven or Hell."
H. W. Gould, October 2007
Question: Do you know what grading papers is all about?
Answer: A, F, Heaven, Hell; A, F , Heaven, Hell; D, Purgatory!
I
shall forever be indebted to Professor Carlitz's infinite patience with my finite
brain!
Remark about being Emeritus: "Now that I am not working I can get more work done!"
- - - - You see, this works out pretty good in practice.
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I once asked Professor Gordon T. Whyburn: "What is mathematics?"
Whyburn's answer: "Whatever a mathematician calls it."
At the University of Virginia circa 1948-54, undergraduates took heavy courses in
real analysis, complex analysis, topology, foundations of geometry, etc. immediately
after one year of calculus. Math. majors were thrown into deep studies right away.
I even took topological group theory (we studied Pontryagin and Chevalley) to
complete my B. A. degree. I asked Professor Whyburn how I could do this since I had
not had group theory or much topology. He replied: "Then you will understand those
subjects the better!"
Upon asking Professor E. J. McShane whether to take real or complex analysis first, he
replied: "It makes no difference; each is a prerequisite for the other!"
McShane had another witty insight: "In real analysis we never know the value of an
integral; in complex analysis, we know because of Cauchy's theorem that all integrals
are zero!"
McShane taught us how to prove the existence of limits by "Athletic Induction", i.e.
"proof by leaps and bounds" . . . if a sequence always leaps but cannot go above a certain
bound than it has a limit. This technique is the familiar "Monotone Bounded Theorem."
Prof. McShane would often stand deep in thought while beating the top of his head with
the palm of his hand . . . he called this "womping up a proof." . . . it worked!
In early spring 1949, he found me in the U.Va. Math. Library deeply engrossed in reading
the American Mathematical Monthly. He was much impressed, and enthusiatically had me
fill out a form, pay five dollars, and become a member of the Mathematical Association of
America, and I have retained membership ever since, and have gained enormous inspiration
for research by reading the journal and solving problems posed in its pages. I owe McShane
a deep debt of gratitude for encouragement.
In the same manner I am indebted to the late Howard Eves for writing to me while I was in
the Army and keeping up my enthusiasm for mathematics. Eves was a great guy. He gave a
talk in Oregon years ago wherein he cited my 1959 work in trying to catalogue binomial identities,
and suggested that it would be equally good if someone could catalogue Fibonacci number
formulas. Verner E. Hoggatt, a student at that lecture, was thereby inspired to found the
Fibonacci Association and asked me to be a charter member and an editor. I have continued
to work with the Fibonacci Quarterly ever since. I consider Eves and McShane to have been
great "Mathematical Catalysts", people who really inspire others to achieve.
The late Professor Ben Zion Linfield, at the University of Virginia, was a man who demanded
much of his students. Whenever I would show him some proof he would compliment me but
then show me that the proof could be made better, more elegant. I learned from him never to be
really satisfied upon proving a result until I analyzed the details and found out what really made
the proof work, and how to get a simpler or better proof and generalize the result.
My second paper (1956) was titled "Some generalizations of Vandermonde's convolution", a
safe enough title. But my third paper (1957) I titled "Final analysis of Vandermonde's convolution".
The editor of the Monthly (I believe it was Carl Allendorfer) asked me "Are you sure that this is
the final analysis?" In my sophomoric glee I thought so, but Allendorfer was right; it was just
the second of many papers I have written on the subject and new ideas even today come to mind.
The moral is that you should never believe you have reached a "final analysis" . . . there is always
more to discover and to understand.
William Stone Weedon was a Professor of Philosophy at Virgina. He held degrees in mathematics.
physics and philosophy. He was a deep student of the work of Alfred North Whitehead. Weedon
taught classes very much by the R. L. Moore method. He loved logic. In one class I took with him,
in 1948, that was concerned with the "Logic(s) of Discovery" he wrote out sets of abstract axioms
on the board and then challenged us to prove theorems. In one instance I was supposed to show from
five curious axioms and definitions, that (p')' = p. I did it, and he congratulated me saying I had proved
that not-not-p is logically equivalent to p, a proposition is either true or false. In other words that
there is no middle third (tertium non datur). Proof by contradiction is based on this. I did not even
know we were working on Boolean algebra, or what it was. Weedon then explained to the class that
these were postulates for Boolean algebra due to some famous logician. It was all very interesting to be
able to derive not-not-p = p from axioms that did not assume this
property. We studied R. D. Carmichael's book "The Logic of
Discovery", which was concerned with the process of scientific
discovery. We
studied Hadamard's "The Psychology of Invention in the Mathematical Field". We studied how
Harvey had discovered the circulation of the blood, whereas Galen failed to figure it out.
As a result of all this, I spent a year cataloguing and studying a hundred postulate sets for Boolean algebra,
and still hope to publish my findings.
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Cryptography:
Remark: Prof. William S. Weedon
was also an accomplished cryptographer! He gave courses at UVA in the
1940's, and when he was a Lt. Commander in the U. S. Navy in the
Pacific in World War II, he once broke a Japanese coded message and
saved his ship from being bombed. He was on the twelfth invasion ship.
Weedon had a vast grasp of spoken Japanese dialects. He remained in
Japan a while after the war to study Japanese thinking, esp. harakiri, seppuku and the Zero bombers, forerunners of recent suicide terrorists.
Weedon had studied in Nazi Germany in the 1930's and told us of
having had cocktails with Hermann Goering, Himmler, and other ranking
Nazis. He was studying the German mentality of obedience to law and the
Hitler Jugenbund (Youth Corps). At last, one day Goering asked Weedon
"Well Herr Weedon, are you with us?" which Weedon took as his
final hint to get out of Germany. Weedon had done his Ph.D.
dissertation at Virginia on 'Persuasion'. Hence his interest in German
and Japanese methods of persuasion.
I had the good fortune to study analytic number theory at the University of North Carolina in 1957 with the late Prof. Hans Rohrbach
who was in charge of Geheimschrift (Cryptography) for the German
Wehrmacht during World War II. Rohrbach was an old friend of number
theorist Alfred T. Brauer at UNC. Alfred and his brother Richard had been students of Issai Schur at Berlin. Rohrbach
deciphered the famous cryptogram that Goldbach sent to Euler, and
explained to us how he did it, and published a paper about this.
He did not receive the prize from the Swiss Euler Commission for
this, because someone else beat him to it. However his method was
remarkable. He wrote a book on cryptography for the OSS (Office of
Strategic Services, predecessor of the CIA). He told me once that he
never gave them all his cryptologic methods because "if I did, no
government would hire a cryptographer who had revealed all
his secrets."
A revealing anecdote about the German view on the Holocaust can be
discerned from a remark made by Rohrbach in 1957. We asked him: "In
your position, being in charge of so much secret information for the
Wehrmacht, did you not know that the Nazis were killing Jews at
Nürnberg. Auschwitz, Maidonek, Treblinka, Dachau, etc.?" His
answer: "Ja, aber das war nicht meine Abteilung!" - - -
"Yes. but that was not my department", the standard German excuse for doing nothing.
The noted English cryptographer Irving Jack Good
was with the "black box" group at Bletchley Park in World War II and
worked with Alan Turing. Jack once offered me a position at Virginia
Polytechnic Institute and State University, but I declined and remained
at WVU.
Many mathematicians at American universties worked with cryptography at
the NSA (National Security Agency), Ft. Meade, MD. The NSA began its
work in 1954. Several of my teachers, colleagues and students worked
for NSA.
I am sorry I never had the good fortune to meet any of the Japanese cryptographers from World War II.
Cryptography is a very interesting subject, involving so much
mathematics and statistics, and today using such sophisticated computer
techniques and electronic technology. In amateur radio we use single side band
(sometimes with suppressed carrier) to broadcast voice using a narrow
bandwidth and less power. Military and some civilian services have used
somewhat similar methods to stack many dozens of voice channels onto a
single carrier. The resulting signal sounds like so much super "Donald
Duck" with an ordinary receiver not equipped with a suitable
descrambler.
A most interesting hobby is to design and build mechnical and electrical cryptographic machines.
I was proud of having once broken a coded paragraph of several hundred
letters enciphered using multiple substitution alphabets. I always
remember it because the deciphered text started somewhat as follows:
"Intuition, like a blinding flash of light, often follows when one has
reviewed in his mind all the useless experiments that have been tried .
. . " This required statistical analysis of digraphs, trigraphs, etc.
and some lucky hunches!
More stories about cryptography must wait until a later date.
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In 1957, at the University of North Carolina, Professor Drury Wall taught us lattice theory and partial order by the R. L. Moore method.
Wall removed every book and paper he could find about these subjects
from the UNC Math. Library, locking them in his office. He then
proceeded to write definitions, axioms and theorems on the blackboard
and challenged us to prove some 100 theorems. We found out later that
he was having us prove most of the theorems in Garrett Birkhoff's 1948
book "Lattice Theory".
Of course I was already familiar with the R. L. Moore method since I had attended a seminar at Virginia
in 1948 taught by R. H. Bing, a famous student of R. L. Moore. Bing had us work from Moore's axioms.
He was heavily concerned with establishing metrizability of a topological space.
It was the relentless emphasis at Virginia under Ed Floyd, Gordon Whyburn(another
of Moore's brilliant students), and others, that led students
to worry about just why compactness (Virginia or Texas kind),
connectedness, separability, etc. are important and what they imply. We
were taught to really understand what makes a mathematical theorem true
and what it means.
About Handicaps:
In the 1950's at Virginia, topological group theory was taught by Professor Ed Floyd.
We studied the book by Lev Semenovich Pontryagin. Five of us battled that book, and
and came in one morning to Floyd's class with bloodshot eyes, from reading it all night.
At that point, Floyd wanted to teach us a special lesson. He calmly explained that although
we had trouble reading the book, the author was blind! Pontryagin lost his sight due to a stove
explosion when he was 14. We were humbled. Moreover, Floyd himself had a handicap:
he had lost a hand in an accident on the farm in Alabama when he was young. He made up
for his handicap: he could write rapidly and clearly several lines on the blackboard ahead of
what he was saying, and it was legendary that he often used his one good hand to beat two
players simultaneously at ping pong. I later learned that Solomon Lefschetz lost both hands
and forearms in an explosion when he was 23, but became a famous mathematician.
Thus, we often learn that a "handicap" may not be a deterrent to success. The old adage
that "You'll never know unless you try" is worth remembering.
I remember that when (1948-57) we trained announcers at WUVA, the student radio
station at the University of Virginia, youngsters with serious speech impediments worked
hard to overcome them, and sometimes became really good speakers, and at least one became
an accomplished speech therapist.
So do not be discouraged by a "handicap". If you do not have a Ph. D. be content to know
that the "unterminated club" has many mathematicians of note. If an administrator has the
audacity to tell you that hiring you would upset his college's "Ph. D. ratio", be assured
there are better and more receptive places to teach and do research.
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"You'll never know unless you try!"
When teaching or mentoring, it is wise to always remember that learning is a two-
way process. At the same time you 'teach' a student you may be surprised to find
that you will learn something in return. I always tell graduate students that "we must
learn together." You'd be surprised how often this leads to joint research work.
My early work with Thomas A. Chapman (1962) came from our joint effort to
understand a problem about a peculiar limit on the Putnam Contest.
I remember that the first time I taught a calculus class, I found that some things I had
taken for granted I just had not really learned to understand myself. One of the best
ways to learn a subject is to try teaching it! Your students will ask you about something
you probably think is trivial, but may be very difficult for them to see. You have to be
humble enough to admit that you do not necessarily understand everything.
The secret to learning is to learn how to learn.
Once you know this secret you may be pleased to find that you can then learn any subject.
And you have to make the effort. Sometimes you may need to find an "elmer" who can
inspire you to try to do something you did not think you could do. The young lady who
taught me to drive in 1985 (when I was 57) was the first person who showed me that I could
actually get behind the wheel of a car and drive the thing. We were on Interstate 250 outside
of Norfolk, Virginia, headed to Morgantown, W. Va. when she turned to me and said:
"Henry, you drive!" I replied that I could not drive, that it would be hard. Diane then said:
"Driving is NOT hard, and besides you showed me that math is easy."
I countered with: "But math is different." She would not accept that argument. Her reply:
"They're not different! You'll never know unless you try!"So . . . I drove all the way back to
Morgantown (some 350 miles), and went out and got a license. Now hundreds of thousands
of miles later, and with five cars at hand, I will drive any place. If you want to find excruciating
fun driving, try the twelve lane belt boulevard around Washington, DC, in pouring down rain
at 3:00 A.M. when you hope to get into the correct lane to find a correct exit. It ain't exactly fun,
it is a challenge, but then again "You'll never know unless you try!" This was the same advice
given to me in 1959 by Professor Hannibal Davis, Chair of Mathematics at WVU, when he
suggested that I apply for an NSF Research grant in combinatorics. I said: "I have only written
a few small papers. How could I get a research grant?"
His answer: "You'll never know unless you try!"
So those are the words to perk you up: "You'll never know unless you try!"
H. W. Gould, 1 August 2007
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I signed up for Philosophy 20 at the University of Virginia in 1948 and spent a rough semester
with Professor William Stone Weedon trying to fathom what Alfred North Whitehead was saying
in his famous book "Concept of Nature." I earned a grade of B. It was tough reading.
When I returned to the University a few years later, after military service, I thought to myself
that "gee that was fun, I think I'll take another philosophy course." So I dutifully signed up
for Philosophy 15, Metaphysics according to my Catalogue. Upon going to the classroom, there
was the same Professor Weedon, assigning the same book, "Concept of Nature" by Whitehead,
and day after day it seemed we were talking and reading the same thing I had had before in the
first course. After the first test I went to the professor's office and asked him: "When do we get to
metaphysics?" Weedon looked at me said: "Well you are taking the same course you had before."
"But" I objected, "that was Philosophy 15 and this is Philosophy 20." "Ah," the professor replied,
"while you were in the army we switched the numbers. But don't drop the course, you are now
beginning to really understand Whitehead!"
That was when I started to learn. After more than a dozen philosophy courses, mostly in logic, I
seriously considered going for a doctorate in philosophy. I had learned in Weedon's classes how
to prove theorems from bare axioms and definitions. He would write axioms and definitions on the
board, then a list of theorems, and then dismiss us. The challenge was to prove something all by
your own self. I remember one day I gave him a proof, and he congratulated me on having deduced
that not-not-p was equivalent to p, i.e. there is no middle third. It is the basis of mathematical proof
that a statement is either true or false, and is the basis of proof by contradiction. It is interesting to
note that tertium non datur (excluded middle third), does not always hold in some Eastern logics and
philosophical systems, such as in Zen Buddhism.
So I really learned what it means to prove something, not in a mathematics class, but in a
philosophy class. That, plus what I learned in topology seminars, and other classes taught in the
manner of R. L. Moore, fixed my ideas of proof. At the University of North Carolina Professor
Drury Wall had our class prove a hundred results from Birkhoff's "Lattice Theory" book with
absolutely no use of the library or any book at all.
It is really vital for a mathematics student somewhere in his/her education to come to grips with
WHY we say something is true or false. It is just not right to treat mathematics as a game where we
take as true some difficult statement with which we have not really wrestled.
In some sense one should, like Jacob (Genesis 32:28), wrestle with G_d, or be as the Hebrew word
has it, become "Israel", he who has wrestled with G_d. We should all be like Jacob, wrestle with
the ultimate logician and change our name to Israel or Yisrael. Maybe that is what Paul
Erdös meant about G_d being the "Ultimate Fascist" who has a book in heaven giving the solution
to our difficult mathematical problems. Certainly Erdös deserves the name Yisrael.
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"Accept it as a universal truth that all men tell lies and add to the truth, or take away from it, just as
it suits their purpose. Especially must we believe nothing which, if known, would add to the reputation
of the speaker or flatter his interlocutor, for that is sure to be false."
- - - Leopold Mozart, the father of Wolfgang Amadeus
Mozart.
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A Hadith often ascribed to the Prophet Muhammad (PBUH):
“OTLOB AL ALM WALAAH FESEEN”
meaning “Seek knowledge, even in China.”
This hadith was not reported by the authoritative Bukhari, but others
have attributed it to the Prophet.
See this in written Arabic on my web page.
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On the title page of my popular book "Combinatorial Identities" I placed a
motto in Latin:
SCIENTIA NON HABET INIMICUM NISI IGNORANTIAM
The correct translation is "Knowledge has no enemies except ignorance."
"Scientia" in old Latin meant just knowledge, not the modern term "science".
I came upon this motto in 1949 on the inside lid of a beautiful Baroque
Harpsichord when Ralph Kirkpatrick (who had been a student of Wanda
Landowska) and Alexander Schneider (violinist) gave a concert of Bach and
Mozart sonatas for violin and continuo in Cabell Hall at the University of
Virginia. Kirkpatrick said he had been told that the motto might have been set
down by Mozart, but we have no evidence for that. However, it is a wonderful
motto and I forthwith adopted it as my own.
Such Latin mottos were common on old harpsichords, pianos, etc. and there is
is now a web site giving lists of such mottos.
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The Armies of War
Dusty roads that I cannot retrace,
Worn deep by the stamp, stamp, stamp of my feet;
Each step to the measure of a funeral beat
Which bore me closer to the last embrace,
Bore me downward to deep despair,
Held me pinioned in dungeons of Hell
While Angels of Death around me fell --
Fell at my feet my grave to prepare,
Fell at my feet to impale me there.
Endless roads that came to an end
With the last, long, lingering throb
Of the weary hearts that had done their job
To the measured cadence that made them descend
From the ranks of the living to the living dead;
Dead to the world that saw them not,
Cared little that they should rot,
Bore them downward to make their bread
And drink their milk from a dead man's head.
Bloodstained roads as old as man,
As old as the stamp, stamp, stamp of the feet
That step to the measure of the funeral beat
Descending downward where time began
Where hearts destroy, condemn and kill;
In measured step the living dead
Make their wars and plod ahead
And the Angels of Death are falling still
To make more wars when e'er they will.
by Henry Wadsworth Gould, 1951.
Written while pondering the insanity of the
Korean "Police Action" War. My poem won
Honorable Mention and was published by the
National Poetry Association, Los Angeles,
California in 1954 in the
collection "America Sings", page 14.
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The famous German poet, dramatist, novelist, and philosopher Johann Wolfgang
von Goethe (1749-1832), has sometimes been called the 'Shakespeare of Germany'.
He is recalled for his many years of close friendship with Johann Christoph
Friedrich von Schiller (1759-1805), author of the novel 'Wilhelm Tell' about the
Swiss Hero. The Sturm und Drang period of German literature (1765-1790),
marked by rebellion against accepted artistic and moral standards includes
the early work of Goethe and Schiller. Goethe also studied science and is credited
with some genuine discoveries, however he had a running battle against Newton's
theory of color.
Goethe was very biased against Frenchmen and mathematicians. He once
declared that 'Die Mathematiker sind eine Art Franzosen; redet man zu ihnen
so übersetzen sie es in ihrer Sprache und dann ist es alsobald etwas anders.'
(Mathematicians are like Frenchmen, If you say anything to mathematicians
they translate it into their own language and then it is something entirely different.)
Shakespeare was probably far more popular in Germany than in France since
the words and meter translated over very neatly into German. By contrast,
the works of Heinrich Heine were more popular in France than in Germany,
partly because he used words taken from French rather than basic Teutonic words.
Thus Heine would write 'componieren' rather than 'zusammensetzen'.
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Sir William Berkeley (colonial Governor of the
Virginia Colony, 1641-1677) was perhaps as
famous, or infamous, for the following statement
as for anything else that he ever did:
"I thank God there are no free
schools or printing presses and I
hope we shall not have any these
hundred years; for learning brought
disobedience and heresy and sects
into the world and printing has
divulged them, and libels against
the best government;
God keep us from both."
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Here is an interesting quotation in Greek:
ΚΑΙ
ΓΝΟΣΕΣΘΕ
ΤΗΝ
ΑΛΗΘΕΙΑΝ
ΚΑΙ Η
ΑΛΗΘΕΙΑ
ΕΛΕΥΘΕΡΩΣΕΙ
ΥΜΑΣ
"Kai gnosesthe ten aletheian kai e aletheia eleutherosei umas."
Translation of the Greek:
"And ye shall know the Truth,
and the Truth shall make you free."
- - - from The Gospel according to Saint John,
Chapter 8, Verse 32
This motto in foot-high letters carved in stone adorns the entablature on the front of
Cabell Hall on the Lawn facing the Rotunda at Mr. Jefferson's 'Academical
Village' (University of Virginia) to remind us of the importance of seeking Truth.
Jefferson had another saying about the University he founded:
"(The) . . . university will be based on the illimitable freedom
of the human mind, for here we are not afraid to follow the
Truth, wherever it may lead, nor to tolerate any Error so long
as Reason is left free to combat it."
This was written in a letter to William Roscoe in 1820.
The motto appears even today in the Cavalier Daily, the UVA student newspaper.
And yet, in this nation, we still have to fight to get information via FOIA (Freedom
Of Information Act).
The idea was embodied in his "Statute of Religious Freedom for the State of
Virginia," guaranteeing our freedom to believe or not to believe. This laid down the
rules of total intellectual freedom for Americans, and is the basis for our academic
freedom in the university. This is really where the separation of church and
state began. Teachers should not be told by the State to preach religious dogma.
Rhode Island, however was the cradle of this freedom. My own original immigrant
ancestor (1635) was so persecuted in Massachusetts for his beliefs that he fled that
state to go to Rhode Island in 1636.
In Newport you will find the oldest Synagogue (Touro), the oldest Quaker Meeting
House, and the place where Roger Williams gave birth to his Baptist Church.
Additionally, there is a statue of a man on top of the State Capitol in
Providence, which is described even today as "the independent man, with no
shackles on his brain." . . . no one to tell him what to believe or not to believe.
Today at the Touro Synagogue you can see the letter from George Washington
guaranteeing the safety of the Jews during the American Revolution.
Think well about these things when you consider "evolution" versus "intelligent
design" the latter of which is totally antithetical to the spirit of science.
I was happy to discover that one of my ancestors (John Coggeshall of Newport)
was the first Governor of
Rhode Island and Providence Plantations May 1647
– May 1648, under the Patent of 1643. Roger Wlliams served as Chief Officer,
before Coggeshall.
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"An honest heart being the first blessing, a knowing head is the second." - - Jefferson
"If a nation expects to be ignorant and free in a state of civilization, it expects
what never was and never will be." - - Jefferson
"I have sworn upon the altar of God eternal hostility against every form of tyranny
over the mind of man." - - Jefferson (to Benjamin Rush in 1800)
"Nature intended me for the tranquil pursuits of science by rendering them my
supreme delight." - - Jefferson
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"The other day upon the stair,
I met a man who wasn't there;
Now how could I this man bave met,
If there were not an empty set?"
- - - - author unknown
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Lewis Carroll used the empty set in some of his humor. I paraphrase the original:
As Alice stood on the chess board, looking down a row (or column), the King asked her
what she saw. Alice replied: "Nothing," to which the King replied: "Ah, I wish I had
such eyes, imagine, to see nothing . . . and at such a distance too!"
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Count Alfred Korzybski, in his seminal book on semantics (1933) Science and Sanity
called attention to the temporal aspect of mathematics when he said the following
(somewhere around page 253):
"A semantic definition of mathematics might run somehow as follows:
Mathematics consists of limited linguistic schemes of multiordinal analysis,
capable of exact treatment at a given date."
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"Someone who thinks by the inch, and talks by the yard, should be removed by the foot."
- - - John Horne, Monongalia County Surveyor, Dec.
2007
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"There is no religion higher than the truth." Dalai Lama
Updated as of 22 May 2009