The Atomic Origins of My Scientific Curiosity
Leading to Mathematics
As a child I was fascinated by electrical
and mechanical things. My father (a molder) taught me to be a good mechanic and carpenter and how to solder wires;
my parents provided me with an Erector Set (to build
mechanical devices), a Lionel electric train, chemistry sets,
microscope sets, radio kits, etc. In 1940, when I was still
11 years old I read the amazing newspaper story below (related to the
work of Otto Hahn and Lise Meitner in splitting the Uranium atom) and
it opened up my total curiosity about the atom! My childhood
imagination was absolutely fascinated by the fact that a small amount
of this U-235 stuff could be so powerful, and I decided that I
needed to learn physics and mathematics.
My first radio set did not work. It was a
mechanical imitation made from Erector Set parts; a blue, rectangular
box with gearwheels for knobs. I built this when I was about 9 or 10
years old. Upon exhibiting it, I was told by the other kids: "But it
doesn't really work!" This made me decide to figure out how to make a
real radio set! I guess my first real vacuum tube radio set was made
when I was 10 years old. I started reading every book in the
public library on crystal and tube radio sets. I broke open
old tubes to find out what the filament, cathode, grid and plate were.
I unwound old electrolytic condensers (now called capacitors) to find
out how they were made, and wound some of my own. I wound my own 5
volt, 20 ampere transformer to show I could do it. By age 17 I had
built Geiger
counter tubes (I learned glass blowing at a local Neon sign shop),
Geissler tubes, an X-ray tube, a cathode ray oscilloscope, scale of 64
Eccles-Jordan flip-flop counting device, a radium spinthariscope, a
giant 1,000,000-volt Tesla coil, Jacob's ladder, a mercury barometer,
and I built my own working oil-drop apparatus a la R. A. Millikan to measure the electron charge, after reading his fascinating book "Electrons Plus and Minus".
After reading about the invention of the
transistor in 1947 by John Bardeen, William Shockley, and Walter
Brattain, I made one myself around 1950 from a 1N34 germanium diode.
This involved picking wax or whatever from a tiny hole in the side of
the ceramic tube case, and using a microscope to insert a "whisker"
wire very close to the existing whisker so as to form base, emitter and
collector. It is this basic need to experiment, whether with wires or mechanical parts or numbers, that is the driving force of curiosity behind research,
whether it be in mechanics, chemistry, biology, electronics or
mathematics. My poor mother had to endure me stinking up our house as I
tried to mix and boil sulfur and other noxious stuff in an attempt to
make rubber. After spilling sufficient hot solder on my bed room
floor, she sent me to our large unfinished attic which became my
laboratory. There I had a 35 by 35 foot space, with a corner for
radio work, a corner for chemistry sets, a corner for microscopes and
biology, and a huge 15 foot bench my Dad made for my mechanical work
and experimentation. This was my Walden, my hideaway, my place to read,
study and think. In 1944-45 my giant Tesla coils, tossing 36 inch
sparks and making terrible noise, and Jacob's ladders running yellow
sparks up the walls, made the whole attic look like Dr. Frankenstein's
lab. The house reeked of ozone. Neighbors could not get any radio
reception when I had all my coils running. A nearby fireman would come
out of his house with an ax and threaten to chop me down. And
how many times did we electroplate old V nickels to look like five
dollar gold pieces! How many small bombs did we make as we tried to
make our own gun powder! And imagine all the nitrogen tri-Iodide we
splashed around on key holes, door knobs and lamps to give everyone a
tiny blast here and there. Maybe that was a little too much, but it was
fun. My Dad was almost totally deaf, and when Mallory came out with the
chunky, small mercury batteries, I saved every one of his old hearing
aid batteries. You see, you drill two small holes in the copper top,
and metallic mercury then gradually oozes to the top which you can
suck out with a medicine dropper. I obtained a small Listerine bottle
of Hg this way (cleaned by use of HCL) and used it to make a mercury
barometer that works. This was raw, insatiable curiosity trying to find
out how things work. It is like my student in a calculus class who tried to divide by zero because as she said, "When you want to figure something out, you will try anything!"
It is interesting, however, that when I
returned from army service in the early fifties, I found that my Mom
had thrown out many of my tanks of acids and stuff for fear the house
would blow up! She had also thrown out all my hundreds of old comic
books, including first edition copies of circa 1938 Superman and the
like (now worth a fortune), all my Astounding Science Fiction (ASF)
magazines, and a hundred toy cap pistols! One excuse she gave was that
science fiction "harms" the mind. There she was quite wrong. I
have even written some science fiction (unpublished) about time
travel. John R. Pierce (1910-2002), who invented the traveling wave
tube for microwave amplification, wrote for ASF under the pseudonym J.
J. Coupling (derived from a quantum J to J coupling effect). In ASF
Magazine he described his traveling wave tube as a true science fiction
device invented before its useful time. But, of course, he wrote
his mathematical proof of the device in the Proceedings of the
Institute of Radio Engineers (IRE). Eminent combinatorialist John
Riordan wrote fiction. Mathematician Eric Temple Bell wrote science
fiction under the pseudonym John Taine. I have met many brilliant
scientists who thrive on real science fiction, stuff about space and
time travel, systems that depend on nitrogen instead of carbon, etc. I
do not think my Mom ever realized what pure science is really about,
any more than President George W. Bush understands stem cell research,
or that global warming is a real threat to our civilization, or
that creationism should not be taught in public schools!
What could be more like "science fiction" than "Flatland"? . . . Nevertheless Mom and Dad supported me.
My mother had started me reading by setting
up a small library for me with hundreds of books on every imaginable
subject, but heavy on languages (Latin, Greek, German) and
science. By age 10 I had learned the Hebrew alphabet, and I began to
study written Chinese. Mom taught me etymology at
home using an old book that traced the origins of English to its
Latin, Greek and Anglo Saxon roots in great detail. I was exposed to
the curious old book by Albert C. Crehore on the theory of the atom,
where I first learned about the infinite series I later learned is
called the Riemann Zeta function. The book introduced me to the Bohr
theory of the atom. I pondered the possibility of the existence of
Element 118. This was in the years before Neptunium and Plutonium. In
college in 1946 I started learning some Chinese and Arabic from
textbooks written in German.
In high school days there were three of us
(Paul Louis Goodfriend, Bernard Brown and myself) who were the genius
science kids. Around 1944-45 a Los Alamos physicist showed us the
(then-classified) Los Alamos Report LA-24 which outlined atomic theory
and gave the differential and integral equations of neutron flow
in an atomic pile. We studied Pollard and Davidson's book "Applied
Nuclear Physics" and Strong's "Experimental Physics". We absorbed the
"Smyth Report". We studied Uranium chemistry, and "We Three" set about
to extract one gram of Uranium from pitchblende ore or Uranium
glass
(used in neon sign manufacture). There we were, three high school kids
mixing hydrofluoric acid with uranium glass in lead crucibles, and
ultimately going through a flow chart to obtain a small quantity of
Uranium Tetrafluoride and ending up with a tiny amount of Uranium
Oxide (U2 O8), which is what Klaproth obtained when he discovered
Uranium
in 1789. In the course of our crude experiments we probably breathed in
a lot of the volatile Uranium Hexafluoride vapors from our crucibles!
Thank our lucky stars we did not suffer! By the way sodium
diuranate (Na2 U2 O7.6H2O), also called yellow cake, and uranyl nitrate
(UO2(NO3)2) don't taste bad at all! Brown, Goodfriend and I studied the
theory but were not able to obtain some critical items to build a
linear accelerator or a cyclotron.
My late friend Jerome Hines (Metropolitan
Opera Star who sang 30 roles at the Met) started his studies in
mathematics, physics and chemistry. But his voice was better, and
he soon became a basso profundo singing sensation. Dad took
me to hear him sing many years ago. I tried singing but gave it up for
science, just the reverse of what Jerome Hines did. My Dad had a really
good voice but no training as his brother William Benjamin Gould had
had. Uncle Bill studied voice at the Toronto Conservatory
after the First World War and sang opera in Europe and especially
in Moscow. He had a White Russian girl friend and wrote a novel
"When White was Red" but this was never published, being critical of
the political system under Lenin.
Curiously, Hines and I had a
mathematics professor in common! William Marvin Whyburn was
Jerry's mathematics mentor at UCLA and many years later
I knew William at UNC in 1957. This Whyburn was the brother of the
famous Gordon Thomas Whyburn, topologist and student of R. L. Moore.
Jerry published about 12 papers in the Mathematics Magazine; one was on
the Stirling numbers. Hines visited West Virginia University several
times, singing and reviewing performances of master's voice students.
One time when Jerry visited here at Morgantown, the Maestro reviewed
voice students in the morning, then we had lunch, and in the
afternoon the Maestro gave a talk in our Department's Colloquium about
"Operator Mathematics" (see the program poster under "Gould's Photo
Album" here on this web site). The first time he sang here, I had
him autograph
reprints of his mathematics papers while everyone else wanted him to
sign the concert program. Jerry and I sat up till 3:00 AM at the
Hampton Inn here in town reminiscing about mathematics, chemistry,
physics and music. He told me that he too had dabbled with uranium and
that he once had his own small bottle of uranium nitrate with which to
play. This man, this great basso profundo, was the epitome of the
Renaissance Man Ne Plus Ultra. He once gave a short course at the
University of Illinois on "Calculus for Poets". Such was his enthusiasm
for the beauty of mathematics. Baritone Joseph Shore has a web page
<http://www.josephshore.com/> giving a marvelous memorial to
Jerome Hines.
Circa 1945, using the greatest integer
function (bracket function), factorials, and other stuff, I was able to
devise a formula for the determination of the binding energy of every
known isotope of every then-known element! I sent it to a famous
physicist but he never answered my poor letter, and I did not know
until years later how accurate my formula was. It remains unpublished.
Here is advice to teachers and professors: Always answer inquiring
letters from budding scientists.
I understand that Einstein (not the
professor to whom I wrote) always answered such letters
from youngsters. I know (from personal observation when he visited
three times here at West Virginia University) that the late number
genius Paul Erdös was always eager to interact with bright
youngsters,
giving maximum support for their curiosity. Genuine curiosity must be
stimulated and encouraged throughout one's lifetime. Without curiosity
and enthusiasm there can be no advance in science and our understanding
of the Universe. Blind and unquestioned obedience to religious myth
stifles the search for knowledge about the universe. This is not say
that religion is 'wrong'; indeed the Bible (whichever of hundreds of
translations you choose) is a document about morality and ethical
behavior, and has no meaningful information about science or how the
physical universe came to be and does not refute the truth of evolution.
There is no conflict between religion and science.
In high school I read in Popular Science
Magazine (July 1944) about the maverick Viennese physicist Felix
Ehrenhaft (1879-1952) and his claim to have discovered a magnetic
monopole. I built models of his "magnetic motor" and pondered what
his claims about a "magnetic current" meant. Of course at that
time I had not studied enough physics to understand this. But doing
these experiments was bold, raw curiosity in action! Ehrenhaft was
very controversial and disagreed viciously with Einstein.
Now, already in 1945 I had published a
circuit diagram for a reflex regenerative radio receiving set in a
radio magazine (Radio Craft Magazine) and again in 1948 a circuit
design for a super-regenerative FM radio receiver.
My first mathematical discovery
(around 1942) was the formula n(n-1)/2 for counting the number of pairs
of pins you must choose to check for continuity when testing a radio
tube with n pins on the base. Of course this also led to the explicit
determination of the actual permutations and combinations of
k items chosen from a set of n items.
Around 1944-45 I developed a formula for
finding the sum of the k-th powers of the first n natural numbers.
Only later did I learn that I had "rediscovered" the Bernoulli numbers.
This was my first experience with recurrence formulas or recursive
algorithms to prove an interesting result by mathematical induction.
I decided that I would make a project in my
life to study everything knowable about sums and products of
numbers. My algebra teacher Miss Elizabeth Culpepper, at Woodrow Wilson
High School in Portsmouth, Va., had let me study factorials and
binomial coefficients while the other kids were studying the quadratic
equation, and I became utterly fascinated by the integer coefficients
when you multiply out x(x - 1)(x - 2)(x - 3) . . . (x - n +
1), and it was in this way that I became intrigued with what I later
learned are the Stirling Numbers of the First Kind. I studied them in
my master's thesis (1956) and later published a simple formula for
them in the Proceedings of the American Mathematical Society (1960).
After joining the faculty of mathematics at West Virginia University in
1958 I was able to obtain NSF grants starting in 1960 to support my
research in combinatorics. Later I found out that these were the
first NSF mathematics research grants in the State of West
Virginia. I made a trip back to my home town to visit my high school
algebra teacher to thank her for encouraging my fledgling interest in
permutations and combinations. It was because of her encouragement that
I stayed with mathematics and I credit her for edging me on toward
success in combinatorics.
Two other high school episodes will show
other reasons for my plunge into science, and especially mathematics. I
remember a time circa 1942-44 when my English teacher asked our class
to write a theme (i.e. an essay) on something we were really
interested in. Ergo! I wrote my essay "Introduction to the Mathematical
Theory of Four-Dimensional Geometry." I had been reading "Flatland" by
Edwin A. Abbot. I had also absorbed Henry P. Manning's "The Fourth
Dimension Simply Explained" (1910/1912) and his "Geometry of Four
Dimensions" (1914). These were among the various books that my
mother had provided me. My teacher gave me a grade of C minus, and I
asked her about the grade: "I can understand the C because I am just an
average kid, but why the minus?" Teacher's answer: "Poor choice of
topic!" Then I understood what she really meant: We could write on any topic we wanted . . . provided she
could understand it! Then I heard a mathematics teacher tell another
teacher: "Don't let that Gould kid in your class; he asks too many
questions!" . . . ah, but that is the only way I learn, by asking
questions. At ten years of age being told that God created the Earth,
my response was: "Then who or what created God?" I asked: "How many
years existed before time was created?" I was told quite summarily that
I couldn't ask such questions. Ah, but you see, I did and do ask such
questions.
I remember a Greyhound Bus trip to visit
Professor Florence Mears at George Washington University because I saw
in a college catalogue that she taught a course on infinite series. She
and others (e.g. Saunders Mac Lane at University of Chicago) soon made
it clear to me that I would have to study a lot more than a simple
course on infinite series in order to achieve my dream of summing all
the series!
Most importantly, I wrote to Leonard Carlitz
at Duke around 1953. He liked some of the formulas I sent him but some
of my stuff had been anticipated long ago by Ramanujan. By analogy,
Carlitz became my G. H. Hardy, and he taught me so much that I
shall forever be indebted to his infinite patience with my finite
brain!
It should be remarked that my mother
left me a letter which I found after her death on 27 July 1964, in
which she explained that she had prayed for a child for 13 years. She
was nearly 39 when I was born, and I was the only child. It is a difficult
task to be an only child and live up to your parents' expectations. The
Bible commands us to honor our father and mother. All that I am I owe
to my parents.
= = = = = = = = = = = =
New Chemical Discovered
New York, May 4, 1940 - (AP) - Laboratory isolation of a new chemical
substance, one pound of which is said to be capable of yielding the
power output of 5,000,000 pounds of coal or 3,000,000 gallons of
gasoline, was described today by the New York Times.
The Times said that the discovery, announced in the
current issue of the Physical Review, scientific journal, had been
hailed by leading scientists as holding the promise of revolutionizing
all present methods of power production, and ushering in the era of
atomic power.
The substance was identified as "U-235", an isotope
or chemical twin of ordinary uranium, which when simply immersed in
cool water releases its energy in a form unseeable by man ---- steam.
A chunk of 5 to 10 pounds of the substance,
plentifully available in many parts of the earth, would drive a
battleship or sea-going submarine around the oceans of the world for an
indefinite period without refueling, it was said.
The Times said that the Nazi government had heard of
American research in this field and had ordered its greatest scientists
to concentrate on the problem of improving the method of extracting the
U-235, one pound of which was said to have the explosive force of
15,000 tons of TNT.
- - - - -
from: The Portsmouth Star newspaper,
Portsmouth, Virginia, Sunday, May 5, 1940, Local Section, page 8.
= = = = = = = = = = = =
Addendum about Number Theory
When I was in Army basic training, I carried around a list of all the
prime numbers less than 10,000. I had factored the serial numbers of
all the other guys in the company, but could not factor my own number.
Whenever we were out on a long march and were given a "five-minute
smoke break", I would dig into my back pack and get out my list of
primes and continue to divide my serial number by the next prime,
always getting a remainder, so I finally decided that I, the only
mathematician in the company, was a PRIME! Years later, first using a
huge mechanical adding machine, then an early desk-top computer, and
finally a big IBM-360 digital computer, I was able in microseconds to
convince myself that I was indeed a PRIME. It was not until around 1962
that all of the first 100 Fibonacci numbers were finally factored or
shown to be prime. There is this big unsolved problem to show whether
there are infinitely many prime Fibonacci numbers. So far as I know
this is still undecided.
In the first issue of the
Fibonacci Quarterly, Volume 1, February 1963, page 46, Leo Moser and
Leonard Carlitz posed the second advanced problem H-2 "Resolve the
conjecture: There are no Fibonacci numbers which are integral squares
except 0, 1, and 144." The same problem was posed in the American
Mathematical Monthly, Vol. 70, No. 2, February 1963, page 46, Problem
5080, by A. R. Rollett, Crediton, England, in the form: "In the
Fibonacci series (F_1 = 1, F_2 = 1, F_n+1 = F_n + F_n-1) the first
second and twelfth terms are squares. Are there any others?" Between
the two journals, I believe about 7 proofs were received (not all published) showing that
144 is indeed the last Fibonacci square. In the meantime Marvin
Wunderlich had run computer tests showing that no other squares existed
in the first million Fibonacci numbers. In time it was been shown that
8 is the last cube in the sequence, and it appears that someone has now
shown that no other perfect powers appear in the Fibonacci sequence.
These are difficult topics.
In a lighter vein, I posed
the very first problem (H-1) in the Fibonacci Quarterly, Vol. 1, Feb.
1963, p. 46: "Find a formula for the n-th non-Fibonacci number, that
is, for the sequence 4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19,
20, 22, 23, . . . " I gave hints to see a paper by Leo Moser and
Joachim Lambek, and later published a paper on this in the Fibonacci
Quarterly, Vol. 3(1965), pp. 177-183. with the title "Non-Fibonacci
Numbers." This was all about complementary sequences whose intersection
is empty but union is all the natural numbers. I used to joke Vern
Hoggatt that I would have to start a Non-Fibonacci Association and
publish a Non-Fibonacci Quarterly. It appears that the study of
complementary sequences began in earnest with Samuel Beatty in Canada,
and was long a popular Canadian topic.
- - - - Henry W. Gould, 26 August 2008