My intention is to provide a “life overview”, so that any readers can have a better sense of who I am, and thus place my writings in some context. It’s a little overwhelming to try summarizing 60+ years of life experiences – but I’ll start with an overview sketch, and elaborate as inspiration guides me in subsequent posts.
I decided to go with the “milestones” approach to hitting highlights:
(and it was my intention to be brief, but it turns out I wasn’t very successful at that — you may want to read in “installments”)
I was born on the 4th of July, 1952. I love having that holiday as my birthday – I like to think that everyone is celebrating with me! I was born in Lebanon (that’s a town in PA, not the country). I was the first-born to my parents, and so was the oldest of 4 siblings growing up. We lived with my grandfather (on my Dad’s side, we called him “PawPaw”). I remember when he died – I was about 3 years old. Only time I ever saw my Dad cry! I never met my father’s mother – she died a few days before I was born.
Move to Hershey, PA
Our family moved to Hershey when I was about 3 or 4 years old. I remember being confused about what was going on. I think we moved to be closer to Harrisburg for several likely reasons: my father worked for IBM (as a Field Engineer) and the “area” he covered was in the vicinity of Harrisburg (Hershey is half-way between Lebanon and Harrisburg), also my mother’s Christian Science Church was located in Harrisburg, and finally Hershey was well-known as having an excellent school system. I started Kindergarten in 1957.
Falling in Love with Music
My father was a talented amateur keyboard player – he played piano, organ, and accordion. I remember his playing his favorite Beethoven sonatas: the Pathetique and the Moonlight, which quickly became my favorites. I also remember an amazing (but less well-known) piece called Suite Gothique by Boellmann (written for pipe organ, but my father played it on the piano in our dining room). I was so struck that I implored my father to show me how to play it myself! I tried to memorize the Toccata riff (my favorite), and soon my Dad agreed to start me on piano lessons. He was the only piano teacher I ever had, and I studied through middle school, by which time I was playing and enjoying the Beethoven sonatas for myself! I’ve been passionate about music ever since, but now my favorite music is the blues (did I mention that I listen to Sunday Morning Blues every week?), and I now try my hand at songwriting and performing at open mic nights. I also studied trumpet from 4th-12th grade, but I didn’t love it like piano. I still enjoy playing keyboards whenever I get the chance!
Falling in Love with Math
In 5th grade, our teacher told us the story of Gauss (in grade-school) figuring out the “short-cut” method for adding consecutive numbers (e.g. 1 through 100). I’m sure she explained his technique, but either I didn’t understand it, or I forgot it. While at Overnight Summer Camp (Nawakwa) the summer after 5th grade, I started thinking about this while lying awake in bed. I started by looking for patterns with small sequences like 1 – 9 (sums to 45) and 1 – 10 (sums to 55). Somehow I guessed the formula: n * (n+1) / 2, and verified that it worked for all the examples I could check. I continued pondering this to try to understand what it meant – finally realizing that every number pairs up with a “twin” so the pair adds to n+1 (e.g. when adding 1-10 the 1 pairs with 10 to add to 11, 2 with 9 is also 11, etc. This makes 10 pairs of 11 (if you order the pairs), but that counts each pair twice, so 10 * 11 is the sum of the ordered pairs, but must be divided by 2 since we only want to count each pair once! I was extremely excited by this insight! I recall being afraid that I might forget it while asleep, and not remember it in the morning — thankfully I did remember it, and my deep love for mathematics was born.
Falling in Love with Puzzles
I don’t know as precisely when this happened, but sometime around 5th, 6th, or 7th grade, my father started bring home puzzles that were shared with him through his work. These were brain-teasers, that programmers (even in the early days of mainframes) and computer engineers loved to challenge each other with. My father shared these puzzles with me, and I have fond memories of working on puzzles together with him. One puzzle I remember in particular was the “Who is the Engineer?” logic puzzle that we found in Reader’s Digest. Also known as the Smith, Jones, Robinson puzzle – they were the fireman, engineer, and brakeman on a train (in some order) and there were 3 passengers named Mr. Smith, Mr. Jones, and Mr. Robinson. As is standard with such “logic puzzles” there were a list of clues having to with who did or did not have certain jobs, live in certain cities, have particular salaries, and so on. The object is to reason out which one was the engineer. I remember not being quite able to solve this – because I didn’t realize that a salary of $20,000 was not exactly divisible by 3. I think the clues were something like: Person X’s salary is exactly 3 times as much as Person Y’s salary, and the person living in City Z earns exactly $20,000. Knowing that $20,000 is not exactly divisible by 3, one could conclude that Person X does not live in City Z. I felt cheated when I learned the answer, but grappling with such puzzles was a lot of fun, and started me on a lifetime passion for brain puzzles in their many forms! In fact, now my professional life revolves around puzzles – primarily designing logic puzzles for smartphones (check out Monorail for iPhone and Android, if you haven’t already).
Discovering that I had some serious mathematical talent
I knew that I loved math, and that I was pretty good at it – I was one of 5 or so “top math students” in my school class (we all got straight A’s in math). I was blessed with terrific math teachers throughout my middle school and high school grades. In 7th grade I remember driving my teacher crazy by always raising my hand to point out technicalities (e.g. “that won’t work if x is 0”). After the math final exam in 7th grade (which I must have finished early since my teacher graded it immediately), I’ll never forget how he came to my desk and asked me to come “out in the hall with him” — I was terrified that I was about to be punished for something — these were still the days when corporal punishment was permitted, and I had received spankings in school. When we got outside the classroom, I was relieved to learn that my teacher wanted to shake my hand and congratulate me for being his first student to get a perfect score on the final!
I didn’t realize I was a “math nerd”, but I did spend the Summer after 7th grade reading all the math books I could find at our local public library. I remember being especially intrigued and perplexed at the notion of imaginary numbers. The first day of math class in 8th grade, I asked my Algebra teacher what was up with these imaginary numbers. He went back to my 7th grade teacher to ask “who is this guy who spends his summer reading math books?”. I suppose I was aware that not everyone shared my passion for math – so my explorations were largely a solitary endeavor (I wish I had friends and co-learners to explore with, but that didn’t come until later).
I absolutely loved Euclidean Geometry in 9th grade. The spatial/geometric aspect combined with formal and rigorous proofs had remarkable appeal for me! It was this year that I learned I had a special and unique talent (at least compared with my immediate peers). Other students started struggling a bit with geometry, but I simply “ate it up”.
In 7th or 8th grade we took the JET “Engineering Aptitude Test”, and I scored 99 percentiles in all the technical subjects (math and science). My guidance counselor met with me to suggest that I should be thinking about colleges like MIT and Carnegie Tech (now CMU). First I had ever heard of either of those, but it made an impression, and years later I did end up at MIT.
A budding electrician / computer designer?
Sometime around 4th or 5th grade, my father gave me a wonderful present: an “electrical kit” consisting of 2 dry cell batteries, several toggle switches, and some small (3 volt) light bulbs, along with wires for connecting them! I loved playing with these, and learning about “parallel” and “series” circuits. My father was an Electrical Engineer (Bucknell College on the G.I. bill). I’m forever grateful that he shared this interest with me! It was my first exposure to Boolean Logic (though I didn’t know it by that name) – series circuits are ANDs, and parallel are ORs. Dad also showed me how to wire 2-way switches (so that toggling either switch will toggle the light (this is how lights in a house can be controlled from 2 different switches). I later learned that this represents the XOR (Exclusive Or) logic function.
In 7th grade my father taught me about electro-magnetic relays. These were used in IBM equipment at the time. I hand-wrapped a nail to make an electromagnet and used a flexible piece of sheet metal as the “switch element”, and did my 7th-grade science fair project on How a Relay Works.
I remember wanting to design my own computer, so I decided to start by designing an “Adder Circuit”. I knew about binary numbers and binary addition. I also had a collection of relays (2-position, 4-throw), discarded parts that my father brought home from IBM. First I had to come up with a design. Naturally I started out as simple as I could by designing a 1-bit adder. Adding 2 single bits can yield a 2 bit answer so I needed 2 lights to display the result. I realized that the unit digit of the result was the XOR of the 2 input bits – on only if exactly 1 of the inputs was on (1 = “on”, 0 = “off”), and off otherwise. The “carry bit” (the 2’s bit of the result) should be on only when BOTH the inputs are on, so it represented an AND of the input bits. Cool – I was off to a great start. Next I set about designing the circuitry for adding the 2’s bits. This suddenly got a lot more complicated — now there were 3 input bits (two 2’s bits from the input numbers, and a possible carry-over bit from adding the 1’s digits. This stage also required 2-outputs (1 to the light for the 2’s bit of the sum result, and another carry output to go to the “next stage” (4’s place). I immediately realized that once I designed this module (3-input, 2-output) I could simply iterate it to obtain and adder for as many bits as I wanted (and had relays to work with). I struggled with the logic and circuitry for this stage, and went through a number of design iterations. I remember feeling stuck at one point, and gave to take a break to watch one of my favorite shows “The Man from UNCLE”. After the show, when I took up the problem afresh – things immediately clicked into place. This was my first memorable experience of the well-known psychological phenomenon of subconscious problem-solving (the can mind continue to work on a problem even when not consciously aware of it). I’ll spare you the remaining details of my design [after all, I did want to keep this post brief! As one of my brothers is fond of pointing out: “Way too late, Glenn!” — brevity is not one of my long suits, as you must be learning by now], other that to say that my key insight was to “feed back” the carry result from stage n+1 to turn off the light if there was a carry, which is almost correct, working great for the first stage, failing to work for the general modules with 3 inputs. In that case I needed to “override the turnoff” and have light stay on if all 3 inputs were 1’s. I implemented my design using the spare relays (my Dad gave me serious help creating the 40-volt power supply with a spare transformer) and did all the wiring myself. When I was ready to turn it on for the first time, my father cautioned me to be prepared for it not working (he knew how easy it was to have bugs in both wiring and logic circuitry). I was puzzled – why wouldn’t it work? My design seemed correct, and I was super careful doing the wiring. Well, I didn’t learn my lesson that day, because it did work correctly the first time! My working device was a 5-bit binary adder. The binary inputs were set by the position of 2 rows of 5 toggle switches, and the output displayed on 6 lights. This became my 8th grade science fair project which won 1st place in my high-school, and an honorable-mention at the Regional level. The judges at the regional fair asked me if I knew about Boolean Algebra, and I had to confess that I didn’t (it should have been obvious from the fact that my designs – presented in a logbook as part of my project – where decidedly non-optimal). Later, when I did finally learn Boolean Algebra, I was able to simplify my design a good bit. Still, I’m pretty proud of the accomplishment.
A budding philosopher?
In middle school I recall thinking a lot about “deep questions”, such as the nature of reality, is there a God?, what happens when we die? is there absolute truth? and can we know it? I was educated during the “Sputnik era”, so I had an intense exposure to Science because there was a big educational push in that direction at that time. So I absorbed the “scientific method” and became a strong “believer” in science and empiricism. My curiosity led me to explore philosophy and I read through Will Durant’s “History of Philosophy”. I found the questions and various philosophical arguments fascinating. I’ll write more about this later, because it overlaps with my spiritual / religious journey, and deserves it’s own post at some point. Suffice it to say that I spent many nights pondering “big questions” before falling asleep. Thinking about math and philosophy were my favorite ways to fall asleep, along with reading by my “dry-cell powered night light”.
In Middle School (we actually called it “Junior High”) I took over a paper route. I’d deliver papers riding my bicycle every day, in order to obtain some income to supplement my meager allowance. I recall being a pretty good saver – and after a few years (on May 27, 1968) I bought my first (and only) share of IBM stock. That single share cost $334.25 + $6.00 trade commission. I still have that share, which has split multiple times so it is now 16 shares. The story I want to tell about my paper route concerns my mother, and illustrates what an amazing mother she is! When I wanted to get involved with Basketball (I was tall, so I felt obligated to give it a try, but I was never very good) after school, I was confronted with the possibility of losing my paper route – if I let someone else take over, I might never get it back. So my mother (bless her heart!) agreed to take over the route for me, during the winter when I was involved with basketball. Being young and naive, I didn’t truly appreciate what a sacrifice she made for me — after all, she was the mother of 4 children who were all active, and I’m sure quite a handful. Mom – thank you so much!
NSF Summer Math Program at Ohio State
I was extremely fortunate and grateful to be selected to attend the Arnold Ross SSTP (Summer Science Training Program) at Ohio State for 2 summers (1969, and 1970). The program was amazing on many levels! I got to study “advanced” mathematics — a Number Theory course which forms the core of Prof. Ross’s program for high-ability high-school students. I later learned that this was actually a graduate course at OSU, and there were math grad students taking the course along with us high-schoolers! It was also my first exposure to other math geniuses (many much brighter than me) who shared my passion for mathematics. Socially, it was my first interactions with any Jews (when I was growing up, Hershey had no minorities, other than the Italian Catholic community – Hershey was predominantly WASP, and extremely conservative). I must say that my initial impression of Jews and Jewish culture was extremely positive. More on all that later when I discuss my spiritual journey. Perhaps the most influential aspect of my experience studying with Prof. Ross is the way he taught Number Theory — by discovery! Every day we had a new problem set, and the typical problem was to “Prove, or Disprove and Salvage”. This was not typical h.s. math where you are given problems to prove. Rather we were given questions to explore — our task was to determine whether a proposition was true or not. If it was true, we needed to prove it. If it was false we were to come up with a counterexample, and try to find a related proposition that was true (a “salvage”) and prove that. Number Theory was a wonderful vehicle for learning this skill of doing mathematics – because the topics are accessible (they don’t require a lot of advanced math knowledge), yet rich and deep. Perhaps Prof. Arnold’s most enduring quote (and most influential advice) was to “Think deeply about simple things!” This idea is itself extremely profound, and we should all think deeply about just how powerful it is! Another smaller (but still influential) aspect of my summer experience, was that of “falling behind” — the daily problem sets were so difficult that I couldn’t finish them all everyday, so I was constantly playing catch up. Later, when I got to MIT, I was fearful that we’d have problem sets like at OSU every day, and I’d be taking 4 or more courses instead of just 1 or 2 (I actual tried to learn Linear Algebra that summer, but it only met 3 times a week). Much to my relief, most problem-sets at MIT were easier than the OSU ones, and they were only given out once per week, not every day. My summers at OSU were perhaps the most intense and most rewarding intellectual experiences of my life.
High School Graduation
I graduated Hershey High in 1970. I was a co-valedictorian (there were 4 or 5 people who had straight A’s through high-school). I also received medals in English and Physics, but I was devastated at not receiving the Math Medal (which was the only one that meant anything to me)! It was my first exposure to “back-room” politics — when I asked my math teacher why the medal went to someone else, he agreed that I deserved it, but said I shouldn’t be “greedy” since I had received 2 other medals. It became obvious that the teachers had “negotiated” who would get which medals, and they probably thought that it would make me appear more “well-rounded” for college applications, and that the math medal would help another student with his college applications. I felt really burned to learn that the medals were not truly “merit-based” as I felt they should have been!
to be continued …
I’ll pick up next week with my College years at MIT.