Math vs. Science
A personal journey of discovery
Over the course of a human lifetime, a small number of critical decisions or events can change the direction of a person’s existence forever. This singular inflection point of fate is a favorite theme for science fiction writers and therapists alike. For me, one of these came at the age of 17 while sitting in math class on the first cool day of autumn.
I had always loved mathematics. The Count was by far my favorite character on Sesame Street. My childhood scribblings more often than not involved a ruler, or a protractor, or any other number other classroom geometry tools. I kept meticulous game-by-game statistical records of the progress of the local soccer team. While I was far from being the smartest kid in class, I had never failed to get an ‘A’ on my school reports in math.
On this particular day, I took my usual seat by the window. My school was built in the bucolic neighborhood surrounding an English cathedral, and the windows of the mathematics department looked out across a grassy courtyard to the great old church beyond. In my mind, the cathedral and mathematics had always been somehow inextricably linked — in the years I had spent sitting in this room, I had calculated all the angles on the vast cathedral’s various intersecting parts, practiced trigonometry to calculate its height and length from my seat in the classroom. I had mnemonically named many angular parts of the building, or its monuments, after mathematical principles that I would need to remember on tests — when exam time came, I simply had to look out of the window at the timeless architecture beyond — my own personal crib sheet. My faith in the indisputable truth of mathematics was as unquestioningly solid as that of the people who had built the great stone edifice long ago.
Calculus. Excellent. The riddle-solving part of math class. The perfect way to spend a chilly Friday morning. Solve a dozen or so questions to find the value of x, and spend the rest of the morning getting ahead on homework. Then bunk off for the whole weekend while everyone else is still studying.
The teacher started writing out an equation on the blackboard. Nuts — this was going to be one of those instructional lessons that I was going to have to pay attention to. The last time we had one of these, he had introduced us to the concept of i and imaginary numbers, which made no sense at all at the time. This would be even worse. This is what he wrote:
a=b
Multiply by a: a^2 = ab
Subtract b^2 from both sides: a^2 -b^2 = ab -b^2
Which is: (a+b)(a-b)= b(a-b)
Cancel out (a-b) on both sides: a+b=b
Since a = b, then: a+a=a
Therefore: 2a=a
Thus: 2=1
It looked right. It certainly didn’t feel right, but I couldn’t work out how. Nothing seemed to violate the logic of calculus as far as I could understand. All of the factors added up correctly. But it couldn’t be right. It just couldn’t! If it were this easy to prove 1=2, then how could anything in math ever make sense again? The teacher told us to work it out and we’ll discuss on Monday. The bastard.
My weekend was a bust. I could not work it out, nor could I banish the riddle from my mind. By the time I sat back in class on Monday morning, I was utterly dejected. I was sure there was a simple trick, but I just could not see it. The teacher wrote out a question on the blackboard:
Calculate (a-b)
Zero! Suddenly, it all made sense. In line 5, we cancelled out the identical (a-b) on both sides of the equation. More accurately — we had divided everything by (a-b). But (a-b) is Zero! We had been tricked into committing a mathematical fallacy by misleading jargon! Simply substituting the phrase “cancel out” for “divide” had been enough to fool the entire class into making a critical error in logic.
For the teacher, this might have been a fun trick, a good introduction to the nuances of calculus and sources of error in logic systems that could be caused by unexpected zero-value calculations. For me, I couldn't stop thinking about which other solid truths I knew might be undermined by erroneous assumptions and misleading jargon. While the other students were beginning to dream of which University they would ideally like to attend after graduation, I was becoming trapped in a paranoia that everything I had ever learned could be wrong, just because someone forgot to account for a zero in the math.
I found a refuge in biology class. I had always enjoyed biology for many of the same reasons as math — the way that a series of relatively simple repeating patterns could interact to create the complex molecular and cellular makeup of all living things fascinated me. While physical scientists would pillory biology as inexact or mathematically weak, it represented a bridge to the natural world that dry calculations could not. The nuanced observations and logical arguments of Darwin’s The Origin of Species came to be more satisfying to my mind than a thousand mathematical proofs. It was a science where I could feel comfortable that what I was studying was real, because it was there, right in front of me. Science, after all, was supposed to be about testing ideas, not just justifying them with fancy logic systems.
Over the next few weeks, our class completed a genetics experiment with fruit flies. Every morning, we had to collect flies from the plastic incubator chamber, using Ethyl Ether to anesthetize them. Then we would carefully count whether they had black abdomens, or stripey ones — all before they woke up and started flying off around the room. Over the course of several weeks we learned three crucial principles: the biological basis of simple dominant/recessive Mendelian genetics; that handling living creatures is difficult, time-consuming, and easy to screw up; and that if you accidentally leave the stopper off a large demijohn of Ether, all the kids who have lessons in that classroom later in the day will get high.
At the end of the month, we tallied up the results:
Black Butts: 2,247 — Stripey Butts: 752
Ratio of Black Butts : Stripey Butts = 2.99 : 1
Mendelian genetic inheritance, as we had been taught, predicted that there should be a 3 : 1 ratio of black-bottomed flies (the “dominant trait”) in our population to flies with a striped pattern on their abdomen (the “recessive trait”). In our actual experiment, in which we looked at almost 3,000 flies, we found a ratio of 2.99 : 1 . Not a very big difference, but how do we know whather it matters or not? After all, very small differences can add up make a very large effect over long periods of time.
Now, barely a month after my faith in the predictive power of mathematics had been rocked by my teacher’s trickery, our biology class needed to turn to math in order to understand our experimental results. A chi-square test is a common statistical tool for determining if data fits into a predicted ratio — plugging our numbers into chi-sqaure gives a “two-tailed p value” of 0.92 . Simply put, this means that there is a 92% chance that our numbers do indeed fit into the predicted 3 : 1 ratio, after accounting for random variance. In high-school biology class, this was considered a significant value, and therefore our observed ratio was “correct” as predicted by the theory of genetic inheritance discovered by Gregor Mendel. That was good for an A.
Years later, and I am still working in biological science. Whereas my 17 year-old self was engaged with deciphering the colors and patterns on the back end of fruit flies, I now help to develop biotech ideas from research labs into new medicines and other products. Just like anyone else evaluating a new technology, my first job is to make a determination if it is feasible. After all, no one wants to invest precious resources to develop a technology that isn't likely to work in the first place. There are many reasons a new biotechnology product might fail — for instance, biological systems can develop resistance, or other molecular pathways might compensate for the activity of a drug, rendering it useless. The most critical issue I have to determine though, is if the scientists’ original research actually shows what they say (or what they think) it does.
Every day, I read scientific papers outlining the research that I am examining. In every case, their papers detail their hypothesis, the experiments they performed to test it, and statistical analysis of their results, confidently capped with a statement that p <0.05 . That is, there is a less than 5% chance that their results are merely coincidentally aligned with their predictions. A skeptic might point out that if that were literally the case, you just need to perform an experiment 20 times for it to show a positive result by chance alone. Then publish that one. In a world where only about 10% of “landmark” cancer research papers could be independently reproduced by the pharmaceutical companies they were licensed to, there is good reason to be skeptical of published reports, even (especially) when their data seems overwhelmingly positive.
In the end, it is always a judgment call. We make our decisions based on where we determine the consensus of scientific knowledge to be at the time. Over the long-term, science is ultimately self-correcting. Unlike every other philosophy or method that has been employed by man to describe the Universe around us, science uniquely allows itself to be rewritten when new evidence shows old models to be incorrect. The scientific literature is not, nor should be, viewed as an inerrant body of knowledge. After all, scientists are people too and make errors in their work, just like anyone else. Rather, it is a continually updated narrative of a massive voyage of discovery that is shared by every person on Earth who asks a scientific question. All journeys take wrong turns and reach dead-ends. Eventually, even the most entrenched assumptions can be shaken by the weight of new evidence against them, freeing us again to explore a new and uncharted domain.
It is easy to be distracted by a headline about a scientific breakthrough, or a tweet about a fantastic new discovery. After all, many of our reward systems in society encourage the idea that a single genius discovery will cure X or fix Y. No one single result, or paper, or model system, should be taken by itself as gospel. Scientific knowledge is the result of the amalgamation of many studies, from many different perspectives, to advance the body of its whole. Even then, it is subject to revision, nuance, and even complete rewriting.
Nullius in verba
The journey is never easy, but should we expect it to be for such an immense cause as the knowledge of the Universe?