I, Scientist | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Afterword
After two years in neuroscience, a new and wider picture about how a theory could be done started to emerge in my mind: “A valuable theory logically connects two interpretable but seemingly unrelated things through a series of interpretable things.” This working definition constrains fairly loosely regarding what can be described as a theory. Though loosely constrained, I think it encourages trying out many different ways to connect whatever pieces of knowledge—no matter how concrete or vague—deeply and extensively.
But this working definition of mine used to be much narrower and stricter before I learnt neuroscience. Below, I summarized some of the things I learnt during these two years about what formulating a theory means in math and physics, and how it might be done in biology. And I hope these realizations could be of use for someone who might be puzzled why a theory in physics and a theory in biology look so different.
A standard physics curriculum starts from teaching a few base quantities: length (m), time (s), mass (kg), electric charge (C), and temperature (K) with which all physical quantities can be constructed:
| physical quantity | name of unit | in terms of base units |
|---|---|---|
| velocity | n/a | m / s |
| acceleration | n/a | m / s^2 |
| angular momentum | n/a | kg*m^2 / s |
| energy | Joule (J) | kg*m^2 / s^2 |
| entropy | n/a | kgm^2 / s^2K (J/K) |
| heat capacity | Calorie | kgm^2 / s^2K (J/K) |
| electric current | Ampere | C / s |
| magnetic field | Tesla | kg / C*s |
| magnetic vector potential | n/a | kgm / Cs |
From the above examples, one might note that some of the quantities have a name, like Joule for energy or Tesla for magnetic field; these quantities usually can be measured in relatively straightforward ways. For those who don’t have a name, either they are too simple to bother—e.g., mile per hour for velocity, or they are not a directly measurable quantity but only serve as an intermediate concept within its native theory. For example, magnetic vector potential $A$—that is related to magnetic field $B$ through a special kind of spatial derivative called *curl—*doesn’t have a dedicated name for its unit:
$$ B\equiv \nabla \times A $$
Nearly all theories in physics are formulated in form of mathematical equations consisting of physical quantities. Take a planet $m$ orbiting a star $M$ as an example:
$$ \frac{1}{2} mv^{2} -\frac{GMm}{r} =E_{planet} $$
This equation claims that the total energy $E_{planet}$ is the sum of the first term: kinetic energy and the second term: gravitational potential energy. From which one can workout the trajectories $r(t)$ of a planet being an ellipse if the total energy is conserved throughout.
When a trained physicist encounters such a theory for the first time, they would perform a diagnostics to check if the equation is legit. Such a diagnostics is called dimensional analysis which, simply stated, is a way to check if individual terms on both side of the equality have consistent unit:
| term in above equation | dimensional analysis | legit |
|---|---|---|
| kinetic energy | kg*(m/s)^2 = kg*m^2 / s^2 = Joule | ✔︎ |
| potential energy | from force: GMm/r^2 (Newton), | |
| one has: GMm/r (Newton*meter = Joule) | ✔︎ | |
| total energy | Joule | ✔︎ |
In my over ten years of physics, a theory was always formulated in such a manner—i.e., 1) being mathematical and 2) built from physical quantities. So much so, I implicitly adapt the idea that, in the process of building a theory, one should first identify relevant physical quantities (or something equivalent to that) as building blocks before putting them together. In a way, the existence of physical quantities is almost synonymous with the existence of reality itself. It is something you simply don’t question, and something above any theories.
I remembered vividly that this idea of “physical quantities above all” was challenged for the first time when I was in a class of Advanced Solid State Physics at UT-Austin. I was particularly excited about this class for I still owed myself a satisfying answer about how superconductivity really works—a mystery that had been with me for more than 7 years ever since a college course of electromagnetism. We had a final presentation on any topics we prefer, and superconductivity was the topic **I chose. In the final presentation of that course, I explained: