When I was an engineering student, I would routinely hear complaints from my classmates about how much mathematics we had to study for chemical engineering. I was genuinely puzzled by this attitude, since perhaps I had a better idea of what engineering school was going to entail. Most people know the practice of science requires a tremendous amount of mathematics, but don’t really know why scientists are so obsessed with mathematics and quantitative measurements.

# A Reasoned Thought Experiment: Projectile Motion

The main reason science is so quantitative in practice, is because mathematics is the only language for which there is exactitude in meaning. It is impossible to formulate physical laws and make predictions on the basis of vague, subjective statements written in an ordinary language. For example, suppose I was back in the 1600s, analyzing the motion of bodies subjected to the earth’s gravity. Which statement is more precise, and more amenable to making predictions?

- A body heaved into the air at an angle will take a curved path down to the ground.
- A body heaved into the air at an angle will take a parabolic trajectory.

“Curved path” could mean anything; heck, a question mark qualifies as a “curved path”. But a parabola has definite geometric properties. Furthermore, we run into the problem of *falsifiability* with statement number 1. It encompasses so many different possible results, that it could always fit the scientific data. However, statement number #2 is directly vulnerable to a counterexample: if someone can produce a trajectory that is not parabolic in nature, statement #2 is refuted.

Let’s go a little further, and compare statements 1 and 2 with statement 3:

3a. .

3b. .

3c. .

3d. .

Is not statement 3 even more exact than statement 2? Both say the path of motion is a parabola, but statement 3 describes *the exact parabola the trajectory will be*. Statement 3 is more exact – **and easier to disprove** – than statement 2. If someone can produce a trajectory that differs from the statement 3 prediction outside of experimental uncertainty, then statement 3 is refuted by counterexample.

Hopefully the line of thinking has come full circle for the reader now as to why scientists so heavily prefer mathematical equations to written languages when attempting to write laws that describe the physical world. With mathematics, there is maximum precision with no ambiguity in meaning – either the equation predicts the experimental data, or it doesn’t.

# Broad Statements Are Easy to Disprove

“*Easier to disprove”* is an important thing to understand. You see, in science, things are never *proven* – they are only *disproven*. A theory that continually resists attempts at disproof is regarded as useful. Good theories are ones that allow you to make quantitative predictions about the behavior of some system.

The best theories are simple, yet extremely broad statements that described the behavior of a system under a large variety of experimental conditions. Sometimes someone comes up with a simpler, broader statement than the current state-of-the-art. A famous example is Kepler’s Laws of Planetary Motion, versus Newton’s Law of Universal Gravitation. Both predict the motion of the heavens – but Kepler’s Laws are mathematically proven to be a direct consequence of the Law of Universal Gravitation. Kepler’s Laws only describe the motion of the planets, and requires three different sub-laws to make a prediction with. Newton’s Law describes also the motion of the moons (and all other heavenly bodies), with a single equation. Kepler’s Laws describe less, and so are harder to disprove. Newton’s Law describes vastly more phenomena, making it easier to disprove.

# The Generality of Scientific Laws

The “broadness” of a statement depends on how much it describes. In rough order, in increasing order of generality, goes like this:

- Algebraic equations
- Ordinary differential equations
- Partial differential equations

I should also note, that these are in increasing level of ease of disproof. Partial differential equations are probably the easiest statements to disprove by counterexample with quantitative data, because the statement is so broad as to be vulnerable to attack from all directions. By contrast, an algebraic equation describes comparatively less about a system, and as such, it is more difficult to disprove due to a much narrower scope.

Good examples of partial differential equations are found in the study of fluids, electromagnetism, and heat transfer applications. Often for practical problems, the only available method of solution is a numerical solution obtained via a powerful computer.

# Quantitative > Qualitative

The partiality toward quantitative data is rather simple to explain: it is much easier for two researchers to compare quantitative results than to compare two different sets of qualitative results. Suppose two researchers were observing the color of light emitted from the flame of a certain chemical reaction. Let us say researcher A is not a very good researcher, and prefers qualitative observations, and his results are presented like this:

“A small quantity of reagents A and B were mixed, and then reacted to produce a yellowish-white flame. Varying the quantities of the reagents caused the flame to shift from bright yellow to pure white.”

Compare with how researcher B conducted his work and has reported his results:

“5 0.1 grams of reagent A were mixed with 1 0.1 grams of reagent B, producing a flame with the spectrum of wavelengths shown in Figure 1. Varying the quantity of reagents as described in Table 1 caused the spectrum to vary from light of wavelength 580 nanometers to a broad spectrum of white light, as shown in Figure 2.”

Researcher B has put considerably more detail into reporting his methodology and presenting his results. Exact wavelengths and spectral data are provided, and the exact amounts of reagents were presented. Should another research group try to reproduce researcher B’s results, they will have a much easier time than trying to reproduce researcher A’s. Furthermore, improved methodology make it easier to contradict researcher B’s results, whereas researcher A’s presentation is so broad there is little hope of anyone contradicting it.

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