How to think like a physicist

There is a tendency by physicists to present their work with jargon, assumption of considerable prior knowledge, and complex, domain-specific language. I think that's a shame, because it obscures the beauty that underlies the physicist's outlook on the world. So I want to try to explain how to really think like a physicist here.

What is doing physics?

Physicist are concerned with understanding things. By understanding, we don't just mean being able to describe the characteristics of something. No, by understanding, we want to know something well enough to predict properties of that thing. And we want to check our understanding by testing if those predictions turn out to be true.

So how do we do this? Well, first, we do inductive reasoning, and this just means reasoning from observation. The observation could be actual measurements, but just as often in the history of physics, it could be a realization. For Newton, that could be "I observe the Moon is somehow bound to the Earth in a regular fashion", for Einstein that could be "I observe that accelerating objects look very similar to gravitationally bound objects". When an observation appears to be general enough that it can be applied broadly, we consider it a principle. For instance, two such principles are Newton's principle of universal gravitation, and Einstein's equivalence principle. Other famous principles include the conservation of energy and the four laws of thermodynamics.

Afterwards, we formulate these principles using mathematics, which allows us to derive conclusions from them. These derivations, along with a certain amount of inspired guesswork, allows us propose a general form of a new theory. A theory should be as simple as possible, but nothing simpler. This step is difficult, but quite satisfying - it involves as much creativity and imagination as scientific and mathematical rigor.

Formalizing a theory through mathematics

Mathematical forms of theories of physics typically describe the relationships between different physical quantities. For instance, a theory of aerodynamic drag might recognize that drag is related to air resistance. When the quantities themselves are defined in terms of derivatives, then the relationships between them take the form of differential equations. Much of physics is formulated using differential equations.

It should be noted that new theories are generally not found from scratch; rather, a general form of the new theory is first found, and then by demanding that the new theory reduces to an existing theory in the appropriate limit, the precise parameters of the new theory can be found. For instance, this is how Einstein's theory of General Relativity was developed from the Newtonian theory of gravity.

But simply having a theory is not much use - the theory must be able to make predictions, and so, now comes deductive reasoning. In simplified terms, mathematical analysis is used to draw testable conclusions from the proposed theory. While tedious at first glance, math is just a way of formalizing logical statements - essentially, if we assume A, then B would also be true, or in other terms, $A \to B$. Physicists like to put a lot of faith in math, but correct math doesn't mean correct physics. A theory is only correct if its conclusions are proven through experiments. Otherwise, the theory has to be refined until it agrees with experimental results.

This whole process is understandably quite nebulous, which is why physicists typically try to use standard formalisms to simplify the process. One frequently used formalism is Newtonian mechanics, as well as its various derivatives. Formalisms used in more advanced physics include Lagrangian and Hamiltonian mechanics, and their extensions to field theories. All of these formalisms have the same end goal - to ease the process of analyzing a system.

Tricks in the physicist's toolbox

After the process of theorizing, we need to complete a theory and make sure it is logically-consistent. This step is very mathematical and frequently abstract in nature, because applying a theory can be just as complicated as creating it in the first place!

Luckily, physicists have some powerful mathematical tools at their disposal. The first is to use proportional relationships - while we might not know exactly how two variables might be related, we can usually guess how changing one would proportionally affect the other. Another frequently-used technique is dimensional analysis. Basically, we take a guess at an equation based on arranging variables in a way that makes the units work out - a surprisingly powerful technique that you can use to derive the Schwarzschild radius of a black hole without needing any General Relativity! In addition, a lot of physics uses a technique called the ansatz. The word comes from German, and it means an educated guess at an answer to a problem. That is, we first start with an inspired guess, and then follow through on what that guess might mean. If we get it wrong, don't worry - we can just guess again!

But perhaps the most powerful technique in physics is to use approximations. For instance, we approximate all manner of things in physics as particles - for example, a basketball, a rocket, or a grain of sand. We approximate continuum things - things composed of lots and lots of little things - as fields, such as fluids, waves, and dust clouds. Particles and fields are mathematical idealizations, yes, but they are close enough to reality that the approximation works! Indeed, being able to use approximations is fundamental to being able to do any physics at all.

The role of mathematics in physics

Physics is formulated in the language of mathematics, but it is important to remember that physics itself is not mathematics. Rather, theories in physics have mathematical models, but that does not mean they are mathematical in nature.

Mathematics is a language of logic, not of words. All mathematical statements are expressions of logic. It cannot express anything outside of logic, without other quantifying information. But it can express rigorous logical statements well. For instance, here is the definition of a limit:

$$ \begin{align*} &\lim_{x \to c} f(x) = L: \\ &\forall \epsilon > 0 ~\exists~ \delta > 0: |x - c| < \delta \rightarrow |f(x) - L| < \epsilon \quad \forall x \neq c \end{align*} $$

In normal English, it means that the limit of a function $L$ is such that we can get as close as we want to $x=c$, such that the value of the function will get as close as we want to $L$. It is a very abstract concept, yes, but precisely-stated using the language of mathematics. Physics takes the abstract concepts of mathematics and applies them to physical phenomena - to real things that we can observe and describe.

The most common use of mathematics in physics is in using equations, and in particular, differential equations that describe the dynamics of a given system and predict its future states. Differential equations are very difficult equations to solve (in fact sometimes impossible to solve), but physicists and mathematicians have figured out many ways to solve complicated differential equations. One approach is to use symmetries - when the system being analyzed looks identical from different directions. Another approach is to use approximations to simplify the differential equations until they can be solved.

But how did physicists do math without calculators?

This is a good question. Generally, physicists in the past use the same techniques we still do: keeping expressions in symbolic form until the very last step when it was time to numerically evaluate them, which reduced rounding errors. They also used numerical approximations (for instance, $\sqrt{2} \approx 1.414$ and $\pi = 3.141$), which were (and are) well-known, and scientific notation to be able to set numbers to those of similar magnitudes to simplify calculations. But afterwards, calculations were usually done by referencing mathematical tables, and often that task was done by hired workers called "computers" (yes, the original meaning of a computer was a job title, and it was a job that was predominantly done by low-paid women, who are the unsung heroes of science). A very famous book of mathematical tables is the famous work by Abramowitz and Stegun, which contains not only the numerical values of functions evaluated at regular points, but formulas for numerical interpolation, differentiation, integration, as well as special functions like the Legendre and Bessel functions.

Later on, the slide rule became a popular mechanical calculator for physicists and reduced (but did not remove!) the need for manual arithmetic; essentially, it provided a mechanical means of reading off precomputed values that allowed for very fast calculations by a skilled user. However, keeping track of orders of magnitude with scientific notation was still necessary, and addition and subtraction had to still be done manually. Today, computers are used extensively in physics and can solve problems that no physicist could do in any reasonable time by hand, but a strong grasp of mathematics is still essential to be a successful physicist.

The philosophy of physics

In antiquity, the idea of "physics" was rather vague: after all, our understanding of the world was fairly rudimentary, and without a rigorous approach it was impossible to ascertain which ideas were true as opposed to which were nonsense. But today, physics is the study of physical phenomena based on proven theoretical models that must satisfy two conditions:

The requirement that physics must be developed with experimental verification in mind is central to physics, and distinguishes physics from simple philosophical inquiry. Our models must correspond to something that we can observe (or more technically, measure) and test. Different ideas in physics are not tested based on which one is more fashionable or well-liked, but by which one fits the experimental data the best.

One might ponder and ask why physics exists at all, or more precisely, why the Universe can be modelled so well by precise mathematical models. I do not know the answer to this question. I suspect it isn't one that is answerable at all; maybe the most likely reason is that if the Universe was chaotic and could not be described by mathematical models, it wouldn't be able to support life, and thus us human beings wouldn't even be here to ask that question. Physics is so deeply entrenched in physical laws and mathematical models that a Universe without at least a reasonable amount of order would be one without physics. We're lucky to at least be in a Universe that does, and that might owe us our very existence.

Why do we do physics?

With an understanding of physics, we can solve problems, gain a greater insight into our world, and create new technologies. Some of us do physics to help solve pressing issues in the world. Others of us do it because we work in applied fields that are based on an understanding of physics. But many of us do physics simply for its own sake - out of the human curiosity deeply embedded in us. Physics is beautiful; its beauty is just waiting to be discovered - maybe, by you!

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