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Physics has something of a reputation for difficulty. It’s not the first university course you sign up for if you’re looking for an easy grade. But there is a paradox lurking behind this reputation. The reason that physics seems hard is because it actually is easy.
To unlock this seemingly self-refuting statement, it’s useful to consider what we might call the “spherical cow philosophy”. The name comes from a joke that physicists like to tell about themselves: A dairy farmer is struggling with milk output at the farm and decides to ask a scientist at the local university for help. For reasons that remain unexplained, they consult with a theoretical physicist. The physicist goes off to do some complicated calculations and returns with an impressive-looking stack of equations. “I’ve solved your problem, I think,” says the physicist. “What is it?” replies the farmer excitedly. “Well, first assume a spherical cow…”
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The humour, in case it’s not immediately obvious, lies in the fact that not only are cows not spherical but also that their non-spherical nature is crucial to what it means to be a cow. A spherical cow wouldn’t be a cow at all. Making that assumption might simplify the calculations a physicist might want to do, but, in doing so, it risks removing us entirely from the realm of insights that would actually be useful to the dairy farmer.
The joke is famous not because it’s side-splittingly funny – nobody ever claimed that – but because the equivalent of “assume a spherical cow” actually does work in physics, and it works incredibly well. It’s an example of a general principle – namely, idealise a difficult problem down to a simple one by ignoring as many complications as you can. Get an answer to the simple problem. Then put the complications back in and calculate how they affect the answer.
Think of the motion of something like a coffee cup, sitting peacefully on a table. Namely, there’s no motion at all: the cup will just sit there unless something pushes on it. And if you do give it a gentle push, it will move in response, but then quickly come back to rest when the push ceases. This led ancient thinkers like Aristotle to posit that “being at rest” was the natural state of material objects, and that they would only move if some kind of force was applied.
Later philosophers, however, fretted about how this picture extended to something like an arrow shot into the air, which could manifestly move through the air even after leaving the bow. They suggested that perhaps the natural state of motion was more like constant velocity in a straight line, not coming quickly to rest.
The problem with that is, almost nothing actually does move at a constant velocity in a straight line. Aha, came the reply, that’s because there are almost always forces acting on things. In the case of a cup on a table, it’s the force of friction that brings it quickly to rest. If you imagine an object on a completely frictionless surface, in an environment with no air resistance or other kinds of dissipation – the kinematic equivalent of a spherical cow – then things would move in straight lines at constant velocity. We can always put the messy real-world complications back in later.
The master of this kind of reasoning was Galileo. He had a genius for figuring out what things were essential and what could be ignored at a first pass. Aristotle had claimed that heavier objects fell faster than lighter ones. Again, not wrong; drop a book and a single piece of paper at the same time and see what your experimental results reveal. But Galileo argued that if there was no air resistance, they would fall at the same rate. That’s an experiment no person could actually perform until centuries later, but Galileo was able to construct ingenious experimental setups that were able to establish the underlying principle.
Galileo’s modern equivalent would be none other than Albert Einstein, who had an uncanny knack for clever thought experiments that revealed the essence of a problem. His theory of special relativity was inspired in part by his thoughts about what it would be like to travel on a beam of light. And general relativity, the follow-up theory that posited curved space-time as the origin of gravity, stemmed from thinking about what a physicist could discover if they were sealed inside an opaque box. They could not discover whether they were in a gravitational field, he reasoned – there would be no way of being sure they weren’t simply accelerating in a rocket ship. Einstein concluded that gravity must be a feature of space-time rather than a conventional force, and from there it was just a matter of mathematics.
The trick, of course, is to figure out which aspects of a problem are centrally important and which can be ignored. Nobody can write down an algorithm that is guaranteed to answer that question. The best physicists just seem to have a knack for distinguishing which aspects of a system can be neglected at first, their effects to be added in later.
As useful as the spherical-cow philosophy is, it’s not a universal truth-finding method. It doesn’t work that well in dairy farming, for example. In many complex systems, such as we often encounter in the macroscopic human-scale world, different aspects of the underlying situation interact in crucial ways with one another, such that you can’t just ignore them one at a time and correct for their effects later. In fields like biology or economics, everything often depends on everything else.
What’s surprising isn’t that the spherical-cow method fails to be useful for certain complex problems, but that it’s so incredibly useful in the systems that physicists like to study. In the real world, anything we look at – from a galaxy to a single atom – is embedded in a rich context of other systems, which are continually bumping together and affecting each other. Billiard balls actually make noise when they collide, and tabletops aren’t frictionless. But nature is set up so that we can temporarily pretend otherwise, and still make progress. I’m not sure if anyone truly understands why that is the case, but we should consider ourselves fortunate that it is.
The reason why physics seems so hard is because it actually is quite easy, compared to other sciences. There are always complications, but they can often be ignored at a first go. As a result, physicists are able to construct enormously simple theories and extend them to realms far beyond what they originally imagined. Isaac Newton wrote down laws of gravity to describe falling apples and planetary orbits; he never envisioned that they would help get rockets to the moon, but his equations were more than up to the task. Einstein himself never contemplated black holes or conditions mere minutes after the big bang, but modern cosmologists describe them with the same equations that he wrote down over a century ago. In physics, our best theories are always smarter than we are.
Which is not to say that physics can’t seem legitimately hard when we try to understand it. Its very simplicity allows us to discover amazing and counterintuitive features of the world, from quantum mechanics to relativity to the big bang. We would never have been able to just guess these things by being clever; we were forced to invent them by trying to fit the experimental data, and the task was made enormously easier by our ability to put irrelevant complications aside. But precisely because these ideas are so counterintuitive, they can be hard to understand, at least at first glance. If the vista before us seems alien, it is because we are looking so very far.
Ěý
is a physicist, philosopher and author. He is based at Johns Hopkins University in Maryland and spends his time trying to answer fundamental questions about quantum mechanics, space-time and cosmology. He also hosts a podcast called and his latest book is .