In my last post I raised a question about the pros and cons of common sense. I left it as a wide-open question, as I was curious to see how readers would react.
Many aspects of common sense affect how we relate to other people, and it’s clear they have considerable value. But the intuitions we have for nature, though sometimes useful, are mostly wrong. These conceptual errors pose obstacles for students who are learning science for the first time.
It’s also interesting that once these students learn first chemistry and then Newtonian-era physics, they gain new intuitions for the natural world, a sort of classical-physics common sense. Much of this newfound common sense also turns out to be wrong: it badly misrepresents how the cosmos really works. This is a difficulty not only for students but also for many adults. If you’ve read about or even taken a class in basic astronomy or physics, it can then be challenging to make sense of twentieth-century physics, where Newtonian intuition can fail badly.
Let’s take just one example. Any child who has tried to move a heavy box by sliding it along a floor knows that if you want to keep it moving at a constant speed, you have to keep pushing it; and the heavier the box, the harder you have to push it. It also seems harder to push it at a high speed than at a low speed. For this reason, the natural expectation of any reasonable person who hasn’t taken a physics course is that the amount of force required to push the box grows with the speed v at which you want to push it and with the weight W of the box.
Such a person might then imagine, incorrectly, that this is true for any object in any circumstance, at least on Earth. It’s common sense.
Misconceptions of this sort (which arise from not recognizing the crucial role of friction) were typical for centuries, even among highly intelligent, thoughtful scholars. It wasn’t until Isaac Newton that the veils were entirely lifted, with his laws of motion, of which the second reads F = m a. This equation implies
- that no force at all is required to make an object move at a fixed speed and direction,
- that the force F required to change the speed or direction of an object is proportional to its mass m (not its weight) and the object’s acceleration a (not its speed,) and
- the direction of the object’s acceleration is the same as the direction of the force applied to the object.
And yet no sooner has a student spent months learning and internalizing this law, developing a detailed intuition for it, than it is undermined by the 20th century, first by Einstein’s relativity and then by quantum physics. That’s unfortunate for people who take only one year of physics, because those advanced subjects are either not covered, or are covered in a cursory way that does not allow for a new intuition to take hold. Worse, incoherent statements about these topics are not uncommon in first-year textbooks.
In Einstein’s theory of motion, the analogue of F = m a is a much more complex equation, as you can find in the 2nd, 4th and 5th sections of https://en.wikipedia.org/wiki/Acceleration_(special_relativity) . Worse, the intuition for what is meant by force and acceleration in this context are not straightforward. Textbooks and courses rarely consider them with care.
Not realizing this, people often apply the following incorrect logic:
- Einstein claimed that no object can move faster than the cosmic speed limit c (also known as the speed of light);
- Therefore, the closer an object’s speed to c, the more difficult it must be to accelerate it (and therefore the more force must be applied to do so, for the same acceleration); otherwise its speed could easily be made to exceed c ;
- And so, an object’s mass must grow with its speed, so that a fixed force produces less and less acceleration.
- In fact, as its speed reaches c , the object’s mass must become infinite to assure that that the acceleration correspondingly becomes zero; otherwise it could accelerate past c.
This sounds perfectly reasonable, but it is somewhere between misleading and inconsistent. It is an effort to retain Newton’s law, and Newtonian common sense, in the context of Einstein’s relativity. But it’s simply not true that F = m a, where F and a point in the same direction in three-dimensional space; it is only approximately true at low speeds. In Einstein’s terms, you can write a generalization of F = m a, but then both force and acceleration have to be generalized to point in four dimensions (i.e. both in space and in time) and the mass m has to be the object’s rest mass, a constant which does not depend on the object’s speed. If you try to maintain your Newtonian intuition, defining F and a as before (in terms of three-dimensional quantities) and defining mass m to increase with speed, ambiguities arise which cannot be resolved. Among other things, the object cannot be assigned a definite mass; any definition you choose will have to depend on the direction in which the force is applied relative to the object’s direction of motion.
What happens to F = m a in quantum physics? Even the idea of such an equation assumes that objects have trajectories — paths across space on which objects travel as time goes by. But what we learned from quantum physics is that real objects in the real world do not, in fact, have trajectories. It’s a long story to see what happens to Newton’s law in that context. (For a glimpse, see https://www.physnet.org/modules/pdf_modules/m248.pdf, but that’s just the beginning.)
In short, not only does our common sense intuition from before first-year physics have to be discarded, so does the intuition built up during first-year physics! This is a serious challenge for students, teachers and writers. And it raises an interesting question: would it be better, both for those who will someday take a first-year physics class and for those who never will, to try to convey some preliminary, qualitative intuition for how the world really works? Only later would we then teach Newton’s physics for what it is: not as a set of ancestral truths, but merely as an approximation that served (and still serves) as a temporary bridge between ordinary common sense and the universe’s underlying reality.
