[LUCID LIVING #2]  The Mass-Force Circle: Mysteries vs. Puzzles

By Piet Hut

In my previous post I wrote that the relationship between matter and information is one of the great mysteries in all of science.  But what counts as a mystery?

When we see something happening, and we have no clue as to why it happened, we call it a mystery.  We like to have clear explanations.  In the past, we explained the way plants grow and animals move by invoking a kind of life force.  Now we have much more specific and precise explanations in terms of underlying physical and chemical reactions.  Growth and movement can now be traced to processes on sub-microscopic scales within the living cells that make up macroscopic organisms.

Yet mysteries remain.  For example, as yet, we still have very little insight in the mechanisms of the origins of life, even though significant progress has been made in the last few decades.  Also, even if we could trace the behavior of each molecule in a living cell, or even in the formation of the first life form on Earth, an enormous database of motions of molecules would still not be much of an "explanation" for what happened, and how and why it happened on macroscopic scales. In other words, to understand emergent properties of a system, even if given complete knowledge of its microscopic behavior, is still a daunting, and fascinating challenge.

The deepest questions in science appear as mysteries, not as puzzles

All these mysteries are by no means expected to be unsolvable.  In that sense, they are perhaps more like puzzles, but with one big difference.  While a puzzle is difficult, it typically comes with a well described challenge: given such and such rules, how to solve a, b, c.  Given these pieces of a jigsaw puzzle, how to put them together.  Given these descriptions of words in a specific crossword puzzle, how to find a letter for each open square.

The deepest questions in science never have the character of puzzles; they always appear as mysteries.  And their solutions always stem from changing the rules in order to transform the old mysteries into new puzzles.  Here is one mystery, that lies at the very base of mathematical physics, Newton's classical mechanics.  It is the mystery of how to operationally define the meaning of force, perhaps the most basic concept that got physics started.

Newton's second law of motion describes the amount of force needed to accelerate an object.  In order to produce an amount of acceleration a, for a body with mass m, the force F is equal to the product of m and a.  In short, F = ma.   To obtain a given acceleration, if we double the mass of an object, we have to apply twice as much force.  And to provide twice as much acceleration to a given mass, we also need to double the force we use.

F = ma

All of that may seem straightforward.  It is one of the first equations we encounter in physics class in high school. But when we reflect on the way in which these three variables are actually determined, in practice, suddenly things are no longer that simple.

In order to test the relation F = ma, we have to measure the values of each of the three variables, under a number of different experimental conditions, to check whether Newton's second law holds for each case.  Only then can we conclude that the law has universal applicability.

Measuring the acceleration a is straightforward.  Once we have a measuring rod to determine distances traversed, and a watch to keep track of time elapsed, we can at any time measure the velocity v of an object, and after that we can compute a as the rate of change of v.

Measuring m is not so simple.  How do we determine the mass of an object?  The easiest way is to put it on a scale, to measure its weight.  The way this works is that we measure the force that the gravitational field of the Earth applies to the object.  And to calibrate a scale, we convert that force into a determination of mass, using F = ma, in the form m = F/a, with a here the acceleration of a free-falling object at the surface of the Earth.  But this already presumes the relationship we were about to test!


With a few other such thought experiments, we quickly come to the conclusion that mass is defined in terms of force, and that force is defined in terms of mass.

Another way of seeing this is to imagine that everything around us would be massless. In that case, we could never develop the concept of force.  No matter how small a force we would apply, we could immediately accelerate any object arbitrarily strongly.

On the other hand, if we imagine that we could not exert any force whatsoever, then we could not push or pull any object.  Because we could not affect the motion of any object, we could never develop the concept of mass as something that makes it harder to change the state of motion of a thing.

The conclusion is that mass and force are defined in a circular way.  They have to be introduced together, in order to make sense.  And there is nothing wrong with that.  What is wrong is that our educational system tends to emphasize an abstract and incorrect picture of how science is built up, step by step, from the ground up.  A typical text book, and a typical teacher, will tell us that F = ma, for any given F and m and a, without asking how each of those three quantities are measured in the first place.  The implicit understanding that is conveyed is that all three values can be measured independently, which is impossible.

A much better way of teaching would be to emphasize that circular reasoning can be a good thing.  And often, in order to get started in solving a mystery, we have to start with circles.  The idea that science gets off the ground, so to speak, by building things up, independently, step by step, is a myth.  The concepts of mass and force are co-arising; they are introduced together, and experimentally verified together, in a very precise and useful way.  There is neither a need, nor a possibility, to introduce them separately.

Piet Hut is President of YHouse (where this blog is hosted), Professor of Astrophysics and Head of the Program in Interdisciplinary Studies at the Institute for Advanced Study in Princeton, and a Principal Investigator and Councilor of the Earth-Life Science Institute in the Tokyo Institute of Technology.

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