July 1, 2014 Leave a comment
I recently talked to a person who was convinced that scientific theories, mathematical theories, mathematical theorems, knowledge, truth, and scientific laws were all basically synonymous. He said that physics could not exist without math, because math defined physics. He also was convinced that believing and agreeing were the same thing. I attempted to remedy these misconceptions using some basic arguments, but I was finally written off as “not understanding anything” and “unwilling to do the math”. When I asked him to define the word truth, he merely kept repeating, “I don’t know what you mean. Truth is just that which is.” When I attempted to explain that the word “truth” was a symbol referring to a concept, and that we couldn’t have a discussion if we were referring to different concepts with the same word, he said “you don’t need to define truth, it just is. It’s very simple.” He couldn’t understand why I kept “bringing up philosophy when we’re talking about simple truths here.”
Sigh. If I can’t break through that kind of rhetoric, I might as well just explain my thoughts here.
Why is it important to know about the philosophy behind knowledge, truth, and science when talking about it? Isn’t it possible to rely on a the natural human consensus of truth? Besides, while it is so hard to explain using language, people intuitively grasp the concept. Right?
Well, let’s give some examples. It’s true that if you drop an object, it falls, right? Well, yeah, that statement is true if you are on the surface of a planet, and not orbiting it. Or if you are underwater and you drop a buoyant object — it goes up! But wait, can you drop something underwater if it doesn’t go down? No, that wouldn’t be dropping it would just be… releasing? Hold on, when an object is in orbit, isn’t it actually just falling in a special way? It’s moving sideways fast enough that it misses the ground by the time it’s fallen far enough. But if an astronaut releases a wrench, and it float right in front of him, you wouldn’t call that “dropping”.
What we see is that the word “drop” has a definition, and we need to know what the definition of “drop” is before we can begin to assess the truth of the statement “if you drop an object, it falls”. As it turns out, “dropping” an object consists of releasing it such that it falls away from you. Uh oh. So yeah, “if you drop an object, it falls” is true, but it doesn’t actually convey any physical knowledge; it just defines a property of the word “drop” in terms of another word, “fall”.
So lets look at some more meaningful examples. Most people would say it’s true that planets orbit the sun in an elliptical manner. Except it isn’t true. It’s true that the movement of the planets can be approximated into ellipses, but in fact there are measurable deviations. “Okay, sure. The movement is actually described by Newton’s laws of motion, and the law of gravitation.” Okay, yes, an N-body approximation gets much, much closer to describing reality. In fact, it perfectly matched the observations Newton was working from. However, it’s still not true the Newton’s laws describe the motion of the planets.
We can look to general relativity to describe the motion of the planets even better. We have launched satellites to observe very minor fluctuations in the path of the Earth that would confirm the prediction made by general relativity. As it turns out, general relativity makes predictions that perfectly match our observations. Woof. Finally, we’ve found some truth. The path of the planets around the sun is described by general relativity.
But wait, can we say this in good conscience? No! Just like Newton, we’ve found a set of laws which create predictions that match our observations. But just like Newton, we cannot measure the motion perfectly. All we can say is that general relativity describes the motion of the planets as far as we can observe. We don’t know if there is some unknown mechanic that affects the motion of planets in a way we can’t measure right now. We can’t say that general relativity is “true”, we can only say that it is confirmed by all of our observations to date, much in the same way that Newton could not say that his laws of motion were true; they merely described the all physical data he was capable of obtaining.
This gets to the root of the problem. While mathematical notions can be “true” because they exist within an entirely constructed framework defined through logic, theories in science can never be “true”. The point of science is not to find things that are true, but to find the best explanation for why the world works the way it does. And just to get one thing clear, theories are explanation of “why”, and laws are explicit definitions of how physical quantities relate. So no, we don’t use “math to define physics”, physics uses math to explain the physical universe. But even without math, we can perform a sort of qualitative physics.
For instance, “things stay still until you push them, and things keep going straight unless you push them.” This phrasing of Newton’s first law of motion is simplistic and uses words like “thing” and “push” without really defining them, but it gets the point across. Similarly, “big things move less when you push them, and small things move more.” This is very simplistic, and doesn’t even mention the fact that acceleration changes linearly with force, but it communicates the basic idea of Newton’s second law of motion, without even getting into what “big”, “small”, and “move” really mean.
The point is that the traditional phrasing of Newton’s second law, F=ma (which, by the way, is more accurately ), merely uses mathematical symbols rather than English symbols, which allows us to manipulate it using the rules of mathematics. But just because we are manipulating arbitrary quantities with math doesn’t mean anything physically. Just because I calculate that an object which masses 1 kg should accelerate at 1 m/s^2 when I apply 1 N of force doesn’t mean the thing is actually going to act that way if I perform the experiment. This is because “mass” is really a simplification of a whole range of things, as is “acceleration”. It doesn’t even account for internal forces, and only describes the movement of the center of mass.
Math may be true, but only within the realm of math. When we translate physical quantities into the mathematical universe, they lose they physical meaning. We may translate them back, but the results we get can only be an approximation, not a truth, not a reality. These approximations can be very useful, but we have to remember the limitations of our theories, and our instruments.