Laws and Definitions: Newton’s Laws of Motion

I find that in a lot of textbooks take definitions for granted and they do not differentiate laws and definitions. Let me be clear by what I mean by definitions. Most things cannot even be defined in the dictionary sense. For example one cannot really define what time is. But we can measure it. Quantitative relationships, which are the core of physics, come from quantitative measurements. We can take a periodic event as our standard measure of time and quantify time. We can also pick a unit of length and define a frame of reference in order to measure distance and position. We can then relate distance and time and find relationships such as centripetal acceleration to tangential velocity.

In order to demonstrate how most source do not properly define entities, I am going to discuss Newton's laws of motion and explore mass and force. Let's take Newton's second law: F = ma. Acceleration is a well-defined quantity which can be observed and measured. But what about mass and force? We might have some conceptual ideas about them but how do we measure them? Most resources say force is the thing that causes an object to accelerate. And mass or inertia is the resistance of the object to acceleration. Those are a good conceptual definitions, but this is mathematical truism. The measurement of force depends on mass and the measurement of mass depends on force. One quickly realizes that this is a useless relationship on its own because it does not tell us anything. When two quantities are undefined in a mathematical statement, its neither a law nor a definition - it's just useless. This is where most textbooks fall short because they do not define these quantities.

So how can we define and measure mass and force and give meaning to Newton's second law? It is clear that we need to define these values based on some measurable entity. Realizing this problem, the Austrian physicist Ernst Mach reformulated Newton's laws of motion based on measurable entities. He summarized Newton's laws into a single law:

"When two compact objects [point masses] act on each other, they accelerate in opposite directions, and the ratio of their accelerations is always the same. "

There is no mention of mass or force in this law; only acceleration. One can then quantitatively define mass and force based on this law. First we define an inertial frame of reference. Take the set of isolated point masses A and R, where R is takes as a reference point mass. At any time, if A is accelerating, R is accelerating in the opposite direction and the ratio of the accelerations is a constant. We define the mass of R as our reference mass with a unit of 1. Therefore, the mass of A is the ratio of the acceleration of A to R. Now mass is a measurable entity. Then we define force as mass times acceleration so that a unit of force (N) accelerates one unit of mass (kg) at 1 unit of of acceleration (m/s^2).

You have probably noticed that this definition is based on Newton's third law, which does not hold in relativity and thus in forces under Lorentz Transformation. Therefore, we need to modify and generalize our definition of mass. This definition is given in the paper "A Rigorous Definition of Mass in Special Relativity" written by E. Zanchini and A. Barletta. There is also a more formal version of the definition of mass in Newtonian mechanics. I highly recommend reading the paper:

http://arxiv.org/PS_cache/physics/pdf/0606/0606167v2.pdf

The Essence of Intuition and Its Role in Physical/Mathematical Discoveries

I have been thinking about the role of intuition in physical discoveries and mathematical reasoning. Intuition is a great tool that we have developed in order to navigate through the world around us. If one stops to think, intuition or common-sense is a crude and simple form of mathematical logic and a set of definitions that are based on what is around us. Our intuition can only comprehend finite and discrete things, as these are the things that we deal with in our lives. Furthermore, we build a conceptual framework by categorizing and naming how the things around us seem to us. Thus, say things like “this is a particle, that’s a wave.” And we use analogies to “understand” things by saying like “The particle acts like a bullet here, etc.” However, the whole universe is a much much more complicated mathematical structure which goes beyond the simple logic that evolution and experience have equipped us with. The remarkable fact, however, is that we have come to learn mathematical logic and discover nature through its most basic logic. Well maybe it is not that surprising after all since we are ourselves one of the products of this mathematical structure – the universe looking back at itself.

As we enter new realms of reality through experimentation or mathematical reasoning, we often discover “counter-intuitive” realities. If one does not have the right approach, the struggle to “understand” becomes really frustrating. One has to realize that our intuitive understanding is only designed for quick navigation for only a small portion of the vast mathematical reality. Therefore, none of our discoveries are counter-intuitive because we simply do not have any intuition for those contexts. For example, some people might find the fact that 1 = 0.99999… very counter-intuitive. But the point is that we do not have any intuition about something that is infinitely continued. I would only call it counter-intuitive if one shows that 0.9 = 1 or that a large sphere can fit in a smaller one, because those are the things that I have developed intuition for. But these things are mathematically false and so I don’t have to worry about them. The point is that intuition is a tool that has been developed out of mathematical logic and our intuitive understanding is limited to our context. Another example of the misuse of intuition is quantum mechanics. One cannot say the behaviour of things (I say “things” for the lack of a better word) is counter-intuitive at the quantum level, because we have not developed any intuition for that scale; its neither intuitive nor counter-intuitive.

Nevertheless, intuition can still be really helpful at times. After all, its our brains’ natural mode of thinking and reasoning. One very simple example of how intuition might be helpful could be analyzing the motion of a ball rolling down an incline. Generally-speaking, circular motion is not the most intuitive concept, but instead of taking the integral of the forces on all the points in and on the ball in its initial state in order to find the torque, we can quickly say that there is no torque due to gravity because of the symmetry of the ball. Note that this is still mathematical reasoning, but it’s the kind of reasoning that we are intuitively good at and it can save us a lot of time and effort.

All in all, it is important to realize the capabilities and limitations of our intuition. When used properly, it can boost our problem-solving and give us directions, and when used improperly it can prevent us from developing our understanding of nature.

The Strange Effectiveness of Mathematics

Lately I have been preoccupied with the rather philosophical question of why we are able to discover physical laws through mathematics; or put another way, why does the universe follow mathematical laws and a consistent line of reasoning. One could also go on to ask if mathematics is a human invention or an independent entity that is being discovered bit by bit by humans

Some say mathematics is a form of language invented for the purpose of formulating and describing the world around us and thus there is no sense in asking why it is so effective. I do not agree with this. Firstly, mathematics is not a language, it is a logic system. Furthermore, what is so interesting and astonishing about the power of math is not that it can describe the world around us, but that it can make predictions. Unlike the other sciences, in physics one can deduce new ideas and principles mathematically. I recommend reading “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” by Eugene Wigner (http://hiperc2.buffalostate.edu/~carbonjo/documents/Wigner.pdf) and its follow-up, “The Unreasonable Effectiveness of Mathematics” by R.W. Hamming (http://nedwww.ipac.caltech.edu/level5/March02/Hamming/Hamming.html) to understand the power of mathematics.

As I was reading different ideas about and nature and relation of mathematics to science, I came across the Mathematical Universe Hypothesis proposed by Max Tegmark. Tegmark’s hypothesis states that our external physical reality is a mathematical structure. Tegmark extends his idea to say every mathematical structure has physical reality and our universe is just one mathematical structure, out of many, that has been complex enough to allow us to evolve in it (anthropic principle). The purpose of this post is not to discuss the details of Tegmark’s ideas (which can be found here:http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0646v2.pdf) but to discuss my own thoughts in regard to the questions proposed in the beginning of this post. The source and effectiveness of mathematics are no longer puzzling if, as Tegmark says, “our universe is mathematics in a well-defined sense.” After all, the universe follows mathematical logic and physicists follow this line of logic to discover how nature works. Maybe developing mathematics is ultimately a journey of discovery – a unique form of discovery which progresses through creativity in the use of the logic that comes to us from nature.

One of the challenging ideas about this school of thought is the relationship of mathematical equations and their interpretations. I find one good example of this to be the equations of quantum mechanics. Our mathematical model of quantum mechanics seems to accurately describe what happens at the quantum level, and their implications have been consistently proven. Yet, there are many interpretations of quantum mechanics that attempt to give some physical meaning to the mathematics of the theory. If the universe is a purely mathematical construct, then there should not be different candidates of reality arising from a single mathematical theory. If this is true, then surely there is more than mathematics to the reality and truth that physicists attempt to understand. However, I do not believe this to be the case. The important point important point is that each scientific interpretation assumes a different and distinguishable mathematical structure. For example, in quantum mechanics the Many Worlds interpretation introduces a unique mathematical system in which the collapse of the wave function is associated with splitting the universe into mutually unobservable, alternate histories—distinct universes within a greater multiverse.Obviously this is a completely distinct mathematical landscape than the Copenhagen interpretation for instance. Therefore, the existence of many interpretations for the same set of mathematical laws is not because mathematics does not completely define reality, but the fact that we do not have the complete mathematical landscape of reality. In fact, conceptual and non-mathematical ideas are just shortcuts for us to examine possible mathematical structures, which will be proven right or wrong through experimentation or further mathematical reasoning as other laws get discovered. All in all, conceptual interpretations are a starting point to get to the ultimate mathematical reality, not the other way around.