# Physical interpretation of mathematical operations

I have difficulty interpreting mathematical operations in a physical way. For example, I see many people reading Newton's second law as "Resultant force equals mass times acceleration", however I now see a theoretical physics professor reading as "Resultant force is the same as giving mass an acceleration". The difference may seem silly, but in my view it is extremely different, in the second sentence you can physically imagine the equation. My question is, how do I extrapolate this "reading" to the other operations? Like division, exponential, logarithm, tensioners, etc.

• Are you a physics student? May 14 at 0:42
• Yeah! First year of bachelor's degree in theoretical physics. May 14 at 1:21
• We don't know why it seems that math is able to describe and predict physical observations. It just seems that it does so we use it. May 14 at 4:37
• In my opinion one of the lovely things about the subject is how one equation can be understood in so many ways. May 14 at 5:43
• Not sure, but you might find Wigner's "Unreasonable Effectiveness of Mathematics" (google that phrase) relevant to your thinking, e.g., en.wikipedia.org/wiki/… May 14 at 11:05

There are two separate things here.

There is the universe. It is a collections of stuff like electrons, light, and rocks, together with repeatable patterns of behavior. For example, if you put electrons near each other, they will repel each other.

Then there is physics. Physics is a mathematical description of the behavior of the universe. It is a set of laws like $$f = ma$$. Each law precisely (or sometimes approximately) describes some piece of behavior.

Physics and the universe are connected through measurement, or more generally experiments. You use a ruler to find the distance between two events, and a clock to find the time interval between to events. From this you can divide to find velocity. And so on.

So you can approach mathematical operations in two ways. First you have numbers taken from measurements that fit a mathematical relationship. Given m and a, you can calculate F.

And second, you can look at how the math describes the behavior. m represents a measurement of the amount of stuff in an object. a represents how quickly the object changes velocity. F represents a force, which might mean how hard two electrons repel each other. The math makes the description of the behavior precise in a way that words cannot.

Or is it this? You can approach $$F = ma$$ different ways.

Given $$m$$ and $$a$$, you can figure out how big is the force that caused that mass to accelerate that hard.

Or given $$F$$ and $$m$$, you can figure out how much acceleration you get from that force acting on an object of that mass.

Or given $$F$$ and $$a$$, you can figure out how much mass is in the object that that force caused to accelerate that hard.

The link in the comment below is to the 3blue1brown video Divergence and curl: The language of Maxwell's equations, fluid flow, and more. This is a good video that gives an intuitive feel for the divergence and curl operators.

Physics is often abstract and counter intuitive. Developing an intuitive feel can make it more understandable. However, don't carry it too far. As Feynman said, you can get farther using the abstract equations than intuition.

Physicists often take it the other direction. They think in terms of math to the point that they confuse the mathematical objects with reality. For example, they say two electrons repel each other because they generate electric fields, which are vector fields. They forget that vectors are mathematical tools used to describe reality.

This is a useful thing to do. Mathematical tools describe reality well. If you know how the math behaves, you know the universe behaves the same way.

So I recommend that you watch the video to get an understanding of what divergence and curl mean and why they are useful tools to describe the universe. But then get familiar with the math. The math will begin to feel natural as you get used to it.

Don't learn to read all equations by intuitive analogies. The analogies are almost always a less detailed description of reality than the math. They are generally approximately right. The incorrectness will bite you when you take it too far. The behavior of the universe is complex, and so the math describing it must get complex. Likewise an analogy would have to get just as complex to be a faithful description. You might as well use the math. It is a better tool.

To answer your question about how he went from the divergence operator equation to the intuitive picture: He skipped a few steps. Divergence is a kind of derivative. He has other videos that describing derivatives and vectors. The missing steps are in them. They are worth watching for both the intuition they provide, and the math in them.

• I think my question is for the second approach you commented on more. My question is more about how I ´´convert´´ the mathematical operations into something physical. I want to look at math equations and ´´read´´ them in a physical way. The best example I can give is from this link: youtube.com/watch?v=rB83DpBJQsE May 14 at 1:29
• Basically, he treats the divergent operator as a "drain", the question is, how did he go from the equation that describes the divergent operator to this way of interpreting it? (it is something more physical to imagine a drain) May 14 at 1:32
• Based on this video I went on to read Maxwell's first and second laws in completely different ways.At minute 6:15 he explains the physical reading of the first law. I want to learn how to do this for all equations. May 14 at 1:35
• Mathematical tools describe reality well. – More poignantly, one could say that (well-known) mathematical structures were conceived that way because reality behaves similarly. For example, mathematical addition is commutative because many real operations like combining weights or the duration of time periods is commutative. May 14 at 9:49
• I was thinking about your conclusion, it makes sense and I agree! R. Feynman said when summarizing the Copenhagen interpretation as "Shut up and calculate!". Basically it's like you said, but it's sad to know that up front even the brightest minds only use mathematics (there is no physical interpretation) May 14 at 18:32

You are asking how to extrapolate the physical reading of some equations to other equations and math operations.

There is no way but to learn more math and more physics. It is like the job of a good translator from one language to another. Only by mastering and practicing both languages, the translator can grasp the nuances of a text. The example of Maxwell's equations and their reading in terms of the description of fluids should make evident this point: it is possible to become proficient in this translation only by knowing the mathematical meaning of div and curl and the way they appear in the equations of hydrodynamics, understanding their meaning in that context.

• Your conclusion makes a lot of sense, but as the other colleagues stated above, there will come a time when I should only base myself on math, right? May 14 at 18:28
• @SrAAlb Maybe I see things in a different way. I would say that there will come a time when your mastering of Physics and Math will be such that you will have a hard time separating them. Your thoughts won't be based on math, but you'll be able to express physical ideas directly and fluently with the language of math. May 14 at 22:24

In my teaching, I distinguish physics-equal-signs, math-equal-signs, special-case-signs, and definition-equal-signs. Essentially, it tries to remind you of WHY something is true.

$$\vec F_{net}\stackrel{PHY}{=}\frac{d\vec p}{dt}$$

$$(a+b)^2\stackrel{MAT}{=}a^2+2ab+b^2$$

$$a_x \stackrel{HERE}{=} 0{\ \rm m/s^2}$$

$$\vec a \stackrel{DEFN}{=} \frac{d \vec v}{dt}$$

So, I would say "Resultant force equals mass times acceleration (because of Newton's Second Law)."

In looser talk, I might say "(In solving a particular problem,) Resultant force is the same as giving mass and acceleration"... but I would use the previous sentence as more fundamental.

In trying to interpret mathematical operations "physically", I often try to use the phrase "as if..." in order to suggest that a particular physical interpretation is merely one (possibly very useful, but not the only one) such interpretation.

• Very cool, I've never seen anything like it. Did you learn this from a book or was it based on lessons? May 14 at 18:34
• @SrAAlb I came up with it by myself. I was my way of seeing the logic of a derivation. May 14 at 18:51