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Is every mathematical concept, even the complex ones, explainable?

As someone who will be needing to explain my line of work for a position to a committee who is very, very, educated, just not in math (I would say over a decade or so years removed from taking a calculus I course, I'm having a hard time optimizing efficiency of my words and impact of my statement. I have a lot of applications knowledge that is relevant to them and might get them interested, but I also have to lay the foundation for what a PDE is.

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    $\begingroup$ Let's face it. You cannot even explain "derivative" to a very educated layman with no math background. So do not describe what a PDE is. Instead describe the types of problems you have solved/could solve. $\endgroup$ – Gerald Edgar Aug 8 '16 at 13:36
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    $\begingroup$ I would suggest being absolutely minimal with technical details. Use your applications to show the relevant issues. $\endgroup$ – Adam Aug 8 '16 at 14:08
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    $\begingroup$ @GeraldEdgar: Why not? I see how it's hard to get someone who does not understand the subtleties/difficulties of defining an instantaneous rate of change to appreciate the concept of a derivative, but getting them to understand the seemingly-glorified-triviality of a concept. $\endgroup$ – R.. Aug 8 '16 at 19:14
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    $\begingroup$ It depends on what you mean by "explainable." $\endgroup$ – John Coleman Aug 9 '16 at 15:17
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    $\begingroup$ I note that the question in the title specifically refers to PDEs, but the body of the question also asks whether every mathematical concept is explainable. $\endgroup$ – J W Aug 9 '16 at 19:45
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I would say something like this: "Often in complicated systems one needs to study multiple quantities, each of which varies at rates that depend on the other quantities and on how fast they are varying. For example the rate at which a drug metabolizes may depend not only on how much of the drug is in the body but also on how much blood sugar is in the body, how much of certain other hormones are present, and so forth, and each of those quantities is also changing at a rate that depends on all of the others. Because everything is interdependent, to understand how one quantity changes over time you need to study the interactions of the whole system. The study of partial differential equations provides a set of tools for studying such systems."

If the example above is for whatever reason inappropriate for the audience, replace it with another one involving multiple interconnected systems: for example, complicated financial markets, or weather patterns, etc.

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    $\begingroup$ +1 As I was reading the question, I began thinking that I might try taking a stab at this, then read your answer . . . $\endgroup$ – Dave L Renfro Aug 8 '16 at 16:20
  • $\begingroup$ I think this is the way to go, the only thing I would possibly change is the example. I like historical examples, especially here where the prominent historical examples are the heat equation and Navier-Stokes equation and given the role of PDE involving laplacian still hold. I would probably go with the heat equation, as temperature is a single quantity that is well understood, and the basic equation is pretty intuitive (the temperature at a point changes to approach the average surrounding temperature). $\endgroup$ – Benoît Kloeckner Jul 19 '17 at 8:16
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Imagine a surface, for example, a horse saddle. Note that the "curvature" of the surface differs not only when you measure it at different points, but also when you measure it along different directions. For example, at the "center" of the saddle, the surface is curving upwards if measured along the "horse's spine" and the surface is curving downwards if measured perpendicular to that.

Tell the very educated layman with no math background that many systems can be modeled as surfaces and that partial differential equations answer questions about the "curvature" of such surfaces.

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Here's my attempt.

It's about building a big picture of a situation when all you know is small-scale relationships. For a simple example, imagine a Mars rover has crashed, so most of it is broken. All it can do is move about, and it can tell how steeply it's going up or down. NASA want to make the best use of the equipment they can, so they send it up and down, back and forth, taking lots of transects recording the up/down information as it goes. Although the rover only knows what's going on immediately around it, NASA can build up a relief map of the whole area.

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    $\begingroup$ On the wider question of whether everything is explainable, I don't think PDEs rank as very complex. At least they relate to things people know about. Some pure topics would be much harder, as there are several layers of definitions and abstraction. $\endgroup$ – Jessica B Aug 10 '16 at 5:37
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Like Mwiess, I would give an example from social sciences or sports or the like of a function with several inputs. I think this is the critical first idea to get across. After this and only after this, you can get to discussing things like interactions of the factors.

I would NOT start with a saddle point as in one answer. This requires thinking in 3D Cartesian space and puts people's back up who are less mathy. Not saying it is not useful example and would use it in a class (it has value, which is why texts use it all the time). But realize than that even with weak students, you are dealing with people who are in a process of learning and used to these structures. With a layman, would not refer to such a structure at first. If you absolutely have to use a visual, just try something like a topo map or talk about mountains. (But even here, I worry that the reference to a Cartesian system (directions and altitude) will bother some people. But worth a shot. Definitely do that before the saddle point though!

Once, you have gotten into useful discussion of the concept of multivariable functions, you have give the person the most useful insights for daily life. Even though you haven't gotten into the diffyq part of it at all. If the person responds well and you want to carry on further, you could mention that PDEs are a method of going from evidence about how the factors change to try to unravel the equation. If the person knows integral calculus, refer to that. Solving an ODE or PDE is kissing cousins to "going backwards" in integration. Asuming they don't, don't refer to it.

The other thing that is extremely useful for a layman is to give them some context of where PDEs occur in a curriculum and who needs to learn them. (Something that I find lacking at times on Stack.) Mention how it is really part of the "calculus track". That it is like "harder calculus". and comes after 3 semesters of calculus and one of ODEs (just say simple DEs if you want). That it is used by engineers and physicists (and of course mathematicians). But I would emphasize the place in the technical curriculum over the math track...after all look at the numbers of students.

P.s. I will even admit to not knowing PDEs well. Have encountered them at times but never really mastered them. But knowing where they "fit in the universe" is still useful. So I have some sympathy for the layman and some feel for what is useful for him to understand.

This is why I struggle with Wikipedia articles about math concepts or math subjects which take a definitional approach (with lots of rigor) as the very first step to understanding. Rather than saying "a lion is sort of like a cat but much bigger. It is yellow and has big teeth. The males have a bunch of hair on their head called a mane. They live in Africa and can kill big animals like deer or zebra and can kill a man no problem." This helps me out way more than some formal taxonomy (with massive amounts of recursions or lookups...in math with an amount of lookups that basically requires learning courses to understand even an overview encyclopedia article!)

An easy example would be Lie groups. I have no idea what they are but would like to know. I don't have a deep math background but way more than the average cat (1580 on old style SATs, math through ODE and some engine math, lots of engineering and science and even some baby group theory in the context of crystallography and IR spectroscopy.) So while perhaps not ready to learn Lie algebra, I am WAY more knowledgeable than a complete layman (say a smart NYT reader who "hated math" and was an English major in college). Yet I can't even just get a basic orientation. So in your discussion with the layperson on PDEs, I would try to give him some orientational description o the field. NOT the first lecture in a course and NOT an attempt to teach most of it an hour (to a person who even lacks prereqs). He just wants to know what kind of animal the PDE is!

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