[Math] Examples of using physical intuition to solve math problems

big-listexamplesintuitionproblem solving

For the purposes of this question let a "physical intuition" be an intuition
that is derived from your everyday experience of physical reality. Your
intuitions about how the spin of a ball affects it's subsequent bounce
would be considered physical intuitions.

Using physical intuitions to solve a math problem means that you are able to
translate the math problem into a physical situation where you have physical
intuitions, and are able to use these intuitions to solve the problem. One
possible example of this is using your intuitions about fluid flow to solve
problems concerning what happens in certain types of vector fields.

Besides being interesting in its own right, I hope that this list will give
people an idea of how and when people can solve math problems in this way.

(In its essence, the question is about leveraging personal experience for
solving math problems. Using physical intuitions to solve math problems is a
special case.)

These two MO questions are relevant. The first is aimed at identifying when using physical intuitions goes wrong, while the second seems to be an epistemological question about how using physical intuition is unsatisfactory.

Best Answer

The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$. The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

$$ Cn \, log(T/G) $$

where $G$ is the geometric mean. It follows that $T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).

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Practical Applications of Set Theory

The purpose of set theory is not practical application in the same way that, for example, Fourier analysis has practical applications. To most mathematicians (i.e. those who are not themselves set theorists), the value of set theory is not in any particular theorem but in the language it gives us. Nowadays even computer scientists describe their basic concept - Turing machines - in the language of set theory. This is useful because when you specify an object set-theoretically there is no question what you are talking about and you can unambiguously answer any questions you might have about it. Without precise definitions it is very difficult to do any serious mathematics.

I guess another important point here is that it is hard to appreciate the role of set theory in mathematics without knowing some of the history behind the crisis of foundations in mathematics, but I don't know any particularly good references.

Your second question is more specific, so I'll give a more specific answer: to thoroughly understand the mathematics behind, say, modern physics does in fact require (among many other things) that you understand the properties of infinite sets because topology has become an important part of this mathematics, and understanding general topology depends heavily on understanding properties of infinite sets. Whether this means that set theory has any bearing on "reality" depends on how much faith you have in topological spaces as a good model for the real world.

As a specific example, the mathematics behind general relativity is called differential geometry. I think it's fair to say the development of general relativity would have been impossible without the mathematical language to express it. Differential geometry takes place on special kinds of manifolds, which are special kinds of topological spaces. So to understand differential geometry you need to understand at least some topology. And I don't think I need to justify the usefulness of general relativity!

[Math] How To Present Mathematics To Non-Mathematicians

I have given talks about mathematics to non-mathematicians, for example to a bunch of marketing people. Supplemental: to see an example of a talk of mine that was given to a general audience, see my TEDx talk "Zeroes" (with supplemental material). The talk lasted 15 minutes and it took me about two weeks to prepare.

In my experience the following points are worth noting:

If the audience does not understand you it is all in vain.
You should interact with your audience. Ask them questions, talk to them. A lecture is a boring thing.
Pick one thing and explain it well. The audience will understand that in 10 minutes you cannot explain all of math. The audience will not like you if you rush through a number of things and you don't explain any one of them well. So an introductory sentence of the form "Math is a vast area with many uses, but in these 10 minutes let me show you just one cool idea that mathematicians have come up." is perfectly ok.
A proof of something that seems obvious does not appeal to people. For example, the proof of Kepler's conjecture about sphere packing is a bad example because most people won't see what the fuss is all about. So Kepler's conjecture would be a bad example.
You are not talking to mathematicians. You are not allowed to have definitions, theorems or proofs. You are not allowed to compute anything.
Pictures are your friend. Use lots of pictures whenever possible.
You need not talk about your own work, but pick something you know well.
Do not pick examples that always appear in popular science (Fermat's Last Theorem, the Kepler conjecture, the bridges of Koenigsberg, any of the 1 million dollar problems). Pick something interesting but not widely known.

Here are some ideas I used in the past. I started with a story or an intriguing idea, and ended by explaining which branch of mathematics deals with such ideas. Do not start by saying things like "an important branch of mathematics is geometry, let me show you why". Geometry is obviously not important since all of mathematics has zero importance for your audience. But they like cool ideas. So let them know that math is about cool ideas.

To explain what topology and modern geometry are about, you can talk about the Lebesgue covering dimension. Our universe is three-dimensional. But how can we find this out? Suppose you wake up in the morning and say "what's the dimension of the universe today?" You walk into your bathroom and look at the tiles. There is a point where three of them meet and you say to yourself "yup, the universe is still three-dimensional". Find some tiles in the classroom and show people how always at least three of them meet. Talk about how four of them could also meet, but at least three of them will always meet in a point. In a different universe, say in a plane, the tiles would really be segments and so only two of them would meet. Draw this on a board. Show slides of honeycombs in which three honeycomb cells meet. Show roof tilings in which thee tiles meet, etc. Ask the audience to imagine what happens in four dimensions: what do floor tiles in a bathroom look like there? They must be like our bricks. What is a chunk of space for us is just a wall for them. So if we have a big pile of bricks stacked together, how many will meet at a point? At least four (this will require some help from you)!
To explain knot theory, start by stating that we live in a three-dimensional space because otherwise we could not tie the shoelaces. It is a theorem of topology that knots only exist in three dimensions. You proceed as follows. First you explain that in one or two dimensions you can't make a knot because the shoelace can't cross itself. It can only be a circle. In three dimensions you can have a knot, obviously. In four dimensions every knot can be untied as follows. Imagine the that the fourth dimension is the color of the rope. If two points of the rope are of different color they can cross each other. That is not cheating because in the fourth dimension (color) they're different. So take a knot and color it with the colors of the rainbow so that each point is a different color. Now you can untie the knot simply by pulling it apart in any which way. Crossing points will always be of different colors. Show pictures of knots. Show pictures of knots colored in the color of the rainbow.
Explain infinity in terms of ordinal numbers (cardinals are no good for explaining infinity because people can't imagine $\aleph_1$ and $2^{\aleph_0}$). An ordinal number is like a queue of people who are waiting at a counter (pick an example that everyone hates, in Slovenia this might be a long queue at the local state office). A really, really long queue contains infinitely many people. We can imagine that an infinite queue 1, 2, 3, 4, ... is processed only after the world ends. Discuss the following question: suppose there are already infinitely many people waiting and one more person arrives. Is the queue longer? Some will say yes, some will say no. Then say that an infinite row of the form 1, 2, 3, 4, ... with one extra person at the end is like waiting until the end of the world, and then one more day after that. Now more people will agree that the extra person really does make the queue longer. At this point you can introduce $\omega$ as an ordinal and say that $\omega + 1$ is larger than $\omega$. Invite the audience to invent longer queues. As they do, write down the corresponding ordinals. They will invent $\omega + n$, possibly $\omega + \omega$. Someone will invent $\omega + \omega + \omega + \ldots$, you say this is a bit imprecise and suggest that we write $\omega \cdot \omega$ instead. You are at $\omega^2$. Go on as far as your audience can take it (usually somewhere below $\epsilon_0$). Pictures: embed countable ordinals on the real line to show infinite queues of infinite queues of infinite queues...

Best Answer

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Practical Applications of Set Theory

[Math] How To Present Mathematics To Non-Mathematicians

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