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(Added an epilogue)

I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching).

In the last session the two people in charge of the workshop said that we should be able to "explain our research field to most people, or at least those with some academic background, in about three minutes". I argued that it might be possible to give a general idea of a specific field in psychology, history, maybe some engineering, and other fields that deal with concepts most people hear on a daily basis. However, I continued, in mathematics I have to explain to another mathematician a good 30 minutes explanation what is a large cardinal.

I don't see how I can just tell someone "Dealing with very big sizes of infinity that you can't prove their existence within the usual system". Most people are only familiar with one notion of infinity, and the very few (usually physicists and electrical engineering students) that might know there are more than one - will start wondering why it's even interesting. One of the three people who gave a presentation that session, and came from the field of education, asked me what do I study in math. I answered the above, and he said "Okay, so you're trying to describe some absolute sense of reality." to which I simply said "No".

Anyway, after this long and heartbreaking story comes the actual question. I was asked to give my presentation next week. I said I will talk about "What is mathematics" because most people think it's just solving huge and complicated equations all day. I want to give a different (and correct) look on the field in 10 minutes (including some open discussion with the class), and the crowd is beginner grad students from all over the academy (physics, engineering of all kinds, biology, education, et cetera...)

I have absolutely no idea how to proceed from asking them what is math in their opinion, and then telling them that it's [probably] not that. Any suggestions or references?

Addendum: The due date was this morning, after reading carefully the answers given here, discussing the topic with my office-mates and other colleagues, my advisor and several other mathematicians in my department I have decided to go with the Hilbert's Hotel example after giving a quick opening about the bad PR mathematicians get as people who solve complicated equations filled with integrals and whatnot. I had a class of about 30 people staring at me vacantly most of the 10 minutes, as much as I tried to get them to follow closely. The feedback (after the micro-teaching session the class and the instructors give feedback) was very positive and it seemed that I managed to get the idea through - that our "regular" (read: pre-math education) intuition doesn't apply very well when dealing with infinite things.

I'd like to thank everyone that wrote an answer, a comment or a comment to an answer. I read them all and considered every bit of information that was provided to me, in hope that this question will serve others in the future and that I will be able to take from it more the next time I am asked to explain something non-trivial to the layman.

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When people ask my specialty, I say "combinatorics." When they ask what combinatorics is, I say "it's a broad subfield, but for example one important area is graph theory" and go on to say a bit about that (despite the fact that I'm not a graph theorist) because I technically am telling the truth, I think it conveys the correct spirit, and I haven't had any luck coming up with a one-sentence definition of enumerative combinatorics. [cot'd] –  JBL Nov 24 '10 at 13:36
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In your case, I would probably say something about how there are many orders of infinity and, while our intuitions about finite sets guarantee certain things about infinite sets, they aren't enough to settle all reasonable questions about infinite sets; in particular, it turns out that infinite sets come in different sizes, but we can choose (for some very large sizes of infinity) whether to include them, and study what the implications of such large sizes are. I would probably expect to not get past the fact that there are infinite sets of different sizes most of the time. –  JBL Nov 24 '10 at 13:39
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Sadly,I think trying to explain real mathematics to nonmathematical people is a little like trying to explain German grammar rules to people that don't speak a word of German. I think the best you can do in either case is motivating what you do,not really explaining it. –  Andrew L Nov 24 '10 at 22:00
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I speak no German at all, but still find tidbits about unusual features of German grammar to be interesting. And it's my understanding that linguists typically don't speak the languages that they study. –  JBL Nov 25 '10 at 1:53
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Mark Twain did a fairly decent job of explaining at least some important rules of German grammar to English-speaking people without using any German words. From a newspaper article about a tavern that burned down: "When the flames the onthedownburninghouseresting stork's nest reached, flew the parent birds away." Running that phrase together into a single word is exaggerated, but otherwise it's how German syntax actually works. (Some other parts of his account were humorous exaggerations and distortions, but certainly based on actual phenomena.) –  Michael Hardy Feb 1 '11 at 0:26

26 Answers 26

I have given talks about mathematics to non-mathematicians, for example to a bunch of marketing people. In my experience the following points are worth noting:

  1. If the audience does not understand you it is all in vain.
  2. You should interact with your audience. Ask them questions, talk to them. A lecture is a boring thing.
  3. 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.
  4. 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.
  5. You are not talking to mathematicians. You are not allowed to have definitions, theorems or proofs. You are not allowed to compute anything.
  6. Pictures are your friend. Use lots of pictures whenever possible.
  7. You need not talk about your own work, but pick something you know well.
  8. 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.

  1. 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)!

  2. 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.

  3. 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...

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+1, but I'm afraid "we can choose color to be the 4th dimension, so if 2 points are different color they can pass through each other" would lose every nonmathematician I know. –  JBL Nov 24 '10 at 13:45
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Well, actually, I asked "what is the fourth dimension?" and someone said "time", and then I explained it need not be. I showed pictures of colored knots and I told people to imagine that the color is "another dimension" just like when they look at a geographic maps and color represents altitude above sea, which is another dimension. I think it went over well. –  Andrej Bauer Nov 24 '10 at 14:38
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Actually, I thought the idea of using color was very neat; the two points never actually touch if they have different color coordinates. –  Todd Trimble Nov 24 '10 at 15:40
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I once made a ceramic model of a Klein Bottle which was colored so that it did not have any self intersections when you considered color as the 4th dimension. –  Steven Gubkin Nov 24 '10 at 16:29
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Ordinals are a great example, I think. I am sure I'm not the only one who's played the "I dare you times infinity!" "I dare you times infinity plus one!" game as a child, and it might be fun for people to learn that this can be made precise. –  Qiaochu Yuan Nov 24 '10 at 20:01

Here is one example of what is mathematics as opposed to "layman reasoning" that I was told about by Yuval Peres. During WWII, the British airplanes were often shot down by Germans because the armor was too weak. The decision was made to put some extra armor on the planes but, since you can add only that much weight to a plane without affecting its performance, only a few selected areas could be enforced. Upon looking at the damaged planes that returned to the base, the engineering committee recommended to put armor to the most frequently damaged areas, as seen on the planes available for investigation. It took a statistician to explain why you should do exactly the opposite.

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I don't think this is what distinguishes mathematics from layman reasoning. It is rather what distinguishes a good model from a bad model. Mathematics takes models as an input. Of course, many mathematicians have some above-average talent at telling good models apart from bad ones, but this is not what mathematics is about. –  darij grinberg Nov 24 '10 at 14:11
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Why should you do exactly the opposite? –  Qiaochu Yuan Nov 24 '10 at 14:20
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@Qiaochu: I am guessing that the reason has to do with sampling bias. The reason that these planes managed to make it back to base is probably because their undamaged regions are critical for their functioning. So, their undamaged regions should be reinforced. –  Tony Huynh Nov 24 '10 at 14:25
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Or, in other words: the most-damaged places on planes that survive are places that are inessential to flight. –  JBL Nov 24 '10 at 14:38
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One can do even better if one has a good estimate on the probability that every square inch of the surface of the plane gets hit with a bullet. The "exact opposite" decision above means they assumed uniform distribution. –  timur Nov 27 '10 at 17:41

My own "go-to" introduction is Euler characteristic. What is nice about it is that you can have tons of audience participation.

First draw a bunch of polyhedra on the board (or have models that you can distribute to the audience). Ask people to count the number of faces, edges and vertices. Write a few of these down on the board. Ask if anyone sees any patterns. Usually at least one person will notice that F+V-E=2.

A combinatorial proof of this, by first reducing to the planar case, triangulating, and then removing triangles from the outside in, showing that at each stage you are leaving F+V-E invariant, is something I have had success with even with highschool students (at least one on one). At each stage you can have someone in the audience confirm the invariance ("What happens to F+V-E when I take a triangle like this away?") Have a triangulated torus already prepared. Observe that F+V-E=0. Tell them that in general an "n-holed" donut has F+V-E=2-2g where g is the number of holes. So somehow this number F+V-E depends on the whether you could stretch one of these shapes into the other, but not on the rigid geometry. Explain that a similar combinatorial proof would be difficult for the n-holed donut, but their is an entire subject called "algebraic topology" which has developed machinery which makes this kind of result easy to see.

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You can also use a deflated soccer ball for this demonstration. This way you can show that the number is measuring something about the ball, but not a rigid thing. It is not exactly its "roundness", but something else. Then whip out the torus. –  Chris Schommer-Pries Nov 24 '10 at 19:50
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You can also talk about how people living in a 2-D universe could tell what `shape' it is - how do we generalise this to the 3-D case. By far the best tool I've ever found for this is the Asteroids game play.vg/games/4-Asteroids.html Anyone who's ever played it will love the fact it's a universe with genus 1. –  Ollie Margetts Nov 25 '10 at 4:26
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@Ollie: then you'll love geometrygames.org/HyperbolicGames . –  Qiaochu Yuan Nov 25 '10 at 19:13

There is this nice quote whose wording I can't quite recall. It is something like "physics is the study of the laws of God. Mathematics is the study of the laws even God must follow."

I think there are some nice elementary examples of this in certain areas of combinatorics. Consider, for example, Ramsey numbers: if $6$ people are at a party, either $3$ of them all know each other or $3$ of them all don't know each other (but this is not true for $5$ or less people). That's something most people don't know, it's really easy to demonstrate by picking six people from the audience, and it is of a totally different flavor from the "equational" mathematics most people are familiar with. (Caveat: I have never actually tried this demonstration.) You can then continue: if $18$ people are at a party, then either $4$ of them all know each other or $4$ of them all don't know each other (but this is not true for $17$ or less people).

Then you continue: the corresponding best number for $5$ people is not known. This is just about the most easily stated open problem I know, and it is a good way to show people that mathematics is not "finished" in any meaningful sense. If you were sufficiently handwavy and included lots of pictures, it might even be possible for you to sketch the proof that all the Ramsey numbers exist.

Another potentially good example is Hall's marriage theorem, especially if you use the marriage-theoretic terminology the entire time. I saw a lecturer do this recently and it was quite funny. Again, if you use enough pictures, this might be manageable to sketch.

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Both the links are missing apostrophes. I can't fix it because (it seems) I don't have enough reputation to put links in answers. –  Max Nov 24 '10 at 15:57
    
The links don't appear properly if I add in the apostrophes. I'm not sure how to escape them. –  Qiaochu Yuan Nov 24 '10 at 17:40
    
Ok, fixed: you can use %27 to escape the apostrophes. –  Max Nov 24 '10 at 18:54
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+1 for "marriage-theoretic terminology" :) –  Nick Salter Nov 24 '10 at 23:35
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I found the source of the quote again! It's from Mumford's foreword to Parikh's The unreal life of Oscar Zariski: "Everyone knows that physicists are concerned with the laws of the universe and have the audacity sometimes to think they have discovered the choices God made when He created the universe in thus and such a pattern. Mathematicians are even more audacious. What they feel they discover are the laws that God Himself could not avoid having to follow." –  Qiaochu Yuan Dec 12 '10 at 2:06

For some reason, many mathematicians have trouble with the idea that when some layman asks them about their work, the appropriate response is not to try to figure out how to describe the latest theorem you've proved. We seem to feel like we're "selling out" unless we try to describe all the technical subtleties of our most recent work. Some other professions seem much better at this: when a psychologist is asked about what they do, they don't reply with their intricate struggles to minimize bias in their latest experiment, even if that's what's occupying most of their attention that week. (On the other hand psychologists can say "I devise experiments to study the long-term effects of alcohol abuse on cognition" and be reasonably confident that this will make at least some superficial amount of sense to a generic college-educated person. If I say "I study period-index problems in the Galois cohomology of abelian vareties" then notwithstanding the considerable syntactic similarities between these sentences, the social effect could hardly be more different: I might as well say "Please go away".)

I have to say that I feel privileged never to have had to prepare a ten minute (or less!) précis of my work to a general audience in any kind of formalized setting. I agree that that does not sound like much fun -- for me or the audience -- and if asked to do so at the point in my life I would raise my eyebrow and begin to question (inwardly at least) the assumptions and goals of the person who wanted me to do so.

However, one of the necessary evils of socializing with people outside of the mathematical sciences is that you are inevitably asked "What do you do?" in very informal settings. Usually my first answer is that I'm a mathematician, and my second answer is either (depending upon my mood?) that I'm a number theorist or that I'm an arithmetic geometer. The second answer is more ambitious, because after the expected "What's that?" I have to explain that I work in sort of a hybrid of two fields, number theory -- the study of properties of the whole numbers like primes and divisibility -- and algebraic geometry -- the study of the curves, surfaces and higher dimensional objects that arise as solution sets to polynomial equations. When I'm on and the other person cares I can get all this out in a couple of minutes without causing any obvious trauma.

If they want to hear more than this I often state Fermat's Two Squares Theorem. I think this is nice because it's specific and it's relatively simple but certainly not obvious: indeed it's a little window into how pleasantly surprising mathematics can be: why should there be such a nice, clean pattern like this? Of course this is number theory of 350 years ago not of today, but this was the theorem that attracted me to number theory in the first place, when I first learned about it at the age of 16. (Actually, in my more recent career I have spent time thinking about different proofs and generalizations of exactly this result. But while speaking to a layman I probably wouldn't even remember that.) I should say that sometimes I get completely cut off in my statement of the two squares theorem -- I mean cut off in the middle of a sentence. And then I have often gone on to have quite a pleasant conversation on something else entirely.

In fact, my most negative experiences in the "What do you do?" game have come from non-math people who have insisted on hearing about exactly what I've been working on, with all the technical terminology. For instance, shortly before I received my PhD I went to a bar with my cousin and somehow found myself at a table full of medical students and residents. The above gambits were not sufficient for them. At one point one of them demanded to know the title of my thesis. "All right: it's `Rational points on Atkin-Lehner quotients of Shimura curves'." His reponse? "Okay. So basically you study points on curves." He said this with the smug pleasure of someone who had demonstrated that once all the big words had been stripped away, the Harvard PhD student was actually studying something very simple and childish. Of course having omitted all the big words, even an expert wouldn't have the slightest clue as to what the title meant. What a jerk.

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Pete, I somehow got to read this again. I can say that my reply to the medical student would have been to diminish him to a butcher or something similar. Being a jerk can be fun, especially when alcohol is involved! :-) –  Asaf Karagila Aug 15 '11 at 12:45
    
If you start with the phrase "Rational Points on Atkin-Lehner Quotients of Shimura Curves" and remove all the words the medical student did not know, you are left with "Points on of Curves" - almost precisely what he echoed back to you. :) –  Austin Mohr Nov 22 '12 at 20:37

A big part of math is about transformations and structure. For a wide audience in a short time, you can give an inkling about this in a short time along the following lines:

  • "Numbers'' are really abstract things, not just "quantities''. They can correspond to transformations --- for example, dilations and translations of the line. Show how this corresponds to multiplication and addition. Negative numbers are flips. This explains "negative times negative is positive," and shows $x^2=1$ has two solutions.

  • Solving $x^2=-1$ corresponds to answering the question "What can you do twice to get a flip?" Likely as not someone will think of rotating 90 degrees. Dilations, translations and rotations of the plane are complex numbers.

  • "What about rotations in 3D?" Demonstrate they don't always commute. A lot of math is about understanding the rules and concepts that govern much more general transformations of complicated kinds of data.

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If you haven't heard of it already, Timothy Gowers has written a lovely little (150 pages) book called Mathematics - A Very Short Introduction http://books.google.ie/books?id=DBxSM7TIq48C

It's a great read and I'm sure you will get some nice ideas from it.

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That said, I know exactly how you feel. I took a "generic skills" course called "Information, Communication, and Literacy". It was at around the same point (when we were asked to convert our research questions into soundbytes) that I finally quit. I'm going to suppress the rant and just suggest that you should stand up to this kind of twitterification. A cool thing to do would be to say "Go read a book" and suggest a reading list. –  Sonia Balagopalan Nov 24 '10 at 10:07
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I think we need to distinguish between popularization of math and real math. Popularization of math is important there should be "soundbytes" that try to explain to ordinary people what math is. If you tell ordinary people "go read a book" they will just think you are an arrogant jerk (which you are if you say that). Of course, when it comes to real math the story is different. Math presented in papers, funding proposals and conferences should be done properly. –  Andrej Bauer Nov 24 '10 at 12:02
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@Andrej: I'm a big fan of math popularization, but it is notoriously difficult to do well; one might even argue that it's enlightened non-mathematicians who do it best, and the books in my library seem to agree. I think that asking a graduate student to summarize their research topic in a soundbite is quite unfair, and has nothing to do with popularizing math (though students should be warned that such requests tends to happen quite a bit during hiring season). –  Thierry Zell Nov 25 '10 at 2:05

I realized recently that one problem with these types of talks is that (in the US at least) the audience often has no idea where mathematics comes from, even in a naive sense. In psychology, physics, engineering, etc, most people have a vague sense of what the roots of the discipline are. But in mathematics, particularly pure mathematics, the sense of purpose of mathematics is what is usually missing from a lay audience member.

I have found the work of Saunders Mac Lane in Mathematics: Form and Function to be very helpful when discussing mathematics with non-mathematicians. Once these ideas are "in the air," then the specific problems like Euler characteristic, graph coloring, etc, have a context in which to be appreciated (I agree that trying to discuss specific research problems at the lay level is typically impossible in pure mathematics). Mac Lane's argument is very roughly the following:

  1. Human cultural activities lead to
  2. Recognition of mathematical ideas, which lead to
  3. Mathematical formalism.

There is a nice table on the wikipedia page about Mac Lane's book with lots of examples of this. If I am giving a talk about what mathematicians do, or what mathematics is about, I discuss Mac Lane's ideas first to set the stage for what is to come. It is remarkably easy to discuss this in just a few minutes, and might be helpful for your situation as well.

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As has been said, the main point is to give up the idea of communicating your actual research topic. Even job seekers giving colloquium talks should usually not attempt this. The best such talks instead teach the audience, even mathematicians, the simplest underlying ideas of their subject, and only mention the direction of their own work in the last few minutes or so.

When discussing infinity with laypersons, I have often used "Hilbert's hotel", in which the infinitely many rooms are all full when another guest arrives. Everyone moves up one room and the new guest goes in room 1, thus showing that infinity plus 1 equals infinity, (as a cardinal). The audience easily figures out how to add 2 new guests or a thousand. Next ask them how to deal with an infinite sequence of new guests, which if course may be all placed in the odd numbered rooms, as the current guests each move from room n to room 2n.

When teaching topology I asked how to tell if an invisible butterfly net actually enclosed the butterfly, by looking at the winding behavior of the visible border of the net. when discussing higher dimensions, it is easy to get people to eventually visualize a 4 dimensional sphere as a family of three dimensional spherical slices, by starting with lower dimensional cases. Noting that one can escape a circle in the plane by jumping over it in three dimensions, point out that if one goes back in time, before the building one is in was built, one can escape a three dimensional room without breaking down the walls or opening the doors.

You will enjoy thinking about your favorite fundamental idea underlying your area.

edit: To convey the idea that knowing "how many" differs from knowing when two sets are "equipotent", I used the example of the cyclops in Ulysses, who knew when all his sheep were back in the cave, by matching them up one to one with a pile of rocks. Nonetheless he did not know "how many" sheep he had. I.e. "what number?" differs from "same number".

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Besides what Sonia suggested, also take a look at the classic What is Mathematics by Courant and Robbins. It is a bit long, so I am not sure how much you can pick out of it in one week, but good luck. For preparing your talk, you may want to consult some of the advice given by VI Arnold.

A good way to start any general discussion about mathematics, I think, is to remind people of the definition of the word mathematics. The original Greek term, μάθημα, means study/learning/science, which I think says a lot about the typical, somewhat idealized, mathematical worldview. That said, your work is significantly harder than that would be mine were I in your position. Large cardinals, set theory, and logic are much harder to explain to the average audience than something pedestrian like PDEs or sexy like mathematical physics.

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The (implicit) message, of course is that the level of abstraction varies in inverse proportion to the ease of explaining it satisfactorily to non-specialists... (not to knock down anything; I've been in that uncomfortable position myself.) –  J. M. Nov 24 '10 at 11:21

Mathematics is about the reasoning that can be made precise. The different branches of mathematics reason about different objects: numbers, shapes (rigid or stretchy), games, arrangements and relations, and other things for which the words do not exist in the everyday language. There are some branches of mathematics exploring the reasoning itself. Pretty much any set of rules of reasoning one can normally think of is equally powerful (we say equiconsistent). The large cardinals is a name of for various rules that are stronger.

To explore reasoning about different things, give a puzzle to the class. Many people like them, and do not think of these as maths. I would consider "hats" puzzles, or some topological puzzle (e.g. is this picture an unknot? are there two antipodal points on the surface of the Earth with the same temperature? etc). There are good puzzles books to get ideas from, such as those by Martin Gardner and Raymond Smullyan.

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You absolutely cannot explain the details of a research question in ten minutes, but you should definitely be able to explain the philosophy -- in the literal sense of "love of wisdom" -- of your problem in that much time. Mathematics lets you take really primal ideas (like continuity, symmetry, smoothness, shape, proof, truth, size, chance and information) and make them precise. This is really wonderful, and it's very much to your advantage to be able to communicate this wonder to others -- this is really what distinguishes mathematics from accounting or chess.

Anyway, in ten minutes, you can't communicate all of this, but you should be able to show off a gem or two. The go-to subject for this kind of demonstration is group theory, but if you want to focus on your own area, you have a big advantage. Large cardinals belong to logic, and so numerous great mathematicians have spent an enormous amount of effort trying to understand and explain them.

For example, you can motivate large cardinals by giving a safe impredicative definition, such as the greatest lower bound of a set, and then contrasting it with some unsafe ones, such as Russell's paradox or the Liar paradox. This tension -- between the enormous practical utility of impredicative definitions and their tendency to open the door to paradox -- can be used to motivate large cardinals via the question of how close to the edge is safe.

Note that the question here is really a basic one: when does a definition actually define something?

A good resource is Poincare's essay "Logic and Mathematics, II", in which he argues for predicativism. Large cardinals are sort of the maximal rejection of his view, but Poincare is such a good writer and thinker that he's worth reading if only to get the most beautiful exposition of the alternative.

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You cannot give a definition, that's way too complicated. Ordinary people are not familiar with the idea that something can be defined at will. –  Andrej Bauer Nov 24 '10 at 11:58
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The poster said his audience would be new graduate students from the sciences and humanities. I would expect them to have an intuitive idea of what a definition is, and to have seen some definitions before in their own studies. If they haven't realized that "what is a valid definition?" could even be an open question, then this is good! The demonstration that it is a nontrivial question which nonetheless has rigorous answers could be an enlightening thing to teach. –  Neel Krishnaswami Nov 24 '10 at 14:48
    
I should add that I think your answer is fantastic, though. –  Neel Krishnaswami Nov 24 '10 at 14:51

You can look at the argument in Mendelian Dynamics and Sturtevant's Paradigm, it combines a nice application

In the 1913 paper ”The linear arrangement of sex-linked factors in Drosophila, as shown by their mode of association” Alfred Sturtevant11 , long before the ad- vent of the molecular biology and discovery of DNA, has deduced the linearity of the arrangement of genes on a chromosome from the statistics of simultane- ous occurrences of particular morphological features in generations of suitably interbred Drosophila flies. Thus he obtained the world’s first genetic map, i.e. he determined relative positions of certain genes on a chromosome, where he used his ideas of linearity and of gene linkage.

and a simple but conceptual maths argument:

On the mathematics side, Sturtevant’s reasoning may seem to be limited to the banal remark saying that if in a finite metric space the triangle inequality reduces to equality on every, properly ordered, triple of points then the metric is linear, i.e. inducible from the real line. But this is not exactly what is truly needed as the Sturtevant’s linearity is more about the order or, rather the ”between” relation, than about metrics.

Another suggestion coming from the same paper is Hardy-Weinberg Principle for Allele Distributions; it is different but possibly more familiar maths to your audience, and you may add a personal touch (Hardy was proud to be a pure mathematician, yet arguably this is his most cited result nowdays).

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I've come across the same issue (that it's difficult to explain math to non-mathematicians) many times, but inspired by this thread I decided to think more precisely about possible causes and solutions to this problem. I was further inspired because my dad happens to be in town, and when I tried to explain set theory to him, one response I got from him was essentially the same one you got, "so you're trying to describe ultimate reality."

First, I think the following important features about mathematics in general (i.e. not just set theory specifically) are unknown to most people:

  1. Math is vast - People somehow know that most academic fields are vast, but they don't know this about math. For instance, someone who has only studied up to classical mechanics has still heard about general relativity, classical mechanics, fluid dynamics, electricity and magnetism, etc. On the other hand, a lot of people honestly think there's nothing more to math than matrices and calculus.
  2. Math is about cool ideas - It's about coming up with cool ideas, exploring them, figuring out facts about them, and proving this facts using both creativity and logic. It's not about crunching out numbers using complicated formulas.
  3. Math is not about the real world - This is an overgeneralization to the point that it's false, but it might be closer to the truth than what your audience thinks math is about. Historically, math has absolutely been about modeling the real world and solving real world problems, but for various reasons, much of modern math is done purely for its own sake, and the content it discusses is very abstract. People will try to relate what you explain to them to something they already know, and this is an entirely natural thing to do, but it's almost surely bound to miss the point.
  4. Math is new - People will often look at you quizzically when you say you study math, and ask you, "what's there left to figure out?" Although the formulas of single variable calculus that they're familiar with have been figured out for centuries, there are constantly new questions arising in math, especially since math is so vast.
  5. Math is a different language - First of all, there's a lot of technical strange-sounding vocabulary. Secondly, there's a lot of technical familiar-sounding vocabulary that means something different in natural language - be careful about how you use the word "axiom" for instance.

When it comes to set theory specifically, if you get a response like, "so you're trying to describe ultimate reality," you have to spend a bunch of time convincing them that current set theory isn't really about anything that they already know or familiar with. If you manage to do this, then you'll be faced with the following question, "then what's the point?" This is where some historical motivation becomes necessary I think.

So you could explain that Cantor started the study of sets almost 150 years ago because people were starting to study the real numbers, and functions on them, in more and more sophisticated ways and talking about infinity in more and more sophisticated ways, and so this "required" a more sophisticated, rigorous framework for talking about these things. This naturally led to asking more precise questions about infinity. Thinking about the real line as set of points and not just a geometric line, and thinking about functions as objects that act on points, was quite a novel idea at the time. You can try to explain the importance of bijections to the notion of size and counting, and then state and vaguely explain that $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ have the same size, but $\mathbb{R}$ is strictly bigger. Emphasize that $\mathbb{Q}$ only seems bigger than $\mathbb{N}$ because the way it's arranged not because it actually has more things. Then introduce Cantor's Problem - is there a subset of the reals that's a bigger infinity than the naturals but a smaller infinity than the whole set of reals?

Then I'd talk about Hilbert and his famous problems, the first two of his problems being about putting math on a rigorous logical foundation. Talk about how truth and provability can be formalized, what it would mean for a formula to be true (in some model) but unprovable (from some axioms), and then talk about Godel's groundbreaking results on incompleteness. I would then come back to set theory, mentioning that a great variety of questions had come up in set theory, a good deal of them had turned out to be undecidable, including Cantor's Problem.

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Set theory is probably not about "describing ultimate reality", but it certainly has to do with describing what can be thought of as the "ultimate building blocks of mathematical reality". Of course, let's not use this as an explanation of math to a nonmathematician (although, as far as I know, that's the usual explanation of what math is about for philosophers -once I told a philosophy professor that I was working on number theory, and he immediately replied asking whether that had to do with coming up with the "right" definition of number-). –  David FernandezBreton Mar 13 '11 at 7:59

I know it is kind of trite, but why not start out by saying that most people think that there is only one kind of infinite set (and all mathematicians thought so until 150 years ago) but in fact there are several kinds. In particular some, like the rational numbers, can be arranged in a list (sequence), while others like the real numbers (infinite decimals) cannot---and then give Cantor's diagonal argument.

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What I've heard several people say is that it is very hard to get most non-mathematicians to accept Cantor's diagonal argument. For example, Dick Lipton and Terence Tao have both blogged about this several times, and Kevin Buzzard says something similar in the question linked to in the comments. So I don't know if this is the best idea. –  Qiaochu Yuan Nov 24 '10 at 17:42
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I think this is because Cantor's diagonal argument is not about real concepts but made up ones. If you take formalism seriously, then you have to start by explaining formalism to your audience. "Mathematics is about figuring the logical consequences of ideas we have made up, according to a specific notion of logical consequence that we have made up." From a layman's perspective, there is no reason to accept infinite sets, or that the existence of a bijection gives a meaningful equivalence relation on sets, or that the equivalence classes ought to be studied. –  Alexander Woo Nov 24 '10 at 21:57
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@Alexander: right. You have to fight against your audience's intuition, in a bad way, to convince them that Q and N have the same "size" but Q and R do not, and 10 minutes is not really enough time to do this... –  Qiaochu Yuan Nov 24 '10 at 23:11
    
Even as a math student at the undergraduate level, I had a hard time understanding Cantor's diagonal argument: it took me at least three explanations (from the textbook, from the lecture and finally from a fellow student) to understand how it works, although now I work with similar arguments all the time (I'm currently working on a Ph. D. in set theory!) However, explaining why $\mathbb N$ and $\mathbb Z$ have the same size, although intuitively $\mathbb Z$ contains "twice" as many elements as $\mathbb N$, would be a really good idea as the argument is not nearly as involved as Cantor's. –  David FernandezBreton Mar 13 '11 at 7:36
    
Probably talking about the set $\mathbb Q$ is also involved, since I have seen first year math undergrads having problems with understanding the concept, but it's possible to talk about, for instance, the set of even numbers, or the set $\{n^2|n\in\mathbb N\}$, both of which have also the same size as $\mathbb N$ although they seem "smaller"... this should be enough to let the audience appreciate how counterintuitive the concept of infinity can be, and we don't need to jump to uncountable sets to generate that effect. That's exactly the approach followed by Hans Magnus Enzensberger in his book –  David FernandezBreton Mar 13 '11 at 7:41

Vi Hart is a master at doing the opposite: Presenting mathematics, which on the surface seems to be aimed at laymen, but is actually aimed at mathematicians in my opinion. Granted, those mathematicians might be young, and may not have realized yet that they actually are mathematicians.

This video from her is actually fairly deep, while following Andrej Bauer's 8 suggestions above almost to the letter.

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Although the speed of her presentations is not really to my taste, I have to say that I find her basic schtick pretty clever: "So you're sitting in your boring trigonometry class doodling while your teacher is droning on about..." and meanwhile there's all this really cool, interesting mathematics in the doodling. So subversive and witty! –  Todd Trimble Oct 25 '12 at 15:12

Though I am not teaching on a regular basis, I often explain what mathematics is to laymen. My explanations tend to converge toward the following lines:

1) Mathematics is poetry: 2 quotes:
Quote 1 : Mathematics is the art of giving two names to the same thing AND the same name to two different things ( HENRI POINCARE).

Quote 2 : In mathematics you have an absolute liberty, the price to pay for this is that you have to be very precise ( YURI MANIN).

2) Maths is made of observations and rendering them with an eventual need to make up a new language, just as anybody would need one in a complex and professional field (say dancing).

3) THE WORLD OF A SURFACE IN ITSELF

Then show them a band a paper,make it a cylinder (2 faces, 2 circle boundaries) . Then link it with a twist and ask them to count the boundaries and then faces. ( They will be astonished and see the difference for themselves ...)

3.a) On a Moebius band a river drawn in the middle ( a blue pencil will do) has only one bank( Let them check it), it is a different world if you live on it ( you are little bugs with no sense of the third dimension)

3.b) Tell them about the game of cylindrical chess (played on a torus, abstracting the game if necessary ( just moving pieces) those who know the rules of chess feel more at ease. Show them the game while remaining flat, then tell them that for someone really dumb you could imagine to produce a real torus by bending the board and using magnetic pieces. As a world (a fighting world for example) make the observation that proximity is changed.... (chess serves as a surrogate topology in this, but you do not have to pronounce the frightening word topology).

3.b) Back on the Moebius band : the game of chess on it is not the same as the cylindrical one: a piece does not move the same way and does not aspects the sames cells...

4) DIFFERENT WORLD AND VIEWS :
Now take a band make it a cylinder with a knot first ( need a band long and thin enough) put it side by side with the normal cylinder and ask them is it the same. After their answer your is of course yes AND no (the ambient or the embedded surface) . A matter of point of view.

5) USEFULNESS OF MATHEMATICS:
Of course there lots of applications but the killing example is : In 400 BC Greeks were doing land regrouping (consolidation),each land was measured by willing geometers. A year later there was plenty of lawyers at work because the pieces of land had been measured by perimeter!! Tell them that it might seem obviously stupid to do so, yet basic school told them about surface concept. Moreover using the perimeter might be a good way to do things if the goal was not farming but showing off with high flags and poles. Again many points of views blablabla...

NOTE : The interactivity is essential at least when checking the boundary of MB with the finger or sight for some. This is a close call for ten minutes, part 5 can be removed.
Try it on some none mathematical friends first after three times you will be probably quite sleek. It is also important to have the right length and width for the band paper 12 inches by less than one roughly usually the side of a sheet of paper...

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I would suggest having a look at the opaque square problem. The story of the problem is the following:

Suppose you own a square piece ofland and you are being told that a phone line runs through it. As you have no phone and internet connection yourself, because the phone company cannot, or will not, provide it, you want to find this line and rig up the thing yourself. Now the question is: how long is the shortest trench you would have to dig to find this phone line?

If you restrict yourself to a connected trench the optimal solution is a Steiner tree for the four corner points. For a trench system consisting of two connected parts, there is also a shortest solution known, which is shorter than the one consisting of one connected set. If you only ask about the shortest trench without restricting yourself to trenches with at most n components this problem is, to the best of my knowledge, still open.

It is actually quite fun and usually leads to a lively discussion if you let the audience guess what these trenches should look like, probably with some support.

This problem can also be taken to the next dimension, by looking at the opaque cube problem, or any other shape you like. Also there are different stories one could choose to introduce this problem.

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Do you mean to imply that the phone line is a straight line? –  Zsbán Ambrus Jan 31 '11 at 21:33

I find that, when working with non-math people, to approach the mathematics from a problem solving perspective. Why did anyone find it so important to develop this in the first place? Where can this be used today? Students tend to believe that mathematicians live some kind of arcane life behind academic walls. In fact, many of the topics covered in undergraduate math programs come from real world problems that someone had to solve. When students are posed with the original issue or a current day issue that uses the mathematics, their interest and willingness to work increase dramatically. The same thing goes for me in another discipline. For example, take history. Knowing what happened at the battle of Verdun in WWI is not all that exciting. However, if you look at the situation from both sides just prior to the battle, you start to see why it is important and why the leaders made the decisions that they did. Math educators need to focus on learning for the long term, not short term 'can you remember this'. Student should accept nothing less.

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The second and third sentences of your answer are things that should be taught also to mathematics students! But then you have think how to assess this knowledge and expertise so that students take it seriously. At Bangor we developed a course in "Mathematics in context"; there is an old debating club tag: "Text without context is merely pretext." –  Ronnie Brown Apr 6 '12 at 21:08

I always explain that math is a way of creating new knowledge from old using two things: deductive logic and abstraction. Then I try to give reasonably nontrivial examples of each. But of course I can't remember right now what exactly I say. For the latter, if there's enough time, I like to use groups as abstracting both numbers and geometric transformations.

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Their are two things I say to people when I am in such a situation:

1) I think of math as being divided into 3 parts: Algebra, Analysis and Topology. Each of these comes from starting with a set and looking at different types of structure on it. Algebra is about taking two things in the set and asking how you can make another element of the set given those two things. (or maybe take 3 things! etc) Analysis comes from taking two things in your set and asking what the distance is between them. Topology is about seeing when two things in your set are close two each other (it gets hard to convince them this is different from analysis, but it is doable).

Now first I should mention that this is a very simplistic description and probably incorrect, but you only have ten minutes and the audience can feel like they learned something about the field, or at least how it is put together, or was at least 100 or more years ago. This also might not appeal to you since I haven't said anything about set theory, but this does give you a jumping off point for talking about what you can do when you don't have any of those more "sophisticated" structures lying around.

2) I study algebraic topology, so I try and talk about how you might try and differentiate two geometric objects or rather determine if they are in the same homeomorphism class. First you have to say when you will be thinking of two things as the same. This seems a bit strange to people, but remind them of congruent triangles and how natural it is to think of congruent triangle as the same thing when they are in different places. Then tell them it is a lot like that, but with different rules: no tearing or untearing/gluing. Next you can talk about rudimentary invariants like things being path connected, if I stay inside the space can I get o every other point. Next, when i remove a point is it still path connected? etc... This gets very strange and hard quickly so you need some new tools to accomplish your goal. So you talk about $\pi_1$ which is not too hard to convince them that they understand. The best space to help them compute the fundamental group of is $\mathbb{R}^2-0$, that is something they can wrap their heads around, but don't mention the group structure. And I talk about how this invariant can detect differences, sometimes, but it doesn't tell you when things are the same.

That takes about 30 minutes or so at least, and it is not about set theory. My recommendation would be to take that model of presentation of a field and use it in the following way. There are certain types of problems in fields, the big problems you know, like classification problems, enumeration, and computation (I am sure there are other big schemes for programs, I just can't think of them, or don't know them). There must be some big program in set theory that you can talk about, and look at early toy problems that people may have tested the theory on. Pick some small examples and use imprecise and soft words that are non technical. Try to draw a picture and don't use mathematical symbols. Maybe you could talk about how $\omega = 1 + \omega \neq \omega + 1$ (where $\omega$ is the first ordinal after all of the finite ones, maybe I have it backwards or wrong). That is kind of cool, and doable... i think.

Anyway, that is what I do when people ask, hope it helps. (I also have a similarly watered down explanation of Spectral Sequences)

Again, apologies for misrepresentations or inaccuracies, no disrespect is meant.

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The Euler Characteristic (Steven Gubkin above) is potentially a good option - it feeds into, for example, how to make a football, and why there are pentagons in the domes at the Eden Project: but that is more than 10 minutes.

Another option is talking about the number line - and showing you can cover the rationals in intervals with intervals of arbitrarily small total length: related to the infinities question above. But I sometimes use this to show that the number line (which is used from primary school upwards) requires some careful thinking - and if I have time I explain how this feeds into that mathematics of continuity and change (a mention of Zeno's paradoxes gets in too).

If you want a 'why do maths' question such stories as the making of nuclear bombs/reactors (making sure that the bombs explode, but the reactors don't), or sending men to the moon and getting them back could provide a narrative (not just how much fuel, but how long does it take, therefore how much food etc) could provide motivation.

But if you are looking for material in relation to schools, the Mathematical Association has a good list of resources, which might also give some ideas.

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Yeah, the production of nuclear bombs is a big motivation to do mathematics .... –  Martin Brandenburg Nov 25 '10 at 12:30
    
Do half-sphere shaped domes need pentagons? –  Zsbán Ambrus Jan 31 '11 at 21:17

Since you deal with higher ordinals, 10 minutes seems like more than enough time to prove that there are more real numbers than natural numbers. I think most people can follow the "list of decimals" proof. After you do this, your audience is ready to believe that there are different infinities. Now, exlain that YOU are interested in the problem characterizing these infinities.

You want to show them that there are limitations on proving the existence of higher ordinals. Of course, you cannot do this. But, for example, you could use Russell's paradox to show that there are subtleties that one must take into account.

Good luck!

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I attended a fantastic lecture by Catherine Roberts of Holy Cross, which was accessible to me as a lowly calculus student. She explained the process of making a mathematical model of rafting trips down the Colorado River without using any "scary" equations. Despite the absence of "scary equations," she demonstrated how useful math is.

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You might like to look at the article "Making a mathematical exhibition"

http://pages.bangor.ac.uk/~mas010/icmi89.html

With regard to content we agreed that the aims were to:

  1. suggest that the making of mathematics is a natural human activity, part and parcel of the usual methods by which man has explored, discovered, and understood the world;

  2. present each item with a purpose and context, and not just because it was something that could be shown or demonstrated;

  3. convey an impression of some of the key methods by which mathematics works;

  4. show mathematics in the context of history, art, technology and other applications.

(In the end, item 4. was much too ambitious! But it led to a collaboration with John Robinson in presenting his Symbolic Sculptures.) The subject of knots is suitable for conveying things about mathematics. You can see the exhibition "Mathematics and knots" at

http://www.popmath.org.uk

In this area we could present the methods of:

  1. Representation
  2. Classification
  3. Invariants
  4. Analogy
  5. Decomposition into simple elements (and I would also include, laws of combination)
  6. Applications

See also articles on my web page on "Popularisation and Teaching"

www.bangor.ac.uk/r.brown/publar.html

This experience of popularisation, and presentations to 13 year olds in Masterclasses, proved very useful when I was invited to give talks to a wide range of scientists, who are very interested in what conceptual advances are being made in mathematics (rather than solutions to "million dollar problems"). See arXiv:math/0306223 , to a conference on theoretical neuroscience.

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If I had to explain large cardinals in 10 minutes my first source of inspiration would be Kanamori's excellent writings about their (whiggish) history. People like history. I wouldn't spend any time trying to justify or prove any result. Maybe 2 minutes to explain the cumulative hierarchy so that everybody understands the typical picture of V. If people guess there is a long coherent tradition of research starting from Cantor, hence of interest in the field, I think that's enough. They really don't need to understand anything, just have a general feeling of a sort of flow of ideas.

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