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This question already has an answer here:

I'm sorry in advance if here is not the suitable place to ask this question, and people can feel free to vote to close this if that's the case. However, since I'm not sure about this issue, I'll ask it in any way:

I was reading Hungerford's Algebra, and my friend working on food science saw the title "Fundamental Theorem of Galois Theory", and asked me what it's about. (Seeing the word "fundamental", he thought it should be a very basic, simple concept.) Then, I said something like "there are structures called groups, fields, and they have some special subsets called subgroup, subfields. In order to understand subfields, mathematicans makes a correspondence between subgroups of some associated group, which is easier to work with and better understood, compared to fields."

He said it made some sense, but I'm sure it was a terribly bad explanation. My question is how do people generally deal with this issue? Is this only my problem (which really might be the case), or is this a general problem with pure math people?

This issue really annoys me in the sense that, for example, when you consider some person working on food science, psychology, chemistry etc., they're generally able to tell roughly what they're trying to do. But, I'm always having very hard time when I'm asked the same question. I'm a PhD student now, and I plan on spending whole my life with mathematics. But, I'm afraid that, say in my 50 (if I live that much), if someone asks me what I did in my entire 50 years, and I fail to answer, that would be a shameful situation for me, I believe.

I tried my best to express my question in a good way, but still I'm sorry if it's not good enough, and don't hesitate to let me know if I should write some parts more clearly/explicitly.

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marked as duplicate by user147263, Micah, Daniel W. Farlow, Community Jul 12 '15 at 22:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ ^^i already have this problem as an undergrad. i tell my mom I'm in abstract algebra and she says "havent you already taken algebra?" so it goes. $\endgroup$ – Elliot G Jul 12 '15 at 6:10
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    $\begingroup$ math.stackexchange.com/questions/683891/… $\endgroup$ – Asaf Karagila Jul 12 '15 at 7:52
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    $\begingroup$ I have the feeling that only the best mathematicians have the ability to explain some ideas from their work without plunging into technical details. This requires a really deep knowledge of one's subject. $\endgroup$ – Giuseppe Negro Jul 12 '15 at 14:49
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    $\begingroup$ I think we should be careful about making claims like "when you consider some person working on food science, psychology, chemistry etc., they're generally able to tell roughly what they're trying to do". I'd say often it only appears that way because we're not well enough acquainted with the field to spot all the oversimplifications they're forced to make. $\endgroup$ – Ben Millwood Jul 12 '15 at 17:15
  • $\begingroup$ Maybe you want to make your question specifically about Galois theory? $\endgroup$ – BCLC Jul 15 '15 at 10:53
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The difficult task of explaining what you do or study as a mathematician in layman terms is a common experience.

I think that the main mistake is to try to be precise nonetheless. In your case of Galois theory, terms as "fields" or "groups" are useless and may even lead to more confusion (I remember once a physicist getting really confused at a talk because he had never been exposed to the algebraic meaning of the term "field"). I would have said something along the following lines:

"There is a large class of numbers called algebraic numbers which need to be considered when solving simple equations in just one unknown. They include most numbers people are familiar with, like usual fractions or their roots, but don't include others, like $\pi$. The set of algebraic numbers has symmetries, which are in some sense like spatial symmetries, but less easy to visualize. Galois theory is the theory of these symmetries."

From here one could move on (for instance telling how the $\pm$ sign in the formula for the quadratic equations--something that everybody has seen in his/her school years--is a manifestation of these symmetries) according to the interest shown by the person we are talking to.

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  • $\begingroup$ Even I, an engineer, know about Galois theory now! $\endgroup$ – gurghet Jul 12 '15 at 16:07
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I am far from being a mathematician, but I deal in other areas of abstract thinking. My model is that all knowledge is based on layers... in math terms we start with numbers, then we add operators, then jump to algebra, then on to calculus... From the above, I surmise that Galois Theory is a few layers below (above) that. [A layman's view!]

The problem comes in when you are trying to explain concepts that are beyond one or two layers from what the person is comfortable with - such as explaining vector calculus to someone who can only do arithmetic. Or routing protocols to a smartphone (only) user. Or the biochemistry of cancer to anyone who thinks you catch a cold by being cold (it's a rhinovirus infection people!).

In these cases, I'll aim to take them to the next level and then say this is an example of {Galois theory, Internet technology, cancer research,...}. This builds on what they know already and usually provides a workable answer to the initial question without confusing or being meaningless words. Then if they ask more questions, they are usually being more specific and you can give more focused questions.

2c (& not worth a penny more).

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I usually explain my girlfriend some cool (for me) concepts I've been learning. It's pretty simple, you need to link mathematical concepts to usual objects, expressions, ideas, ... .

For example, I explained her that Sets were like "bags" where you could store things (usually numbers). From here, I told her that if you give to a set the ability to add it's members, in any order (I'm refering to associativity), then you add a zero and an inverse for every number, so every given number plus it's inverse gives 0, then you have a group. I know that this is just an example of group, but you need to explain concepts to other people who have few to non-existent mathematical background in simple examples, and then you can build up to a general case. Then I kept adding things to that set untill I got to a field. And this is how my gf learnt about algebraic structures without needing any kind of definitions nor formal explanaitions. (Just for a note, I teached her some of the basics of modular arithmetic some time ago and she really liked it. Now, from time to time, she asks for more number theory :) ).

I find that explaining mathematical concepts to people who's mathematical knowledge isn't really there, is satisfying and it also lets you learn a lot more, because you need to dissect every piece of information that you have in order to translate it to a more common language.

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  • $\begingroup$ Exactamente... ;) $\endgroup$ – Masacroso Jul 12 '15 at 18:17
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If you're studying algebra, I usually explain to people that the algebra they studied in high school and perhaps college was a very old and in fact very specific part of algebra that has been known for centuries. Tell them that the type of algebra you study is a much more modern, generalized and advanced approach to the subject and has a wide variety of applications and open problems that you hope to contribute to.

If it's topology you study, just tell them that a coffee mug and a donut are the same shape and enjoy the look on their face.

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    $\begingroup$ I guess that If you are applying for a grant and your interviewer is not a mathematician your enjoyment will be very short-lived $\endgroup$ – Andrea Mori Jul 12 '15 at 8:13
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Don't try to impress them with complicated words. Don't explain things using words they don't know in a way that you think is simple. Use examples they know. If they're interested, they'll ask more.

And then, there's always the classic "Oh, you know, boring number stuff... Not very interesting" that lets you move the conversation on into a different, more interesting direction.

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Try beginning with a real-life application or something fundamental.

Likely, the average person will not understand any given theorem in any given mathematical theory. Instead, you can explain the motivation or fundamentals of the theory and then if your friend is still interested, you can explain some of the underlying concepts.

For example, if I were asked to explain the strong law of large numbers, I would probably summarize it as a result of what happens if an infinite sum of random variables satisfies a certain property.

Then, I explain what a random variable is. It can be thought of as a payoff from a game. For example, I bet against someone else on the price of a stock or the flip of a coin. A random variable could denote how much I won from the game (eg +1 if I win and -10 if I lose).

I'm hoping the one who asked me had some basic probability in highschool.

Similarly, you can begin with algebra in highschool and explain why fifth root whatevers don't have solutions.

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My six pence:

Tell them that you are studying how the four elementary operations ($+,-,\times,\div$) generalize to other things than ordinary numbers. For instance, when you are limited to a range like $[0,255]$. It turns out that this theory is extremely rich and powerful, and we are still far from understanding all of it. I has numerous real-world applications like cryptology (avoiding that your data be stolen on the cloud) or error correction (ensuring that your scratched DVD is still readable).

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For your example is better to explain the use of Galois theory as a kind of high level language over calculus (that is what really is) that started with the proof that you cant evaluate with simple formulas roots of polynomials due to "geometrical" structure of the polynomials. You dont need to go to technical language.

If you understand really in deep the topic about you want to talk you can explain the basic concepts easily in a normal language without, some times, any relation to mathematics.

So the question is not "how to deal with", it is more like "when you will know enough about some topic to express in natural language and easily the basic ideas that are behind".

Explain mathematics (or any kind of other technical science) your best friend is conscience that it comes not only from direct experience with the topic, it comes too from the history of the develop of maths (or any other technical science), i.e. from a broad context and knowledge about the topic you want explain in it basic ideas.

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