What is it like to understand advanced mathematics?
I'm interested to hear what very talented mathematicians and physicists have to say about "what it's like" to have an internalized sense of very advanced mathematical concepts. As someone who only completed college calculus and physics, and has some basic CS background, but who is very intrigued by mathematics, I've always been curious about this. Does it feel analogous to having mastery of another language like in programming or linguistics? Any honest, candid insights will be appreciated!.
40 Answers
348.1k Views • Upvoted by Ori Gurel-Gurevich, Level 17 Mathematician • Nikhil Garg, Taught Combinatorics at INMO training camp • Prashant Saxena, PhD in Mathematics
- You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if a question is answerable by one of a few powerful general purpose "machines" (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small -- see http://www.tricki.org/tri
cki/map for a partial list, maintained by Timothy Gowers. - You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
- You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion. More on this in the next few bullets.
- Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a "fixed point" which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don't know that they shouldn't be straining to visualize things for which they don't seem to have the visualizing machinery.) Instead...
- When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights. For example, you might imagine two- and three-dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don't have the desired property. Then you think about what was important to the examples and try to distill those ideas into symbols. Often, you see that the key idea in the symbolic manipulations doesn't depend on anything about two or three dimensions, and you know how to answer your hard question.
As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the "simple case" you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly. - To me, the biggest misconception that non-mathematicians have about how mathematicians work is that there is some mysterious mental faculty that is used to crack a research problem all at once. It's true that sometimes you can solve a problem by pattern-matching, where you see the standard tool that will work; the first bullet above is about that phenomenon. This is nice, but not fundamentally more impressive than other confluences of memory and intuition that occur in normal life, as when you remember a trick to use for hanging a picture frame or notice that you once saw a painting of the street you're now looking at.
In any case, by the time a problem gets to be a research problem, it's almost guaranteed that simple pattern matching won't finish it. So in one's professional work, the process is piecemeal: you think a few moves ahead, trying out possible attacks from your arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas you understand. This is the same way that you solve difficult problems in your first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try. Sometimes, during this process, a sudden insight comes, but it would not be possible without the painstaking groundwork [ http://terrytao.wordpress.com/ca... ].
Indeed, most of the bullet points here summarize feelings familiar to many serious students of mathematics who are in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to "bigger" things but have the same fundamental flavor. - You go up in abstraction, "higher and higher". The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves -- that is, you "zoom out" so that every function is just a point in a space, surrounded by many other "nearby" functions. Using this kind of zooming out technique, you can say very complex things in short sentences -- things that, if unpacked and said at the zoomed-in level, would take up pages. Abstracting and compressing in this way makes it possible to consider extremely complicated issues with one's limited memory and processing power.
- The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians", where _____ is something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a knowledge of maths and a facility with abstraction.
Learning the domain-specific elements of a different field can still be hard -- for instance, physical intuition and economic intuition seem to rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.
- You move easily among multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment.
Indeed, some of the most powerful ideas in mathematics (e.g., duality, Galois theory, algebraic geometry) provide "dictionaries" for moving between "worlds" in ways that, ex ante, are very surprising. For example, Galois theory allows us to use our understanding of symmetries of shapes (e.g., rigid motions of an octagon) to understand why you can solve any fourth-degree polynomial equation in closed form, but not any fifth-degree polynomial equation. Once you know these threads between different parts of the universe, you can use them like wormholes to extricate yourself from a place where you would otherwise be stuck. The next two bullets expand on this. - Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days figuring out why a result follows easily from some very deep and general pattern that is already well-understood, rather than from a string of calculations. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. (Nevertheless, detailed calculation of an example is often a crucial part of beginning to see what is really going on in a problem; and, depending on the field, some calculation often plays an essential role even in the best proof of a result.)
In A Mathematician's Apology [http://www.math.ualberta.ca/~mss..., the most poetic book I know on what it is "like" to be a mathematician], G.H. Hardy wrote:
"In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way."
[...]
"[A solution to a difficult chess problem] is quite genuine mathematics, and has its merits; but it is just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly) which a real mathematician tends to despise." - You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. Mathematicians don't really care about "the answer" to any particular question; even the most sought-after theorems, like Fermat's Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a mathematician seeks is finding a new dictionary or wormhole between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Timothy Gowers has some interesting comments on this issue, and disagreements within the mathematical community about it [ http://www.dpmms.cam.ac.u
k/~wtg1... ].
- Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: "First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams..." All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated. (I should say, "without feeling unbearably lost and frustrated"; some amount of these feelings is inevitable, but the key is to reduce them to a tolearable degree.)
Andrew Wiles, who proved Fermat's Last Theorem, used an "exploring" metaphor:
"Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them." [ http://www.pbs.org/wgbh/nova/phy... ]
- In listening to a seminar or while reading a paper, you don't get stuck as much as you used to in youth because you are good at modularizing a conceptual space, taking certain calculations or arguments you don't understand as "black boxes", and considering their implications anyway. You can sometimes make statements you know are true and have good intuition for, without understanding all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation. (I first saw these phenomena highlighted by Ravi Vakil, who offers insightful advice on being a mathematics student: http://math.stanford.edu/
~vakil/... .) - You are good at generating your own definitions and your own questions in thinking about some new kind of abstraction.
One of the things one learns fairly late in a typical mathematical education (often only at the stage of starting to do research) is how to make good, useful definitions. Something I've reliably heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at proving difficult propositions that were stated in a textbook exercise, but would be lost if presented with a mathematical structure and asked to find and prove some interesting facts about it. Concretely, the ability to do this amounts to being good at making definitions and, using the newly defined concepts, formulating precise results that other mathematicians find intriguing or enlightening.
This kind of challenge is like being given a world and asked to find events in it that come together to form a good detective story. You have to figure out who the characters should be (the concepts and objects you define) and what the interesting mystery might be. To do these things, you use analogies with other detective stories (mathematical theories) that you know and a taste for what is surprising or deep. How this process works is perhaps the most difficult aspect of mathematical work to describe precisely but also the thing that I would guess is the strongest thing that mathematicians have in common. - You are easily annoyed by imprecision in talking about the quantitative or logical. This is mostly because you are trained to quickly think about counterexamples that make an imprecise claim seem obviously false.
- On the other hand, you are very comfortable with intentional imprecision or "hand-waving" in areas you know, because you know how to fill in the details. Terence Tao is very eloquent about this here [ http://terrytao.wordpress
.com/ca... ]:
"[After learning to think rigorously, comes the] 'post-rigorous' stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the 'big picture'. This stage usually occupies the late graduate years and beyond."
In particular, an idea that took hours to understand correctly the first time ("for any arbitrarily small epsilon I can find a small delta so that this statement is true") becomes such a basic element of your later thinking that you don't give it conscious thought. - Before wrapping up, it is worth mentioning that mathematicians are not immune to the limitations faced by most others. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes. Above, I have tried to summarize how the mathematical way of thinking feels and works at its best, without focusing on personality flaws of mathematicians or on the politics of various mathematical fields. These issues are worthy of their own long answers!
- You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. (The theoretical computer scientist Richard Lipton lists some examples of potentially "deep" ignorance here: http://rjlipton.wordpress
.com/20...) This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being very intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.
This is from a physicist point of view. I've found that mathematicians and physicists think in *very* different ways. Basically physicists think in terms of analogies, whereas mathematicians think in terms of proofs, because they are often exploring "strange new worlds" in which analogies don't work.
Snowflakes.
If you look at a glass of water, you just see water. However, once you know some advanced math, whenever you look at water, you can imagine snowflakes.
Now imagine describing what a snowflake looks like to a blind man.
There's also the story of Paul on the road to Damascus, when he heard the voice of God. I've had moments like that when I'm in a classroom, and something just works out, and I suddenly have something similar to a religious experience. For a few moments, you are floating in the middle of the universe, rather than in a classroom, and you are just sitting there amazed that everything just fits together.
Also sometimes it feels like seeing a unicorn. You are in the middle of a deep forest. You turn around, and in the corner of you eye, you see something. It looks like a unicorn, but when you follow it, it disappears. But then you see it somewhere else. And it disappears. Sometimes you follow the trail and it leads you into an undiscovered world. One example, when I was in college, I was taking a class in tensor calculus while I was taking a basic algorithms class, and I spotted a unicorn. It just felt the same. It took me a decade and a half to follow the unicorn, and I ended up in the world of category theory.
The thing about advanced math is that I know that I understand it when I "feel" it. If you throw an apple to me, I know what path the apple is going to take, and I know where I have to put my hand to catch it, and I don't have to consciously think about this. For some mathematical things, it's the same way. I know what piece fits where, and I don't have to consciously think about it. For the stuff that I don't understand well, I have to think, and this point it's like a blind man feeling his way around a room.
The other thing is that sometimes in advanced math, simple things become complex. The religious experience that I had was when the instructor was talking about Clifford algebras. These are used in describe rotations, and one thing that you figure out is that rotation in 3-d space turns out to be quite interesting and complicated.
Some other things, I've found that the more math that I know the *dumber* I feel.
The more you know the more you realize your ignorance. Once you learn something, this can open up entire new worlds that you didn't realize existed before.
Also, you realize how much other people are just better at this sort of thing. Imagine being deaf. You really can't tell the different between an expert concert pianist and someone that is just banging on the keys, since it all looks the same. If you can even hear a little bit, then you start to realize how good mathematicians are, and you realize that you'll probably never be that good.
There are also different levels of "good." There is a difference between listening to Mozart, being able to play Mozart, or being able to compose like Mozart. Once you understand higher level math, that's sort of equivalent of being able to play Mozart. Being able to play Mozart doesn't make you Mozart, and if you are someone that can play Mozart, you start being more in awe of people that are better composers or pianists that are much better you are, and so I've found that the more I understand math, the stupider I feel.
It's also frustrating once you understand something, not to be able to explain it to someone else. On the other hand, sometimes you see something, it becomes obvious to you, and when you explain it, it becomes obvious to the person you are explaining it to.
Snowflakes.
If you look at a glass of water, you just see water. However, once you know some advanced math, whenever you look at water, you can imagine snowflakes.
Now imagine describing what a snowflake looks like to a blind man.
There's also the story of Paul on the road to Damascus, when he heard the voice of God. I've had moments like that when I'm in a classroom, and something just works out, and I suddenly have something similar to a religious experience. For a few moments, you are floating in the middle of the universe, rather than in a classroom, and you are just sitting there amazed that everything just fits together.
Also sometimes it feels like seeing a unicorn. You are in the middle of a deep forest. You turn around, and in the corner of you eye, you see something. It looks like a unicorn, but when you follow it, it disappears. But then you see it somewhere else. And it disappears. Sometimes you follow the trail and it leads you into an undiscovered world. One example, when I was in college, I was taking a class in tensor calculus while I was taking a basic algorithms class, and I spotted a unicorn. It just felt the same. It took me a decade and a half to follow the unicorn, and I ended up in the world of category theory.
The thing about advanced math is that I know that I understand it when I "feel" it. If you throw an apple to me, I know what path the apple is going to take, and I know where I have to put my hand to catch it, and I don't have to consciously think about this. For some mathematical things, it's the same way. I know what piece fits where, and I don't have to consciously think about it. For the stuff that I don't understand well, I have to think, and this point it's like a blind man feeling his way around a room.
The other thing is that sometimes in advanced math, simple things become complex. The religious experience that I had was when the instructor was talking about Clifford algebras. These are used in describe rotations, and one thing that you figure out is that rotation in 3-d space turns out to be quite interesting and complicated.
Some other things, I've found that the more math that I know the *dumber* I feel.
The more you know the more you realize your ignorance. Once you learn something, this can open up entire new worlds that you didn't realize existed before.
Also, you realize how much other people are just better at this sort of thing. Imagine being deaf. You really can't tell the different between an expert concert pianist and someone that is just banging on the keys, since it all looks the same. If you can even hear a little bit, then you start to realize how good mathematicians are, and you realize that you'll probably never be that good.
There are also different levels of "good." There is a difference between listening to Mozart, being able to play Mozart, or being able to compose like Mozart. Once you understand higher level math, that's sort of equivalent of being able to play Mozart. Being able to play Mozart doesn't make you Mozart, and if you are someone that can play Mozart, you start being more in awe of people that are better composers or pianists that are much better you are, and so I've found that the more I understand math, the stupider I feel.
It's also frustrating once you understand something, not to be able to explain it to someone else. On the other hand, sometimes you see something, it becomes obvious to you, and when you explain it, it becomes obvious to the person you are explaining it to.
You can start appreciating and understanding the inner beauty of mathematics at a new, emotional level.
It is like art - having some knowledge and understanding of it will enable you to appreciate masterpieces even more.
Mathematics is, unfortunately, considered as dull and boring in (most of) the popular culture. Big part of that is due to the way it has been taught, almost exclusively as a seemingly endless sequence of rote memorizations of abstract formulae and objects most people have trouble relating to.
Understanding mathematics is really about understanding proofs, i.e. why theorems are true. Instead of needing to remember bunch of statements, you know they are true because you know how to prove them.
Despite popular misconceptions, proofs of the most important theorems are often simple and elegant, things of beauty.
Consider Pythagora's theorem, there are many simple and elegant proofs of it - http://www.cut-the-knot.o
Another example, a personal favorite, is Cantor's diagonalization proof http://en.wikipedia.org/w
This fundamental proof, discovered surprisingly late in 1891, is the basis for such groundbreaking and seemingly unrelated works such as Turing's proof of undecidability of the halting problem and Goedel's incompleteness theorems.
Personally, I find it rather amazing that before 1891, the world thought there was only one infinity (as I did, before understanding Cantor's proof for the first time).
Of course, the above examples are but a very small set showing the beauty of mathematics.
Having even the most basic understanding, and interest, in advanced mathematics will start uncovering this amazing world to you.
It is like art - having some knowledge and understanding of it will enable you to appreciate masterpieces even more.
Mathematics is, unfortunately, considered as dull and boring in (most of) the popular culture. Big part of that is due to the way it has been taught, almost exclusively as a seemingly endless sequence of rote memorizations of abstract formulae and objects most people have trouble relating to.
Understanding mathematics is really about understanding proofs, i.e. why theorems are true. Instead of needing to remember bunch of statements, you know they are true because you know how to prove them.
Despite popular misconceptions, proofs of the most important theorems are often simple and elegant, things of beauty.
Consider Pythagora's theorem, there are many simple and elegant proofs of it - http://www.cut-the-knot.o
Another example, a personal favorite, is Cantor's diagonalization proof http://en.wikipedia.org/w
This fundamental proof, discovered surprisingly late in 1891, is the basis for such groundbreaking and seemingly unrelated works such as Turing's proof of undecidability of the halting problem and Goedel's incompleteness theorems.
Personally, I find it rather amazing that before 1891, the world thought there was only one infinity (as I did, before understanding Cantor's proof for the first time).
Of course, the above examples are but a very small set showing the beauty of mathematics.
Having even the most basic understanding, and interest, in advanced mathematics will start uncovering this amazing world to you.
I must start with the disclaimer that I am merely a physics undergrad with no real understanding of mathematics (and certainly nothing close to that of Anonymous). But this is the picture I've managed to piece together of what an advanced understanding of mathematics feels like through countless late-night conversations with maths majors.
For the moment, let us assume that we are living in an alternative reality of some sort in which people know nothing of geometry, but are comfortable handling natural numbers, integers and rational numbers. Elementary results of combinatorics, number theory and algebra are things everyone is familiar with, including the notion of algebraic closure in the context of field extensions.
Now imagine, a revolutionary genius comes along pulling out seemingly nonsensical constructions out of thin air. She starts with infinite sequences of numbers, which are deemed within the realm of sanity by her contemporaries, but everyone loses her when she begins fixate upon the notions of numbers in a sequence getting arbitrarily close to one another and to a new number she calls the 'limit' and go on about how nice it would be if all nice it would be have all numbers arise as 'limits'. She proves that this isn't the case and goes on to extend the notion of numbers by what appears to be a very convoluted construction. What on earth is she trying to do? Why bother with all this in the first place?
She says, wait, here's the magic. Using the notion of limits developed by her, she proves certain results about 'fixed points' and 'winding numbers' and shows that a particular field naturally arising out of her constructions is algebraically closed. Everyone else is left scratching their heads over how she is even managing to see something like that through all these layers of abstraction.
As you may have realised, none of the mathematicians of this alternative reality except her have a picture of objects such as lines and planes. To her, the extension of rational numbers to include the limits of sequences of them corresponds to points on a line. Including a single number [math]i[/math] allows her to extend this to a plane whose points transform in certain ways under addition and multiplication, and demonstrating the algebraic closure of this field becomes relatively trivial. Without the language of geometry to fall back onto, she has no way to concretely convey the notion of a line and plane to her fellow mathematicians who find her clear-as-day (to us!) methods to be nothing short of sorcery.
It is perhaps an accident of history or of the way our brains are wired that we are able to think of geometry in an intuitive manner. So, many of the results about real numbers seem painfully obvious to us. But as anyone who's ever come across the concept of a p-adic number would understand, real numbers come with a lot of structure that allows us to connect their algebraic and analytic properties in a seamless way.
It is conceivable that on some planet far way, intelligent beings capable of doing mathematics in the sense we understand would find their intuition better equipped to handle notions such as that of a spectrum of a ring and that certain deep and nontrivial results in algebraic geometry would be utterly obvious to them. So having an advanced understanding of mathematics is a lot like getting to walk awhile in the skins of differently wired beings inhabiting an alternative reality.
For the moment, let us assume that we are living in an alternative reality of some sort in which people know nothing of geometry, but are comfortable handling natural numbers, integers and rational numbers. Elementary results of combinatorics, number theory and algebra are things everyone is familiar with, including the notion of algebraic closure in the context of field extensions.
Now imagine, a revolutionary genius comes along pulling out seemingly nonsensical constructions out of thin air. She starts with infinite sequences of numbers, which are deemed within the realm of sanity by her contemporaries, but everyone loses her when she begins fixate upon the notions of numbers in a sequence getting arbitrarily close to one another and to a new number she calls the 'limit' and go on about how nice it would be if all nice it would be have all numbers arise as 'limits'. She proves that this isn't the case and goes on to extend the notion of numbers by what appears to be a very convoluted construction. What on earth is she trying to do? Why bother with all this in the first place?
She says, wait, here's the magic. Using the notion of limits developed by her, she proves certain results about 'fixed points' and 'winding numbers' and shows that a particular field naturally arising out of her constructions is algebraically closed. Everyone else is left scratching their heads over how she is even managing to see something like that through all these layers of abstraction.
As you may have realised, none of the mathematicians of this alternative reality except her have a picture of objects such as lines and planes. To her, the extension of rational numbers to include the limits of sequences of them corresponds to points on a line. Including a single number [math]i[/math] allows her to extend this to a plane whose points transform in certain ways under addition and multiplication, and demonstrating the algebraic closure of this field becomes relatively trivial. Without the language of geometry to fall back onto, she has no way to concretely convey the notion of a line and plane to her fellow mathematicians who find her clear-as-day (to us!) methods to be nothing short of sorcery.
It is perhaps an accident of history or of the way our brains are wired that we are able to think of geometry in an intuitive manner. So, many of the results about real numbers seem painfully obvious to us. But as anyone who's ever come across the concept of a p-adic number would understand, real numbers come with a lot of structure that allows us to connect their algebraic and analytic properties in a seamless way.
It is conceivable that on some planet far way, intelligent beings capable of doing mathematics in the sense we understand would find their intuition better equipped to handle notions such as that of a spectrum of a ring and that certain deep and nontrivial results in algebraic geometry would be utterly obvious to them. So having an advanced understanding of mathematics is a lot like getting to walk awhile in the skins of differently wired beings inhabiting an alternative reality.
