Math is Everywhere

A number of coincidences led to this post -- first, I got an email from one of the Mikes, who encouraged me to digress more from strictly technical topics. Thus I was going through some old email looking for anything off-the-wall and I ran into this old rant about mathematics that I wrote a while back. Not five minutes after re-reading it, I saw Gretchen's recent post on how her high school math teacher always told her that algebra would come in handy, and sure enough, it did. Go, algebra!


A while back a high school math teacher acquaintance of mine threw out a question to a mailing list I was on: does your job involve 'real world' applications of mathematics?

Sure, absolutely, all the time. I mean, basic arithmetic goes without saying. I don't think we need any examples of that. Let's think about harder stuff. Like, I could not do my job without a good working knowledge of logarithms. I often have to test the performance of my product in real-world server environments and that means understanding what happens when you throw 10, 100, 1000, 10000 and 100000 clients against a server farm. Being able to categorize, graph, and understand that data requires math ranging from the ability to do logarithmic graphs to understanding statistical concepts like variance and standard deviation. And that's not the only reasonably buff math that Microsofties use, not by far. I use linear algebra less in my own work, but anyone doing any kind of low-level three-d graphics needs to understand matrix transformations inside-out. My friends in the Windows Media team have a deep understanding of the mathematics of entropy, to make their compression schemes work better. I could go on, but you get the point.

And moreover, that stuff is just fancy arithmetic . Mathematics is all about designing algorithms and writing proofs that they work, and those skills go deep to the heart of computer programming. It's nigh impossible to understand recursive programming without also understanding proof by induction; they go together like hand in glove.

But there's a broader question lurking here. Why is a math teacher asking this question in the first place?

"When are we ever going to use this in real life?" has been the cry of bored math students since time immemorial. Though the vast majority mightn't a clue what "real life" entails, it's still a fair question that deserves an answer. Coming up with practical examples of math in real life is a reasonable approach.

But I would go further. I would deny the very implicit premises that the question is based upon : first, that the only really legitimate knowledge worth having is practical "real life" knowledge, second that anything which lacks an immediate, direct application is by definition impractical .

They seem like reasonable premises, no? After all, couldn't we all live perfectly good lives without knowing:

  • whether the sun goes around the earth or the earth goes around the sun
  • the causes of the French Revolution
  • how to compute the area under a curve
  • what continent Guyana is in
  • the difference between acids and bases
  • how the digestive system works
  • how to play the piano
  • how to throw a shot put
  • how to speak Latin

Proof by example -- billions of people have lived their lives knowing none of these things. But I can't help but think that the world would be a better place if more people did know these things, whether it was ever of direct, practical use or not . But that's just me: Eric Lippert, effete snob at large. To answer the practical question, we need to come up with practical reasons to know impractical things. Seven immediately come to mind.

First, learning impractical things is just plain good for you mentally. It teaches you new ways to learn other things, some of which may be practical. It teaches you new ways to solve problems, new ways to think. That is eminently useful! Mathematics in particular can be a wonderful way to combine intuitive and formal reasoning. Mathematics, contrary to popular belief, is a mostly intuitive. It's a creative process that is then justified by formal proofs -- which themselves require intuition and cleverness to create.

Show me a job where you don't have to think, you don't have to solve new problems in new ways, and I'll show you a job that is deathly boring. You want a deathly boring job, fine by me, but most people do not.

Second, human experience does not actually come to us neatly compartmentalized into "Biology" and "History" and "Physics" -- these are human-made compartments that we impose to try and put some order on the vast, tangled web of all knowledge. Those compartments are as artificial as lines of longitude. If you only know the bare necessities to keep yourself alive then your ability to traverse that vast web of knowledge is greatly diminished. There are connections . And the reason why mathematics is second only to writing as a tool for manipulating knowledge is because mathematics is applicable to almost every single human endevour.

Third, high school students today are probably going to live to be 120 and change jobs every couple years. How can anyone possibly know what skills are going to be useful someday? Limiting oneself to the "obviously practical" cuts off whole avenues of possible occupations. Maybe the only math you'll ever need is the math necessary to be a bank teller. That's fine -- there's nothing wrong with being a bank teller. But to willfully choose to know only that much is to restrict your choices in the future. Banking just might get dull after a few years, and you know what? It's easier to learn huge amounts of stuff when you're younger, believe me.

Fourth, it's a competitive world out there. Is the fact that the United States' information technology sector is about to have its lunch eaten by India's in any way correlated to the shrinking number of Americans getting math and computer science degrees? Enrollment has dropped again as the tech sector has languished. Does anyone honestly believe that there is no relationship between the value a society places on excellence in education and its viability in the information economy?

Fifth, having lots of "useless" skills is a way to manage risk. Think of some worst cases. What if you get fired? What if you get blinded? Lose use your legs? Having a few "useless" skills can really come in handy if suddenly you can't make a living on any of your so-called "useful" skills. This guy is a germane example.

Sixth, limiting yourself to only the obviously practical is essentially marginalizing yourself. Mathematics, particularly statistics, is everywhere in our technological world. To deny the practicality of mathematics is to say " I choose to be powerless in a world where a technological/political/scientific aristocracy understands and controls the systems upon which my life depends. I choose to be unable to understand those systems and unable to communicate effectively with the people who create them. I choose to be a person who cannot make a difference. "

Seventh, maybe you don't, but I live in a representative democracy. Americans are always going on about their beautiful system of checks and balances -- well, the ultimate check and balance from which all political power derives in the American philosophy is, at least in theory, the will of the people. Citizens in representative democracies are asked to make choices about mind-numbingly complex issues. Scientific, ethical, geopolitical, economic, you name it. Making informed choices is everyone's responsibility. And that means having at least a working knowledge of a vast quantity of "useless" information, much of which will be expressed in percentages, rates, etc. How can you manipulate those numbers correctly to arrive at meaningful conclusions without a strong working understanding of math?

Comments (26)

  1. sebmol says:

    Right on. And I haven’t even finished college. Education could be the silver bullet to almost all of society’s problems.

  2. jeffcal says:

    you should check out bertrand russell’s essay collection "in praise of idleness" – in it, russell takes a whole essay to provide a compelling argument for learning "useless information". the title of the particular essay is "useless knowledge"…


  3. Eric Lippert says:

    Re: Russell: Indeed, I have a copy, though I have not read it for some time. I’ll take another look tonight.

    Re: Silver bullet: No, I don’t believe education is a panacea. I do believe that widespread ignorance is disastrous though. Education is necessary but not sufficient; it’s no magic bullet.

  4. Zac says:

    🙂 Eric, you’ve lost your way if you can sign your name to posts that include sentences like "Mathematics is all about designing algorithms …". No, it ain’t, neither! But I like your post today. Let me respond:

    Point 1 – Tru dat.

    Point 2 – Agreed! I’ve got a book called "The Human Factor" by Kim Vicente that you should read, which scathingly criticises the compartmentalization of knowledge.

    Point 3 – Yah, not so much. I’m increasingly beginning to agree with this guy here on that point: <;

    Point 4 – I don’t think there is much of a correlation there, actually … and if anything, this is an argument for learning more practical skills, not more abstract ones.

    Point 5 – Those unused skills, like unused muscles, atrophy with time. Therefore, so does their ability to manage risk. I wish I had studied less French in high school — not because I haven’t used and benefited from the French education; I have. But I could have learned other things instead, that I would have used more.

    Point 6 and 7 – Good points both.

  5. Eric Lippert says:

    Well, maybe not ALL about, but certainly lots of professional mathematics is about coming up with algorithms. Ask Professor Orbifold sometime about how much fun it is to come up with algorithms to prove, uh, whatever the thing he was proving about the hyperbolic bananas connected to the other hyperbolic bananas. I didn’t really get the whole idea of the proof, but I recall the banana thing as being pretty clever.

    Re: atrophying — I’m helping a friend study for his Physics 101 midterm and boy, is it ever amazing how rusty you get after ten years. I’m like, OK, there’s work and there’s momentum and there’s kinetic and potential energy, and the work-energy theorem says… uh… I knew that once…

  6. Miles Archer says:

    It’s noble to learn for the sake of knowledge, but for me, it really helps to have a way to apply something to really learn it. For example, I could learn a programming language by reading a book, but I won’t really know it until I apply it to solve some real problem.

    Maybe that’s why I was such a poor student.

  7. Legion says:

    I decided at the age of 19 that I was smarter than all that and quit college and entered the workforce. At the age of 26 I went back to college because I found that people really did care whether or not I had a degree, even if they knew I could do the job. I had already been a professional programmer for a few years when I finally got around to taking my Calculus class. One of the students in my class made a big point of asking why in the world we needed to learn this stuff … and asked me in front of the entire class if I ever used Calculus while writing code. My reply was, "No, but I didn’t really have the option either because I didn’t know any Calculus. I don’t know how many problems would have been made infinitely easier just by having better tools."

  8. I study Applied Math in college and I live with it.

    I’m agree with you 100%

  9. sara ford says:

    A fellow mathematics major once told me that Mathematics is the study of learning how to solve problems. Hence, Mathematics applies to everything in real life.

    Check out my MAA career profile on how Mathematics relates to Computer Science and my job as a Microsoft SDET at

  10. A good post by Eric Lippert titled "Math is Everywhere" in which he posits seven reasons why the study of any subject, and particularly math, is never really a waste of time. His thesis in a nutshell:"When are we ever going to use this in real life?" h …

  11. mschaef says:

    "Fourth, it’s a competitive world out there. Is the fact that the United States’ information technology sector is about to have its lunch eaten by India’s in any way correlated to the shrinking number of Americans getting math and computer science degrees?"

    I think it’s more related to the glut of low cost talent overseas. Given the number of people willing to undercut the cost of an american CS grad by 80-90%, I’m not at all suprised that less people are trying to enter the field. The IT sector would hardly be historically unique in that regard.

  12. Tim Smith says:

    I have my B.S. in Math. I have always stated that the reason I did this was to learn how to think.

    Early on (1983ish), it was obvious to me that picking up computer programming was not the issue. I do regret missing out of some of the computer theory, but I will never regret learning how to solve problems. My two favorite courses where Analysis and Logic. In Analysis we started from the basic axioms of math and "proved" math. I learned so much about problem solving in that course.

    Eric is right. (Ack, I nearly wrote Raymond. Too many blogs early in the morning.) Much of proving math is about learning techniques or algorithms. Given what is known, you usually end up carting out your standard set of techniques. If those don’t work, you start searching for other techniques that will work. Sometimes you have to invent your own techniques.

    If you look at the proof of Fermat’s Last Theorem, there isn’t proof. The problem is divided up into sub-problems where different algorithms are used to prove those sub-problems.

  13. Tim Smith says:

    –I think it’s more related to the glut of low cost talent overseas.–

    It is a combination of both. You look at test scores and Indians do significantly better than you average Joe in the U.S..

    India has a proud history of some amazingly talented mathematicians.

  14. Eric Lippert says:

    I’ve been helping a friend study for his Physics 101 midterm the last couple days, and I think part of the frustration that students feel comes from the contrived, unrealistic, context-free problems.

    Physics problems are particularly bad — I mean, take the problem we were doing last night. There are no carts with wheels that require no energy to turn, and no one would really build a 24-metre diameter roller-coaster loop that depended on the centrifugal "force" to keep the cart on the track. That means that from the beginning we know that the answer will be wrong and the solution is irrelevant, so why bother?

    Now, obviously there is a reason why textbooks give problems like this — because you have to be able to solve simple, unrealistic problems first. Real-world problems get way complicated way fast. All applied mathematics is about idealizing a problem, abstracting away unnecessary details.

    But still, as the techniques get more and more abstract, it gets harder to come up with a justification.

    Case in point: I recall a day about ten years ago when Thingo, Orbifold and I were in Professor Stan "the man" Burris’s algebra class and someone — Thingo, I think — DEMANDED to see an example of a particular matrix manipulation with honest-to-goodness NUMBERS OTHER THAN ZERO AND ONE in it, dammit!

    Eventually someone said something like "this Jordan normal form stuff is all well and good, but what ever is it for?"

    Once Professor Burris explained that the normal form was highly useful when solving certain systems of differential equations, and that we were in fact just laying the groundwork for later topics in analysis, I felt much better about doing the work.

  15. mschaef says:

    "It is a combination of both. You look at test scores and Indians do significantly better than you average Joe in the U.S.. "

    What group of Indians are you comparing to the "average Joe in the U.S..":

    a) The top performers

    b) The average Indian taking the test

    c) The average Indian in school

    d) The average Indian

    My hunch (with no real substantiation) is that the relationship |b|<|c|<|d| is a lot more true for India than it is for the U.S., which could lead to higher test scores in and of itself.

  16. mschaef says:

    "India has a proud history of some amazingly talented mathematicians. "

    BTW, I in no way dispute the high level of talent overseas.

  17. CB says:

    I personally agree with just about every point you make. But it’s important to realise that most western societies (especially the anglo ones) have now comprehensively rejected a non-utilitarian approach to knowledge and education since around the mid-70’s. Most universities are now essentially job-training centres, and education in any true sense is hardly a part of their remit any longer.

    The citizens of our societies, again especially the English-speaking ones, repeatedly vote for extremely narrow right-wing voices who reject evidence-based policy in favour of blind utilitarian recalculation of GNP.

    Unfortunately it’s too late to do anything about all this now — oil’s nearly run out, food production is in crisis, land and marine ecosystems are damaged almost beyond repair, and global warming is about to bring many chickens home to roost. So, in truth, those of us who believe in non-utilitarian knowledge gathering have lost the ultimate battle.

  18. Eric Lippert says:

    This is kind of weird actually — I made this post on Wednesday, and the next day there’s a new study released that says that American high schools aren’t teaching enough math, and the cover story of USA Today is that 22 states now require algebra for a high school diploma, an all-time high. I did not realize that algebra is traditionally taught only to the college-bound in the United States.

    I certainly agree with you that universities have lost their earlier function — to expose students to the whole universe of human knowledge, to make better human beings. The contract now is more about "don’t show me anything I don’t need to know in order to get a better job", which is a shame.

    However, I don’t want to get too rosy in my look back at the past. The organized education system has been classist since the Platonic "Philosopher Kings should rule the world" Academy.

    But I’m not following your point — yes, wholesale damage to the global ecology has proceded apace for some time now, but I have a hard time blaming that on the lack of widespread liberal arts education.

    (To nit pick — the world now produces more than enough food to feed everyone comfortably, for the first time in history. The crisis lies in bad distribution due to corruption, war and other badness, not due to production.)

  19. One of my proffessors in college made all his students read a great book just on this very matter. It was called ‘Innumeracy: Mathematical Illiteracy and Its Consequences’ by John Allen Paulos. Fantastic book.

  20. Elbie says:

    Besides, if you don’t have a good background at math, you’ll never be accepted socially.

    What kind of backwards misfit would you be if you don’t get the punchline to the timeless classic, "What’s purple and commutes?"

  21. Dr. Orbifold says:

    Again, I go away from the internet for two whole days and I miss good posts like this…

    Good post Eric, although I have to agree with Zac about the mathematics vs. algorithm design part. Keep in mind that my own research uses computers more than the average mathematician, so algorithms play a larger than average part in my research.

    If I had to answer the question "What is mathematics" in one sentence, I’d say "Mathematics is composition". Mostly that would be a ploy to annoy the questioner into giving me more sentences in my follow-up, but it’s still an honest answer. Mathematics is about describing things, either the world around us or the world inside our heads, in the language of algebra and logic. It’s much closer to the liberal arts than any other branch of science.

  22. Dave Anderson says:

    • "the world now produces more than enough food to feed everyone comfortably, for the first time in history. The crisis lies in bad distribution due to corruption, war and other badness, not due to production"

    An enligtening corollary: "If the solution for AIDS were to bring a glass of clean water to everybody in the world, we would not be able to do it." — Dr. Joseph Decosas, 1996 International Conference on AIDS

    • Didn’t Paulos also write "A Mathematician Reads the Newspaper"?

    • As for the Abelian grape, I’m still looking for a good math joke that begins with: What’s xxx and telecommutes?

    • Great topic, Eric.

  23. CB says:

    My point relating what you (Eric) said to ecological crises wasn’t that one caused the other (in either direction). It’s more that I wonder if *both* come from a widespread swing away from the messy complexities of rationality, with all its requirements of evidence, probability, and balance, towards simple utilitarian calculus as a primary decision-making tool. In education: as you say, only teach me what will tend towards a higher-paid job. In global politics: do whatever we believe results in GNP growth (regardless of other consequences).

    A small nitpick re your nitpick: global food stocks are now at a living-memory low (about 3 months, the last I read). Average per-hectare yields have been declining now for over 20 years as modern farming techniques have been unable to continue to compensate for soil degradation, reduced water availability, and climate change in many parts of the world. China, self-sufficient in food for many years, is now a net importer because it’s wheat belt is being swallowed by the Gobi desert which is growing at 1000’s of square km’s per year. Distribution was and is probably still the primary problem, but production is becoming increasingly an issue as world population increases at the same time as the amount of available productive land, in decent condition, and in areas where the climate conduces to large-scale production, decreases.

  24. Eric Lippert says:

    Ah, I understand your point now.

    And indeed, no matter how you slice it, food and resource production and distribution are going to be a mess as the world population doubles one last time.

  25. Cassandra Thompson says:

    You make some good points. Still I understand why the math teacher asks this question. There are many more reasons then just the ones that you address.

    I would the chance to ask her what sort of responses she recieved (, as I am in the process of gathering the same sorts of things.

    We use a pile of bus timetables to help students understand networks and spanning trees. See an application is a great memory tool. A teacher knowing an application exists and being able to implement it in the classroom is also a great learning tool.

  26. ines says:

    my teacher gave me like a homework to find out some info for coding in math.i didnt know nothing about it because im still in the 10th grade in collage.but im very interested to know about this topic and i wish ill do the best for it!

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