Good Math, Bad Math

What is math?

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I’ve got a bunch of stuff queued up to be posted over the next couple of days. It’s
been the sort of week where I’ve gotten lots of interesting links from
readers, but I haven’t had time to finish anything!

I thought I’d start off with something short but positive. A reader sent
me a link to a post on Reddit, with the following question:

Throughout elementary and high school, I got awful marks in math. I always
assumed I was just stupid in that way, which is perfectly possible. I also
hated my teacher, so that didn’t help. A friend of mine got his PhD in math
from Harvard before he was 25 (he is in his 40′s now) I was surprised the
other week when I learned he isn’t particularly good at basic arithmetic etc.
He said that’s not really what math is about. So my question is really for
math fans/pros. What is math, really? I hear people throwing around phrases
like “elegant” and “artistic” regarding math. I don’t understand how this can
be. To me, math is add, subtract, etc. It is purely functional. Is there
something you can compare it to so that I can understand?

This hits on one of my personal pet peeves. Math really is a beautiful
thing, but the way that math is taught turns it into something
mechanistic, difficult, and boring. The person who posted this question
is a typical example of a victim of lousy math education.

So what is math? It’s really a great question, and not particularly
an easy one to answer.

You’ll get lots of different answers depending on just who you
ask. It’s a big enough thing that you can describe it in a lot of
different ways, depending on your perspective. I’m going to give
my own, and you can pipe in with your own in the comments.

To me, math is the study of how to create, manipulate, and understand
abstract structures. I’ll pick that apart a bit more to make it more
comprehensible, but to me, abstract structures are the heart of it. Math
can work with numbers: the various different sets of numbers are
examples of one of the kinds of abstract structures that we can work
with. But math is so much more than just numbers. It’s numbers, and
sets, and categories, and topologies, and graphs, and much, much more.

What math does is give us a set of tools for describing virtually
anything with structure to it. It does it through a process
of abstraction. Abstraction is a way of taking something
complicated, focusing in on one or two aspects of it, and eliminating
everything else, so that we can really understand what those one
or two things really mean.

For example, look at topology. Topology is basically a way of
understanding shapes. But it does it in a completely abstract way. It throws
away everything except the concept of closeness. You have a
collection of points, and you’ve got a concept of things that are
close to one another, defined in terms of neighborhoods. By
playing with different notions of what things are close to each other, you can
create any shape you can imagine, and some that you probably can’t. But you
don’t really need numbers at all: you can just create and play with shapes in
topology – as long as you’ve got the set of points, and you’ve got set
relations, you can figure out what it really means for something to be a
torus. You can see what’s really strange about a moebius strip. You can
take the moebius strip, and add a dimension to it, and see exactly how you
produce a klein bottle.

For another example, look at category theory. It’s a way of understanding
function. What’s a function? At it’s core it’s a mapping from one
thing to another. But what does that really mean? What can you do
with that basic idea? What can you make with it? The answer is:
virtually anything you can imagine.

But math is more even than just those abstract things. Why does music
sound good to us? Because it’s got an underlying structure. That structure
can be described mathematically. Personally, I’m a huge Bach fan. I believe
that he was the greatest composer of music that ever lived. His music
is magnificently beautiful, and incredibly moving. But to really understand
it, to really grasp all of what he was doing in his music, you need to understand
that it’s structure on structure on structure on structure. That structure
is mathematical. If you’re really understanding the structure of Bachs music -
if you sit down and analyze it, you’re doing math

When you look at a cubist painting, you’re looking at a strange kind of
projection of something. The artist has taken the subject of the painting
apart, viewed it from different perspectives, different points of views,
different ways of understanding it or seeing it, and assembled them together
into a single image. When you look at a cubist painting, and try to understand
what the artist was seeing, how they were seeing it, and how the pieces
of the final image really fit together – you’re doing math.

When a scientist tries to analyze something about the world, to understand
how it works, and describe it in a way that tells us something important about
how things behave – they’re doing math. They’re abstracting the
world to come up with a precise, formal, descriptive way of stating what
they’ve learned.

When you look at a road map, and figure out how to get from one place to
another – you’re doing math. The map is an abstract representation of
the world that allows you to do certain useful things with it. (And frankly,
this is one example that I’ve never been able to understand. I can’t read
maps. Quite literally a bit o’ brain damage – some scar tissue in the left
frontal lobe of my brain.)

When a jazz musician improvises, part of what they’re doing is
math. For an improvisation to make sense, for it to sound good, and fit
with what’s going on around it, there are a set of constraints on it:
on pitches, pitch progressions, rhythm, chords. Those are all abstract
properties of the music, which are mathematical!

Math is unavoidable. It’s a deeply fundamental thing. Without math,
there would be no science, no music, no art. Math is part of all of those
things. If it’s got structure, then there’s an aspect of it that’s
mathematical.

Comments

  1. #1 James Sweet
    December 7, 2009

    I excel at memorization and at mental arithmetic — and I have nothing but contempt for pedagogical approaches which emphasize these.

    The ability to do some basic mental arithmetic is useful, because you don’t want to have to pull out your calculator to do 8 + 6… but much beyond that, and who freaking cares? And memorization, jeez… with a handful of exceptions, anything worth memorizing will be something you use often enough to memorize it naturally.

    My gift for mental arithmetic has actually helped me on my job maybe two, three times, tops, and even then I cannot say I might not have come to the conclusion eventually anyway. Mostly it is just a parlor trick, and a boring and tedious one at that. Which is too bad, because I’m really quite good at that useless and boring parlor trick! :D

  2. #2 Zombie
    December 7, 2009

    Math is a language for reasoning, is the way I’ve tended to describe it.

    (As for memorization, anything worth memorizing is worth learning to derive…)

  3. #3 woupiestek
    December 7, 2009

    As mathematical logician I must disagree that analyzing Bach and jazz and cubist painting is mathematics. Math is the science of laws and concepts, and their logical consequences. A mathematician does not study ideas, but works out the ultimate consequences of the concepts and laws he derives from them. This is the only way, that mathematics can ever be what it is named for: that which we can know.

  4. #4 Alan B
    December 7, 2009

    James: I mostly agree with you – I’ve never memorized any arithmetic stuff intentionally (only naturally), but I think there is some use to it. Maybe its a byproduct of all those math competitions in grade school and high school.

    I can recite (or recognize) small squares, cubes and integer powers (I know up to 2^16 off the top of my head thanks to being a programmer – some people are absolutely astounded by this. I could compute them almost as fast as reciting them, and that would still astound people), as well as decimal expansions for most fractions with denominators 16 or less. I also know how to think in log space as opposed to linear space, and I can multiply 2*8192 by decomposing it into 2*(8200 – 8).

    Just these three things (recognizing easy numbers, thinking in log space, multiplying any numbers to 3-4 digits) probably account for about 95% of the mental arithmetic I can do. Combining them allows me to do a lot more, still quickly. In particular, I can make quick calculations in my head, in the middle of a (technical) conversation. Or while writing out long answers on an exam. I’m done with college, so multitasking on a test isn’t a big issue anymore, but I can guarantee that it helped my GPA by a few tenths of a point. Maybe it’s more of a parlor trick now that I’m out of school, but I’m definitely glad I had the ability back then.

  5. #5 t3knomanser
    December 7, 2009

    I regret not learning more math when I was younger; the more I develop as a programmer, the more I wish I had a better foundation in mathematics.

    And your pet-peeve is bang on: math is taught as a process, a formula, a routine. This ties into my complaints about education in the US in general, specifically the idea that it’s essentially a factory process, and all the education reforms haven’t really done all that much to change that.

    I find it incredibly frustrating that attempts to popularize mathematics among youth try to find something kids like and then cram numbers into it. “(American) Football counts yards, let’s contrive some problems about moving around the field.” The same rote process, in a different outfit. Kids may be young, but they’re not stupid.

    Ideally, mathematics should be taught as abstract problem solving, and the details of arithmetic should be taught to serve that end. And times tables should all be burned and consigned to the dustbin of history.

  6. #6 Paul W. Homer
    December 7, 2009

    Looking at math in terms of its structure is a reasonable approach, but I still tend to see it in the way that Hofstadter presented it: as a formal system.

    And a formal system is an axiomatic approach to modeling something in abstraction. Ultimately we create these ‘formalized’ intellectual models and then relate them back to the real world somehow. Our models can be perfect (in the sense that they are complete and not self-contradictory) but our mappings back and forth to reality may be faulty. Still we cannot say anything conclusively about reality unless we map it onto some type of formal system first.

    From this perspective I tend to see the various branches of mathematics as just being large contained sets of axioms. Independent formal systems. You map reality onto the axioms, mix and match a few more, and then map that back to reality to make a statement. 1 + 1 = 2 in that sense is an arithmetic model using three (or four) axioms that helps me find the symbolic name for the number of apples I now have. To me the fascinating part is the formal system, and how it is mapped. The actual number of apples is irrelevant.

    Paul.

  7. #7 Maya Incaand
    December 7, 2009

    “Geometry without algebra is dumb and algebra without geometry is blind”.

    This is the same sort of thing, a dumb calculator (programmer) or a blind abstractor (or is it the other way around?).

    You need both skill sets if you want to call yourself a mathematician.

  8. #8 brazzy
    December 7, 2009

    School teaches mostly *practical* math, i.e. math as a tool to solve real-world problems, with the underlying theories as an footnote.

    And I see absolutely nothing wrong with this – because for 98% of the kids, this is the only kind of math they’ll ever need, the kind that might actually help them in their daily lives.

    The whole abstract stuff – definitions, theorems, and proofs – may be so much more satisfying if you’re really into it, but it can also be deeply frustrating if you’re not.

    I started out studying math at university because I’d always gotten top grades in school. To my horror, I found that A) university math was all about proving theorems, with practical applicability often not mentioned even once during an entire semester, and B) I sucked at proving theorems. Fortunately, I also found that C) computer science, my secondary subject, was different.

    So I switched primary and secondary subject. And during my remaining math courses I had one very interesting experience: a lecture in group theory described an algorithm (for enumerating cosets, IIRC). I later mentioned to the prof that this was my favorite part of the course because as a CS major, algorithms were more familiar to me than theorems. He laughed and said that many math majors had problems understanding that part…

  9. #9 Dan M
    December 7, 2009

    Edward Nelson once wrote (or maybe he wrote it several times) that “Mathematics is the invention and investigation of formal patterns, and good mathematics is the invention and investigation of deep and beautiful formal patterns.” I would add also the “participation in formal patterns” to cover activities like folding origami, painting, playing music, and the other activities you mentioned that have a formal component.

  10. #10 Scott Hanley
    December 7, 2009

    Some years ago I read about a junior high school math teacher who taught “magic formulas,” a la Dungeons & Dragons. That was how he portrayed equations for R,T, & D problems, etc.: “Now apply the magic formula!” I recall thinking that it was a pretty spicy take on rote formulas and may have promoted interest in some of his students, but it was still rote and I wonder how many of the students ever understood why those formulas worked.

  11. #11 Joe Shelby
    December 7, 2009

    woupiestek [#6] – I don’t think you really understand how music is mathematics. I fought that concept for years but I do understand it now. Music composition IS a formal system. A formal system with a lot of options, and there is never a sense of a “final”, except within rules we’ve invented for ourselves. Diatonicism is one of those rules – that every “chord” must progress to the next chord by having at least one “voice” move by the major fifth and that the music should end with a chord based on the home tonic key. These are rules, and yes they are arbitrary, but they are based on evidence: it sounds good, doesn’t it?

    The circle of fifths and the chromatic scale that results is another product of a formal system, as well as a base for producing a new formal system (or many – diatonicism is one, pure chromaticism is another, serial composition a third). But it is not the only result of acoustic mathematics, as you can ask anybody from a culture that only worked in pentatonic scales, or a culture that developed music around micro-intervals, or a culture (gamelan orchestras, for example) where the overtones of a note are more explicitly the intention of a sound and the real music is in the interaction of these overtones rather than the interaction of the intended note.

    In fact, Schoenberg who tried so much to finally free music from the limitations of diatonicism, that composition became impossible: there were simply too many choices. To resolve this, he had to introduce the basics of serialism in order to give himself some formal constraints. Again, it wasn’t the only new rule set one could have done in that circumstance (Varese went in another direction, and Stravinsky in yet another, Cage in yet another, Reich in yet another), but it was a rule set that gave a mathematical order to composition.

    The constraints of music composition are rules, and they are mathematical, both as a result of acoustic science, and as a formal system for others to build works, or more rules, or even alternative rules, upon.

    Music “analysis” which Mark mentioned, is the study of composers, compositions, and those rules. This is different from music “appreciation”, which is more a study of it from an aesthetics approach. Appreciation is the study of the ideas of music, but analsys is explicitly the study of the maths of music.

  12. #12 Flex
    December 7, 2009

    I’m often amazed when I find people who’ve gone through college and haven’t figured out that a lot of the math is duplicated.

    What I mean is, the same bell curve found in, say, failure analysis is found in biology, anthropology, metallurgy, crystallography, etc. Any time you use statistics you find bell curves.

    The same is true with the other basic curves. The linear curve and power curves show up all over the place in all fields. The s-curve is the same in electrical hysteresis as toxicology.

    And yet plenty of people have never figured it out. I can’t understand how someone could be great at the statistics used in say, plant growth variation, but can’t get their head around the statistics found in the metal casting variations. The math is the same!

    I don’t know if it would help, but I’ve often thought that a semester long course on the basic curves found in natural phenomena, i.e. linear, power, exponential, bell, and ‘S’, would enable people to recognize when these curves pop up during everyday life. Maybe if people could see how ubiquitous these simple curves are they wouldn’t be quite so afraid of them.

  13. #13 Mary
    December 7, 2009

    I agree with the person above who said math is a language, which happens to be good for reasoning in. I took a “Formal Logic” class in the philosophy dept in college. I believe math is just an extention of Formal Logic. The only advantage is that while there may be a million ways of saying the same thing in English, and a million ways of saying something different which only sounds the same, in “Math” or “Formal Logic” you can usually tell whether you’re repeating yourself or contradicting yourself just by performing some symbol manipulation. It reduces ambiguity and makes it more obvious what statements follow from one another, and what statements contradict one another. That’s a powerful tool.

  14. #14 Chris Whitman
    December 7, 2009

    I went for drinks recently with some other folks from the math department here, and the topic of ‘the mathematics of music’ came up. The general consensus was that the math involved in music is generally pretty trivial and uninteresting.

    There are a few ratios involved in which intervals sound more consonant together, and there are some predictable patterns in the resolution of chords, etc., but the latter are really more a matter of musical convention and don’t particularly arise from some deep mathematical concept. Rhythmic conventions like polyrhythm still don’t really use anything more complex than ‘counting.’

    So it irks me a bit that it always comes up as an example. I imagine it’s the same feeling a practised painter would have when someone says, “Oh, you’re an artist? Can you draw a picture of my cat?”

  15. #15 Chris Whitman
    December 7, 2009

    Just to clarify here: I’m not suggesting that music itself isn’t an interesting thing to study from an academic standpoint; I’m simply saying that the math behind it is nothing to get particularly excited about.

  16. #16 Robert
    December 7, 2009

    “What is mathematics?” by Richard Courant, Herbert Robbins, Ian Stewart

  17. #17 Pete
    December 7, 2009

    I’m with Brazzy. Teaching abstract math would be great for some, frustrating for most.

    I have two daughters, aged 11 and 13. The 11 year old resists math, not because she’s bad at it but because she just doesn’t see why this is useful.

    If we tried starting with theorems and proofs, she’d be even more in the “Dad, why are we bothering with this?” camp. It wouldn’t be beautiful or fascinating, it would be pointless and irrelevant.

  18. #18 Paul Murray
    December 7, 2009

    Mathematics and art have in common that they are the exercise of creativity within constraints – in mathematics, those constraints are logical truth. In art, it’s the medium you are working in. Maths, coding, painting, composing and performing – they are all about knowing the rules and then using those rules to

  19. #19 Mark C. Chu-Carroll
    December 7, 2009

    @14:

    I think that the real picture of math and music is rather the opposite of what you said. That is, a lot of mathematicians look at the shallow structure of music, and assume that they’ve grasped the whole thing, and conclude that therefore it’s trivial.

    Things like the fact that the harmonic fifth is 1 1/2 times the frequency of the tonic? That’s trivial? That the musical scale can be understood as a mod-12 algebra? Slighly
    less trivial, but still not terribly interesting. The fact that the development section of most symphonic forms builds on symmetric transformations of the melodic and chord structure? That’s quite a bit more interesting – especially because some of the symmetries are really interesting.

    Look at the musical structure of a piece like Bach’s endlessly rising canon, and tell me that it’s not interesting. You could easily spent weeks studying all of the structure of that piece.

  20. #20 Gray Gaffer
    December 7, 2009

    What you all said. Another one of those “everybody is right” things.

    Except about rote. At least, one particular rote, and that is the times tables. Along with orders of magnitude and units of measurement, they form the basic tripod of back-of-envelope calculations on anything and everything, from target cross-section to tipping your waiter and making change, and should be ingrained at a young age. All the rest is pretty much derivable on the spot from first principles should more be needed, but the quantification aspects of math are not, and having them mentally available as lookup tables is invaluable. I got mine via 1/2 hr chanting every morning 1×1 to 20×20 from ages 5 to 8.

    The other stuff I think needs to be in everybody’s curricula before the age of 15 are predicate calculus and proof systems. Knowing how to abstract into symbol sets and then how to reason about them is essential if you are to have a chance to avoid getting seriously scammed later in life. Or even just how to work debates or arguments. Only perhaps some friendlier names are needed. Not difficult stuff at all – at my HS we all got that at age 12 – it just got a bad rep.

    Later stuff not so fundamental for general life. But this as early stuff is essential. I think the fundies are making such headway these days because too many people got shorted on early logic and don’t know what they should about thinking for themselves. So they don’t know how to spot fallacies, how to spot false conclusions or derivation steps, or even to recognize there are missing steps. So to those kids, yes, there is a use for it in life, pretty much every day, whenever somebody asserts something as “true” as justification for some behavior, or enters into a financial transaction with you.

  21. #21 Richard Leon
    December 7, 2009

    @19:

    Conversely, some mathematicians think music is all structure, when in fact it’s all experience – just like all of the other arts. Which is why we’ve had a century of mathematically precise formal composition, after Schoenberg and the rest, which has abstracted itself into irrelevance, because so many of the experiences it creates are sterile and unrewarding.

    In art, you don’t truly understand what you’re doing until you begin creating memorable, engaging experiences. Deconstructing Bach’s canons tells you next to nothing about Bach’s music, because the experience inhabits the structures, but isn’t exclusively defined by them.

    Trying to understand Bach mathematically is like learning the grammar of a language without knowing what any of the words mean.

  22. #22 cfranc
    December 7, 2009

    Here is a wonderful article of John Baez discussing math and music.

  23. #23 Sara
    December 7, 2009

    I wish Math were something that it was more ok to just learn. I find it fascinating and I loved doing it in school but got pushed in a not hard science way and am sad to miss it. It is surprisingly difficult to work math skills and learn more in a day to day life. Basic math (heLLO budget spreadsheets!) sure but abstract math? Apparently it isn’t a sexy hobby. So I practice it quietly and in tiny bits in knitting and writing sci-fi and MMORPG’s (mmm theory crafting weirdly the only thing I miss since I’ve quit WoW) and failing to learn to program. But I wish there were as many math groups meeting weekly as there are knitting groups. Some day it will be hip!

  24. #24 Abie
    December 7, 2009

    This reminds me of “Lockhart’s Lament“.

    PLT and Bootstrap claim to have had success in motivating students and teaching algebra via functional programming and animation.

  25. #25 rimpal
    December 7, 2009

    Mark,

    I would go further to say that Math(s) is about creating tools, rules, axioms etc to find or lend structure to anything.

    Of course music is mathematical. Here’s a link to an article on integer sequences in the Indian system of musical scales or Ragas
    http://bit.ly/6huTiw

  26. #26 Joe Shelby
    December 7, 2009

    Richard [19]: Saying music is all experience no math is just as bad as saying music is all bad and no experience, and neither is true. Art and aesthetics is all about layers of meaning, the depths to which one can look at a piece. In some cases the depths, the layers, are intentional at all levels by the creator, especially in matters of literature and poetry.

    In other cases, the mathematics can provide a means of, for lack of a better term, “automating” layers. It is this understanding that made serialism even possible, but just like those seemingly (and *only* seamingly) arbitrary rules of Schoenberg and Boulez make one realm of composition possible, the rules of diatonic tonality, perfected by Bach, worked to provide composers from his time through today with a toolset for incredible variety and complexity out of those exact same 12 notes. Composers work to a formal system, limited by the rules of sound and the abstractions to handle it as much as painters are limited by the rules of color – you can’t paint what can not be seen.

    The music listener, and perhaps the mathematician who is a music listener, need not know or study the maths of composition to appreciate the results, and nobody has asked them to or required it of them.

    But the musician DOES need to know those rules, and learns those rules whether by learning the maths directly (through classes) or indirectly (through just listening to enough works to get the patterns).

    Music listening is often pattern recognition, as is most forms of folk music composition (including Rock). Pattern recognition is itself another form of mathematics, as anybody in programming and A.I. can tell you.

  27. #27 Joe Shelby
    December 7, 2009

    correcting first sentence (it looked fine when i read it…amazing how blind we are to what we type until it is final…):

    Saying music is all experience no math is just as bad as saying music is all math and no experience, and neither is true.

  28. #28 David
    December 7, 2009

    Great post. In a way, our awful rote approach to math education (and to science education generally, but not as extreme) is what’s responsible for science illiteracy in the US.

    A while ago I was talking about something like this with my Mother-in-Law, who claims to be a “people person” (meaning she thinks she hates math). She doubted me when I claimed that great math is beautiful. So I showed her Euclid’s proof that there are infinitely many primes. What’s lacking in early math education is the opportunity for that “aha!” moment, which (IMHO) is what motivates most math nerds.

  29. #29 lucky7
    December 8, 2009

    The way I describe math is the fractal-H-diagram. Math is the study of objects (points on the diagram), their relationships and/or pattens (lines linking points), and recursively applying that rule (lines linking lines and lines linking those lines, and so forth…) The diagram looks like a fractal H where each end point is a smaller copy of the H-diagram.

  30. #30 Lucky7
    December 8, 2009

    just 2 more cents. For me the pillars of math, is Abstraction, Reasoning, Calculation. What I call ARC. In school we mostly learn is Calculation, and some reasoning and preciously little abstracting. So I often meet people and students that have very little idea of abstraction. We need to have students that are well rounded enough to be equally familiar with all three. And not think calculation is end all be all.

  31. #31 Pareidolius
    December 8, 2009

    I just turned 50, so “new math” was just coming into vogue when I was entering grade school in the mid 60s. The teachers were uneasy with it, my parents, while very educated, were intimidated by it and I just hated it. By the end of high school, my English SAT scores were the highest in the county, my math, among the lowest. I really think I might have gone into science if I’d been introduced to math as something beautiful, something that was the language underlying all the natural sciences that I was so passionate about as a child.
    Any great books for adults who want to learn math but always hated it?

  32. #32 Sangcheol
    December 8, 2009

    Keith Devlin, The Math Guy at Stanford, called math as “science of patterns.” Yes. Math is quite different from other scientific disciplines as it’s quite detached from real world. However, I think the phrase really conveys the sense of what the heck it is in most intuitive, yet less wrong way.

  33. #33 Martin
    December 8, 2009

    slightly OT – but I wonder why you lot across the pond call it “math” and we Brits call it “maths”. It’s short for “mathematics”, which is plural – so “maths” should be plural too. At least, that’s how it seems to me.

  34. #34 woupiestek
    December 8, 2009

    When someone presents the analysis of a mathematical structure, like real analysis is an analysis of real numbers and real valued functions, and proves a counterintuitive theorem, then I will say that my intuition is wrong, and that the real numbers are what the axioms imply. When someone presents the analysis of a musical structure, the opposite happens: a counterintuitive conclusion would lead to the rejection of the analysis, not of the music. This is how mathematics is different from music, or anything else. And even though music can be interesting for it’s structure, it isn’t mathematics.

    I don’t like simple minded criticism of Schoenberg, Selby. Try listening to his music first, and then tell me something about it I haven’t already heard.

  35. #35 Joe Shelby
    December 8, 2009

    I wasn’t criticizing Schoenberg – I was more giving a somewhat simple introduction his approach as an example of a composer introducing a new formal system, a new “math” to music.

    Thanks to how “middle” maths (geometry and calculus) are taught, we get very used to the idea that proofs, axioms and rules, necessarily must lead to a specific goal – the theorem to be proved. It is what we’re asked to do on every test we take.

    Music is a formal system with *no expressed conclusions* – rules are applied to axioms (those axioms might be a tone row, or a motif, or a pentatonic scale, or something not limited by the 12 chromatic tones we currently use) to produce a work, but though the work is the composer’s intent, the final nature of the work might not – music can go in different directions. A composer’s work is often to undo music already written because it “isn’t right” – it doesn’t quite reach the idea the composer is trying to express, or it “just doesn’t sound good”. Bernstein’s discussion of Beethoven’s false starts in the 5th Symphony [The Joy of Music, broadcast on CBS's Omnibus series back in the 50s] can be quite illuminating in this.

    In fact, much of the criticism of post-Boulez serialism (especially of Carter) is often about how their efforts in making a piece of music work “mathematically” often got in the way of creating a work of any perceivable beauty. Analyzing the work reveals more beauty than actually listening to the damned thing. :)

    Other composers rejected serialism expressly because, as one put it [paraphrased, and I can't recall where I read this], “I didn’t want to be a computer programmer.”

    BTW, I happen to like Schoenberg’s work. I don’t “love” it (among serial composers, i prefer Berg, Takemitsu’s middle years, and Stravinsky’s late output), with the exception of Survivor in Warsaw, where his sprechstimme fits most appropriately.

  36. #36 James Sweet
    December 8, 2009

    I know up to 2^16 off the top of my head thanks to being a programmer – some people are absolutely astounded by this.

    Before I became a programmer, when I was just a kid in elementary school or junior high or something, I memorized the powers of 2 up to 2^20 or so just for fun. Little did I know it would come in handy so much years later…

  37. #37 James Sweet
    December 8, 2009

    It’s funny, Alan B, except for the thinking in log space, those are exactly the same tricks I use.

    The trick for the decimal expression of fractions with 7 in the denominator’s a pretty cool one, eh? That always impresses the shit out of people, and yet you only have to memorize six numbers. Awesome.

  38. #38 AJ
    December 8, 2009

    Thanks Mark, for the interesting topic. To me, the beauty and interesting part of mathematics is not the numbers, computations, or even the analysis, but rather in recognizing the patterns and structures. As some have pointed out, our education often falls short of this creative element of math.
    To contrast and show our faults in math education, consider language arts. Here we teach grammar and spelling then consider the creative side with composition and literature.
    It wouldn’t be very interesting to diagram all the sentences of a great novel, or to describe a great piece of music with numbers and arithmetic. The beauty and interest are in the structures and patterns.

  39. #39 Thony C.
    December 8, 2009

    When I studied maths my exercise partner was a more than somewhat genial Yugoslavian who had attended a special school for gifted mathematicians in Yugoslavia. He told me of a guest Czechoslovakian professor who said in a lecture that mathematics was the only science that had no need of numbers.

    Having spent ten years formally studying history and philosophy of mathematics and many more years informally investigating the same I can only say that trying to define mathematics is a hopeless task. However having said that I find that the best simple definition for normal use is that mathematics is the science of order.

  40. #40 cm
    December 8, 2009

    To all those who are saying that innumeracy or lack of scientific thinking in students is the result of “rote” approaches: I doubt it.

    I have a feeling that if one were to try to teach the “beauty” and elegance of math (and science), it would work out just as poorly. Students are uninterested not because it is taught wrong (though sometimes it is, or poorly), but because they are limited in their ability to grasp any difficult concept, and find schoolwork generally a drag. This is because they are children, they have XBox waiting for them at home, have a text message coming in, etc., etc.

  41. #41 Beth Welsh
    December 8, 2009

    Mark CC, this was a fascinating article and comes at a time that I find myself looking for a way to teach math to my son without a battle. My son is an HG kid and figured out the basics of multiplication one night while playing with legos (he was 5). But he detests math worksheets. While he can figure out addition or subtraction for practical purposes, he won’t do the repetitious practice that makes us memorize certain basic calculations. My spouse and I have gone rounds about this because hubby loves math and does calculations in his head for entertainment. I sort of understand our son because while I can do basic math more quickly than most of my co-workers can grab their calculators, I wouldn’t call myself great at math. There are times I detest it; too much work with numbers frustrates me and too much frustration will make me cry.

    So I pose this question to you: How does someone who received the typical boring math education, and who sees math as number-based, change the mind-set enough to fully comprehend what you’re saying here? I mean, you made it sound beautiful and fascinating and all, but I frankly don’t really understand or have any practical reference other than the beauty of the language — which is something you even admitted you’d have difficulty with. What is more, how do I help my son receive a math education that isn’t boring or dull or driven by rote operation when it is a. all I know, b. the primary offering of every public school and therefore most boxed curriculum, and c. a requirement of law that governs education in our state that he be able to do those basic functions? If you can answer that and make me understand your answer, I’ll be forever indebted to you.

  42. #42 SharonO
    December 8, 2009

    I have always been an enthusiastic musician, but I was never comfortable with math. I didn’t ‘get’ it, and only through sheer determination and cussedness memorized and repeatedly solved problems enough to manage through to calculus.

    But then, I was delighted to find interesting ‘mathy’ concepts in my music theory classes in college; finally a reason to care! Followed by wonder at the revelation in graduate school for speech pathology that lab analysis of the resonance properties of voice is quite useful for evaluating speech therapy treatment protocols! Turns out, all those resonances are based on the physics (and thus the explanatory math) of how sound travels, how our auditory structures are built to collect it, and how our neurology perceives voice… reproducible and predictably influenced under experimentation. All this math and science that I was destined to get excited about…unforetold by nary a glimmer of the ‘aha’ experience in twelve years of uninspiring secondary math education. I’m sure that disconnect discourages many a potential.

  43. #43 Furcas
    December 8, 2009

    Is there anything that doesn’t have structure?

    What does it even mean to talk about something structureless?

  44. #44 JMatt
    December 8, 2009

    I agree. The way most people understand what math is and the way we teach math is horrible. The math curriculum is a disservice to everyone that’s been exposed to it. I think mathematicians and scientists alike are realizing this and beginning the long process of fixing it in schools and eventually the culture. Some of the most creative people in the world today are not musicians or artists but rather mathematicians. (and as a corollary computer scientists, engineers, etc.)

    I recommend “A Mathematician’s Lament” by Paul Lockhart [1]. He further investigates what math is, how we can better teach it and how it’s portrayed in our culture.

    [1] http://www.maa.org/devlin/LockhartsLament.pdf

  45. #45 abb3w
    December 8, 2009

    File me in the “language of structured abstraction” region; it might also be worth adding “self-consistent”. Mathematics usually avoids the areas where FALSE and TRUE mean the same thing.

    Mary: The only advantage is that while there may be a million ways of saying the same thing in English, and a million ways of saying something different which only sounds the same, in “Math” or “Formal Logic” you can usually tell whether you’re repeating yourself or contradicting yourself just by performing some symbol manipulation.

    Unfortunately, Rice’s Theorem says that telling if you’re repeating yourself is mostly equivalent to the halting problem; you can sometimes find solutions in particular, but not in general.

    Beth Welsh: What is more, how do I help my son receive a math education that isn’t boring or dull or driven by rote operation when it is a. all I know, b. the primary offering of every public school and therefore most boxed curriculum, and c. a requirement of law that governs education in our state that he be able to do those basic functions

    The basic functions are important; however, there’s more to math.

    Some possibly suggested reading:

    Godel, Escher, Bach; still in print, and most used bookstores can turn up a copy.
    Asimov on Numbers; lamentably out of print, but your local library might be able to help you borrow a copy.

    Also, sometimes exposure to some of the odder results of mathematics sometimes will pique an interest, leading to further independent study. Beyond Godel’s Theorem (covered at a casual level in GEB above), other examples I can readily think of: Arrow’s Impossibility Theorem, the Banach-Tarski sphere dissection “paradox”, why angle trisection is generally not possible via compass+straightedge, the Four Color Theorem, Penrose aperiodic tilings, Bertrand’s Paradox, and Russell’s Paradox. Wikipedia has decent coverage on many of these.

  46. #46 Robin S.
    December 8, 2009

    Thanks for this article. I think the much-maligned Investigations curriculum for elementary math was an attempt to reach toward this level of understanding, but in my experience it has not been successful.

    To Sara — check out http://www.mathcircles.org/ to find people who love math and want to play with it.

    To Beth Welsh — my family has had some success with Math Dice, a cheap ($5) game that gave our competitive daughter a reason to learn her arithmetic facts. There’s an online version at http://www.mathdice.com/ — it doesn’t move you toward the goal of showing your son why math is elegant and interesting, but might provide some incentive for exploring a variety of operations and calculations. And, you might want to look at the Math Circles site as well — you might find a group appropriate for your son.

  47. #47 Tony Warnock
    December 8, 2009

    I would argue that there are quite a few interesting consequences to the idea that tones a fifth apart have a ratio of 3/2 and those an octave apart have a ratio of 2/1. As no power of 3/2 equals a power of 2, the entire problem of temperament arises, and there is no perfect solution.

  48. #48 Tony Warnock
    December 8, 2009

    Just one more thing, I have never seen a mathematical analysis of things such as the “shape” of a theme. For example, a “tonal” answer to a fugue subject has the same “shape” and is considered thematically “identical” but does not consist of the same sequence of intervals. An answer with the identical intervals is called a “real” answer. Both are essentially identical and both are often used in the same piece of music.

    The question of whether a theme (melody or figure or whatever) is “close” to another theme has implications for copyright. The different types of transformations of a theme isn’t easily described with simple mathematical operations. There are many classifications of “theme and variations” and lots of examples that fit none of the classifications.

  49. #49 Jonathan Vos Post
    December 8, 2009

    Related thread was about Sudoku-guy. To repost, slightly compressed and edited…

    Math, in its heart of hearts, is only about THREE things:
    (1) Quantity (most people only think of this);
    (2) Structure (some Geometry is badly taught in school);
    (3) Change (pre-Calculus, schools only teach Motion and Compound Interest).
    And all permutations (Enumeration of Structures; Structure of Change; …)

    I’ve used that with students as early as Middle School, and it usually startles them that no teacher had ever said anything of the kind before.

    Others would add to the list:
    (4) space;
    (5) relation;
    (6) pattern;
    (7) form;
    (8) entity.
    The others being as referenced below.

    However, I can still explain:
    (4) space, in terms of structure (for example, Euclidean space, hyperbolic space, topological space, Minkowski space);
    I can still explain:
    (5) relation, in terms of structure, using axiomatic development of relations and function;
    I can still explain:
    (6) pattern, as a type of structure in space, giving examples of
    integer sequences, tessellations, symmetries, orbifolds, …;
    I can still explain:
    (7) form, as I did pattern, within the more general “structure” — given that the structure may be very esoteric or abstract; and
    I can still explain:
    (8) entity, if only by saying that Mathematics is not about any specific entities at all, with the evasive “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” — Bertrand Russell

    Orthogonal to that, so far as I see, most teachers do not understand
    multiple levels of abstraction. Even in Math, which so much depends on this. Most teachers are clueless about how the human brain works. I keep saying to my students that science and math are about:
    (1) stuff,
    (2) structures of stuff, and
    (3) properties of structures of stuff.
    (that being a deep insight from Category Theory). And relentlessly connecting that to their own lives. And making it hands-on, collaborative, and fun.

  50. #50 Hephaestus
    December 8, 2009

    I managed to survive forty-seven credits of math in college and grad school (physics will do that to you). I’ve told people for decades that I’m terrible in math; I just happen to be terrible in forms of math that most people have never heard of.

    The problem that I have is not with abstraction. That’s my strong point. I have trouble with the formalisms. I can remember taking statistics and thinking that the semester was nothing but one long proof and at random intervals someone would slap a name on whatever point you were at in the proof.

    Standard deviation? Slap!
    Regression? Slap!
    Student’s t-test? Slap!

    There was no grace or elegance to be found anywhere. I’m sure that there is beauty in math, but no one ever managed to make it visible to me. Was that my fault? Probably.

    Oddly enough, I am good at the same kind of mental math that James Sweet talks about, with the additional trait of being very good at estimation. It impresses otherwise highly educated people to no end when you can say how thickly eight hundred gallons of liquid will cover a parking lot to within two percent without using a calculator. Apparently, even basic number manipulation is no longer taught.

  51. #51 The Dissenter
    December 9, 2009

    I enjoyed math in school up until I came across irrational numbers. They spoiled it for me. Here I was thinking that these relatively simple sets of rules essentially spelled out the secrets of the universe until one day my teacher showed me a number that went on forever. How can this be a number when it has no known value? Obviously “numbers” is a highly flawed concept if this was so. I had the feeling of the floor dropping away from beneath my feet. I can’t trust numbers at all now.

    Math is to reality what digital is to analog. Its a logical structural approximation that almost maps to reality but not quite. That’s why I don’t really agree with the “Bach is math” philosophy. Bach was writing music that he enjoyed and thought others would enjoy as well and that is why I like it, not because it might coincidentally have some mathematical looking scaffolding holding it up.

  52. #52 fractalwoman
    December 9, 2009

    – just found your site – love it! What math is and what makes it beautiful (at least in the eyes of some) is a lot to try to answer, but I’ll jump in and give a bit of what I see.

    Long story how I ended up majoring in mathematics – really didn’t like it that much in college – began to like it once I started teaching and began helping students prepare for competitions. We explored genuine problem solving, not just the working of exercises such as class was filled with, and I realized that mathematics is a big game or puzzle. In fact, it is the biggest game there is! That’s one thing that now draws me to it (though it often drove me to tears in college).

    In terms of beauty, to me mathematics and poetry have a lot of similarities. This won’t help someone who doesn’t like poetry to see the beauty of it, but it appeals to me. Once you get beyond arithmetic and algebra, math really is pretty much all about proof. Just as there are some compelling types of poetry that hold irony or surprise, there are some extremely creative and surprising ways to go about proof. It is a creative process. As with poetry, the idea (typically) is to express the concept elegantly, briefly, effectively and powerfully.

    I read a great deal of poetry and have tried to write some, and, now that I’ve gone on and gotten an advanced degree, I’ve done a lot of proving, and – for me – they have the same feel to them – a striving after creativity, clarity and the “aha” moment.

    There are so many beautiful and intriguing topics in mathematics – knot theory, graph theory, fractal geometry, logic, set theory, and on and on – and it saddens me that all most children see is number crunching and some algebra. As some other commenters have pointed out, this is taught because it is what most people need in order to function – to balance checkbooks and be home-owners and so on, and that’s very true. That is the math everyone needs – just as everyone needs to learn grammar and spelling and punctuation. But, thankfully, in English class, we get to read literature and do some writing as well. We get a chance to see the potential that exists there – many genres, many options: from newspaper journalism to classic literature to poetry to love letters to trashy novels. There is lots to be explored and enjoyed in the world of literature.

    I just wish that alongside the necessary grammar of mathematics (times tables and solving for x) that students were also given at least a glimpse of fractals or the 4-color map theorem or knot theory or codes or . . . something, anything, to give them the bigger picture so that there can be appreciation, and perhaps interest, rather than just the necessities of life.

  53. #53 Roger Witte
    December 9, 2009

    I agree with much that has been said here but there is one other way of defining maths which is to define it in terms of objects and operations/transformations. Maths is the study of what operations on a space leave invariant and particular branches of maths study the invariants of particular sets of operations. This definition comes from geometry and works towards algebra wheras the structural definitions start at algebra and work towards geometry. (The definition in terms of formal systems starts at linguistics). I think that the three are equivalent.

    Moving to pedagogy: since my father retired he volunteers as a maths teaching assistant in a local school. He always tries to relate the problems to the pupils’ everyday life and the solution methodologies to solution methodologies they already know. In short he teaches ‘maths’ as using intellectual tools to solve problems. The other teachers tell us that after a term at my Father’s maths class the pupils improve in every subject: it isn’t just maths that benefits from learning to think effectively.

  54. #54 PT
    December 9, 2009

    Re: #14 I think the point of bringing music into discussions that start with the question of “how can math be beautiful” is self evident.

    Re: #8 I agree that most people only need practical tools, but I think more people would be attracted to the deeper beauty of math if they were exposed to it as something beautiful instead of merely as a tool. This truth seems to be reflected in many anecdotes like the one that began the article: math phd’s who are inept at the practical tool-level math that most people were exposed to exclusively.

    It seems to be that no one gets a phd in math because they love arithmetic. If this is true, or somewhat true, certain conclusions should follow…. Such as: Why should we hope to discover/support/develop future mathematicians in arithmetic classes? I would argue that we kill, not develop, our future mathematicians in these classes. The ones that survive surely do not outnumber the ones we destroy. And this is, I believe, the genesis of the reason for this article. People do not understand the beauty of math because we only expose it to those few who survive the boring, rote, uninspired ugliness of math for years and years, and who perhaps only discover the latent beauty of the subject by accident, long after so many creative, energetic minds have gone elsewhere.

    Math is just as much a part of the humanities as it is of the sciences. The history of the number sense, from the integers, to the irrationals and the imaginary and beyond is the history of human progress and should be taught as a part of Western Civilization 101. Likewise, Maxwell’s field equations are the crowning event of the 19th century and should be taught in every history book.

    We make a fundamental error in thinking of math in terms of science merely.

  55. #55 JimV
    December 9, 2009

    Math is thinking. When you decide whether it is better to go to the supermarket first and then the gas station, or vice-versa, you are doing math.

  56. #56 Nomen Nescio
    December 9, 2009

    I enjoyed math in school up until I came across irrational numbers. They spoiled it for me. [...] How can this be a number when it has no known value?

    i was about to point out that a lot of irrationals have very well known values, you just can’t write them out in decimal form and get a finite string, but then Mark has already talked about the indescribables.

    the set of reals is just weird in general when you start looking at it closely. whether you’re trying to approximate them in some format a computer can understand (and have to deal with the headaches of IEEE floating point), or just working with them abstractly, there be dragons in between the rational numbers.

  57. #57 Def-Star
    December 9, 2009

    Simply put, a great post.

  58. #58 Def-Star
    December 9, 2009

    @14 the trivial nature of the math in music is only as trivial as you want to make it.

  59. #59 Jonathan Vos Post
    December 9, 2009

    To give a small part of a long paper that I wrote:
    http://csmp.ucop.edu/downloads/cmp/math.pdf

    In general, these is much to praise and much to provoke annoyance in Mathematics Framework 2005. Mathematics Framework for California Public Schools, Kindergarten Through Grade Twelve (2005). Complete Mathematics Framework …
    ISBN 0-8011-1474-8…

    Developed by the Curriculum Development and Supplemental
    Materials Commission

    Adopted by the California State Board of Education

    Published by the California Department of Education

    http://www.cde.ca.gov/ci/ma/cf/

    JVP: I’d like to make a few quick comments, as to the seriously flawed definition of Mathematics in the opening sentence of the Framework, and then some philosophical comments (quoting David Corfield, Michael Polyani, George Polya, Terry Tao, Yau, Witten, Feynman, and others) on the relationship between Mathematics and Science, and then finally append one of the better papers (as an appendix, pp.11-28 below) which agrees with what I have observed about the difficulty of following the advice of the Framework with the urban Spanish-speaking students similar to those whom I have encountered in the Pasadena Unified School District’s middle schools and high schools.

    DEFINITION OF MATHEMATICS
    ========================

    “Mathematics—using abstract symbols to describe, order, explain, and predict—has become essential to human existence.”
    [Framework, p.4]

    I simply disagree with this definition. It is shocking to see such foundational confusion in the opening sentence.

  60. #60 Michelle
    December 9, 2009

    I’m a tutor as part of the ‘No Child Left Behind Act’ and am terrible at math. Except I’m great at basic math–which I have taught myself as an adult because the school system failed to detect a learning disability (they did kick me out of Algebra 2, literally walked me out, as a lost cause. Then I repeated Algebra I with a good teacher and performed like a well trained monkey but never understood what the hell I was doing).

    However, as much as I ‘do’ fractions, I still don’t understand the point of math. Aside from cooking and perhaps some construction projects, what is the point of fractions? And why does cross canceling work?

    What the hell does algebra do for me in my world?

    I asked a math teacher with a BS in Math and they didn’t know.

    So I still can’t say what math is because I’m still not sure exactly how useful it is once it gets more abstract.

    But I’m glad I found this blog. :)

    Michelle

  61. #61 bcoppola
    December 9, 2009

    paredolius #30 coulda took the words out of my mouth. I have to go somewhere now but I’d be interested as well in responses to his question and observations.

  62. #62 Jonathan Vos Post
    December 10, 2009

    {I’m guessing that my previous comment is stuck in the queue because of its two PDF links?}

  63. #63 Jonathan Vos Post
    December 10, 2009

    I agree, about half the time, that Math is MERELY a tool “to describe, order, explain, and predict”. The other half the time I partly believe the alternative philosophical position: Realism/Platonism (that Mathematical objects exist, but not in the same sense that stars and people exist), or that Math is a vast game/conspiracy/social construct for people great at Math to get money, sex, and power.

    SUBJECTIVELY it sometimes seems that I glimpse Platonic reality, and “see” a 4-dimensional shape or the like, but that is merely a psychological quirk — I think. As to social construct, one can say that a Triangle or a Divergent Series is real the qualified way that Washington State Law or a Black Queen in Chess is real, i.e. follows certain rules, which people can break by cheating.

    The deepest mystery is that Math has some use in solving physical problems, which reference to the evolution of our brains does not fully explain. Wigner famously inquired as to the “the unreasonable efficacy of mathematics in explaining the physical world.”

    Does Math — as a tool — “govern reality”? No, it does not. It is brainwashing by an academic elite to say that it does. My mentors, including Feynman, knew better, and rejected String Theory on that and other bases.

    I often adhere to my mentor’s dictum about Physics being about experimental fact, not algebraic slight of hand.

    The greatest error in that dreadful definition is the conflating of Mathematical Truth (axiomatic truth) with Empirical Truth (Scientific truth).

    Let me amplify on that by first quoting from David Corfield, a Philosopher of Mathematics with whom I correspond.

    Michael Polanyi and Personal Knowledge
    Posted by David Corfield
    June 28, 2008

    There was a discussion over at the Secret Blogging Seminar about the differences between mathematics and the natural sciences, which interested me greatly as someone who has frequently looked to the philosophy of science for ideas about how to treat mathematics. By and large Anglophone philosophy has chosen to treat these disciplines very
    differently, and has overlooked opportunities to elaborate their similarities, such as furthering George Polya’s Bayesian treatment of mathematics.

    One way to lessen the difference between the disciplines is to bring to centre stage the personal involvement of scientist and mathematician in their respective theories. In that each is a member of a tradition of long standing, each has to struggle against some intransigent reality, and to convince their colleagues that their perspective on this reality is a good one, they can be seen to have
    much in common. For some, however, the distinction between empirical evidence and whatever support a mathematician receives trumps any such consideration.

    Michael Polanyi in Personal Knowledge, written in 1958, while reflecting on this latter difference, seeks to understand it in the context of a general account of participation in a wide range of practices:

    The acceptance of different kinds of articulate systems as mental dwelling places is arrived at by a process of gradual appreciation, and all these acceptances depend to some extent on the content of relevant experiences; but the bearing of natural sciences on facts of experience is much more specifiable than that of mathematics, religion or the various arts. It is justifiable, therefore, to speak of the verification of science by experience in a sense which would not apply to other articulate systems. The process by which other systems than science are tested and finally accepted may be called, by contrast, a process of validation.

    Our personal participation is in general greater in a validation than in a verification. The emotional coefficient of assertion is intensified as we pass from the sciences to the neighbouring domains of thought. But both verification and validation are everywhere an
    acknowledgement of a commitment: they claim the presence of something real and external to the speaker. As distict from both of these, subjective experiences can only be said to be authentic, and authenticity does not involve a commitment in the sense in which both verification and validation do. (p. 202)

    Some participants in the Secret Blogging Seminar debate, such as Terence Tao, were keen to provide a continuum between mathematics and physics. An important point to note in this respect is that physics is guided by other than empirical considerations. Earlier in the book, pp. 9-15, Polanyi discusses how Einstein came to relativity theory more by way of what we have heard him call above validation than by verification. He continues:

    When the laws of physics thus appear as particular instances of geometric theorems, we may infer that the confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are
    cultivated.(p. 15)

    Again we are returned to the passionate, personal engagement of the scientist with their field. I’ll leave you with his diagnosis of the mistake which encourages us to discount this engagement:

    We cannot truly account for our acceptance of such theories without endorsing our acknowledgement of a beauty that exhilarates and a profundity that entrances us. Yet the prevailing conception of science, based on the disjunction of subjectivity from objectivity, seeks–and must seek at all costs–to eliminate from science such
    passionate, personal, human appraisals of theories, or at least to minimize their function to that of a negligible by-play. For modern man has set up as the ideal of knowledge the conception of natural science as a set of statements which is ‘objective’ in the sense that its substance is entirely determined by observation, even while its presentation may be shaped by convention. This conception, stemming from a craving rooted in the very depths of our culture, would be shattered if the intuition of rationality in nature had to be acknowledged as a justifiable and indeed essential part of scientific theory. That is why scientific theory is represented as a mere economical description of facts; or as embodying a conventional policy for drawing empirical inferences; or as a working hypothesis, suited to man’s practical convenience–interpretations that all deliberately overlook the rational core of science….

  64. #64 Kristine
    December 10, 2009

    When I dance, I literally envision the xyz axis and conic sections. Plus, I love maps. I can stare at a map for hours. I also studied music, stage movement, and physics. All are related! (I had to play one too many Bach inventions, though. I appreciate their genius, but I don’t need to hear them again.)

    I recommend John Allen Paulos’ book Innumeracy for those who struggle with comprehending math, because it’s not about math but concepts and relationships. Paulos has the same complaint about how mathematics is taught.

  65. #65 Veky
    December 11, 2009

    Math is a collection of true statements about imaginary objects.

  66. #66 LilMick
    December 11, 2009

    I hated math when I was in school–boring, slow, repetitive–and started skipping math class altogether in junior high. Since then, I have attended math lectures at my university and have discovered a love for math, especially number theory, Riemannian geometry, and analysis. It opens a beautiful world of imagination, which sometimes produces tools for studying physical phenomenon.
    That being said, in my own experiences at least, this world of possibilities never was opened to me by my teachers in grade school and high school. Math is so much more than rote memorization or 30-minute drills. It would be nice to see more ideas from topology, analysis, number theory, logic, geometry indroduced earlier in school, so as to engage young minds in thinking critically and imaginatively about problems, rather than spewing out answers.

  67. #67 Jonathan Vos Post
    December 11, 2009

    Let me point to the expert who explained “stuff, structure of stuff, and property of structure of stuff” to me, as the heart of the heart of Mathematics.

    http://math.ucr.edu/home/baez/cohomology.pdf#page=15
    [2.3, p.15]

    Stuff, structure, and properties. What’s all this nonsense about? In math we’re often interested in equipping things with extra structure, stuff, or properties, and people are often a little vague about what these mean. For example, a group is a set (stuff) with operations (structure) such that a bunch of equations hold (properties).

    You can make these concepts very precise by thinking about forgetful functors. It always bugged me when reading books that no one ever defi ned `forgetful functor’.

    Some functors are more forgetful than others. Consider a functor p: E –> B (the notation reflects that later on, we’re going to turn it into a fibration when we use Grothendieck’s idea). There are various amounts of forgetfulness that p can have:

    * p forgets nothing if it is an equivalence of categories, i.e. faithful, full, and essentially surjective. For example the identity functor AbGp –> AbGp
    forgets nothing.

    * p forgets at most properties if it is faithful and full. E.g. AbGp –> Gp, which forgets the property of being abelian, but a homomorphism of abelian groups is just a homomorphism between groups that happen to be abelian.

    * p forgets at most structure if it is faithful. E.g. the forgetful functor from groups to sets, AbGp –> Sets, forgets the structure of being an abelian group, but it’s still faithful.

    * p forgets at most stuff if it is arbitrary.
    E.g. Sets^2 –> Sets, where we just throw out the second set, is not even faithful.

  68. #68 Miguel
    December 11, 2009

    There are many other connections between mathematics and music besides the ones mentioned in this thread like the following

    1º The fundamental bass of a chord, which is two octaves lower than the tonic. There are some beautiful mathematics behind harmony.

    2º The sound of a violin has a triangular shape and that of a piano is a square wave. Again, there is a lot of Fourier Analysis behind this and I wouldn’t call it a triviality.

  69. #69 Jonathan Vos Post
    December 12, 2009

    http://mathgradblog.williams.edu/?p=478

    … I see two main reasons to expect graduate students to be confused.

    (1) Mathematics is much like a language. Would you expect to immerse yourself in a culture that spoke a new language without there being a period of discomfort? Moreover, would you feel comfortable claiming that you were fluent if you hadn’t been immersed? So what I suggest, and what I have suggested to my mentees for years, is to go to seminars and classes that you may not understand. Try hard to understand, but when that breaks down, simply try to understand how the experts are using the terms, what value they are attaching to certain goals or processes, and something basic about how they fit together.

    (2) Although you have taken “Math” classes since you were in elementary school, you have only recently been introduced to Mathematics as is is known to practitioners. The material you have learned is complete revisionist history. It was neither created in the form in which it is currently taught to young people (0-22 years old), nor is it created in the order in which it is taught. There are hundreds of thousands of people who have devoted their lives to structuring basic mathematical ideas in a way that all citizens can learn and use it. But now you are hoping to answer a new question (or heck, find a new question), and the old skill set is close to inapplicable. In short, sometimes I miss tests: straight-forward goals set by someone who is sure that I have exactly the skills needed to succeed (and has primed me to chose those skills instead of others).

    I’m in no position to tell anyone if grad school is right for him/her, but I hope it helps you to realize that you are functionally entering a new discipline that uses a language only slightly familiar to you, and as such you should adjust the standards that you have set for yourself.

  70. #70 Jonathan
    December 12, 2009

    All well and good (not really. I’m not with the Investigations guy, or the Lockhart people)

    But arithmetic is still arithmetic. And there are patterns, and beauty, and of course, a few applications.

    But don’t be fooled. Number and arithmetic are complete abstractions. We can’t taste 3, touch 3, feel 3. We don’t know what it looks like. But kids figure it out pretty fast. Definite bridge to abstract thinking.

  71. #71 Kevin
    December 13, 2009

    Beth Welsh, may I recommend teaching your child how to program a computer. The language “Scratch” (free from scratch.mit.edu) is particularly appropriate for teaching children. I’ve taught it to kids as young as 8 years old, and with patience could teach it to younger kids. (It is aimed more at teens and preteens.)

    A lot of the formal reasoning beloved by mathematicians becomes externalized in computer programming. Notions of elegance and correctness translate almost exactly.

    Programming the movements of sprites on the screen leads naturally to learning about simplified notions of physics (gravity is one of the first things kids learn to simulate in many Scratch classes). It can also be used to motivate high school math (like trig, algebra, and calculus).

    Personally, I switched from pure math to computer science in grad school. Initially it was because all the math I found beautiful was discrete math, practiced more in CS departments than math departments in those days. Over the decades, I’ve become more and more applied, but I still retain enough pleasure in math to coach a math team at my son’s school.

  72. #72 abb3w
    December 13, 2009

    Jonathan Vos Post: Math, in its heart of hearts, is only about THREE things

    As I understand the vantage of ZF set theory, both (1) and (3) are forms of (2).

    PT: Such as: Why should we hope to discover/support/develop future mathematicians in arithmetic classes? I would argue that we kill, not develop, our future mathematicians in these classes.

    As I understand, that was part of the point of the “new math” approach attempted in the late 1960s. It was less effective at the short-term goal of instilling ability for basic arithmetic (although not completely ineffective), but much more effective at the long-term goal of starting the development of budding mathematicians.

    Michelle: Aside from cooking and perhaps some construction projects, what is the point of fractions?

    Allowing rigorous construction of the “real numbers”; IE, decimals. More esoteric uses include the FRACTRAN programming language.

    Jonathan Vos Post: Mathematics is much like a language.

    Actually, it is a language; or rather, family of languages. The intimate relation between them is covered in the Automata and Formal Language theory (EG, Linz; ISBN 0763714224) leading to theory of computation. The main difference between mathematics and other human languages is the higher degree of internal self-consistency and reduced ambiguity.

  73. #73 Jonathan Vos Post
    December 13, 2009

    abb3w raises some good points. I don’t want to get into hair-splitting debate, but things are more subtle than that commenter or I have explained, partly because of keeping reasonable length, and partly because of the heterogeneous level of people reading this blog.

    (1) I drew on Category Theory (and Topos Theory) precisely so as not to be constrained by ZF (Zermelo-Fraenkel set theory). Not that I dislike ZF; far from it. But Category Theory, n-Caregory Theory, and Topos Theorey are, t me, more right-brained, holistic, gestalt than the Bourbaki left-brain analytic axiomatic approach.

    (2) New Math, worked for me, but I’m a statistical outlier. My mother was a 3rd grade teacher in Brooklyn, her brother played cards with Feynman, I had a school teacher in common with Feynman. I was a beta-test subject for New York City’s
    first New Math textbooks in 2nd or 3rd grade (I breezed through all the texts through the 6th grade ones in the first day, and they took that as a good sign), and years later I discussed this whole schoolbook matter
    with Feynman.

    Feynman explained the monstrosity of California science and Math textbooks when he was a consultant.

    “‘John and his father go out to look at the stars.
    John sees two blue stars and a red star. His father
    sees a green star, a violet star, and two yellow
    stars. What is the total temperature of the stars seen
    by John and his father?’ — and I would explode in
    horror….”

    I (JVP) once heard a docent at a science museum “explain” to
    a crowd that the “surface” temperature of the Sun is
    about 11000 Farenheit. I was okay with that, since the
    Sun’s outer visible layer is called the photosphere
    and has a temperature of roughly 6000°C = 11000°F.
    But then he said:

    “That’s more than 50 times the temperature of boiling
    water.”

    I objected, politely. He defended himself with the
    fact that 11000 / 212 = 51.8867925.

    I explained that neither water nor the Sun were
    personally familiar with humans with temperature scales
    named after them. Yet, in a sense, they acted as if
    they knew about Kelvin. I pointed out that water boils
    at water boils at 373.16K and since the Sun’s
    photosphere was at roughly 6000K it was more
    scientific to say that the Sun’s photosphere is
    roughly 16 times as hot as boiling water:

    6000 / 373.16 = 16.0788938

    He didn’t get it. He repeated his 50+ line to the
    puzzled crowd. My girlfriend (this was years before I
    was married) physically pulled me away from the crowd.

    (3) Yes, fractions lead us to the Reals, and FRACTRAN is very cute, and I think was discussed on a different thread of this blog. But yes, chefs and carpenters use Math every day. And if a child does not grasp 2/3 then she or he is unlikely to grasp X/Y, nor (x^2 – 1)/(x+1)

    (4) In my humble opinion, it is misleading to say, out of context, that Mathematics “is a language; or rather, family of languages.”

    The problem being the connections between syntax, semantics, and pragmatics. Yes, Automata and Formal Language theory clarify the syntax/semantics matters. But such theories fall flat as they try to explain what actually goes on in the mind of a professional Mathematician; let alone in the social dynamics of the academic, industry, or government worlds of theory and application.

    Still, as I say, some good points made.

  74. #74 James Sweet
    December 14, 2009

    JVP, the surface temperature of the sun story makes me hang my head in despair. Argh.

    Re: Carpenters using math… especially if they are amateur “carpenters” with very few tools but a master’s in engineering.

    When installing some flooring last year, I discovered I needed to know an angle to set my miter saw, but I had no protractor. No problem — trigonometry to the rescue! heh…

  75. #75 A Reader
    December 14, 2009

    Mathematics is a “language” (tagging on nature (signposts/stickers) & finding relationships thereof). Nothing more, nothing less. Because it triggers stuff in our brain that ‘seems’ “less vague” (than spoken language), it allows for “better”(less vague) constructs than spoken language, and consequently allows “better” (w.r.t predictive ability e.g.) tagging, and thus is marketed as “language of nature” etc. Also, logic, mathematics, etc are “feelings” (stuff in our brain) and not really out there i.e. out there its just some activity. If it solves our problems, we “elevate” it to the status of logic etc. i.e. its a “human-centric” thing. (could be more, but we cannot say anything about that)

  76. #76 Kev
    December 19, 2009

    @Michelle

    I think you don’t see any uses for mathematics because you don’t understand it or know what it is. Would you even sit down with me and work hard to learn why cross canceling works? Do you even care? I have a feeling you would accuse me of being too theoretical when you just wanted the ‘practical’ application. Are you even curious as to why it works? If you really want to know, try getting help at this good site http://mathforum.org/dr/math/

    Anyways, teachers and tutors like you are what cause the confusion, so if you *really* can’t understand why cross multiplication works I encourage you to learn why it works as quick as you can. What you are doing is like someone tutoring English when they can’t write complete sentences. I’m not trying to be mean, I’m just trying to give you an accurate idea of how deep your misunderstanding must be (both basic grammar and basic algebra are taught in 6 and 7th grade). When you get help, make sure the person who helps you likes math a lot. If they don’t find math interesting, they probably don’t understand it.

    Also, I googled “why math is important” and the first two search results are good.

  77. #77 TruePath
    December 26, 2009

    This hits on one of my personal pet peeves. Math really is a beautiful thing, but the way that math is taught turns it into something mechanistic, difficult, and boring. The person who posted this question is a typical example of a victim of lousy math education.

    Yes, but it’s the kind of lousy math education that we (mathematicians) perpetuate. Sure, mathematicians bitch all the time about non-majors not understanding what math is really about but what do we do whenever we teach non-major classes? Concentrate on the the formulaic, procedural junk and complain if they come to us without strong enough knowledge of computational tricks.

    Of course some degree of computational ability must be taught for practical reasons but most of what we drill into students heads is taught for the same (bad) reasons that everyone used to learn latin: a belief that you don’t count as really educated without the skill simply because everyone educated in the last generation learned it. The underlying problem is that no one wants to stand up and voice a theory of what math education is trying to accomplish, perhaps because they are afraid of the consequeces.

  78. #78 math[o]phile
    December 27, 2009

    this post deals with the opening volley of questions from Philosophy of Mathematics. from the point of view of computer science, this field probably makes very little sense. the best i can imagine, a “philosopher of computer science” might find something interesting. after all, i see computer science as largely a software question wrapped up in a hefty layer of design-engineering. there are a lot of physical “hands on” aspects of computer science that make no appearances in mathematics.

    Hence, i suggest an alternative interpretation:
    considered historically, mathematics and religion originated, roughly, co-evolutionarily. i’m making a wild assumption here, by claiming that Pythagoras and Thales were not, in fact, the originators of mathematical art and the philosophy therein, but instead, like Newton, themselves stood on the shoulder of giants that preceded them, but whose names have escaped the long arm of our collective recorded histories.

    thus we see what mathematics is, only in relation to the answer of what religion is. this is profound, in my opinion. people of this earth seek answers, often for questions involving the material world. but there are also the questions that involve that which is unseen. this is a VERY tricky area to be in. it is very very easy to fall sideways into any one of the multitude of pseudosciences and cheaply manufactured para-epistemological knowledge fields.

    finally, a question: this earth, even in the modern, post-modern, trans-modern, anti-modern sense, is still populated by creatures who channel religious convictions. on the whole! en masse! and yet humanity claims that “Science,” with a capital C, indicates clearly and logically to, e.g. creationists, that their claims are false. yet this does not stop the creationists, instead it multiplies them! how can this be? prehaps, it is not so much clarity and logic that is needed, but instead a more fine-tuned understanding of the large and less-than-empty spaces that surround us.
    religion and mathematics are two existent things (structures) that humanity responds to on an individual level. both structures can provide an inquisitive individual with answers to “large” questions. religion provides its answer in the form of supreme otherly authority. this authority, as witnessed from the point of view of individuality, is calming and makes mundane un-technological life bearable.
    mathematics delivers its answer, on the other hand, via the most singular act of individuality: the proof. This simple idea is at the heart of my own philosophy of mathematics; namely, the proof is most singularly focused at the individual level. no outside authority required. EVERY person can ask themselves: did i truly, really, and honestly follow the proof that, e.g., the square root of 2 is not rational? what about wiles’ work with modular elliptic curves? one “piece of mathematics” is well known, and takes up one paragraph, while the latter “piece of mathematics” is less known and occupies some 260 pages, however in both cases we have examples of the “meat” of mathematically-ontological objects.
    an objection should now arise: how can i be comparing the existence of a real number, or an elliptic curve, with real day-to-day questions that people all over the world have, questions that could not possibly be answered via any mathematical methodology: my child is sick, how can i help? i hunger, how can i feed myself? there is injustice, and i am found insufficient: in all of this, there is no answer from mathematics herself. only silence. but there is always change. mathematics, by being the science that conjures and describes the un-imaginable and non-existing, can also conjure for us the solutions to our common and collective pain as a young and sapling civilization. this, i must believe. for otherwise, humanity runs the more than common risk of outgrowing the desire to pursue pure knowledge, and will end up as just another landmark fossil, sitting comfortably between the dinosaurs and those who will have outgrown us.

  79. #79 Jonathan Vos Post
    December 28, 2009

    “…questions that could not possibly be answered via any mathematical methodology: my child is sick, how can i help? i hunger, how can i feed myself?”

    Good point. It is said that someone asked the Buddha: “is the universe finite or infinite?”

    He is said to have answered: “I don’t know and I don’t care. I want to end all suffering.”

  80. #80 Curt Sampson
    January 4, 2010

    I too previously thought of music as a mathematical discipline as well. However, I think that woupiestek has changed my views on this.

  81. #81 James Wu
    January 5, 2010

    So, we all agree that math education is lousy (this goes for worldwide)! But, how do we change it? The solution seems impossible at best. The basics are, not only we need to have better teaching materials, we also need to have better teachers… now, good luck with that; the teachers are Unionized, well, need I say more!

    Saying “math education is lousy” is as effective as saying “pigs can’t fly.” Frankly, it may be easier to genetically engineer a flying pig than to make math education “effective.” What we need to do is to prepare children better for the “lousy math education.” The steps are actually ‘psychological.’

    Most people are not good with arithmetic, that is because human minds can comprehend ‘relationships’ better, and numbers (the writing of 1, 2, 3) actually have no “relationships” with each other. For example, we started learning math by counting balls, one ball, two balls, three balls… so, our little 4 year old brain understood that one-ball is one, and two-balls is two, and three-balls is three; but when it comes to writing it, we write “1,” okay, easy, that means “1.” But, shouldn’t two be “11?” And three? If you have little toddlers in your house, you will always get that little confused face! This is where ‘disassociation’ builds.

    Each little ‘disassociation’ made our perception of math less friendly. And we all know what we do with the less friendly things… our minds turn away from “math” in general, when math class starts, children’s physical bodies are there, but their minds are out on the playing fields.

    In my years of hobby of wanting to enhance education, I came across soroban abacus and met many of its student/children who think math is fun… and from my interviewing the parents, many students were formerly “math convicts.” But after receiving math learning with abacus, these students’ math improved… many went from D and F’s to B and A’s. In my research, many teachers attested that a few students’ grades went from F to B within a few weeks of receiving abacus learning. But, how can this be? Abacus teaches only basic arithmetic, and these are fifth or higher grade children…

    What I found was astounding, even I, myself, was amazed: All abacus training had done was to remove the “math mind block” of students–abacus and its “pearls” creates the “relationship” between all the numbers! By turning arithmetic into graphical symbols, abacus actually made math easier to comprehend. And with faster calculation speed, (video example here http://www.youtube.com/watch?v=PdJSljryuxw&feature=PlayList&p=1E6F4E420C8C891D&index=0&playnext=1 ) children has new found confidence in math! No longer do our children feel vulnerable about math, they actually start to enjoy math… and with that attitude change, these bad students start to listen to the boring math teachers in class! And with that, it is just simple domino effect that when students pay attention in class, usually they can get great grades.

    Changing math education as a whole maybe too complex, but enhancing children to better take-on the crappy math education is certainly the way to go! MathSecret Education Foundation (http://MathSecret.org) has a FREE learning system for learning math via soroban abacus. You can also check out the many positive ripple effects of learning math via abacus at http://www.mathsecret.com/e/ripple_effect.html

    It is worth a try!

  82. #82 Jim Kelley
    January 8, 2010

    Just stumbled on your site as I was poking around for some info on slide rules, which I used during my undergraduate engineering days back in the late 60′s and early 70′s. Your site is terrific–please keep up the good work.

    Jim

  83. #83 AKE
    January 14, 2010

    Great post! I think you’ve accurately described contemporary pure mathematics. But mathematics has not always been this way. In fact, prior to the mid 1800′s it was different in both form and its own self-characterization.

    I think an interesting question arises if we try to find a single definition that captures the meaning of Mathematics across the entire spectrum of its evolution. Is there anything that unites the practice of Mathematics throughout its long history and that persists into the present time?

    1. What is Mathematics?

    2. The Development of Mathematics in a Nutshell

    3. Characteristics of Modern Mathematics

    I’d be interested in your comments!

  84. #84 MPL
    January 18, 2010

    I take a somewhat deflationary view: mathematics is what mathematicians (in particular) tend to do. And I say that not just to be smart, but because it has a way of changing over time. Poincare, for instance, had some mighty nasty things to say about set theory. He wasn’t even all that big a fan of formal logic. But undoubtedly a mathematician, doing mathematics.

  85. #85 Rashed Khalid
    January 26, 2010

    How about this definition of Math.
    Any system of knowledge where the verification of statements about the system are done by a rigorous proof method is math.

  86. #86 Anonymous
    January 27, 2010

    this is to long all u should put is this is BORING hahahahahah lol lol lol lol lolri was here

  87. #87 Julio de la Yncera
    March 10, 2010

    I agree with post 16 one of the best ways to get familiar with what is math by reading the book

    “What is mathematics?” by Richard Courant, Herbert Robbins.

    The mathematical education that most people get is devoid of mathematical proof. To enjoy Math without proofs is like to develop a taste for music without sound.

  88. #88 How Si Yu
    April 12, 2010

    Maths is about logic, about understanding. You don’t need to memorize anything about maths because if you can’t remember it, just derive it from the basics.

  89. #89 ian childs
    June 14, 2010

    i dont like math or understand it. i understand how you say its abstract but then again whats abstract about it. in the math world 2+2 will always equal 4. i feel like math is more like society trying to figure out an abstract world more than they need to be. why do things make since? The world is trying to answer every question, even if answers don’t exist.

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