I am not sneering; I am genuinely asking.
For the last few months I have been education blogging. I’ve never been much good at working out site stats, and things are made harder by my education blog sharing its numbers, or all the ones that I see, with my personal blog. But, going only by how the comment rate has gone from zero to detectable, my education blog is now showing occasional but definite signs of life. I reckon that education blogging is rather like teaching. To begin with you often achieve very little, but if you stick at it, good things may eventually start happening.
In connection with my education blog, and in connection with the helping out that I am now doing once a week at one of the supplementary schools run by the think tank Civitas, I find myself asking: what is the point of learning maths? I entirely accept that there is a point, in fact many points. It’s just that I don’t know much about what these points are. Some of the boys at the supplementary school – two in particular spring to mind – strike me as showing real mathematical talent, at any rate compared to the others. What can I say to them that might encourage them – and encourage their parents to encourage them – to get every bit as far in maths as they can? What use is maths? For lots of people, especially for lots of teachers and lots of children, that is surely a question worth knowing answers to.
I don’t need to be convinced about the usefulness of arithmetic. People cheating you out of change in a shop, or loading you with debt obligations that you did not understand when you made the deal – working out floor areas and carpet costs – getting enough nails and screws and planks when you are DIYing about the house – just generally keeping track of work. I get all that. And, I find, I’m pretty good at teaching arithmetic to young boys and girls, partly because I do indeed understand how important it is.
But what about the kind of maths that really is maths, as opposed to mere arithmetic, with lots of complicated sorts of squiggles? What about infinite series, irrational numbers, non-Euclidian geometry, that kind of thing? I, sort of, vaguely, know that such things have all manner of practical and technological applications. But what are they? What practical use is the kind of maths you do at university? I hit my maths ceiling with a loud bump at school, half way through doing A levels and just when all the truly mathematical stuff got seriously started, and I never learned much even about what the practical uses of it all were, let alone how to do it.
I also get that maths has huge aesthetic appeal, and that it is worth studying and experiencing for the pure fun and the pure beauty of it all, just like the symphonies of Beethoven or the plays of Euripides.
But what are its real world applications? Please note that I am not asking how to teach maths, although I cannot of course stop people who want to comment about that doing so, and although I am interested in that also. No, here, I am specifically asking: why learn maths?
Occasional Samizdatista Michael Jennings works as a Something in the City, analysing things like technological trends. Not at all coincidentally he has a PhD in maths. He is the ideal sort of person to answer such questions, and he and I have fixed to record a conversation about the usefulness of mathematics later this week. But I am sure that a Samizdata comment thread on this subject would help us both, if only by helping me to ask some slightly smarter questions.
Proper maths [by which I mean statistics, calculus – second-order differential equations, integration and the like] is deeply useful in that it lets those of us who understand such stuff deploy our knowledge to analyse trends in stocks and commodities, thereby helping us make the truly advantageous and enduringly profitable trades.
Those that fail to appreciate this are doomed to a retirement dependent on the penurious horror known as ‘the state pension’.
It’s a valid point to ask what good is higher mathematics as compared to basic arithmetic. But the problem is, in careers that really do demand it, if you haven’t got the basics of further mathematics by the time most people enter the workforce, it’s too late. If we assume that we do need physicists and engineers (a pretty valid assumption I would say), then we have to accept that their apprenticeship is a very lengthy and arduous one.
It takes, on average, continuous schooling from the age of about five to 21 or 22 years old to turn out a competent physics graduate. At this point, one hopes, the graduate will have been exposed to, inter alia: fairly advanced trigonometry and algebra; matrix mechanics; differential and integral calculus; differential equations; statistical methods; statics; dynamics and thermodynamics. But a BSc just marks you down as a larval scientist. Deferring the teaching of the more abstruse elements would mean deferring entry of people into a profession to a ridiculous degree. You can’t do maths ‘on demand’. And the higher reaches of mathematics are fundamentally underpinned by the abstract, but elementary concepts one learns beyond the absolute basics. For example, rapid and accurate manipulation of algebraic equations is essential, but this is a learned skill which takes a lot of time and effort to master.
If you don’t have a command of some maths beyond arithmetic by 18, you will struggle to catch up. I have first hand experience if this by dint of my exposure to first year undergraduates in the late 90’s. Many of the new intake were woefully under-prepared compared to my cohort a decade previously. Emergency remedial work was needed, but even so some of the more densely mathematical elements of some courses had to be abandoned or modified. We were accepting students who, on being asked to simplify sin x/cos x cancelled the s and the x to give in/co. Pulling someone who is that far behind up to the required level without neglecting his more able peers is a very hard thing to do,
So, in summary, I think one reason at least that maths is important is in providing the feedstock for the knowledge-based technical professions.
In my former work in a manufacturing company, we spent a lot of time trying to work out from our October sample sales, how much of each product we were going to sell of each product in the main season in the spring. This ended up with calculating exponential trends and using the R^2 numbers to get a handle on how much trust we should put in the answer.
So yes, real maths does have real world applications for ordinary people. It’s just that in my case none of this statistical knowledge was learned at school – it was all picked up through reading Climate Audit.
Almost everthing from finance to engineering, computer programing.
The point of learning math (forgive me, but I’m an American, and we don’t say “maths” over here) is not useful. It’s aesthetics, same as learning classical music or mythology. Math is beautiful, and if you don’t know it, your life is aesthetically impoverished.
That said, math does have one important quality that has value for certain people: It teaches you to think abstractly, and about issues where your emotional biases have no relevance at all. This is a useful ability to have. There’s something refreshing about a subject where there are clearly right and wrong answers, and neither current intellectual fashion nor the teacher’s opinions nor the feelings of your classmates have the slightest relevance.
How abstract are we talking here?
One of the main USPs of probability theory these days is the pricing of derivative securities such as options. This is the work for which Black and Merton won the Nobel Prize and is normally referred to (in shorthand) as just the Black-Scholes formula.
To decipher the formula requires some knowledge of probability/statistics.
To really understand where it comes from and how to adapt it to price something other than a plain vanilla European call option requires a knowledge of
1) Analysis (1st year BSc Maths)
2)Measure Theory (final year BSc Maths)
3)Measure-theoretic probability theory (final year BSc Maths or post-graduate level)
4) Stochastic Calculus (Masters Maths degree)
I should add that you can do a postgraduate degree in Financial Maths which will cover this but you’ll need to have a highly numerate BSc under your belt to start.
Also, these one-year courses cost c £20,000.
Then again, we could talk about the transportation algorithm …
I’ll try to come up with some more examples later, if you want.
You cannot be an engineer without it, and the world is short of engineers.
Engineering as a profession is generally rewarding. For my part, no two days are ever quite the same and one is often involved with projects that give one a sense of satisfaction when successfully completed.
The shortage of engineers mean that one can be reasonably picky about where and for whom one works and the pay is often above average (sometimes well above). To get the plum jobs and the best rewards, one does have to have some flexibility with regard to being willing to move (and have a spouse of equal willingness).
Most engineers will only directly apply a small proportion of the higher maths they learnt at uni, but you cannot know in advance which you will and won’t use in advance. Some engineers, especially in research, will use a lot more, and other engineers will make use of this work.
At 49, I can sincerely say that engineering has given me (and continues to) an interesting and rewarding life and if I was 17 again, I’d take the same course again.
Not so sure about ‘computer programming’.
One of my formative experiences as an ‘adult’ was as a graduate in the early 80’s leaving college and getting my first salaried position in a Teacher Education Center in South London.
Job #1 was working with some education specialists who were to implement a gubmint program that was intended to cross train mathematics teachers as computer science teachers. It looked awfully like a Soviet Five Year Plan which consisted of moving 10,000 engineers from Ball Bearing City #23 to Domestic Appliance City #17. I had the gratifying experience of having my old ‘A’ Level Math teacher calling me up at home asking me if I could help him with some rudimentary Z80 assembler language.
The net result was that a significant part of the borough’s maths teachers being dragooned into becoming CS teachers. The statistics that came back after a few years led my colleagues to the conclusion that good math teachers of the early 80’s made (on average) pretty dismal CS teachers, and the net result was that the quality of math teaching was correspondingly degraded because competent math teachers had been harvested as CS personnel. (see, there’s always a way to insert a crumb of libertarian propaganda)
I’d place ‘advanced numeracy’ as a valuable skill for a computer programmer, but I’m not sure that more than 1 in 20 coders now would ever need something as advanced as ‘A’ Level math (1970’s-1980’s vintage syllabus) That may be even less relevant now in that most development languages have extensive number crunching abilities.
Of course, this doesn’t help the practitioner (notably in the case of statistics) if he doesn’t know the limitations of the techniques he intends to employ (sampling, significance etc).
However, having followed a circuitous route, I’m in my late 40’s now, working in the investment management business. I’ve never made better use of my math skills than I am now, although, I have to say I really only leverage the equivalent of a ‘good’ GCE + 2 semesters of stats.
Coming from an accounting graduate who, for some reason, found math/calculus/diffy-q’s very entertaining (I ended needing 4.5 instead of 4 years to get my degree due to the 2 extra caculus classes and differential equations class, ultimately one course shy of a minor if I so chose) and used very little of it (directly) since, I can say that math is simply another language to describe the world. Shapes, curves, infinites, sums, etc etc all merely describe the world.
So, just as prose has a place, to describe our world in a more elegant way, with flair, and in an artistic way, math reduces it to a factual description. In so doing, one gains an appreciation both for the subjective and abstract AND the objective and real.
Also, it helps in some philosophical senses as well. Most observable data typically follows a bell shaped curve. Normalcy clusters around the mean. The exceptional, good and bad, exists within the tails. So many people, Statists all, try and con people into thinking everyone and everything can be above average (if you’re below the mean, or squarely on it, you’ve been screwed by someone). Anyone who has taken statistics knows this to be false. In a nutshell, when the world is consistently seen through the dispassionate prism of math, a more realistic and rational understanding of the material world is made. That’s not to say that there isn’t a time and place for art, the discipline of confounding the real, material world for pleasure. But art, creativity, and fiction should never replace a rational perception of the real world.
So while I might not being doing renderings in spherical coordinates, or endlessly writing Sigma from 0 to infinity, having spent so much time with math/calculus has left an impression on how I continue to view the world, just as any other language can affect the mindset of the individual. A person who speaks a language built from a passive point of view versus a person who speaks a language built around an active point of view are bound to be different. I suppose the magical difference, in learning any non-native language, is whether one emerses oneself in it voluntarily, or has it forced on them as a chore. Most people treat math as a chore.
The use of math is best demonstrated by people who lack the ability. There’s a big fuss over literacy, and an even bigger fuss over the illiterate.
But if you want real trouble, just see what can happen when you turn the innumerate loose on a budget, an election turning on economics, or risk analysis.
Long periods of time – geological periods of time – have a quality all their own. You can grow a bit of a gully or gulch in your backyard over the course of a year; but the innumerate cannot multiply that gulch by tens of millions to get the Grand Canyon. And thus they posit Divine intervention or Noah’s Flood.
Bummer.
As a student in a statistics course who has recently read Popper’s Logic of Scientific Discovery, I see how directly maths come into play when one needs to quantify the plausibility of any assertion with margins of error and such. The global warming debate is a prime example of the need for this kind of knowledge- many of the studies purporting to link human activity and climate change are shoddily designed and have low levels of correlation, but the average maths-illiterate bloke wouldn’t have the slightest about the reasons these studies aren’t credible.
In a similar vein, the quantitative social sciences, economics particularly, make very good utilitarian arguments for the free market.
Ever play stuff like D&D? Ever wanted to be a Magic User?
All that eldritch stuff, all those scrolls and potions and whatnot?
Math is like magic but real.
Try this.
If that don’t float your boat then…
Math is real (except when it’s complex) and so astonishingly rich and beautiful.
I pity the fool!.
I am aghast at Brian’s question. However, it is actually a difficult one to answer where there is a gulf of understanding (as I think there must be between him and me).
Mathematics is the language of science, engineering and other technology. Without maths, almost all of the machines we rely on for our modern lifestyle would not exist, and we would be a primative society.
In order to demonstrate this, to the sceptic, one way would have to take each particular bit of technology that we use and analyse how it works and how it was developed. This could be very time-consuming, and difficult for the ‘recipient’ who would probably struggle with why everything is so complicated.
Let’s make a start. We communicate here using modems; I have done quite a bit of work on modems. Signals are sent down a transmission channel to convey digital data. The signal sent must be suitable for the channel – it must get through. This requires an understanding of frequency; that depends on the algebra of complex sinusoids; sinusoids are built on trigonometry; the complex part relies on ‘imaginary numbers’ – those using the square root of minus one (which we [all] know does not exist, except as an abstract concept). This is extended using the theory of Fourier analysis and Laplace transforms, which are based on integral calculus.
Algebra itself is the manipulation of symbols that substitute for numbers (ie might have numeric values put in place of the symbols. much later on) in order to allow more general manipulation of the meaning of the numbers. A possible equivalence with language would be to try and define grammar without the concept of gramatical classes such as nouns, verbs and adjectives.
Back to modems. Modern high-speed modems use Adaptive Channel Equalisation (ACE) of various sorts. ACE embodies a mathematical model of the actual communications channel in use, and how it ‘distorts’ the transmitted signal. This is built up (using channel probe signals sent when the modem link is first established) and then modified as it changes with time. Matrix algebra is commonly used to define such ACE models, with matrix inversion being a common computational technique used within modems. The matrix algebra is itself usually derived using differential calculus to find the optimum solution for the actual ACE model to be used for the actual transmission channel.
It is also the case that most modems send the same data stream more than once, so that there is some redundancy to overcome occasional transmission errors, perhaps caused by a short burst of noise on the channel or just from the overall quality and statistical fluctuations in noise. Complicated techniques are used for this, commonly including Viterbi decoding – this is an efficient way of decoding the received redundant bitstream to make the best use of the repeated transmission of the same data (where repetion can, cleverly, be done say half a time rather than once or twice). The efficient encoding is usually based on a bitstream derived from but different to the original; this uses polynomial transformation based on a sliding window and arithmetic done using Boolean operations (particularyly exclusive OR) rather than more conventional arithmetic. The combination of these techniques is sometimes refered to as Forward Error Correction (FEC). Similar techniques are used in checksums, which indicate whether a whole packet of data has been received correctly (after demodulation and application of the FEC), or whether a re-transmission should be requested.
I think I’d better stop there, at least for the time being, on modems.
Another concept to think about, based on maths, is navigation: say in crossing an ocean. As I understand it, it all starts with geometry, and goes on through trigonometry.
And what is the escape velocity required to launch a rocket into space; I remember that as one of my early uses of integral calculus to derive something tangible. And what elevation is required by artillery to hit that target over there at 8,000 metres, given the known windspeed, and Corriolis force. And just how does one design an electric generator, to give the required voltage and current without overheating.
I hope the above goes some way towards what Brian needs to convince himself. I doubt it will be much use in convincing children of the age he is talking about, though a different set of examples might well be suitable. On that, again, I have this recollection of ball-bearings being rolled down slopes. We measured the times with photocell switches, and the length and gradient with rulers, and then checked whether our theory, according to Newton’s Laws (expressed as algebraic equations) matched the experimental observations.
It will be interesting to see what others come up with to help Brian.
Best regards
‘non-Euclidian geometry’
that’s non-Euclidean geometry.
Admit it, you’re talking out of your ass (arse), right?
Maths is all very well, but it’s no substitute for logic 🙂
Some areas of practical computing (rather than computer science) use maths extensively, most interestingly in computer games. I always thought it was a good route in to maths, especially for children – everything from simple trig for calculating simple collision detection to all sorts of matrix maths, fractals, solid geometry, and so on.
There is also just starting to be some commercial interest in logic, but you’ll have a harder time getting people enthusiastic about that.
I have regularly used two bits of maths that are not “basic arithmetic” in my life.
The first is trigonometry. I have designed a number of triangle-based leaflets and flyers and, unfortunately, DTP programmes are not good at dealing with anything that’s not rectangular. So, trigonometry has to be employed.
The second is quadratic equations. I used these when I was producing amateur theatre shows, with multiple ticket prices and audience distributions (yes, there are probably better ways to do this than with quadratics, but quadratic equations were what I remembered).
I would used the results to calculate budgets and ticket prices. I have produced 40 productions (including 12 in the Edinburgh Fringe) and never lost money, so that has been very useful.
DK
Consider what google does:
PageRank relies on the theory of random walks and markov chains as well as eigenvectors of linear systems. Hard to get more practical than page-rank.
Similarly, most collaborative filtering systems.
Any system that relies on GPS is relying on relativistic corrections (Lorentz metrics), sophisticated calculus and trigonometry. That’s a lot of systems.
Similarly, the world runs on Kalman filters which depend on calculus, matrix analysis, and probability theory.
As a maths graduate myself, I can say that advanced mathematics has been of nil use in my career (IT), and A-level maths, plus some statistics learnt after Uni have been sufficient for all my maths needs.
When I left Uni, there were two jobs for mathematicians as such: cryptography, and nuclear design. Nowadays the City also uses a few mathematicians as quantitative analysts.
Obviously a certain level of maths is necessary for all types of engineers, statisticians, and scientists (although you could also ask what use is a science degree), but you don’t need a maths degree to be a good engineer statistician or scientist.
So you should study maths only if you are drawn to it, you will have to learn something else as well to make a living.
SethK
No. Assuming you’re right about the spelling, my apologies. But I distinctly remember doing some non-EuclidEan geometry at school. But although I remember it as having been rather pretty, I never saw any other point to it, and never have since.
Are you suggesting that there is no such thing? If you are, then you are talking out of your … rear end.
Besides which, my spelling error would only be seriously undignified if I were claiming to know a lot about this topic, and three quarters of my point here is that I don’t. Have you been paying attention?
To everyone else: thanks, and keep them coming. Very useful.
The study of mathematics is an intellectual adventure, growing progressively more challenging and we know that challenged brains grow, while unchallenged brains shrink. My schtick 50 years ago was the behaviour of very hot gas streams and the materials containing them, including other gas streams. I don’t use the maths these days but believe the mental agility and curiosity developed during those years does help in writing my weekly sermon.
A lot is dependant on the enthusiasm of the teachers and the programmes they use. Only one of our grandchildren enjoys maths and is therefore good at it.
The books the others have are appaliingly disorganised, unintelligible and therefore BORING.
Guess what!! Rev. Peter Woods, Niagara on the lake, Ontario, Canada.
“… it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case. …..
… I do not think it is possible, because mathematics is not just another language. Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.”
From: Feynman, R.P. (1965), The Character of Physical Law
At my secondary school I remember us having to get to grips with lots of detailed, squiggly equations in Maths. Christ it was boring. Well, a few of us rebels decided that it was all pointless. Why aren’t they teaching us things that matter, things that may actually be of some practical use in the real world?
We finally plucked up the courage to confront our maths teacher on the issue halfway through the class. I’ll always remember his reply. “Boys, you are learning something that will be of tremendous help to you all. You are learning how to learn.”
Math is the one academic subject where you have to learn how to reason in a logical manner. In fact, the one course I ever had in formal logic was my seventh grade math class.
It’s true that I have very little use for anything I’ve learned above basic arithmetic, and have forgotten most of what I learned in high school and college math courses–retaining enough only to follow the argument in whatever articles I need to read–but it was the basic skill of thinking rationally that I learned, and practiced, in all my math classes up through the last one in sophomore year of college.
A few answers and a few questions:
A. Math(s) is the filtering subject of choice for nearly all careers which combine a high level of compensation with a high probability of employment. I’m not saying math is used in those careers, just that you can’t get into them without it. IT is probably the most prominent exception.
A. Innumeracy is a truly terrible fate. I am not convinced that higher math is needed to avoid it, but that’s a question of where one draws the line.
Q. Why is mathematics consistently the most poorly taught subject? My own mathematics education was a horror show. Closely tied to question #2…
Q. Why is math such a macho subject? Crew cut professors barking orders, questions ridiculed, explanations refused… “intuitively obvious” “obvious to the most basic intellect” What is up with that? No other subject in my experience is taught with half the contempt for the student or half the “we’re all tough guys here” kind of attitude. Even the female calculus teacher I had lined everyone up in alphabetical order, barked instructions, and ridiculed people who didn’t understand something.
Q. Where does this “language” language come from? Is is simply exaggeration? I can see a computer programming language as a kind of limited special purpose kind of language, but when I hear fans of mathematics go on, I keep expecting one can write sonnets in it… or even just good solid prose. Can you say “excuse me where’s the restroom”, “I’ll have another beer”, or “I love you” in math?
Thanks.
To my observation over (mumble-mumble) years, the overall level of ones education determines which economic quintile one winds up in:
High-school dropout
High school graduate
Bachelors
Masters
Doctorate
Your competence at math is an excellent predictor of how much money, relative to your peers, you’ll wind up making.
Wanna make more money, kid?
Well I walk my dog in the company of a Professor of Maths from Bristol Uni on occasion, and he describes Maths(Pure) as poetry.
I can dig that, but was never very good at the subject myself.
I did Economics to A level, and would have liked to do it to degree level, but the subject became increasingly stats and math orientated and frankly I couldn’t handle it. Numbers being cold to me, rather than words with their emotive instead of absolute meaning.
I have enough maths to count my millions though!
But I’m willing to give it another go!
I have my shoes and socks off, and at a stroke have doubled my digital computing capacity-
What was the question again?
Well, everybody else has already done the main real-life application areas – science, technology, engineering, and statistics. I thought I might add a bit from the other end, those particular examples you mentioned – infinite series, irrational numbers, non-Euclidean geometry.
Infinite series are used by computers and calculators to calculate all those fancy functions like sin and square roots and raising to powers. Has it ever occurred to you to wonder how you go about finding a square root? Most people just rely on the calculator, or their favourite computer language’s built in maths library, but sometimes you need to roll your own (and some people write maths libraries for a living) where series come in very useful.
(And people have been known to ask such questions when there’s been no calculator handy.)
Another simple example of an infinite series is positive feedback. You load up your aeroplane with cargo, which needs so much more fuel to carry it, which needs more fuel to carry the fuel, which needs more fuel to carry that. If every 100 extra kilos requires 20 kilos of extra fuel, and you add 100 kilos of cargo, how much extra fuel do you end up needing?
An interesting bit of trivia on irrational numbers concerns the most irrational number there is, the Golden Ratio. This number has the property that if you try to approximate it as a fraction, you have to use bigger numbers to get a certain accuracy with the Golden Ratio than you do with any other number. This makes it useful in areas where you don’t want things lining up or resonating and introducing mechanical weaknesses. It’s used by many plants to avoid their leaves lining up and shading one another from the sun – the angle the leaf buds turn around the step is very close to the Golden Ratio.
And the best example of non-Euclidean geometry is the original one, the measurement of angles and distances on the surface of the Earth. (Geo-metry being the measurement of the Earth.) When orienteering or sailing, it’s usual to figure out the bearing of your next waypoint and just walk/sail along the constant heading. Most people are under the impression that the result is a straight line, the shortest distance between points, but in fact it isn’t. The difference is minor if you aren’t going very far, but if you’re planning long-haul aircraft routes, or doing long-distance surveying the old-fashioned way (i.e. pre-GPS), then the curvature of the Earth becomes quite important. Spherical and ellipsoidal geometries are non-Euclidean.
I could go on and on about maths and what you can do with it, but all the best examples take quite a bit of maths to explain, and it’s late here.
The maths that is truly beautiful, the language that the universe speaks when it tells the fundamental pieces of our being how to behave, is tricky. because to understand it you have to see the beauty, like you have to feel music. It has to be in your bones, you can’t teach someone who is tone deaf how to sing, like any art, the mad world of pure mathematics requires innate talent.
A good way to teach maths to seemingly disinterested teenagers is to apply it to things that they are interested in. Geometry = Snooker. ballistics = golf, football etc. Talk in beats per minute and frequencies to a wannabe DJ. Apply their interests to the math not the other way around, you may surprise them by making them realise that they know an awful lot of maths already but lack the language to articulate it.. It would probably be frowned upon but a basic understanding of probability can be taught by teaching them how to count cards.
It needs to start young if maths is to be given any importance in someone’s life. Like any language it is easier to learn while the brain is still making connections. Anyone who tells you that they are no good at maths should be asked if they can catch a ball, the maths involved in even that simple action is incredibly complex and they do it without even thinking about it. They merely lack the language to express it.
“…because to understand it you have to see the beauty, like you have to feel music. ”
Yes! Math is really just about rigorously expressing an idea. I remember that incredulous delight I felt when I finally understood that E=mc^2 was a simple statement regarding the equivalence of energy and matter.
In one simple formula, it captures the *implications* of viewing energy as matter. There’s nothing magical about the math itself; knowing how to parse the math won’t make the underlying idea any more or less valid.
And yet, I would have been unable to generalize to understand the crucial term – that the kinetic energy component of mass-energy at light speed has to be accounted for, or, c^2 – had I not already had a few semester units of college algebra under my belt.
That, coupled with all the reading I’ve done over the years from my layman’s enthusiasm for physics, had to be in place first before I could make the connection. In other words I had to learn how to reason first, as others have discussed here.
I do think that math is more than a “language.” It strikes me as more of a succession of intellectual toolkits. E.g., you can do arithmetic perfectly accurately with integers, but in order to work with fractions, you’ll want to know something about the set of real numbers.
I’ve begun to suspect that a lot of the difficulty of math lies in the fact that each “toolkit” has to be mastered in its entirety before one can begin thinking clearly with it.
(Yes, I’m a Yank, I use the singular noun. Beg pardon for any confusion.)
A posteriori reasoning?
Can one of you guys explain to me how these two statements can be anything but contradictory…
How can “rigorous” (one of the most commonly used descriptions of mathematics) and the way it is apparently practiced once you pass trig, which is to say intuitively, based upon aesthetic judgments, by feel, like music, be reconciled. How can it be at once subjective and objective? How can intuition and beauty be used to find mathematically correct answers? Isn’t it supposed to be about rigor, algorithm, and cold hard facts?
John. It is down to truth, balance and beauty.
Many can get the truth and balance, but few get the beauty.
It’s like when I asked my French tutor how she knew she really understood French.
When I dreamed in it!
She said.
As well as the areas mentioned above, a good grasp of statistics is invaluable for understanding a lot of what goes on in the world. It is useful for working out if an argument someone is using makes sense or if they are trying to pull the wool over your eyes (or if they are simply mistaken). This applies in areas of practical importance like finance and medicine, and also in social policy where governments or lobby groups are trying to sell you some measure.
Mostly this is agreeing with several above posts.
In general, math is a skill which almost uniquely forces a person to learn difficult thinking skills. There’s only one right answer, for purely logical reasons, and there’s zero room for fudging for reasons of political correctness or trendy sociology. That alone is enough to make me think it’s worth teaching. And those critical thinking skills carry over into other mental disciplines.
True, many people will never need math beyond a checkbook, but then again most people will never technically need to know who George Washington was either in their occupations.
And many professions do require substantial math. It’s not possible to be a decent engineer without calculus. Computer programming, lots of economics stuff, and most of the (non-biological) natural sciences require substantial math, probably through calculus. Biological sciences require at least a mastery of algebraic techniques. Calc and discrete math wouldn’t hurt either.
So basically, quit teaching math and you’ve quit teaching one of the few useful things people learn in school these days apart from science. And most science is not doable in the real world without advanced math.
Here might be an interesting question in the course of your discussion — Why does the English language have no equivalent for “illiterate” in the field of numbers?
Ah! Many will say, there is “innumerate”. But “innumerate” as the parallel to “illiterate” is a very modern & limited usage. The word is not found in some dictionaries at all, while in others its meaning is given as “very numerous”, “countless”.
Does the lack of a pedigreed word equivalent to “illiterate” suggest an ancient cultural bias against the ability to count & measure & understand numbers? Or is the absence of a word for the inability to handle numbers and math because that inability was traditionally seen as being of no account?
Are we talking about the uses of the applications of forms of mathmatics; or, are we talking about the uses of the study of and values of learning processes derived from that study of mathmatics?
How many are familiar with the history of Thomas Hobbes; whether you found his conclusions acceptable or not? What led him to that pattern of thought?
Though perhaps an oversimplification, obtaining knowledge consists of working out the relationships of bits of information, and testing the results (it’s red & glows, is it really hot?). The study of mathmatics is one route to mental training in how to obtain knowledge.
A great deal of the application of mathmatics, and obtaining the knowledge for that application, serves that same function in the actual conversion of information into most knowledge; or at least into what we take to be “knowledge.”
“As to studies, they should be taught everything useful and everything ornamental. But art is long, time short. I propose, therefore, they learn those things that are the most useful and the most ornamental.”
Benjamin Franklin, Proposal for the Pennsylvania Academy
Advanced math is most likely one or the other, depending on your view. It is certainly not moderately useful or moderately ornamental.
I’m an electronic engineer, and let me tell you, you wouldn’t have a mobile phone without advanced maths.
Any kind of digital communications relies on Fourier transforms, matrix algebra, and mathematical simplifications in order to implement a formula in the digital domain.
As someone with a Master’s in economics . . . wow, what a question!
Aside from all the arguments others have made (i.e., professional success being strongly correlated with mathematics training, and a technology-enabled lifestyle being impossible without it), rigorous training in math forces you to learn how to think in strictly logical ways, which is invaluable in every area of your life. When you have to prove something with formal logic, you don’t have the luxury of lazy or inconclusive thinking. Which ultimately is good for you.
Not many people need to be taught higher level maths or science. But some do. The UK education system fails the country by failing to ensure that we have a highly scientifically and mathematically educated minority.
Aside from a pure intellectual pursuit, it also helps us describe and understand the world around us and without the calculus we wouldn’t have many of our buildings. Without even more abstract maths we wouldn’t have many of the materials we take for granted either.
My day to day work is founded upon ideas from mathematics and I use maths in a non-formal manner all the time – I’m a software developer.
Very occasionally I do resort to formal maths to work out efficiency of an algorithm and statistics is essential when producing reports for customers, mostly the maths is only perceivable because I studied a lot of maths at university and see the ideas in software design and some programming language constructs.
Perhaps its because I am in part a mathematician by training that I see it, but maths is all around us.
With children who show promise of mathematical ability – the best thing is to show them both how interesting pure maths can be, and how useful it is in describing the world around us.
Patterns in maths often intrigues children, showing them how maths explains things in everyday life is also good (in outline at least).
Not knowing maths can reduce your worries by about 200%.
Mathematics is much more than just arithmetics of calculations for practical purposes. It is an integral part of our culture. If taught properly, it also teaches the student conceptual thinking in way that purely verbal explanations cannot possibly achieve.
Nick E
Economics is one area where maths, and in particular calculus, is massively overused.
If you wish to do any seriously advanced engineering you need to know some advanced mathematics. Yes, it is possible to program a computer with only a rudimentary knowledge of mathematics. But you will never really be a computer programmer, you will never really be able to design new systems, to create and analyse algorithms, to do all the things that highly paid and happy computer programmers do unless you have a good grasp of mathematics. The same applies to every other form of engineering: you will never progress beyond the tedious basics unless you have the necessary mathematical ability.
If you want to be one of those pioneers who spend their lives finding out how the universe really works, then you will need to understand seriously advanced, cutting edge mathematics. Without those pioneers, we would still be a pre-technological society.
You might this relevant: XKCD
And, finally, somebody ought to quote Heinlein:
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.
I’m a graphic designer and typographer, and I use maths quite a bit in my day to day work. It’s mostly trigonometry and geometry, but I’ve used matrices, stats, imaginary numbers, number series etc. in my work. One of the most useful arithmetical/mathematical skills a designer can learn is how to factor numbers – how else can you work out the space between 5 irregular items that need to fit within a given width with no space at the edge? By eye? Pfft! Amateur!
The first observation, from a practical perspicative, is that, according to education writer Alison Woolf, if you look at any profession, from plumbing to law, people in that profession who have a maths A levels earn more than people who do not. This may not be down to maths per se – maybe smarter people get maths A levels. It is a pracitical encouragment, though, to give maths a try.
I also believe that if you are mathematically illiterate, you cannot understand the news. For example, are suicides in wales – the latest scare – a huge social problem or are they a statistical cluster? Can huge multivariate models accurately predict future climate and can they be used as the basis for big tax increases? Has crime gone up or down in Britain? Is there an obesity academic in Britain?
On a day to day level, I think that anyone who is mathematically illiterate would struggle to understand how a mortgage works, to choose the cheapest one or to optomise mortgage repayments. (This is more than arithmetic, it is geometric series and present value). They would have little chance of assessing an investment, or figuring out if their investments have truly outperformed or if luck was involved.
All of the above does not even start to touch on understanding the trends and history of subjects such as physics, biology, economics et.
Amen to that! In fact the usefulness of any economic theory tends to have an inverse relationship to the amount of maths used to express it 🙂
Can you tell I am an ‘Austrian’?
Thanks to Anomenat, who linked http://xkcd.com/179/: that is seriously appropriate for here and now.
On that website there are lots of great cartoons, several of which I barely understand – which is surely good for me. However http://xkcd.com/386/ rather took my fancy, even though it’s too general to be properly on-topic for this thread.
Best regards
Perry wrote:
It is perhaps the case that many people are over-impressed, and perhaps also over-awed, by lots of maths. If there was greater numeracy, this would surely happen less often to less people, so all that use of pretentious irrelevant equations would be less useful in ‘winning’ arguments.
Best regards
There is a book, Innumeracy: Mathematical Illiteracy and Its Consequences, that might answer your questions. I have had it on my bookshelf for some time now but I have not yet got around to reading it.
You can buy it new from Amazon:
http://www.amazon.co.uk/dp/0809058405/
Or you can buy a second-hand copy of the previous edition on the Amazon Marketplace for only 34p:
http://www.amazon.co.uk/dp/0140122559/
Sure, get rid of higher maths… provided you’re willing to let go to almost all branches of engineering, the entire finance sector, physics and the chemicals industry.
Have fun driving over a major suspension bridge designed with only simple arithmetic.
A number of excellent points have already been made about the value of maths in areas such as engineering, technology, and so forth. In the investment world – which I am a part of – a good understanding of basic statistical relationships, understanding of things like correlations, compounding, percentages, is essential if an investor wants to avoid being screwed by an incompetent fund manager. For example, if a hedge fund charges an annual management fee of 2%, based on the assets in the fund, and a performance haircut of 20%, that can take a huge bite out of the returns the investor makes. How many school-leavers, to take a different example, have the fluency in maths to realise that if you get paid an annual equivalent rate (or interest) of 6% on a internet savings account, that a top-rate income tax of 40% will cut the return to 3.6%, which is actually less than headline retail price inflation, which means the investor actually loses money on the savings, unless inflation falls or whatever. And yet this is basic stuff. The same goes for things like loans, mortgages, bonds, understanding of things like equity returns, yields, etc.
Another area that I have found maths to be handy in, despite my being crap at maths at school, is navigation. Even in the age of a hand-held satellite navigation kits and the like, trigonometry and some understanding of vectors is vital for navigating in tidal waters, checking bearings and positions, and understanding speeds over the ground. As an amateur yachtsman, I use maths quite a lot. By being able to use it to practical effect, it also makes maths much more satisfying.
There are many reasons to learn maths to A level standard. Whether you should go further is less clear but, as many people above have noted, a grounding in Maths will get you a better job and probably help you avoid being scammed.
Maths is not just useful in itself it is also, when properly taught (and some US Math syllabi seem to be especially bad here), an excellent way to teach children basic troubleshooting and problem solving skills. Tricks like how to choose which tool from a mental toolbox, tricks like estimation that tell you when your detailed work has led you off the true trail etc.
Personally I think everyone should take a course in how to lie with statistics because that would help innoculate the population from the more ridiculous scammers and scares.
Finally if you want to inspire teenager boys to learn maths point out that Natalie Portman and Danica Mckellar are mathematicians (it might also inspire young girls too in a spirit of emulation).
In addition to the myriad practical uses detailed above, learning higher maths makes you a better teacher of maths in my view – you see the links that make maths a coherent whole (at least at GCSE/ A Level) and furthermore you can teach that to kids rather than the j.
I’m a trainee secondary maths teacher as of now, and one of my key concerns is trying to show my kids how it all links up and makes sense – if they see that, a lot of the questions such as ‘Well, why did you do x here, and not y?’ will disappear. Far better than learning disjointed methods for every little thing without any reason to believe them.
That said, I would not encourage anyone to follow my footsteps into teaching, I myself cannot see myself staying in this profession more than one year – too much bureaucracy, too much regimentation, too many fools (kids and colleagues).
While virtually all fields of maths have practical applications, most people will never need to use them. Designing an elegant bridge requires maths; walking over it doesn’t.
However, despite this, most people would benefit from learning maths, preferably while young. Just as physical exercise benefits the body, so mental exercise benefits the brain. In adults, tackling challenging but solvable puzzles helps keep the mind sharp; in children, it actually boosts intelligence, a desirable end.
of course, maths isn’t the only possible source of mental stimulation – Latin grammar or chess would both work about as well – but unlike the alternatives, some people will actually find it useful.
A knowledge of higher maths is essential for science and much engineering.The small fraction of people who go into those fields need higher maths, 15 years worth of it. In principle everyone else need never know anything beyond basic arithmetic – they could spend their school days doing Sodoku instead – but there is no good way of identifying at the age of five the people who will actually need higher maths, so it’s necessary to teach everyone, just in case. That this also helps make the children brighter is a bonus.
Again, thanks to everyone. This is turning out better than I dared to hope.
That latest point, from MathsBA, is troubling though, isn’t it? All must be taught maths, for the sake of the minority who are really going to need it.
I don’t like that.
As others have already articulated the comment that I intended to write, I’ll just echo that I’ve always thought the value of math was not in direct application of it (although in bits and pieces, of course it proves useful again and again) but in learning new ways to think. As basic arithmatic teaches us the concept of quantities and the precision in manupulating them, geometry teaches us how to develop problem-solving processes based on logical rules, algebra teaches us to think symbolically in those processes, and so on.
I regard these ways of working with ideas as even more than a foundation for critical thinking. They spill over into expressing ideas, communication – something I regard as our most critical skill. Critical thinking never built much without the ability to express it.
As for how to hook kids, that’s a trickier question, and I think you’ve been given good counsel already. In my own education, I have to admit that I did poorly with trig from 7th grade to 12th. In fact, I sort of rebelled against learning it. However, in college I finally worked out that those wretched tables of inscrutable numbers were actually the ratios of the sides, and it was the thing that clicked – once they lost their arbitrarious pallor, my previous enthusiasm for advancing in math resumed. It started making sense again.
Can you say “excuse me where’s the restroom”, “I’ll have another beer”, or “I love you” in math?
Not without some anatomy and biological chemistry.
It’s only like the classic light house, which has to illuminate all the ocean nearby, not just the bits where the ships are. It’s a grossly wasteful procedure, but it has to be done that way, since no one knows where the ships are.
In education, the obvious alternative is to decide at the age of five what each child’s career will be, then teach them only what they need to know for that career – but that’s not an acceptable solution. Compared to that, teaching children maths they don’t strictly need, with the incidental benefit of making them slightly smarter, looks pretty good.
All must be taught maths, for the sake of the minority who are really going to need it.
I don’t mean to be rude but that’s completely the wrong way to look at it. Being taught maths – indeed, being taught anything – is not a punishment or a burden. You teach everyone maths so that everyone – subject to natural ability, which presumably plays some role – has the opportunity to become an engineer, a scientist, a mathematician, a financial analyst or any of the other professions that require mathematical knowledge.
Or would you prefer to plan people’s lives for them? Would you pick out a young child and say, “No, that one should be an artist, no maths for him.” I doubt it. I suspect you would prefer everyone to have the freedom to excel, or to slack, in their own chosen career. I can’t imagine you would really dislike that idea, given that the alternative is for some bureaucrat to pre-assign a person’s role in society.
Would you deny music lessons to a child that shows no signs of being a professional musician?
Brian wrote:
A few things on this. Firstly, I think MathsBA made it pretty clear that he was talking about giving an introduction to maths (obviously, I think, in addition to teaching arithmetic starting around age 5) so that those who are interested can notice that. Currently in the UK, this continues up to age 16: the end of compulsory education.
Secondly, in what way is this different from many other subjects, including those more favoured by Brian (whatever they are). I remember my worst subjects to O-level were foreign languages and English literature: I understood they have value and that I had to learn some things that I found neither easy nor particularly enjoyable; I chose them over history and Latin, rather than doing none of them. I would have added history too, if it had been practical to timetable it.
Thirdly, in addition to the vocational benefits and societal needs already mentioned several times above, is not an important part of education to understand a great deal about the universe, society, etc. This is beyond mere vocational need.
Fourthly, as has also already been mentioned in comments above, for those needing higher ability in maths, for whatever reason, there is an aweful lot to learn. This learning needs to be spread over time, mixed with all the other things that need to be learned.
I don’t know Brian personally; however, from what I have read over the years on Samizdata (and a bit elsewhere), I would class him as an educated person – by the standards we normally apply. I don’t think he is alone in having a bit of a down on the relative importance of numeracy, though I am rather against those of that view. It seems to me that being educated, an intellectual or whatever should not be limited to the higher levels of literature, the arts, philosophy, etc. It should clearly include an understanding of science, engineering, technology, medicine, etc, including an adequate sprinkling of maths, with some knowledge of their scope and particularly of their importance to our modern world and of their influence on the whole of human development.
It seems to me rather strange that people should consider themselves educated when they not only lack that wider knowledge, but also consider it unnecessary and something that should be discouraged, or at least not encouraged. However, I do accept that this is pretty much the norm in the UK – it has been so for as long as I can remember. It is definitely not the case in Germany, nor in France, nor (I believe) in the USA and Italy.
Am I being too hard of Brian? And on other British intellectuals?
Best regards
Maths is what (along with high order language) separates us from the other multicellualr (higher) animals.
It’s a way of modelling what goes on in the universe, or even in their minds, and making sense of the world for them, these kids.
I could say so much more but I have to go. had to stop by even just for now.
I think people are put off studying maths because maths doesn’t initially appear logical. For example, add two negative numbers together and the answer is a negative number. Multiply two negative numbers together and the result must be a positive number. The non-mathematically inclined of us are apt to scratch our heads at that point and “This is rubbish.”
“Hear! Hear!” to what other commenters have pointed out about the macho math teaching approach. My observation is that as a teaching subject, it does seem to attract outliers, mostly on the kooky side.
An exellent stream of comments. I enjoyed it tremendously, and all I wished to say has already been said.
Brian,
Here is a suggestion for your next post: why teach poetry, who needs it ? (Same with history, lit, Latin, etc.).
I knew maths teaching was in a state, but until I read that I didn’t realise just how bad things had got. I have come across people who cannot calculate a simple mean or work out a percentage but I thought that was their particular problem not a generic one.
On a recent Newsnight program Paxman expressed surprise – in his usual sneering way – when someone describes science as being part of our culture. It is of course and mathematics is the absolutely critical underpinning for all science past ‘nature study’. I can’t lay claim to any great skill and my A level maths was over 40 years ago, but I recall we covered algebra, geometry, coordinate geometry, differential calculus and mechanics – unfortunately not statistics. most of that appears not to be taught anymore.
What is more up to O level I had very little choice – English literature and grammar (as separate subjects), french, history, geography, chemistry, physics and maths.
Never mind removing oral tests for languages because it is stressful – if kids have the capacity they need to be stretched – and that includes maths even if you can see no use for it.
Let/help those who love math learn it to the nth degree. They are the ones who will build better bridges and lead us into space. Those of us who do not love math are busy loving and doing other things of equal value.
I did more than enough math because I ddn’t seem to have a choice about it in school. They put me into physics cuz I was good at science, but I lacked the math background to get good grades at it. I worked hard at it anyhow, did poorly, yet in subsequent decades I’ve found that I picked up more physics than I thought given the judgement of failure put upon me.
Now you tell me that I have not been having a rich life experience because of my lack of math, when I have been vastly satisfied all along. I’ve learned (and still learning) enough about finance and economics and statistics and whatever else is relevent to my life to as I’ve needed it, some in classes but mostly information searched out on my own or with advice from people who know more about it whom I trust.
I’m an advocate of children following their interests for learning. Those who are not fans of formal study of math still learn alot in the course of living their lives and following their passions.
How much of math needs to be pulled out and made explicit, for those who are not interested in doing so? A person immersed in computer/video games is learning problem solving and a host of other things some of which is mathematical, as is a person immersed in playing music or drawing or reading or fishing or cooking or animal husbandry or growing things or watching cartoons.
Is understanding things implicitly/intuitively good enough, for living a rich life? I suspect it is. In a libertarian world view, allowing people the freedom to learn these things or not, as their interest dictates, seems like a given.
Some of the more esoterical fields of math with practical applications I can think of offhand:
– General relativity is nearly all non-Euclidean geometry.
– Irrational numbers don;’t have any great use as a concept, but individual ones(specifically, e and pi) are used pretty much everywhere.
– Infinite series have some small amount of application in many parts of economics – for example, calculating the present value of environmental policy(Spend $100 billion today to prevent global warming, or $1 billion a year forever dealing with it?). Also, some more advanced versions are used in algorithms for calculators and assorted math computer programs.
– Imaginary numbers are heavily used in certain equations for electrical currents, making them a must-know for electrical engineers of all sorts. They’re also a real convenience when getting at the fundamentals of many other concepts
– If you count calculus as serious math, it’s used pretty much everywhere – economics, physics, chemistry, all sorts of fields.
– Statistics isn’t properly taught until university in my experience, and it has even more practical applications than calculus – just try doing clinical trials, opinion polling, thermodynamics, or really anything experimental outside of physics, without it.
There’s got to be more I’m not thinking of, but count that as a as tarter.
You might not? It’s all I’m good at. Way to break my ego. 🙁
‘Calculus Made Easy’ by Silvanus Thompson, published in the late nineteenth century, is as good a basic introduction as one could wish for, if anyone is interested in it.
My A level maths was quite recently completed, and I studied all those subjects. Whether to the same extent as in the 1960s I know not, but it was there.
I’d point out that even in physics, statistics are vital to the experimentalist. It’s discouraging to note the increasing number of engineering professors these days who work exclusively with deterministic computer simulations for their research, and thus don’t have a clue when it comes to statistical variations we see in the real world. That doesn’t even touch the power statistics have for debunking junk science of all varieties, but particularly in biology/medicine. Stats should be a required freshman math course in higher ed, even more than differential calculus.
Susan, no one is saying that kids should be forced to learn math – or anything else, for that matter. All we are saying is that they should be exposed to it, and given an opportunity to learn it, given they show no persistent aversion to it. If it is done by competent teachers, such aversion should be very rare.
Furthermore!
Maths (that is to say, the machinery for discovering how to model all possible relationships between all the possible sets of observations or events in the Universe) is the instrument allowed us by God
(Who Is “Order”, see the Gospel of John, I-John, v-i)
for us to be able, once His Idea had got to a certain point, to look back at Him, and possibly even into His Mind.
For me as a scientist, it is an extraordinary privilege, to not be a tree or a trilobite, or even a dinosaur, but to be living in the morning of Time, and to be able to contemplate the possibility that we might be able to do this.
Without Maths, this is impossible to do, and I am so sad and vexed, that all the billions of human beings are not given this opportunity and insight into Creation, and the glory of God, as Isaac Newton himself put it, when groping through a glass, darkly, in “Principia”.
The reason that many think Maths to be useless, is that the State (what else!) has stripped out all that is beautiful and/or mind-wrenching. It has to ba “accessible”….effing bollocks. Take them, aged 8 or so, on a tour of the Cosmos, with real numbers, and a few good Hubble pictures, explaining place value, powers and indices, and a bit about what atoms are and how big, and you’ll never lose them.
I make 2.4 scientists a year this way. THEY ARE WORTH IT!!!!!
I don’t use much of the so-called “higher” math in my life.
Once a quarter, I sit down and try to figure out what my 401k is doing relative to the rest of the market. Hardly advanced stuff. And sometimes I need to take tire marks and figure out how fast a car was going. Algebra, plugging stuff into a formula that someone else already worked out.
However, my original training was in biology. And math was far more important then. Statistical methods were our bread and butter. (Among my classmates, organic flatbread and cruelty-free non-bGH butter, anyway). And a little calculus was important for understanding what the statistical methods were actually doing: statistics is a toolbox with wrenches and torx-bit screwdrivers. Using them tells you nothing of how screws actually hold things together.
The real killer, though, is that math is usually a surrogate for logic. No public (US-definition) schools require logic classes anymore. As a result, we get a world full of fuzzy-headed people who simply cannot think worth a damn. Hell, I went to a major US state university and in the mid-1990’s when I attended, Philosophy was the only degree program to actually require a logic class. (Or Ethics, but that’s a far more profanity-laced rant for another time).
And my high school…well, they didn’t always offer it. And when they did the instructor was the same person as the US History AP teacher, a collectivist shrew[1] who ran a class such that being baked out of my gourd was the only way that year was bearable.
[1] Not the traditional stalinist type. More like the guilty white suburban really nice person who treats the entire world like a bunch of first-graders and frankly bothers me even more.
Ah, please forgive my assumption that mandatory math was being considered a greater good by some commenters here. Some statements seem to imply that, it being necessary to teach math to everyone just in case, and seeing as one cannot look into ‘the mind of god’ without it, a lovely poetic statement with which I don’t agree. I see many paths leading to a glimpse of ‘the mind of god’, and myriad people living great successful money-earning lives without a great deal of explicit mathematical knowledge.
We all benefit from those who love math and apply it in the many ways they do. Having to learn it, at the expense of quality of life, ain’t worth it for those not interested in using math. Other avenues to the ‘mind of god’ are availalbe and worthy of respect. There are other ways to learn about how to think critically and analyze stuff.
Sunfish: I loved biology at school. I really wanted to go to a medical school, but chickened out, and ended up in engineering. Big mistake (too much math, incidentally…).
So how does one go from biology to policing?
Susan, this is a libertarian site, so almost nothing is mandatory:-)
A lot of good points have been made, but I think there are still some outstanding.
Group theory hasn’t been mentioned. It is the mathematical study of symmetry, and consequently can be applied to a vast number of areas from fundamental physics, to tracking evolution through DNA, to understanding some of the perversities of voting systems. The related subject of semi-group theory is of fundamental importance to computer science.
Combinatorics is the study of counting things, and much richer and more interesting than that description might suggest. It has applications in information theory.
Economics has been mentioned already, and Perry and others have commented on the overuse of calculus. This is a very good point, but I don’t think it means that maths is useless in economics. I think that something like cellular automata could be used to study economics from an austrian view point. This *might* improve insight into the problems, given sufficiently powerful computers, and technical development of the maths. But the problem is that the problem is exceptionally complex.
Related to economics, are fractals. Stock price charts are fractal in nature; as are many other natural phenomena.
Also nobody has mentioned Bletchley Park and Alan Turing yet.
But the main reason for studying mathematics – particularly pure mathematics – is that it is all about deepening our understanding of what originally were practical rules of thumb, discovered and improved by trial and error. (Nobody just writes down some axioms and tally-hos off into the logical wilderness; well nobody worth paying attention to.)
Maths is about formalising, and deepening such practical ideas, and finding connections between them. Without this intellectual excavation maths would be little more than a bunch of seemingly random but useful tricks and recipes. Recalling Arthur C. Clarke’s comments about advanced technology maths would seem like magic. The driving motivation for research into maths is that if there is something missing, or something that appears odd, then we do not understand, and if we don’t understand then we might be doing something wrong, or sub-optimal; studying maths means we can improve our understanding of and control over the world.
This is one way to understand the number system. We start with counting and get the natural numbers 1, 2, 3, 4, …; we add a symbol to denote the lack of something – 0; we then write down equations like (x+2)^2 = 16; we realise that some of those equations apparently don’t have solutions, so we get negative, irrational, algebraic, and imaginary numbers. We also realise that there is a link between numbers and geometry, but the two don’t tie up because we have gaps in the number system so we get real numbers, and then the full set of complex numbers which is both algebraically and analytically (i.e. geometrically) complete. We think we seem to understand these number things.
Then we start to abstract the notions of addition and multiplication and discover that the numbers have a lot in common with some other structures and we get groups and fields and rings that we never imagined might exist. And with these new objects we discover new and deeper connections between algebra and geometry and we start doing algebraic geometry, and combinatorial group theory, and then someone invents category theory and creates a language for translating between some of these highly abstract mathematical objects. All the time our understanding is deepening and improving and we are finding newer better ways to do solve old problems we could solve, and new solutions to problems we couldn’t solve and new problems we didn’t know we had.
Several people have mentioned statistics and probability, including correlation. A correlation matrix must be “positive semi-definite”. That’s usually just stated in statistic text books without much explaination. But why is it true, and what does “positive semi-definite” mean? Understanding this takes you into linear algebra that is not trivial.
Why is the integral of 1/x = log(x)? I’ve never understood that.
I think a key thing to get across about mathematics is that is a live evolving continuously developing part of our culture and it influences and is influenced by the wider culture. I’ve mention Bourbaki before on Samizdata, and Bourbaki is an example of modernist mathematics, as much part of mid-century european culture as fascism, communism, or welfare state democracy; the belief that the world can be made anew and all life reduced to a few simple rules and their rational consequences. Kurt Godel provided the mathematical refutation of that.
I think that there is a good case for studying maths from a historical perspective. What problems were mathematicians studying and why? How did their choice of problems, and manner of study relate to the wider culture, and to the work of previous mathematicians? How did maths grow, develop and deepen and improve our understanding of the world and allow us to come up with better solutions to our problems.
Some suggestions for doing this:
* Development of computers and algorithms. Starting with the Greeks, and moving though Babbage, to Turing, and Church and beyond.
* Development of Financial Mathematics. Starting with mortality tables and early actuarial science, though Bachelier, throught Harry Markowitz to Black-Scholes and beyond.
* Formalism in maths -Hilbert, Bourbaki, Godel, Turing – and its relationship with modernist culture.
* The development of geometry from Babylonian land measurement up to General Relativity
* Non-Western maths – Indian, Chinese, Arabic, Mayan, Babylonian
* Current problems and maths – statistics and science, computer models and climate change, maths of communication (internet, mobile phones, computers), financial markets
Its important to note that this is not to take a “relativist” view point. There is a universal criteria for whether maths if true or not.
How about that most state lotteries are a way to tell who didn’t pay attention in early Statistics classes ?
Personally, I used what I learned in mathematics on a regular basis, mostly in constructing some form of mathematical model to deal with some aspect of Life …
Mostly, it ends up being Arithmetic-based and Statistics-based … a certain amount is Geometric or Algebraic …
I find Venn diagrams *very* useful – and I was shocked to learn that they seem no longer to be part of the Scots education in Mathematics at the Secondary School level … over here, in the US, I use Venn diagrams to show inter-relationships amonst groups … and, that, combined with Boolean logic, ends up helping to explain all sorts of interactions …
I use vector arithmetic when I teach beginning sailing …
One thing which has ‘dumbed down’ Scottish (and British) Math teaching is the switch from Imperial to Metric measures …
It used to be that kids could count in base 10 (decimal), base 12 (pennies to shillings), base 14 (pounds (weight) to stones), base 16 (ounces to pounds), and base 20 (shillings to pounds (sterling)) … now, with decimal and calculators/computers, kids can barely handle change when they buy something …
Of course, there can be a downside to mathematical learning …
You have probably heard of the problem for the Computer Science Numerate folk … they can have great difficulty telling Christmas from Halloween … mostly because, for them, Dec 25 = Oct 31 …
Well, for one thing, if everyone had a higher level of math as a baseline it would make it much easier for me to explain things like why the Chinese shooting at a satellite in high orbit is a problem where as the US shotting at a satellite in a low and decaying orbit is not. You can not talk Physics without math. I will go further. You cannot understand the world around you or even ask t (let alone answer) he big questions about reality without math. Yes, I know some people claim to do so, but they are just fooling themselves and those gullible enough to believe them…
Math is the best tool we have for understanding reality.
Dear Brian:
I find Robert Scarth’s explanation to be the best of all, added to David Davis’ post. Which is pretty much what I wanted to say also.
Mathematics is a descriptive language, and a beautiful one, once you unpack it all. Since arithmetic forms the basis of all maths, and you agree that arithmetic should be taught, why not go further?
But more importantly, as many, many others have said, in maths the answers are clear and unambiguous. Learning how to solve equations – more, learning how to apply theories and functions – teaches critical thinking that cannot otherwise be conveyed.
More, maths is life. We all start with a set of axioms and basic functions, which we then apply to everything around us. Whatever decisions we make in life is based on some mathematical principle or another. We can quibble over whether that principle is the right one to apply (or whether the values we assign certain variables/constants are the right ones), but the fact remains. We can even simplify a person’s thought processes to general functions (for example, a lefty simply values a jihadi’s life more than his fellow right-winged counterpart’s).
By the way, you can, in fact, say I love you in Maths. Well, maths and logic, but you should not separate the two.
A = me (I)
B = you
Let f(x) be the function that calculates the life/overall value of x. Let Z = minimum value of f(x) below which I do not care what happens to x.
I love you can then be expressed as f(B) >= f(A) > Z
Which is to say, you have at least as much worth as I do, and I care what happens to you.
If that is not love, what is? I’m sure someone can come up with a better expression than I can.
>Can one of you guys explain to me how these two statements can be anything but contradictory…
>”…because to understand it you have to see the beauty, like you have to feel music. ”
>Yes! Math is really just about rigorously expressing an idea.
>…How can it be at once subjective and objective? How can intuition and beauty be used to find mathematically correct answers? Isn’t it supposed to be about rigor, algorithm, and cold hard facts?
**
Math works when it allows you to distinctly and clearly perceive an idea. That’s all.
Call it a test of recognition. You may object that this is irrelevant ; formally you are correct. No scientist worth her salt would ever ask you, “but how did you feel about this result?” as way of ascertaining experimental adequacy.
But we use emotional cues all the time – because depending on pure logic is irrational, without some system of cataloging emotional cues, we’d wouldn’t be able to take care of ourselves, let alone solve math problems.
I contend that if the result you’re getting doesn’t allow you to clearly and distinctly formulate an idea, that whatever math you’ve thrown at the question hasn’t helped you. But if it has helped you, then you understand the basic idea in such sharp relief that it has triggered whatever synaptic pattern you happen to have acquired that makes you shout “Eureka!” when the evidence of your eyeballs triggers it.
Put it this way: the presence of truth does not require the communication of beauty; but the experience of beauty is a necessary way for us humans to discover truth, if not communicate it.
Again: necessary because it allows us to efficiently prioritize our (problem solving) actions. Call it an argument from evolution; the experience of beauty must somehow be a positive one in a evolutionary context.
Not to totally hammer this point home, but it just occurred to me that someone who has used math over their career has a whole catalog of emotional experiences with it, which is cognitively vital in recognizing promising problem-solving strategies.
So it would make sense that whatever confidence that person has with their math knowledge, would readily transfer to whatever answer that person has managed to wrangle from a problem that’s entirely new to their experience.
Just my 2 cents. (for integral values of 2 greater than zero…)
Gambling is a tax on those who can’t do maths
The simple answer to your question is: The reality of scarcity means we need to measure things. If the world was static, arithmetic would do the job. But the real world is dynamic, and measuring that takes mathematics.
Math is life. Or Life is math.
Pythagoras of Samos (wikipedia)(between 580 and 572 BC–between 500 and 490 BC):
“We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus, Pythagoras once said that “number is the ruler of forms and ideas and the cause of gods and demons.” ”
Proper maths works as exercise for the mind. Latin and to a lesser extent all foreign languages do the the same thing, as do lot of other things (talmud, propositional logic, dialectical materialism for all I know) just so long as you don’t devote an overwhelming proportion of your time to a single discipline.
On the whole I think Maths has more value in this regard than anything else I’ve done (though other disciplines have diverse benefits that maths does not). I did Further Maths for A level and I’m absolutely certain that I was a lot smarter then across the board than after I went to university.
“How can it be at once subjective and objective? How can intuition and beauty be used to find mathematically correct answers? Isn’t it supposed to be about rigor, algorithm, and cold hard facts?”
Beauty is a human instinct which can be triggered by many things. For a mathematician, beauty is about the simplicity underlying apparent complexity, about the unification of what seemed to be different subjects suddenly realised to be different aspects of the same subject, about elegance and efficiency in methods, and about the vast expansion of our intuitive understanding of how things work when we find weird, amazing things that turn out to be true.
It’s very difficult to explain without using serious maths, but I’ll have a go.
For the first example, you may have seen a thing called the Mandelbrot set. Images of it were very popular a few years back, and you can explore it yourself using free programs like ChaosPro. It consists of the most fantastic swirls and spirals and chains and lightning bolts, infinitely detailed, each part subtly different from every other, and containing an infinity of embedded images of the whole in each of its parts. As above, so below. Even the layman can appreciate that it is beautiful. And yet it is all derived from a formula that can fit on one line. You define a sequence of complex numbers x(0,C) = 0, x(n+1,C) = x(n,C)2 + C, and colour any point C with the first n for which |x(n,C)|>4. That’s it. And when you understand that formula as a mathematician understands it, you are able to derive all those spirals and lightning bolts, to see why they are as they are and how they all interlock, just from that simple formula. It’s like the difference between admiring the precision engineering of an intricate steam engine and actually being able to design and build one.
A second very simple example is one I sometimes use to explain the difference between arithmetic and mathematics. There is a story about a classroom of seven year olds that the teacher wanted to keep busy for an hour or so. And being back in the days when ‘boring’ was not seen as a bad thing for maths classes to be, he set the students to add up all the numbers from 1 to 100. Now this is within the capabilities of everyone, you can see how to do it, and while laborious, the process is straightforward. It’s just a matter of arithmetic.
However, the teacher was surprised to see one boy scribble for less than a minute, and then sit back, apparently done. Being of a cruel disposition, the teacher said nothing, intending to shame the lazy student at the end of the lesson. Imagine the teacher’s surprise when the answer the boy had written down turned out to be correct!
How could a seven year old add up a hundred numbers so fast? Was he a child genius? Well yes, it was Karl Friedrich Gauss, possibly the greatest mathematician ever, but the method he used is simple enough for anyone to do.
He reasoned thus. You can write down the sum in a long line one way.
S = 1+2+3+4+5+….+96+97+98+99+100
And then you can write it down in reverse order.
S = 100+99+98+97+96+….+5+4+3+2+1
And now you can add the two lines up, in pairs.
2 x S = (1+100)+(2+99)+(3+98)+(4+97)+… +(97+4)+(98+3+99+2)+(100+1)
And you can see that each pair gives the same total: 101. There are a hundred of them, so 2 x S = 10100, and S = 10100/2 = 5050.
Adding the numbers up laboriously, one by one, is arithmetic. Being able to spot the patterns and more importantly use them to short-circuit all that labour, and incidentally solve any other similar problem similarly fast, is mathematics. You can add the numbers from one to a million, or a billion, or whatever, and do it in seconds.
The mathematician considers the second method more elegant, more beautiful, than the first. And while the calculation is just as rigorous, it is was the intuitive understanding of the pattern and how it might be exploited, the intuitive leap that suggested writing the sum again in reverse order, that made Gauss’s feat possible.
I don’t know if that helps – or if it seems just a trite trick. Unfortunately all the really interesting stuff takes a bit longer to explain.
John – a Lottery is usually a tax on those who cannot do maths …
For someone skilled in the relevant branches of mathematics, gambling can be a very good source of income – until Guido makes you an offer you cannot refuse …
wm –
– I would suggest to you that a respected scientist (of either gender) could well use those very words, to ask is the person questioned has any subconscious response to said result … if a result doesn’t ‘feel’ right, that can be a cue for a skilled scientist to try to find out what about the result doesn’t ‘feel’ right – and that delving will tend to be done rigorously … it’s not an emotional ‘feel’, it’s a subconscious response, instead …
Sean: brilliant. I’d just add:
I wanted to say thanks for the responses to my questions, especially from RAB, Doug, Gregory, wm, Pa Annoyed, and maybe unintentionally from Robert Scarth.
I’m afraid I still don’t really understand, but I don’t want to risk hijacking, or appearing to highjack, Brian’s thread. It really is a different topic or topics. If anyone wants to discuss it further you can drop me an email at john@cargocult.biz and we can either carry on the conversation by email or I’ll create a blog post to comment on.
Thanks again folks.
John
I’ll second John: this has been a fascinating thread and it has got me fired up again about math. I keep coming back to these comments and learning new things – thanks to everyone!
Alisa,
I thought I was going to be a park ranger. Then I discovered that a, those jobs are ridiculously selective and b, I really liked the policing parts of the job. (Usually) Oh, and c, in my field there are no jobs without graduate degrees and I don’t want to go back to college right now.
At least where I went to school, premedicine would not save you from math. Maybe it’s not as bad as engineering, but it seemed like the pre-med folks were constantly taking either another calculus class or another chemistry class.
Alasdair –
>> No scientist worth her salt would ever ask you, “but how did you feel about this result?” as way of ascertaining experimental adequacy.
>I would suggest to you that a respected scientist (of either gender) could well use those very words…
>…it’s not an emotional ‘feel’, it’s a subconscious response, instead …
I see what you mean, and I agree – thanks for clarifying the point.