Thursday, March 24, 2011

What is mathematics good for?

Some weeks ago I asked my midwife what made her chose her job. She told me she had actually wanted to study medicine, but didn't meet the numerus clausus. Rspt she ranked place thirtythousandsomething. With an apologetic look at the shelves full with physics and maths books behind me, she added maths was her problem. She couldn't figure out what is was supposed to be good for.

She has a point there, I thought through endless repetitions of my pelvis floor exercises, and though it's hardly the first time I've heard this remark I started to wonder what role mathematics does really play in every day life. (Okay, I admit, what I really thought was it would make a good topic for a blog post.) Arguably, I need a lot of maths in my life because otherwise I'd be unemployed. But how much maths does the average person really need? And what do they need? And does school teach it?

You don't need to learn maths to survive. Otherwise mankind would have gone extinct long ago. Amazingly enough though, your brain performs some basic mathematics all the time, such as extrapolating the motion of moving objects. In an interesting experiment measuring the activity of neurons in rhesus monkeys, researchers from the University of TÃ¼bingen have found that different sets of neurons fire in response to the monkey seeing sets with different numbers of elements. Basically, there's neurons that are (primarily) activated by specific numbers. (See Bongard and Nieder, PNAS 107, 2277 (2010)). And it is known that people with certain brain injuries lose the ability to understand, compare, and deal with numbers, a disability known as acalculia. It does seem plausible then that dyscalculia, difficulties in learning and comprehending mathematics, is so some extend due to wiring instead of motivational problems. However, that's estimated to affect only a small percentage of the population. Most people who don't understand maths don't understand it because they've never really made an effort. Which brings us back to the question what's it good for?

Basic arithmetics is so universally useful that it benefits your selective advantage. Whether you want to know if you've enough money to fill up the tank, are worried that the baby didn't drink enough, or need to know how many bottles of sparkling wine to order for your graduation party, it haunts you everywhere. Beyond that, if you want to understand your average magazine or newspaper, you better know how to read a graph. And unless you want to blindly trust your financial adviser, percent calculation should be on your list.

Having come to this point, I Googled for "mathematics in every day life." The first hit was a long deserted blog with a handful of entries that, next to percent calculation, discusses symmetries in car logos and flowers. However, one doesn't need to know the mathematical definition of a group to plant a flower. Google further brought up a document I couldn't open, a file not found, a power point representation on photoshopping, and a Tutorvista question "How is maths used in everyday life?" with the reply "Math is used in time calculation, shopping, traveling, cooking, and all other important activities." All together not an impressive result. What is maths good for if not even Google knows?

School mathematics tends to drown pupils in 'real life' examples that no normal person will ever use in their real life. Yes, I sometimes add up the prices of items in the supermarket just for distraction, but it's arguably a pretty pointless exercise. Yes, it helps to know some trigonometry to figure out if the new furniture will actually fit through the door, but then you can rent furnished. And who really cares what's the volume of that piece of cake.

The real value of mathematics isn't that you can calculate what 500 sq ft is in international units, because Google does that for you. That, incidentally doesn't have much to do with maths anyway. Sadly, school doesn't teach children much about the beauty of maths, the value of logic, and the power of proofs. You don't need mathematics to live, but you need it to understand - for example Google's PageRank. What is mathematics good for? Mathematics is at the basics of science, including physics, computer science, and economics, examples are omnipresent in your every day life. Without mathematics, you're left in the fuzzy realm of storytelling. How can one understand the world without knowing what a differential equation is, without knowing what optimization is?

No, you don't need to know maths to plant a flower, to admire a night sky, or to like a crystal. But as in the arts, getting to know the artist and his techniques add to the appreciation and understanding of her work - may that be the Fermat's principle, data compression, self-organization, Noether's theorems or chaos. Mathematics is the language of Nature and learning it is your connection to the universe. No more and no less.

Since I acknowledge that the selection of maths taught at school is, sadly, suboptimal to this end, I set out to explain to my midwife that statistics is essential to understand the studies she's been telling me about and a doctor should indeed know what a standard deviation is. And being familiar with the exponential function might explain the funny face I made when she recommended some homeopathic remedy in D10. Things went downhill from there.

Joseph Smidt said...

I was thinking the other day about cosmology in a similar context. I was in the hospital and a doctor told me he didn't believe we (scientists) really knew enough about the big bang and so he didn't know if he believed it.

At first I defended the big bang by giving several reasons we believe what we do but he remained unconvinced. And then I had this thought:This doctor is actually very good at what he does, even though he dismisses cosmology, and so started wondering how much of cosmology do people need to know/believe in order to do a wide variety of technical jobs very well.

The answer, unfortunately for people in my business, is probably not much if any. In fact, if a correct knowledge of cosmology was needed for advanced societies to function they would never have formed in the first place.

Nidnus Rep said...

In my opinion mathematics is great to learn to recognize when you really understand something. It might seems trivial, if you know you know right, but my experience is that this is not the case. In fact, its almost opposite. Most people are very sure about this or that, but how sure can you be about being sure? Having studied mathematics, and learned complicated stuff, the process of doing this (which can be summed up with, now I understand, oh, no I didn't, NOW I understand, ups again, I didn't, and so on) gives one a taste of what it means to 'understand' something. This is very useful in every day life.

Thanks, Per

Zephir said...

2 Bee: What your specialization (I presume, it's the quantum gravity) is actually able to compute? I mean to predict in real numbers - not only qualitatively with formal equations with dozens of missing parameters...

GMP said...

I think the greatest asset that math provides is connecting seemingly disparate phenomena. Here's an example: I teach intro quantum mechanics to engineering students. They are very disinterested in a particle-in-a-box or the free particle, "who cares about standing or plane waves blah blah", and why do I torture them with coordinate and momentum representation, etc. However, none of them question the relevance of digital signal processing for their education -- e.g. processing of video and audio signals for facial and voice recognition, which they can envision as giving them jobs in security or IT industry etc. That's where I can insist it's all the same math (Fourier transforms) and if they can learn it my class they will be better off in classes they consider more applied/useful. I keep reminding them that there is a relatively small number of overarching mathematical concepts that will permeate all of their courses, and if they focus on these math commonalities, they will have a much easier time and get more out of their courses, and see how common math ties all the physical world together and be better engineers for it.

Uncle Al said...

I remember public school word problems in which two kinds of nuts, expensive and cheap, must be mixed to sell for a given price. If you gave the positive root, you passed. If you gave the negative root, you failed. If you presented complex roots, you became an economist.

Given mathematics, sulci and gyri become minima and maxima. Medicine doesn't have saddle points or minimal surfaces

http://www.indiana.edu/~minimal/essays/index.html

yet pornography exists. Go figure. A major use of calculus in US universities is to remove freshmen from matriculation. It is a carefully calculated increment of decrement.

Steven Colyer said...

The numerus clausus, one of many methods used to limit the number of students who may study at a university, is currently used in Germany to address overcrowding and protect specific occupations—while the number of students has increased by 100% to two million since 1980, the number of professors has increased by only 25% in the same time period.

Interesting. Why is that? Is it because the first generation of reunified Germany are attending college and university? How many colleges are there in what was once East Germany compared to what was West Germany, and per capita?

Anyway, education is of course the responsibility of the student, but the teacher's passion for the subject can be infectious. Teaching Math is a challenge to be sure, but you can lose the better students if you teach too slow, which in America anyway given the slow pace per subject, adds to the boredom.

Also, some mathematical notation is bizarre, and textbooks seem to written to be as dull as possible. Education is a wide open field, and not all teachers like their job. If so, that comes through to the student too and may be the kiss of death of their liking let alone loving a subject.

So here's to all the great teachers who know what I'm talking about, and are doing something about it, by teaching with great gusto.

Robert L. Oldershaw said...

"Mathematics is the language of Nature..."

Another version of this statement might be as follows.

Mathematics is the very human language with which we humans describe the orderliness (and disorderliness) of nature - its laws, relationships and patterns.

Take your most powerful microscopes and telescopes and search for integrals, numerical symbols or coordinate systems.

You will find none. You will find relationships that can be described with this abstract symbolic human language, but you will also conclude that nature can get along just fine without any of it. Just by itself.

The closest we get to "the language of nature" is probably conformal geometry, but nature doesn't need a text.

I am not downplaying the importance of mathematics to us humans. We could not do science without it. With mathematics we can approximately understand the laws, relationships and patterns of nature.

But the key idea is that nature is not mathematics and vice versa.

A corollary is that mathematics is a human language, not nature's language (if we are speaking carefully and accurately).

RLO

Oh dear, have I opened Pandora's Box?

Bee said...

Hi Robert,

I didn't say nature is mathematics or vice versa. Best,

B.

Bee said...

Hi Steven,

Why is that? Stupidity. More people studying but not more people being hired to teach them. Law, economics and medicine are typically the fields that are overrun by students. I don't know what the situation is today, but I vividly recall 800 law students trying to squeeze into a lecture hall made for 500. Studying math was very comfortable actually. At least you always had a place to sit.

More seriously, part of the problem is that studying in Germany is cheap. While I do think higher education should be accessible for everybody and am not a friend of high tuition fees, it has the side effect that many people go to university after high school even though they have little or no motivation. This clogs the lecture halls till these people finally figure out they won't pass the exams. In math they notice very quickly, but in other fields it takes longer. For this reason I think the numerus clausus is not a bad idea. It would be better though to have a tougher selection during the first semesters at the university. It would also help if there was more information provided to high school students what to expect and what alternatives there are.

Best,

B.

Phillip Helbig said...

"And being familiar with the exponential function might explain the funny face I made when she recommended some homeopathic remedy in D10. Things went downhill from there."

Obviously, the answer is: mathematics is important in everyday life because it demonstrates that homoeopathy is bullshit.

Having said that, in a D10 solution there is actually some of the active ingredient left in it. Of course, whether it does any good is another question, and in any case it doesn't work in the way that it is believed to.

Is there more, less, or about the same belief in homoeopathy in Sweden as compared to Germany? What about the percentage of people who believe that vaccinations are bad etc?

Bee said...

Hi Phillip,

Yeah, D10 isn't nothing. If it was a plant extract with a low dose of active ingredients to begin with however, one wonders though what's the point of diluting it away.

I have no clue how many people believe in homeopathy in Sweden or Germany. Also, it's not that homeopathic remedies per se don't work, it depends on the dilution. Among midwifes in particular sugar pills are very widely spread though. Tongue in cheek, there's some wisdom to it as pregnant and breastfeeding women should avoid medication if anyhow possible. Best,

B.

Bee said...

Hi Zephir,

Mathematics isn't the same as calculation. As to your question, see this paper and references therein. Best,

B.

Bee said...

Hi Joseph,

It's depressing they think it's a matter of believe to begin with. Maybe the big bang isn't a good example though, as there's physicists who don't believe in it either. But I've made similar experiences with the expansion of the universe. It's like people don't see how one can have evidence for something so detached from every day life. Best,

B.

Phil Warnell said...

Hi Bee,

“What’s mathematics good for?”

For those having a true appreciation of mathematics the answer is it’s good for the soul. That is it allows you to explore and recognize the world as a logically ponderable thing, to find its not simply arbitrary, yet having parameters and relationships within it by which it can defined, understood and yet most importantly appreciated as wonderful.

So although I would agree a lot of the problem rests with the way it’s taught, yet like with art there’s no beauty to be found unless you first have a desire to look. However I do agree it teachers could do better in exposing the truth and beauty to be found in mathematics, but if its quality still goes unrecognized there is little that can be done after that.

“Mathematics possesses not only truth, but also supreme beauty” ”

-Bertrand Russell

Best,

Phil

Plato said...

I agree with Phil in that it is is good for the soul in that such an expression of supreme beauty has to have arrive from "a place."

Symmetry would have us believe that such a place is possible and to where such a statement above in essence is derived and explainable. To go away from such a definition it would have a hard time explaining that beauty:)

Best,

Plato said...

Like nature I am sure you can say what has beauty to do with anything?

Let's say Beauty is "simplicity" and that is what you are looking for in the maths?

Michael Atiyah: Beauty in Mathematics

Best,

Phil Warnell said...

Hi Plato,

Yes with mathematics recognizing its quality is the key. As for instance, although nearly everyone knows, as to have been told, that a circle is the set of points in a plane that are equidistant from a given point, I would ask how many have been told, as to know, it is more simply the shortest line able to enclose the greatest area. That is the former only relates to a possible method for its construction, while the latter to both its truth and beauty, such rendering the circle having a utility found only by appreciating its quality.

Best,

Phil

Georg said...

@ Steven Colyer, Bee,
that numerus clausus in Germany is not from
unification, that is much older (in the western part) Medicine was long en vogeue because "Doctors" uded to earn a lot of money, but that is no longer the case.
Law, was decided by politics to be the field which should stay open at any circumstance to take in the students which were excluded by numerus clausus.
This leads to incredible numbers of failed students, because the exams in the end are very rigorous. (In one state eg there is the rule, that so many students pass the first "state exams" as there are positions for them.)
On top of those failed law students we have a lot of lawyers, wich makes it nearly impossible to live on that, if You dont happen to be the son/daughter of someone in an old and well established lawyer business.
And economics, that is study if You do not know what to do and You know about the risks of law study.

Georg said...

Bee,
one of my daughters refused to learn addition and multiplication in first or second year elementary school, because she "won't need that".
When I asked her, what she would do when grown up, she said she would go cleaning (Putzfrau) as she knew of an woman in neighborhood.
I said then that such cleaners need multiplication to calculate the area they had worked on, to get the money they deserved. This point convinced her. Today, as a judge she is often asked by other judges for help on "that formula" which is used to calculate distribution of cost in series accidents on highways, (when maybe a dozen cars crash into each other one behind the other). She made math in high school quite good, but with real esteem when it came to statistics, she liked it because it helped her in design of games, which she liked to do.
Such examples make me think that You can't make a math genius by motivation, but it helps for everyday skills.

Steven Colyer said...
This comment has been removed by the author.
Steven Colyer said...

Here's a kid who is the opposite of Bee's midwife. At the age of 3, he was taking on Planetarium lecturers, on Astrophysics.

There are probably 2-3000 kids like young Jake spread across the planet, but most likely in abject poverty, so we'll never know. And thinking of that leaves me with no sympathy whatsoever for undergraduate slackers. If the exams are too tough for them, better for society that they failed.

One funny reply to that article:

"Has Jake ever made a mistake? I thought I made a mistake once. But I was wrong."

Zephir said...

/** Mathematics isn't the same as calculation..*/

Of course, but without calculations and real numbers comparable with real values the whole math becomes useless - especially in physics, which relies on testable hypothesis.

So, my question remains the very same: which real physical value are you able to replicate with your math, please? You article doesn't refer any such a value. You reply even demonstrates, you apparently don't understand the meaning of my question.

JimV said...

I have also encountered people who disparage math as being abstract and unnecessary. My counter, which I believe, is that math is just thinking, rationally and logically, so that when you have three errands to do and you pause to determine what the best order to do them is, you are doing math.

The study of math, on the other hand, gives you practice in thinking and solving problems, which should be generally useful, and techniques and special tools for handling some particular types of problems, the latter of which may or may not be useful.

Best Regards,

Jim

Steven Colyer said...

The problem with Math isn't Mathematics (which everyone does in their head naturally), the problem with Math isn't even the obscure notation, which can be very much improved and IS being improved (it's called Computer Science and Algorithms ... "Maths" that actually works i.e., gets things done).

No, the problem with Math is the horrible way in which it is taught, i.e., as a chore.

Mathematics is a thing of beauty, and the best Math teachers know this and pass that joy on. In college. In high school, the "State' dictates how it is taught, and how it is taught is WAY TOO SLOW !

And why is is taught way way too slow? Because, as I said, because of the "State" and their stupid regulations. The "State" is basically Education majors who know nothing about Math.

In America, we teach our pre-schoolers, kindergartners, and 1st and 2nd graders well. Beginning in 3rd grade, there is a huge dropoff until college.

And why is that?

It's because ... the people who dictate programs in Education from grades 4-12 couldn't pass a 4th grade Math test if you paid them. The idiots.

Kaleberg said...

We use mathematics for planting flowers. We usually have a range of dates for when it is safe to put our plants out. We adjust this based on how the season is progressing and our particular micro-climate. (Some areas get more sun than others.) Since we start our seeds in flats, then move them to small pots before moving them outside, we have an estimated indoor growing period which, with the germination time, tells us when to plant the seeds. It's all addition and subtracting, and some range arithmetic and some probability, but we do grow some flowers.

If we were serious and wanted to make money selling flowers to florists or perfume makers, we'd use more math to compute how many plants we could fit in our beds, how much fertilizer they'd need, how to run the irrigation drip lines, and so on. There are a number of trade offs, and they involve costs, yield and risk reduction. We can also use statistics to estimate our selling price and get a sense of whether we'll be making money.
In the early 80s, I worked for a company that produced the first spreadsheet program, and a lot of our customers were grain farmers with Apple IIs running our spreadsheet while they browsed farm newsletters and listened to the Department of Agriculture on the radio. They have all sorts of trade offs on spacing, fertilizer, fuel, insecticides, seed types, subsidies, futures and depreciation.

I gather you can also use math for stuff besides planting things, but without planted food, a lot fewer of us would be doing any mathematics at all.

Bee said...

Hi Phil,

"it’s good for the soul."

I like that. It aptly sums up all the words I clumsily aligned in the above post. Best,

B.

Bee said...

Hi Zephir,

B.

Bee said...

Hi Kaleberg,

I stand corrected ;-) Note however that I only said you don't need it, not that it's not useful. I actually wanted to add a paragraph elaborating on why I think a wider spread knowledge of mathematics would make the world a better place, but then what one considers 'better' is already subjective, so there wasn't really a good point to make. One could maybe agree though that it would make the world run more efficiently. Best,

B.

Phil Warnell said...

Hi Bee,

I’m glad you liked it and as you noted it mainly served just to have you know I was paying attention:-) As from my own perspective, the aspect(s) of mathematics I like to explore is the same that I do in physics, being its foundations and fundamentals. That’s because I find when approached from this direction one needs not to remember so many rules, as its the reason behind its reasonableness becoming more readily apparent as being more important .

I’ve often thought if teachers approached the subject more from this side, it wouldn’t be so laborious for so many students, with having its true methodology and purpose revealed. That is this might have more become disciples of mathematics themselves or at the very least appreciative beneficiaries. That is they may come to discover mathematics being simply an ongoing development of the expression of thought, supported by a scaffolding of logic.

The thing that is most overlooked however, is this development continues, not only by its language of expression being expanded, yet every once in a while also its scaffolding. This leaves it having a richness that is found in few other things and yet also possessing a mystic being equally rare; leaving it as something not totally defined, as not being complete, yet rather seeking completion.

“Ask a philosopher ‘What is philosophy’ or a historian ‘What is History’ and they will have no difficulty in giving an answer. Neither of them, in fact, can pursue his own discipline without knowing what he is searching for. But ask a mathematician ‘What is mathematics’ and he may justifiably reply he does not know the answer but that does not stop him from doing mathematics.”

-Francois Lasserre (Quote noted in “Pi in the Sky: Counting Thinking and Being, by John D. Barrow” (page 1)

Best,

Phil

P.S. I would have noted to your midwife that being able to count contractions and with noting the significance of the diminishing period between them leaving mathematics for her at least having some utility:-)

Steven Colyer said...

GMP wrote:
I teach intro quantum mechanics to engineering students. They are very disinterested in a particle-in-a-box or the free particle, "who cares about standing or plane waves blah blah", and why do I torture them with coordinate and momentum representation, etc. However, none of them question the relevance of digital signal processing for their education -- e.g. processing of video and audio signals for facial and voice recognition, which they can envision as giving them jobs in security or IT industry etc. That's where I can insist it's all the same math (Fourier transforms) and if they can learn it my class they will be better off in classes they consider more applied/useful.

Wow. Why weren't you MY Intro to QM teacher? The Physics professor who taught us Engineers Spec Rev, Gen Rev, and Quantum Physics was the nastiest professor we ever had. He was competent and intelligent, but he had an ego the size of Mt. Everest, and we got the impression he hated teaching "mere Engineers." He totally turned us off when we questioned things like Heisenberg's Uncertainty Principle. He was completely patronizing.

We didn't know what his problem was. Either tenure denied, bad home life, would rather be researching, or just plain nasty. Anyway we were glad when the semester was over. He left us with the impression that Physicists were a bunch of jerks.

We were of course, in error. Not all Physicists have that same personality. All of our Mathematics professors were excellent, however. Very nice men every last one, very willing to back up and answer our questions until we got it.

You're awesome GMP, I love your blog and I added you to my feed. Thanks.

Kay zum Felde said...

Hi Bee,

my math teacher at school told me: "you need math at university", when I asked her what math is good for. As a theoretical physicist math is very important for me, but it is not the only thing. Since one cannot solve every equation, because of its complexity, we are dealing with models. These models are sometimes hard to understand. That is due to intuitive assumptions that are not so intuitive.

Best, Kay

Martin said...

I was asked the same question by a 7th grader when I was his tutor.

My answer: so you can tell when the grown-ups are lying.

Arun said...

I don't need to learn any particular skill today. All I need is - to live in a society where some people have the particular skills, and - the wherewithal to be able to hire them or buy their product.

I don't know farming, but somebody does; I don't know medicine, but somebody does, I don't know law, but somebody does, etc. That is how I survive.

There is no particular skill that is essential to survival and that is true of mathematics as well.

"You don't need to learn maths to survive. Otherwise mankind would have gone extinct long ago" is exactly as true or as false as "You don't need to know agriculture to survive. Otherwise mankind would have gone extinct long ago."

A person ought to understand why he or she should prefer to live in a society where there are good mathematicians than in a society where there are not. That is the minimum of understanding of mathematics that is obligatory on a person as a member of society.

Steven Colyer said...

I was asked the same question by a 7th grader when I was his tutor.

My answer: so you can tell when the grown-ups are lying.

And just to show how great that explanation is, Martin's student went on to win the Fields medal.

Plato said...

`Bees then, know just this face which is of service to themselves, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material used in constructing the different figures. We, however, claiming as we do a greater share in wisdom than bees, will investigate a problem of still wider extent, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest plane figure of all those which have a perimeter equal to that of the polygon is the circle.'

Sir Thomas Heath, A History of Greek Mathematics, Oxford, 1921.Background on the Hexagonal Honeycomb conjecture

While not of the mathematical breed, to my thinking the deductive/inductive powers are still an important feature of being reasonable toward such a simplicity phrased mathematically?

This could be called philosophical in it's beginnings, it is a necessary part of framing such simplicity as to arriving at the basis of let's say social interactions, economically framed as to reveal such a basis as having inherent features of let's say Game theory.

A flock of birds in movement or a school of fish? More then just Sheople:)

Best,