Life Lessons from
Mathematics
Here’s a question for you:
Suppose that one in a thousand people have a serious disease and there
is a medical test which determines whether or not a person has it. However, as
with all tests, it is not 100% accurate. If you have the disease, it will give
the correct diagnosis 100% of the time. If you don’t have the disease it will
give the correct diagnosis 97% of the time. You take the test and it indicates
you have the disease. Should you be concerned? What’s the probability that you
actually have it?
You may be surprised to learn that you can breath a sigh of
relief. Chances are only about 3% that
you have the disease! Surprised? Think of it this way: If 1000 people get
tested, just one of those will likely have the disease and will be diagnosed as
such. However, 3% of the 1000 – that is, 30 people - who are healthy will also be
diagnosed as diseased. Thus there are a
total of 31 people who test positive, but only one of them – about 3% - is
actually sick.
OK – now that you’re thinking more clearly, try this one:
How many people need to be in a room before you can be absolutely certain that
at least two of them have the same birthday (just the day, not the year)? Well, allowing for leap year, it’s possible
(though highly unlikely) that 366 people can each have a different birthday –
each on a different day of the year. However, as soon as one more person enters
the room, s/he HAS to share a birthday with one of the 366 already there. Thus
367 people GUARANTEES that two will have the same birthday.
Sorry, that wasn’t the question. Here it is: How many people
need to be in a room before the probability is greater than one-half that two
will have the same birthday? That is, if
you had to bet, you’d do better to guess that somewhere in the room two people
share a birthday. What do you
think? A common guess is 183 (half of
366). However, you’re likely to be just
as surprised as before. It turns out that if the room has just 23 people in it,
chances are better than 50% that two will have the same birthday. Don’t believe
me? The math is too involved for this column, but try it out. Have everyone in a room write down their
birthday (and relatives as needed) so that the total number of birthdays is,
say, over 35. Then go around and have
everyone give the dates - - you’ll very likely find a match.
I have spent the last fifteen years going to high schools,
colleges, and universities around the country telling about Elvis, my (late) Welsh
Corgi. When I threw a ball down the beach and into the water, he ran down the
beach and jumped into the lake at just the right point to retrieve the ball in
minimum time. (For details, google: elvis, corgi, calculus.)
My favorite part of the talk was when I asked what Elvis
should do if we backed up another 20 yards, but still threw the ball to the
same place in the water. Most everyone
guessed that Elvis’s water-entry point should also back up a bit. Using calculus, I showed them that their
intuition was wrong. Elvis’s entry point remained the same.
These examples show the power of mathematics. They also show the value of a formal
education. As I tell students, one can
learn much about life by getting a job, or traveling the world, or joining the
military, or just living out on the street. However, a formal education
provides valuable and life-changing insights in mathematics, literature, and
science – insights that one won’t likely learn on one’s own. That is why a liberal arts education is truly
liberating.
These examples provide another lesson as well. Our gut intuition
and beliefs – our common sense – may well be dead wrong, even when others are
in agreement with us. Think about that.
There may be things that you and those around you are convinced are true, but
in fact are not. In mathematics it’s relatively easy to prove the error. In
other areas we have no choice but to hold our beliefs with a dose of humility realizing
that maybe – just maybe – we’re all mistaken.
That’s not a bad lesson to learn – at least that’s what my
gut tells me.
On some level, though, doesn't mathematics also sometimes rely on your gut? I remember that Nova special on Fermat's Last Theorem you showed us. For years no one had solved it, and some mathematicians had declared it unsolvable, that Fermat had missed something in his long-lost proof to his own theorem, etc. Then someone "solved" it, and all mathematicians rejoiced (and maybe got a bit jealous)! Then someone else pointed out a flaw in his logic, so the theorem regained its unsolved status. Then someone else came along and actually solved it (we think). The amazing part of this whole scene was that there was very little argument at any point. The theorem was agreed solved by most or all mathematicians until someone pointed out the flaw, so all mathematicians now agreed it was unsolved. Then someone else came and fixed the flaw, and all mathematicians agreed it was now solved. Sometimes our gut instincts are wrong; but sometimes our guts are right, and our training can actually work against us. If you haven't yet, check out Malcolm Gladwell's book "Blink" for some interesting examples of gut instincts and first impressions.
ReplyDeleteGood point. Yes, at a deep level, we don't know anything for certain because our senses and experience are faulty at times - thus not 100% reliable, and we get tricked in mathematics/logic.
ReplyDeleteBut more practically, we attain "essential" certainty after enough people have carefully thought through the proof of a math theorem that we take it for truth. That is different from a quick "here's my gut reaction" response. So even though qualitatively the same, I'd argue that one is much more sure than the other.
That's probably true, though I also believe in the danger of "overthinking" and that sometimes first impressions are right, especially if you have experience to back it up. The story of the Getty kouros, which opens that Gladwell book "Blink," is very interesting. This master forger had fooled all those American curators, people at the top of their field. They (curators) used every tool in their training: stylistic analysis, technical analysis, and even carbon dating to confirm the statue's authenticity. When some of their European counterparts came over to witness the statue, their first reaction was revulsion. One of them even said within two minutes of laying his eyes on it, "I hope you didn't pay for this yet."
ReplyDeleteThat said, it's important to note that those European curators were equally trained in their craft. All that training we get in order to think through things logically also enables us to have better-educated first impressions and gut reactions. Good cops can tell right away when someone's lying, good therapists know right away what to say to their patients, etc. Gladwell's thesis, and he has a fair bit of science to back it up, is that the careful analysis we do more often confirms our first impressions rather than upsets them.
This comment has been removed by the author.
ReplyDelete