Chapter I. The adventure begins.
Contents: 1. Daddy, what's the biggest number?
2. What are the prime numbers, and what do they have to do with space aliens?
3. There ain't no biggest prime neither.
4. A brain like a sieve.
5. Twin primes, and proofs Sudoku style.
6. Your cellphone is crawling with prime numbers.
1. Daddy, what's the biggest number?
Anyone who's been around small children much has probably heard a
question like this one. Your first answer is probably ``actually there
isn't a biggest number'', to which the child will likely reply ``why
not?''. Well, you say, suppose you had any enormous number, like a
gazillion. If you just add one to it you get a gazillion and one, which
is even bigger. So no matter how big a number you take---the number of
stars in the sky, the number of grains of sand on the beach---you can
always add one to make it bigger.
Another way to say it (not recommended when explaining things to a little
person) is that there are infinitely many whole numbers 1,2,3,..., or
``positive integers'' as we like to call them. So it's a subtle concept,
involving as it does the notion of infinity. In any case, your
explanation to the child is a simple example of a ``proof by
contradiction''. A bit more formally, it would go like this:
I claim there is no biggest number. Proof: Suppose (in order to derive a
contradiction) that there was a biggest number. Let's call this
presumptuous number ``N''. (You could call it ``Finley'' or ``Hannah'' or
``thingamagig'', but N is short and sweet and serves to remind us that it
stands for a number.) So, Mister N, you claim to the biggest number?
Hah! I'll just add one to you to get a bigger number N+1. This
contradiction proves you are a fraud, a liar, and a menace to
society. No ``biggest number'' exists!
This might seem like making a mountain out of a molehill. But it's
always good to warm up, to stretch out your brain a bit before moving
on, so as to avoid blowing out any neurons. We'll use a more subtle
version of the same idea later to prove there is no biggest prime
number.
2. What are the prime numbers, and what do they have to do with space
aliens?
Let's think about some things you can do with numbers. You can add them,
for instance, or you can multiply them. Let's indicate multiplication
with an asterisk: 2*3=6, 3*5=15, etc. (I do this only because of the
limited symbols available on this blog page). One interesting
observation is that some numbers can be ``factored'' and others
cannot. For example 15=3*5, but there is no way to factor 7 other than
the trivial factorization 7=1*7. This state of affairs even has a nice
geometric interpretation: If you want to put 15 chocolates in a box you
can arrange them in three rows of five each:
x x x x x
x x x x x
x x x x x
But you can't do it with 7, other than by putting them all in one boring
row; no matter how you do it there are rows of different lengths, for
instance.
x x x x
x x x
So we've discovered some very special and interesting numbers, namely
those numbers that cannot be factored into smaller numbers. If we were
the first to discover these numbers we could name them whatever we like.
We could call them ``thin numbers'', ``groovy numbers'', or ``apple
fritters'' if we so desired. But we're a few millenia late to have
naming rights, so we'll stick to the standard term ``prime number'' or
simply ``prime''. While we're at it, though, it behooves us to define
our terms precisely. The description we just gave ``numbers that can't
be factored into smaller numbers'' is precise enough for our purposes,
but if taken literally it includes the number 1 as a prime. There is
nothing wrong from a logical standpoint with counting 1 as a prime, but
if you do so you'll soon discover that to say anything interesting about
primes you'll constantly have to insert disclaimers such as ``suppose p
is a prime other than 1''. Experience shows that it's better to just exclude
it by fiat. So here's our official definition:
A prime number is a number greater than 1 that cannot be factored into
smaller numbers.
You can (and should!) check that the primes less than 30 are the
following:
2,3,5,7,11,13,17,19,23,29
Except for 2, all prime numbers are odd. Why?
In one of my favorite movies, ``Contact'' (based on the book by Carl
Sagan and starring Jodie Foster), beings from beyond our galaxy send out
radio pulses in prime groups: first two, then three, then five and so
on. The point is that prime-ness is an intrinsic property of numbers,
completely independent of what symbols are used to denote them, whether
you use base ten as we do, what language you speak, and so on. The
extra-galactic reasoning therefore is that any intelligent form of life
will recognize the sequence of primes and realize that another
intelligent life-form is attempting to make contact.
Prime numbers are important because they are the building blocks for
making all whole numbers, via multiplication. If you take any such
number greater than one, you can factor it as a product of prime
numbers. For example, 30=2*3*5, 72=2*2*2*3*3.
But hold on just a doggone minute! Giving a couple of examples doesn't
prove anything. How do we know that ALL numbers admit such a
factorization into primes?
Well, suppose someone walks up to you on the street and hands you a
positive integer. Let's call this number N. If N happens to be a prime
number, then N is its own factorization (with just one factor). If it
isn't prime, then N can be factored into two smaller numbers N=a*b. If a
and b are primes, then we have our factorization. If not, then they in
turn can be factored into smaller numbers. If we continue this process,
it has to stop after a finite number of steps, because positive integers
(just good old whole numbers, remember) can't keep getting smaller
forever. When the process stops, we have written N as a product of
smaller numbers, none of which can be factored further and so are primes
by definition. Here ends the (convincing, I hope) explanation.
Let's illustrate with a specific example. Say you started with
N=60. There are multiple ways you might do the first step, but probably
the most obvious would be 60=6*10. Neither 6 nor 10 is prime, so we
factor them further: 6=2*3, 10=2*5. We luck out and are done after two
steps: 60=2*3*2*5 is our prime factorization. However, and this is a
crucial point, an example does not constitute a proof. A proof is a
convincing argument that works in ALL cases under consideration. It
should hold up in a court of law under rigorous cross-examination. The
person-on-the-street argument given above is an example of a proof. In
a textbook you might see something like this:
Theorem. Every number greater than 1 can be factored into primes.
Proof: (Insert here a cleaned-up, more precisely worded version of the
version above.)
There's nothing wrong with this style, but some people are put off
and/or intimidated by the words ``Theorem'' and ``Proof''. So if you
prefer, rewrite it as follows:
Cool Fact: Every number greater than 1 can be factored into primes.
Here's why it's true, dude: Suppose someone walks up to you on the
street...etc.
Also important to note is that we didn't START by writing down Theorem:
blah blah blah Proof: blah blah blah. We started with a question. We
found an answer and convincingly demonstrated its truth. The more formal
``theorem'' and ``proof'' come later, after the smoke has cleared and
the dust has settled. The reason that we mathematicians often write out
our proofs very pedantically, dotting every i and crossing every t, is
that the deeper the mathematics, the easier it is to make subtle
mistakes. Very careful reasoning is required. But that's not the form in
which the proof is discovered.
Note also the following corollary of the theorem.
Corollary: If N is bigger than 1, then there is at least one prime that
divides N.
(By ``divides N'' I mean the same thing as ``is a factor of N''. In
mathematics, as in other languages, there are often different words for
the same thing.)
In mathematics a ``corollary'' of a theorem is a second theorem (cool
fact!) that follows easily from the first. What constitutes ``easily''
is of course a matter of opinion, but in the present case it's like,
totally easy: The theorem says N can be factored into primes, and any of
these prime factors will divide it.
3. There ain't no biggest prime neither.
What's the biggest prime? This would be good to know, as then we'd have
a finite list of the primes, the basic building blocks for all numbers.
Well, there ain't no such thing. The Greek mathematician Euclid (who
lived around 300 BC and is most famous for his work in geometry), found
a clever argument to prove that there are infinitely many primes. His
proof is by contradiction. Such a proof really is like a
cross-examination. Imagine that Mr. P is on trial for mathematical
fraud, having sold stock based on his claim to be the largest prime
number. The defense has called him as a witness, and Euclid, the
prosecuting attorney, now proceeds to the cross-examination.
Euclid: Mr. P, you claim to be the largest prime, is that correct?
P (with a disgusted look): Yes.
Euclid: I see. (He pauses, shuffling a few papers.) Now Mr. P, suppose I
multiply together the first few primes---let's say 2,3 and 5, and then
add 1. Is this number divisible by 2, 3, or 5?
Defense: Objection, your honor! Irrelevant!
Judge: Overruled. I'll give you some leeway, Mr. Euclid, but this better
be leading somewhere. The witness may answer the question.
P (sarcastically, as though talking to a child): Well, 2 times 3 times 5
is 30, and if I add one I get 31. That isn't divisible by 2, 3 or 5. If
that was too fast for you I can go over it again.
Euclid: All right, but isn't it true that you really didn't need to do
the multiplication at all? I mean, if you multiply 2,3 and 5 together,
then without even writing down the answer you know that adding 1 will
produce a number whose remainder, after division by 2,3 or 5, is 1.
Defense: Objection! I fail to see what this has to do with---
Judge: Sit down, counselor! I'll allow it, but I warn you, Mr. Euclid,
that my patience is wearing thin.
P: Well yeah, sure. I could take 2,3,5,7,11, multiply them together,
add 1 and know in advance that if I divide any of those five primes into the
new number the remainder will be 1. That's how remainders work. But
maybe they didn't cover that when you went to school.
Euclid (continuing to ignore P's sarcasm): So if you're the biggest
prime, there is a finite list of ALL primes and we can multiply them all
together. Call this number N. Then N+1 is not divisible by any of the
aforementioned primes, because if you divide by any prime on the list
you'll get a remainder of 1.
P: Right. So what?
Euclid: I call the attention of the court to the earlier testimony of
expert witness Herr Professor Doctor Mitchell, who testified that for
any number bigger than 1, there is SOME prime that divides it. So there
is a prime---I'll call it q, if it please the court---such that q
divides N+1. In other words, the remainder after division by q is 0, not
1. So this prime q is not on the list, it is bigger than you, and your
claim is a lie!
(Pandemonium erupts in the court. Mr. P is found guilty and sentenced to
a year of watching Dragnet re-runs.)
More formally, say for publication in the Proceedings of the North
Kirkland Philosophical Society, one might write things out like so:
Theorem. There are infinitely many primes.
Proof: Suppose (in order to reach a contradiction) that the theorem is
false, i.e. that there are only finitely many primes. Multiply them all
together (this is possibile because there are only finitely many); call
the resulting number N. Then dividing any of the primes into N+1
produces a remainder of 1, so no prime divides N+1. But since N+1 is
greater than 1, there IS a prime that divides. We have therefore arrived
at a contradiction, proving that the theorem is true.
4. A brain like a sieve.
When I was a boy, my father used to say I had a ``brain like a sieve'',
a reference to the absent-mindedness that apparently was already
evident. Even sieves can be put to good use, however.
To expand our above list of primes more efficiently, we can use a method
picturesquely known as the ``Sieve of Eratosthenes''. Eratosthenes (276
BC-195BC) was a Greek man of many talents---mathematician, poet,
``Father of Geography''. The Wikipedia article on him includes an
color-coded animation of the sieve being used to find all primes less
than 121. But let's see how you might discover this method yourself to
find all primes below 121. Start by writing down all the numbers from 2
to 120. Or at least imagine that you've done so; as you'll see, there
are many shortcuts to speed up the process.
1. Cross out all even numbers greater than 2 on your list, since these
have 2 as a factor and so can't be prime. This is superty-duperty easy,
as Kaia would say, because in our customary base 10 a number is even if
and only if its last digit is even. But hey, even without knowing this,
you could just start at 4 and cross out every other number 4,6,8,....
2. Cross out all numbers with 3 as a factor, other than 3 itself. So you
cross out 6, 9, 12, 15,...ah, but actually all the even ones 6, 12, 18
are already gone.
3. Skip to the next number that hasn't yet been crossed out. It has to
be a prime (why?), in this instance 5. Cross out all multiples of 5,
which is very easy in base 10 because they are the numbers whose last
digit is 0 or 5 (why?). All such multiples that also multiples of 2 or 3
have already been crossed out, speeding up the process further.
4. Repeat step 3. The next number not already crossed out is 7, which
indeed is prime as claimed. Proceed as before to cross out all multiples
of 7 (there won't be many left).
5. Repeat step 3. The next number not already crossed out is 11...but
wait a minute, we're already done! There's no need for Step 5! We know
without even looking that every multiple of 11 has already been crossed
out. Do you see why? (Hint: 121=11*11.) If you can answer this then you
see why we can stop the process at Step 4. All remaining numbers on the
list are primes!!
For future reference, here's the list of all primes below 121:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,
107, 109, 113.
5. Twin primes, proofs Sudoku style, and other diversions.
If you ponder the above list of primes, you notice some things about the
way they're ``distributed''. For example, every now and then one
encounters ``twin primes'', meaning two consecutive odd numbers that are
both prime, such as 3,5 and 29,31. Taking our cue from Euclid, we ask:
Are there infinitely many such pairs of twin primes? No one knows! This
is an example of an unsolved problem in mathematics. If you can answer
it you will be famous, and probably will rake in some big bucks too.
Okay, what about ``triplet primes'', meaning three consecutive odd
numbers that are prime, such as 3,5,7? Staring at the list above, you'll
see that 3,5,7 are the only triplet primes below 121. It's a fun problem
to prove that these are the ONLY triplet primes. I'll give some hints
after a digression on Sudoku (which you can skip if you're not familiar
with the game).
When you play Sudoku, you are doing mathematical proofs. The proofs in
question could be styled ``proofs by contradiction'' or ``proofs by
process of elimination''. In the simplest case it works something like
this: Maybe you have a row with only three squares remaining to be
filled in, so that you know that one of the squares has to be, let's
say, a 3,5 or 8. But that square already has a 3 in its column, and an 8
in its 3 by 3 block. So by process of elimination you have PROVED that
it has to be a 5. Sudoku is pure mathematics!
Now, back to the triplet prime problem. Usually I don't like to give
hints, because they interfere with your creativity. This is true at all
levels of mathematics. But since we're just getting started, and you
might still be uncertain about what a ``proof'' is, I'll make some
suggestions. First of all, experiment with small numbers to get an idea
of what's going on. Examples don't constitute a proof, but they can
SUGGEST a proof. For instance, 5,7,9 aren't triplet primes because 9
isn't prime; it's divisible by 3. Nor does 13, 15, 17 work, as 15 isn't
prime (it's divisible by 3). Or try 27, 29, 31. Nope; 27 is divisible by
3. In fact it appears that in every case, one number in the triplet
is divisible by 3. This suggests proceeding as follows: We need names
for our triplets; let's call the smallest one n. Then the other two are
n+2 and n+4. Try proving that one of the numbers n,n+2,n+4 has to be
divisible by 3 and therefore can't be prime (except in the special case
n=3). Here's where ``Sudoku style'', or process of elimination, comes
in. If n or n+2 is divisible by 3 we can quit; our theorem is proved. So
suppose that n and n+2 are NOT divisible by 3 and prove that this forces
n+4 to be divisible by 3. Further hints available on request!
Here's another interesting question, at the opposite extreme from twin
primes. How big can the gaps between consecutive primes be? If you look
at the list of primes below 121 given above, you'll see that for a long
time there is no gap bigger than 6, e.g. 23 and 29, and then we suddenly
get a gap of 8 from 89 to 97. In fact the gaps can be arbitrarily large.
Once you have the right creative idea, the proof of this fact is
surprisingly ``easy''. But of the course that little idea is the catch!
Here it is: Let's use the traditional, centuries-old notation n! to mean
the product of all the numbers from 1 to n. Thus
3! =1*2*3=6
4! =1*2*3*4=24
5!= 1*2*3*4*5=120
and so on. I don't know who first came up with the exclamation point
notation, but it was long ago. You could just as well use a heart, a
smiley-face, or a skull-and-crossbones, but n! is the standard (to be
read as ``n factorial''). Now consider the following list of
consecutive numbers:
n!+2, n!+3, n! +4,...,n!+n.
Here the ellipses ... are a standard way of saying ``keep going in this
way until you get to n!+n''. Then no number in this list is prime. Do
you see why? Prove this to yourself! There are n-1 numbers on the list,
so the gap between the biggest prime less than n!+2 and the next prime
is at least n-1. Since n can be any number here---a thousand, a million,
a gazillion, whatever---this proves that there can be arbitrarily large
gaps between consecutive primes.
The distribution of primes has been intensively studied, and is
currently a very active area of research. The biggest unsolved problem
in this area, often regarded as the biggest unsolved problem in
mathematics, is the ``Riemann hypothesis''. It was first conjectured by
the brilliant German mathematician Bernhard Riemann (1826-1866). The Clay
Research Institute has offered a million dollars for its solution.
(Unfortunately, I can't explain it here; you'd need a solid background
in undergraduate mathematics.)
6. Your cellphone is crawling with prime numbers.
In his 1940 essay ``A mathematician's apology'' (the word ``apology'' is
used in the sense of justification), the prominent British mathematician
G.H. Hardy (1877-1947) claimed that number theory would never have any
practical applications. Number theory was his field, and he regarded the
lack of applications as a good thing, as then it could never be applied
to war. Within a generation or two he was proved spectacularly wrong, as
number theory---this includes the prime numbers we've been
discussing---is absolutely crucial for our current technology: the
internet, CD's, DVD's, cellphones, anything to do with computers and
data transmission. There is an ``elliptic curve'' in your cellphone, for
example, and it is crawling with prime numbers.
The main applications that I'm aware of (and I've only scratched the
surface) have to with (1) encryption, and (2) error-correcting
codes. Secure data transmission---of your bank information or your
credit card number, of military secrets---requires that it be
encrypted. An ingenious way of doing this using prime numbers was
invented about thirty years after Hardy's statement that number theory
would never be applied. Later a colleague of mine in the UW Math
department, Neil Koblitz, invented a new approach based on mathematical
objects called ``elliptic curves''; this too involves much number
theory. You literally have an elliptic curve in your cellphone.
Setting aside security, there is still the issue of accuracy. All data
transmission is subject to error. Minor mistakes could erase your bank
account, turn CD's and DVD's into gibberish, and cause a spacecraft to
land in Hoboken instead of on Mars. To prevent this, sophisticated
schemes have been developed that allow the machine receiving the signal
to not only recognize transmission errors, but also correct them in
real-time. These ``error-correcting codes'' are of mathematical origin,
and again involve primes and other aspects of number theory.
As I said at the outset, it's not my goal to talk about applications.
But there are millions of applications out there. If you want to learn
more, try searching ``public-key cryptography'', ``elliptic curve
cryptography'' and ``error-correcting codes''. I can't promise that it
will be easy reading, however.
Monday, June 8, 2015
Saturday, May 23, 2015
A mathematician explains himself. Introduction.
It happens all the time. I'm at a social gathering, I meet someone new,
and the standard question comes up: ``So, what do you do?'' I'm tempted
to limit my reply to ``I'm a professor at the university'', but this
only postpones the inevitable. It's better to just get it over with:
``I'm a math professor at the university''.
At this point there are several common responses:
1. The person steps back in horror, as though I had just announced I had
the bubonic plague. Then, perhaps recalling that mathematics is not
contagious, or at least not fatal, they recover and reply: ``I was never
any good at math.''
I've never understood this. If I'd said I was a biology professor, would
the response be ``I was never any good at biology''? If I played the
bassoon in the orchestra, would they reply ``I was never any good at the
bassoon''? I doubt it. Yet a surprising number of people seem compelled to
immediately proclaim their incompetence in mathematics. I myself am
hopeless at any form of visual art, but I don't feel the need to
announce this every time I meet an artist. But what I really want to say to
the person, and sometimes do, is this: ``What makes you think you're no good
at math? I don't believe it!'' The same goes for any reader of this blog
who thinks they are ``no good at math'' or claims not to like math. In
this series of posts, I'm going to try to prove you're wrong.
2. ``oh, you must balance the checkbook in your family''. That's
hilarious, especially for Wendy. Even worse:
``So do you know Professor so-and-so in the Accounting Department?''
Huh? No, I don't know Professor so-and-so, nor do I know Professor
Whatshername in the Department of Finnish Literature. Well, I don't
actually say that, recognizing all too well what prompts the question:
For many people, ``math'' is the same thing as ``arithmetic''. I was
astonished to find a website asserting that ``there is no math in
Sudoku''. On the contrary, this popular game is PURE mathematics. What
the website author meant is that there is no arithmetic in Sudoku, which is
true, but the unfortunate choice of language insinuates that if there
was math in it, it couldn't be fun. On the contrary, if you like Sudoku
then you are a mathematician! I'll explain this in a later post.
In any case, I have nothing against accounting, but it has nothing to do
with the mathematics I love. Music would be much closer.
3. ``So you teach math?'' Well, yes. Often I leave it at that, which
leads to an easier, more pleasant conversation in which I can talk about
the courses I teach, what the students are like, and so on. Only in my
more optimistic moods do I venture to add: ``And I do research, which in
fact is about half of my job.''
This statement is often met with outright incredulity. ``You can do
research in MATH???'' A cousin of my wife's, in his early 20's at the
time, found the idea so absurd that he literally burst out laughing,
and asked: ``What do you do, look for new ways to find...[here he
paused, evidently searching for the most advanced, esoteric mathematical
concept he'd heard tell of]...square roots?''
I can understand why it comes as a surprise to most people that there is
such a thing as mathematical research. We mathematicians have to take
some of the blame for this state of affairs; evidently we've done a bad
job of explaining ourselves to the general public. What I don't
understand is this attitude of willful ignorance, all too common in our
country: ``If I can't imagine it, then it doesn't exist''.
Yes, you can do research in mathematics. People have been doing it for
three thousand years, and tens of thousands of people around the world
are doing it at this moment. And when I say you, I mean YOU! In the
course of these posts I'll mention some unsolved problems that anyone
can work on, or at least play around with.
4. ``Does it (mathematical research) have any practical applications?''
This is certainly a reasonable question, but a very frustrating one to
answer. On the one hand, there is no field of knowledge that has more
practical applications than mathematics. It has been pervasive for
centuries, and especially since the advent of computers. In fact, a
striking trend over the last fifty years or so is that EVERY branch of
pure mathematics, no matter how abstract or esoteric it might appear,
finds applications sooner or later. On the other hand, this is not why I
love mathematics, and therein lies the frustrating point. I love
mathematics for its intrinsic beauty; the applications are just a bonus.
I could talk about practical applications, many of which are quite
amazing, but that's not what I personally think about. So in these essays
I'll mention an application or two when I can, but my main goal is to
give some inkling at least of why mathematics in and of itself is so
fascinating. And fun!
The fun will begin in the next post. Meanwhile, if you had any bad
experiences with math in school, banish all such memories from your
brain. Did you find it boring to recite multiplication tables, solve
long lists of algebra equations, analyze equilateral triangles or memorize
trig identities? Excellent! This indicates good taste, and that you have
a future as a mathematician. Or perhaps you enjoyed geometry but not
algebra, or vice-versa. This too is excellent; mathematics is a vast
subject and there is nothing wrong with preferring one branch over
another. Most important of all, get rid of the idea that mathematics was
all written in stone a thousand years ago, and the Big Mathematical
Cheese in the Sky handed it down to a few select prophets on the
mountaintop. Mathematics is a vibrant, living, human enterprise. WE are
the mathematicians, and this includes YOU. Relax your mind. Return to
the innocent, unfettered curiosity that all children have. And enjoy!
and the standard question comes up: ``So, what do you do?'' I'm tempted
to limit my reply to ``I'm a professor at the university'', but this
only postpones the inevitable. It's better to just get it over with:
``I'm a math professor at the university''.
At this point there are several common responses:
1. The person steps back in horror, as though I had just announced I had
the bubonic plague. Then, perhaps recalling that mathematics is not
contagious, or at least not fatal, they recover and reply: ``I was never
any good at math.''
I've never understood this. If I'd said I was a biology professor, would
the response be ``I was never any good at biology''? If I played the
bassoon in the orchestra, would they reply ``I was never any good at the
bassoon''? I doubt it. Yet a surprising number of people seem compelled to
immediately proclaim their incompetence in mathematics. I myself am
hopeless at any form of visual art, but I don't feel the need to
announce this every time I meet an artist. But what I really want to say to
the person, and sometimes do, is this: ``What makes you think you're no good
at math? I don't believe it!'' The same goes for any reader of this blog
who thinks they are ``no good at math'' or claims not to like math. In
this series of posts, I'm going to try to prove you're wrong.
2. ``oh, you must balance the checkbook in your family''. That's
hilarious, especially for Wendy. Even worse:
``So do you know Professor so-and-so in the Accounting Department?''
Huh? No, I don't know Professor so-and-so, nor do I know Professor
Whatshername in the Department of Finnish Literature. Well, I don't
actually say that, recognizing all too well what prompts the question:
For many people, ``math'' is the same thing as ``arithmetic''. I was
astonished to find a website asserting that ``there is no math in
Sudoku''. On the contrary, this popular game is PURE mathematics. What
the website author meant is that there is no arithmetic in Sudoku, which is
true, but the unfortunate choice of language insinuates that if there
was math in it, it couldn't be fun. On the contrary, if you like Sudoku
then you are a mathematician! I'll explain this in a later post.
In any case, I have nothing against accounting, but it has nothing to do
with the mathematics I love. Music would be much closer.
3. ``So you teach math?'' Well, yes. Often I leave it at that, which
leads to an easier, more pleasant conversation in which I can talk about
the courses I teach, what the students are like, and so on. Only in my
more optimistic moods do I venture to add: ``And I do research, which in
fact is about half of my job.''
This statement is often met with outright incredulity. ``You can do
research in MATH???'' A cousin of my wife's, in his early 20's at the
time, found the idea so absurd that he literally burst out laughing,
and asked: ``What do you do, look for new ways to find...[here he
paused, evidently searching for the most advanced, esoteric mathematical
concept he'd heard tell of]...square roots?''
I can understand why it comes as a surprise to most people that there is
such a thing as mathematical research. We mathematicians have to take
some of the blame for this state of affairs; evidently we've done a bad
job of explaining ourselves to the general public. What I don't
understand is this attitude of willful ignorance, all too common in our
country: ``If I can't imagine it, then it doesn't exist''.
Yes, you can do research in mathematics. People have been doing it for
three thousand years, and tens of thousands of people around the world
are doing it at this moment. And when I say you, I mean YOU! In the
course of these posts I'll mention some unsolved problems that anyone
can work on, or at least play around with.
4. ``Does it (mathematical research) have any practical applications?''
This is certainly a reasonable question, but a very frustrating one to
answer. On the one hand, there is no field of knowledge that has more
practical applications than mathematics. It has been pervasive for
centuries, and especially since the advent of computers. In fact, a
striking trend over the last fifty years or so is that EVERY branch of
pure mathematics, no matter how abstract or esoteric it might appear,
finds applications sooner or later. On the other hand, this is not why I
love mathematics, and therein lies the frustrating point. I love
mathematics for its intrinsic beauty; the applications are just a bonus.
I could talk about practical applications, many of which are quite
amazing, but that's not what I personally think about. So in these essays
I'll mention an application or two when I can, but my main goal is to
give some inkling at least of why mathematics in and of itself is so
fascinating. And fun!
The fun will begin in the next post. Meanwhile, if you had any bad
experiences with math in school, banish all such memories from your
brain. Did you find it boring to recite multiplication tables, solve
long lists of algebra equations, analyze equilateral triangles or memorize
trig identities? Excellent! This indicates good taste, and that you have
a future as a mathematician. Or perhaps you enjoyed geometry but not
algebra, or vice-versa. This too is excellent; mathematics is a vast
subject and there is nothing wrong with preferring one branch over
another. Most important of all, get rid of the idea that mathematics was
all written in stone a thousand years ago, and the Big Mathematical
Cheese in the Sky handed it down to a few select prophets on the
mountaintop. Mathematics is a vibrant, living, human enterprise. WE are
the mathematicians, and this includes YOU. Relax your mind. Return to
the innocent, unfettered curiosity that all children have. And enjoy!
Wednesday, May 13, 2015
CT Report, coming attractions
Friday I had another CT-scan and today we saw the oncologist. There is no measurable increase in the cancer, and I am considered "in remission". The doc was careful to emphasize that remission does not mean cure, but still this is great news. Oddly, however, my reaction has been somewhat subdued, for the simple reason that it's what I expected all along anyway! I'll have another scan in 3 months, at which time I'll expect the same result, in which case the next scan would be 6 months later, and so on until after three years or so they consider it gone.
In fact an actual cure, i.e. complete elimination of the cancer, is very rare (he says), especially when it has already spread from the bladder, but does happen. But I've been an exceptional case from the beginning, for instance he commented again on how unusual it is not to have experienced any pain from the cancer. So I just assume I will continue to be exceptional and will totally kick the cancer's butt, sangue della Madonna! (This last is one of many interesting curses from 19th-century Sicily, "blood of the Madonna", "sangue di un cane!" (blood of a dog; sorry Zoe), etc.. I've been learning quite a few from the novels of Giovanni Varga.) And once again, I'll give no thought whatever to the matter until the next scan, as I have much more interesting things to do!
Coming attractions for Dr. Steve's blog: I started another installment of "la bella lingua", but it's getting too technical for those who don't already know some Italian. So I'll quit on that. Instead I'm going to start a new series "A mathematician explains himself", loosely based on an essay I originally wrote in Italian. I hope to get a few more readers for the English version! It is intended for readers who haven't had any math beyond high-school, and for whom even the high-school math may be a distant, hazy, or painful memory. I hope it will be fun, and just remember, if you enjoy Sudoku then you are a mathematician whether you know it or not!!!!
In fact an actual cure, i.e. complete elimination of the cancer, is very rare (he says), especially when it has already spread from the bladder, but does happen. But I've been an exceptional case from the beginning, for instance he commented again on how unusual it is not to have experienced any pain from the cancer. So I just assume I will continue to be exceptional and will totally kick the cancer's butt, sangue della Madonna! (This last is one of many interesting curses from 19th-century Sicily, "blood of the Madonna", "sangue di un cane!" (blood of a dog; sorry Zoe), etc.. I've been learning quite a few from the novels of Giovanni Varga.) And once again, I'll give no thought whatever to the matter until the next scan, as I have much more interesting things to do!
Coming attractions for Dr. Steve's blog: I started another installment of "la bella lingua", but it's getting too technical for those who don't already know some Italian. So I'll quit on that. Instead I'm going to start a new series "A mathematician explains himself", loosely based on an essay I originally wrote in Italian. I hope to get a few more readers for the English version! It is intended for readers who haven't had any math beyond high-school, and for whom even the high-school math may be a distant, hazy, or painful memory. I hope it will be fun, and just remember, if you enjoy Sudoku then you are a mathematician whether you know it or not!!!!
Monday, April 20, 2015
La bella lingua, part IV, and Kaia as herpetologist
``There's a misunderstanding about snakes,'' Kaia told me, ``when they
wrap around your arm it doesn't mean they want to squeeze you and eat
you. Because if people didn't have arms, how would they hug?''
That last statement might seem obscure--unless you spend a lot of time
around 5-year olds, in which case the meaning is clear: If the snake had
arms, it would give you a hug. And without arms, people too would have
trouble hugging, so it isn't fair to blame the snake. (Kaia was recently
at some kid event or another where they got to handle real snakes.)
Sunday Jessie, Kaia, Finley and I went on an expedition up Denny Creek,
a wonderful little ravine near our house that very few people frequent
(it's part of Denny Creek Park, but almost everyone stays on the other
side of the road, by the lake). Among other things it features the
biggest tree in King County, 26 feet in circumference. Unfortunately it is
now only 40 feet high, the rest having been broken off in a storm years
ago. But it's still pretty impressive.
If we dawdled too long, Finley would urge us on: ``come on, let's go on
our way!'' At the end he announced that ``we are at the end of a very
long journey''. Kaia took great delight in finding ways to cross the
creek, via stepping stones, logs etc. After a tricky crossing she would
say, contemplating the return, ``this was not such a good idea'', and
then after getting back, ``I have saved my own life!''.
On to things Italian. Or French, as the case may be. I finally finished
the 60-hour audiobook of ``I miserabili''. It was quite an experience,
even apart from listening to it in Italian. Anyone else out there read
it, in whatever language? I found the story to be very involving, even
gripping at times. But to maintain interest and suspense one has to skip
past Hugo's strange historical digressions. An extreme example: Valjean
has rescued the unconscious Marius at the barricades, carrying him on
his back and fleeing into the labyrinthic Paris sewers. Just at this
suspenseful moment, Hugo digresses for several chapters on the history
and design of the Paris sewer system. Later Valjean encounters
quicksand in the sewers, at which point we get a lecture on the
different types of quicksand and where they can be found.
The core of the novel still has some flaws that might diminish its
appeal to some readers, notably the many absurd coincidences that are
contrived to reconnect various threads of the plot. When Valjean and
Cosette are running for their lives in Paris, with relentless policeman
Javert hot on their heels, they are able to climb over a high wall into
a large garden. Desperate for a hiding place, Valjean takes the risk of
approaching the elderly gardener to ask for help. And what luck! The
gardener is none other than Fauchelevant, a man whose life
Valjean saved years earlier (and far from Paris). There are many more
coincidences of this type.
Nevertheless, I enjoyed the plot and the characters, and in particular the
relationship between Valjean and the orphaned Cosette. Hugo lost one of
his own beloved daughters in a boating accident (her fiance also
drowned, attempting to save her), a tragedy visible between the lines of
``I miserabili''.
My current Italian reading project is ``I Malavoglia'' by Giovanni
Verga, a 19th century author of Sicilian origins. I'm listening to the
audiobook too, but in this case I find it essential to have the book, as
the vocabulary and the plethora of characters make it
difficult to follow. Although a tragedy, its dialogue is often amusing;
those Sicilians sure were good at insulting one another: Ladro!
Assassino! Nemico di Dio! (``Thief! Assassin! Enemy of God!'') Plus
there is an interesting connection of the sort I love discovering: The
opera Cavalleria Rusticana by Pietro Mascagni is based on some short
stories by Verga, with a similar setting. I'll probably read those next.
Meanwhile, returning to grammar, I offer a brief introduction to the
subjunctive in Italian (for the long version, see my essay ``The
subjunctive: theory and practice'' in ``A mathematician looks at
Italian'').
If you rely on a typical textbook or grammar book, the subjunctive
can appear daunting and mysterious. One such book gives ten different
categories of main clause verbs that require the subjunctive in the
subordinate clause, plus eight further categories with 24 examples. But
this is the wrong way to look at it; there is a single, simple, elegant concept
that governs the use of the subjunctive. Let's first look at a few
examples. Since I don't want to assume familiarity with Italian, I'll
present the examples in English, putting an asterisk after a verb that
would be ``subjunctified'' in Italian. For instance:
1. I suppose your brother arrives* by train.
If English functioned like Italian, the conjugation ``arrives'' would be
replaced by something else; imagine for example ``I suppose your brother
arriveth by train''. (And in fact the verb ``functioned'' in the
sentence I just wrote would be in the subjunctive, if we had one.) Some
other examples:
2. I'm afraid you didn't* understand me.
3. I demand that you tell* me the truth.
4. I doubt that Fabrizio shows* up with Paola.
5. It's rather unlikely that they have* already sold that house.
6. It seems that Julia is* better.
7. I'm happy that my friends had* fun.
8. The fact that he rejected* it doesn't mean anything.
In some of the examples, such as no. 5, it is the auxiliary verb
``have'' that gets the subjunctive, rather than ``sold''. That's just a
rule of Italian, and is not part of the concept.
So what the heck IS the concept? As my favorite grammar book (by Maiden
and Robustelli) points out, the first question to ask is not ``what does
the subjunctive mean?'' but rather ``what does the indicative
(i.e. normal conjugation) mean?''. The indicative means that the verb
in question is actually realized: You didn't understand me. They have
(or haven't) already sold the house. Julia is better. My friends had
fun. He rejected it. And so on. The subjunctive, on the other hand,
refers not to actual realization of the verb, but to its abstract
possibility or abstract essence. In example 6 one is not asserting that
Julia is better, only that it seems that she is better. Example 8 is the
purest illustration of this abstraction: Even though ``he rejected it''
is a realized fact, the sentence refers to the ABSTRACT fact that he
rejected it. Similarly in example 7 my friends did in fact have fun, but
what I'm saying is really ``the (abstract) fact that my friends had
fun'' makes me happy and for that reason the subjunctive is used.
Needless to say, every language has its exceptions and its funny,
arbitrary rules. In particular, the Italian subjunctive doesn't follow
the above logic perfectly. But it follows this ``Realization
Principle'' to a remarkable degree, and that's one of the things I love
about it. It's interesting to note too that the subjunctive seems to have
evolved independently in completely unrelated languages, suggesting that
the Realization Principle is somehow wired into our grammatical
brains. In English we do have a pitiful, vestigial remnant of the
subjunctive, e.g. ``if I were king'', ``I demand that he tell me the
truth''. The latter example is a good case in point, because it only
works in the third person singular; in e.g. ``I demand that you tell me the
truth'' the subjunctive conjugation is the same as the indicative and
therefore effectively non-existent.
You have to see lots of examples in Italian before it really starts
making sense. But just to reassure any would-be Italian learners, in
speech you can get by perfectly well without the subjunctive; if you use
the indicative it is unlikely that you will be misunderstood. On the
other hand, you do need to at least be able to recognize the subjunctive
conjugations, not for the above conceptual reasons but simply to
recognize what the verb is (when listening or reading). To take a simple
example, ``I don't think Paolo knows German'' would be rendered in
Italian as ``Non credo che Paolo sappia il tedesco''. If you only know
the indicative ``Paolo sa'' for ``Paolo knows'', the appearance of
subjunctive conjugations like ``sappia'' will get confusing. And there
are some cases where you'll get the meaning wrong. Take for example the
famous aria ``Nessun dorma'' from Puccini's opera Turandot (even those
who are not opera fans would recognize the aria if they heard
it). Without knowing the subjunctive, you would think it means ``Nobody
sleeps''. But that would be rendered in the indicative: ``Nessun
dorme.'' The subjunctive ``dorma'' changes the meaning completely to
``Let nobody sleep!''
Well, by this point all but the most hardcore grammar fans are probably
already asleep, so I will quit. Those who are interested, however, might
take a look at the essay cited above.
Buona notte a tutti!
wrap around your arm it doesn't mean they want to squeeze you and eat
you. Because if people didn't have arms, how would they hug?''
That last statement might seem obscure--unless you spend a lot of time
around 5-year olds, in which case the meaning is clear: If the snake had
arms, it would give you a hug. And without arms, people too would have
trouble hugging, so it isn't fair to blame the snake. (Kaia was recently
at some kid event or another where they got to handle real snakes.)
Sunday Jessie, Kaia, Finley and I went on an expedition up Denny Creek,
a wonderful little ravine near our house that very few people frequent
(it's part of Denny Creek Park, but almost everyone stays on the other
side of the road, by the lake). Among other things it features the
biggest tree in King County, 26 feet in circumference. Unfortunately it is
now only 40 feet high, the rest having been broken off in a storm years
ago. But it's still pretty impressive.
If we dawdled too long, Finley would urge us on: ``come on, let's go on
our way!'' At the end he announced that ``we are at the end of a very
long journey''. Kaia took great delight in finding ways to cross the
creek, via stepping stones, logs etc. After a tricky crossing she would
say, contemplating the return, ``this was not such a good idea'', and
then after getting back, ``I have saved my own life!''.
On to things Italian. Or French, as the case may be. I finally finished
the 60-hour audiobook of ``I miserabili''. It was quite an experience,
even apart from listening to it in Italian. Anyone else out there read
it, in whatever language? I found the story to be very involving, even
gripping at times. But to maintain interest and suspense one has to skip
past Hugo's strange historical digressions. An extreme example: Valjean
has rescued the unconscious Marius at the barricades, carrying him on
his back and fleeing into the labyrinthic Paris sewers. Just at this
suspenseful moment, Hugo digresses for several chapters on the history
and design of the Paris sewer system. Later Valjean encounters
quicksand in the sewers, at which point we get a lecture on the
different types of quicksand and where they can be found.
The core of the novel still has some flaws that might diminish its
appeal to some readers, notably the many absurd coincidences that are
contrived to reconnect various threads of the plot. When Valjean and
Cosette are running for their lives in Paris, with relentless policeman
Javert hot on their heels, they are able to climb over a high wall into
a large garden. Desperate for a hiding place, Valjean takes the risk of
approaching the elderly gardener to ask for help. And what luck! The
gardener is none other than Fauchelevant, a man whose life
Valjean saved years earlier (and far from Paris). There are many more
coincidences of this type.
Nevertheless, I enjoyed the plot and the characters, and in particular the
relationship between Valjean and the orphaned Cosette. Hugo lost one of
his own beloved daughters in a boating accident (her fiance also
drowned, attempting to save her), a tragedy visible between the lines of
``I miserabili''.
My current Italian reading project is ``I Malavoglia'' by Giovanni
Verga, a 19th century author of Sicilian origins. I'm listening to the
audiobook too, but in this case I find it essential to have the book, as
the vocabulary and the plethora of characters make it
difficult to follow. Although a tragedy, its dialogue is often amusing;
those Sicilians sure were good at insulting one another: Ladro!
Assassino! Nemico di Dio! (``Thief! Assassin! Enemy of God!'') Plus
there is an interesting connection of the sort I love discovering: The
opera Cavalleria Rusticana by Pietro Mascagni is based on some short
stories by Verga, with a similar setting. I'll probably read those next.
Meanwhile, returning to grammar, I offer a brief introduction to the
subjunctive in Italian (for the long version, see my essay ``The
subjunctive: theory and practice'' in ``A mathematician looks at
Italian'').
If you rely on a typical textbook or grammar book, the subjunctive
can appear daunting and mysterious. One such book gives ten different
categories of main clause verbs that require the subjunctive in the
subordinate clause, plus eight further categories with 24 examples. But
this is the wrong way to look at it; there is a single, simple, elegant concept
that governs the use of the subjunctive. Let's first look at a few
examples. Since I don't want to assume familiarity with Italian, I'll
present the examples in English, putting an asterisk after a verb that
would be ``subjunctified'' in Italian. For instance:
1. I suppose your brother arrives* by train.
If English functioned like Italian, the conjugation ``arrives'' would be
replaced by something else; imagine for example ``I suppose your brother
arriveth by train''. (And in fact the verb ``functioned'' in the
sentence I just wrote would be in the subjunctive, if we had one.) Some
other examples:
2. I'm afraid you didn't* understand me.
3. I demand that you tell* me the truth.
4. I doubt that Fabrizio shows* up with Paola.
5. It's rather unlikely that they have* already sold that house.
6. It seems that Julia is* better.
7. I'm happy that my friends had* fun.
8. The fact that he rejected* it doesn't mean anything.
In some of the examples, such as no. 5, it is the auxiliary verb
``have'' that gets the subjunctive, rather than ``sold''. That's just a
rule of Italian, and is not part of the concept.
So what the heck IS the concept? As my favorite grammar book (by Maiden
and Robustelli) points out, the first question to ask is not ``what does
the subjunctive mean?'' but rather ``what does the indicative
(i.e. normal conjugation) mean?''. The indicative means that the verb
in question is actually realized: You didn't understand me. They have
(or haven't) already sold the house. Julia is better. My friends had
fun. He rejected it. And so on. The subjunctive, on the other hand,
refers not to actual realization of the verb, but to its abstract
possibility or abstract essence. In example 6 one is not asserting that
Julia is better, only that it seems that she is better. Example 8 is the
purest illustration of this abstraction: Even though ``he rejected it''
is a realized fact, the sentence refers to the ABSTRACT fact that he
rejected it. Similarly in example 7 my friends did in fact have fun, but
what I'm saying is really ``the (abstract) fact that my friends had
fun'' makes me happy and for that reason the subjunctive is used.
Needless to say, every language has its exceptions and its funny,
arbitrary rules. In particular, the Italian subjunctive doesn't follow
the above logic perfectly. But it follows this ``Realization
Principle'' to a remarkable degree, and that's one of the things I love
about it. It's interesting to note too that the subjunctive seems to have
evolved independently in completely unrelated languages, suggesting that
the Realization Principle is somehow wired into our grammatical
brains. In English we do have a pitiful, vestigial remnant of the
subjunctive, e.g. ``if I were king'', ``I demand that he tell me the
truth''. The latter example is a good case in point, because it only
works in the third person singular; in e.g. ``I demand that you tell me the
truth'' the subjunctive conjugation is the same as the indicative and
therefore effectively non-existent.
You have to see lots of examples in Italian before it really starts
making sense. But just to reassure any would-be Italian learners, in
speech you can get by perfectly well without the subjunctive; if you use
the indicative it is unlikely that you will be misunderstood. On the
other hand, you do need to at least be able to recognize the subjunctive
conjugations, not for the above conceptual reasons but simply to
recognize what the verb is (when listening or reading). To take a simple
example, ``I don't think Paolo knows German'' would be rendered in
Italian as ``Non credo che Paolo sappia il tedesco''. If you only know
the indicative ``Paolo sa'' for ``Paolo knows'', the appearance of
subjunctive conjugations like ``sappia'' will get confusing. And there
are some cases where you'll get the meaning wrong. Take for example the
famous aria ``Nessun dorma'' from Puccini's opera Turandot (even those
who are not opera fans would recognize the aria if they heard
it). Without knowing the subjunctive, you would think it means ``Nobody
sleeps''. But that would be rendered in the indicative: ``Nessun
dorme.'' The subjunctive ``dorma'' changes the meaning completely to
``Let nobody sleep!''
Well, by this point all but the most hardcore grammar fans are probably
already asleep, so I will quit. Those who are interested, however, might
take a look at the essay cited above.
Buona notte a tutti!
Subscribe to:
Posts (Atom)