To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.
- William Blake
I have a master in pure mathematics (it also includes a substantial amount of applied mathematics and physics). People often wonder what is it that we do? And they visualize
the most complicated mathematics they know, and they try to take it
to the next level mentally. But it is very hard to try to understand
something you know nothing of. People who only have a minimum of
mathematical knowledge guess "Oh, so you multiply really large
numbers", those who had math in high school/college guess "Oh,
so you differentiate and integrate difficult functions?", those
who have a university education where they needed advanced
mathematics guess "Oh, so you solve hard differential equations on
difficult grids/spaces?". Today I will endeavor to give a
picture of how I feel when doing pure math.
For inspiration, I recommend the
following videos. After we have seen them, we will try to do some of
this mathematics. Note that we mention Cantor several times.
(Part 2 if you're interested)
Pure mathematics is that of unlimited
abstraction and precision. We define everything as precisely as
humanly possible, and we try to make everything into abstract
concepts. Understanding something in pure mathematics is often like
walking a tightrope over an abyss, no room for small deviations or
imperfect intuition.
A lot of the problems I think about when working, you would need 4 years of university education just to be able to ask. But now I will use
some examples that you need almost no mathematics education to
understand.
First we will look at sizes of infinity
(this is very close to the paradoxes talked about in the videos
above). To talk about size, we have to define it, and size of what?
Definition
A set contains elements, and an element is contained in a set.
Example
{1,2,3,5} is a set containing the elements 1, 2, 3, and 5.
Example 2 {a,
b, car, boat, 99} is a set containing the elements a, b, car, boat
and 99.
Definition 3
The size of a set is the number of elements it contains
Example 4 The
size of {1,2,3,5} is 4
Example 5
The size of {a, b, car, boat, 99} is 5
Using these definitions, we could have
defined a set to be infinite size if it has no finite size. But to be
able to speak of different infinity sizes, we have to use a more fine
tuned definition.
Definition 6
Two sets are equal if there is a bijection between them
Definition 7 A
bijection is a function which maps all the elements of a set to the
elements of another set, and where no two elements are mapped to the
same.
To
understand this last definition, you should see some examples. So
look at wikipedias page bijection.
For
more information, see function and Bijection-Injection-Surjection.
Now
we can state the first question
Question 1
Are there as many even (positive) numbers as there are (positive)
numbers in total?
Answer 1 (with proof)
Yes. Let E={2, 4, 6, 8, 10, ...} be the set of even numbers, and let
N={1, 2, 3, 4, 5, ...} be the set of all numbers. Then we can
construct a function, f, from E to N by f(x)=x/2. This function is a
bijection. Hence the size of E is the same as the size of N.
How
can this be? Clearly the set N contains the set E, and then some, how
can they be equal? Well we just proved that they were. The only thing
we can conclude form the fact that N contains E is that the size of E
is smaller or EQUAL to the size of N.
What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests.
You should look at the link. This is NOT
a paradox, this is a well established mathematical fact. What seems
to be a paradox is only because our intuition is not used to the
concept infinite.
Calculations with infinite
If
you use a modern computer program, it often has inf (infinite) as a
kind of number. It will normally give the following computations:
inf +
1000 = inf
inf –
1700 = inf
inf +
inf = inf
inf*1000
= inf
inf*inf+inf^inf
= inf
1/inf
= 0
1/0 =
inf
inf-inf
= NAN
inf/inf
= NAN
(-1)^inf = NAN
Here
NAN means Not A Number. That is because you are not allowed to do
these operations, the result is undefinable (if you chose a definition
you would end up with a contradiction). The problem with inf-inf is
that we don't know which inf is "largest". In some
applications you may get the answer 0, in others 31, in yet others
you may get inf.
Countable and Uncountable infinity
The
size of the set Z (all finite numers), or the set N (all positive
finite numbers), is called countable infinity (the sizes of Z and N
are equal). It is easy to prove that the set of all fractions, the
rational numbers Q, also has countable size (see
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php).
But
what about the set R of real numbers (all numbers, including pi and
the square root of 2, but not including imaginary numbers), is this
also the same size as Z and N and Q? No. This proof is quite deep,
and took me several days to understand (several years ago). If you
want a challenge, see wikipedia's page on Cantors diagonal argument.
The Length of the Rationals (the set
of fractions Q)
There is another
very much used notion of size. This is what we use for integration,
and to avoid confusing it with the size of sets from before, we call
this new thing for length.
Definition
The length of an interval on the real line, is the right endpoint
minus the left endpoint.
Example
We write [-3,7] for all the numbers between -3 and 7 including -3 and
7. The length of this interval is 7-(-3) = 10.
We
can generalize this concept of length to other sets than intervals,
for example to the union of intervals. Not surprisingly we get:
Theorem
If one set is contained in an interval, the length of the set is
smaller or equal to the length of the interval.
Supertheorem:
The size of Q is countably infinite, but the length of Q is 0 (on
the real line).
Proof: We
have already seen that the size of Q is countably infinite. What
about its length? Well, write Q as a sequence Q={q1, q2, q3, q4, ...}
where all qi are fractions. Let K>0 be any arbitrary number (for
example 0.000000001). Then the first fraction, q1, is contained in an
interval of length K, namely [q1-K/2, q1+K/2]. The second fraction is
contained in an interval of half that length (namely [q2-K/4,
q2+K/4]). The third fraction q3 is contained in an interval with half
that length again. Let us sum up this:
q1
contained in an interval of length K
q2
contained in an interval of length K/2
q3
contained in an interval of length K/4
q4
contained in an interval of length K/8
q5
contained in an interval of length K/16
...
So Q
must be contained in the union, which will be an set with size
smaller than (smaller because some of the intervals may overlap):
K +
K/2 + K/4 + K/8 + K/16 + ... = 2K
How
to calculate this? This is what we call a geometric series.
What
do we now know? Q is contained in something of length 2K (you can
choose any K>0). Hence the length of Q is smaller or equal to 2K
(you can choose any K>0). The only possibility is that the length
of Q is 0. QED.
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Comment by Nok er Nok:
I'll keep beating the same horse as always, I guess. We have spent some time discussing these questions and generally agree, but I can't help but object to the following paragraph. I have mostly the same knee-jerk reaction to it as the Newcomb's paradox, Hangman's paradox, etcetera. On the whole, I'll even illustrate my point with an XKCD strip (gasp!), http://xkcd.com/169/ , though I'm not sure if the miscommunication is intentional or not.
"What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite."
This is most certainly a paradox. The mathematical formalism you describe and which mathematicians use is consistent and useful, yes, that's not what I'm trying to deny. Using that formalism to claim there are -as many- even numbers as natural numbers, however, is dishonest; using layman's terms in that manner -creates- the paradox. Nobody protests against the bijection-wizardry which is firmly belonging to alternate math-dimension. They do, however, have issues with mathematicians redifining common English words to mean something entirely different, and then marveling at how counterintuitive it is - particularly when they a few moments later set the trap by mixing real world example, impossible mathematical constructs, normal English and indistinguishable mathematical definitions.
Let's deconstruct Hilbert's Hotel to start with, and let's pretend we are not familiar with the mathematical formalism for dealing with infinities. Hilbert then claims the following:
- hotel with infinite number of rooms
- infinite number of guests at the hotel
- such that all rooms are occupied, each by a single guest
which is all fine and dandy, but sets up for the counterintuitive conclusion
- the hotel can still house another group of guests, exactly as large as the number which currently occupies every single room
Now, if you look at that without consulting your English<->Mathematics dictionary, you will surely conclude that this is perfect nonsense. Somebody is pulling a cheap parlour trick on you, one of these words have to mean something else that is appears. You can easily construct equally (more?) valid arguments than Hilbert presents, to show that the conclusion is impossible. For instance, it is easy to visualise that no room will be left unoccupied after you swap any two guests between their respective rooms, any number of times, and even though Hilbert does this an infinite number of times, this shouldn't change anything.
The parlour trick here is, of course, that occupied does not mean occupied at all, it has to do with bijections, and infinity does not mean a number you can increment arbitrarily many times, it is instead some mathematical construct with such properties that it cannot possibly have anything to do with any actual hotel. You might say the point of Hilbert's Hotel is that infinity cannot be treated as just any large number, you claim that our intuitions are not prepared to deal with infinity, but I strongly disagree. Hilbert's Hotel only shows that the mathematical lingo he ends up translating to 'occupied' and 'infinity' has nothing to do with a normal understanding of these words.
Taken as a story to accompany the mathematical formalism, to illustrate how it handles infinities, cardinality and size as something to do with bijections, it does an okay job. Without that context, it is not the slightest bit clever or enlightening, but just a load of gibberish. Hilbert's Hotel says -nothing- -whatsoever- about how hotels of arbitrary size work; it intentionally mixes mathematical formalism with a real world example which it then -fails to describe-!
Of course, all of this comes from the same sort of people who with a straight face will call f(x) = constant an increasing function - and a decreasing one, at the same time. Nevermind that increasing is a code word for non-decreasing, which you cannot know without consulting your Google-translate English<->Mathspeak, or being familiar with the tradition of inclusive definitions in mathematics.
I think inclusive definitions are useful, and I'm not quite decided on whether using somewhat familiar but inaccurate and misleading terms is better than inventing new, arbitrary ones. However, I'm certainly not going to give mathematicians any credit for clever paradoxes which does nothing but illustrate that the mathematicians themselevs do not understand that their redefined words cannot be inserted into common English prose without appropriate and careful translation.
As an endnote, I feel fairly certain that it would be very possible to develop a formalism in which the natural numbers, the even numbers, the prime numbers and so on and so forth were -not- the same size. Of course, these alternate defintions would not develop fruitfully into integration and cardinality, like the current one does, but this alternate mathematics would be able to present the exact same Hilbert Hotel and the exact opposite conclusion; the hotel -cannot- accomodate even one more guest, much less another infinity of them. Or perhaps they would balk the moment you suggested that every one of the infinite number of rooms is currently occupied. And if this is true, it should be all the more obvious why Hilbert's Hotel is, indeed, a paradox of sorts.
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