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Imagine you're in a bar, or a club,

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and you start talking, and after a while,
the question comes up,

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"So, what do you do for work?"

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And since you think
your job is interesting,

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you say, "I'm a mathematician."
(Laughter)

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And inevitably, during that conversation

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one of these two phrases come up:

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A) "I was terrible at math,
but it wasn't my fault.

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It's because the teacher
was awful." (Laughter)

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Or B) "But what is math really for?"

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(Laughter)

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I'll now address Case B.

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(Laughter)

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When someone asks you what math is for,
they're not asking you

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about applications
of mathematical science.

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They're asking you,

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why did I have to study that bullshit
I never used in my life again? (Laughter)

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That's what they're actually asking.

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So when mathematicians are asked
what math is for,

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they tend to fall into two groups:

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54.51 percent of mathematicians
will assume an attacking position,

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and 44.77 percent of mathematicians
will take a defensive position.

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There's a strange 0.8 percent,
among which I include myself.

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Who are the ones that attack?

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The attacking ones are mathematicians
who would tell you

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this question makes no sense,

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because mathematics
have a meaning all their own --

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a beautiful edifice with its own logic --

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and that there's no point

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in constantly searching
for all possible applications.

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What's the use of poetry?
What's the use of love?

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What's the use of life itself? 
What kind of question is that?

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(Laughter)

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Hardy, for instance, was a model
of this type of attack.

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And those who stand in defense tell you,

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"Even if you don't realize it, friend,
math is behind everything."

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(Laughter)

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Those guys,

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they always bring up
bridges and computers.

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"If you don't know math,
your bridge will collapse."

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(Laughter)

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It's true, computers are all about math.

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And now these guys
have also started saying

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that behind information security
and credit cards are prime numbers.

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These are the answers your math teacher
would give you if you asked him.

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He's one of the defensive ones.

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Okay, but who's right then?

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Those who say that math
doesn't need to have a purpose,

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or those who say that math
is behind everything we do?

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Actually, both are right.

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But remember I told you

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I belong to that strange 0.8 percent
claiming something else?

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So, go ahead, ask me what math is for.

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Audience: What is math for?

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Eduardo Sáenz de Cabezón: Okay,
76.34 percent of you asked the question,

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23.41 percent didn't say anything,

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and the 0.8 percent --

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I'm not sure what those guys are doing.

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Well, to my dear 76.31 percent --

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it's true that math doesn't need
to serve a purpose,

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it's true that it's
a beautiful structure, a logical one,

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probably one
of the greatest collective efforts

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ever achieved in human history.

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But it's also true that there,

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where scientists and technicians
are looking for mathematical theories

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that allow them to advance,

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they're within the structure of math,
which permeates everything.

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It's true that we have to go
somewhat deeper,

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to see what's behind science.

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Science operates on intuition, creativity.

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Math controls intuition
and tames creativity.

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Almost everyone
who hasn't heard this before

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is surprised when they hear
that if you take

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a 0.1 millimeter thick sheet of paper,
the size we normally use,

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and, if it were big enough,
fold it 50 times,

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its thickness would extend almost
the distance from the Earth to the sun.

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Your intuition tells you it's impossible.

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Do the math and you'll see it's right.

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That's what math is for.

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It's true that science, all types
of science, only makes sense

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because it makes us better understand
this beautiful world we live in.

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And in doing that,

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it helps us avoid the pitfalls
of this painful world we live in.

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There are sciences that help us
in this way quite directly.

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Oncological science, for example.

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And there are others we look at from afar,
with envy sometimes,

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but knowing that we are
what supports them.

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All the basic sciences
support them,

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including math.

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All that makes science, science
is the rigor of math.

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And that rigor factors in
because its results are eternal.

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You probably said or were told
at some point

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that diamonds are forever, right?

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That depends on
your definition of forever!

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A theorem -- that really is forever.

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(Laughter)

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The Pythagorean theorem is still true

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even though Pythagoras is dead,
I assure you it's true. (Laughter)

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Even if the world collapsed

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the Pythagorean theorem
would still be true.

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Wherever any two triangle sides
and a good hypotenuse get together

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(Laughter)

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the Pythagorean theorem goes all out.
It works like crazy.

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(Applause)

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Well, we mathematicians devote ourselves
to come up with theorems.

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Eternal truths.

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But it isn't always easy to know
the difference between

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an eternal truth, or theorem,
and a mere conjecture.

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You need proof.

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For example,

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let's say I have a big,
enormous, infinite field.

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I want to cover it with equal pieces,
without leaving any gaps.

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I could use squares, right?

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I could use triangles.
Not circles, those leave little gaps.

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Which is the best shape to use?

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One that covers the same surface,
but has a smaller border.

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In the year 300, Pappus of Alexandria
said the best is to use hexagons,

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just like bees do.

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But he didn't prove it.

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The guy said, "Hexagons, great!
Let's go with hexagons!"

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He didn't prove it,
it remained a conjecture.

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"Hexagons!"

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And the world, as you know,
split into Pappists and anti-Pappists,

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until 1700 years later

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when in 1999, Thomas Hales proved

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that Pappus and the bees were right --
the best shape to use was the hexagon.

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And that became a theorem,
the honeycomb theorem,

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that will be true forever and ever,

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for longer than any diamond
you may have. (Laughter)

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But what happens if we go
to three dimensions?

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If I want to fill the space
with equal pieces,

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without leaving any gaps,

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I can use cubes, right?

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Not spheres, those leave little gaps.
(Laughter)

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What is the best shape to use?

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Lord Kelvin, of the famous
Kelvin degrees and all,

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said that the best was to use
a truncated octahedron

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which, as you all know --

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(Laughter) --

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is this thing here!

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(Applause)

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Come on.

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Who doesn't have a truncated
octahedron at home? (Laughter)

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Even a plastic one.

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"Honey, get the truncated octahedron,
we're having guests."

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Everybody has one!
(Laughter)

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But Kelvin didn't prove it.

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It remained a conjecture --
Kelvin's conjecture.

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The world, as you know, then split into
Kelvinists and anti-Kelvinists

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(Laughter)

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until a hundred or so years later,

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someone found a better structure.

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Weaire and Phelan
found this little thing over here --

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(Laughter) --

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this structure to which they gave
the very clever name

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"the Weaire-Phelan structure."

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(Laughter)

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It looks like a strange object,
but it isn't so strange,

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it also exists in nature.

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It's very interesting that this structure,

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because of its geometric properties,

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was used to build the Aquatics Center
for the Beijing Olympic Games.

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There, Michael Phelps
won eight gold medals,

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and became the best swimmer of all time.

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Well, until someone better
comes along, right?

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As may happen
with the Weaire-Phelan structure.

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It's the best
until something better shows up.

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But be careful, because this one
really stands a chance

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that in a hundred or so years,
or even if it's in 1700 years,

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that someone proves
it's the best possible shape for the job.

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It will then become a theorem,
a truth, forever and ever.

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For longer than any diamond.

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So, if you want to tell someone

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that you will love them forever

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you can give them a diamond.

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But if you want to tell them
that you'll love them forever and ever,

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give them a theorem!

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(Laughter)

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But hang on a minute!

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You'll have to prove it,

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so your love doesn't remain

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a conjecture.

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(Applause)