Mathematics and us | Takehiko Nakama | TEDxDoshishaU

Professor Takehiko Nakama, who obtained two PhDs, one in mathematics and one in neuroscience, from Johns Hopkins University, discusses the essence of mathematics and its connection to our perception of beauty.

This talk was given at a TEDx event using the TED conference format but independently organized by a local community. Learn more at https://www.ted.com/tedx

Today, I'd like you to join me
in examining what mathematics is.

In elementary, middle, and high school,
everyone studies mathematics,

and I am sure that all of you have too,

but have you ever asked yourself,
"What is mathematics?"

Many people say they are not good at math,

and quite a few people also seem to think

that mathematics is about memorizing
and using formulas for computation.

How about you? What do you think?

There is no single answer to the question:
"What is mathematics?"

shared by all mathematicians.

However, I am certain
that no mathematician thinks

mathematics is about memorizing
and using formulas for computation.

For those who have such
a misconception about mathematics,

studying math may be quite painful.

If you consider mathematics as something
far from what it actually is about,

then you will have trouble
studying mathematics properly.

Therefore, it is significant

to delve into and understand
the essence of mathematics.

Today, we are going to explore mathematics

by reflecting on the words
of Einstein and Hardy

and analyzing what we pursue
in the study of mathematics.

This will help us understand
the essence of mathematics.

Einstein said,

"Mathematics is the poetry
of logical ideas."

I like this remark

and would like you
to think about what it means.

First, as suggested
by the expression "logical ideas,"

mathematics is logically rigorous.

On the other hand,

there is another important
aspect of mathematics,

illustrated in Einstein's remark
that "mathematics is poetry,"

suggesting that mathematics
is rich in imagination and creativity

and that it is also artistic.

You may think that something
that is logically rigorous

cannot also be creative

and that logic and creativity
contradict one another.

However, in mathematics,

the two elements highly complement
and reinforce each other.

As we move through the rest of this talk,

I hope you will start appreciating
this profound remark by Einstein.

Next, let's discuss
what we pursue in mathematics.

You may think that mathematics
is the investigation of numbers,

but some branches of mathematics
do not concern numbers.

To understand the essence of mathematics,

we must look at a little more universal
and fundamental aspects of mathematics.

The mathematician Hardy said,

"A mathematician, like a painter or poet,

is a maker of patterns."

He also said mathematics
is about pursuing structures.

This illuminates the foundational
importance of creating, finding,

and using structures in mathematics.

You may not have a good idea

about what kind of structures
can be investigated in mathematics.

Let's consider Pascal's triangle,
which is shown here,

to examine several examples
of mathematical structures

and find some of the essential
characteristics of mathematics.

This triangle appears
in many mathematical problems.

For instance, you probably
learned this in school:

(x＋y)² ＝ x² + 2xy + y²

Notice that the coefficients
form the third row of the triangle.

Well, let's not go into technical details.

A surprising number of structures
can be found in this triangle,

and they can help us understand
many important mathematical concepts.

One of them is fairly easy to find.

I wonder if you can find it.

Perhaps the most obvious structure
is that of vertical symmetry.

The numbers appearing on the left half

also appear on the right half
in the same manner.

As you will see later,
symmetry is an important structure.

Can you figure out how to determine
the value of each element in the triangle?

The numbers along the right
and left edge are all 1.

Each of the other numbers is the sum
of the two numbers directly above it.

For instance, we have 2 = 1 + 1 ,

3 = 1 + 2,

and 6 = 3 + 3.

This triangle extends endlessly,

but we can construct it
without memorizing the numbers

but by performing simple computations

once we understand
the underlying structures.

This simple example

shows the importance of understanding
and applying mathematical structures.

Next, let's examine a structure created
by the odd numbers in Pascal's triangle.

This figure shows the positions
of odd numbers in the first nine rows:

1, 3, 5, 7, and so on, in white.

By focusing on where odd numbers
appear in the triangle,

we can find an interesting structure.

Using your imagination,

picture a huge Pascal's triangle
consisting of 128 rows.

This one consists of only nine rows.
The triangle of 128 rows must be enormous.

This figure shows in white the positions
of the odd numbers in the first 128 rows.

Clearly, there is a structure.

Can you describe it?

There is one big triangle.

And if you look closely,

you can see that it consists
of three smaller triangles.

And if you take another look,
you will see that each of these triangles

are again made up
of three smaller triangles.

This process repeats itself.

Technically,

this is an example of "fractals,"

in which the structure of the whole
is the same as that of its parts.

This property is called "self-similarity."

Fractals can be found
in coastlines, plants,

crystals, and intestinal walls,

to name a very few examples
of fractals found in nature.

Remember the fractal structure.

Near the end of this talk, you will
unexpectedly see another example of it.

We can't possibly look at all
the mathematical structures today,

but as a mathematician,
there is another structure

that I can't leave the room
without showing you.

If we draw these diagonals,

and for each of them,
we sum the numbers on it,

the sum is 1 for the first
and second diagonals,

and 2 for the third diagonal.

By continuing this, we obtain
the sequence of numbers shown here.

This is called the Fibonacci sequence.

It has great mathematical significance

appearing in a lot
of mathematical analyses.

Like fractals,

the Fibonacci sequence is also effective
in characterizing structures in nature.

For example, we arrange squares

whose side lengths are set
to the Fibonacci numbers as shown here.

They can be arranged so neatly,
don't you think?

Why do these squares
fit together so neatly?

Think about it when you get home tonight.

Using these squares,
we can create a spiral

that is effective in characterizing
various objects in nature:

a seashell,

a galaxy,

or a hurricane, for instance.

As you can see,

Pascal's triangle can open the door
to many different structures

just by exploring
its mathematical applications.

There are a wide variety
of fields in mathematics,

but in each of them,

we create, find, or apply structures

in order to understand something
and establish mathematical truths.

We must understand
this aspect of mathematics

to properly study its nature.

We can also find some of the important
characteristics of mathematics.

By analyzing Pascal's triangle, we found

the fractal, the Fibonacci sequence,
and the Fibonacci spiral.

Many other structures can be found
in this triangle as well.

In mathematics, we often discover
that diverse concepts and structures,

which ostensibly have nothing
to do with each other,

are intricately intertwined
at profound levels.

To understand these various
concepts or structures

as well as their connections
with each other,

we need both rigorously logical thinking
and a rich imagination.

No matter how imaginative you are,

with imagination alone,

you will not be able to recognize
the links between these concepts.

On the other hand,

you cannot come up with these concepts
with logical thinking alone.

I believe that Einstein's remark

that mathematics
is the poetry of logical ideas

is starting to resonate more with you.

We can also find a rather mysterious
feature of mathematics here.

It is that mathematics

is astonishingly effective
in describing structures in nature.

Galileo said,

"The book of nature is written
in the language of mathematics."

Feynman said,

"To those who do not know mathematics,
it is difficult to get across

a real feeling as to the beauty,
the deepest beauty, of nature."

Furthermore, Wigner said

that mathematics is
"unreasonably effective"

in the natural sciences.

It is no wonder that mathematics
is indispensable to sciences.

Now, we are going to move on

to the last topic of this talk:
mathematics and beauty.

Hardy, the mathematician
whom I mentioned earlier, said -

when we evaluate mathematical ideas -

"Beauty is the first test:

there is no permanent place
in this world for ugly mathematics."

This remark suggests

that the pursuit of beauty
is a paramount element of mathematics.

Let's investigate this further.

First, we must examine beauty itself.

What do you find beautiful?

Picture something you consider beautiful.

Mt. Fuji holds a special place
in the hearts of Japanese people.

But why do we think it is beautiful?

It is indeed beautiful, isn't it?

Distinctively, this mountain shows
an almost perfect vertical symmetry

no matter from what angle
it is viewed from.

Mathematically,
the feature is referred to

as "symmetry with respect
to the central vertical axis."

Also distinctive is the very smooth
contour of the mountain,

which we can characterize mathematically

using a function expressing the curve
and its differentiability.

These are slightly
difficult technical terms,

but they are all mathematical structures,
which we discussed earlier.

Do these features or concepts
have something to do with beauty?

Recently, various scientific studies
have been conducted

to investigate our aesthetic sense,

and they have helped us better understand
how mathematics is linked to beauty.

One such study showed

that among images
having similar amounts of data,

those that were perceived as beautiful
had high data compressibility.

Let's think about it.

First, data is compressible when it can be
downsized without loss of information.

Pascal's triangle,
which we've just examined,

has vertical symmetry.

Numbers appearing on the left half
reappear on the right half.

Therefore, to draw this triangle,

we don't need all the numbers
but only those on the left half.

We can copy the left half, flip the copy,
and paste it on the right side

to create the whole triangle.

In this case, it means the original data
can be compressed by half,

so Pascal's triangle
has high data compressibility.

The same is true of the Mt. Fuji image.

By copying the left half of the picture
and pasting the flipped copy on the right,

we can create a picture that is virtually
indistinguishable from the real image.

Therefore, the picture of Mt. Fuji
also has high data compressibility,

and the study has demonstrated that images
like this are perceived as beautiful.

What makes this picture interesting

is the horizontal symmetry due to the
reflection of the mountain on the lake.

Don't you think it enhances the beauty?

We've just established that images
that are perceived as beautiful

have high data compressibility.

Now, in general,

what kind of image
has high data compressibility?

Like the Mt. Fuji example,

the data compressibility
of an image is high

when it has a mathematical structure.

This is one of the images perceived
as beautiful in the aforementioned study.

A mathematical structure was used

to draw this face very effectively,

and so the data compressibility
of this image is very high.

Can you figure out what sort
of mathematical structure was used?

What do you think?

Actually, we already discussed it earlier.

It's a fractal that is used here.

Considering what the study has shown,
we can say that in mathematics,

we pursue what we consider beautiful

or those things that constitute them.

A recent study in neuroscience
also supports this view.

The brain region shown in green here,
called the medial orbitofrontal cortex,

which has been known to respond
to natural scenes, paintings, and music

that we perceive as beautiful,

was shown in the study

to responds similarly
to mathematical concepts or structures.

For the human brain,

the beauty pursued in mathematics

shares similarities
with the beauty in nature and art.

It is time to wrap up this talk.

"Mathematics is the poetry
of logical ideas."

"We pursue and create structures
in mathematics."

"We pursue 'the beauty' in mathematics."

These characteristics might have been

a little unexpected or surprising to you.

You may have gained a new perspective

on what we perceive as beautiful.

Next time you find something beautiful,

you might think about mathematics.

Maybe, maybe not.

If you do, then don't you think
that is a substantial change?

Thank you for listening.

(Applause)