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)