0:00:14.861,0:00:20.383
Today, I'd like you to join me[br]in examining what mathematics is.

0:00:20.923,0:00:27.014
In elementary, middle, and high school,[br]everyone studies mathematics,

0:00:27.014,0:00:31.218
and I am sure that all of you have too,

0:00:31.503,0:00:36.420
but have you ever asked yourself,[br]"What is mathematics?"

0:00:37.420,0:00:41.098
Many people say they are not good at math,

0:00:41.618,0:00:44.503
and quite a few people also seem to think

0:00:44.503,0:00:47.888
that mathematics is about memorizing[br]and using formulas for computation.

0:00:47.888,0:00:50.788
How about you? What do you think?

0:00:51.058,0:00:55.505
There is no single answer to the question:[br]"What is mathematics?"

0:00:55.675,0:00:57.900
shared by all mathematicians.

0:00:58.667,0:01:01.545
However, I am certain[br]that no mathematician thinks

0:01:01.545,0:01:05.697
mathematics is about memorizing[br]and using formulas for computation.

0:01:06.202,0:01:10.118
For those who have such[br]a misconception about mathematics,

0:01:10.118,0:01:13.090
studying math may be quite painful.

0:01:13.660,0:01:19.538
If you consider mathematics as something [br]far from what it actually is about,

0:01:19.538,0:01:22.888
then you will have trouble[br]studying mathematics properly.

0:01:23.792,0:01:26.071
Therefore, it is significant

0:01:26.071,0:01:29.945
to delve into and understand[br]the essence of mathematics.

0:01:31.285,0:01:34.009
Today, we are going to explore mathematics

0:01:34.009,0:01:37.155
by reflecting on the words[br]of Einstein and Hardy

0:01:37.155,0:01:42.166
and analyzing what we pursue[br]in the study of mathematics.

0:01:42.636,0:01:46.157
This will help us understand [br]the essence of mathematics.

0:01:47.891,0:01:49.612
Einstein said,

0:01:49.612,0:01:54.554
"Mathematics is the poetry[br]of logical ideas."

0:01:55.115,0:01:57.247
I like this remark

0:01:57.247,0:02:01.180
and would like you[br]to think about what it means.

0:02:02.431,0:02:06.171
First, as suggested[br]by the expression "logical ideas,"

0:02:06.881,0:02:10.572
mathematics is logically rigorous.

0:02:11.568,0:02:13.006
On the other hand,

0:02:13.006,0:02:16.138
there is another important[br]aspect of mathematics,

0:02:16.138,0:02:20.465
illustrated in Einstein's remark[br]that "mathematics is poetry,"

0:02:20.997,0:02:25.271
suggesting that mathematics[br]is rich in imagination and creativity

0:02:25.641,0:02:28.194
and that it is also artistic.

0:02:28.995,0:02:33.803
You may think that something [br]that is logically rigorous

0:02:34.433,0:02:38.471
cannot also be creative

0:02:38.471,0:02:41.448
and that logic and creativity [br]contradict one another.

0:02:41.856,0:02:43.858
However, in mathematics,

0:02:43.858,0:02:48.958
the two elements highly complement[br]and reinforce each other.

0:02:50.383,0:02:52.468
As we move through the rest of this talk,

0:02:52.468,0:02:57.527
I hope you will start appreciating[br]this profound remark by Einstein.

0:02:59.818,0:03:05.138
Next, let's discuss[br]what we pursue in mathematics.

0:03:06.286,0:03:11.199
You may think that mathematics[br]is the investigation of numbers,

0:03:11.579,0:03:14.989
but some branches of mathematics[br]do not concern numbers.

0:03:16.086,0:03:18.643
To understand the essence of mathematics,

0:03:18.643,0:03:22.442
we must look at a little more universal[br]and fundamental aspects of mathematics.

0:03:23.613,0:03:25.614
The mathematician Hardy said,

0:03:26.198,0:03:29.440
"A mathematician, like a painter or poet,

0:03:29.790,0:03:32.973
is a maker of patterns."

0:03:33.186,0:03:36.976
He also said mathematics[br]is about pursuing structures.

0:03:37.590,0:03:42.856
This illuminates the foundational[br]importance of creating, finding,

0:03:42.856,0:03:46.324
and using structures in mathematics.

0:03:48.784,0:03:50.507
You may not have a good idea

0:03:50.507,0:03:55.043
about what kind of structures[br]can be investigated in mathematics.

0:03:55.563,0:03:58.834
Let's consider Pascal's triangle,[br]which is shown here,

0:03:58.834,0:04:01.377
to examine several examples[br]of mathematical structures

0:04:01.377,0:04:05.597
and find some of the essential[br]characteristics of mathematics.

0:04:06.846,0:04:10.044
This triangle appears[br]in many mathematical problems.

0:04:10.357,0:04:12.801
For instance, you probably[br]learned this in school:

0:04:12.801,0:04:19.279
(x＋y)² ＝ x² + 2xy + y²

0:04:19.958,0:04:23.915
Notice that the coefficients[br]form the third row of the triangle.

0:04:24.587,0:04:27.998
Well, let's not go into technical details.

0:04:29.318,0:04:33.474
A surprising number of structures [br]can be found in this triangle,

0:04:33.474,0:04:37.216
and they can help us understand[br]many important mathematical concepts.

0:04:37.492,0:04:40.612
One of them is fairly easy to find.

0:04:40.935,0:04:42.985
I wonder if you can find it.

0:04:43.223,0:04:48.143
Perhaps the most obvious structure[br]is that of vertical symmetry.

0:04:48.353,0:04:50.932
The numbers appearing on the left half

0:04:50.932,0:04:53.259
also appear on the right half[br]in the same manner.

0:04:53.619,0:04:57.771
As you will see later,[br]symmetry is an important structure.

0:04:59.298,0:05:04.358
Can you figure out how to determine [br]the value of each element in the triangle?

0:05:05.237,0:05:09.714
The numbers along the right [br]and left edge are all 1.

0:05:10.157,0:05:14.265
Each of the other numbers is the sum [br]of the two numbers directly above it.

0:05:14.387,0:05:16.631
For instance, we have 2 = 1 + 1 ,

0:05:17.168,0:05:18.521
3 = 1 + 2,

0:05:18.681,0:05:20.452
and 6 = 3 + 3.

0:05:21.129,0:05:24.757
This triangle extends endlessly,

0:05:25.172,0:05:29.222
but we can construct it [br]without memorizing the numbers

0:05:29.222,0:05:32.265
but by performing simple computations

0:05:32.265,0:05:34.655
once we understand[br]the underlying structures.

0:05:35.595,0:05:37.151
This simple example

0:05:37.151,0:05:40.690
shows the importance of understanding[br]and applying mathematical structures.

0:05:42.354,0:05:47.554
Next, let's examine a structure created[br]by the odd numbers in Pascal's triangle.

0:05:48.294,0:05:51.712
This figure shows the positions[br]of odd numbers in the first nine rows:

0:05:51.736,0:05:56.495
1, 3, 5, 7, and so on, in white.

0:05:57.331,0:06:01.742
By focusing on where odd numbers[br]appear in the triangle,

0:06:02.168,0:06:04.504
we can find an interesting structure.

0:06:04.904,0:06:07.654
Using your imagination,

0:06:08.268,0:06:14.793
picture a huge Pascal's triangle[br]consisting of 128 rows.

0:06:15.624,0:06:20.981
This one consists of only nine rows.[br]The triangle of 128 rows must be enormous.

0:06:22.267,0:06:29.060
This figure shows in white the positions [br]of the odd numbers in the first 128 rows.

0:06:29.844,0:06:32.678
Clearly, there is a structure.

0:06:32.678,0:06:36.245
Can you describe it?

0:06:37.111,0:06:39.856
There is one big triangle.

0:06:39.856,0:06:41.062
And if you look closely,

0:06:41.062,0:06:46.335
you can see that it consists[br]of three smaller triangles.

0:06:46.728,0:06:52.558
And if you take another look,[br]you will see that each of these triangles

0:06:52.842,0:06:56.685
are again made up[br]of three smaller triangles.

0:06:56.988,0:06:59.590
This process repeats itself.

0:07:02.336,0:07:03.666
Technically,

0:07:03.666,0:07:06.376
this is an example of "fractals,"

0:07:06.649,0:07:11.618
in which the structure of the whole[br]is the same as that of its parts.

0:07:11.618,0:07:13.881
This property is called "self-similarity."

0:07:14.942,0:07:18.868
Fractals can be found[br]in coastlines, plants,

0:07:18.946,0:07:21.623
crystals, and intestinal walls,

0:07:21.709,0:07:25.235
to name a very few examples[br]of fractals found in nature.

0:07:25.535,0:07:28.374
Remember the fractal structure.

0:07:28.617,0:07:32.303
Near the end of this talk, you will[br]unexpectedly see another example of it.

0:07:33.598,0:07:36.602
We can't possibly look at all[br]the mathematical structures today,

0:07:36.602,0:07:39.548
but as a mathematician,[br]there is another structure

0:07:39.548,0:07:41.785
that I can't leave the room[br]without showing you.

0:07:41.950,0:07:44.666
If we draw these diagonals,

0:07:44.666,0:07:48.268
and for each of them,[br]we sum the numbers on it,

0:07:49.480,0:07:54.153
the sum is 1 for the first[br]and second diagonals,

0:07:54.478,0:07:56.775
and 2 for the third diagonal.

0:07:57.135,0:08:02.327
By continuing this, we obtain[br]the sequence of numbers shown here.

0:08:02.643,0:08:05.629
This is called the Fibonacci sequence.

0:08:05.629,0:08:08.162
It has great mathematical significance

0:08:08.162,0:08:10.442
appearing in a lot[br]of mathematical analyses.

0:08:11.325,0:08:13.513
Like fractals,

0:08:13.749,0:08:20.024
the Fibonacci sequence is also effective [br]in characterizing structures in nature.

0:08:20.621,0:08:23.002
For example, we arrange squares

0:08:23.002,0:08:28.692
whose side lengths are set[br]to the Fibonacci numbers as shown here.

0:08:29.380,0:08:31.919
They can be arranged so neatly, [br]don't you think?

0:08:32.776,0:08:36.841
Why do these squares[br]fit together so neatly?

0:08:37.302,0:08:40.040
Think about it when you get home tonight.

0:08:40.931,0:08:44.188
Using these squares, [br]we can create a spiral

0:08:44.218,0:08:47.748
that is effective in characterizing [br]various objects in nature:

0:08:48.028,0:08:50.574
a seashell,

0:08:51.172,0:08:52.275
a galaxy,

0:08:52.478,0:08:55.908
or a hurricane, for instance.

0:08:57.949,0:08:58.955
As you can see,

0:08:58.955,0:09:03.151
Pascal's triangle can open the door[br]to many different structures

0:09:03.151,0:09:05.734
just by exploring[br]its mathematical applications.

0:09:06.221,0:09:10.527
There are a wide variety[br]of fields in mathematics,

0:09:10.882,0:09:13.046
but in each of them,

0:09:13.046,0:09:18.100
we create, find, or apply structures

0:09:18.100,0:09:22.198
in order to understand something[br]and establish mathematical truths.

0:09:22.874,0:09:26.445
We must understand[br]this aspect of mathematics

0:09:26.445,0:09:27.986
to properly study its nature.

0:09:29.667,0:09:34.442
We can also find some of the important[br]characteristics of mathematics.

0:09:35.403,0:09:38.764
By analyzing Pascal's triangle, we found

0:09:39.073,0:09:44.007
the fractal, the Fibonacci sequence,[br]and the Fibonacci spiral.

0:09:44.609,0:09:47.947
Many other structures can be found[br]in this triangle as well.

0:09:48.265,0:09:52.200
In mathematics, we often discover[br]that diverse concepts and structures,

0:09:52.220,0:09:55.083
which ostensibly have nothing[br]to do with each other,

0:09:55.093,0:09:59.513
are intricately intertwined[br]at profound levels.

0:10:00.981,0:10:05.386
To understand these various[br]concepts or structures

0:10:05.386,0:10:08.152
as well as their connections[br]with each other,

0:10:08.729,0:10:14.034
we need both rigorously logical thinking[br]and a rich imagination.

0:10:15.388,0:10:17.723
No matter how imaginative you are,

0:10:17.723,0:10:19.704
with imagination alone,

0:10:19.962,0:10:24.217
you will not be able to recognize[br]the links between these concepts.

0:10:24.935,0:10:26.424
On the other hand,

0:10:26.424,0:10:31.412
you cannot come up with these concepts[br]with logical thinking alone.

0:10:32.042,0:10:33.597
I believe that Einstein's remark

0:10:33.597,0:10:36.780
that mathematics[br]is the poetry of logical ideas

0:10:36.780,0:10:39.909
is starting to resonate more with you.

0:10:41.610,0:10:46.748
We can also find a rather mysterious[br]feature of mathematics here.

0:10:47.346,0:10:49.431
It is that mathematics

0:10:49.431,0:10:55.794
is astonishingly effective[br]in describing structures in nature.

0:10:57.222,0:10:58.641
Galileo said,

0:10:58.641,0:11:02.800
"The book of nature is written [br]in the language of mathematics."

0:11:03.350,0:11:05.066
Feynman said,

0:11:05.066,0:11:08.601
"To those who do not know mathematics,[br]it is difficult to get across

0:11:08.601,0:11:12.738
a real feeling as to the beauty,[br]the deepest beauty, of nature."

0:11:13.108,0:11:14.754
Furthermore, Wigner said

0:11:14.754,0:11:19.623
that mathematics is [br]"unreasonably effective"

0:11:19.623,0:11:20.991
in the natural sciences.

0:11:21.591,0:11:25.937
It is no wonder that mathematics[br]is indispensable to sciences.

0:11:25.937,0:11:27.643
Now, we are going to move on

0:11:27.643,0:11:31.170
to the last topic of this talk:[br]mathematics and beauty.

0:11:31.513,0:11:34.512
Hardy, the mathematician[br]whom I mentioned earlier, said -

0:11:35.042,0:11:38.612
when we evaluate mathematical ideas -

0:11:38.722,0:11:45.287
"Beauty is the first test:

0:11:45.906,0:11:50.245
there is no permanent place[br]in this world for ugly mathematics."

0:11:50.515,0:11:51.750
This remark suggests

0:11:51.771,0:11:57.874
that the pursuit of beauty[br]is a paramount element of mathematics.

0:11:58.319,0:12:00.532
Let's investigate this further.

0:12:01.859,0:12:06.310
First, we must examine beauty itself.

0:12:07.200,0:12:12.286
What do you find beautiful?

0:12:13.250,0:12:16.907
Picture something you consider beautiful.

0:12:18.033,0:12:22.229
Mt. Fuji holds a special place[br]in the hearts of Japanese people.

0:12:22.704,0:12:28.243
But why do we think it is beautiful?

0:12:30.024,0:12:32.521
It is indeed beautiful, isn't it?

0:12:33.547,0:12:37.365
Distinctively, this mountain shows [br]an almost perfect vertical symmetry

0:12:37.365,0:12:39.580
no matter from what angle[br]it is viewed from.

0:12:39.821,0:12:42.650
Mathematically, [br]the feature is referred to

0:12:42.650,0:12:45.640
as "symmetry with respect[br]to the central vertical axis."

0:12:46.096,0:12:49.502
Also distinctive is the very smooth[br]contour of the mountain,

0:12:49.502,0:12:51.725
which we can characterize mathematically

0:12:51.725,0:12:56.845
using a function expressing the curve [br]and its differentiability.

0:12:57.321,0:13:00.927
These are slightly [br]difficult technical terms,

0:13:01.116,0:13:07.719
but they are all mathematical structures,[br]which we discussed earlier.

0:13:08.689,0:13:14.800
Do these features or concepts[br]have something to do with beauty?

0:13:16.905,0:13:21.352
Recently, various scientific studies[br]have been conducted

0:13:21.352,0:13:23.914
to investigate our aesthetic sense,

0:13:23.914,0:13:27.877
and they have helped us better understand[br]how mathematics is linked to beauty.

0:13:28.451,0:13:30.468
One such study showed

0:13:30.671,0:13:33.525
that among images[br]having similar amounts of data,

0:13:33.525,0:13:39.058
those that were perceived as beautiful[br]had high data compressibility.

0:13:39.326,0:13:41.597
Let's think about it.

0:13:42.449,0:13:48.167
First, data is compressible when it can be[br]downsized without loss of information.

0:13:48.512,0:13:50.965
Pascal's triangle,[br]which we've just examined,

0:13:50.965,0:13:52.314
has vertical symmetry.

0:13:52.406,0:13:57.182
Numbers appearing on the left half[br]reappear on the right half.

0:13:57.590,0:13:59.797
Therefore, to draw this triangle,

0:13:59.984,0:14:06.560
we don't need all the numbers[br]but only those on the left half.

0:14:07.715,0:14:11.220
We can copy the left half, flip the copy,[br]and paste it on the right side

0:14:11.220,0:14:13.011
to create the whole triangle.

0:14:13.431,0:14:18.683
In this case, it means the original data[br]can be compressed by half,

0:14:19.360,0:14:23.178
so Pascal's triangle[br]has high data compressibility.

0:14:24.676,0:14:27.968
The same is true of the Mt. Fuji image.

0:14:28.470,0:14:34.056
By copying the left half of the picture[br]and pasting the flipped copy on the right,

0:14:34.056,0:14:38.068
we can create a picture that is virtually[br]indistinguishable from the real image.

0:14:38.106,0:14:41.744
Therefore, the picture of Mt. Fuji[br]also has high data compressibility,

0:14:42.099,0:14:47.647
and the study has demonstrated that images[br]like this are perceived as beautiful.

0:14:48.971,0:14:50.750
What makes this picture interesting

0:14:50.934,0:14:56.903
is the horizontal symmetry due to the[br]reflection of the mountain on the lake.

0:14:57.996,0:15:01.383
Don't you think it enhances the beauty?

0:15:03.331,0:15:07.202
We've just established that images[br]that are perceived as beautiful

0:15:07.202,0:15:10.150
have high data compressibility.

0:15:10.349,0:15:11.702
Now, in general,

0:15:11.702,0:15:15.860
what kind of image[br]has high data compressibility?

0:15:16.805,0:15:19.451
Like the Mt. Fuji example,

0:15:19.653,0:15:23.032
the data compressibility[br]of an image is high

0:15:23.032,0:15:25.572
when it has a mathematical structure.

0:15:26.813,0:15:32.425
This is one of the images perceived[br]as beautiful in the aforementioned study.

0:15:33.510,0:15:37.801
A mathematical structure was used

0:15:37.801,0:15:40.625
to draw this face very effectively,

0:15:40.846,0:15:43.769
and so the data compressibility [br]of this image is very high.

0:15:44.031,0:15:47.956
Can you figure out what sort[br]of mathematical structure was used?

0:15:49.645,0:15:51.464
What do you think?

0:15:52.463,0:15:56.164
Actually, we already discussed it earlier.

0:15:56.619,0:16:00.460
It's a fractal that is used here.

0:16:02.014,0:16:08.867
Considering what the study has shown,[br]we can say that in mathematics,

0:16:09.122,0:16:12.690
we pursue what we consider beautiful

0:16:12.690,0:16:16.719
or those things that constitute them.

0:16:19.320,0:16:24.087
A recent study in neuroscience [br]also supports this view.

0:16:25.004,0:16:30.405
The brain region shown in green here,[br]called the medial orbitofrontal cortex,

0:16:30.648,0:16:36.851
which has been known to respond[br]to natural scenes, paintings, and music

0:16:36.851,0:16:39.714
that we perceive as beautiful,

0:16:39.944,0:16:42.923
was shown in the study

0:16:42.923,0:16:46.490
to responds similarly[br]to mathematical concepts or structures.

0:16:47.186,0:16:49.072
For the human brain,

0:16:49.072,0:16:51.450
the beauty pursued in mathematics

0:16:51.450,0:16:57.089
shares similarities[br]with the beauty in nature and art.

0:16:59.425,0:17:01.299
It is time to wrap up this talk.

0:17:02.165,0:17:05.368
"Mathematics is the poetry[br]of logical ideas."

0:17:06.471,0:17:11.063
"We pursue and create structures[br]in mathematics."

0:17:12.010,0:17:16.498
"We pursue 'the beauty' in mathematics."

0:17:16.591,0:17:20.103
These characteristics might have been

0:17:20.103,0:17:24.255
a little unexpected or surprising to you.

0:17:25.517,0:17:31.142
You may have gained a new perspective

0:17:31.142,0:17:35.345
on what we perceive as beautiful.

0:17:36.701,0:17:39.853
Next time you find something beautiful,

0:17:39.853,0:17:43.642
you might think about mathematics.

0:17:44.487,0:17:46.254
Maybe, maybe not.

0:17:46.464,0:17:49.475
If you do, then don't you think[br]that is a substantial change?

0:17:50.193,0:17:52.220
Thank you for listening.

0:17:52.739,0:17:54.887
(Applause)