Mathematics and us | Takehiko Nakama | TEDxDoshishaU
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0:15 - 0:20Today, I'd like you to join me
in examining what mathematics is. -
0:21 - 0:27In elementary, middle, and high school,
everyone studies mathematics, -
0:27 - 0:31and I am sure that all of you have too,
-
0:32 - 0:36but have you ever asked yourself,
"What is mathematics?" -
0:37 - 0:41Many people say they are not good at math,
-
0:42 - 0:45and quite a few people also seem to think
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0:45 - 0:48that mathematics is about memorizing
and using formulas for computation. -
0:48 - 0:51How about you? What do you think?
-
0:51 - 0:56There is no single answer to the question:
"What is mathematics?" -
0:56 - 0:58shared by all mathematicians.
-
0:59 - 1:02However, I am certain
that no mathematician thinks -
1:02 - 1:06mathematics is about memorizing
and using formulas for computation. -
1:06 - 1:10For those who have such
a misconception about mathematics, -
1:10 - 1:13studying math may be quite painful.
-
1:14 - 1:20If you consider mathematics as something
far from what it actually is about, -
1:20 - 1:23then you will have trouble
studying mathematics properly. -
1:24 - 1:26Therefore, it is significant
-
1:26 - 1:30to delve into and understand
the essence of mathematics. -
1:31 - 1:34Today, we are going to explore mathematics
-
1:34 - 1:37by reflecting on the words
of Einstein and Hardy -
1:37 - 1:42and analyzing what we pursue
in the study of mathematics. -
1:43 - 1:46This will help us understand
the essence of mathematics. -
1:48 - 1:50Einstein said,
-
1:50 - 1:55"Mathematics is the poetry
of logical ideas." -
1:55 - 1:57I like this remark
-
1:57 - 2:01and would like you
to think about what it means. -
2:02 - 2:06First, as suggested
by the expression "logical ideas," -
2:07 - 2:11mathematics is logically rigorous.
-
2:12 - 2:13On the other hand,
-
2:13 - 2:16there is another important
aspect of mathematics, -
2:16 - 2:20illustrated in Einstein's remark
that "mathematics is poetry," -
2:21 - 2:25suggesting that mathematics
is rich in imagination and creativity -
2:26 - 2:28and that it is also artistic.
-
2:29 - 2:34You may think that something
that is logically rigorous -
2:34 - 2:38cannot also be creative
-
2:38 - 2:41and that logic and creativity
contradict one another. -
2:42 - 2:44However, in mathematics,
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2:44 - 2:49the two elements highly complement
and reinforce each other. -
2:50 - 2:52As we move through the rest of this talk,
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2:52 - 2:58I hope you will start appreciating
this profound remark by Einstein. -
3:00 - 3:05Next, let's discuss
what we pursue in mathematics. -
3:06 - 3:11You may think that mathematics
is the investigation of numbers, -
3:12 - 3:15but some branches of mathematics
do not concern numbers. -
3:16 - 3:19To understand the essence of mathematics,
-
3:19 - 3:22we must look at a little more universal
and fundamental aspects of mathematics. -
3:24 - 3:26The mathematician Hardy said,
-
3:26 - 3:29"A mathematician, like a painter or poet,
-
3:30 - 3:33is a maker of patterns."
-
3:33 - 3:37He also said mathematics
is about pursuing structures. -
3:38 - 3:43This illuminates the foundational
importance of creating, finding, -
3:43 - 3:46and using structures in mathematics.
-
3:49 - 3:51You may not have a good idea
-
3:51 - 3:55about what kind of structures
can be investigated in mathematics. -
3:56 - 3:59Let's consider Pascal's triangle,
which is shown here, -
3:59 - 4:01to examine several examples
of mathematical structures -
4:01 - 4:06and find some of the essential
characteristics of mathematics. -
4:07 - 4:10This triangle appears
in many mathematical problems. -
4:10 - 4:13For instance, you probably
learned this in school: -
4:13 - 4:19(x+y)² = x² + 2xy + y²
-
4:20 - 4:24Notice that the coefficients
form the third row of the triangle. -
4:25 - 4:28Well, let's not go into technical details.
-
4:29 - 4:33A surprising number of structures
can be found in this triangle, -
4:33 - 4:37and they can help us understand
many important mathematical concepts. -
4:37 - 4:41One of them is fairly easy to find.
-
4:41 - 4:43I wonder if you can find it.
-
4:43 - 4:48Perhaps the most obvious structure
is that of vertical symmetry. -
4:48 - 4:51The numbers appearing on the left half
-
4:51 - 4:53also appear on the right half
in the same manner. -
4:54 - 4:58As you will see later,
symmetry is an important structure. -
4:59 - 5:04Can you figure out how to determine
the value of each element in the triangle? -
5:05 - 5:10The numbers along the right
and left edge are all 1. -
5:10 - 5:14Each of the other numbers is the sum
of the two numbers directly above it. -
5:14 - 5:17For instance, we have 2 = 1 + 1 ,
-
5:17 - 5:193 = 1 + 2,
-
5:19 - 5:20and 6 = 3 + 3.
-
5:21 - 5:25This triangle extends endlessly,
-
5:25 - 5:29but we can construct it
without memorizing the numbers -
5:29 - 5:32but by performing simple computations
-
5:32 - 5:35once we understand
the underlying structures. -
5:36 - 5:37This simple example
-
5:37 - 5:41shows the importance of understanding
and applying mathematical structures. -
5:42 - 5:48Next, let's examine a structure created
by the odd numbers in Pascal's triangle. -
5:48 - 5:52This figure shows the positions
of odd numbers in the first nine rows: -
5:52 - 5:561, 3, 5, 7, and so on, in white.
-
5:57 - 6:02By focusing on where odd numbers
appear in the triangle, -
6:02 - 6:05we can find an interesting structure.
-
6:05 - 6:08Using your imagination,
-
6:08 - 6:15picture a huge Pascal's triangle
consisting of 128 rows. -
6:16 - 6:21This one consists of only nine rows.
The triangle of 128 rows must be enormous. -
6:22 - 6:29This figure shows in white the positions
of the odd numbers in the first 128 rows. -
6:30 - 6:33Clearly, there is a structure.
-
6:33 - 6:36Can you describe it?
-
6:37 - 6:40There is one big triangle.
-
6:40 - 6:41And if you look closely,
-
6:41 - 6:46you can see that it consists
of three smaller triangles. -
6:47 - 6:53And if you take another look,
you will see that each of these triangles -
6:53 - 6:57are again made up
of three smaller triangles. -
6:57 - 7:00This process repeats itself.
-
7:02 - 7:04Technically,
-
7:04 - 7:06this is an example of "fractals,"
-
7:07 - 7:12in which the structure of the whole
is the same as that of its parts. -
7:12 - 7:14This property is called "self-similarity."
-
7:15 - 7:19Fractals can be found
in coastlines, plants, -
7:19 - 7:22crystals, and intestinal walls,
-
7:22 - 7:25to name a very few examples
of fractals found in nature. -
7:26 - 7:28Remember the fractal structure.
-
7:29 - 7:32Near the end of this talk, you will
unexpectedly see another example of it. -
7:34 - 7:37We can't possibly look at all
the mathematical structures today, -
7:37 - 7:40but as a mathematician,
there is another structure -
7:40 - 7:42that I can't leave the room
without showing you. -
7:42 - 7:45If we draw these diagonals,
-
7:45 - 7:48and for each of them,
we sum the numbers on it, -
7:49 - 7:54the sum is 1 for the first
and second diagonals, -
7:54 - 7:57and 2 for the third diagonal.
-
7:57 - 8:02By continuing this, we obtain
the sequence of numbers shown here. -
8:03 - 8:06This is called the Fibonacci sequence.
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8:06 - 8:08It has great mathematical significance
-
8:08 - 8:10appearing in a lot
of mathematical analyses. -
8:11 - 8:14Like fractals,
-
8:14 - 8:20the Fibonacci sequence is also effective
in characterizing structures in nature. -
8:21 - 8:23For example, we arrange squares
-
8:23 - 8:29whose side lengths are set
to the Fibonacci numbers as shown here. -
8:29 - 8:32They can be arranged so neatly,
don't you think? -
8:33 - 8:37Why do these squares
fit together so neatly? -
8:37 - 8:40Think about it when you get home tonight.
-
8:41 - 8:44Using these squares,
we can create a spiral -
8:44 - 8:48that is effective in characterizing
various objects in nature: -
8:48 - 8:51a seashell,
-
8:51 - 8:52a galaxy,
-
8:52 - 8:56or a hurricane, for instance.
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8:58 - 8:59As you can see,
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8:59 - 9:03Pascal's triangle can open the door
to many different structures -
9:03 - 9:06just by exploring
its mathematical applications. -
9:06 - 9:11There are a wide variety
of fields in mathematics, -
9:11 - 9:13but in each of them,
-
9:13 - 9:18we create, find, or apply structures
-
9:18 - 9:22in order to understand something
and establish mathematical truths. -
9:23 - 9:26We must understand
this aspect of mathematics -
9:26 - 9:28to properly study its nature.
-
9:30 - 9:34We can also find some of the important
characteristics of mathematics. -
9:35 - 9:39By analyzing Pascal's triangle, we found
-
9:39 - 9:44the fractal, the Fibonacci sequence,
and the Fibonacci spiral. -
9:45 - 9:48Many other structures can be found
in this triangle as well. -
9:48 - 9:52In mathematics, we often discover
that diverse concepts and structures, -
9:52 - 9:55which ostensibly have nothing
to do with each other, -
9:55 - 10:00are intricately intertwined
at profound levels. -
10:01 - 10:05To understand these various
concepts or structures -
10:05 - 10:08as well as their connections
with each other, -
10:09 - 10:14we need both rigorously logical thinking
and a rich imagination. -
10:15 - 10:18No matter how imaginative you are,
-
10:18 - 10:20with imagination alone,
-
10:20 - 10:24you will not be able to recognize
the links between these concepts. -
10:25 - 10:26On the other hand,
-
10:26 - 10:31you cannot come up with these concepts
with logical thinking alone. -
10:32 - 10:34I believe that Einstein's remark
-
10:34 - 10:37that mathematics
is the poetry of logical ideas -
10:37 - 10:40is starting to resonate more with you.
-
10:42 - 10:47We can also find a rather mysterious
feature of mathematics here. -
10:47 - 10:49It is that mathematics
-
10:49 - 10:56is astonishingly effective
in describing structures in nature. -
10:57 - 10:59Galileo said,
-
10:59 - 11:03"The book of nature is written
in the language of mathematics." -
11:03 - 11:05Feynman said,
-
11:05 - 11:09"To those who do not know mathematics,
it is difficult to get across -
11:09 - 11:13a real feeling as to the beauty,
the deepest beauty, of nature." -
11:13 - 11:15Furthermore, Wigner said
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11:15 - 11:20that mathematics is
"unreasonably effective" -
11:20 - 11:21in the natural sciences.
-
11:22 - 11:26It is no wonder that mathematics
is indispensable to sciences. -
11:26 - 11:28Now, we are going to move on
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11:28 - 11:31to the last topic of this talk:
mathematics and beauty. -
11:32 - 11:35Hardy, the mathematician
whom I mentioned earlier, said - -
11:35 - 11:39when we evaluate mathematical ideas -
-
11:39 - 11:45"Beauty is the first test:
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11:46 - 11:50there is no permanent place
in this world for ugly mathematics." -
11:51 - 11:52This remark suggests
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11:52 - 11:58that the pursuit of beauty
is a paramount element of mathematics. -
11:58 - 12:01Let's investigate this further.
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12:02 - 12:06First, we must examine beauty itself.
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12:07 - 12:12What do you find beautiful?
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12:13 - 12:17Picture something you consider beautiful.
-
12:18 - 12:22Mt. Fuji holds a special place
in the hearts of Japanese people. -
12:23 - 12:28But why do we think it is beautiful?
-
12:30 - 12:33It is indeed beautiful, isn't it?
-
12:34 - 12:37Distinctively, this mountain shows
an almost perfect vertical symmetry -
12:37 - 12:40no matter from what angle
it is viewed from. -
12:40 - 12:43Mathematically,
the feature is referred to -
12:43 - 12:46as "symmetry with respect
to the central vertical axis." -
12:46 - 12:50Also distinctive is the very smooth
contour of the mountain, -
12:50 - 12:52which we can characterize mathematically
-
12:52 - 12:57using a function expressing the curve
and its differentiability. -
12:57 - 13:01These are slightly
difficult technical terms, -
13:01 - 13:08but they are all mathematical structures,
which we discussed earlier. -
13:09 - 13:15Do these features or concepts
have something to do with beauty? -
13:17 - 13:21Recently, various scientific studies
have been conducted -
13:21 - 13:24to investigate our aesthetic sense,
-
13:24 - 13:28and they have helped us better understand
how mathematics is linked to beauty. -
13:28 - 13:30One such study showed
-
13:31 - 13:34that among images
having similar amounts of data, -
13:34 - 13:39those that were perceived as beautiful
had high data compressibility. -
13:39 - 13:42Let's think about it.
-
13:42 - 13:48First, data is compressible when it can be
downsized without loss of information. -
13:49 - 13:51Pascal's triangle,
which we've just examined, -
13:51 - 13:52has vertical symmetry.
-
13:52 - 13:57Numbers appearing on the left half
reappear on the right half. -
13:58 - 14:00Therefore, to draw this triangle,
-
14:00 - 14:07we don't need all the numbers
but only those on the left half. -
14:08 - 14:11We can copy the left half, flip the copy,
and paste it on the right side -
14:11 - 14:13to create the whole triangle.
-
14:13 - 14:19In this case, it means the original data
can be compressed by half, -
14:19 - 14:23so Pascal's triangle
has high data compressibility. -
14:25 - 14:28The same is true of the Mt. Fuji image.
-
14:28 - 14:34By copying the left half of the picture
and pasting the flipped copy on the right, -
14:34 - 14:38we can create a picture that is virtually
indistinguishable from the real image. -
14:38 - 14:42Therefore, the picture of Mt. Fuji
also has high data compressibility, -
14:42 - 14:48and the study has demonstrated that images
like this are perceived as beautiful. -
14:49 - 14:51What makes this picture interesting
-
14:51 - 14:57is the horizontal symmetry due to the
reflection of the mountain on the lake. -
14:58 - 15:01Don't you think it enhances the beauty?
-
15:03 - 15:07We've just established that images
that are perceived as beautiful -
15:07 - 15:10have high data compressibility.
-
15:10 - 15:12Now, in general,
-
15:12 - 15:16what kind of image
has high data compressibility? -
15:17 - 15:19Like the Mt. Fuji example,
-
15:20 - 15:23the data compressibility
of an image is high -
15:23 - 15:26when it has a mathematical structure.
-
15:27 - 15:32This is one of the images perceived
as beautiful in the aforementioned study. -
15:34 - 15:38A mathematical structure was used
-
15:38 - 15:41to draw this face very effectively,
-
15:41 - 15:44and so the data compressibility
of this image is very high. -
15:44 - 15:48Can you figure out what sort
of mathematical structure was used? -
15:50 - 15:51What do you think?
-
15:52 - 15:56Actually, we already discussed it earlier.
-
15:57 - 16:00It's a fractal that is used here.
-
16:02 - 16:09Considering what the study has shown,
we can say that in mathematics, -
16:09 - 16:13we pursue what we consider beautiful
-
16:13 - 16:17or those things that constitute them.
-
16:19 - 16:24A recent study in neuroscience
also supports this view. -
16:25 - 16:30The brain region shown in green here,
called the medial orbitofrontal cortex, -
16:31 - 16:37which has been known to respond
to natural scenes, paintings, and music -
16:37 - 16:40that we perceive as beautiful,
-
16:40 - 16:43was shown in the study
-
16:43 - 16:46to responds similarly
to mathematical concepts or structures. -
16:47 - 16:49For the human brain,
-
16:49 - 16:51the beauty pursued in mathematics
-
16:51 - 16:57shares similarities
with the beauty in nature and art. -
16:59 - 17:01It is time to wrap up this talk.
-
17:02 - 17:05"Mathematics is the poetry
of logical ideas." -
17:06 - 17:11"We pursue and create structures
in mathematics." -
17:12 - 17:16"We pursue 'the beauty' in mathematics."
-
17:17 - 17:20These characteristics might have been
-
17:20 - 17:24a little unexpected or surprising to you.
-
17:26 - 17:31You may have gained a new perspective
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17:31 - 17:35on what we perceive as beautiful.
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17:37 - 17:40Next time you find something beautiful,
-
17:40 - 17:44you might think about mathematics.
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17:44 - 17:46Maybe, maybe not.
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17:46 - 17:49If you do, then don't you think
that is a substantial change? -
17:50 - 17:52Thank you for listening.
-
17:53 - 17:55(Applause)
- Title:
- Mathematics and us | Takehiko Nakama | TEDxDoshishaU
- Description:
-
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
- Video Language:
- Japanese
- Team:
- closed TED
- Project:
- TEDxTalks
- Duration:
- 17:56
Riaki Ponist approved English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Hiroko Kawano accepted English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Riaki Ponist rejected English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Riaki Ponist edited English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Rhonda Jacobs edited English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Riaki Ponist edited English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Hiroko Kawano accepted English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU | ||
Hiroko Kawano edited English subtitles for 数学と私達|仲間 壮彦|TEDxDoshishaU |