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The Education of T.C. Mits: What modern mathematics means to you
"A delightful book."—New York Times
"I have studied with pleasure [this] new book…Beautiful examples…Illuminating. I am convinced that [Lieber's] original enterprise will get the recognition it so richly deserves."—Albert Einstein
"The Liebers have written an ingenious, entertaining, and illuminating book."—Saturday Review of Literature
"The book should be 'required reading' especially for non-mathematicians."—E.T. Bell, author of The Development of Mathematics
First published in 1942, this whimsical exploration of how to think in a mathematical mood continues to delight math-lovers of all ages.
Do you know that two times two is not always four; that the sum of the angles in a triangle does not always equal 180°; that sometimes it is possible to draw two parallel lines through the same point? InThe Education of T. C. MITS, Lillian Lieber opens the door to the wonder of mathematical thinking and its application to everyday life. Lieber uses simple language and fanciful illustrations drawn by her husband, Hugh, to present fundamental mathematical concepts with a deft touch.
The new foreword by Harvard University mathematics professor Barry Mazur is a tribute to the Liebers' influence on generations of mathematicians.
Lillian Lieber was the head of the Department of Mathematics at Long Island University. She wrote a series of lighthearted (and well-respected) math books in the 1940s, including The Einstein Theory of Relativity, Infinity, and Mits, Wits & Logic.
Hugh Gray Lieber was the head of the Department of Fine Arts at Long Island University. He illustrated many books written by his wife Lillian.
Barry Mazur Barry Mazur is a mathematician and is the Gerhard Gade University Professor at Harvard University. He is the author of Imagining Numbers (particularly the square root of minus fifteen). He has won numerous honors in his field, including the Veblen Prize, Cole Prize, Steele Prize, and Chauvenet Prize.
"I have studied with pleasure [this] new book…Beautiful examples…Illuminating. I am convinced that [Lieber's] original enterprise will get the recognition it so richly deserves."—Albert Einstein
"The Liebers have written an ingenious, entertaining, and illuminating book."—Saturday Review of Literature
"The book should be 'required reading' especially for non-mathematicians."—E.T. Bell, author of The Development of Mathematics
First published in 1942, this whimsical exploration of how to think in a mathematical mood continues to delight math-lovers of all ages.
Do you know that two times two is not always four; that the sum of the angles in a triangle does not always equal 180°; that sometimes it is possible to draw two parallel lines through the same point? InThe Education of T. C. MITS, Lillian Lieber opens the door to the wonder of mathematical thinking and its application to everyday life. Lieber uses simple language and fanciful illustrations drawn by her husband, Hugh, to present fundamental mathematical concepts with a deft touch.
The new foreword by Harvard University mathematics professor Barry Mazur is a tribute to the Liebers' influence on generations of mathematicians.
Lillian Lieber was the head of the Department of Mathematics at Long Island University. She wrote a series of lighthearted (and well-respected) math books in the 1940s, including The Einstein Theory of Relativity, Infinity, and Mits, Wits & Logic.
Hugh Gray Lieber was the head of the Department of Fine Arts at Long Island University. He illustrated many books written by his wife Lillian.
Barry Mazur Barry Mazur is a mathematician and is the Gerhard Gade University Professor at Harvard University. He is the author of Imagining Numbers (particularly the square root of minus fifteen). He has won numerous honors in his field, including the Veblen Prize, Cole Prize, Steele Prize, and Chauvenet Prize.
230 pages, Paperback
First published June 1, 2007
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October 10, 2023This classic work of math popularization, addressed to "The Celebrated Man in the Street," opens with an impish preface:
This book is very much about "nowadays": not our nowadays, but Lieber's. Reading it in 2023, her writing remains as fresh and approachable as ever, but you can tell she has been breathing 20th-century air. I felt myself transported through time.
Take her description of the research process. It's a metaphorical five-story building, which she likens to the five Platonic solids:
After that comes the "Second Floor, the ICOSAHEDRON":
This floor of "practical," industrious men is followed by "the OCTAHEDRON," where "Pure Science" unfolds:
Above them is the fourth floor, "the DODECAHEDRON," where we find applied mathematics:
And above them, at the apex of abstraction and uselessness, is the "TETRAHEDRON," where pure mathematics unfolds:
So far, this is just the standard account of progress: we chase ideas, which later yield applications, which eventually birth technologies. But then Lieber goes someplace different, someplace I struggle to imagine a 21st-century writer going: toward a celebration of the institutions of science as a model for democracy.
Look at any technology, she says (say, the radio or the airplane) and you will see that its story spans not only all five floors, but a variety of nations. As Lieber puts it:
The book came out in 1942, with a handsome jacket blurb from Albert Einstein. A few years later, science would birth the atom bomb, a technology manifested from Einstein's own work--a perfect example of the process Lieber describes, and a mighty challenge to her optimistic interpretation of it.
My own feeling (a cynical, 21st-century feeling, I fear) is that math and science are amoral. There is no inherent goodness in the research process, no reason to think that every technology will better humanity, no reason to believe that science escapes all the pitfalls of human institutions.
And it is precisely for this reason that we need Lieber and those like her. We need optimists, humanists, teachers, poets, and writers of impish prefaces. I am glad to see this book back in print, with a foreword by the fabulous Barry Mazur.
Sometimes the freshest voices are those from the past.
This is not intended to be
free verse.
Writing each phrase on a separate line
facilitates rapid reading,
and everyone
is in a hurry
nowadays.
This book is very much about "nowadays": not our nowadays, but Lieber's. Reading it in 2023, her writing remains as fresh and approachable as ever, but you can tell she has been breathing 20th-century air. I felt myself transported through time.
Take her description of the research process. It's a metaphorical five-story building, which she likens to the five Platonic solids:
The First Floor, the CUBE,
contains all the scientific gadgets
with which we are all so familiar:
automobiles and refrigerators
and radios and airplanes...
After that comes the "Second Floor, the ICOSAHEDRON":
Here we find
a great industrial laboratory--
this is where the gadgets are
invented, tried out, manufactured...
This floor of "practical," industrious men is followed by "the OCTAHEDRON," where "Pure Science" unfolds:
They are not concerned with
any practical applications of
their ideas.
They are the theoretical men--
they ask the most "useless" questions.
Above them is the fourth floor, "the DODECAHEDRON," where we find applied mathematics:
The fourth-floor mathematicians are
the ones who know the
Classical Mathematics of the past
and apply it to
the scientific findings...
And above them, at the apex of abstraction and uselessness, is the "TETRAHEDRON," where pure mathematics unfolds:
On that top floor,
they draw geometric figures
on doughnuts and pretzels
(no fooling!)
and on rubber sheets.
And they have up there
Algebras and Arithmetics in which
twice two is NOT four!
So far, this is just the standard account of progress: we chase ideas, which later yield applications, which eventually birth technologies. But then Lieber goes someplace different, someplace I struggle to imagine a 21st-century writer going: toward a celebration of the institutions of science as a model for democracy.
Look at any technology, she says (say, the radio or the airplane) and you will see that its story spans not only all five floors, but a variety of nations. As Lieber puts it:
SCIENCE IS INTERNATIONAL...
it is trying to tell us that
Hitler's racial theories are
utterly false.
It is also trying to tell us--
if we would only listen--
that co-operation is essential
for accomplishing things...
Thus we see that
Science is NOT AMORAL,
but has a PHILOSOPHY
to offer us...
the activities on the top floor
are trying to tell us that
HUMAN NATURE HAS
INFINITE POSSIBILITIES.
The book came out in 1942, with a handsome jacket blurb from Albert Einstein. A few years later, science would birth the atom bomb, a technology manifested from Einstein's own work--a perfect example of the process Lieber describes, and a mighty challenge to her optimistic interpretation of it.
My own feeling (a cynical, 21st-century feeling, I fear) is that math and science are amoral. There is no inherent goodness in the research process, no reason to think that every technology will better humanity, no reason to believe that science escapes all the pitfalls of human institutions.
And it is precisely for this reason that we need Lieber and those like her. We need optimists, humanists, teachers, poets, and writers of impish prefaces. I am glad to see this book back in print, with a foreword by the fabulous Barry Mazur.
Sometimes the freshest voices are those from the past.
July 30, 2020
When I taught at Kofa High School in Yuma, Arizona, my best friend, a fellow English teacher, called me a “book slut” because I never saw a book I didn’t want to own. Actually, I preferred to think of it as having Faust syndrome—I wanted to know everything, or at least be able to locate any information if I needed or wanted it (this was before the ubiquitousness of the Internet and the World Wide Web). Because of this, every year when the school librarians would discard old library books to make room for new ones, I would greedily go through the discards and carry home (or to my classroom) a large portion of them.
One year, I carted off an odd little volume entitled The Education of T. C. Mits: What modern mathematics means to you (“T. C. Mits” being an acronym for “the celebrated man in the streets”). In the twenty-five or thirty years from the time I acquired it, I had never read it, but today, for some reason, I pulled that book down from the shelf. From the subtitle, it sounds as if it would be a primer on basic mathematics and its usefulness, but it’s really a primer on logic and flexible and elastic thinking. The book was originally published in 1942—what I had was the revised and enlarged edition published in 1944. I was prepared to find the contents quaint and amusing (i.e., outdated), but eerily enough, I found that in today’s insane political climate where some people denigrate sound, logical thinking as a tool of the elitist white male patriarchy instead of embracing it as a valuable tool for dealing with the world, this book may be more relevant than ever. In fact, I don’t know who now owns the rights to The Education of T. C. Mits (Dr. Lieber died in 1986, at the age of 99), but the time is ripe for an updated edition.
One year, I carted off an odd little volume entitled The Education of T. C. Mits: What modern mathematics means to you (“T. C. Mits” being an acronym for “the celebrated man in the streets”). In the twenty-five or thirty years from the time I acquired it, I had never read it, but today, for some reason, I pulled that book down from the shelf. From the subtitle, it sounds as if it would be a primer on basic mathematics and its usefulness, but it’s really a primer on logic and flexible and elastic thinking. The book was originally published in 1942—what I had was the revised and enlarged edition published in 1944. I was prepared to find the contents quaint and amusing (i.e., outdated), but eerily enough, I found that in today’s insane political climate where some people denigrate sound, logical thinking as a tool of the elitist white male patriarchy instead of embracing it as a valuable tool for dealing with the world, this book may be more relevant than ever. In fact, I don’t know who now owns the rights to The Education of T. C. Mits (Dr. Lieber died in 1986, at the age of 99), but the time is ripe for an updated edition.
May 7, 2012
Interesting little book which introduces mathematics as both useful and beautiful. I really appriciated the 'totem pole' disscussion -- five levels of mathematical inquiry and why they are each important. Pretty good stuff. Although, I have to say I didn't like the illustrations.
(Sorry to everyone who raves about them.)
(Sorry to everyone who raves about them.)
April 15, 2008
I found this book looking for something else in the Norristown library. It's a quick read; I'm slow and I finished it in a bit more than two hours. There is no math in it that you didn't learn in high school, but it's not really about math. Rather, it's about the human abilities of generalizing and abstraction, the connections between theory and application, and the relationship between old and modern knowledge; all this and it's rendered in just-past-Dick-and-Jane language that's reminiscent of Shel Silverstein. It was written during World War Two and addresses the use and misuse of technology vis a vie Nazi Germany as well as the nature of democracy and how the philosophy of science relates to both. By the Bye, T.C. Mits is "The Celebrated Man In The Street", not some esoteric mathematician. It would be a good companion to Paulos's "Innumeracy".
October 28, 2014
Easy to read, informative, and quirky. T.C. Mits (the celebrated man in the street) is written for the every-man to illuminate the beauty and art of the math (and sometimes it's practical uses) in our lives that we take for granted. Math is the common language of our universe and this book strives to present it in such a way that we will finally realize it is so much more than mere numbers.
September 29, 2019
What a deliciously informative read! I found myself at Khan Academy during this read and doodling on my white board. Splendid and refreshing!
January 6, 2024
함수란 뭘까?
중학교에서 이미 배운 내용이지.
교과서에는
두 변수 x, y에 대하여 x의 값이 하나 정해지면
그에 따라 y의 값이 오직 하나씩 정해질 때,
y는 x의 함수이다.
라고 적혀 있어.
좀 어려운가? 하긴, 수학 교과서는 좀 딱딱하지.
흠. 쉽게 얘기하자면
마치 자판기 같은거야.
너는 목이 말라서 길가의 자판기로 갔어.
자판기에서 '콜라' 버튼을 눌렀더니
쿠르릉 하고 코카콜라가 배출구로 나왔어.
그게 바로 함수야.
X가 Y에 대응되는 것!
나는 1학년 5반 5번이거든?
학번 '1505'에 대응되는 나, 김예진.
이것도 함수라고 할 수 있지.
봐, 별 거 아니지?
수학도 별거 아니야. 괜히 겁먹지 말라구!
아무튼, 이제 함수가 뭔지 알겠지?
함수를 더 정확히 알려면 다른 것들도 알아야 해.
두 집합 X, Y에 대하여
X의 각 원소에 Y의 원소가 오직 하나씩 대응할 때,
이 대응을 집합 X에서 집합 Y로의 함수라 하고
이 함수 f를 기호로
f: X → Y
와 같이 나타내.
어렵다구? 하지만 수학에서 정말 중요한 게
바로 정의인걸.
정의는 딱딱하고 어렵지만 정확히 해야해.
그래야지 정의를 기반으로
'정리'라는 탑을 탄탄히 쌓아나갈 수 있거든.
그래, 그렇지. 머리가 좀 지끈지끈하지만.
나름 재밌지 않아?
수학적으로 사고한다는 거. 꽤 매력있는 일이거든.
한 번 빠지면 헤어나오지 못할 걸?
큼. 아무튼 말이야,
이때 집합 X를 함수 f의 정의역,
집합 Y를 함수 f의 공역이라고 해.
또 함수 f의 함숫값 전체의 집합을 함수 f의 치역이라고 하지.
이 정도는 알아야 함수에 대해 이야기할 수 있어.
더 자세한 건 교과서를 찾아보라고!
함수에는 여러가지 종류가 있어.
중학교 1학년에 가장 먼저 배운 일차함수,
이차함수. 우리가 최근에 배운 유리함수, 무리함수.
그리고 나중에 우리를 마구 괴롭힐
삼차, 사차.. 다항함수. 삼각함수. 로그함수, 지수함수....
하지만 이 함수들이 마냥 우릴 괴롭히기만 하는 건 아냐.
정말 많은 곳에 쓰이니까. 그래서 우릴 편리하게 해주니까!
또 꼭 그 뿐만은 아니지.
예전엔 제대로 계산도 못했던 식을 그래프로 그려낸다니!
나중에 미분이란 걸 배우면 정말 많은 그래프를 그릴 수 있게 되거든?
정말 기대되지 않아?
내가 다 아는 것들만 이야기했다고?
하하, 그렇구나. 미안.
너도 내 이야기를 이렇게 오래 들어주는데 뭐 하나라도 얻어가는 게 있어야지.
역함수에 대한 이야기를 좀 해볼게.
아, 이것도 이미 안다고 해도 뭐라하진 마!
속으로만 불평하라구.. 나도 힘들단 말야..
역함수란 함수 f: X → Y 에서
Y의 각 원로 y에 y=f(x)인 X의 원소 x를 대응시켜
Y를 정의역, X를 공역으로 정의한 새로운 함수를 말해.
간단하게 말해서 그냥 거꾸로 간단 소리야
g라는 함수가 선생님이 "5반 5번!"이라고 불렀을 때 내가 "네, 김예진입니다!"라고 대답하는 것이라면
g의 역함수는 내가 "김예진입니다!"하고 말했을 때 선생님이 "5반 5번이구나!"하고 대답하는 거지.
대충 무슨 말인지 알겠지?
내가 말해주고 싶은 건 이거야.
철저하게 생각하라는 거.
학원이나 학교에서
함수 f와 f의 역함수의 교점은 함수 f와 직선 y=x의 교점과 같다,
라고 배웠을 테지? 그리고 넌 그냥 고개를 끄덕였을 거야.
이렇게 문제를 풀면 잘만 풀리니까 말야.
마치 어렸을 적
절댓값을 씌운 값은 앞에 -를 뗀 값과 같다
라고 믿은 것처럼 말이지.
하지만. 내가 문제를 내볼게.
I-aI=? (단, a<0)
어린 넌 이렇게 대답했을 거야.
에이~ 너무 쉽다. 답은 a! 앞에 마이너스만 떼면 되잖아!
하지만 그게 아니란 걸 이제는 알겠지?
답은 -a야. 절댓값 안의 부호가 양수면 그대로 나오고,
절댓값은 안의 부호가 음수면 마이너스가 붙어서 나오기 때문이지.
이게 바로 철저하게 생각해야하는 이유야.
무턱대고 공식만 외우지 말라는거.
그럼 이제 내가 무슨 말을 할지 감이 오겠어?
또 하나의 문제를 낼게. 바로 역함수 문제야.
y=루트(1-x)와 역함수의 교점을 모두 구해봐.
너가 배운대로라면 직선 y=x와 연립해야겠지?
흠, 교점이 하나 나오네. 하지만 이게 끝일까?
아니야.
(1.0)하고 (0.1)이 더 있어. 분명 y=x위에 있지 않은 점이지?
이건 그래프를 그려보면 분명해져.
그리고 알 수 있지.
감소함수에서는 함수와 역함수의 교점이 y=x 이 외에서도 생긴다!
일반화하자면,
감소함수에서는 임의의 실수 k에 대하여 y=-x+k위에 함수와 역함수의 교점이 존재해.
자, 이제 왜 한 번 더 생각해봐야하는지 알겠지?
수학이란게 원래 그래.
세월이 아무리 흘러도 자명한 정리를 만들기 위해선
아무리 당연해 보이는 것일지라도 묻고 또 되물어야 하지.
모든 걸 수학으로 증명해야 해. 그래야 수학이 수학으로써 있을 수 있어.
그런 수학을 배우는 네가, 좀 더 '수학스럽게' 생각했으면 좋겠어.
단지 문제를 하나 더 맞추자고 초등학교 때부터 지금까지
지겹다 싶을 정도로 배운 건 아닐거야.
그래, 분명 이유가 있다고.
우리가 수학을 배우는 이유. 배워야만 하는 이유!
그걸 항상 잊지 말고, 수학적이자고!
*책은 창의적이고 흥미로운 관점으로 수학의 세계에서 생각하는 방법을 알려줍니다. 1942년 처음 출간되었고, 지금까지도 전 세대의 독자들에게 꾸준히 읽히고 있는 수학 분야의 고전 필독서입니다. 저는 이 책의 말투로 행을 나눠 쉽게 읽고 이해하도록 했습니다. 책에서 알려주는 것을 바탕으로 현재 배우고 있는 수학적 내용을 재구성했습니다.
중학교에서 이미 배운 내용이지.
교과서에는
두 변수 x, y에 대하여 x의 값이 하나 정해지면
그에 따라 y의 값이 오직 하나씩 정해질 때,
y는 x의 함수이다.
라고 적혀 있어.
좀 어려운가? 하긴, 수학 교과서는 좀 딱딱하지.
흠. 쉽게 얘기하자면
마치 자판기 같은거야.
너는 목이 말라서 길가의 자판기로 갔어.
자판기에서 '콜라' 버튼을 눌렀더니
쿠르릉 하고 코카콜라가 배출구로 나왔어.
그게 바로 함수야.
X가 Y에 대응되는 것!
나는 1학년 5반 5번이거든?
학번 '1505'에 대응되는 나, 김예진.
이것도 함수라고 할 수 있지.
봐, 별 거 아니지?
수학도 별거 아니야. 괜히 겁먹지 말라구!
아무튼, 이제 함수가 뭔지 알겠지?
함수를 더 정확히 알려면 다른 것들도 알아야 해.
두 집합 X, Y에 대하여
X의 각 원소에 Y의 원소가 오직 하나씩 대응할 때,
이 대응을 집합 X에서 집합 Y로의 함수라 하고
이 함수 f를 기호로
f: X → Y
와 같이 나타내.
어렵다구? 하지만 수학에서 정말 중요한 게
바로 정의인걸.
정의는 딱딱하고 어렵지만 정확히 해야해.
그래야지 정의를 기반으로
'정리'라는 탑을 탄탄히 쌓아나갈 수 있거든.
그래, 그렇지. 머리가 좀 지끈지끈하지만.
나름 재밌지 않아?
수학적으로 사고한다는 거. 꽤 매력있는 일이거든.
한 번 빠지면 헤어나오지 못할 걸?
큼. 아무튼 말이야,
이때 집합 X를 함수 f의 정의역,
집합 Y를 함수 f의 공역이라고 해.
또 함수 f의 함숫값 전체의 집합을 함수 f의 치역이라고 하지.
이 정도는 알아야 함수에 대해 이야기할 수 있어.
더 자세한 건 교과서를 찾아보라고!
함수에는 여러가지 종류가 있어.
중학교 1학년에 가장 먼저 배운 일차함수,
이차함수. 우리가 최근에 배운 유리함수, 무리함수.
그리고 나중에 우리를 마구 괴롭힐
삼차, 사차.. 다항함수. 삼각함수. 로그함수, 지수함수....
하지만 이 함수들이 마냥 우릴 괴롭히기만 하는 건 아냐.
정말 많은 곳에 쓰이니까. 그래서 우릴 편리하게 해주니까!
또 꼭 그 뿐만은 아니지.
예전엔 제대로 계산도 못했던 식을 그래프로 그려낸다니!
나중에 미분이란 걸 배우면 정말 많은 그래프를 그릴 수 있게 되거든?
정말 기대되지 않아?
내가 다 아는 것들만 이야기했다고?
하하, 그렇구나. 미안.
너도 내 이야기를 이렇게 오래 들어주는데 뭐 하나라도 얻어가는 게 있어야지.
역함수에 대한 이야기를 좀 해볼게.
아, 이것도 이미 안다고 해도 뭐라하진 마!
속으로만 불평하라구.. 나도 힘들단 말야..
역함수란 함수 f: X → Y 에서
Y의 각 원로 y에 y=f(x)인 X의 원소 x를 대응시켜
Y를 정의역, X를 공역으로 정의한 새로운 함수를 말해.
간단하게 말해서 그냥 거꾸로 간단 소리야
g라는 함수가 선생님이 "5반 5번!"이라고 불렀을 때 내가 "네, 김예진입니다!"라고 대답하는 것이라면
g의 역함수는 내가 "김예진입니다!"하고 말했을 때 선생님이 "5반 5번이구나!"하고 대답하는 거지.
대충 무슨 말인지 알겠지?
내가 말해주고 싶은 건 이거야.
철저하게 생각하라는 거.
학원이나 학교에서
함수 f와 f의 역함수의 교점은 함수 f와 직선 y=x의 교점과 같다,
라고 배웠을 테지? 그리고 넌 그냥 고개를 끄덕였을 거야.
이렇게 문제를 풀면 잘만 풀리니까 말야.
마치 어렸을 적
절댓값을 씌운 값은 앞에 -를 뗀 값과 같다
라고 믿은 것처럼 말이지.
하지만. 내가 문제를 내볼게.
I-aI=? (단, a<0)
어린 넌 이렇게 대답했을 거야.
에이~ 너무 쉽다. 답은 a! 앞에 마이너스만 떼면 되잖아!
하지만 그게 아니란 걸 이제는 알겠지?
답은 -a야. 절댓값 안의 부호가 양수면 그대로 나오고,
절댓값은 안의 부호가 음수면 마이너스가 붙어서 나오기 때문이지.
이게 바로 철저하게 생각해야하는 이유야.
무턱대고 공식만 외우지 말라는거.
그럼 이제 내가 무슨 말을 할지 감이 오겠어?
또 하나의 문제를 낼게. 바로 역함수 문제야.
y=루트(1-x)와 역함수의 교점을 모두 구해봐.
너가 배운대로라면 직선 y=x와 연립해야겠지?
흠, 교점이 하나 나오네. 하지만 이게 끝일까?
아니야.
(1.0)하고 (0.1)이 더 있어. 분명 y=x위에 있지 않은 점이지?
이건 그래프를 그려보면 분명해져.
그리고 알 수 있지.
감소함수에서는 함수와 역함수의 교점이 y=x 이 외에서도 생긴다!
일반화하자면,
감소함수에서는 임의의 실수 k에 대하여 y=-x+k위에 함수와 역함수의 교점이 존재해.
자, 이제 왜 한 번 더 생각해봐야하는지 알겠지?
수학이란게 원래 그래.
세월이 아무리 흘러도 자명한 정리를 만들기 위해선
아무리 당연해 보이는 것일지라도 묻고 또 되물어야 하지.
모든 걸 수학으로 증명해야 해. 그래야 수학이 수학으로써 있을 수 있어.
그런 수학을 배우는 네가, 좀 더 '수학스럽게' 생각했으면 좋겠어.
단지 문제를 하나 더 맞추자고 초등학교 때부터 지금까지
지겹다 싶을 정도로 배운 건 아닐거야.
그래, 분명 이유가 있다고.
우리가 수학을 배우는 이유. 배워야만 하는 이유!
그걸 항상 잊지 말고, 수학적이자고!
*책은 창의적이고 흥미로운 관점으로 수학의 세계에서 생각하는 방법을 알려줍니다. 1942년 처음 출간되었고, 지금까지도 전 세대의 독자들에게 꾸준히 읽히고 있는 수학 분야의 고전 필독서입니다. 저는 이 책의 말투로 행을 나눠 쉽게 읽고 이해하도록 했습니다. 책에서 알려주는 것을 바탕으로 현재 배우고 있는 수학적 내용을 재구성했습니다.
March 20, 2021
my rating - overall Score: 4.6/5.0
- quality of writing (5/5)
- quality of the content (5/5)
- impact on my perspective (5/5)
- personal resonance (3/5)
- rereading potential (5/5)
Lieber explores the fundamentals of the scientific mind through mathematics and makes a convincing case on how to apply said fundamentals to create moral virtue/morality in oneself.
here are moral lessons for each of his chapters
1. don't be a Conclusion-Jumper
2. wake up and LIVE! and follow your hunches and check them!
3. Your head CAN go farther than your feet!
4. Streamline your mind with Mathematics.
5 & 6. Oh, listen to the Totem Pole!
7. Be a man—not a mouse.
8. Progress is made by respecting tradition without slavishly following it 100 per cent!
9. Why not read some of these stories in your spare time?
10. Learn to study ON THE WING!
13. Be REASONABLE by bringing to light your "tacit" ideas.
14. Don't be NAIVE—use the methods of Mathematics.
15. Beware of superficial appearances!
Get behind them with a clear head,
and find out what sis back of that good old propaganda.
This process may lead you to some strange, "dismembered,"
modernistic things; but do not let the strangeness scare you;
DEEP-SET PREJUDICE MAY BE WORSE THAN STRANGENESS!
16. At the end of Chapter 7 we said:
"Be a man—not a mouse!"
And now we add
"Be a man—but do not try to play God!"
In short, T.C., "BE YOURSELF!"
17. Go MODERN;
learn to APPRECIATE THE ABSTRACT.
19. The modern viewpoint demands greater flexibility of mind and preparedness for change.
Pull your mind out of those muddy old ruts!
And adapt yourself to a continually CHANGING world.
- quality of writing (5/5)
- quality of the content (5/5)
- impact on my perspective (5/5)
- personal resonance (3/5)
- rereading potential (5/5)
Lieber explores the fundamentals of the scientific mind through mathematics and makes a convincing case on how to apply said fundamentals to create moral virtue/morality in oneself.
here are moral lessons for each of his chapters
1. don't be a Conclusion-Jumper
2. wake up and LIVE! and follow your hunches and check them!
3. Your head CAN go farther than your feet!
4. Streamline your mind with Mathematics.
5 & 6. Oh, listen to the Totem Pole!
7. Be a man—not a mouse.
8. Progress is made by respecting tradition without slavishly following it 100 per cent!
9. Why not read some of these stories in your spare time?
10. Learn to study ON THE WING!
13. Be REASONABLE by bringing to light your "tacit" ideas.
14. Don't be NAIVE—use the methods of Mathematics.
15. Beware of superficial appearances!
Get behind them with a clear head,
and find out what sis back of that good old propaganda.
This process may lead you to some strange, "dismembered,"
modernistic things; but do not let the strangeness scare you;
DEEP-SET PREJUDICE MAY BE WORSE THAN STRANGENESS!
16. At the end of Chapter 7 we said:
"Be a man—not a mouse!"
And now we add
"Be a man—but do not try to play God!"
In short, T.C., "BE YOURSELF!"
17. Go MODERN;
learn to APPRECIATE THE ABSTRACT.
19. The modern viewpoint demands greater flexibility of mind and preparedness for change.
Pull your mind out of those muddy old ruts!
And adapt yourself to a continually CHANGING world.
January 1, 2018
This approachable book is really an introduction to the complex interaction of mathematics and society. Mathematics does not exist in a vacuum but reflects the interests and mores of society. Here, the author builds a case about the ways that mathematics "teaches" the value of interdisciplinarity, multi-culturalism, and democracy. A book of its time, for sure, both in its exultation of democracy and style of art. However, the lessons are still valuable even if the language, art, and format betray their age.
August 17, 2019
Какая современная книга! Вообще на все времена, честно говоря. Автор не боится быть немножко морализатором и занимать четкую позицию, не боится менять сам подход к тому как нужно набирать текст в книге, не боится связать математику и современное абстрактное искусство. Вообще это смелый, честный и очень понятный подход, но при этом без упрощений. Читается за несколько часов вдумчивого чтения. Для людей всех возрастов и мировозрений.
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January 19, 2010read
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