(During a lecture)
This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
Quoted in J E Littlewood, A Mathematician's Miscellany, 1953.
This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
Quoted in J E Littlewood, A Mathematician's Miscellany, 1953.
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.
Reader's Digest. Oct. 1977.
I don't believe in mathematics.
Quoted in Carl Seelig. Albert Einstein.
Quoted in Carl Seelig. Albert Einstein.
Imagination is more important than knowledge.
On Science.
On Science.
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
What I Believe.
What I Believe.
The bitter and the sweet come from the outside, the hard from within, from one's own efforts.
Out of My Later Years.
Out of My Later Years.
Gott würfelt nicht.
Common sense is the collection of prejudices acquired by age eighteen.
Quoted in E T Bell Mathematics, Queen and Servant of the Sciences. 1952.
Quoted in E T Bell Mathematics, Queen and Servant of the Sciences. 1952.
God does not care about our mathematical difficulties. He integrates empirically.
Quoted in L Infeld Quest, 1942.
Quoted in L Infeld Quest, 1942.
How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?
(About Newton)
Nature to him was an open book, whose letters he could read without effort.
Quoted in G Simmons Calculus Gems (New York 1992).
Nature to him was an open book, whose letters he could read without effort.
Quoted in G Simmons Calculus Gems (New York 1992).
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Quoted in J R Newman, The World of Mathematics (New York 1956).
Quoted in J R Newman, The World of Mathematics (New York 1956).
What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
Quoted in H Eves Mathematical Circles Squared (Boston 1972).
[i.e. Brouwer vs. Hilbert]
Quoted in H Eves Mathematical Circles Squared (Boston 1972).
Raffiniert ist der Herr Gott, aber boshaft ist er nicht.
God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.
God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.
Nature hides her secrets because of her essential loftiness, but not by means of ruse.
The human mind has first to construct forms, independently, before we can find them in things.
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).
Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).
Do not worry about your difficulties in mathematics, I assure you that mine are greater.
The truth of a theory is in your mind, not in your eyes.
Quoted in H Eves Mathematical Circles Squared (Boston 1972).
Quoted in H Eves Mathematical Circles Squared (Boston 1972).
These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward.
Quoted in H Eves Mathematical Circles Adieu (Boston 1977).
Quoted in H Eves Mathematical Circles Adieu (Boston 1977).
A human being is a part of the whole, called by us "Universe," a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the resta kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.
Quoted in H Eves Mathematical Circles Adieu (Boston 1977).
Quoted in H Eves Mathematical Circles Adieu (Boston 1977).
The world needs heroes and it's better they be harmless men like me than villains like Hitler.
Quoted in H Eves Return to Mathematical Circles (Boston 1988).
Quoted in H Eves Return to Mathematical Circles (Boston 1988).
It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiousity of inquiry.
Quoted in H Eves Return to Mathematical Circles (Boston 1988).
Quoted in H Eves Return to Mathematical Circles (Boston 1988).
Everything that is really great and inspiring is created by the individual who can labor in freedom.
Quoted in H Eves Return to Mathematical Circles (Boston 1988).
Quoted in H Eves Return to Mathematical Circles (Boston 1988).
The search for truth is more precious than its possession.
The American Mathematical Monthly 100 (3).
The American Mathematical Monthly 100 (3).
If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.
Address at the Sorbonne, Paris.
Address at the Sorbonne, Paris.
We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately.
"Space-Time." Encyclopaedia Britannica, 14th ed.
"Space-Time." Encyclopaedia Britannica, 14th ed.
Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.
The Evolution of Physics.
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.
Reader's Digest, Nov. 1973.
(To a student)
Dear Miss --
I have read about sixteen pages of your manuscript ... I suffered exactly the same treatment at the hands of my teachers who disliked me for my independence and passed over me when they wanted assistants ... keep your manuscript for your sons and daughters, in order that they may derive consolation from it and not give a damn for what their teachers tell them or think of them. ... There is too much education altogether.
The World as I See It, (New York, 1949), 21-22.
Dear Miss --
I have read about sixteen pages of your manuscript ... I suffered exactly the same treatment at the hands of my teachers who disliked me for my independence and passed over me when they wanted assistants ... keep your manuscript for your sons and daughters, in order that they may derive consolation from it and not give a damn for what their teachers tell them or think of them. ... There is too much education altogether.
The World as I See It, (New York, 1949), 21-22.
(Written in old age)
I have never belonged wholeheartedly to a country, a state, nor to a circle of friends, nor even to my own family.
When I was still a rather precocious young man, I already realized most vividly the futility of the hopes and aspirations that most men pursue throughout their lives.
Well-being and happiness never appeared to me as an absolute aim. I am even inclined to compare such moral aims to the ambitions of a pig.
Quoted in C P Snow, Variety of Men, (Harmondsworth 1969) 77.
I have never belonged wholeheartedly to a country, a state, nor to a circle of friends, nor even to my own family.
When I was still a rather precocious young man, I already realized most vividly the futility of the hopes and aspirations that most men pursue throughout their lives.
Well-being and happiness never appeared to me as an absolute aim. I am even inclined to compare such moral aims to the ambitions of a pig.
Quoted in C P Snow, Variety of Men, (Harmondsworth 1969) 77.
The relativity principle in connection with the basic Maxwellian equations demands that the mass should be a direct measure of the energy contained in a body; light transfers mass. With radium there should be a noticeable diminution of mass. The idea is amusing and enticing; but whether the Almighty is laughing at it and is leading me up the garden path -- that I cannot know.
When I am judging a theory, I ask myself whether, if I were God, I would have arranged the world in such a way.
Great spirits have always encountered violent opposition from mediocre minds.
.. common sense is nothing more than a deposit of prejudices laid down in the mind before you reach eighteen.
Quoted in E T Bell, Mathematics: Queen and Servant of Science
Quoted in E T Bell, Mathematics: Queen and Servant of Science
Thus the partial differential equation entered theoretical physics as a handmaid, but has gradually become mistress.
The World as I See It
The World as I See It
But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed.
Quoted in H R Pagels, The Cosmic Code
Quoted in H R Pagels, The Cosmic Code
But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, withut mathematics, they could not attain.
Quoted in E T Bell Men of Mathematics
Quoted in E T Bell Men of Mathematics
One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts.
Sidelights on Relativity
Sidelights on Relativity
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Sidelights on Relativity
Sidelights on Relativity
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?
Sidelights on Relativity
Sidelights on Relativity
Mathematics are well and good but nature keeps dragging us around by the nose.
Quoted in A P French, Einstein: a Centenary Volume
Quoted in A P French, Einstein: a Centenary Volume
Education is that which remains when one has forgotten everything learned in school.
Ideas and opinions (New York, 1954).
Ideas and opinions (New York, 1954).
A hundred times every day I remind myself that my inner and outer life depend on the labours of other men, living and dead, and that I must exert myself in order to give in the same measure as I have received.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
Before God we are all equally wise - equally foolish.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
Each of us visits that Earth involuntarily and without an invitation. For me, it is enough to wonder at its secrets.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
If at first the idea is not absurd, then there is no hope for it.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
It is my contention that killing under the cloak of war is nothing but an act of murder.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slightest details we are able to perceive with our frail and feeble minds.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
Reading after a certain time diverts the mind too much from its creative pursuits. Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Quoted in Des MacHale, Wisdom (London, 2002).
[EFR: For a modern view replace reading by watching television.]
Quoted in Des MacHale, Wisdom (London, 2002).
[EFR: For a modern view replace reading by watching television.]
Sometimes one pays most for things one gets for nothing.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
The most incomprehensible fact about the universe is that it is comprehensible.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
The world we have made, as a result of the level of thinking we have done thus far, creates problems we cannot solve at the same level of thinking at which we created them.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
There are two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.
Quoted in Des MacHale, Wisdom (London, 2002).
Quoted in Des MacHale, Wisdom (London, 2002).
大変興味深く読みました：
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。
Einstein's Only Mistake: Division by Zero
再生核研究所声明353（2017.2.2） ゼロ除算 記念日
2014.2.2 に 一般の方から100/0 の意味を問われていた頃、偶然に執筆中の論文原稿にそれがゼロとなっているのを発見した。直ぐに結果に驚いて友人にメールしたり、同僚に話した。それ以来、ちょうど３年、相当詳しい記録と経過が記録されている。重要なものは再生核研究所声明として英文と和文で公表されている。最初のものは
再生核研究所声明 148（2014.2.12）: 100/0=0, 0/0=0 － 割り算の考えを自然に拡張すると ― 神の意志
で、最新のは
Announcement 352 (2017.2.2): On the third birthday of the division by zero z/0=0
である。
アリストテレス、ブラーマグプタ、ニュートン、オイラー、アインシュタインなどが深く関与する ゼロ除算の神秘的な永い歴史上の発見であるから、その日をゼロ除算記念日として定めて、世界史を進化させる決意の日としたい。ゼロ除算は、ユークリッド幾何学の変更といわゆるリーマン球面の無限遠点の考え方の変更を求めている。― 実際、ゼロ除算の歴史は人類の闘争の歴史と共に 人類の愚かさの象徴であるとしている。
心すべき要点を纏めて置きたい。
１） ゼロの明確な発見と算術の確立者Brahmagupta (598 - 668 ?) は 既にそこで、0/0=0 と定義していたにも関わらず、言わば創業者の深い考察を理解できず、それは間違いであるとして、１３００年以上も間違いを繰り返してきた。
２） 予断と偏見、慣習、習慣、思い込み、権威に盲従する人間の精神の弱さ、愚かさを自戒したい。我々は何時もそのように囚われていて、虚像を見ていると 真智を愛する心を大事にして行きたい。絶えず、それは真かと 問うていかなければならない。
３） ピタゴラス派では 無理数の発見をしていたが、なんと、無理数の存在は自分たちの世界観に合わないからという理由で、― その発見は都合が悪いので ― 、弟子を処刑にしてしまったという。真智への愛より、面子、権力争い、勢力争い、利害が大事という人間の浅ましさの典型的な例である。
４） この辺は、２０００年以上も前に、既に世の聖人、賢人が諭されてきたのに いまだ人間は生物の本能レベルを越えておらず、愚かな世界史を続けている。人間が人間として生きる意義は 真智への愛にある と言える。
５） いわば創業者の偉大な精神が正確に、上手く伝えられず、ピタゴラス派のような対応をとっているのは、本末転倒で、そのようなことが世に溢れていると警戒していきたい。本来あるべきものが逆になっていて、社会をおかしくしている。
６） ゼロ除算の発見記念日に 繰り返し、人類の愚かさを反省して、明るい世界史を切り拓いて行きたい。
以 上
追記：
The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world:
Division by Zero z/0 = 0 in Euclidean Spaces
Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1-16.
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
0 件のコメント:
コメントを投稿