About Research
Know thyself!
— Attributed to various philosophers [ link ]
Heed these words, You who wish to probe the depths of nature: If you do not find within yourself that which you seek, neither will you find it outside. If you ignore the wonders of your own house, how do you expect to find other wonders? In you is hidden the treasure of treasures. Know Thyself and you will know the Universe and the Gods.
— An inscription attributed to Pythia, an oracle at the Temple of Apollo in Delphi, Greece
The highest form of pure thought is in mathematics.
— Plato, likely after Pythagoras
Number rules the Universe.
— Pythagoras
Do not say a little in many words, but a great deal in a few!
— Pythagoras
Sit down before fact like a little child, and be prepared to give up every preconceived notion, follow humbly wherever and to whatever abyss Nature leads or you shall learn nothing.
— Thomas H. Huxley
...the least initial deviation from the truth is multiplied later a thousandfold.
— Aristotle, On the Heavens
[People think] a scientist is cold... and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong... The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly.
— Isaac Asimov
New scientific ideas never spring from a communal body, however organized, but rather from the head of an individually inspired researcher who struggles with his problems in lonely thought and unites all his thought on one single point which is his whole world for the moment.
— Max Planck
What would you attempt to do if you knew you could not fail?... Doubt and fear always creep in. We think someone else, someone smarter than us, someone more capable, someone with more resources will solve that problem. But there isn't anyone else. There is just you.
— Regina Dugan [ link ]
All courses of action are risky, so prudence is not in avoiding danger (it's impossible), but calculating risk and acting decisively. Make mistakes of ambition and not mistakes of sloth. Develop the strength to do bold things, not the strength to suffer.
— Machiavelli, The Prince
Code of Ethics for Researchers:
- Act with skill and care in all scientific work. Maintain up to date skills and assist their development in others.
- Take steps to prevent corrupt practices and professional misconduct. Declare conflicts of interest.
- Be alert to the ways in which research derives from and affects the work of other people, and respect the rights and reputations of others.
- Ensure that your work is lawful and justified.
- Minimize and justify any adverse effect your work may have on people, animals and the natural environment.
- Seek to discuss the issues that science raises for society. Listen to the aspirations and concerns of others.
- Do not knowingly mislead, or allow others to be misled, about scientific matters. Present and review scientific evidence, theory or interpretation honestly and accurately.
— Sir David King [ link ]
I find in these thoughts so many things which alarm me, and which almost all men, if I am not mistaken, will find so shocking, that I do not see of what use a writing can be, which apparently all the world will reject.
— A response by Arnauld, a Catholic Jansenist, to correspondence from Leibniz
Few, but ripe!
— Carl Friedrich Gauss
I frame no hypotheses.
— Isaac Newton
I don't mind your thinking slowly, I mind your publishing faster than you think.
— Wolfgang Pauli
Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.
— Isaac Newton
In order to be effective truth must penetrate like an arrow — and that is likely to hurt.
— Wei Wu Wei
A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.
— David Hilbert
Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.
— Nikola Tesla
He liked the encouragement of physics or of concrete thinking of mathematics. He always said that if mathematics is left to itself, and has to develop out of itself, it becomes baroque and uninteresting; and his liking not only for physics but also for other branches of knowledge, in particular economics, was based on this circumstance.
— Eugene Wigner in "John von Neumann — A Documentary"
To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.
— Pafnuty Chebyshev
Any intelligent fool can make things bigger and more complex... It takes a touch of genius and a lot of courage to move in the opposite direction.
— Albert Einstein
The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift.
— Albert Einstein
A mathematician experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
— Henri Poincaré
From his tireless quiet efforts, he often took a rest by visiting me. He never talked about anything in advance, and kept silent even of his finished work. I only saw some fleeting happiness pass over him once, when he gave me the little board on which he showed the constructability of the septadecagon.
— Farkas Bolyai of his friend Carl F. Gauss [translated by me]
I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half a proof is zero, and it is demanded for proof that every doubt becomes impossible.
— Carl Friedrich Gauss
Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself — and you are the easiest person to fool. So you have to be very careful about that. After you've not fooled yourself, it's easy not to fool other scientists. You just have to be honest in a conventional way after that.
— Richard Feynman
If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things.
— René Descartes
[Doubting the great Descartes] was a reaction I learned from my father: Have no respect whatsoever for authority; forget who said it and instead look at what he starts with, where he ends up, and ask yourself, "Is it reasonable?".
— Richard Feynman
In questions of science the authority of a thousand is not worth the humble reasoning of a single individual.
— Galileo Galilei
He who desires to philosophize must first of all doubt all things. He must not assume a position in a debate before he has listened to the various opinions, and considered and compared the reasons for and against. He must never judge or take up a position on the evidence of what he has heard, on the opinion of the majority, the age, merits, or prestige of the speaker concerned, but he must proceed according to the persuasion of an organic doctrine which adheres to real things, and to a truth that can be understood by the light of reason.
— Giordano Bruno, De triplici minimo et mensura
Nothing exists. Even if something did exist, nothing can be known about it; and even if something can be known about it, knowledge about it cannot be communicated to others. And, finally, even if it can be communicated, it cannot be understood.
— Gorgias
We have to remember that what we observe is not nature in itself, but nature exposed to our method of questioning.
— Werner Heisenberg
If you knew how much work went into it, you would not call it genius.
— Michelangelo about the Sistine Chapel
New ideas are not easy to find. If you are lucky enough to be working on an idea which is actually right, it can take a long time before you know that it's right. Conversely, if you are going up a blind alley, it can also take a long time before you find out. You can end up saying "Oops, I've been working for years on something wrong." A good mathematician must have the courage to take a lot of work and throw it away.
— Charles L. Fefferman
We hope that Professor Langley will not put his substantial greatness as a scientist in further peril by continuing to waste his time and the money involved, in further airship experiments. Life is too short, and he is capable of services to humanity incomparably greater than can be expected to result from trying to fly... For students and investigators of the Langley type, there are more useful employments.
— New York Times, Dec. 10, 1903
It is the mark of an educated mind to be able to entertain a thought without accepting it.
— Aristotle
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
— G. H. Hardy
Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... From his to our own day, the finest intellects of each generation in turn attacked these problems, but achieved broadly speaking nothing... Weierstrass, Dedekind, and Cantor... have completely solved them. Their... solutions are so clear as to leave no longer the slightest doubt or difficulty. This achievement is probably the greatest of which our age can boast.
— Bertrand Russell
Although this may seem a paradox, all exact science is dominated by the idea of approximation.
— Bertrand Russell
Yet it sometimes happens that the liberator of one age becomes the oppressor of the next. His reputation becomes so great — he becomes so revered and worshipped — that the followers in his name attack the hero who endeavors to take another step in advance.
— Robert G. Ingersoll, "Eulogy on Lincoln"
When I am working on a problem, I never think about beauty. I think only how to solve the problem. But when I've finished, if the solution isn't beautiful, I know it's wrong.
— Buckminster Fuller
Mathematics needs both birds and frogs. Mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details.
— Freeman Dyson
Advice to a Young Mathematician [ link ]
— Michael Atiya, Béla Bollobás, Alain Connes, Dusa McDuff, Peter Sarnak
Like all men of wealth, you make the mistake that money gives you the right to tell an artist how to create, rather than know your proper place... on your knees, in awe.
— Cesare Borgia, Borgia (Season 3 Episode 5)
About Teaching
The mind is not a vessel to be filled but a fire to be kindled.
— Plutarch
Teach our kids doubt so they can build strong, reliable minds. Teach them to avoid the cosmetic, the craven, the jingle of self-satisfaction.
— Bruce Pascoe
The greatest truth must be the recognition that... in every child is the potential for greatness.
— Robert Kennedy
We should seize every opportunity to give encouragement. Encouragement is oxygen to the soul.
— George M. Adams
The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious.
— Henri Poincaré
With a little effort, one can present the material by honestly revealing how one arrived or could have arrived at the results, and then proceeding to mold the theory into its final form. It's fine, even great, if the students end up thinking that it wasn't that big of deal, they could have figured it out on their own.
— László Kalmár [translated by me]
I constantly meet people who are doubtful, generally without due reason, about their potential capacity. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other subjects need mean nothing; much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.
— John E. Littlewood
If leadership does not mean coercion in any form, if it does not mean controlling, protecting or exploiting, what does it mean? It means, I think, freeing. The greatest service the teacher can render the student is to increase his freedom — his free range of activity and thought and his power of control.
— Mary Parker Follett, "The Teacher-Student Relation"
Reciprocity insists that in the pedagogical scale of values, first place be reserved for every individual student's autonomous intellectual development, whether that is in the direction we approve, or not.
— Kenneth Westhues, "On Trying Not to Be a Kierkegaardian Professor"
...As mentors, you should err in the direction of encouraging your mentees to pursue their goals, not discouraging them. They receive more than enough messages from skeptics and critics about how they can’t do it, won’t make it, and should be practical and think small... It’s okay to mention potential challenges, but let mentees try anyway and make their own mistakes.
— The Mentoring Group, "First, Do No Harm..."
A Mathematician's Lament [ link ]
— Paul Lockhart
About Life
A mind that is stretched by a new idea can never go back to its old dimensions.
— After O.W. Holmes, Sr.
In this world of practical realism one must constantly remind himself that our only limitations are those which we set up in our minds or permit others to establish for us.
— Napoleon Hill, You Can Work Your Own Miracles
Without deeper reflection, one knows from daily life that one exists for other people...
— Albert Einstein [ essay ]
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
— Carl Friedrich Gauss
For one swallow does not make a summer, nor does one day; and so too one day, or a short time, does not make a man blessed and happy.
— Aristotle
That was a memorable day to me, for it made great changes in me. But it is the same with any life. Imagine one selected day struck out of it, and think how different its course would have been. Pause you who read this, and think for a moment of the long chain of iron or gold, of thorns or flowers, that would never have bound you, but for the formation of the first link on one memorable day.
— Charles Dickens
Bethink you, then, of old age which cometh all too soon, and not an instant will you lose. While yet you may, and while you yet enjoy the spring-time of your years, taste of the sweets of life. The years flow on like to the waters of a river. The stream that fleeteth by, never returns to the source whence it sprang. The hour that hath sped returns again no more. Make the most of your youth; youth that flies apace. Each new day that dawns is less sweet than those which went before.
— Ovid
Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less.
— Marie Curie
Don't waste your life in doubts and fears: spend yourself on the work before you, well assured that the right performance of this hour's duties will be the best preparation for the hours or ages that follow it.
— Ralph Waldo Emerson
Here’s to the crazy ones. The misfits. The rebels. The troublemakers. The round pegs in the square holes. The ones who see things differently. They’re not fond of rules. And they have no respect for the status quo. You can quote them, disagree with them, glorify or vilify them. About the only thing you can’t do is ignore them. Because they change things. They invent. They imagine. They heal. They explore. They create. They inspire. They push the human race forward. Maybe they have to be crazy. How else can you stare at an empty canvas and see a work of art? Or sit in silence and hear a song that’s never been written? Or gaze at a red planet and see a laboratory on wheels? We make tools for these kinds of people. While some see them as the crazy ones, we see genius. Because the people who are crazy enough to think they can change the world, are the ones who do.
— Apple Inc.
The greatest achievement is selflessness.
The greatest worth is self-mastery.
The greatest quality is seeking to serve others.
The greatest precept is continual awareness.
The greatest medicine is the emptiness of everything.
The greatest action is not conforming with the world's ways.
The greatest magic is transmuting the passions.
The greatest generosity is non-attachment.
The greatest goodness is a peaceful mind.
The greatest patience is humility.
The greatest effort is not concerned with results.
The greatest meditation is a mind that lets go.
The greatest wisdom is seeing through appearances.
— Atisha
One's conduct should be modeled after a tree,
the trunk is rooted while its leaves flourish.
Whereas one's attitude should be like a coin,
square on the inside, while smooth on the edges.
— Ip Man
Men and women range themselves into three classes or orders of intelligence; you can tell the lowest class by their habit of always talking about persons; the next by the fact that their habit is always to converse about things; the highest by their preference for the discussion of ideas.
— Henry Thomas Buckle
One great part of every human existence is passed in a state which cannot be rendered sensible by the use of wideawake language, cutanddry grammar and goahead plot.
— James Joyce about Finnegans Wake in a letter to Harriet Shaw Weaver on Nov. 24, 1926
The noise of the external world is muting the sound of the internal world, and therefore our intuition pays the price for it.
— Dr. Malidoma Somé (elder of the Dagara people, W-Africa & author) in the film "InnSæi - the Power of Intuition"
Until you make the unconscious conscious, it will direct your life and you will call it fate.
— Carl Jung
