The mathematician and Fields medalist Vladimir Voevodsky on using automated proof assistants in mathematics:

[Following the discovery of some errors in his earlier work:] I think it was at this moment that I largely stopped doing what is called “curiosity driven research” and started to think seriously about the future.

[...]

A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.

[...]

It soon became clear that the only real long-term solution to the problems that I encountered is to start using computers in the verification of mathematical reasoning.

[...]

Among mathematicians computer proof verification was almost a forbidden subject. A conversation started about the need for computer proof assistants would invariably drift to the Goedel Incompleteness Theorem (which has nothing to do with the actual problem) or to one or two cases of verification of already existing proofs, which were used only to demonstrate how impractical the whole idea was.

[...]

I now do my mathematics with a proof assistant and do not have to worry all the time about mistakes in my arguments or about how to convince others that my arguments are correct.

From a March 26, 2014 talk. Slides available here.

"It is one thing for you to say, ‘Let the world burn.' It is another to say, ‘Let Molly burn.' The difference is all in the name."

-- Uriel, Ghost Story, Jim Butcher

It is easier to fight for one's principles than to live up to them.

-- Alfred Adler

ADDED: Source: http://en.wikiquote.org/wiki/Alfred_Adler

Quoted in: Phyllis Bottome, Alfred Adler: Apostle of Freedom (1939), ch. 5

Problems of Neurosis: A Book of Case Histories (1929)

Philosophers often behave like little children who scribble some marks on a piece of paper at random and then ask the grown-up "What's that?"- It happened like this: the grown-up had drawn pictures for the child several times and said "this is a man," "this is a house," etc. And then the child makes some marks too and asks: what's this then?

• Wittgenstein, Culture and Value

Slartibartfast: Perhaps I'm old and tired, but I think that the chances of finding out what's actually going on are so absurdly remote that the only thing to do is to say, "Hang the sense of it," and keep yourself busy. I'd much rather be happy than right any day.

Arthur Dent: And are you?

Slartibartfast: Well... no. That's where it all falls down, of course.

Douglas Adams, Hitchhiker's Guide to the Galaxy

Now, one basic principle in all of science is GIGO: garbage in, garbage out. This principle is particularly important in statistical meta-analysis: because if you have a bunch of methodologically poor studies, each with small sample size, and then subject them to meta-analysis, what can happen is that the systematic biases in each study — if they mostly point in the same direction — can reach statistical significance when the studies are pooled. And this possibility is particularly relevant here, because meta-analyses of homeopathy invariably find an inverse correlation between the methodological quality of the study and the observed effectiveness of homeopathy: that is, the sloppiest studies find the strongest evidence in favor of homeopathy. When one restricts attention only to methodologically sound studies — those that include adequate randomization and double-blinding, predefined outcome measures, and clear accounting for drop-outs — the meta-analyses find no statistically significant effect (whether positive or negative) of homeopathy compared to placebo.

• Alan Sokal, What Is Science

"Throughout the day, Stargirl had been dropping money. She was the Johnny Appleseed of loose change: a penny here, a nickel there. Tossed to the sidewalk, laid on a shelf or bench. Even quarters.

"I hate change," she said. "It's so . . . jangly."

"Do you realize how much you must throw away in a year?" I said.

"Did you ever see a little kid's face when he spots a penny on a sidewalk?”

Jerry Spinelli, Stargirl

It is, in fact, a very good rule to be especially suspicious of work that says what you want to hear, precisely because the will to believe is a natural human tendency that must be fought.

- Paul Krugman

Specifically, [these recent books that deal with parallel universes] argue that if some scientific theory X has enough experimental support for us to take it seriously, then we must take seriously also all its predictions Y, even if these predictions are themselves untestable (involving parallel universes, for example).

As a warm-up example, let's consider Einstein's theory of General Relativity. It's widely considered a scientific theory worthy of taking seriously, because it has made countless correct predictions -- from the gravitational bending of light to the time dilation measured by our GPS phones. This means that we must also take seriously its prediction for what happens inside black holes, even though this is something we can never observe and report on in Scientific American. If someone doesn't like these black hole predictions, they can't simply opt out of them and dismiss them as unscientific: instead, they need to come up with a different mathematical theory that matches every single successful prediction that general relativity has made -- yet doesn't give the disagreeable black hole predictions.

-- Max Tegmark, Scientific American guest blog, 2014-02-04

How much of a disaster is this? Well, it’s never a disaster to learn that a statement you wanted to go one way in fact goes the other way. It may be disappointing, but it’s much better to know the truth than to waste time chasing a fantasy. Also, there can be far more to it than that. The effect of discovering that your hopes are dashed is often that you readjust your hopes. If you had a subgoal that you now realize is unachievable, but you still believe that the main goal might be achievable, then your options have been narrowed down in a potentially useful way.

-Timothy Gowers, on finding out a method he’d hoped would work, in fact would not.

He said:

When you play bridge with beginners—when you try to help them out—you give them some general rules to go by. Then they follow the rule and something goes wrong. But if you'd had their hand you wouldn't have played the thing you told them to play, because you'd have seen all the reasons the rule did not apply.

from The Last Samurai by Helen DeWitt

Richard Feynmann claimed that he wasn't exceptionally intelligent, but that he focused all his energies on one thing. Of course he was exceptionally intelligent, but he makes a good point.

I think one way to improve your intelligence is to actually try to understand things in a very fundamental way. Rather than just accepting the kind of trite explanations that most people accept - for instance, that electricity is electrons moving along a wire - try to really find out and understand what is actually happening, and you'll begin to find that the world is very different from what you have been taught and you'll be able to make more intelligent observations about it.

http://www.reddit.com/r/askscience/comments/e3yjg/is_there_any_way_to_improve_intelligence_or_are/c153p8w

reddit user jjbcn on trying to improve your intelligence

If you're not a student of physics, The Feynman Lectures on Physics is probably really useful for this purpose. It's free for download!

http://www.feynmanlectures.caltech.edu/

It seems like the Feynman lectures were a bit like the Sequences for those Caltech students:

The intervening years might have glazed their memories with a euphoric tint, but about 80 percent recall Feynman's lectures as highlights of their college years. “It was like going to church.” The lectures were “a transformational experience,” “the experience of a lifetime, probably the most important thing I got from Caltech.” “I was a biology major but Feynman's lectures stand out as a high point in my undergraduate experience … though I must admit I couldn't do the homework at the time and I hardly turned any of it in.” “I was among the least promising of students in this course, and I never missed a lecture. … I remember and can still feel Feynman's joy of discovery. … His lectures had an … emotional impact that was probably lost in the printed Lectures.”

Being wrong about something feels exactly the same as being right about something.

-- many different people, most recently user chipaca on HN

“If only there were irrational people somewhere, insidiously believing stupid things, and it were necessary only to separate them from the rest of us and mock them. But the line dividing rationality and irrationality cuts through the mind of every human being. And who is willing to mock a piece of his own mind?”

(With apologies to Solzhenitsyn).

– Said Achmiz, in a comment on Slate Star Codex’s post “The Cowpox of Doubt”

“Even if it's not your fault, it's your responsibility.”

• Terry Pratchett, A Hat Full of Sky

Today is already the tomorrow which the bad economist yesterday urged us to ignore.

-- Henry Hazlitt, Economics in One Lesson

“I propose we simply postpone the worrisome question of what really has a mind, about what the proper domain of the intentional stance is. Whatever the right answer to that question is—if it has a right answer—this will not jeopardize the plain fact that the intentional stance works remarkably well as a prediction method in these other areas, almost as well as it works in our daily lives as folk psychologists dealing with other people. This move of mine annoys and frustrates some philosophers, who want to blow the whistle and insist on properly settling the issue of what a mind, a belief, a desire is before taking another step. Define your terms, sir! No, I won’t. That would be premature. I want to explore first the power and the extent of application of this good trick, the intentional stance. Once we see what it is good for, and why, we can come back and ask ourselves if we still feel the need for formal, watertight definitions. My move is an instance of nibbling on a tough problem instead of trying to eat (and digest) the whole thing from the outset. “Many of the thinking tools I will be demonstrating are good at nibbling, at roughly locating a few “fixed” points that will help us see the general shape of the problem. In Elbow Room (1984a), o compared my method to the sculptor’s method of roughing out the form in a block of marble, approaching the final surfaces cautiously, modestly, working by successive approximation. Many philosophers apparently cannot work that way and have to secure (or so they think) the utterly fixed boundaries of their problems and possible solutions before they can venture any hypotheses.”

-Daniel Dennett, Intuition Pumps and Other Tools for Thinking, Chapter 18 "The Intentional Stance" [Bold is original]

Reminded me of the idea of 'hacking away at the edges'.

“Anything outside yourself, this you can see and apply your logic to it." She said. "But it’s a human trait that when we encounter personal problems, those things most deeply personal are the most difficult to bring out for our logic to scan. We tend to flounder around, blaming everything but the actual, deep-seated thing that’s really chewing on us.”

Jessica speaking to Thufir Hawat in Frank Herbert's Dune

I have long ceased to argue with people who prefer Thursday to Wednesday because it is Thursday.

G. K. Chesterton, attributed.

...the utility of a thought experiment is inversely proportional to the size of its departure from reality.

-- Daniel Dennett, Intuition Pumps and Other Tools for Thinking

A BS detection Heuristic.

You can tell if a discipline is BS if the degree depends severely on the prestige of the school granting it. I remember when I applied to MBA programs being told that anything outside the top 10 or 20 would be a waste of time. On the other hand a degree in mathematics is much less dependent on the shool (conditional on being above a certain level, so the heuristic would apply to the differene betwewn top 10 and top 2000 schools).

The same applies to research papers. In math and physics, a result posted on arXiv (with a minimum hurdle) is fine. In low quality fields like academic finance (where almost all academics are charlatans and all papers some form of complicated storytelling), the "prestige" of the journal is the sole criterion.

Nassim Taleb

There is an important difference between “We don’t know all the answers yet” and “Do what feels right, man.” These questions have answers, because humans have biochemistry, and we should do our best to find them and live by the results.

~J. Stanton, "The Paleo Identity Crisis: What Is The Paleo Diet, Anyway?"

Encoded in the large, highly evolved sensory and motor portions of the human brain is a billion years of experience about the nature of the world and how to survive in it. The deliberate process we call reasoning is, I believe, the thinnest veneer of human thought, effective only because it is supported by this much older and much more powerful, though usually unconscious, sensorimotor knowledge. We are all prodigious olympians in perceptual and motor areas, so good that we make the difficult look easy. Abstract thought, though, is a new trick, perhaps less than 100 thousand years old. We have not yet mastered it. It is not all that intrinsically difficult; it just seems so when we do it.

Hans Moravec, Wikipedia/Moravec's Paradox

But understanding human limitations does not mean we can overcome them. It only means we can’t pretend they don’t exist. It should point us toward humility, not hubris.

Yuval Levin in the National Review

There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot be put into equations — because it is nonsense.

Clifford Truesdell

If the best minds were in charge of designing a bridge, I would expect the bridge to hold up well even in a storm. If the best minds were in charge of designing an airplane, I would expect it to fly reliably. But if the best minds were in charge of something no one really knows how to do, I would be ready for a failure, albeit a failure with superb academic credentials.

Terry Coxon

I assume that the reader is familiar with the idea of extrasensory perception, and the meaning of the four items of it, viz., telepathy, clairvoyance, precognition and psychokinesis. These disturbing phenomena seem to deny all our usual scientific ideas. How we should like to discredit them! Unfortunately the statistical evidence, at least for telepathy, is overwhelming.

Alan Turing (from "Computing Machinery and Intelligence")

'There is of course the question of public safety,' said Vetinari. 'Did I hear you say earlier you have blown up... "one or three" I think was the phrase?"

'I made those explode a-purpose, to see exactly how it 'appened. That's the way to get knowledge, you see, sir.'

Raising Steam, Terry Pratchett

Regarding the first steam engine in Pratchett's fictional world.

it's like arguing that fairies are coming out of my toaster in the middle of the night. You can't prove to me that there aren't fairies in my toaster, but that doesn't mean you should take me seriously. What I have a problem with is not so much religion or god, but faith. When you say you believe something in your heart and therefore you can act on it, you have completely justified the 9/11 bombers. You have justified Charlie Manson. If it's true for you, why isn't it true for them? Why are you different? If you say "I believe there's an all-powerful force of love in the universe that connects us all, and I have no evidence of that but I believe it in my heart," then it's perfectly okay to believe in your heart that Sharon Tate deserves to die. It's perfectly okay to believe in your heart that you need to fly planes into buildings for Allah.

--Penn Jillette in "Penn Jillette Is Willing to Be a Guest on Adolf Hitler's Talk Show, Vanity Fair, June 17, 2010

Edited OP to make it clear that you can provide a link to the place you found the quote, rather than needing to track down an authoritative original source.

The representatives of the scientific world-conception resolutely stand on the ground of simple human experience. They confidently approach the task of removing the metaphysical and theological debris of millennia. Or, as some have it: returning, after a metaphysical interlude, to a unified picture of this world which had, in a sense, been at the basis of magical beliefs, free from theology, in the earliest times.

The increase of metaphysical and theologizing leanings which shows itself today in many associations and sects, in books and journals, in talks and university lectures, seems to be based on the fierce social and economic struggles of the present: one group of combatants, holding fast to traditional social forms, cultivates traditional attitudes of metaphysics and theology whose content has long since been superseded; while the other group, especially in central Europe, faces modern times, rejects these views and takes its stand on the ground of empirical science. This development is connected with that of the modern process of production, which is becoming ever more rigorously mechanised and leaves ever less room for metaphysical ideas. It is also connected with the disappointment of broad masses of people with the attitude of those who preach traditional metaphysical and theological doctrine. So it is that in many countries the masses now reject these doctrines much more consciously than ever before, and along with their socialist attitudes tend to lean towards a down-to-earth empiricist view. In previous times, materialism was the expression of this view; meanwhile, however, modern empiricism has shed a number of inadequacies and has taken a strong shape in the scientific world-conception.

Thus, the scientific world-conception is close to the life of the present. Certainly it is threatened with hard struggles and hostility. Nevertheless there are many who do not despair but, in view of the present sociological situation, look forward with hope to the course of events to come. Of course not every single adherent of the scientific world-conception will be a fighter. Some, glad of solitude, will lead a withdrawn existence on the icy slopes of logic; some may even disdain mingling with the masses and regret the ‘trivialized’ form that these matters inevitably take on spreading. However, their achievements too will take a place among the historic developments. We witness the spirit of the scientific world-conception penetrating in growing measure the forms of personal and public life, in education, upbringing, architecture, and the shaping of economic and social life according to rational principles. The scientific world-conception serves life, and life receives it.

• Hans Hahn, Otto Neurath and Rudolf Carnap, The Scientific Conception of the World: The Vienna Circle, 1929.

It has come to be accepted practice in introducing new physical quantities that they shall be regarded as defined by the series of measuring operations and calculations of which they are the result. Those who associate with the result a mental picture of some entity disporting itself in a metaphysical realm of existence do so at their own risk; physics can accept no responsibility for this embellishment.

Sir Arthur Eddington, 1939, The Philosophy of Physical Science

Our recent research into team behavior [...] reveals an interesting paradox: Although teams that are large, virtual, diverse, and composed of highly educated specialists are increasingly crucial with challenging projects, those same four characteristics make it hard for teams to get anything done. To put it another way, the qualities required for success are the same qualities that undermine success. Members of complex teams are less likely—absent other influences—to share knowledge freely, to learn from one another, to shift workloads flexibly to break up unexpected bottlenecks, to help one another complete jobs and meet deadlines, and to share resources—in other words, to collaborate. They are less likely to say that they “sink or swim” together, want one another to succeed, or view their goals as compatible.

Eight Ways to Build Collaborative Teams by Lynda Gratton and Tamara J. Erickson

This seems applicable as the LessWrong community is "large, virtual, diverse, and composed of highly educated specialists" and the community wants to solve challenging projects.

It wasn’t easier, the ghost explains, you just knew how to do it. Sometimes the easiest method you know is the hardest method there is.

It’s like… to someone who only knows how to dig with a spoon, the notion of digging something as large as a trench will terrify them. All they know are spoons, so as far as they’re concerned, digging is simply difficult. The only way they can imagine it getting any easier is if they change – digging with a spoon until they get stronger, faster, and tougher. And the dangerous people, they’ll actually try this.

Everyone who will ever oppose you in life is a crazy, burly dude with a spoon, and you will never be able to outspoon them. Even the powerful people, they’re just spooning harder and more vigorously than everyone else, like hungry orphan children eating soup. Except the soup is power. I’ll level with you here: I have completely lost track of this analogy.

What I’m saying, giant talking cat, is that everyone is stupid. They attain a narrow grasp of reality and live their life as though there is nothing else. But you, me, creatures with imagination – we aren’t constrained by our experiences. We’re inspired by them. If we have trouble digging with a spoon, we build a shovel. If we’re stopped by a wall, we make a door. And if we can’t make a door, we ask ourselves whether we really need an opening to pass through something solid in the first place.

You point out that you’re not a ghost, and that you do need an opening to pass through solid objects.

No – that’s your mistake, he replies. That’s why you’re still not thinking like a witchhunter. You’re trying to do things right, and that’s wrong. Mysticism means taking a step back – accepting that the very laws of reason and logic you abide by are merely one option of many. It means knowing you only see half the picture in a world where everyone else thinks they see the whole thing. It means having the sheer arrogance to have humility.

That’s why I’m saying you have to think like a witchhunter. You have to be a little wrong to be completely right – to abandon truth in favor of questioning falsehood. If you think something’s the easiest way, you have to know you’re wrong. You have to understand how to stand against the very stance of understanding! You have to know you are inferior; that your knowledge and perceptions will never stand up to the true scope of all possible reality. You have to be a little further from perfect, and embrace that notion.

Source: http://www.prequeladventure.com/2014/05/3391/