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When you ask adults what they got wrong at that age, nearly all say they cared too much what other kids thought of them.

When someone contradicts you, they're in a sense attacking you. Your instinct when attacked is to defend yourself. But like a lot of instincts, this one wasn't designed for the world we now live in. Counterintuitive as it feels, it's better most of the time not to defend yourself. Otherwise these people are literally taking your life.

Cultivate a habit of impatience about the things you most want to do. Don't wait before climbing that mountain or writing that book or visiting your mother. You don't need to be constantly reminding yourself why you shouldn't wait. Just don't wait.

Relentlessly prune bullshit, don't wait to do things that matter, and savor the time you have. That's what you do when life is short.

Not all cities send a message. Only those that are centers for some type of ambition do. New York tells you: you should be richer. Boston is that the message there is: you should be smarter. The message Berkeley sends is: you should live better. The message Silicon Valley sends is: you should be more powerful. Cambridge as a result feels like a town whose main industry is ideas, while New York's is finance and Silicon Valley's is startups.

Unless you're sure what you want to do and where the leading center for it is, your best bet is probably to try living in several places when you're young. You can never tell what message a city sends till you live there, or even whether it still sends one.

Some people know at 16 what sort of work they're going to do, but in most ambitious kids, ambition seems to precede anything specific to be ambitious about. They know they want to do something great. They just haven't decided yet whether they're going to be a rock star or a brain surgeon. There's nothing wrong with that. But it means if you have this most common type of ambition, you'll probably have to figure out where to live by trial and error. You'll probably have to find the city where you feel at home to know what sort of ambition you have.

Suppose you're a college freshman deciding whether to major in math or economics. Well, math will give you more options: you can go into almost any field from math. If you major in math it will be easy to get into grad school in economics, but if you major in economics it will be hard to get into grad school in math. Flying a glider is a good metaphor here. Because a glider doesn't have an engine, you can't fly into the wind without losing a lot of altitude. If you let yourself get far downwind of good places to land, your options narrow uncomfortably. As a rule you want to stay upwind. So I propose that as a replacement for "don't give up on your dreams." Stay upwind.

If it takes years to articulate great questions, what do you do now, at sixteen? Work toward finding one. Great questions don't appear suddenly. They gradually congeal in your head. And what makes them congeal is experience. So the way to find great questions is not to search for them-- not to wander about thinking, what great discovery shall I make? You can't answer that; if you could, you'd have made it. The way to get a big idea to appear in your head is not to hunt for big ideas, but to put in a lot of time on work that interests you, and in the process keep your mind open enough that a big idea can take roost. Einstein, Ford, and Beckenbauer all used this recipe. They all knew their work like a piano player knows the keys. So when something seemed amiss to them, they had the confidence to notice it.

Almost anyone would rather, at any given moment, float about in the Carribbean, or have sex, or eat some delicious food, than work on hard problems. The rule about doing what you love assumes a certain length of time. It doesn't mean, do what will make you happiest this second, but what will make you happiest over some longer period, like a week or a month.

Prestige is especially dangerous to the ambitious. If you want to make ambitious people waste their time on errands, the way to do it is to bait the hook with prestige. That's the recipe for getting people to give talks, write forewords, serve on committees, be department heads, and so on. It might be a good rule simply to avoid any prestigious task. If it didn't suck, they wouldn't have had to make it prestigious.

"Always produce" is a heuristic for finding the work you love. If you subject yourself to that constraint, it will automatically push you away from things you think you're supposed to work on, toward things you actually like. "Always produce" will discover your life's work the way water, with the aid of gravity, finds the hole in your roof.

Don't decide too soon. Kids who know early what they want to do seem impressive, as if they got the answer to some math question before the other kids. They have an answer, certainly, but odds are it's wrong. You won't want to have a life chosen for you by a high-school kid. Unless you're fairly sure what you want to do, your best bet may be to choose a type of work that could turn into either an organic or two-job career. That was probably part of the reason I chose computers.

I think what religion and politics have in common is that they become part of people's identity, and people can never have a fruitful argument about something that's part of their identity. The more labels you have for yourself, the dumber they make you.

The way most fortunes are lost is not through excessive expenditure, but through bad investments. The most dangerous way to lose time is not to spend it having fun, but to spend it doing fake work. With time, as with money, avoiding pleasure is no longer enough to protect you.

When experts are wrong, it's often because they're experts on an earlier version of the world.

Instead of trying to point yourself in the right direction, admit you have no idea what the right direction is, and try instead to be super sensitive to the winds of change. The way to come up with new ideas is not to try explicitly to, but to try to solve problems and simply not discount weird hunches you have in the process.

Surround yourself with the sort of people new ideas come from. If you want to notice quickly when your beliefs become obsolete, you can't do better than to be friends with the people whose discoveries will make them so.

These seem to me what philosophy should look like: quite general observations that would cause someone who understood them to do something differently.

Getting to general plus useful by starting with useful and cranking up the generality may be unsuitable for junior professors trying to get tenure, but it's better for everyone else, including professors who already have it. This side of the mountain is a nice gradual slope. You can start by writing things that are useful but very specific, and then gradually make them more general. Joe's has good burritos. What makes a good burrito? What makes good food? What makes anything good? You can take as long as you want. You don't have to get all the way to the top of the mountain. You don't have to tell anyone you're doing philosophy.

This argument seems to me like someone in 1500 looking at the lack of results achieved by alchemy and saying its value was as a process. No, they were going about it wrong. It turns out it is possible to transmute lead into gold (though not economically at current energy prices), but the route to that knowledge was to backtrack and try another approach.

Physics seems to us a promising thing to work on, and alchemy and theology obvious wastes of time. But that's because we know how things turned out. In Newton's day the three problems seemed roughly equally promising. Newton made three bets. One of them worked. But they were all risky.

If we were talking about the taste of apples, I'd agree that taste is just personal preference. Some people like certain kinds of apples and others like other kinds, but how can you say that one is right and the other wrong?

The thing is, art isn't apples. Art is man-made. It comes with a lot of cultural baggage, and in addition the people who make it often try to trick us. Most people's judgement of art is dominated by these extraneous factors; they're like someone trying to judge the taste of apples in a dish made of equal parts apples and jalapeno peppers. All they're tasting is the peppers. So it turns out you can pick out some people and say that they have better taste than others: they're the ones who actually taste art like apples.

In fact, one of the reasons artists in fifteenth century Florence made such great things was that they believed you could make great things. So the most important consequence of realizing there can be good art is that it frees artists to try to make it. There is such a thing as good art, and if you try to make it, there are people who will notice.

When a technology is young, the existing solutions are usually terrible; which means it must be possible to do much better; which means many problems that seem insoluble aren't. It's important to remember we're trying to solve a new problem, because that means we're going to have to try new things, most of which probably won't work.

I like to find (a) simple solutions (b) to overlooked problems (c) that actually need to be solved, and (d) deliver them as informally as possible, (e) starting with a very crude version 1, then (f) iterating rapidly.

I think the problem with many, as with people in their early twenties generally, is that they've been trained their whole lives to jump through predefined hoops. They've spent 15-20 years solving problems other people have set for them. And how much time deciding what problems would be good to solve? Two or three course projects? They're good at solving problems, but bad at choosing them.

What appears to be a fault, often, by a change of viewpoint, turns out to be one of the greatest assets you can have. But you are not likely to think that when you first look the thing and say, "Gee, I'm never going to get enough programmers, so how can I ever do any great programming?'' And there are many other stories of the same kind; Grace Hopper has similar ones. I think that if you look carefully you will see that often the great scientists, by turning the problem around a bit, changed a defect to an asset. For example, many scientists when they found they couldn't do a problem finally began to study why not. They then turned it around the other way and said, "But of course, this is what it is'' and got an important result. So ideal working conditions are very strange. The ones you want aren't always the best ones for you.

"Knowledge and productivity are like compound interest.'' Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime.

There's another trait on the side which I want to talk about; that trait is ambiguity. It took me a while to discover its importance. Most people like to believe something is or is not true. Great scientists tolerate ambiguity very well. They believe the theory enough to go ahead; they doubt it enough to notice the errors and faults so they can step forward and create the new replacement theory. If you believe too much you'll never notice the flaws; if you doubt too much you won't get started. It requires a lovely balance. But most great scientists are well aware of why their theories are true and they are also well aware of some slight misfits which don't quite fit and they don't forget it. Darwin writes in his autobiography that he found it necessary to write down every piece of evidence which appeared to contradict his beliefs because otherwise they would disappear from his mind. When you find apparent flaws you've got to be sensitive and keep track of those things, and keep an eye out for how they can be explained or how the theory can be changed to fit them. Those are often the great contributions. Great contributions are rarely done by adding another decimal place. It comes down to an emotional commitment. Most great scientists are completely committed to their problem. Those who don't become committed seldom produce outstanding, first-class work. Now again, emotional commitment is not enough. It is a necessary condition apparently. And I think I can tell you the reason why. Everybody who has studied creativity is driven finally to saying, "creativity comes out of your subconscious.'' Somehow, suddenly, there it is. It just appears. Well, we know very little about the subconscious; but one thing you are pretty well aware of is that your dreams also come out of your subconscious. And you're aware your dreams are, to a fair extent, a reworking of the experiences of the day. If you are deeply immersed and committed to a topic, day after day after day, your subconscious has nothing to do but work on your problem. And so you wake up one morning, or on some afternoon, and there's the answer. For those who don't get committed to their current problem, the subconscious goofs off on other things and doesn't produce the big result. So the way to manage yourself is that when you have a real important problem you don't let anything else get the center of your attention - you keep your thoughts on the problem. Keep your subconscious starved so it has to work on your problem, so you can sleep peacefully and get the answer in the morning, free.

Over on the other side of the dining hall was a chemistry table. I had worked with one of the fellows, Dave McCall; furthermore he was courting our secretary at the time. I went over and said, "Do you mind if I join you?'' They can't say no, so I started eating with them for a while. And I started asking, "What are the important problems of your field?'' And after a week or so, "What important problems are you working on?'' And after some more time I came in one day and said, "If what you are doing is not important, and if you don't think it is going to lead to something important, why are you at Bell Labs working on it?" I wasn't welcomed after that; I had to find somebody else to eat with! That was in the spring. In the fall, Dave McCall stopped me in the hall and said, "Hamming, that remark of yours got underneath my skin. I thought about it all summer, i.e. what were the important problems in my field. I haven't changed my research,'' he says, "but I think it was well worthwhile.'' And I said, "Thank you Dave,'' and went on. I noticed a couple of months later he was made the head of the department. I noticed the other day he was a Member of the National Academy of Engineering. I noticed he has succeeded. I have never heard the names of any of the other fellows at that table mentioned in science and scientific circles. They were unable to ask themselves, "What are the important problems in my field?'' If you do not work on an important problem, it's unlikely you'll do important work. It's perfectly obvious. Great scientists have thought through, in a careful way, a number of important problems in their field, and they keep an eye on wondering how to attack them. Let me warn you, 'important problem' must be phrased carefully. The three outstanding problems in physics, in a certain sense, were never worked on while I was at Bell Labs. By important I mean guaranteed a Nobel Prize and any sum of money you want to mention. We didn't work on (1) time travel, (2) teleportation, and (3) antigravity. They are not important problems because we do not have an attack. It's not the consequence that makes a problem important, it is that you have a reasonable attack. That is what makes a problem important. When I say that most scientists don't work on important problems, I mean it in that sense. The average scientist, so far as I can make out, spends almost all his time working on problems which he believes will not be important and he also doesn't believe that they will lead to important problems.

I saw that computers were transforming science because I spent a lot of time asking "What will be the impact of computers on science and how can I change it?'' I asked myself, "How is it going to change Bell Labs?'' I remarked one time, in the same address, that more than one-half of the people at Bell Labs will be interacting closely with computing machines before I leave. Well, you all have terminals now. I thought hard about where was my field going, where were the opportunities, and what were the important things to do. Let me go there so there is a chance I can do important things.

The great scientists, when an opportunity opens up, get after it and they pursue it. They drop all other things. They get rid of other things and they get after an idea because they had already thought the thing through. Their minds are prepared; they see the opportunity and they go after it.

I notice that if you have the door to your office closed, you get more work done today and tomorrow, and you are more productive than most. But 10 years later somehow you don't know quite know what problems are worth working on; all the hard work you do is sort of tangential in importance. He who works with the door open gets all kinds of interruptions, but he also occasionally gets clues as to what the world is and what might be important.

You should do your job in such a fashion that others can build on top of it, so they will indeed say, "Yes, I've stood on so and so's shoulders and I saw further.'' The essence of science is cumulative. By changing a problem slightly you can often do great work rather than merely good work. Instead of attacking isolated problems, I made the resolution that I would never again solve an isolated problem except as characteristic of a class. Now if you are much of a mathematician you know that the effort to generalize often means that the solution is simple. Often by stopping and saying, "This is the problem he wants but this is characteristic of so and so. Yes, I can attack the whole class with a far superior method than the particular one because I was earlier embedded in needless detail.'' The business of abstraction frequently makes things simple. Furthermore, I filed away the methods and prepared for the future problems. To end this part, I'll remind you, "It is a poor workman who blames his tools - the good man gets on with the job, given what he's got, and gets the best answer he can.'' And I suggest that by altering the problem, by looking at the thing differently, you can make a great deal of difference in your final productivity because you can either do it in such a fashion that people can indeed build on what you've done, or you can do it in such a fashion that the next person has to essentially duplicate again what you've done. It isn't just a matter of the job, it's the way you write the report, the way you write the paper, the whole attitude. It's just as easy to do a broad, general job as one very special case. And it's much more satisfying and rewarding!

As a result, many talks are ineffective. The speaker names a topic and suddenly plunges into the details he's solved. Few people in the audience may follow. You should paint a general picture to say why it's important, and then slowly give a sketch of what was done. Then a larger number of people will say, "Yes, Joe has done that,'' or "Mary has done that; I really see where it is; yes, Mary really gave a good talk; I understand what Mary has done.'' The tendency is to give a highly restricted, safe talk; this is usually ineffective. Furthermore, many talks are filled with far too much information. So I say this idea of selling is obvious.

The people who do great work with less ability but who are committed to it, get more done that those who have great skill and dabble in it, who work during the day and go home and do other things and come back and work the next day. They don't have the deep commitment that is apparently necessary for really first-class work.

Many a second-rate fellow gets caught up in some little twitting of the system, and carries it through to warfare. He expends his energy in a foolish project. Now you are going to tell me that somebody has to change the system. I agree; somebody's has to. Which do you want to be? The person who changes the system or the person who does first-class science? Which person is it that you want to be? Be clear, when you fight the system and struggle with it, what you are doing, how far to go out of amusement, and how much to waste your effort fighting the system. My advice is to let somebody else do it and you get on with becoming a first-class scientist. Very few of you have the ability to both reform the system and become a first-class scientist.

Now self-delusion in humans is very, very common. There are enumerable ways of you changing a thing and kidding yourself and making it look some other way. When you ask, "Why didn't you do such and such,'' the person has a thousand alibis. If you look at the history of science, usually these days there are 10 people right there ready, and we pay off for the person who is there first. The other nine fellows say, "Well, I had the idea but I didn't do it and so on and so on.'' There are so many alibis. Why weren't you first? Why didn't you do it right? Don't try an alibi. Don't try and kid yourself. You can tell other people all the alibis you want. I don't mind. But to yourself try to be honest.

In summary, I claim that some of the reasons why so many people who have greatness within their grasp don't succeed are: they don't work on important problems, they don't become emotionally involved, they don't try and change what is difficult to some other situation which is easily done but is still important, and they keep giving themselves alibis why they don't. They keep saying that it is a matter of luck. I've told you how easy it is; furthermore I've told you how to reform. Therefore, go forth and become great scientists!

Commencement speakers are always telling young people to follow their passions. Be true to yourself. This is a vision of life that begins with self and ends with self. But people on the road to inner light do not find their vocations by asking, what do I want from life? They ask, what is life asking of me? How can I match my intrinsic talent with one of the world’s deep needs? Their lives often follow a pattern of defeat, recognition, redemption. They have moments of pain and suffering. But they turn those moments into occasions of radical self-understanding — by keeping a journal or making art. As Paul Tillich put it, suffering introduces you to yourself and reminds you that you are not the person you thought you were. The people on this road see the moments of suffering as pieces of a larger narrative. They are not really living for happiness, as it is conventionally defined. They see life as a moral drama and feel fulfilled only when they are enmeshed in a struggle on behalf of some ideal. This is a philosophy for stumblers. The stumbler scuffs through life, a little off balance. But the stumbler faces her imperfect nature with unvarnished honesty, with the opposite of squeamishness. Recognizing her limitations, the stumbler at least has a serious foe to overcome and transcend. The stumbler has an outstretched arm, ready to receive and offer assistance. Her friends are there for deep conversation, comfort and advice. External ambitions are never satisfied because there’s always something more to achieve. But the stumblers occasionally experience moments of joy. There’s joy in freely chosen obedience to organizations, ideas and people. There’s joy in mutual stumbling. There’s an aesthetic joy we feel when we see morally good action, when we run across someone who is quiet and humble and good, when we see that however old we are, there’s lots to do ahead. The stumbler doesn’t build her life by being better than others, but by being better than she used to be. Unexpectedly, there are transcendent moments of deep tranquillity. For most of their lives their inner and outer ambitions are strong and in balance. But eventually, at moments of rare joy, career ambitions pause, the ego rests, the stumbler looks out at a picnic or dinner or a valley and is overwhelmed by a feeling of limitless gratitude, and an acceptance of the fact that life has treated her much better than she deserves. Those are the people we want to be.

The role of Science is to explain and predict the world around us; it ‘converts’ information into knowledge. The role of Engineering is to apply scientific knowledge to the development of solutions for empirical problems; it ‘converts’ knowledge into utility. The role of Design is to produce embodiments of solutions that maximize function and augment human experience; it ‘converts’ utility into behavior. The role of Art is to question human behavior and create awareness of the world around us; it ‘converts’ behavior into new perceptions of information, re-presenting the data that initiated the KCC in Science. At this ‘Cinderella moment’—when the hands of the KCC strike midnight—new perception inspires new scientific exploration. For example, in As Slow as Possible, John Cage transports the listener into a state where space and time are stretched, offering a personal interpretation of time dilation and questioning the nature of space-time itself.

Art is “all the stuff that people make”, and this includes our beliefs (which we like to call “reality”). Most people don’t think of science or technology as Art, but all three of these areas are actually art forms. One way to look at this vast area is to consider the ultimate critics of each art form. Most of what people call “Art” is the shaping of forms, and the ultimate critics are human beings. The forms are quite arbitrary and have no connection to the physical universe. For example, we can say: “It is the case that: blah blah blah”, and we can also insert a “not” into every sentence so we can say: “It is not the case that: blah blah blah”. So we can say anything which is almost the same as saying nothing. At the other extreme, on the right hand side, we have the sciences whose ultimate critic is Nature. Our opinions and hopes don’t matter over here, because Nature is just the way it is, not the way we’d like it to be. The arts of science are to find ways not to be fooled, to make the invisible more visible, and to create theories that are the best maps we can make of what we can’t get at directly. Science is very tricky because we have to use representation systems like mathematics, stories and computing that have no intrinsic relation to what’s out there in Nature (and we have to use our own easily fooled brains!). In the middle we have creations that have to heed Nature – such as bridges and airplanes – that we’d like to not break easily! But they also have visual forms that we’d like to be pleasing. The Technologies are very interesting art forms: they combine the traditional Arts with the new scientific Arts!

The importance of science is partly how well it is able to do with careful approximations. The representations are still story-like but a qualitatively new kind of story. If we look at this in the larger sense, it means that for efficiency sake, evolution set us up to think that our perceptions and beliefs are reality and we act that way. In the last few hundred years using science we’ve found over and over again that our perceptions are not accurate: we are constantly fooling ourselves. This means that a very good strategy for life, is to insert slow thought between perception and quick action because our initial perceptions and reactions are often wrong and dangerous. The ability of the Scientific Arts to “make the invisible a little more visible” has been quite remarkable. By the 18th century, people in Europe delighted in carrying around pocket globes that depicted the Earth as it would be seen from space even though the internal combustion engine and airplanes had not been invented yet.

The most important invisible thing today is ourselves. Most people live in stories made up by them and their societies, and they call these stories “reality”. We are the most dangerous force on Earth, to ourselves and the environment. It is the main aim of education to provide not information or techniques, but a better set of perspectives for better seeing the invisible. You can’t learn to see until you realize you are blind. Education is to help people realize they are blind and show them how to see a little.

I had been thinking of computers as tools, but this made me realize that the computer is a medium of expression – like reading and writing amplified by the printing press.

The real printing revolution was a qualitative change in thought and argument that lagged the hardware inventions by almost two centuries. The special quality of computers is their ability to rapidly simulate arbitrary descriptions, and the real computer revolution won't happen until children can learn to read, write, argue and think in this powerful new way. We should all try to make this happen much sooner than 200 or even 20 more years! This got me started designing computer languages and authoring environments for children, and I've been at it ever since.

The children think they are playing (and they are), but they are playing in an environment that has 21st century toys that embody 21st century ideas. They play for their own reasons – and children differ in why they play, and what they want to play at – but they all learn the powerful 21st century ideas, and even more important: they start learning the most powerful ways of thinking about the ideas.

Cesare Pavese said: “To know the world, one must construct it”. We can see that Computing is a new kind of Romantic Art Form where we make our ideas as Art, and the understanding of these ideas is Art. The Greeks said that the Fine Arts were the imitation of Life – but we see that the Fine Arts of Computing are the Imitation of Creation itself! It is this Romance that attracts children to build their ideas and helps them learn to think better than most adults do today.

In every tragedy, there is a gift, if we are able to see and accept it. From my brother, I received a personal understanding of death, and a constant reminder to live my life as though it may end at any moment. From my daughter, I learned what it means to love unconditionally, without expecting anything in return, a true gift. These gifts were delivered at great cost, but still I often struggle to retain them. Life gets busy, and I forget what matters. But the reminders are all around us, if only we can open our eyes.

"Products are vivid arguments about how we should live our lives." Whatever we put out there as a piece of design, into the world, has a persuasive component. It tries to affect people. It puts a certain vision of the good life out there in front of us. No matter whether we as designers intend it or not, we materialize morality. We make certain things harder and easier to do. We organize the existence of people. We put a certain vision of what good or bad or normal or usual is in front of people, by everything we put out there in the world.

Here’s a phrase I like: “Peace is happiness at rest; happiness is peace in motion.” Someone who’s peaceful at rest will end up happy when they do an activity. While a happy person sitting idle will be peaceful. The ultimate goal is not happiness, even though we use that term a lot. The goal is peace.

You cannot achieve peace directly or even work toward it. Rather, you can work toward understanding. There’s an old Sikh saying, “The name of God is truth.” When you understand certain things and they become a part of you, you naturally become a more peaceful person.

Up to that time the discrete number system and the continuous geometry flourished side by side with little conflict. The crisis of incommensurability tripped off the Euclidean approach to mathematics. It is a curious fact that the early Greeks attempted to make mathematics rigorous by replacing the uncertainties of numbers by what they felt was the more certain geometry. It was a major event to Euclid, and as a result you find in The Elements a lot of what we now consider number theory and algebra cast in the form of geometry. Opposed to the early Greeks, who doubted the existence of the real number system, we have decided that there should be a number that measures the length of the diagonal of a unit square, and that is more or less how we extended the rational number system to include the algebraic numbers. It was the simple desire to measure lengths that did it. How can anyone deny that there is a number to measure the length of any straight line segment?

They say that familiarity breeds contempt, and we are all more or less familiar with the real number system. Very few of us in our saner moments believe that the particular postulates that some logicians have dreamed up create the numbers — no, most of us believe that the real numbers are simply there and that it has been an interesting, amusing, and important game to try to find a nice set of postulates to account for them.

To summarize, from simple counting using the God-given integers, we made various extensions of the ideas of numbers to include more things. Sometimes the extensions were made for what amounted to aesthetic reasons, and often we gave up some property of the earlier number system. Thus we came to a number system that is unreasonably effective even in mathematics itself; witness the way we have solved many number theory problems of the original highly discrete counting system by using a complex variable. From the above we see that one of the main strands of mathematics is the extension, the generalization, the abstraction — they are all more or less the same thing — of well-known concepts to new situations.

The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai. It is necessary to emphasize this. We begin with a vague concept in our minds, then we create various sets of postulates, and gradually we settle down to one particular set. In the rigorous postulational approach the original concept is now replaced by what the postulates define. This makes further evolution of the concept rather difficult and as a result tends to slow down the evolution of mathematics. It is not that the postulation approach is wrong, only that its arbitrariness should be clearly recognized, and we should be prepared to change postulates when the need becomes apparent. The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way.

I am ready to strongly suggest that a lot of what we see comes from the glasses we put on. Of course this goes against much of what you have been taught, but consider the arguments carefully. You can say that it was the experiment that forced the model on us, but I suggest that the more you think about the four examples the more uncomfortable you are apt to become. They are not arbitrary theories that I have selected, but ones which are central to physics. Thus my first answer to the implied question about the unreasonable effectiveness of mathematics is that we approach the situations with an intellectual apparatus so that we can only find what we do in many cases. It is both that simple, and that awful. What we were taught about the basis of science being experiments in the real world is only partially true.

Just as there are odors that dogs can smell and we cannot, as well as sounds that dogs can hear and we cannot, so too there are wavelengths of light we cannot see and flavors we cannot taste. Why then, given our brains wired the way they are, does the remark "Perhaps there are thoughts we cannot think," surprise you? Evolution, so far, may possibly have blocked us from being able to think in some directions; there could be unthinkable thoughts.

I think that we-meaning you, mainly-must continue to try to explain why the logical side of science-meaning mathematics, mainly-is the proper tool for exploring the universe as we perceive it at present. I suspect that my explanations are hardly as good as those of the early Greeks, who said for the material side of the question that the nature of the universe is earth, fire, water, and air. The logical side of the nature of the universe requires further exploration.

I don't think students are made to understand how hard it is to do research. And how very, very hard it is to do important research. It's a lot harder than taking even very demanding courses. What makes it difficult is that research is immersion in the unknown. We just don't know what we're doing. We can't be sure whether we're asking the right question or doing the right experiment until we get the answer or the result.

We don't do a good enough job of teaching our students how to be productively stupid – that is, if we don't feel stupid it means we're not really trying. I'm not talking about 'relative stupidity', in which the other students in the class actually read the material, think about it and ace the exam, whereas you don't. I'm also not talking about bright people who might be working in areas that don't match their talents. Science involves confronting our 'absolute stupidity'. That kind of stupidity is an existential fact, inherent in our efforts to push our way into the unknown.

Productive stupidity means being ignorant by choice. Focusing on important questions puts us in the awkward position of being ignorant. One of the beautiful things about science is that it allows us to bumble along, getting it wrong time after time, and feel perfectly fine as long as we learn something each time. No doubt, this can be difficult for students who are accustomed to getting the answers right. No doubt, reasonable levels of confidence and emotional resilience help, but I think scientific education might do more to ease what is a very big transition: from learning what other people once discovered to making your own discoveries. The more comfortable we become with being stupid, the deeper we will wade into the unknown and the more likely we are to make big discoveries.