Two Cultures of Number Theorists

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There is a famous distinction in prime number theory between the number theorists who like to multiply primes, and the number theorists who like to add primes. As the primes are very heavily multiplicatively structured, the mathematics of multiplying primes is very algebraic in nature, in particular involving field extensions, Galois representations, etc. But the primes are very additively unstructured, and so for adding primes we see the tools of analysis used instead (circle method, sieve theory, etc.).

Mentioned by Terry Tao in a comment at n-category cafe.

Feynman on Reading Difficult Things

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Well, I asked him, “How can I read it? It’s so hard.” He said, “You start at the beginning and you read as far as you can get, until you are lost. Then you start at the beginning again, and you keep working through until you can understand the whole book.”
—Joan Feynman, Richard Feynman’s sister, recalling a discussion with her brother

Taken from No Ordinary Genius.

Scott Aaronson on Mathematics

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From Luke’s recent interview of Scott Aaronson (theoretical compsci guy at MIT, who blogs here):

Things like linear algebra, group theory, and probability have so many uses throughout science that learning them is like installing a firmware upgrade to your brain — and even the math you don’t use will stretch you in helpful ways.

Deciphering Core Human Values In A Society of Mind

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Know thyself? If I knew myself I would run away.
—Johann Wolfgang von Goethe

Humans are evolutionary hacks. I’m often not of one mind, or even two, but of four and sometimes more. Our brains seem to be locked in an eternal struggle, a constant clash of warring preferences. Consider the would-be comedian who, instead of working on his act, spends the day watching Family Guy reruns. He is of two minds: one wishes to watch Family Guy while another wants to brainstorm new routines.

Many-Self Model

You are not the king of your brain. You are the creepy guy standing next to the king going “a most judicious choice, sire”.
—Steven Kaas

It’s interesting to listen to the way that people use language when talking about the self. People say things like, “I had to talk myself into going to the gym.” This is a normal phrase. I hear it all the time. Unremarkable.

But exactly who is talking to who? The self had to convince the self into doing something? Or the popular maxim, “Just be yourself.” Who else are you going to be? All self-talk has this sort of strangeness to it. Why would myself need to counsel myself about anything?

You’re a brain, but there’s not just one you, and these many-you are most evident when they’re in conflict. Consider the overweight man who finds himself in a familiar dilemma: chocolate cake, to eat or not to eat? In one corner, there is a piece of him who wants to eat the cake. In the other corner, there is a piece of him who wants to lose weight. The bell sounds. Fight!

Or take creative alarm clocks. There is one, Clocky, that, like a roomba, moves about the room while it goes off, so that you have to chase it down in the morning. There is another recent one for Android phones that requires you to solve a math problem before it will stop ringing. A friend told me about this. He said he’s been “using” it, but instead of solving it in the morning, he just turns off his phone.

I’ve read, too, about people — adults — who want to stop biting their nails, so they’ll coat them with something bitter. It doesn’t work, though. They just end up finding some clever way to wash it off.

I love these because they characterize the absurdity of the human condition. The present-me installs an alarm clock with every intention of getting up on time, only to be thwarted by morning-me. These two versions of me might as well be different people, each trying to control the other. Our experience is this constant struggle, every part of our brains pulling and pushing us in two or three or a thousand different directions.

Many-Selves and Many-Goals

Imagine that you throw a party for New Year’s Eve and, as part of a game, everyone must write down their resolutions for the incoming year, which you then combine on one sheet of paper. You then go around and guess which resolution belongs to which person.

Now, consider the sheet of everyone’s goals. There’s no reason for them to be consistent with each other. One person might want to save money while another might want to buy a house.

And that’s fine. It’s no problem for these people if their goals conflict. They’re different people, each pursuing rapper Gudda Gudda’s maxim of “You do you. I’ma do me.” It is a problem for you and me, though, because we’re a lot more like a body shared by an entire party of selves (or agents or modules if you prefer) than one consistent identity. A mind is not one individual, but a society. Our goals are as contradictory as a list of the goals of a dozen or so people.

Explicit and Implicit Goal-Keeping

And our list of woes grows longer, because the type of goals that one is willing to write on a list are not the same as the desires of each self inside of us. Our selves have differing time preferences, for example, some preferring instant gratification while others want to plan for the future. I would not write “eat whatever takes the least effort to make” on a list of New Year’s resolutions, but you can be damn sure that there’s a chunk of my mind that prefers convenience over health.

The point is that the human mind is complicated, conflicted, inconsistent, and not so much one unit, but more of a group of competing modules, and this insight forces us to think differently about our goals.

Maybe this is clearer with a thought experiment. Imagine that you’re presented with a genie who is willing to grant you one wish and you wish for a complete list of your goals. This list is going to look a whole lot different than a list that you make by sitting down and thinking about what it is that you want out of life. A list of your explicit and implicit goals is different than a list of just explicit goals.

Let’s make it concrete. Maybe you’re familiar with “Movember,” which is where men grow facial hair during the month of November, in order to raise awareness for men’s health issues, like prostate cancer. This all sounds very nice, yes? But what does growing facial hair have to do with prostate cancer? Nothing. Raising awareness about something doesn’t do much good at all, certainly not as much good as a direct donation. It’s more about appearing caring, convincing other people of your virtue, than about actual helping. Or maybe it’s just about funny facial hair. Either way, not about helping.

Most of us carry around this explicit goal of helping people, while the reality seems to be more sinister. The way we behave seems to be more along the lines of convince-other-people-I’m-virtuous. This is clear whenever some tragedy strikes and my Facebook feed is filled with people posting “My prayers go out to the families of those involved.” First of all, even under the assumption that prayers work, there’s no reason to post on the internet telling everyone about you praying and, second of all, prayers might be nice but a five dollar donation is a lot nicer.

Knowing Thyself

The point I’m developing, then, is:

  • Human value is complicated and often contradictory.
  • Our wants and desires are not obvious.

This leads us to the question of: How can we determine what it is that we want? As a litmus test, do you think an exercise like, “Imagine you’re looking back on your life trying to decide what was important and what wasn’t,” is going to be enough to figure out your goals? The answer is no, although thinking about such a question might give you a starting point.

What we’re after, then, is accurate means of understanding ourselves, techniques that will give us some measure of clarity if it’s to be had. We would like to — where possible — eliminate reliance on subjective experience and inject a measure of rigor into knowing ourselves. We’d like some certainty.

Understanding Why

It’s instructive to step back and survey our surroundings. Why does it matter whether or not we pursue the right goal? There are a whole lot of people at colleges across the country who are right now cramming for finals. They are soon going to forget everything. They’ve replaced the goal of learning with the goal of getting a passing grade.

We care about pursuing certain goals and not others because some will better achieve our values — for the same reason that we prefer eating cheeseburgers to eating dirt: we like cheeseburgers and not dirt.

We can continue down the rabbit hole and ask, “Why ought I prefer one thing to another?” I used to worry about this, but the question is confused. Maybe there is no good reason why you ought to prefer cheeseburgers to dirt, but it’s the case that yous do. Our brains ensure that we have preferences.

The point of a goal, then, is to achieve whatever it is that these preferences are. Over the summer, I did a literature review of the current state of the art of happiness research, because I value happiness. The trouble with the wrong goal is that it moves us towards something we don’t value. It could be the case that people care not so much about doing good than about convincing other people that they’re good. The two values suggest different goals. If I want to help people, I could apply for a consultation at 80,000 hours, while if I want to convince people that I’m a good person, I could work on becoming more charismatic.

Values as Bedrock

People, by and large, act as if goals are nebulous things that appear out of nowhere, as if whispered to them by the gods. Their striving is chaotic, less the product of thoughtful reflection and more the result of the media’s near constant attack on our senses.

Consider the man who decides to become a lawyer because he believes doing so will make him happy. If he had first considered that his ultimate value was happiness, he might first decide to research on what it is that makes people happy and the happiness of lawyers. In the process, he might stumble on Forbes reporting “associate attorney” as the unhappiest job in America, and save hundreds of thousands of dollars and years of striving towards the wrong goal.

The point I’m making, then, is that with an accurate list of your own values, you can come up with plans for achieving those values. If life has a meaning or purpose, this is the closest I’ve come to finding it.

In compelling recipe format, the meaning of life:

  1. Know what it is that you want.
  2. Plan out the best way to get it.
  3. Implement that plan.

Our trouble begins, as I developed earlier, with the first step. It’s not obvious what it is that we want. We need some way to figure it out and, given that this is the foundation on which any goal is built, it’s hard to overstate the importance of some clarity as to our values.

Identifying Values

The most direct route to understanding your own values seems to be by figuring out those of others — at least in part — and then assuming that you also value those things. One example: I don’t have much explicit interest in romantic relationships, but whenever I find myself reading about male-female mate preferences and clash-of-the-sexes-type articles, I notice that I’m fascinated. Given that most people are interested in understanding the opposite sex and that millions of generations worth of evolution has dedicated significant portions of my brain to that task, I find myself forced to update in the direction that, no, I’m not special snowflake who don’t want no woman.

In fact, that anecdote has another point. We can often illuminate ourselves by understanding how evolution has shaped our desires and motives. Indeed, there’s no need to limit ourselves to evolution. Any knowledge that illuminates humankind is useful in furthering our understanding of ourselves, whether it be neuroscience, artificial intelligence, psychology, economics, or politics, which is sort of empowering. There are many routes to self-knowledge.

Given this, what can we say about human values? Well, core human values are straightforward. Most everyone wants:

  • Happiness, positive emotions
  • Freedom from pain, good health, and an absence of negative emotion
  • Fulfilling interpersonal relationships, romantic and otherwise
  • A sense of meaning and purpose in life
  • A conviction that our actions make a difference and that we matter
  • The respect and admiration of other people
  • Personal growth and self-improvement, increases in our own skill and competence

Beyond these, its less obvious. I looked over the New York Times Best Sellers list, but didn’t find it all that illuminating, except I will note that people seem more interested in reading about “proof” of heaven and history in general that I would have thought.

Some values are more idiosyncratic, though. In psychology, there is a personality trait of “openness to experience,” which sort of captures how interested someone is in learning new things, trying new foods, that sort of thing. Creative types score high on openness and this trait varies among individuals. You probably know people who are not interested in reading books or any intellectual pursuits. These people are low on openness.

We could think of this value as “the exploration of one’s interest” or the value of learning about the world. This one is a bit odd because we can ask, “well, do we really care about the exploration of interests or are we interested in the exploration of interests as a means to an end?” It’s a little bit of both. Sometimes we’re interested in things because of what they can do for us, but it’s also enjoyable in and of itself to explore something interesting. This could be grouped under “positive emotions” above, but I think it’s a useful distinction. To be fair, I ought to point out that “positive emotion” and “negative emotion” cover a broad swath of human experience: awe, excitement, interest, anxiety, sadness, dread, contentment, and more.

But what else do we value and want out of life? In economics, there’s this notion of signalling. You might volunteer at a homeless shelter not because you care about the homeless, but because you care about signalling to other people that you’re caring. A fair amount of human activity seems to revolve around looking good rather than anything of substance. Robin Hanson writes quite a bit about this topic.

More troubling are those things that we value — as big, smart monkeys — that we aren’t “supposed” to value. If you’re familiar with Nietzsche, he writes a bit about the enjoyment of cruelty and vengeance. I’m reminded of a scene from Conan the Barbarian, when Conan is asked, “What is best in life?” He responds, “To crush your enemies, see them driven before you, and to hear the lamentation of their women.”

In this vein, you’ll note that people seem, on the whole, more interested in winning arguments than getting to the truth of whatever it is that they’re discussing. It’s more about domination, more battle than discovery. Winning battles against opposing tribes is satisfying (politics!) and, while I have never crushed an enemy — at least not physically — I suspect it feels pretty good.

Stockpiling Self-Knowledge

To know oneself, one should assert oneself.
—Albert Camus

The general undercurrent here, then, seems to be that — in order to identify values — one ought to amass knowledge about oneself and humans in general, develop a certain sensitivity to what it that people value and desire and an accurate understanding of our own idiosyncrasies.

I have a couple ideas about how to go about this, but no silver bullet. There is no royal road to self-knowledge. It’s hard work.

  • Read and learn about different fields that shed light on what it is that humans want. There are a lot of possibilities here, as I mentioned earlier, from psychology, cognitive science, artificial intelligence, ethics, and more. Really, any field that deals with some aspect of humanity has something to offer. (This looks like a good place to start.)
  • Careful observation of ourselves and others. How do we act and feel in different situations? What do our actions and words suggest about our goals and values? When do we clash with other people? What are others striving for? What do people spend money on? Notice what’s popular and why. (Cultivating mindfulness might be useful for this.)
  • It’s instructive to consider what chimpanzees want and how humans are similar and different.
  • Take a different point of view when considering someone. If a stranger did the same things that you do, what would you think about them? If someone does something that you don’t understand, ask why you would do that in their situation. Everyone feels normal from the inside.
  • Try getting honest feedback from others. What do they think about you? How does this differ from your self-concept. One study found that other people were better at predicting the length of a relationship than those in the relationship.
  • You might try keeping a journal, or any other of the thought experiments that people suggest when considering values. What do you want for your children? What makes you jealous? All of these suggest possible values.
  • Some research suggests if we know about our biases, we may be better able to control for them.
  • Reflect. How much would you pay to prevent a chicken from being tortured? Would you rather have more technology or less? Would a happiness pill be a good thing? Construct counterfactuals and intuition pumps. Ask yourself, “Is this my true motive? Is there something deeper here?”
  • In an uncertain world, there’s a great deal of value in preserving your options and hedging your bets. Maybe you don’t think you care about social status or money, but — given that there’s a not insignifcant chance you could be mistaken — invest in something that’s either transferable or that will move you towards many different things simultaneously. Paul Graham writes about majoring in math instead of economics, since a math major can get a PhD in economics, but an economics major can’t get a PhD in mathematics.

The Life Satisfaction of Economists

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Tyler Cowen has written a post on a paper about the life satisfaction of economists. It’s a horrible paper. I don’t like it at all.

Here are some reasons:

  • The authors use satisfaction and happiness interchangeably. They are not the same construct and it will confuse those not familiar with the existing literature.
  • The sample is taken from a few mailing lists of European economists.
  • The study measures life satisfaction with a single question.
  • The life satisfaction question is part of a broader survey focused on scientific misbehavior, which means that such information is going to be primed before the life satisfaction question.
  • Meditation: Imagine that you conduct two studies: one in which you ask participants to reflect on all the good things that have happened to them in the past three months, followed by a life satisfaction question, and one in which you just ask them about life satisfaction. Is there a difference?
  • I remain unconvinced that numeric ratings of life satisfaction can be meaningfully compared across populations. The French rate themselves as less healthy than Americans, but live an average of 3 years longer.

Still not convinced? The Maasai are a semi-nomadic African people in Kenya and the average Maasai is as satisfied as the economists in this study.

World War II: Not a Moral Triumph

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I’ve started reading through some of the works of Mencius Moldbug. I figure that politics is a waste of cognition but, hey, it’s entertaining and I enjoy being exposed to different points of view. Moldbug is a bit long-winded (understatement), but the guys at More Right have put together PDF copies of the “essential Moldbug,” which makes reading it a whole hell of a lot easier.

I guess I better set this up. Mencius has been churning out millions of words criticizing progressivism, a movement which is perhaps best exemplified by the drivel that gets thousands of upvotes on Reddit. It’s the sort of political ideals embodied by academia and all those who describe themselves — without a hint of self-loathing — as an intellectual.

Anyways, I’ve been reading through Moldbug, and was struck by a passage where he points out an important piece of unquestioned and no doubt revisionist history: World War II was a struggle of good versus evil and good won. The popular narrative goes something like, “Hitler was an evil man, committing gross human rights violations by rounding up and slaughtering Jews. The Allied powers wouldn’t stand for this, so we put a stop to it.”

But this ignores:

Makes you wonder about this whole moral progress thing.

The Terrible Future Isn’t: Drone Delivery Edition

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Last night, Amazon’s CEO Jeff Bezos appeared on 60 Minutes and spoke about the possibility of using drones to deliver small packages (“drone delivery“). My gut reaction is the engineer’s natural skepticism: I’ll believe it when he’s rolled out something that works instead of touting vaporware. (Remember hyperloop? Yeah, that’s never going to happen.) That’s not so fun to read about, though, so let’s instead talk about the promise of drone delivery.

The setup: imagine that anything a drone can carry can be delivered to your door within 30 minutes of purchase, provided that someone has it within ten miles of your location.

My initial thought, given that I have yet to eat breakfast, is food delivery. You don’t order meals through Amazon because you don’t want to wait two days when you’re hungry. Given that there are already businesses — pizza delivery, for one — in this market, I could see it happening. Indeed, if drones become easy and cheap enough that small businesses can afford them, we could see scenarios where women working from home make fresh food (cookies or lasagna or what-have-you) to order via the internet. Chefs working from home and delivering meals via drone could disrupt the restaurant industry.

More prosaically, if you’re out of bagels or you need some fresh oregano for the dish you’re planning on making for dinner, you could have a drone fly you some from the grocery store. Or maybe if you need a suit dry-cleaned or a package shipped, a drone could come and pick up the goods.

I could see this changing the operation of libraries, too, where you can order a library book online and then have it delivered to you via drone. The same goes for prescription medication.

There are certain things that people want to replace as quickly as possible: television remotes, specialty drill bits, and internet routers. If you have a pet reptile, you’ll know that when their sunning lamp burns out, you want to replace it pronto. These sort of things would be fine candidates for drone delivery.

Another interesting potential is the ability to deliver goods to places other than houses. If you’re out camping and forgot to bring hot dogs, a drone could fly out to your location. Maybe something like this would work for broken down cars, too, although I’m not sure what you might order. A blanket or hot chocolate if your car is stuck in the snow, maybe.

I’ve seen some talk of this changing the way drug deals go down because it would eliminate the risk of a face-to-face transaction. I’m not completely convinced of this, though, because a police officer could just order drugs online and then follow the drone back to wherever it came from after delivery. It might still be safer, though.

Anyways, the focus here has been on incremental improvement and the disappearance of chores. If drones do pan out, though, the interesting effects are going to be in how they disrupt markets and enable new types of trade. We can imagine trying to predict the consequences of the internet or the telephone when they were first introduced. The changes wrought were no doubt much larger than anyone expected. I don’t think drones are as fundamental as either of those, but I do think that — should the technology come to fruition — the results will be quite different from what we anticipate.

Criticism of Economics

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I don’t have the requisite expertise to lay out compelling criticism of economics as a whole, and I suspect such an endeavor would be profitless (heh). The thrust of such an argument, though — its quintessence — is captured in this quote by Richard Feynman:

See, I have the advantage of having found out how hard it is to get to really know something, how careful you have to be about checking the experiments, how easy it is to make mistakes and fool yourself. I know what it means to know something and therefore, I see how they get their information and I can’t believe that they know it.

A useful heuristic I employ (among others) when evaluating the trustworthiness of knowledge is to ask myself the question, “How confident am I that this will still be true in fifteen years?” Note the progress and effectiveness of the natural sciences, e.g. physics, when compared with the softer sciences, e.g. psychology. The results of physics are solid. You can build on the knowledge. Building on psychology, on the other hand, is building on top of a swamp; constructing a home on quicksand.

Now, economics is more like quicksand than concrete, and most economic reasoning is vulnerable to dismissals based on this. You might say something like: well, such and such assumption is untenable. The world is too complex to be boiled down and understood via simple economic models.

The problem with such arguments, with the skeptic dismissal, is that they’re a means of filtering out just the things that you don’t like. If you read an economic argument that you don’t like, you say, “Boo! Economics!” and dismiss it outright, instead of engaging with it.

Further, even though economics doesn’t work all that well, there are no good alternatives. You can have either not terribly effective models or no models at all, and you’re going to be much better off trying to reason with some model than no model.

Right, so what I’m getting at: it’s a failure mode to take economics too seriously, but it’s another failure mode to dismiss it entirely. There is a middle way; one must reach a certain Zen understanding of the limits of knowledge.

“Not A Real Christian” Is Real Rhetoric

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Perhaps Sarah Palin will actually read about what that Jesus guy kept talking about and her head will explode.
—Reddit user Popcom

I have spent the past couple of weeks spending 15 minutes chunks on Reddit, leaving when I become disgusted enough to get back to working through my topology textbook. The experience has, however, been enlightening insofar as it sheds light on the opinions of the masses. Hence, this post.

Discussions will often go something like this: some religious group deviates from Reddit-brand progressivism (which is almost a caricature of the United States left). Then, some commenter denounces them as not a real Christian and receives several hundred upvotes. The Westboro Baptist group, infamous for protesting funerals, is a common punching-bag in these discussions.

Now, I hope the brain damage here is already evident, but the reasoning seems to go that real Christians cannot believe anything that the typical Redditor finds morally repugnant. Otherwise, they’re not following Jesus! If it’s not my personal brand of Christianity, then it’s not Christianity.

You’ll often see something like, “Any real Christian supports equality for gays,” but let’s be honest here. The Bible’s position on homosexuality is about straightforward as these things get. So, if you’re going to argue that the One True Christianity supports homosexuality, you’ll find yourself scrambling for justifications as to why certain parts of the Bible don’t count.

But such an enterprise is circular. People who are arguing about what makes a real Christian are not interested in what the Bible says. It’s rhetoric. They’re interested in picking and choosing pieces of the Bible that support the belief system that they already have in place.

How do people construct their initial belief system, the one they’re using to fence off the real Christians from the fake ones? Through osmosis. It’s a patchwork of influences of peers, parents, the media, etc. The user who writes, “Not a real Christian!” means something more like, “Boo! Not a member of my tribe!” and is not making a point about what does or does not make a Christian.

Unreasonable Doubt: Legal Certainty is Impossible

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The line about “what if he’s guilty?” made me almost throw my laptop across the room. Who the fuck cares? It’s better to ensure 0 innocent men face punishment than to ensure all those guilty go to prison.
—User RPIAero on Reddit, 172 upvotes

You’ve been called for jury duty and you’re sitting in on a murder trial. The evidence is strong and you feel that, even after correcting for overconfidence and coming at the issue from every possible point of view you can think of, there’s a 99 percent chance that “he done it.”

Is 99 percent confidence enough to send a man to prison for the rest of his life? What if it’s a death penalty case? Then how much evidence do you need? I suspect, for most, 99 percent sounds pretty good. It sounds like enough to put a man in prison or even sentence a man to death. 99 percent is beyond a reasonable doubt.

But, of course, it’s never that simple and, like the quote above, there will be intelligent-but-confused people who will claim that one needs absolute certainty in order to sentence a man to death. You need to be 100 percent certain, or else you must vote not-guilty.

When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man. —Bertrand Russel

They have a point. After all, on the scale of bad-things, sending an innocent man to his death is up there: worse than cheating on your husband, but not “literally worse than Hitler.” It’s easy to imagine standing before Saint Peter at the pearly gates, and Saint Peter going through the list of your sins and saying, “Oh, yeah, Troy Davis was innocent, and section 3.a.III.j of the Heavenly code says that sentencing an innocent man to death is unforgivable, unless you’re a 7th Day Adventist. Tell Judas I said hey,” because, you know, heaven is full of bureaucrats and what-do-you-know, now you’re spending eternity in hell.

But these people are confused. There is no absolute certainty. There’s always some absurd, highly improbable way in which you might be wrong about everything. When people say that absolute certainty is necessary for sentencing a man to death, they really mean that you need to be closer to 99.999% certain than 99% certain of guilt in a death penalty case. If someone insists on absolute certainty of the 100% variety, no one can be convicted of anything. We’ll have to let all prisoners go. There’s that one in a trillion chance that everything you think you know is the product of a vast conspiracy, like in the movie The Truman Show.

All is not lost, though. Presumably, people don’t care so much about absolute certainty, but rather about not convicting people of crimes they did not commit. You don’t have to be absolutely certain, just certain enough that only criminals go to jail.

How certain do you need to be to prevent an innocent man from being imprisoned? What about 99 percent? Well, that doesn’t work because, on average, one out of a hundred prisoners is going to be innocent and, remember, our goal is no innocent people in prison whatsoever.

We want a level of certainty such that, given all of the people in the prison system, we can be confident that none of them are innocent. As of 2011, there are 2,266,800 adults incarcerated in the United States. With a bit of math, we find that even if, on every single trial, the jury was 99.999956% (1 in 2,266,800) certain of the defendant’s guilt, there’s a 37% chance that at least one innocent man will be sent to prison.

Now, I don’t know if you’ve ever met a human before, but none of them are 99.999956% right about anything. There is no way for someone to reach that level of certainty in every single trial. It’s an unpleasant truth that it’s more-or-less impossible to prevent innocent people from being convicted.

Instead, it’s a tradeoff. If we only require 85% certainty before convicting someone, more innocent people go to prison, but we also catch more guilty people. Or we can push the necessary evidence in the other direction: if we want 99% certainty, fewer innocents will go to prison, but more criminals will walk. The only possible system where no innocent men go to prison is one where no one at all goes to prison.

Well, we have to end apartheid for one. And slow down the nuclear arms race, stop  terrorism and world hunger. We have to provide food and shelter for the homeless, and oppose racial discrimination and promote civil rights, while also promoting equal rights for women. We have to encourage a return to traditional moral values. Most importantly, we have to promote general social concern and less materialism in young people.
—Patrick Bateman, American Psycho

But despair not, gentle reader! There is hope yet on the horizon. Why is it wrong to send innocent people to prison? Because it leads to suffering. The wrongness of something is decided by the amount of harm that it does to the world. While you can’t prevent every innocent person from going to jail, it’s within your power to prevent someone from dying of malaria. Hell, a kind word is enough to reduce suffering. The possibilities for doing good and improving the world are endless, and no doubt there are easier ways to go about it than fixing the justice system. Go find them.