Powerful proof techniques

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There's proof by contradiction. There's proof by construction. For the brave of heart, there's proof by induction, and for the untiring, proof by exhaustion.

But sometimes you need something more. And for those cases, Strata of the World proudly presents the following collection of powerful proof techniques.

Proof by the algebra is boring: "... and the result follows after some uninteresting algebraic manipulations." An excellent choice when you're running out of space, need to cut words, or don't remember the intermediate steps.

Proof by boring algebra. Let's be honest, if your proof involves (or can be modified to involve) thirty lines of algebra, how many people are going to go through the details? To maximize the effect, write out the steps with a small font and no line spacing, use pixellated text, and separate it from the rest of the document by putting it in a textbox or appendix. If you suspect this to be insufficient, sprinkle remarks like "the details are not relevant to the main discussion", or "the intrepid reader is warned that the proof is arduous and somewhat tricky" into the surrounding text. Use this if you need to conceal some algebraic sleight-of-hand to "prove" your not-quite-a-theorem, or if you simply want to avoid annoying questions about it.

(Note: extremely determined readers may still decide to figure it out, but they are likely to spend at least an hour trying to work it out before they think to question its validity. This is when you make your getaway.)

Proof by circular reference: Write "We prove B by seeing that it follows from A, which we know to be true" in one place, and "We prove A by seeing that it follows from B, which we know to be true" somewhere else. This is especially effective when at least one of the statements is outside the main body of the text (appendix, textbox, etc.).

Proof by the proof is complicated: "To prove A, we invoke theorem X, the proof of which is beyond the scope of this discussion."

A special case of the above: proof by the proof involves calculus and is therefore omitted. This is a time-honored tactic used by many high-school physics and chemistry books.

Proof by elegance: "One possible solution to problem A is $$X$$. $$X$$ is an extremely elegant solution because [reason], and therefore is correct." A good "reason" might be that $$X$$ is the first thing you thought of. If all else fails, mumble something about "symmetry".

Proof by making a hella assumptions: "This can be shown rigorously if we first assume that $$\Delta x$$ is small, $$s^2 ≈ s$$, $$\sin (\theta) ≈ \theta ≈ 0$$, $$\alpha < \beta$$, and there is a full moon tomorrow."

Is your result shaky even after making all possible assumptions and simplifications? Simply appeal to proof by usefulness: "Though A is not proved here, assuming it allows us to solve/prove B, C, D, and E."

Proof by authority: "This result holds because [the teacher]/[the professor]/[the Mathematical Establishment]/[God]/[Richard Feynman] says so."

If proof by authority is too crass, replace it with a proof by impressive citation: "Theorem A holds (Einstein 1905b; Euclid 302 BCE; Euler 1779e; Hawking, et. al. 1997; Gödel 1931a; Turing 1953; Wiles 1993; Euler 1764g; Erdös 1964; Tao, et. al. 2005; Euler 1778$$\omega$$)".

If mathematical induction is not up to the task, why not try proof by electromagnetic induction? "This result holds. Doubters of this result will be administered electric shocks until they desist their doubting."

A more peaceful means of alternative induction is proof by cult induction: "This proof is the truth, and the whole truth. All those who want to be saved must agree, immediately transfer all their funds to [bank account details], and join us in the Kalahari Desert, where we await the alien messiah."

And, of course, the classic: proof by the proof is left as an exercise for the reader.

Inspired by conversations with friends, as well as the process of writing a longer math paper for school. More examples of this genre of humor can be found online, for example here. See also this article, specifically page 28 (page 5 of the PDF).

Review: Enlightenment Now (Steven Pinker)

Book: Enlightenment Now, by Steven Pinker (2018)
5.0k words (≈17 minutes)

 

Enlightenment Now makes a convincing case that the state of human civilization has improved massively over the course of history, particularly the last few centuries, and that this progress is continuing today.

More generally, Pinker argues for a set of principles, which he succinctly sums up as “reason, science, humanism, and progress”. These are the ideals formulated during the Enlightenment: reason over superstition, knowledge over ignorance, humanism over religion, and a proactive, rational approach to solving problems. These are also the ideals that have guided civilization towards good, and with which our greatest achievements are built.

But first: the idea that life, on average, is improving is a surprising idea (at least to people answering surveys with this question). So is it?

 

How has the world improved? Let me count the ways

The majority of the book is about statistics on long-term trends in the world. Let’s take a look.

(Much of the data in Enlightenment Now and lots more can be found on the incredible Our World in Data website.)

Material progress

• Human lifespans have increased, reaching 71.4 years in 2015.
  
  

• Child mortality has declined from 30-50% even in the richer European countries in the 1800s to a fraction of a percent in developed countries (the global average is 4%).
• The lifespan increase is not just a result of decreasing child mortality: the life expectancy for people of all ages has increased, and if anything the trend is accelerating.
• Deaths from diseases have declined.
• The percentage of undernourished people in the developing world has fallen from 35% in 1970 to less than 15% in 2015.
• World GDP, well …

In many of these cases, it is not just the total progress that it shocking, but how much of it has been recent. Until around 1970, the number of people living in extreme poverty increased as the population rose, and then started falling, and then went into a massive fall starting in 2000.

Safety

I discussed the decline of warfare in a previous post. Meanwhile Steven Pinker has written an entire book, The Better Angels of Our Nature, focused on the decline of conflict. For more on this topic, see one of those resources.

Other aspects of safety have also improved:

• Car-related deaths per kilometer in the US have fallen to about 1/20th of their 1920 value and 1/5th of their 1950 value (though total car deaths have risen, and are a significant global cause of mortality with over 1 million deaths per year).
• Pedestrian deaths have also fallen: in the US, 5 000 pedestrians died in 2014, compared to 15 500 in 1937. This is especially surprising when you consider the difference in the number of cars on the road and even total population.
• Aircraft deaths have fallen from about 5 per year per million passengers in 1970 to a value so close to the x-axis that I can’t read it off the graph (0.1 or less) in 2015.
• Deaths from falls, fire, and drowning have fallen steadily during the 1900s.
• Workplace deaths have declined by about a factor of over 10 since 1910.
• Natural disaster deaths, though, can’t be helped - except, wait, they can. Natural disaster deaths have fallen from about 8 per 100 000 people to 1 per 100 000 since the 1950s.
• Deaths from lightning strikes have declined by a factor of over 50.

Pinker reiterates the standard - though seemingly much needed - litany that terrorism deaths are insignificant compared to practically any other cause of death. Pinker lists lightning, attacks by non-dog mammals, bathtub-related drownings, and, I quote, “contact with hot tap water” as causing more deaths per year in the US.

It is true that terrorist acts are high-variance events; if 2001 had been selected as the year instead of 2015, it is likely that “contact with hot tap water” might have to settle for a lower ranking. And there is a very small chance of statistically significant terrorism - a nuclear attack on a large city could kill millions. But even taking all the existing and potential outliers into consideration, the fact that ISIS tops the list of threats on a recent opinion poll is a colossal failure of reason.

Pinker notes that terrorist groups have utterly failed to accomplish long-run strategic objectives, and their prominence in the media is mostly due to their being so few real security threats. He quotes Yuval Noah Harari (author of Sapiens): “If in 1150 a few Muslim extremists had murdered a handful of citizens in Jerusalem, demanding that the crusaders leave the Holy Land, the reaction would have been ridicule rather than terror. If you wanted to be taken seriously, you should have at least gained control of a fortified castle or two.”


Freedom & democracy

Perhaps civilization has enjoyed a brief respite from starvation and violence, while temporarily beating back illness and death. Perhaps the fragile international order has bestowed us a brief respite from conflict and genocide. But surely the world is once again sliding towards autocracy as our democratic institutions crumble, after which all the material advances will be rolled back and the tanks rolled in?

(The graph shows the average score of countries on a democracy vs autocracy index from -10 to 10. The arrow is 2008, the last year plotted in the 2011 The Better Angels of Our Nature, to show that, no, Pinker has not jinxed the world with his optimism)

It is notable that the upwards trend in this graph was reversed for the time period from 1920 to 1980; progress is far from smooth or constant. But even in the 2010 to 2015 period, the latest shown in the graph, democracy marched steadily onward.

Pinker takes care to point out that this is not because democracy works like the “civics-class ideal” of rational voters electing reasonable, uncorrupt politicians. Instead, perhaps the greatest strength of democratic systems is their ability to foster peaceful power transfers. Pre-modern monarchs would have looked with envy on, say, the United States, which has transferred executive power 44 times without it causing a single coup, revolution, or peasant uprising.

It is also true that many recent trends (including ones in the United States, Turkey, Russia, China) are not positive; it will be interesting to see what the data looks like once it’s updated to 2018. But the long-run trend offers hope that democracy is far from fragile.

Once again, the simple fact that the world has improved so much makes it harder to realize the extent of progress. Human rights watchdogs find more and more evidence of wrongdoing, not always because there is more of it, but because we are - rightly - getting better at finding and eradicating it.


Equality

Alright, maybe civilization is a lot more safe and prosperous and free and democratic, but surely this just allows humanity’s worst tribal instincts to show in the form of discrimination?

In what seems to be a recurring trend, media reports on the latest wrongdoings and failures around the world can give the impression that sexism and racism are on the rise. But surveys show that attitudes are improving, and fast.

Perhaps people are simply concealing their hidden prejudices? In a clever piece of statistics, Pinker looks at trends in Google search terms (which people don’t exactly take pains to filter, at least judging by some of the most popular ones): searches for sexist, racist, and homophobic jokes have all declined by a factor of over 3 since 2004. This is likely understating the effect, since during this time Google has gone from being the platform for young technophile liberals to being the platform for everyone and everything (except a few ardent DuckDuckGo users).

There are dozens of other statistics to back up this trend: hate crimes are decreasing; partner-inflicted violence in the US is down to one-third of the number in the 1990s; in 1950, half of the world’s countries had laws that discriminated against minorities, whereas in 2003 the share had fallen to a fifth; more countries are decriminalizing homosexuality each year; a 2008 survey found that in all countries polled, a majority of people were in favor of ethnic and religious equality.

In general, discrimination decreases as countries grow richer; rich, Western countries are the most tolerant, and developing ones the least. But everywhere, the trend is upwards. The World Values Survey’s Emancipative Value Index shows a remarkably steady upwards increase in every region from 1960 onwards. For instance, the Middle East in 2005 ranks as high as 1960s Western Europe or 1970s United States.

By far the best predictor of liberal, emancipative values, however, is knowledge. The type of knowledge in question is the World Bank Knowledge Index, which explains 73% of variation in emancipative values (much more than per capita GDP). Who would have guessed?


Knowledge

Sure, let’s grant that people are healthier and safer and freer and more equal, but is the world more knowledgeable?

Yes. For example, in 1950 35% of the world population was literate, compared to 85% in 2014. Enlightenment Now has an entire chapter on knowledge full of similar statistics.


Happiness

Fine. Civilization is a lot better, and rapidly becoming even better, at keeping people healthy and safe, at the same time as people become freer and more equal and less affected by discrimination and more knowledgeable. But surely all this material progress and increased comfort and safety and wellbeing and equality and understanding merely sets the stage for crushing existential ennui that leaves people more sad, depressed, and lonely than in the simple old times?

There’s something called the Easterlin paradox which states: within countries, richer people are happier, but richer countries are not happier than poorer ones.

This paradox is easy to resolve, because the claim is false.

(The arrows on the country dots represent the correlation between income and life satisfaction within that country)

Firstly, the trend is so clearly upwards that you can barely ask for any better when you’re comparing 131 different countries. Secondly, all the arrows are upwards; in no country is there a negative correlation between income and happiness.

Note, however, that the x-axis scale is logarithmic, so the straight line would actually look like heavily diminishing returns on a linear plot. In other words, the absolute increase in GDP per capita required to make a country x units happier increases as GDP per capita increases (the formulation of the Easterlin paradox was a result of noisy data making diminishing returns look like no returns).

Do you want more statistics? Another study found that in 8 out of the 9 European countries studied happiness trends tracked changes in GDP per capita. The World Values Survey found that 45 out of 52 countries were happier in 2007 than 1981. Outliers exist - for example, life satisfaction has remained roughly constant in the United States for the past few decades, and what’s up with Bulgaria in the graph above? - but the general trend is clear.

All this means that as developing countries develop, they will become happier. As developed countries continue to grow, they will also become happier. This is great news: there is a relatively straightforward way to make the people happier, and it is already happening and will continue to happen, absent a major disaster, for the foreseeable future.

What about loneliness? Despite the “loneliness epidemic” that sensationalist articles claim is ravaging the West, studies have found that loneliness rates among high school and college students in the United States have steadily trended downwards since the 1970s. The few studies on loneliness in the general population show no increase or a slight increase, probably due to more people being single. Despite social media’s supposed destruction of face-to-face interaction, it turns out people enjoy real social interaction and have not traded it away: on average, people report having as much quality friendships and emotional support as in previous decades. Though avid internet users have less face-to-face contact with people, they spend more time with friends and on average report that social media has a positive affect (once again, increasing averages do not imply a uniform effect on everybody and say nothing about outliers).

Of course, not all modern social trends are positive. But the message in the data is that the net effect of modernity on mental health and wellbeing is positive. Increasing quality of life is not causing epidemics of anything. People tend to be happier and more satisfied with life when they are healthier, safer, richer, freer, and live longer. This is not quite breaking news.

In defense of progress

 

The case for the case for progress

Enlightenment Now is not a dispassionate, academic overview of the effects of modernity, progress, and Enlightenment; it reads more like a lawyer defending a client.

This is not a bad thing. An outright defense of progress is much needed.

Our World in Data published an excellent article about people’s perceptions of worldwide trends. In most surveyed countries, a majority of people think global poverty is increasing. In no country does a majority think it’s decreasing; only China is close, with 49% correctly answering that it is decreasing (and fast; see the graph above). In many countries, particularly developed ones, only 10-15% answered correctly.

As the article argues, this is important because people are far too pessimistic about the future.

(From the same article)

Look at the bottom of that graph. In France and Australia, only 3% of people think the world is getting better. 3% of people is a ridiculously low fraction to get for any survey question, especially one with only three answer choices. That’s probably close to the percentage of people who clicked the wrong box by accident. I would be surprised to see such pessimism if I ran a survey with this question in an underground zombie apocalypse survival shelter that ran out of canned food last week.

So there is need for perspective recalibration, especially in the developed world. If you look at the graphs in that article, you will see that China is always near the top, along with places like Kenya, Senegal, Nigeria, and Indonesia. It is easy to forget how much modernity has done for us when you’re not experiencing it first-hand.

Perhaps modernity has run its course, and though developing countries can still benefit from it, the developed world is about as saturated with progress as it can get without ill-effects? This is the conclusion that each generation seems to come to, and so far they’ve all been proven wrong by the next.


Progress: 2/5, would not recommend

Though the fruits of progress are as popular as ever, the idea that civilization advances is remarkably unpopular. As Pinker illustrates:

An entire lexicon of abuse has grown up to express [scorn about the idea of progress]. If you think knowledge can help solve problems, then you have a “blind faith” and a “quasi-religious belief” in the “outmoded superstition” and “false promise” of the “myth” of the “onward march” of “inevitable progress.” You are a “cheerleader” for “vulgar American can-doism” with the “rah-rah” spirit of “boardroom ideology,” “Silicon Valley,” and the “Chamber of Commerce.” You are a practitioner of “Whig history,” a “naïve optimist,” a “Pollyanna,” and of course a “Pangloss,” a modern-day version of the philosopher in Voltaire’s Candide who asserts that “all is for the best in the best of all possible worlds.”

Several biases contribute to a distorted view of progress. Intuitive ideas of frequency often rely on the availability heuristic (judging frequency based on how readily examples come to mind). Negative examples feature more often in the media and remain in memory for longer, tilting people’s perceptions.

Meanwhile, the “Optimism Gap” is a phenomenon where people tend to assume that their own circumstances and future outlook are good, while those of society as a whole are bad. The most flagrant example of this is a survey which found that across the world, people underestimate the percentage of other people who say they’re happy by 42 percentage points.

But Pinker argues that resistance to the idea of progress runs deeper.

First, there’s an asymmetry between bad outcomes and good outcomes. A random change to something that works is much more likely to do harm than good, simply because of how many things need to go right for something to work. Therefore it is rational to assume that, given random change in a complex system, the trend would be down.

Second, the world is big, so questions of whether there has been progress are almost bound to be quantitative ones. The way to counteract availability bias is to look at numbers, and assigning enough weigh to mere numbers to cancel out vivid examples is something humans are notoriously bad at.

Third, cynicism looks intellectual. Pinker cites a study showing that critics who praised a work were rated as less competent than those who criticized it. If one journalist writes “The Crisis in Crime: How the latest crime epidemic reveals the sickness of modern society”, and another writes “Crime Trends Pretty Okay, Actually: Some further incremental improvement still needed”, which seems like serious, quality journalism?

How might this perception come about? Skepticism is a key attitude when discussing claims and arguments. Most sophisticated, fancy claims rely on a lot of skepticism; this might lead to the impression that cynicism and skepticism are required properties of a sophisticated idea. But these are just surface-level appearances; it does not follow that all skeptical ideas are correct. It’s easy to hear “critical thinking” and think it means “critical thinking”.

Fourth, Pinker argues that during the 20th century, pessimism and skepticism towards Western progress were the Mongol hordes of the intellectual landscape, sweeping out of their far-off lands to conquer the entire continent (he cites Arthur Herman’s The Idea of Decline in Western History and Robert Nisbet’s History of the Idea of Progress). In particular Pinker, bemoans Nietzsche’s resurgent popularity and argues against his misanthropic views. It’s hard not to agree with him. A philosopher who calls for the need of a “declaration of war on the masses by higher men” to annihilate the “decaying races” and the “humbug called ‘morality’”, or says “Man shall be trained for war and woman for the recreation of the warrior. All else is folly” is not a philosopher whose views should be admired by people who value human life or wellbeing.

In addition to philosophers who are diametrically opposed to any reasonable humanist morality, philosophies that seek to undermine knowledge and objectivity are surprisingly popular; Pinker lists critical theory, existentialism, postmodernism, and poststructuralism as examples. His tour of anti-objectivity philosophies also outlines the influence of Nietzschean romantic heroism and the denial of objectivity on the ideologies of such historical heroes as Benito Mussolini, the Nazis, and the “theoconservatism” of Steve Bannon and Donald Trump. Pinker perhaps paints a bit too broadly here; the type of anti-objectivity and romanticism that gives you millions of people sacrificing themselves for the glory of the fatherland is different from the type that philosophy majors debate in coffee shops or Trump voters coddle themselves with. But he does have a point. No society has ever failed due to being too humanist or too rational, while example after example tells us that in the other way lies danger.

Houston, we … don’t have a problem?

So, is everything in the world improving? Is everyone who talks about negative trends wrong? Should we just sit back and watch the inevitable march of progress?

This is not the point. Civilization advances when people work to advance it. Emphasizing the progress we have made is not an argument for laziness and blind faith; it is an argument for doing more of whatever it is we did right to achieve that progress.

And, of course, there are many negative trends in the world, perhaps most notably inequality, climate change, and existential risks.


Inequality

First, inequality. GINI coefficients have been rising in the United States and some other developed countries. While this is a problem, Pinker points out several facts that should be taken into account. The world as a whole is becoming more equal, and fast, as developing countries catch up. In large part due to the rise of China and India, the world income distribution is less like a two-humped camel and more like a single distribution, as vividly illustrated by the graphs on this page.

Even within developed countries like the US and UK, the GINI coefficient is still below the level it was in the 1700s and 1800s. At the same time, social spending as a percentage of GDP has been on a near-continuous rise in developed countries:

While Thomas Piketty writes in Capital in the Twenty-First Century that “The poorer half of the population are as poor today as they were in the past, with barely 5 percent of total wealth in 2010, just like in 1910”, Pinker notes that this is confusing inequality with poverty. Though incomes have, in recent decades, risen slower for the poor, they have still risen. Government statistics on poverty often understate improvements because the definition of poverty is readjusted; using the 1980 definition, consumption poverty in the United States fell from 30% in 1960 to below 15% in 1980, and has since then declined at a slower rate to about 4% in 2015 (disposable income trends are mostly similar, though have stagnated instead of falling slowly since 2000).

There is a lot of room for improvement, but the trend is far from apocalyptic.


Sustainability

One of the greatest challenges facing civilization, is, of course, sustainability. There are a few good trends here: CO2 emissions per dollar of GDP have been declining in developed countries like the US and UK since 1950, and in the world as a whole as 1980, due to the economy’s carbon intensiveness falling and environmental concerns being accorded more weight as countries develop. Predicted resource shortages have failed to occur, international cooperation has worked in slowing ozone depletion and protecting ever greater areas on both land and sea, and the population explosion scenario has largely been averted.

However, at the same time global CO2 emissions have quadrupled since 1960 and, despite staying roughly steady in recent years, there is no sign of the required rapid decrease.

What should we do? Actually, let’s take an easier question: what should we not do? Pinker correctly identifies some things that we definitely should not do, yet which have somehow become associated with the environmental movement.

Part of the mainstream environmental movement has developed an ideology Pinker calls “greenism”, which holds as its basic principle some variation of sacred nature being defiled by human greed. The only solution is to protect nature at all costs, switch to “natural” ways of life and production, scale back industrialization, and, according to more extreme ideologues, reduce human population in rapid, unspecified ways.

I think the central mistake in greenist ideology is selecting shallow terminal values. We care about the environment because of its necessity for human and animal welfare. We care about biodiversity because it is our greatest heritage and, as far as we know, unique in the universe. Greenist ideology forgets the reasons we need to fight for the environment, replacing them with worship of a semi-coherent ideal of nature as an end in itself.

Needless to say, this sometimes goes wrong. Take nuclear power. Pinker writes: “It’s often said that with climate change, those who know the most are the most frightened, but with nuclear power, those who know the most are the least frightened.” The consensus on nuclear power is that it is the most scalable zero-carbon energy production method we have, as well as the safest power source (even solar and wind cause more deaths per watt-hour, though admittedly this is because the installation of rooftop solar and wind turbines dangerous (see here and here). Another study estimates that the use of nuclear power has saved 1.8 million lives, and replacing it worldwide with fossil fuel power could cost from 420 000 (if it’s replaced with only natural gas) to 7 million (if it’s replaced with coal) lives by 2050. The rational, science-based climate policy is to massively ramp up nuclear power production alongside renewables to maximize carbon-free energy generation. (For more depth on nuclear power, see my post here.)

But the end-goal of greenists isn’t solving climate change or preventing deaths from pollution. It’s about “nature”, whatever that means, and everyone agrees that “big complex machine that uses dangerous artificial materials” is not really natural.

And so we get absurd situations, such as Germany, a country more likely to be hit by an asteroid than a tsunami, deciding to phase out all nuclear power plants after the Fukushima disaster (which caused exactly 0 deaths (EDIT: a week after this post was published, the total was updated to 1 death), compared to 11 000 air pollution -related deaths per day), cancelling out a lot of the country’s considerable progress in carbon-free energy generation.

A similar situation, where complaints of something being unnatural hinders real progress on environmental issues, exists for GMOs.

(This is not to say that there are no valid arguments against GMOs or nuclear power; just that in both cases the scientific consensus is that, on balance, they are safe, practical, and their correct use could be an essential part of solving environmental problems.)

Pinker’s alternative to “greenism” is a movement called ecopragmatism, ecomodernism, or bright green environmentalism. Instead of railing against any human effect on nature, ecomodernism acknowledges that some amount of environmental impact is both unavoidable and desirable. Instead of calling for degrowth and the dismantling of modern civilization or, alternatively, bemoaning the inevitable eco-Armageddon, ecomodernism advocates for rapid and constructive technological and societal innovation to solve sustainability problems. And finally, like (one of) its name suggests, it is pragmatic: let’s not waste potentially climate-saving innovations just because they look unnatural.

To sum up: various sustainability problems are the greatest issues facing 21st century civilization. The way forwards, like with any other problem, is not preaching doom and gloom, or pursuing the wrong goal, but working to solve it using knowledge and reason.


Existential threats

Pinker makes good points about world-ending disasters: predictions of apocalypse have a long and unsuccessful history, society is surprisingly resilient to disasters, many existential threats are immensely low-probability events, well-known ones often doubly so, and alarmism is counterproductive (variants of “we are all going to die and there’s nothing you can do about it” are strangely ineffective at inspiring action).

Other points are less convincing. Pinker raises questionable objections to AI threats; he critiques the paperclip maximizer thought experiment by saying that a superhuman AI would not make such “elementary blunders of misunderstanding”. But the point of the though experiment is that very few value systems, executed executed to their fullest by a powerful agent, will align well enough with human values for the results to be good. Pinker seems to implicitly assume that any intelligent system will have goals that are reasonable by human standards.

More generally there is a danger in assuming that the future is assured. This is not to say that Pinker directly endorses such a view, but it is an easy misinterpretation to make. There is no law of nature that forbids civilization-ending disasters. There is no benevolent force making sure that nothing too bad happens, or if it does, it will at least not be for a stupid reason. Maybe Trump wakes up in a bad mood tomorrow morning, starts a nuclear war, and that’s it.
 

Enlightenment when?

To modify the opening of book 2 of The Hitchhiker’s Guide to the Galaxy series:

The story so far:
In the beginning [modernization was started].
This has made a lot of people very angry and been widely regarded as a bad move.

Enlightenment Now argues that it has, in fact, turned out to be a very good move. On practically every metric of human wellbeing and satisfaction, the world is doing better than it was two centuries ago, let alone two millennia ago.

This might be hard to believe, at least until you look at the data. But is progress really so improbable? There are 7.6 billion people in the world, the vast majority of whom want things like life, safety, happiness, comfort, and freedom for themselves and others. Even factoring in all of humanity’s worst tendencies and institutional failures and whatnot, there is still an enormous amount of goodwill being directed towards solving these problems.

A key theme in Enlightenment Now is that, contrary to the opinions of edgy philosophers, civilizational progress is neither an oxymoron nor unattainable nor undesirable nor ill-defined. Good things, it turns out, are good. Futhermore, they are interrelated: education, GDP per capita, life satisfaction, freedom, health, and safety correlate with and bolster each other.

This is great news. It means that we are not locked in a trap of sacrifices: in general, increasing wealth does not leave us listless and decadent; longer life does not make us bored; being educated rather than superstitious and ignorant does not make us sad.

Finally, Enlightenment Now is a reminder of how humanity has wrought the progress it has achieved: rational problem-solving, guided by humanist morality. To continue the upwards rise of civilization and to fix its problems, these Enlightenment ideals are what we need.

RELATED:

• Technological progress
• Review: Foragers, Farmers, and Fossil Fuels – how human values evolved over time

Review: The Baroque Cycle (Neal Stephenson)

Books:
Quicksilver, by Neal Stephenson (2003).
The Confusion, by Neal Stephenson (2004).
The System of the World, by Neal Stephenson (2004).

2.8k words (≈ 10 minutes)

The Baroque Cycle is nothing if not ambitious: three volumes, subdivided into eight books, with chapters covering the 60 years from 1655 to 1715.

And what riveting subject has Neal Stephenson chosen for this grand work? Nothing less than the rise of science and finance, and with them, the modern world. If the proceedings of the Royal Society and machinations of VOC stockbrokers do not sound like the most gripping subjects for a novel:

1. Stephenson succeeds in building an engaging plot out of them (though it takes nearly three thousand pages).
2. There are also pirates, sword fights, and even a duel fought with cannons.

Plot

Quicksilver starts in 1713 with the ever-mysterious Enoch Root arriving to summon Daniel Waterhouse back to Europe to try to settle the dispute between Isaac Newton and Gottfried Wilhelm Leibniz (or, perhaps more accurately, the dispute inflicted on Leibniz by Newton). Daniel leaves his home in Massachusetts, where he had founded a small, quaint institute called the Massachusetts Bay Institute of the Technologickal Arts to continue Leibniz’s research on mechanized computation and information storage.

The first part of Quicksilver jumps between the story of Daniel getting to know fellow student Isaac Newton and becoming involved with the fledgling Royal Society in 1660s to 1670s England, and the story of Daniel’s sea voyage in 1713, which, of course, proceeds without any complications, meandering, or pirates.

In the next part we are introduced to the two other main protagonists: Jack Shaftoe, a London-born urchin who comes to acquire a truckload of sobriquets on his adventures (including King of the Vagabonds, L’Emmerdeur, Quicksilver, Ali Zaybak, and Sword of Divine Fire), and Eliza, a former slave who uses her financial acumen to build a fortune and gain a slew of titles herself.

In The Confusion, the second volume, Daniel is mostly relegated to the sidelines, and the story focuses on Jack’s globe-spanning adventures and Eliza’s political and financial machinations in Europe. Whereas the plot of Quicksilver often trudges along slowly, The Confusion is structured better and moves a lot faster. And where Quicksilver focused on science, The Confusion deals heavily with money.

The System of the World returns the focus to Daniel, beginning with Daniel’s arrival in England in the January of 1714. If The Confusion is where Stephenson masters the art of plot, The System of the World is where he masters the art of character. While previous books had touching moments of character development (particularly the scene in The Confusion where Daniel confronts a demented Newton, and the climax of Eliza’s storyline in the same book), The System of the World has all three main characters undergo change, confront their inner demons, and reach the conclusion of their story arcs.


The System of the Cycle

Overall, The Baroque Cycle does not have the structure of a trilogy, but of a single, monstrously intricate novel. Quicksilver acts as an introduction to the characters that sets up the setting and some of the later conflicts, The Confusion is the adventurous mid-section of the novel where side plots are explored and conflicts develop, and The System of the World is a 900-page climax, complete with no less than five different epilogues.

Therefore I recommend that if you choose to read Quicksilver, you should aim to read all three volumes, since the first or even the first two volumes don’t form a satisfying whole on their own. After Quicksilver, I was ambivalent about reading the next volume because of the way that the book felt like a long prologue. However, the ideas in the book stuck with me, and after I had read the remaining parts, I realized that that was exactly what the book was.

(A word of warning: Quicksilver might be an introduction, but it is an essential one; given the information density of the books, skipping straight to The Confusion would be a bit like starting your pilot training by landing a 747 in heavy crosswinds. Also, Quicksilver focuses on the science more than the other books, and its descriptions of Newton’s and Hooke’s work and the spirit of the Royal Society are themselves worth reading the book for.)


Grand themes

The Baroque Cycle is a loose prequel to Cryptonomicon, Stephenson’s earlier novel about World War II codebreakers and 1990s internet entrepreneurs trying to start a data haven and an internet bank. Reading Cryptonomicon is definitely not required, though there are several minor connections.

What Cryptonomicon and The Baroque Cycle share are themes. Stephenson says that the idea for The Baroque Cycle came when he read about Leibniz’s early work on computation and realized that the themes of science, computation and money in Cryptonomicon were also present in late-1600s and early-1700s Europe.

These themes are in turn related to the overarching theme of that time period: the Enlightenment, and the gradual rise of modern scientific, financial, and political institutions amidst the lingering medieval background that came with it.


Science & alchemy

Stephenson has clearly done his research on the period. Almost every natural philosopher in Europe at the time makes an appearance, including Isaac Newton, Gottfried Wilhelm Leibniz, Christiaan Huygens, Edmond Halley, Robert Hooke, Robert Boyle, John Locke, John Wilkins, John Flamsteed, Nicolas Fatio de Duillier, and Thomas Newcomen. This group of people, all of whom were contemporaries, are responsible for inventing and/or discovering calculus, the inverse square law, the law of elasticity, Halley’s comet, the rings of Saturn, pendulum clocks, the mathematics of probability, the wave theory of light, the particle theory of light, the steam engine, the binomial series, the theory of extinction, the Newtonian telescope, cells, matrices and the method for solving them, dynamics, Boolean algebra, mechanical calculators, and, last but definitely not least, physics as a science.

There are parallels, no doubt intentionally enforced by Stephenson, between the science of Newton’s time and the digital revolution of the late 1900s. In both cases, a new tool (the scientific method / the computer) opened up a new world, which a horde of bright but often eccentric people promptly began exploring, leaving a blazing trail of results and discoveries for later generations to build on. The spirit of discovery and experimentation of this early group of explorers is captured well, particularly in Quicksilver, where we see, for instance, Hooke trying to figure out the law by which gravity diminishes with altitude by lowering a weight into a well.

This period is doubly interesting because even as the vanguard of science made progress by leaps and bounds, earlier superstitions and beliefs remained in favor. At one point, Daniel and Newton discuss how gravity works, with Daniel saying: “The inner workings of gravity, you seem to be saying, are beyond the grasp, or even the reach, of Natural Philosophy. To whom should we appeal, then? Metaphysicians? Theologians? Sorcerers?” Newton replies: “They are all the same to me, and I am one.”

Alchemy, one of Newton’s main interests in addition to physics, theology, and saving England’s economy from ruin, features heavily in the book. In the universe of The Baroque Cycle, alchemy partially works, though it requires knowledge and ingredients that are vanishingly rare. The character of Enoch Root, who also appears 400 years later in Cryptonomicon, is strongly hinted to be immortal (this is not much of a spoiler; the first two chapters of Quicksilver should already make this clear).

The slight speculative touch might seem to be a contradictory addition to a work concerned with the rise of scientific thinking. However, it ingeniously captures something about the spirit of the time. Perhaps the golden age of the supernatural was the dawn of science, the time between the rise of rational inquiry and the discovery of nature’s laws through this inquiry, when theology and alchemy seemed only one experiment away from confirmation. Today we know that we live in the universe of physics, but the people who first charted the border between the natural and supernatural did not have the benefit of this certainty, and often they could not have known which side of the border they were exploring.


Leibniz & Newton

The central scientific figures in the book are perhaps the most important: Hooke, Leibniz, and Newton. All three were polymaths who probed entirely new realms, yet one of those names stands out more than the others.

“If I have seen further than others, it is by standing on the shoulders of giants,” Newton famously wrote in a letter to Hooke, showing the depth of his humility. Or was he discretely insulting Hooke, who is said to have been hunchbacked and short, and thus a poor giant to stand upon? No one knows, because during Newton’s presidency of the Royal Society, all portraits of Hooke, as well as much of his equipment, were removed.

During his long career, Newton waged bitter disputes over priority against Hooke and Leibniz. In particular he used his role as the president of the Royal Society to get the entire English scientific establishment on his side in the war against Leibniz.

Stephenson incorporates this evidence into his portrayal of Newton as capricious, harsh, unforgiving, and unstable. At the same time, he is ruthlessly obsessive in pursuit of his studies, thinking nothing of personal sacrifices like half-blinding himself to study sunspots, sticking things into his eye to probe the nature of optics, tasting chemicals in his quest for alchemy, and starving himself and foregoing sleep to tend to an experiment.

Newton also has a reverence for the old and the sacred, which drives his obsession with theology (particularly the Old Testament), his pursuit of alchemy, and his disdain of even his own calculus in cases where Euclidean geometry is sufficient.

Leibniz, on the other hand, is portrayed as a far gentler, more genial figure. He is also markedly more forward-looking, particularly in his vision that computation and even thought can, in principle, be mechanized.

One of the main overarching plots in The Baroque Cycle is the dispute between Newton and Leibniz over not just the invention of calculus, but also over the broader differences in their philosophical views. Ultimately, this plot line is somewhat overshadowed by other conflicts, which is somewhat disappointing given the buildup. The System of the World does have one chapter where Newton and Leibniz fight it out over their philosophical and theological views, but it gets cut short by an interruption.


Paying for it all

In addition to his scientific, alchemical, and theological work, Newton was also the Master of the Mint for the latter half of his life. He presided over a recoinage of the English currency that helped stave off disaster, and persecuted counterfeiters with his typical ruthless zeal.

That The Baroque Cycle’s themes happen to coincide so well in this time period and its key figures is no accident. The rise of science happened at the same time, and probably for at least some of the same reasons, as the rise of modern financial institutions.

Stephenson gives plenty of examples of the backwardness of doing business in the 1600s. People were only beginning to trust banks, cheques, and other indirect means of payment. One chapter goes to some length in describing the difficulties of shipping timber across the borders of a few dozen fragmented states and provinces, each with its own complex and expensive system of tariffs. Even in the relatively modern England, the inconsistency of the coins means that paying with them is more like bartering than using money; this, and counterfeiters, are the main problems that face Newton in his role as Master of the Mint.

Just like with science, The Baroque Cycle portrays a world in transition from one system another. And also just like with its depiction science, parts of the older system refuse to budge: Jack’s dealings in The Confusion center heavily on gold, the eternal store of value.


Politics

The Baroque Cycle also deals heavily with the politics of the time. This is probably partly due to plot reasons: historical events are the core of the plot of many a historical novel, and historical events tend to be driven by politics. But another factor is that many of the political changes of the period are part of the grander narrative of the Enlightened world order that The Baroque Cycle charts. The dispute between Whigs and Tories, or between the Roundheads and Cavaliers before them, features heavily, as do the schemes of Louis XIV.

As with natural philosophers, countless political figures of the time make an appearance, including William III of England, John Churchill, Duke of Marlborough, James Stuart, Sophia Charlotte of Hanover, Caroline of Ansbach, James Scott, Duke of Monmouth, and Peter the Great, who makes an “incognito” visit to London during which he tears apart half the city.

Even the Spanish Inquisition makes an unexpected appearance. But though Jack bemoans that “This Inquisition is as patient as Death. Nothing can stop it”, another character replies: “Nothing […]except for the Enlightenment”.


Baroque is the soul of wit

Upon being handed a draft of Isaac Newton’s Principia, a character in Quicksilver complains about its length and exclaims: “Some sharp editor needs to step in and take that wretch in hand!” I can’t help but think that Stephenson was reflecting on his own experiences when he wrote that line.

Neal Stephenson is known for complex, lengthy works, and The Baroque Cycle is the lengthiest and most complex. While each page may not be quite as significant as a page of Newton’s Philosophiœ Naturalis Principia Mathematica, it is not nearly as bloated as it may seem.
 
In any fictional work of this scale, there will be slow parts. Especially in Quicksilver, the plot occasionally meanders a bit too much, though The Confusion and The System of the World are a lot better in this regard.

On the smaller scale, Stephenson explains historical background in great detail, often hijacking two characters into having a somewhat stilted conversation that conveniently reveals relevant historical details. But this also means that readers do not have to do their history homework before reading to follow the plot. In addition to historical details, Stephenson delights in relating the particulars of different mechanisms, whether mines or phosphorus purification plants. How interesting these are depends heavily on the reader.

Like Stephenson’s other novels, The Baroque Cycle features plenty of his enviable dry wit. Stephenson is also not afraid to freeze the narration for a paragraph, or even half a page, to go off on some tangent that eventually circles back like Halley’s Comet after setting up the perfect metaphor or humorous remark. The Baroque Cycle does not have any of the hilarious multipage tangents found in Cryptonomicon, and thus has less laugh-out-loud moments, but on the other hand the tangents and humor in The Baroque Cycle tend to be more topical and better integrated with the atmosphere and setting. Some may find the idiosyncrasies of his style distracting, but I found Stephenson’s prose to be very enjoyable.


Winds of change and engines of change

The Baroque Cycle is certainly baroque. The number of characters, subplots, and events - let alone pages - nears the astronomical. But from this mess Stephenson manages to erect a surprisingly compelling image of a world in transition.

Consider medieval society. The standards of human welfare are atrocious and stagnant. The world remains at least as great a mystery as it was a thousand years prior. The Spanish Inquisition is knocking on doors. And then consider the modern world, with its unprecedented and ever-expanding level of well-being, depth of knowledge, and individual freedoms.

What’s more important - at least for a novelist - is that somewhere in between things changed, and the world was caught in the turmoil of that change. Somewhere in between were the people who first thought of science and liberalism, and the forces that supported them, and the people and forces that were against them.

The Baroque Cycle is a story about this change - specifically, about the moment when the roots of the modern world began to sprout. For a while, alchemy coexisted with science, swords with guns, pirates with stock-markets, and slaves with steam engines - and in each case it was still unknown which way the scales would tip. The new mixed with the old, and the old with the new. It is this moment that The Baroque Cycle captures so well, and from which it forms its soul.

Review: The Internationalists (Oona A. Hathaway, Scott J. Shapiro)

Book: The Internationalists, by Oona A. Hathaway and Scott J. Shapiro (2017).

3.2k words (≈ 11 minutes)

 
Today war is not only rare, but also illegal and condemned.

Things have not always been that way. The Bible gives an idea about what was seen as just in ancient times. When Moses heard about the campaign of vengeance on the Midianites, he berated his officers for allowing women and children to live (all men had already been massacred). When King Saul merely imprisoned the last of the Amalekites instead of killing them, Prophet Samuel had to finish the job by hacking the chained prisoners to pieces. Saul’s heinous crime even caused God to announce that He had made a mistake in appointing Saul as king. As Hathaway and Shapiro put it: “In the Hebrew Bible, it was often a war crime not to kill civilians.”

Accepting the mass murder of civilians is not just a feature of biblical times, but a feature of international law until the mid-1700s. For instance, Couhnt Pappenheim, after sacking the city of Magdeburg in 1631, let his soldiers run loose through the city, killing, pillaging, and stealing as they saw fit. He described the sacking in a message to the Holy Roman Emperor: “I believe that over twenty thousand souls were lost … All our soldiers became rich. God with us.”

Something has definitely happened in the past few centuries. Many explanations have been offered. War has become more expensive, trade has become cheaper, and economic growth means that prosperity is not a zero-sum game. Nuclear weapons have prevented war. Countries have become more democratic over time. Perhaps the cause is the gradual advancement of morality.

The Internationalists does not dismiss the importance of these factors, but makes a case for the importance of another: the legal status of war has changed, and that these changes have marked shifts in the international order. Hathaway and Shapiro identify two key events: Hugo Grotius’s construction of a legal basis for war in 1625, and the Peace Pact’s (AKA the Kellogg-Briand pact) outlawing of war in 1928.



The Old World Order

Hathaway and Shapiro refer to the legal framework of the international order from 1625 to 1928 as the Old World Order. Its rules were laid out by Grotius in De Jure Belli ac Pacis (On the Law of War and Peace), who began writing it as an effort to justify the capture of a Portuguese frigate by Dutch ships near Singapore in order to avoid legal troubles for the VOC (Dutch East India Company). This is not quite the most virtuous of motives, and it shows.

The essence of Grotius’s argument is the principle that might makes right. Since there is no arbiter for disputes between nations, nations have the right to defend their interests with war, and neutral nations, who do not have the authority to judge which nation’s claims are right, should simply accept the outcome of the war as right. Even if a nation had an unjust cause to fight a war, the nation, its leadership, and military must not be prosecuted or in any way punished for it; they have the right to take their claims to the ultimate court (war).

In some ways, Grotius was ahead of his time. He recognized poison, rape, and “treacherous” assassination as war crimes, and argued that soldiers who committed these crimes could be tried and punished for them. However, Grotius explicitly stated that torture, enslavement, pillaging, execution of prisoners of war, and the killing of unarmed civilians could not be prosecuted. Grotius did consider the needless slaughter of civilians to be unjust, but in his view it was not a criminal offence.

Grotius was also against “hygienic” wars, in which the principal aim is to cleanse a land of a particular group, often for religious reasons. In this way, even the Old World Order was a step above the biblical extermination campaigns and holy crusades of previous centuries.


Follow the rules

One of the most striking aspects of the Old World Order is how deeply and strictly nations adhered to it.

Under the Old World Order, war truly was, as Clausewitz said, “the continuation of politics by other means.” Slaughter and killing were just another part of a complex, formalized system of rules and procedures for settling differences between nations.

To justify wars, war manifestos were written, often commissioned by the state from esteemed intellectuals (for example, Gottfried Wilhelm Leibniz was recruited to write a defense of the Holy Roman Empire’s participation in the War of the Spanish Succession).

War manifestos and complex traditions for declaring war go back far before Grotius. Consider the (frankly hilarious) Roman practice:

During the Roman Republic, a priest would approach the border of any territory Rome intended to attack wearing a wool veil. Once there, he would announce a legal grievance, called the clarigatio. If the other side ignored the clarigatio for thirty-three days, the Senate could authorize war. War would officially commence when the priest returned to the border and, in the presence of three military-age men, threw an iron-tipped spear over the line. As their empire expanded overseas, Rome had to alter the process, for no priest could fling his spear over the ocean. In the revamped system, a priest would take a prisoner from the soon-to-be enemy’s territory and force him to buy a small plot of land on the Campus Martius, the field of Mars, just outside the ancient city walls of Rome. When the prisoner proved unable to meet the priest’s demands, the priest could then throw the spear into “foreign soil,” thereby starting the war, and still be home for supper.

However, the Old World Order differed from past times in standardizing the rules across all “civilized” (i.e. European) countries, and making sure that these rules were followed with a rigor and consistency that would make many modern international institutions envious. The Internationalists offers countless vivid examples, of which a few are summarized below.

Napoleon’s wars killed 7 million people over 11 years across Europe. He was vilified by the press of Britain and many other countries, and thought of by much of Europe as the physical manifestation of evil.

Yet after his defeat in 1813, the Allies made no attempt to execute or imprison him. By the rules of the Old World Order, as the leader of a country he had the right to wage war, and therefore could not be put on trial since he had done nothing wrong. He retained the title and role of Emperor (albeit of Elba, not France), where he lived in comfort until escaping to France and starting another war.

When he lost yet again in 1815, he surrendered to the English, who once again were unable to prosecute and punish him because of legal difficulties. This was not due to lack of trying. One solution was to treat him as a pirate (a criminal offense) because he had escaped Elba on a ship. However, the British Parliament had recognized the conflict as a war, which foreign nations had the right to wage, and so this idea had to be dropped. Ultimately the British Parliament had to pass a special law to justify detaining Napoleon on St. Helena.

In 1891, a Sioux Native American called Plenty Horses shot, without provocation, a US lieutenant delivering a message. He was tried in court, and acquitted after it was established that there was a state of war between the Sioux and the US. War was not just a justification for murder, it was a strong enough justification that the courts of a country acquitted murderers of its own citizens on that basis (though conveniently enough, the ruling that there was a state of war between the Sioux and US also justified a mass murder committed earlier by US forces).

The Old World Order also stated that neutral countries have to treat belligerents in a war equally, a stark change from classical Greece and Rome (ancient Greek and Latin do not even have a word for neutrality). This, too, tended to be followed to the letter.


Old World Disorder

Significant changes to Grotius’s model began to appear only a century after his seminal work.

By the mid-1700s, a distinction started being made between civilians and soldiers, and the slaughter of the former was increasingly frowned upon. The international lawyer Emer de Vattel even declared: “[civilians have] nothing to fear from the sword of the enemy” (I assume the civilians themselves were not polled).

In the 1800s and early 1900s, war was further regulated by international treaties against killing the wounded, fragmenting small arms ammunition, and the execution of surrendered prisoners of war.

Why? Enlightenment ideals and a growing appreciation of the value of life were certainly a part of it. Other possible reasons that Hathaway and Shapiro cite are the influence of chivalrous aristocrats who considered wars to be formal duels between nations, as well as the horror of long wars in the 1600s and 1700s.

Whatever the reason, by the late 1800s the trend was towards war being made ever more humane by regulations. There is a limit to how humane war can be made, however, and so the natural next step was to ask whether war itself should be outlawed.


The New World Order

World War I was the culmination of the Old World Order. It was a colossal waste of human life and potential, triggered by nations sleepwalking into war to defend against perceived wrongs. It was fought in precise accordance with Grotian rules: nations had issues with each other, and took them up “by other means” once normal politics had failed.

World War I set in motion various plans and movements to prevent another world war. The solution of the League of Nations was to prevent war by threatening war on those who waged war (though it still allowed nations to go to war after three months if they were unsatisfied with the verdict of the League’s tribunal). This was, to put it mildly, somewhat self-defeating. Hathaway and Shapiro do not mince words: “The League of Nations did not herald the end of the Old World Order. The League was its reprieve.”

The book details the story of how several people, most importantly James T. Shotwell and Salmon O. Levinson, gradually built momentum for the outlawry of war movement in the 1920s, leading to the signing of the Peace Pact in 1928 by fifteen nations. Kellogg, the US secretary of state, took much of the credit, and an undivided Nobel prize, for the achievement. This was mostly thanks to his attempts to discredit Levinson: to the Norwegian ambassador who had much influence in the Nobel Committee, he wrote, among other more direct attacks: “… a man by the name of Levinson in Chicago - I have forgotten his first name - … claims to be the originator of the idea”.

(This is not the only instance of dubious credit being given in the field international law: Grotius has often been honored for his contributions to peace, and, is, ironically, the patron saint of the building that houses the International Court of Justice.)

The Peace Pact, as we all know, succeeded perfectly, and never again was a war fought on Earth.


Happily ever after … ?

Not three years after the signing of the Peace Pact, Japan invaded Manchuria. Japan, a quickly modernizing power, had had a rather sudden introduction to the Old World Order in 1852 when four US ships arrived on its coastline and engaged in classic gunboat diplomacy to get the country to open its trade. Just as the country had begun exploiting the Old World Order to its own benefit, the world order had changed.

The League succeeded in not recognizing Manchukuo (Japan’s puppet state in Manchuria) as a state. However, it is telling that the League’s greatest success in the situation was to not do something. Many countries, foremost the US, resisted imposing sanctions because of concerns about profitability and lingering Old World Order beliefs about mandated neutrality.

Some sanctions were, however, imposed on Italy following its annexation of Ethiopia, but nations were not willing to set up sanctions strong enough to be effective. The use of sanctions against a warring nation was a revolution; under the Old World Order, it would have violated the duty of neutral nations to treat the aggressors equally.

Thus the 1930s were marked by nations being caught halfway between the Old and New World Order, unable to effectively use enforcement methods from either. At the same time, other nations, most prominently ascendant industrial powers like Germany and Japan, began to increasingly shun the New World Order of non-violent conflict resolution, partly out of a (not entirely unjustified) belief that it was an attempt to lock the world’s borders in place to the benefit the ruling colonial powers.

If World War I was the culmination of the Old World Order, World War II was the battle of the Old against the New. The Axis powers acted according to the older model - the principle that might makes right - and thus sought to increase their power through conquest. The genocide perpetrated by the Nazis drew upon an even older basis for war: the concept of hygienic war, waged for the purpose of ethnic cleansing.

On the other hand, the Allies belived, to varying extent, that this is not a valid basis for international law. Of course, any side in a war, particularly the ones who write history afterwards, tends to believe they were motivated by some higher purpose. But the end of the war shows that the idealism of the Allies was genuine. Surviving Nazi leaders were given a trial at Nuremberg, in accordance with the principal that they had committed a crime. The book gives an excellent overview of the trials and the key legal figures on both sides: Carl Schmitt (a Nazi), and Hersch Lauterpacht (a Jew).

The end of World War II also saw the formation of the United Nations and other international organizations. The rise of supranational organizations ameliorated the lack of arbiters for international disputes that had formed the bedrock of the Grotian model. (It was not without hitches, though: negotiations for the formation of the UN almost stalled when Stalin demanded separate votes for all 16 Soviet republics. In the end, he settled for instituting the veto system.)

Finally, in an act that would have made most of their predecessors for the past four hundred years turn in their graves like propellers, Allied leaders largely refrained from dividing the spoils and claiming territory for themselves. Germany was split into East and West, but otherwise borders largely reverted back to an earlier date.


Law & order & changing values
Which earlier date? 1928. This is the basis for the authors’ claim that the Peace Pact was a seminal event; though it did not quite end war, it marks the date before which most conquests were recognized, and after which most conquests were not recognized and later reversed.

This is illustrated in the following graph of conquered territory (the units on the y-axis are millions of square kilometers):




(The spike in the late 1800s is partly due to colonial powers claiming big chunks of Africa.)

In charting the story of the outlawry of war, The Internationalists argues for the importance of the 1928 Peace Pact. While it provided a legal basis for post-war reversions of territorial changes and for the Nuremberg trials, it is still a legal document. Thus, half the question of why war has declined remains unanswered: yes, it was made illegal, but laws, especially toothless international treaties, tend to be reactions to changes in values rather than causes of it.

Somehow, Levinson and Shotwell managed to create enough momentum to outlaw the most fundamental part of international law for the past three centuries. They did not achieve this in a vacuum. World War I certainly played a major role in forcing a reconsideration of war. The transition from brutal hygienic conflicts to a more codified form of mass slaughter to an increasingly regulated form of massacre, and finally to the complete outlawing of armed conflict, indicates that some sort of background pattern of accumulating humanism must also be at play. In the post- World War II era, the economic benefits of trade are another factor. But what are the relative importances of these factors? Are the factors keeping the world at peace different from the ones that inspired the Peace Pact in the aftermath of World War I? The Internationalists is ultimately a legal history of the Peace Pact - though a surprisingly fascinating legal history - and so these questions remain outside its scope.


Peace, what is it good for?

Whatever the reasons for the decline of war, the decline has been massive. As Shapiro’s and Hathaway’s statistical analysis points out, before 1928, an area of territory equal to eleven Crimeas was seized by conquest every year (almost one per month), though admittedly the seizing of massive parts of Africa in the late 1800s pulls up the average. The average state before 1928 was invaded once in a lifetime; today, the average is once in a millennium. And, of course, conflict deaths have also fallen massively.

The outlawry of war has, however, had one major side-effect. Whereas frequent wars previously pruned the ranks of nations (in fact, modern nations emerged mainly to fight these frequent wars), in today’s world strong nations do not invade the weak and therefore failed nations can continue to exist. Thus one main category of armed conflict in the modern world: civil wars and violence in failed states (and, of course, terrorist activity operated out of these states, but the resultant deaths are massively outnumbered by deaths within the states themselves).

The second category is cases where there is legal dispute over ownership. Since 1928, states cannot gain territory by occupying it with troops; the only way to get land is to get the rights to it. If the legal status of a territory was unclear in 1928, and the claimants have not reached a diplomatic solution, it largely remains unclear today. Most cases are the result of vague or even contradictory agreements by retreating colonial powers (like Palestine) or uninhabited islands no one bothered to claim, (like those in the South China Sea - or is it the West Philippine Sea?).

In the latter case, Shapiro and Hathaway note that part of the reason why the islands are so contested is not because international law has failed, but precisely because it has succeeded so well: the main perk in owning the islands is the 430 000 square kilometer (370 km radius) exclusive economic zone that is granted by the UN to owners of coastal territory, which is only valid if recognized by the international community.

Whatever the failures of international law and organizations in keeping peace, they are massively outweighed by the benefits. On any conceivable metric, the post- World War II era is more peaceful than any before.

The book ends with a plea for upholding the vision of the “Internationalists” who organized the outlawing of war. The real danger is not that the New World Order fails, but that the world forgets how much it has achieved.

Review: Permutation City (Greg Egan)

Book: Permutation City, by Greg Egan (1979).
2.2k words (≈ 8 minutes)

One way to classify science fiction works is by the scope of the speculative concepts in the work.

For example, the first tier could contain works in which the only speculative elements are things with non-Earth-shattering consequences. Maybe dinosaur DNA could somehow remain intact for over sixty million years. Maybe an evil organization is plotting to create a pandemic.

Then there would be works in which the speculative element is something paradigm-shifting. What if humans made contact with aliens? What would an artificial intelligence do? What if genetic engineering were cheap and widespread?

The level after that would be works that ask similar questions, but go deeper into their consequences, especially by exploring what they say about human nature. How do you tell whether reality is simulated? Can humans even understand sufficiently advanced aliens? What does the possibility of artificial intelligence say about consciousness?

And the last tier is works in which the whole point is speculating about the ultimate nature of the universe itself. Isaac Asimov’s short story "The Last Question" is a classic example. Greg Egan’s 1994 novel Permutation City is another.

(It is hard to limit spoilers in this review, since the plot of the book is very tightly wound to the questions it explores. You have been warned.)


Copies everywhere

The first paragraphs are as pedestrian as it gets: our protagonist, Paul Durham, wakes up in a room and looks around.

Oh, and Durham is inside a computer (literally, though not too literally). He is a “copy”; the “original” had a brain scan made of himself, and started running that in a computer. The simulated reality isn’t an exact copy: only brains are simulated in any detail, while the rest of the environment is an approximation, though a photorealistic one, to save on running costs. Even on economy mode, though, copies in the 2050 world of Permutation City run at best at less than one tenth the speed of the real world.

In the novel’s world, many rich clients have their brains scanned before their biological death, and the copies started after they die. However, they cannot retreat into their virtual worlds, since copies can be affected by real-world events, particularly because legally, they are software, not people.

The wealthiest copies run on private computers managed and paid for by a trust fund. Less wealthy ones run much slower, and their running speed depends on the price of computing power changes, which is traded on a global market (note that the book was published in the 1990s; cloud-based computing power as a service was probably not a very common idea).

What this means for the world of Permutation City is that in addition to private copies running relatively fast, there are also virtual slums of slow-running copies that can afford computing power only when it’s cheapest, and cannot generate new income because their slow running speed (tens or hundreds of times less than the real-world) makes them useless for most jobs. A subculture of poorer copies, calling themselves the "Solipsist Nation", tries to reject external reality completely.

Egan’s bleak vision of copy inequality is not one I have encountered before, and one that seems a bit too credible for comfort.

All the standard brain emulation -related questions are also given some space. To what extent is a copy, based on a brain scan done some time before a person’s death, really a continuation of the life of that person? What about the legal and moral status of a copy of a copy? In general, Egan keeps the sledgehammer on the wall when exploring moral questions, leaving readers to draw their own conclusions and instead focusing on characters’ reactions and attitudes to these questions.


Assemble from the dust?

Egan does, however, present a sustained speculative argument about the nature of copies and therefore consciousness. He presents a thought experiment, which, within the novel, is a literal experiment on thought.

Durham’s copy is conscious. That much can be granted; if consciousness is a byproduct of brain activity, there is no reason for it to not be present in a simulation of the brain.

(Egan is fairly conservative with the technical specifications of the copies; it is mentioned that the brain simulation is not accurate down to the quantum level, and that time for a copy proceeds in discrete time intervals of one millisecond of subjective time. Some might argue that continuous time and/or quantum-level simulation might be necessary for consciousness, but this is a topic that Egan wisely avoids.)

Next, the speed of copy-Durham’s subjective time can be slowed down and sped up by will, simply by changing the rate at which the simulation computes itself. This can be taken even further - if Durham’s copy were simulated only in bursts once every day, or at random intervals, or one frame today and the next frame in a thousand years, copy-Durham would feel no difference.

We can easily stretch the relative times of copies and the real world, but can we break any connection whatsoever between the two experiences of time? Permutation City assumes yes: the frames of the copy’s simulation can be sliced up and rearranged, and the thread of subjective experience will still continue from the perspective of the copy as if nothing were happening even when the simulation is hopping back and forth from one time to another when viewed from the external world.

This is the critical step, and, I think, the weak link. To be able to slice up the copy’s time, the simulation must be able to set itself to, say, the frame at time t=2, and then later to the frame at t=1.

But how can the simulation know the contents of the frame at t=2? It must first compute all preceding frames. After it has done so, of course, there is no obstacle to the simulation loading one frame into memory as the current state, then another and another, all out of order. But is this arbitrary procedure of loading frames into the simulation’s memory what causes the experience of consciousness for the person being simulated? Or would the initial computation of the states be the key?

At one point copy-Durham wonders what his subjective experience of consciousness really is: the current time slice loaded in the simulation, the computation of those time slices, or something else? The novel’s answer seems to the first option.

This raises some very interesting questions. If we reject this premise, the other alternative seems to be that it is the process of computation of frames that causes them to “occur” in the simulation, at least from the perspective of conscious beings in the simulation. This is an interesting topic, and raises many questions. Might the real world, then, be thought of as a computational procedure, in the sense that it is the “computation” of the next moment that makes it happen?

If that sounds like too much to accept, consider what it would mean for the view in the book to be correct: consciousness can thread its way through the disconnected slices of the simulation and therefore the subjective time within the simulation is entirely independent of real-world time. Egan then adds a dose of solipsism: the thread of consciousness of the simulation is real, for the experiencer, despite the fact that it has nothing to do with the “real” time of the outside universe. It has, in other words, somehow “assembled itself from dust”, as Egan likes to repeat, including the italics.

If copy-Durham can assemble itself from the dust despite the time slices of his simulation being scrambled, then why couldn’t other things assemble themselves from the dust?

Extrapolate this further. Imagine a universe of nothing more than an arbitrarily large space of random fluctuations; some of them would, by chance, form sequential, coherent timelines containing conscious entities, which would then be experienced by those entities. Our reality, in the solipsist universe of Permutation City, would just be one of these sequences.

And that’s only the first fourth of the book.


Immortality = cellular automata + solipsist cosmology

What applications could the idea of assembling from the dust ever have? Eternal life and near-omnipotence, apparently.

The other main character, Maria Deluca, is a software engineer who spends a lot of the time she should be working on playing around with the “Autoverse”, a massively complex cellular automata with its own system of chemistry that mirrors real-world chemistry, except without quantum effects (I get the feeling Egan is not fond of quantum physics). Egan spends a lot of pages on the Autoverse, but it is worth it; I found myself wishing for a real one.

After Maria gains some success with getting Autoverse bacteria to mutate, Durham enlists her to design a program to produce an entire planet, complete with primitive bacteria, in the Autoverse. There is not nearly enough computing power, even in Egan’s world of 2050, to run an entire Autoverse planet, but that’s not the point.

Durham’s idea is to simulate the first few minutes of a self-replicating cellular automata computer on a computer, and then stop. The continuation of the self-replicating computer represents a coherent timeline and by the logic of the novel’s solipsist universe, it will simply assemble itself from the dust and continue to exist from the perspective of the wealthy clients who paid to have their copies put on the thing. Thus Durham, and his clientele of billionaires, escape our reality into an alternative universe consisting of an ever-expanding computer that simulates their copies in addition to the Autoverse planet.


Solipsism!

If this weren’t speculative enough, Egan turns the solipsism up to eleven in the second part of the book.

Without revealing too much, the basic idea is that the Autoverse planet has developed intelligent life, which has its own theories for the origin of their universe that do not include being a simulation inside a simulation that was launched by a simulation made by a crackpot theorist and a dozen billionaires hoping for eternal life.

Reality in the universe (or should I say, space of random states) of Permutation City is a subjective thing, and so the logically coherent theories of the simulated lifeforms eventually become more real than the version of reality Durham and the other copies believe in, with destabilizing effects on their apparently-not-quite-eternal universe.

Brain emulation is already a topic with plenty of philosophical questions to explore. Egan, though, is not content with remaining in that territory, and instead takes the reader on a philosophical roller coaster through the consequences of ever wilder and wilder solipsism.


I was told you have to mention literary features when discussing literature …

… but Permutation City was written more for its concepts than its literary merit.

Egan does portray a reasonably diverse cast of characters. We have an eccentric and determined theorist, a software engineer with a terminally ill mother and time-consuming hobbies she cannot bring herself to quit, a remorseful billionaire struggling with past crimes, and a survival-oriented virtual slum -dweller. Many of them struggle in a genuine way with questions of identity and morality in the copy-filled world of Permutation City, and some scenes were touching, but none of the characters were particularly memorable.

Many of Egan’s chapters (not all are named) have names that are anagrams of "Permutation City", an allusion to the slicing of copy-Durham’s simulation. “Remit not paucity” is the most common chapter name. As far as I can tell, it seems to be a warning against trying to eliminate all scarcity from life, as Durham’s flawed universe does. There is also a disconcerting heavily anagrammatic poem at the front of the book, indirectly attributed to the main character Paul Durham. If it has meaning besides building atmosphere, I can’t figure it out.


Meaningful answers?

Permutation City is far from the only work of science fiction to explore esoteric philosophical themes. Peter Watts’ Blindsight (main point: how useful is consciousness; what if a space-faring civilization did not have it?) and Neal Stephenson’s Anathem (main point: uhhh …) also deal with the philosophical questions surrounding consciousness.

However, Permutation City is exceptional in the extent and scope of its speculation. It is also structured well in this regard; Egan gradually ramps up the level of speculation throughout the work, allowing the reader to update their knowledge of how the novel’s world works after the introduction of each speculative leap, and helping to maintain immersion by showing the internal consistency. It also, probably not coincidentally, lays bare Egan’s chain of reasoning, exposing it to readers for easy analysis. The book definitely succeeds in provoking questions.

As to whether the book’s big ideas are anywhere close to being correct, I think Isaac Asimov’s fictional computer in The Last Question put it best:

"THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER"

Review: Gödel, Escher, Bach: An Eternal Golden Braid (Douglas Hofstadter)

Last updated 2018/05/16 with clarifications to the incompleteness proof and links to resources on formal systems.

Book: Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter (1979).  

3.8k words (≈ 15 minutes)

What are the commonalities between Kurt Gödel, Johann Sebastian Bach, and Maurits Cornelis Escher?

I don’t know, because that’s not what the book is about. Gödel, Escher, Bach (GEB) is about a deeper topic: how can minds be formed from simple components? How do mental processes arise from non-mental ones?

Many authors might have handled this topic by writing a dry, dense book on the matter. Not Hofstadter. The tagline of the book is “a metaphorical fugue on minds and machines in the spirit of Lewis Carroll”.

Between each chapter is a dialogue between fictional characters (the three main ones are Achilles, the Tortoise, and the Crab), which serves to foreshadow the themes in the following chapter. Each dialogue mirrors the structure of a piece by Bach, though sometimes the link is very tenuous. Works of art by Escher appear frequently, also linked more or less strongly to the subject being discussed.

Hofstadter particularly enjoys having the structure of the dialogue mirror the subject, self-reference, and simply messing with the structure purely for the sake of it.

Perhaps the most impressive example is his “Crab Canon”, which reads (almost) the same when read backwards line by line. In the dialogue, both characters mention works of art with this property, including Bach’s Crab Canon on which the dialogue is based. This acts as a link to the topic of indirect self-reference in formal systems. And if you think this is a lot of layers of metaphor and meaning, Hofstadter pours on another truckload when referencing the dialogue in a later chapter.

Another example is the dialogue “Contracrostipunctus”. The name is a portmanteau of “contrapunctus” and “acrostic”. When the first letter of each paragraph is written down the result is “Hofstadter’s contracrostipunctus acrostically backwards spells J S Bach”, which, acrostically backwards and treating “J S Bach” as one word, gives “J S BACH”.

How did I realize this? Because Hofstadter points it out. There are countless parts in the book where the author says something to the effect of: “Behold! This seemingly arbitrary choice I made earlier actually reveal a deep analogy between these two seemingly disparate subjects, and now I will use this as a springboard to jump to the next topic”. And then the next topic is unfailingly linked in twenty different ways to other topics, and each link is symbolically represented by a Bach and/or Escher piece.

I assume that there are countless things that Hofstadter does not directly point out that I missed while reading. Or maybe there’s the equivalent of a Schwarzschild radius for cleverness and trying to concentrate any more in a single book would make the book collapse into a black hole.

I thought that this aspect of the book would get old by the time I got to the end, but instead I found myself as entertained as ever when the final dialogue included phrase after phrase from the author’s Ridiculously Inventive Collection of Extremely Recursive and Charming Acronyms for Ricercar (a ricercar is similiar to a fugue, and two ricercars form the backbone of Bach’s Musical Offering, which Hofstadter references heavily). This sparkling of word play, self-reference, metaphor, and mini-insights makes the book a great joy to read, while also setting up the groundwork for the book’s heftier insights.

Informal Formal Systems

So what, then, are the heftier insights? The book starts by introducing the concept of formal systems and discussing the connection between formal systems and meaning.

A formal system consists of two things: axioms and rules of inference. Axioms are the strings of symbols that you start with, and rules of inference are the rules for how you are allowed to change a string, split it into other strings, or combine many strings into one.

Most of the time (such as in formal logic), the symbols are taken to mean something, and the object of using the rules of inference to get to other strings is to try to prove that they follow from the axioms of the system.

Strings within a formal system can be divided into categories:

• Non-well-formed strings. These use the same symbols as the formal language, but they do not make grammatical sense. For instance, “∧))p” makes no sense in the context of formal logic.
•  Well-formed strings. These are grammatically valid.
•  True strings.
•  Decidable true strings: strings that state true things and their truthhood can be determined within the system.
• Axioms: strings that are taken to be true as part of the system.
•  Derived true strings: strings that can be determined to be true by applying the system’s rules of inference to its axioms.
• Undecidable true strings. These strings state true things, but there is no derivation for them within the system and thus they cannot be proved. Guaranteed to exist because of Gödel’s theorems.
• False strings
• Decidable false strings
• Negations of axioms
• Derived false strings
• Undecidable false strings 

These categories are elegantly summed up by one of Hofstadter’s diagrams:

I will not discuss formal systems at great length since GEB does an excellent job of giving the reader a very intuitive understanding of formal systems in a gradual, easy-to-follow manner. The proof of Gödel's theorem given later assumes some familiarity with formal systems, however. If you want to learn more about formal systems, I suggest looking into first-order logic (also called predicate logic or, less specifically, formal logic) or Peano arithmetic.

One more thing about formal systems: an important distinction Hofstadter makes is between reasoning inside a system versus outside a system. It might be obvious that a certain theorem can never be reached within the constraints of a formal system, but if we restrict ourselves to thinking only within the system, we can only churn out theorem after theorem.

BlooP to FlooP but not to GlooP

Hofstadter introduces three hypothetical programming languages to illustrate the differences in what can be expressed with different types of recursion: BlooP, FlooP and GlooP.

In BlooP, only bounded loops are allowed (hence the name): whenever the program runs a loop of any kind, the number of times the loop will run has to be known when it is started. A BlooP-compatible factorial function could therefore be written in pseudocode as:

define factorial (n):
   p <– 1
   for i in range n:
      p <– n * p
   return p

(Hofstadter gives specific syntax for BlooP, but I feel that modern readers will find Python-esque pseudocode more legible)

However, the following definition of factorials would not be allowed in BlooP:

define factorial (n):
   if n = 1:
      return 1
   else:
      return n * factorial(n - 1)

Neither would a function like this:

define nextPrimeAfter (n):
   d <– 1
   while notAPrime(n + d):
      d <– d + 1
   return n + d

Why? Because the loops in both examples do not run a predefined number of times. Why does this matter? When writing a factorial function, not being able to write the recursive version above is a mere inconvenience. With nextPrimeAfter, writing a BlooP version would require knowledge of some mathematical tricks (for example, it has been proved that for n > 3, there is always a prime greater than n and smaller than 2n; thus, the number of times you have to loop can be known before the loop is started).

But are all algorithms Bloopable? Or are there programs that cannot be written in BlooP?

Yes. It can be proved (and it is proved, in the book) that there exist algorithms which cannot be expressed in BlooP. For those, we must allow unbounded recursion, such as in the FlooP (Free Loop) language. Both of the unBloopable cases above are perfectly Floopable.

Is there a language, call it GlooP, even more powerful than FlooP? The answer is no. Some programs, such as a program for determining whether another program will halt, have been proved to be impossible (much to the chagrin of programmers trying to stop their programs from freezing because of accidental loops). But if a procedure is possible, it can be written in FlooP.

Hofstadter notes that all widely-used programming languages are equivalent in power to FlooP. This is less because of the ingenuity of their designers and more because unless limits are deliberately placed on a language for didactic purposes, any reasonable programming language can do everything that FlooP can (in theory - some, of course, are far more difficult to work with).

A language for expressing algorithms therefore tends to be capable of expressing all possible algorithms, especially once recursion is allowed.

This fact will come back to haunt us when it comes to formal systems and Gödelian incompleteness.

Incompleteness

The book walks through a proof of Gödel’s incompleteness theorem. It is fascinating enough that I will try to explain it in full.

Gödel’s incompleteness theorems (there are two) are based on two key ideas.

1.  Gödel numbering. This is simply the process of replacing all characters used in a formal system by numbers. We can completely arbitrarily define, for instance, that “(“ is “10”, “)” is “01”, “p” is “14”, and “∧” is “17”, and then write “1014171401” instead of “(p∧p)”. If, in the formal system we are dealing with, the interpretation of “(p∧p)” is “p is true and p is true”, then “1014171401” has the exact same interpretation. Thus, all statements, axioms, or even derivations within a formal system can be written as a single number (most of the time, a ridiculously large number, but what matters is that it is theoretically possible).
2.  If we have a formal system for arithmetic, Gödel numbering means that we can convert statements of that formal system into numbers. Furthermore, we can convert the rules of that formal system into (very complex) arithmetic operations. What this means is that the formal system can talk about itself.

Self-reference opens the door to a whole host of potential paradoxes. Can the statement “this statement is false” be made within F?

Unfortunately, or perhaps fortunately, a formal system like Hofstadter’s TNT (Typographical Number Theory, based on Peano arithmetic) cannot encode a paradox as simple as “this statement is false”, since the property of a statement being “true” in TNT (or any other comparable system) cannot be stated in a Bloopable form, and it turns out that TNT can only express Bloopable properties.

(For more on TNT, see the Wikipedia article. For the purposes of this proof, it is sufficient to know that TNT statements can express arithmetic operations, contain variables, and be qualified with "there exists an X such that [statement]" or "for all X, [statement]")

So the question is: is there a way in which TNT can be forced to say something paradoxical about itself and thus be proved incomplete?

We might not be able to write a test in TNT for whether or not a TNT statement is true, but we can check the validity of a proof. For instance, an algorithm for checking a proof might be as follows:
- For each line L in the proof (extracted by splitting the proof's Gödel number at each line break symbol):
    - If L is not an axiom:
        - For all pairs of prior statements S in the proof:
            - For all rules of inference R:
                - Does applying rule R to one or both of S result in L (“one or both” because some rules will be about combining statements, others about modifying single statements)? This can be tested in TNT because when dealing with the Gödel numbers of statements, all rules for shuffling the symbols of those statements can be expressed as arithmetic operations, which is what TNT is all about.

If each line is either an axiom or the result of applying one of the (finite) rules of TNT to the lines that come above it, and the last line of the derivation is S, then we can conclude that the derivation P is a valid proof of S and S is a theorem of TNT.

There are no unbounded loops, so therefore it is expressible in BlooP and therefore TNT (though the TNT expression will be enormous). And since it is an expression of TNT, there is an (even larger) Gödel number that corresponds to it.

What we have now shown is that there exists a number which, when interpreted as a Gödel number of TNT, expresses a statement for checking whether another Gödel number is a valid proof for a third Gödel number. Call this statement $$ \text{TNT-PROOF}\{A, B\} $$, where A is a free variable for the Gödel number of the proof and B a free variable for the Gödel number of the statement. For any value of A and B, this statement will yield either true or false.

Therefore we can write the sentence “there exists a number A such that A is the Gödel number of the derivation of statement S” as $$ \exists A: \text{TNT-PROOF}\{A, S\} $$. This is equivalent to saying "there exists a way to derive S within TNT" - all within the strict confines of TNT!

The final trick is related to a process Hofstadter calls “quining” after its inventor Willard Van Orman Quine. An example in English given in the book is the following paradox: “‘yields a falsehood when preceded by its own quotation’ yields a falsehood when preceded by its own quotation”. This is the fundamental structure behind the TNT statement that results in the proof of Gödel's theorem.

The general idea is to substitute the value of a string into some place inside the string. To “arithmoquine” is an implementation of quining inside TNT: take the Gödel number of sentence S, and substitute that in place of a free variable in S. Testing whether or not one number is the arithmoquinification of another is BlooPable and therefore expressible TNT; we will call the function that checks whether Y is the result of substituting the Gödel number of X into the place of the free variable in X $$ \text{ARITHMOQUINE}\{X, Y\} $$.

We are now ready to introduce a statement (call it U):

$$$ \neg \exists A: \exists B: (\text{TNT-PROOF}\{A, B\} \wedge \text{ARITHMOQUINE}\{C, B\}) $$$

What this sentence means is:

There does not exist a number A such that ($$ \neg \exists A : $$) there exists another number B such that ($$ \exists B : $$):

• A is the Gödel number of a valid derivation of B ($$ \text{TNT-PROOF}\{A, B\}) $$)
  AND (∧)
• B is the arithmoquinification of C ($$ \text{ARITHMOQUINE}\{C, B\} $$)

U has exactly one free variable (C), so it can be arithmoquined. To do this, we take the Gödel number of U (call it N), and replace C with N to yield sentence G:

$$$ \neg \exists A: \exists B: (\text{TNT-PROOF}\{A, B\} \wedge \text{ARITHMOQUINE}\{N, B\}) $$$

G, the infamous Gödel’s string, has the following meaning:

There does not exist an A such that there exists another number B such that:

• A is the Gödel number of a valid derivation of B
  AND
• B is the arithmoquinification of N

Since it is guaranteed that N has an arithmoquinification (it has one free variable), the interpretation can be rephrased as:
There is no number A such that A is the Gödel number of a valid derivation of the arithmoquinification of N

But what is the arithmoqunification of N? It is Gödel’s string, G! So we can further simplify our interpretation as:
There is no number A such that A is the Gödel number of a valid derivation of G

In other words, G says that there does not exist a derivation of G in TNT. If there is no derivation of it, it cannot be a theorem of TNT.

Using the concepts of Gödel numbering and arithmoquinification, we created a sentence of TNT which says “this sentence is unprovable in TNT”. Alright, but why should we listen to this sentence? We can make a sentence in TNT to say anything (about numbers) that we want, but that does not mean that 1 = 2.

So let’s step back from Gödel’s string. Is it a theorem of TNT? If G is a theorem, then what G says must be true, and G says that G is not a theorem.

At first glance, this seems just as paradoxical as “this statement is false”. But we’ve only considered the case that G is a theorem - what if it isn’t?

If G is not a theorem, then there is no contradiction: G says “I am not a theorem”, and it is not. Thus, we are forced to accept that G is a true statement, but it is not a theorem: there cannot exist a chain of proof that starts from TNT’s axioms, proceeds by TNT’s rules of inference, and arrives at G.

What Gödel’s incompleteness theorem (or, more precisely, Gödel’s first incompleteness theorem) shows is that within the framework of Typographical Number Theory (or any other framework equivalent to Peano arithmetic), there are statements of the formal system that cannot be proved to be true, even if they actually are true.

Why the focus on formal systems of arithmetic? So what if certain extremely complex properties of numbers cannot be proved with only a handful of axioms and inference rules?

It turns out that any formal system complex enough to embed arithmetic (and therefore complex enough to express all Bloopable properties) is vulnerable to Gödel’s proof. This is because Gödel numbering can be used to convert a statement in any formal system into a number, and because all formal rules for manipulating these numbers can then be expressed as arithmetic properties. Any sufficiently complex formal system can be written in terms of TNT, and has unprovable truths because of the possibility of self-reference.

Applying Gödel: Landscapes and Machines

What if we add Gödel’s string as an axiom of TNT? It wouldn’t help, since it would be possible to construct a sentence that has the same function in the new TNT as G has in standard TNT. It would, however, imply the existence of a class of numbers that Hofstadter terms “supernaturals”, which would have a whole bunch of interesting properties.

But are they real? This begs the question of whether imaginary numbers are real, or whether natural numbers are real, or even whether real numbers are real. Hofstadter points out that choosing which mathematical system to use requires “stepping out” of the mathematical framework and making a judgement about what is applicable to the problem at hand. New number systems will not wreck the banking industry since imaginary numbers, for instance, are not applicable to reasoning about financial systems (despite what many might wish).

At the beginning of the 20th century, there was a movement in mathematics, lead by David Hilbert, to find a set of central axioms and rules of inference from which all of mathematics could be derived. Gödel’s theorem ruined this dream, saying that no matter what the axioms and rules, there will always be unprovable truths. New axioms, and therefore new systems and concepts, will always be needed. I think one of the broader insights here, one which Hofstadter seems to hint at, is a view of mathematics not as a single, godly machine spitting out all truths, but instead as a landscape of overlapping regions, each applicable in different circumstances and with their own local rules, customs, and terrains.

Applying Gödel: Machine Intelligence?

When scientific or mathematical results become famous, there is a tendency for its conclusions to go through a game of “Telephone”. Examples include how the theory of relativity has been interpreted as a statement about how everything is relative, and probably most of what is written about quantum mechanics. Gödel’s incompleteness theorem is no exception: “sufficiently complex formal systems imply truths which cannot be proved within the system itself” easily morphs into far less precise statements.

Hofstadter, who was already working on artificial intelligence in the 1970s, takes some time to refute the idea that Gödel’s theorem refutes the idea of AI. Why would this be the case? To quote Hofstadter quoting J. R. Lucas:

However complicated a machine we construct, it will, if it is a machine, correspond to a formal system, which in turn will be liable to the Gödel procedure for finding a formula unprovable-in-that-system. This formula the machine will be unable to produce as being true, although a mind can see it is true. And so the machine will still not be an adequate model of the mind. We are trying to produce a model of the mind which is mechanical-which is essentially "dead"-but the mind, being in fact "alive," can always go one better than any formal, ossified, dead system can. Thanks to Gödel’s theorem. the mind always has the last word.

Hofstadter argues against this interpretation from various angles at some length, but acknowledges that C. H. Whitely might have done it best by inventing the sentence: “Lucas cannot consistently assert this sentence”. The statement is true, Lucas cannot assert it without being inconsistent, and the information we gain about his mind is zero.

What, then, is the relevance of Gödel’s theorem to the central question of minds? It is not so much an “answer” as a fundamental fact about sufficiently complex formal systems that comes up in many contexts.

For example, Hofstadter makes an analogy between formal systems and an idealized version of DNA replication. The equivalent of Gödel’s theorem within this framework is that for every cell (formal system) there exists a strand of DNA (a string of that formal system) that is unreproducible (cannot be proved) within that cell.

Gödel, Escher, Bach: An Eternal Giant Book

At the start of the review, I mentioned that primary topic of GEB is how minds can arise out of simpler pieces. I then went on to talk for over 3000 words about formal systems, their incompleteness, and Floopability.

Where is the answer to the book’s central question? How do countless disparate topics come together into a unified whole that answers the question of how countless disparate components come together into a unified whole that can think and reason?

A common complaint about GEB is that it spends 800 pages saying what could be condensed into 300. And it is true: if you take the essence of everything Hofstadter says, it boils down to maybe 300 pages. 300 pages, that is, of dry, dense material.

But the book is not about the exact mechanisms of how minds work, or how to build an AI, but more about the general flavor of what systems any intelligent being will have in common.  Where GEB shines is its ability to explore these topics in a way that makes the conclusions seem natural and intuitive: you can know as a fact that mental activity comes down to the firing of unconscious neurons, but that is not the same as having an intuition of how this might happen.

Self-Referential Offering

I will conclude with an attempt to distill some of GEB’s main points into a few paragraphs. This is less of a summary and more of an attempt to separate one voice from GEB’s metaphorical fugue.

Self-reference is an inherently dangerous business, yet also extremely hard to avoid in any formal system or language worth talking about. What often results are what Hofstadter calls “strange loops”, in which different layers of meaning interact in a surprising way. Because of this, there is a fundamental tendency of any complex reasoning system, whether mathematics or brains, to be in some sense incomplete, messy, and non-linear (this has shades of chaos theory, but hopefully the dinosaurs won’t escape).

Hofstadter proposes that consciousness, and other “emergent” phenomenon in brains, are the result of the interaction of different levels in strange and complex ways.

Is this an anti-reductionist position? No. As Hofstadter explains: ”In principle, I have no doubt that a totally reductionistic but incomprehensible explanation of the brain exists; the problem is how to translate it into a language we ourselves can fathom."

The power of GEB is that it shows the beauty of this view.