(e+1)/2(e-1) = Σ_n=-∞^∞ 1/(1 + 4 π² n²)

π/(1-exp(-π)) - π/2 = Σ_n=-∞^∞ 1/(1+4 n²)

The way to prove it is to use the Poisson summation formula, which states that for appropriate functions:

Σ_n=-∞^∞ f(n) = Σ_n=-∞^∞ F(n)

where F is the Fourier transform of f. The "appropriate" part is of course where tricky things may happen, since one or both sums might not converge.

Now, f(x) = exp(-a|x|) has the transform F(ξ) = 2a/(a² + 4 π² ξ²) for a>0.

So we get this possible equality:

Σ_n=-∞^∞ exp(-a|x|) = Σ_n=-∞^∞ 2a/(a² + 4 π² n²)

These sums are both well behaved, since both can easily be bounded by nicely convergent integrals, so convergence is assured.

The left hand side can be rewritten as [2 Σ_n=0^∞ exp(-ax)] - 1 since exp(-a|x|) is an even function (imagine "folding" the number line around n=0; the -1 term corrects the double counting of the n=0 term with value 1). And the sum inside the brackets is a nice geometric series summing to 1/(1-exp(-a)). So the original sum is

2/(1-exp(-a)) - 1 = Σ_n=-∞^∞ 2a/(a² + 4 π² \n²)

Some rearrangement produces

1/a(1-exp(-a)) - 1/2a = Σ_n=-∞^∞ 1/(a²+4π² n²)

Now, we can set a=1 and get

1/(1-e^-1) - 1/2 = Σ_n=-infty^∞ 1/(1 + 4 π² n²)

a=π gives

π/(1-exp(-π)) - π/2 = Σ_n=-∞^∞ 1/(1+4 n²)

Maybe not terribly useful, but beautiful. I remember how delightful it was to learn how to calculate integrals using residues or exploit Fourier series to get cool sums and integrals in closed form. After all, the coolest fact about

π/3 = Σ_n=1^∞ sin(nπ/3)/n

is that it is so easy to derive - once you have the powerful machinery behind it.

Addendum 15/5/12:

Just found another nice identity the same way, this time using the transform pair f(x)=x H(x) exp(-αx), F(ξ)=1/(α+2πξi)² where H(x) is the Heaviside step function. In the same way as above this leads to:

4π²Σ_n=0^∞ n exp(-2πn) = Σ_n=-∞^∞ (1-n²)/(1+n²)²

A neat thing about it is that the sum on the left is all positive, while the sum on the right has just one positive term.

In looking for information I came across this page: Kings Trough, Under Sea Features Prayer Times . The site calculates prayer times for Muslims, and apparently the location database includes underwater locations too. I assume Muslim divers and submarine crews do not have quite the same complications as astronauts in defining the direction to Mecca or proper prayer times.

A hilarious detail: apparently there is a mosque on the California Seamount. Or maybe a data entry error. But I find the vision of a grand mosque built on top of a seamount in the middle of the Pacific lovely. Deep sea corals grow along the walls. Fish swim in the faint blue beams of light crossing the central dome. The minaret proudly rises towards the sun far away, occasionally confusing a passing shark.

First I noticed Io9: Could a single pill save your marriage? - it is my old love enhancement paper reappearing. Of course, the title falls afoul of the rule that you shouldn't title something with a question that can be answered with "no". (With some luck an updated popular version of our paper will appear in New Scientist. (here it is))

Then Neatorama brought up Quantum Gravity Treatment of the Angel Density Problem - my old paper in Annals of Improbable Research on angles dancing on pins.

Unlike the angel paper (and my green dwarf paper) the love paper is entirely serious.

Next TEDx with me will be in Tallinn.

Personally I would not discount quantum tunnelling or wormholes, both which might explain the gift-giving abilities of Santa.

OK, it is a April fool's paper.

The goal of the exercise is to estimate how much information is needed to reconstruct an individual “well enough” without having direct access to their nervous system. The reconstructing agency is assumed to have arbitrary computational powers, but is limited by available information.

In the following I will be using the Stirling approximation of factorials (log(n!) ≈ n log(n)– n) to calculate binomial coefficients: log(N over k) ≈ N log(N) – (N-k)log(N-k) – k log(k). Also, remember that the number of bits of information you need to supply to find a particular object among N is log₂(N), and that log₂(10^x) ≈ 3.32x.

Record personality

Record experience

Other estimates can be made. Human retinas have a bandwidth of 8.75 megabits per second, providing the brain with about 20Mb/s in total across the 2 million axons in the optic nerves. The spinal cord similarly has a couple of million fibers and we can lump it all together by guessing that the maximum input is on the order of 200 Mb/s. 200 Mb/s over 80 years is 5*10¹⁷ bits. So by this approach, there might be up to 2^5*1017=10^1.5*1017 possible persons.

Unfortunately, while storing a few hundred petabits might soon be doable and life recording might allow us to document our lives very well, it doesn’t follow that we record the *right* information. My experience of a piece of music depends on complex details of my auditory physiology and mental processing: replaying it to another system is unlikely to produce the same experience.

The life recording approach forgets about initial conditions. Two pieces of software given the same input can behave utterly different, including storing different information. Chaotic dynamical systems (and we certainly have some of them inside us) diverge exponentially when given different initial conditions, even when perfectly deterministic. And babies already demonstrate personality differences. So the above numbers need to be multiplied by the number of distinct starting states.

There are about 1.5 gigabytes of information in the genome but much of this is shared between different humans; the genetic specification of a person relative to a baseline human genome is likely about 20 megabytes, 2^1.6*108 possibilities.

Genetics also doesn’t specify much of our brains: we have far fewer genes than neural connections, and they are generated in a complex semi-random process influenced by the environment in utero. So we have to take a look at the number of possible human brains. The calculations below show that this pumps up the information need enormously, at least by 15 orders of magnitude. And this information is not externally available (unlike the genetics, which is left in every skin flake).

So it appears unlikely that documenting the information in our environment plus initial conditions will be feasible within the conceivable future. So we need to get information from the brain to pin down what particular brain it is.

Human information output

Even if we were to visibly move our ~639 muscles at 10 Hz (about as fast as they can twitch) that provides just a few kilobits of information per second.
Spoken and written words are even worse: the average entropy per English text character is about one bit, and the entropy rate of spoken dialogue is a few bits per second.

The average daily email production is about 15 emails containing about 30 kilobytes each, corresponding to an information production of 41 bits/s – much of it of course header information generated by computer.

A high resolution video and audio recording of our lives might have a far high bit rate, we do not contribute much new information to each frame.

This leads to a first tentative argument against reconstruction based on external data: we are acquiring potentially personality-affecting information at a fairly high rate during our waking life, yet not revealing information at the same high rate. The ratio seems to be at least 1000:1.

Still, a reconstruction enthusiast might be undeterred. Most of those input bits are discarded: we learn and change far more slowly than what we sense. If the number of possible distinguishable human minds is small enough, we should be able to determine which one inhabits a certain brain by inferring it from its behavior.

Number of brains

A human brain contains 10¹¹ neurons with a few thousand connections each, giving us around 10¹⁵ synapses.
A simple argument for the number of possible persons would be that the 10¹⁵ synapses of a brain can be in either a potentiated or unpotentiated state, leading to 2¹⁰¹⁵ possible states. Or, more simply, we need 10¹⁵ = 1 petabit of information to specify which of them a given individual is. There is also the issue of how many ways the 0.5*10²² neuron pairs can be connected by these synapses. (0.5*10²² over 10¹⁵) is around 10^7*1015. So we need 23 petabits to specify which brain connectivity a person has.

But most of these brains are indistinguishable. We do not become entirely different persons because one pixel on a TV screen 10 years ago was different or because a synapse just got removed. Just like macrostates in statistical mechanics contain *lots* of different microstates, our personal identity macrostates have room for plenty of microvariations. The fact that we remain identifiable (mostly) day from day demonstrates this.

Many neural disorders can progress undetected until a few tens of percent of neurons in affected areas are gone. So let’s make the optimistic (in the sense that it makes reconstruction easier) assumption that brains with 90% the same connections produce the same person. This is likely not too far out: neural networks are robust to the deletion of a few connections. But it of course ignores that certain focal deletions can have big effects.

If we randomize 10¹⁴ out of the 10¹⁵ connections, we can select them in (10¹⁵ over 10¹⁴) ways, or 10^1.4*1014. We can then reconnect them in (0.5*10²²-10¹⁵ over 10¹⁴) ways each, 10^8.1*1014. So the total number of brain connectivities giving persons indistinguishable from the original is the product, 10^9.5*1014. So out of the 10^7*1015 possible brains only 10^6.0*1015 are distinguishable. That requires 2*10¹⁶ bits, or 20 petabits.

20 petabits is staggering, yet not unheard of - there are certainly bigger data centers around today. However, assuming that we produce a 10 Kb/s of personal data per second by moving or not moving, it would take 63,377 years to get enough to specify a mind uniquely enough to construct a brain close enough connectivity. If we actually produce one megabit per second of personal data it can be done in 633 years. In order to get it down into the range of a human lifetime we need tens of megabits per second: this can likely not be done using external means, but requires interfacing directly to the nervous system.

Note that this only counts incompressible, relevant bits – information that actually helps determine the structure of the mind. In reality our activities are of course highly compressible and noisy: after having observed one of my verbal or kinetic tics for the first time, seeing repetitions are not very informative.

Conclusion

Even with arbitrarily powerful computational power, inferring which mind a given human contains appear infeasible given the limited amount of information about its internal state revealed in normal activity. Accurate documentation of the environment may provide helpful constraints, but since the relevant question is how environmental information is processed internally emitted information will always have far higher constraining value.

It might be simpler for arbitrarily powerful future entities to just simulate all possible past humans than try to reconstruct particular ones based on personal information.

In the absence of arbitrary computational power, we will likely have to make do by preserving the brains themselves.

Do you see a space-faring civilization as a way of defending humanity against a catastrophe on Earth?

Absolutely. We would be backing up the biosphere. We wouldn't just be preserving humanity, we would be preserving much of life. It is certainly possible for some calamity to come along — as we see in the several major extinction events in the fossil record. Humanity has obviously developed the means of destroying itself, so I think we need planetary redundancy to protect against the unlikely possibility of natural or man-made Armageddon.

It is important that we take action now to make life multi-planetary, because this is really the first point in the 4-billion-year history of Earth that it has been possible. That window of possibility will hopefully be open for a long time, but it may only be open for a short time. That's why I think urgent action is required on making life multi-planetary.

It is nice to see someone get it. Especially someone in a position to take positive action.

Reductions of existential risk have an amazingly large importance, making even low-probability of success projects worthwhile.

This is why we need something better than the shuttle.

Here is a plot of the Elo ratings of chess systems from the Swedish Chess Computer Association tournaments:

Of course, the data is noisy - I don't really think chess software has become worse over the last few years. In this case it might just have been a narrowing of the field.

Plotting them as a function of clock speed produces the following intriguing graph, showing some jumps in software performance on the same hardware:

Combining the first graph with List of FIDE chess world number ones - Wikipedia, the free encyclopedia we get the following relative graph:

It shows nicely how humans have not been getting much better, while the machines improve from nearly hopeless to superhuman.

Of course, this is slightly iffy: while SSDF has tried to calibrate against humans one can always question the commensurably.

One can also calculate the probability of winning against players of different ranks (again assuming commensurability):

Looking at 6 game matches the probability of winning the majority looks like this, a bit sharper:

Human amateurs were already having trouble at the start, but grandmasters could be confident at the start of the 90s... and were outclassed by the end.

Observant readers might wonder why Kasparov lost to Deep Blue in 1997. The reason is that this tournament data uses consumer machines, not the supercomputer IBM pitted against him. For 1997 the winner in the tournament was HIARCS 6.0, running on a 49 MB P200 MMX. Deep blue was a 30 node system with a 120 MHz processor per node, plus 480 special purpose chess chips. It ran at 11.38 GFLOPS on numerics. A pentium MMX at 200 MHz is around 23.53 MFLOPS. So Deep Blue was at least 484 times more powerful (and likely quite a bit more, thanks to the special chess chips and parallelism).

So the mystery of why it won so early is solved: the brute force allowed IBM to get extra performance equivalent to perhaps a decade development.

Some other reactions or comments to my "green dwarves" paper:

TheGuardian: Bioengineer humans to tackle climate change, say philosophers. Leo Hickman gave us a chance to respond to the criticisms, and printed them at length. This is your opportunity to see how we deal with them individually.

Slate: Is "Human Engineering and Climate Change" Paper a Case of Academic Trolling? - Slate thinks we may be trolling.

Io9: Four ways we could hack human bodies to save the environment - a science fiction site sceptical of the feasibility of our ideas? What next, criticism of the practicality of steampunk? :-)

Planet of the (Little) Apes | Practical Ethics - Steve thinks about the children. How fun would it be to grow up short if your parents are environmentally conscious?

Jeffrey Bossert Clark: Re-making Man by Choice and Decree has a bunch of criticisms at the Library of Law and Liberty (some which I entirely agree with).

The Bard of Murdock: Inventing Children A "poetical cartoon" based on the paper.

Small dead animals had a poll, finding that 98% of participants favoured "Grind Up climate alarmists into tiny particles and blast them into the lower troposphere" over our proposals. Don't they know that climate alarmists being green have a suboptimal reflection spectrum, and will not be efficiently lofted into the stratosphere before they rain out?

I expect that the story of the paper will be the usual one: in a few months it is a paper among other papers in an ethics journal, perhaps cited a few times if we are lucky (if only as a warning example of a stupid idea - but that still counts!). Meanwhile the paper will take on another life among the undergrowth of the noosphere, mutating into the story that scientists are working on (or have already done) re-engineering people to be green (probably using chemtrails or brain implants). It will be yet another piece of evidence of the Big Conspiracy. You doubt it? Look at all those blog posts, that paper was real!

What struck me the most was that the big outcry came from the climate-sceptics rather than the traditional greens (although having Bill McKibben tweet that the paper contained the "worst climate change solutions of all time" warms my heart). I guess this is because they tend to hold conservative views (psychologically if not ideologically) and hence the paper pressed the "messing with human nature" hot button besides the "something radical must be done about the climate" hot button. Among the greens and libertarians only one button was pushed.

I'll work on that for my next paper. Just how many hot buttons can I innocently press?

The fun part is that Slashdot tagged it 'troll'. Which I personally find accurate: when I contributed, I felt I was partially trolling.

The basic argument is that climate change and many other environmental problems have upstream and downstream solutions. For example, 1) human consumption leads to 2) a demand for production and energy, which leads to 3) industry, which leads to 4) greenhouse gas emissions, which lead to 5) planetary heating, which leads to 6) bad consequences. One solution might be to try to make less emissive industry (fix the 3-4 link). Another might be to remove greenhouse gases from the atmosphere (reduce 4), geoengineering that cools the planet (reduce 5) or adapt to a changed world (handle 6). The latter are downstream solutions.

When geoengineering is suggested many people think it is better to use an upstream solution for a variety of reasons (controllability, scepticism of technological fixes etc). Consuming less would be really upstream (ignoring the practical problems of actually doing it to the extent necessary, which are pretty major, and the countervailing aspects of human psychology). But if it is better to have upstream solutions, why not go for cause 0, human desires for various things? If people do not want meat, plenty of grassland could be reforested and emissions reduced. If people want to have fewer children, resources become less scarce and so on. If people are smaller, they need less resources.

So we argue that it might be a good idea to look at reengineering humans to be green. Obviously we can change ourselves through culture and rational convincing... to some extent. But it is tough to change lifestyle this way. So might there be tricks to modify ourselves in order to better behave in ways we want to want. Biological means to make our second order desires dominate first order desires.

The methods we mention are mostly examples and likely *far* too wussy to amount to much. But there might be better methods if we were to investigate them properly. Of course, in the long run I think the real sustainable choice is to become postbiological. But that is a bit further away. This is also why I doubt genetic engineering is going to be that effective: it takes so long time, parents will be very cautious about it, and the things that can be done ethically are fairly limited.

To some degree I think our paper is green design fiction of the kind I criticised earlier. But I suspect that the ethical irritant effect of bioengineering humans might be enough to trigger some thinking about to what lengths we actually do want to go about fixing the environment. Sometimes downstream solutions might be more ethical and humane. But we should not imagine that our biological nature is exempt from being part of a potential solution.

My basic argument is that Doctorow's point that censorship is inseparable from surveillance likely goes both ways: surveillance systems tend to impose censorship.

During the panel I ended up arguing a bit with Rodney Brooks about the problem of AI security. He responded to an audience question with a "total Hollywood nonsense"-response very similar to others I have heard in other disciplines. And he does have a good point - most robotic systems are utterly safe and the things to worry about are entirely separate from robot uprisings. However, that rapid dismissal also tends to throw out the baby of AI safety with the bathwater: implementing proper AI safety has just as much problem with simplified story renditions (actual claim heard from someone who ought to have known better: "Oh, we can always use a computer virus to stop it!")

My biggest problem with the idea is just that eating whole fishes (even when small zebrafish) doesn't appeal to me. I prefer fillets. But that is just a matter of either skilled microsurgery, or making bigger fluorescent fishes.

Overall, I think making the biosphere more colourful is a lovely idea. Sure, there are glorious beetles, flowers and corals out there, but mammals are usually fairly drab.

and the Q&A:

Maybe well-trodden ground for me and most of my readers. Tomorrow I will give a different kind of talk, on commercialising robotics. Then back to transhumanism and the singularity for TEDx Vasastan, cognitive enhancement in Washington and the far future in Paris. I like talking.

It is based on a paper we are writing together that analyses how much harder the Fermi question (because it is not really a paradox, just a question with answers we tend to dislike/disagree on) becomes once you take modern ideas about self replication and exploratory engineering into account. The main finding is that intergalactic expansion is likely doable using local resources and a very high branching factor, and that makes the solar neighbourhood accessible to at least millions of times more potential alien civilizations. So either alien civilizations have to be even rarer than we think, they have to approach some non-visible behavioural attractor with very high fidelity, or they are here and hiding efficiently (in this case likely because the first expanding civilization used its probes to enforce some set of rules for everybody else).

My friends who happen to be members of the Enceladus Protection Society will be happy to know that no moons of Saturn were harmed in this analysis.