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A Mathematical Proof of the Singularity

Proof theory Disciplines ConceptThe author, inventor and futurist Ray Kurzweil has written about the technological singularity, a time when he predicts things will change so rapidly that he likens it to a mathematical singularity. In particular, he postulates the invention of an artificial intelligence capable of re-designing itself, which will inevitably lead to ever-faster progress.

To back up his theory, Kurzweil likes to show exponential curves representing the ever-faster development of computer processor performance, the price of transistors, DNA sequencing costs and such like.

But just how mathematically rigorous is this theory? Is it right to speak of a singularity occurring some time in the middle of the 21st century?

Students of cosmology know that genuine singularities are places where scary stuff happens. Especially naked singularities, which I cannot illustrate here for the sake of public decency. It is not a term to be treated lightly. In this article, I am going to prove that if Kurzweil’s prediction of strong AI is true, then a genuine singularity will occur, and in a surprising way, quite unlike popular thinking on the matter.

Let us suppose that there are an infinite number of inventions that people could ever invent. And by people I mean not just humans, but also alien civilisations, robots and any other class of inventive being. Let us number these inventions starting with 1 (a method for starting a fire, perhaps), then 2 (a method for avoiding burning your fingers), and so on.

Now let us assume, as Kurzweil’s exponential graphs indicate, that the time interval between inventions becomes ever smaller as more inventions are made. This is logical, because the more inventions that are available to an inventor, the easier it is to invent something new. Also, the smarter an AI becomes, the easier it is for it to invent an even-smarter version of itself. This is the key assumption in Kurzweil’s theory.

Now, here comes the tricky mathematical part. If the time between inventions becomes ever smaller as the number of inventions increases, then the total time taken to invent all possible inventions is finite. I will call this time T.

The proof of this is analogous to the resolution of Zeno’s paradox of Achilles and the tortoise. Zeno, the Greek philosopher, who believed that change is an illusion, outlined the following thought experiment. The great warrior Achilles is in a race against the tortoise and the tortoise is given a head start. If both runners start at the same time, then by the time Achilles reaches the starting point of the tortoise, the ponderous tortoise will have moved forward some distance less than Achilles, but will still be in front. By the time Achilles reaches the tortoise’s new position, the tortoise will have moved forward again. It will require an infinite number of steps for Achilles to catch up with the tortoise.

Of course, the paradox is easily resolved by realising that the time taken to complete each one of these infinite steps grows progressively shorter.  Simple calculus shows that Achilles will overtake the tortoise after a finite time T.

The situation is precisely analogous to the question of invention. Although the number of possible inventions is infinite, if the time taken between inventions becomes progressively shorter, then after a time T, everything that can be invented will have been invented. This includes all possible books, all conceivable works of art, an infinite number of cat memes and even the flying car. And the time T is not infinitely far in the future, but is finite and in principle calculable.

Such a time has been prophesied by various cultures and religions throughout history. It has been called Ragnarok, The Twilight of the Gods and the End of Days, but in this article I will call it “T time”.

So, after T time there will be literally nothing to do, as everything interesting will already have been done. This is the true singularity, and it is not a time when things are changing ever more rapidly, but when they have changed so much that no further change is possible. In a neatly ironic way, it is a time when Zeno’s belief that change is impossible will become true.

Some people may think of this as a utopia, but it is really just time for taking a quiet nap after T.


Steve-Morris-thumbAbout the author: Steve Morris studied Physics at the University of Oxford but discovered that writing about other people’s ideas is easier than having original ones yourself. He now writes about awesome technology at S21 and shares random thoughts at Blog Blogger Bloggest.

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  • SHaGGGz

    Kurzweil’s chart pertains to information processing capability, the increasing degree of which corresponds with certain inventions. It doesn’t imply, as I’ve understood it, the invention of “all possible inventions”, a quite dubious and ill-defined notion.
    Also, the absence of change isn’t utopia, it’s death.

  • Fishead62

    There’s an incorrect assumption: that the number of inventions that can be made is infinite. This is impossible. Give any finite amount of space, there is only a finite number configurations that atoms can have in that space. Any invention must occupy a finite amount of space. Since an invention must occupy a finite amount of space and a finite amount of space has a finite number of configurations available, then there is only a finite number of inventions that can be created.

  • Dear friend, just have in mind this article was actually published on April 1st! Happy fools day!!! 😉

  • SHaGGGz

    Touche. But I must admit that this notion that “inventions” are quantifiably definable is uncomfortably similar to Kurzweil’s similarly ill-defined notion of “next paradigm shift”.

  • Eray Özkural

    You sound like you don’t know who first came up with the mathematical theory of “singularity” like most of the folk who think Kurzweil or Vinge found the notion.

    1985 Ray Solomonoff, “The Time Scale of Artificial Intelligence: Reflections on Social Effects,” Human Systems Management, Vol. 5, pp. 149-153, 1985


    It’s a little more complicated than you think it is.

  • Again, today is April 1st – Fools’ Day, so don’t take it too seriously Eray!

  • CM Stewart

    I wonder how many doodads, doohickeys, gizmos, thingamabobs, thingamajigs, and whatchamacallits I can invent per day. Does it still count as an invention if it’s a “useless piece of crap”? 😉

  • Fishead62

    HFD. You got us!

  • Thanks for commenting, Fishead62. But is an invention a configuration of atoms in space or an idea in someone’s mind?

  • Thank you, CM. I’m sure that the overwhelming majority would be useless crap. Of course, if you market it effectively it would still have value.

  • Christian Holata

    I have argued many times on the AGI list that a sufficiently intelligent agent might end up with nothing to do, nothing new to learn, nothing new to discover, nothing new to surprise it, etc. … (with “nothing” here i mean sufficiently interesting enough)

    Running simulations of new universes might not be of sufficiently high enough complexity/interest to be entertaining? What if arriving at AGI/superintelligence renders the mere notion of existence obsolete? 😉

  • Sally Morem

    I can really relate to this analysis. I did some very simple figuring, assuming doubling of science and technology and data over a given period of time. And then halving that increment of time each time the doublings took place. I came up with doublings that would take place in months, weeks, days, hours, minutes, seconds… The Singularity would then jump towards infinities in an infinitesimal period of time.

    We can ask this potent question: Will we ever see the year 2050? We may not, not because we will die of old age before then and not because our civilization will be destroyed by human-made or natural disaster. But because the Singularity may occur before then and we may in some unimaginable way transcend time and space.

    Weird, but just possible.

  • Unless we’re getting into magic, an idea in someone’s mind IS a configuration of atoms (and/or other building blocks).

  • CM Stewart

    Now THAT would be awesome.

  • This in fact has already happened. We became “God,” and gave ourselves amnesia. ^_^

  • So it’ll happen at six o’clock… Good to know. I’ll try to keep my schedule empty and get plenty of biscuits for T time 🙂

    Now, taking the hypothesis seriously, you should add the possibility of a “doomsday” to your growth model because of self-destruction of humans, a meteorite, the discovery of a cosmological limit or any other joke the Universe plays on us before we invent the way to escape from it. Even if human extinction does not arrive, a periodic catastrophe destroying all scientific records -and happening before we invent the way to skip it- would take us back in time again and again, preventing the arrival of the T time. The name is great, though!

  • Thanks for your comments Nacho. You are right about the possibility of doomsday, and you could in fact argue that the invention of a doomsday device is inevitable given the hypothesis. That doesn’t necessarily mean that it will be used, fortunately.

  • Lu Lu

    It seems that many other civilizations in the universe had chose to transcend time and space. Thus we observe no alien transmissions, and hence the Fermi’s Paradox.

  • Lu Lu

    Also, the absence of change isn’t utopia, it’s death.
    To the autists and some old people, a unchanging world is favorable.
    Not utopia (in which people live in constant elation), but certainly much better than a changing one.

  • Lu Lu

    I believe no invention is useless. It takes a very intelligent being to fully exploit the usefulness of any object. Just like Vernor Vinge said: “To me, superhuman-ness is the essence of Singularity. Without it, we will find ourselves in a glot of technological riches, but can never properly absorb them.”
    Everything is useful. They just need a intelligent “user” to discover their potential.

  • Lu Lu

    How about inventions of infinite size, such as the infinite-lengthed Tipler Cylinder (proposed by physicist Frank J. Tipler, as a possible form of time machine), or the Cosmic Omega Point (involving crunching the whole universe, which may be infinite in physical size, down to a physical singularity)?

  • SHaGGGz

    But we aren’t really talking about a metaphorical notion of change that is open to human interpretation as to what constitutes enough qualitative change to qualify as a distinct new thing. “Invention” is such an impossibly broad term that it essentially covers any conceivable change in the arrangement of matter and energy in a system. So if we ever reached a stage where any such further change is impossible, it is indistinguishable from death, a cosmic death of the most total proportions.

  • Lu Lu

    I personally believe that one will never run out of new knowledges to gain.

    The more science you know, the more further you may know (derive, using mathematics and logics, from existing knowledge).

    As well, it is said by many scientists that the most fundamental science is mathematics.

    And, there are certainly infinite number of knowledge waiting to be explored in mathematics.

    For example, prime numbers are infinite, the digits in irrational numbers are infinite.

    Carl Sagan, in his novel Contact, talked about the possibility that there may be a “hidden message” inside Pi (the ratio of circumference to diameter).

    Mathematics-minded sentiences may spend an infinite (or at least very long) time exploring such subjects.
    Whether such AI can exist for infinite time and perform infinite calculation, depends on whether the Universe can be kept alive forever (whether Big Rip, Big Crunch and Heat Death can be avoided, and whether infinite amount of energy is available for calculation).
    Given that our universe apparently came from nothing, there may be ways to create “free-energy”. (or, according to Loop Quantum Theory, entropy is reversible, therefore we may recycle energy and keep the Universe and ourselves(intelligent lifeforms) alive forever).

  • Lu Lu

    The Google Nose is also quite entertaining.
    But, will computerized nose remain a fantasy for long?
    I doubt so. IBM is now working on “IBM 5 in 5”. In 5 years (before 2018), they will add the 5 senses of humans to smartphones. Smartphones can recognize (with their greater intelligence, thanks to Moore’s Law) what they see and hear via their cameras and microphones. As well, smartphones will possess a digital chemical detector (nose and tongue) that can detect chemicals in air and on surfaces. They will also be able to have tactile senses and simulate touch on their screens. It will be an interesting future.

  • Lu Lu
  • Lu Lu

    I meant the “humanistic” definition of unchanging. Thank you for pointing out, anyways.
    I think it would be great if we develop some kind of VR in which the world continues to repeat itself, and upload the statists (that means the people who prefer such a unchanging world) into it. To be truly repetitive, the repetition will involve even the mental function of the uploadees (so that they think the same thoughts repetitively).

  • Lu Lu

    We do have such devices: USA, Russia and China possess thermonuclear weapons that are totally capable of completely annihilating the Earth many times.

    Advanced nanotechnological weapons (replicators) will become reality in a few decades, adding to the possibility of our extermination.

    I believe that the effort to colonize space need more momentum.

    The wise advice of Carl Sagan should be remembered by all of us: “Any intelligent species either become space-faring, or go extinct.”

    Please show your support to space colonization now.

  • SHaGGGz

    So Nietzsche’s eternal recurrence, then? Sounds… pointless.

  • Lu Lu

    In some of Vinge’s novel, the Singularity is depicted as a “termination event” after which and trace of intelligent lifeforms and their artifacts disappears, and the Earth is restored to its pre-human state.
    I seriously doubt that this will be the case in real world. The post-Singularity world may look radically different and imcomprehensible to us pre-Singularity beings, but certainly a “restoration” can be ruled out.

  • Lu Lu

    I as well find it preposterous for us pre-Singularity humans to extrapolate what a post-Singularity being may think and do. Spreading life (that would mean any self-repairing and self-replicating structures, and not necessarily possess intelligence) to the Universe is our responsibility, and certainly the responsibility of greater-than-human superintelligences.
    If a person look up into the nightsky, he/she will find out that the majority of the volume of Universe is unoccupied emptiness. Life is not a Cosmic Norm. Life has to work hard, by itself, to spread itself out and maximize its own likelihood to survive.
    Now, as we humanity have the intelligence and the technology to spread life into the Universe, we should do so.

  • Lu Lu

    When cogntive enhancers, such as medicines and cybernetic implants, become advanced and widely available, many people will be benefited. Especially the scientists, technologists, engineers and mathematicians.

    The greater memory, greater thinking speed and better collaboration (via multisensory conference, or even technological telepathy), will cause the speed of innovation to increase. Ultimately, we will see a “Scientific Explosion” (similar to Vinge’s concept of “Intelligence Explosion”), during which major scientific, technological, engineering and mathematical breakthroughs are made every few hours (judging from that major inventions are made every ew days in today’s world, it is safe, or even over-conservative, to put the estimated timespan as the “hour” scale.)

  • Lu Lu

    As well, I wish to point out that any “finite volume” within our universe is an un-separatable part of the whole universe (which may be infinite). And our universe may exist within a “universe of universes”, also known as a multiverse.
    Such multiverse may be much larger than our universe, if not infinite in proportion.

    Thus, any objects that are finite in size can potentially interact with a infinite universe and the infinite other objects within it. And the number of possible interactions and the resulted new configuration of atoms are infinite.
    Regarding infinity, I guess the “Transfinite Numbers” theory of Georg Cantor, the 19th century mathematician, will be interesting for everyone.

  • LuLu you are shocking me not only with the number of your comments but also with the depth and the spectrum of your knowledge. Hat off to you my friend!

  • LuLu, thanks for your many comments on this article. I can see that you, like me, are fascinated by ideas like the infinite and the singularity. This article was intended to be playful and to provoke, and I am pleased to see that it has provoked many different responses from many people.

  • Interesting ongoing discussion of the Fermi paradox here:

  • Guest

    Thank you. I just wish to contribute some information to fellow people who are curious about the future and Singularity. The person who deserve most kudos remains you, Socrates,

  • I am willing to bet you can write a killer guest blog post if you wanted to… So, why don’t you give it a try?! 😉

  • Lu Lu

    I wish to share my feelings toward technology. To be honest, when I sit in front of a desk with pencil and paper, I won’t write anything half as good as the comment I post here. My eloquence (I wish to inform people who read my comments that English is my second language) is boosted when I use a computer and write with keyboard. Any vocabularies that I want to use can be quickly looked up on Google Dictionary (Google is very helpful, at least to me, thanks to its ability to complete words and correct spellings, plus giving facts and related links). Any facts I want to include as example can, again, be looked up on Internet (using Wikipedia or other websites).

    As well, I am taking free computer science courses on Coursera. The courses offered are of great quality, plus the fact they are free (not just these courses themselves are free, as well such spendings like transportations and on-campus housing fees become unnecessary), contains video lectures (which can be played when the professor and other student is absent, allows rewinding and even slowing down if you miss some parts, plus such effiecient features like subtitles), offers forums allowing collaboration with students from many different parts of the world.

    I read KurzweilAI on a daily basis, and can see that technology is indeed advancing very fast. The world is being improved in incredible ways (and, of course, I hope people can be aware of what wonderful luxury we are enjoying, and can try to contribute to furtherly improve it) Kurzweil’s many predictions are coming true, if one pays attention to daily news in science, technology, engineering and mathematics.

    I do feel that some great, rapture-like event will be faced by the humanity, and am awe-struck by it.

  • Lu Lu

    Michio Kaku, the American physicist and science writer, said: “All children are geniuses. The society crushes them.”

    I believe that it takes the same society to pay what it owe to these children. We should recognize that human beings are, before we make the first Artificial General Intelligence, the only General Intelligences on the Earth (and probably, if there are no aliens, the only General Intelligences in the Universe). All human beings have the responsibility to contribute to the Singularity, because they all carry the most advanced supercomputers in their a-few-cubic-centimeters skulls.
    The most relevant projects on simulating human brains (which is probably necessary for creating Artificial General Intelligence) are the USA’s BAMP ( Brain Activity Mapping Project, also known as BRAIN Initiative, http://en.wikipedia.org/wiki/BRAIN_Initiative , official website cannot be found) and the Europe Union’s HBP (Human Brain Poject http://www.humanbrainproject.eu/).

    Of course, regarding the USA’s BAMP, USA spend too little money on that project (only 100 million per year), compared to the 1 trillion
    and 700 billion dollars USA spend on the two recently wars (I am quoting http://www.kurzweilai.net/how-would-you-like-to-invest-in-immortality/comment-page-1#comment-127050 ).

    USA should spend less money on military and focus their resource on The Most Imporatant Project In The Universe (that is, reverse-engineering the Universe’s most-advanced supercomputer (the human brain)).
    If BAMP and HBP (or at least one of them) make great breakthroughs, the SIngularity can happen much earlier (than Kurzweil’s prediction of 2045).

    As well, there are rumors of that D-Wave Systems’s quantum computers, a model of which is recently sold to Lockheed-Martins (one of the world’s top company military vehicles and weapons) for performing computations on designing their new military aircrafts, are indeed true quantum computers. Quantum computers allow computational capacity that are far larger than any traditional computers can provide ( in other words, quantum computers break the Moore’s Law). If a quantum computer (not necessarily one made by D-Wave, but any one) is used in the BAMP and related projects, the likelihood of great breakthroughs will be even bigger.
    Don’t want to sound over-excited, but I believe that the Singulartiy will come sooner than most people anticipate.

  • Great, you can use the above strategy to do a guest blog article too 😉 Plus, if you do that you’d have the extra special benefit of Socrates for an editor 😉

  • Lu Lu

    I admit that I am not the originator/inventor of that idea (the idea of repeated existence). It came from the Orion’s Arm Science Fiction Worldbuilding Project ( http://www.orionsarm.com ).

    The original idea is known as “No-tee” or “Zero-tee”.


  • SHaGGGz

    It’s unclear whether these notees can be said to be alive or aware at all, having no discernible sensory experience, other than a looping “thought” that is not interacting with the observable universe in any way. Without change there is no possibility of personal growth or any of the things we purport to value about conscious experience.

  • CM Stewart

    “Everything is useful. They just need a intelligent ‘user’ to discover their potential.”

    Except on April Fools’ Day. 😉

  • Cute story, but completely unrealistic because it fails to take reality into consideration. The time between inventions doesn’t shrink indefinitely. Even using a huge number of super-fast super-brains, the time to develop an invention cannot drop below a certain minimum. Quantum mechanics and Amdahl’s law guarantee this. Therefore, the time to invent an infinite number of inventions using finite resources and a performance floor remains infinite. Your proof holds less water than a sieve. Sorry.

  • Dear Claudio, have in mind that this article was published on April 1st – so don’t take it so seriously and happy belayed April Fools’ day… 😉

  • Craig DeMartini

    Given the age and size of the universe, isn’t it probable that singularity has already occurred, and our existence is the result of someone else’s singularity?

  • What happens when the law of accelerating returns meets the law of diminishing returns? Is it the same as when an unstoppable force meets an immovable object?

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