Singularity review repost

The discussion of my repost on the silliness of generational tropes produced a surprising amount of agreement on the main point, then a lot of disagreement on the question of technological progress. So, I thought I’d continue reprising my greatest hits with this review of Kurzweil’s singularity post, which I put up in draft from at Crooked Timber and my own blog, producing lots of interesting discussion.  Again, seven years old, but I don’t see the need to change much – YMMV



I’ve finally received my copy of Ray Kurzweil’s Singularity, which was posted to me by its American publisher six weeks ago. The title refers to the claim that the process of technological change, notably in relation to computing, biotechnology and nanotechnology is accelerating to the point where it will produce a fundamental, and almost instantaneous, change in what it means to be human, arising from the combination of artificial intelligence and the use of biotechnology to re-engineer our bodies and minds.

The term Singularity, used to describe this event, apparently arose  in discussions between the mathematicians Stanislaw Ulam and John von Neumann. The idea of the Singularity was popularised in the 1980s and 1990s by mathematician and science fiction writer Vernor Vinge, and later by  Kurzweil, a prominent technologist and innovator.

Kurzweil’s argument has two main components. The first is the claim that continuing progress in microelectronics and computer software will, over the next few decades, realise the ultimate ambitions of the artificial intelligence (AI) program, producing computers that first equal, and then dramatically outstrip, the reasoning capacities of the human mind.

The key to all this Moore’s Law. This is the observation, first made by Intel CEO Gordon Moore in the mid-1960s, that computer processing power, roughly measured by the number of transistors on an integrated circuit, doubles every eighteen months to two years. Over the intervening forty years, the number of transistors on a typical integrated circuit has gone from less than a thousand to hundreds of millions.

The end of the expansion described in Moore’s Law has been predicted on many occasions, often with reference to seemingly unavoidable constraints dictated by the laws of physics. The constraint most commonly cited at present relates to the size of components. On present trends, transistors will be smaller than atoms within 15 years or so; this does not appear to be feasible, and current industry plans only extend to two or three more generations of progress, enough for perhaps a 100-fold increase in computing power.

Kurzweil dismisses such talk, arguing that just as transistors displaced vacuum tubes and integrated circuits displaced discrete transistors, new computing paradigms based on quantum effects will allow continued progress along the lines of Moores Law for most of this century, and well past the point at which computers are powerful enough to permit functional emulation of human brains. He concedes that there are limits on computing power, but argues that they do not represent an effective constraint on the advent of the Singularity. And Gordon Moore himself has noted that industry plans have never extended more than two or three generations ahead.

The second part of Kurzwei’s argument is based on three overlapping revolutions in genetics, nanotechnology and robotics. These revolutions are presented as being in full swing today but iin any case it is assumed that AI will smooth out any rough spots. Between them, Kurzweil argues, developments  in these three fields will transform medicine, science, finance and the economy.  Although all sorts of miracles are promised, the most dramatic is human immortality, achieved first through dramatic extensions in lifespans delivered by nanorobots in our bloodstreams and, more completely, by the ability to upload ourselves into infinitely-lived computers.

Not surprisingly, Kurzweil has attracted a passionate support from a small group of people and derision from a much larger group, particularly within the blogosphere which might have been expected to sympathise more with techno-utopianism. The wittiest critique was probably that of Daniel Davies at the Crooked Timber blog (disclosure: I’m also a blogger there) who modified Arthur C Clarke’s observation about technology and magic to produce the crushing ‘Any sufficiently advanced punditry is indistinguishable from bollocks’. Riffing off a link from Tyler Cowen on the expected value of extreme forecasts, and a trope popularised by Belle Waring, Davies outbid Kurzweil by predicting not only that all the Singularity predictions would come true, but that everyone would have a pony (“ Not just any old pony by the way, but a super technonanopony!”).

Before beginning my own critical appraisal of the Singularity idea, I’ll observe that the fact that I’ve been waiting so long for the book is significant in itself. If my great-grandfather had wanted to read a book newly-published in the US, he would have had to wait six weeks or so for the steamship to deliver the book. A century later, nothing has changed, unless I’m willing to shell out the price of the book again in air freight. On the other hand, whereas international communication for great-grandad consisted of the telegraph, anyone with an Internet connection can now download shelves full of books from all around the world in a matter of minutes and at a cost measured in cents rather than dollars.

This is part of a more general paradox, only partially recognised by the prophets of the Singularity. Those of us whose lives are centred on computers and the Internet have experienced recent decades as periods of unprecedently rapid technological advance. Yet outside this narrow sector the pace of technological change has slowed to a crawl, in some cases failing even to keep pace with growth in population. The average American spends more time in the car, just to cover the basic tasks of shopping and getting to work, than was needed a generation ago, and in many cases, travels more slowly.

Progress in many kinds of services has been even more limited, a point first made by US economist William Baumol in the 1960s. The time taken to give someone a haircut, or cook and serve a restaurant meal, for example, is as high as it was 100 years ago. As a result, the proportion of the workforce employed in providing services has risen consistently.

The advocates of the Singularity tend either to ignore these facts or to brush them aside. If there has been limited progress in transport, this doesn’t matter, since advances in nanotech, biotechn and infotech will make existing technological limits irrelevant. Taking transport as an example, if we can upload our brains into computers and transmit them at the speed of light, it doesn’t matter that cars are still slow. Similarly, transport of goods will be irrelevant since we can assemble whatever we want, wherever we want it, from raw atoms.

Much of this is unconvincing. Kurzweil lost me on biotech, for example, when he revealed that he had invented his own cure for middle age, involving the daily consumption of a vast range of pills and supplements, supposedly keeping his biological age at 40 for the last 15 years (the photo on the dustjacket is that of a man in his early 50s). In any case, nothing coming out of biotech in the last few decades has been remotely comparable to penicillin and the Pill for medical and social impact (a case could be made that ELISA screening of blood samples, was crucial in limiting the death toll from AIDS, but old-fashioned public health probably had a bigger impact.

As for nanotech, so far there has been a lot of hype but little real progress. This is masked by the fact that, now that the size of features in integrated circuits is measured in tens of nanometers, the term “nanotech” can be applied to what is, in essence, standard electronics, though pushed to extremes that would have been unimaginable a few decades ago.

Purists would confine the term “nanotechnology” to the kind of atomic-level engineering promoted by visionaries like Eric Drexler and earnestly discussed by journals like Wired. Two decades after Drexler wrote his influential PhD thesis, any products of such nanotechnology are about as visible to the naked eye as their subatomic components.

Only Kurzweil’s appeal to Moore’s Law seems worth taking seriously. There’s no sign that the rate of progress in computer technology is slowing down noticeably. A doubling time of two years for chip speed, memory capacity and so on implies a thousand-fold increase over twenty years. There are two very different things this could mean. One is that computers in twenty years time will do mostly the same things as at present, but very fast and at almost zero cost. The other is that digital technologies will displace analog for a steadily growing proportion of productive activity, in both the economy and the household sector, as has already happened with communications, photography, music and so on. Once that transition is made these sectors share the rapid growth of the computer sector.

In the first case, the contribution of computer technology to economic growth gradually declines to zero, as computing services become an effectively free good, and the rest of the economy continues as usual. Since productivity growth outside the sectors affected by computers has been slowing down for decades, the likely outcome is something close to a stationary equilibrium for the economy as a whole.

But in the second case, the rate of growth for a steadily expanding proportion of the economy accelerates to the pace dictated by Moore’s Law.  Again, communications provides an illustration – after decades of steady productivity growth at 4 or 5 per cent a year, the rate of technical progress jumped to 70 per cent a year around 1990, at least for those types of communication that can be digitized (the move from 2400-baud modems to megabit broadband in the space of 15 years illustrates this).  A generalized Moore’s law might not exactly produce Kurzweil’s singularity, but a few years of growth at 70 per cent a year would make most current economic calculations irrelevant.

One way of expressing this dichotomy is in terms of the aggregate elasticity of demand for computation. If it’s greater than one, the share of computing in the economy, expressed in value terms, rises steadily as computing gets cheaper. If it’s less than one, the share falls. It’s only if the elasticity is very close to one that we continue on the path of the last couple of decades, with continuing growth at a rate of around 3 per cent.

This kind of result, where only a single value of a key parameter is consistent with stable growth, is sometimes called a knife-edge. Reasoning like this can be tricky – maybe there are good reasons why the elasticity of demand for computation should be very close to one. One reason this might be so is if most problems eventually reach a point, similar to that of weather forecasting, where linear improvements in performance require exponential growth in computation.

If the solution to a problem involves components that are exponential (or worse) in complexity, initial progress may be rapid as linear or polynomial components of the problem are solved, but progress with the exponential component will at best be linear, even if the cost of computation is itself declining exponentially.

So far it seems as if the elasticity of demand for computation is a bit greater than one, but not a lot. The share of IT in total investment has risen significantly, but the share of the economy driven primarily by IT remains small. In addition, non-economic activity like blogging has expanded rapidly, but also remains small. The whole thing could easily bog down in an economy-wide version of ‘Intel giveth and Microsoft taketh away’.

In summary, I’m unconvinced that the Singularity is near. But unlike the majority of critics of Kurzweil’s argument, I’m not prepared to rule out the possibility that information technology will spread through large sectors of the economy, producing unprecedently rapid economic growth. Even a small probability of such an outcome would make a big difference to the expected returns to investments, and would be worth planning for. So it’s certainly worthwhile reading Kurzweil’s book and taking the time to consider his argument.
At this stage, though, the Singularity is still best considered as science fiction. If you really want to get a feel for the kind of thinking that drives the Singularity, read Ian McDonald’s River of Gods or, better still, Charles Stross’ Accelerando.

Kurzweil, Ray (2005), The Singularity is Near, Viking, New York.

Posted via email from John’s posterous

40 thoughts on “Singularity review repost

  1. See note at Crooked Timber:
    The *speed* claim just isn’t true and hasn’t been for a decade.
    Individual CPUs have not been getting much faster as quickly, but we get more of them per chip. Intel got to ~3GHz around 2002. We are not at 3 * 2^5 = 96GHz or anything like that.
    A current Intel Core i7 gets to 3.7GHz, but most run less.

  2. @John Mashey

    Agreed Re: speed. Moore’s law ended, the wall was hit. But Moore’s law doesn’t need to make anymore contributions for the machine learning/AI singularity to happen don’t you think?

    Big data, that is, big datamining, is one of several far more important things feeding into that singularity. (Which is why certain companies have been grabbing data left right and centre and would like you to put your data in their ‘cloud’.) Incredible “intelligence ” is embodied in humanity’s accumulated data, and that is very probably a very important part of the ultimate creation of a real thinking machine that will leave us behind.

    Norvig, for one, has explained how important big (massive) data is and the tradeoffs between more data and cleverer algorithms. With lots of data you can produce something amasingly clever with algorithms that certainly are not.

    I was hardly the only person certain in the early 1980s that chess programs would be developed stronger than any human. What really amazed me was not so much how quickly it happened but that it had been achieved by algorithms none too smart, and with programs that really have less ‘intellect’ than a toe nail. (In the early ’80s computer programs played at a level maybe slightly cleverer than a toenail but before 2000 anyone could buy a program for maybe $40 that would play better than a human.)

  3. Meant any human rather than “a” human.

    Chess program got there through moore’s law, tuning , and simple algorithms (simply in embodied intellect, although clever in terms of the human accomplishment in thinking them up).

    Interestingly, Herbert Simon is one of the ones credited with dreaming up the alpha-beta algorithm.

  4. John mashey beat me to it. Data and data transformation are the new black. I see it in several eminently practical science areas. It is one reason I am passionate about the nbn.

  5. There I was, believing it is the financial securities innovators that have the expertise in producing a singularity (‘the market freezes’ in the language of practitioners).

  6. Don’t forget advances in 3d printing that threaten to completely replace manufacturing.

    Projects like google glass could be huge for productivity. Imagine changing a spark plug by following on screen prompts instead of having to learn before hand. We will have access to the entire knowledge of the world as an integrated experience. The act of looking something up could become a thing of the past.

    The book analogy is also flawed, technology has made dead tree books obsolete.

  7. Parallelism is another factor that’s enabling much more complex tasks to be undertaken. Furthermore, data repositories such as google maintain admit new solution methods to old problems. Think of place recognition via image matching, something once unimaginable, but now part of the augmented reality landscape.

    These massive dynamic databases also provide solution by lookup, where someone has solved a computational problem which then becomes available to others via lookup.

    Basically, Moore is only one part of the computing picture, and certainly not the bottleneck it once may have been.

  8. Although I don’t have much doubt about the singularity happening on the timetable those in the know seem to think, that is without the human world falling to bits first, the without possibly can’t be dismissed.
    The book though doesn’t sound worth the effort to read. Sounds a bit ‘cargo cult’. Imagine it’ll be a best seller!

  9. I don’t think it will be too difficult to develop strong AI. But this is not because I am confident we will continue to make technological break throughs, it’s more because I don’t think humans are as intelligent as many of us think we are. Some people see the absurd difficulty we have in trying to make a robot do something like fold clothes and assume if we can’t even get it to do that properly we must be a vast distance from making a machine that can discuss Shakespeare. But for the human brain, folding clothes and discussing Shakespeare are roughly comparable in difficulty. (Personally I find folding clothes more difficult. It’s much harder to fake being good at folding clothes than it is to fake being good at Shakespeare.) Currently computers are lousy at understanding normal speech, but they are definitely improving. And once they can understand normal speech, well that’s it, that’s strong AI. There is nothing else that needs to be done. And it doesn’t matter how it does it, whether it is “cheating” or not. Animal brains cheat all the time. So will an improved version of Siri or Watson be strong AI? Yeah, it will. Of course, there will be no end of people who will say it isn’t, and many will even point out that their mobile phones agree with them.

  10. > I’m not prepared to rule out the possibility that information technology will spread through large sectors of the economy, producing unprecedently rapid economic growth.

    The evidence so far is that the spread of information technology tends to cannibalise other economic activity. The latest anecdote to be added to the pile is from a piece in The Atlantic Cities, “Young People Aren’t Buying Cars Because They’re Buying Smart Phones Instead”.

    The estimate is that 2 million car sales per year (out of 17 million) in the USA are not happening because of changes enabled by the technology. Even if it’s only a quarter true, the global loss of car sales is about equal to Apple’s market capitalisation (PPP). The loss in construction would dwarf that figure.

    There are inherent limits to the demand for most services – the chief exceptions being the demand for medical attention during the last six months of life, and, possibly, more kinds of insurance. It’s difficult to see how rapid productivity improvement in services can bring about rapid economic growth.

  11. Ho-hum – yet another riff on the old ‘dream of Enlightenment Reason’. In this case apparently its not the secular utopia produced by the inevitable historical march of scientific reason but the instantaneous transformation of humans replicating the logics of modern technology?

    Sadly for those espousing such apocalytpic visions the affective aspect of human nature has remained highly resilient and resistant to such scientific logics, a situation which appears unlikely to change in the forseeable future.

  12. @Ronald Brak

    Good points. Doesn’t matter if its fake or not, and our exalted place in the animal kingdom is not nearly as, in a class by itself, as we comfortably delude ourselves. (And that can also be said for the differences between smarter specimens relativities to the average, also, I’d speculate.) Apparently Bonobos and Chimps are both much further into the stone age, than previous suspected (see PNAS).

  13. @Greg vP

    My maths could be wrong, but I would have thought that if the quantity of human labour used per unit of output became vanishingly small in a perfectly competitive economy, then (human) labour productivity would tend to infinity. (Didn’t do any maths, actually.but if someone cared to set up a little model, with reasonable assumptions I imagine it would all drop out nicely. If not so, no doubt I’ll be corrected!)

  14. In simpler English, as humans are less required, in the economy, to produce output, labour productivity measured as output divided by labour input goes up, and up, and up. Of course, following Keynes famous essay, we may just lounge around (working vanishingly few hours) instead of drowning outselfs in a flood of goods and services. But the truth will be what really happens will depend on who owns the output. Will it be a small number of super opulent pigs while the rest are not greatly better off, or much more equally.

  15. @Freelander
    Yes, productivity should increase with increases in technology, process refinement, efficiency evolution, knowledge and skills capacity increase etc etc. The reality is that the highest productivity will follow the cheapest labor and lowest running costs eg regulatory, rental, workplace relations etc. That’s the reality of free market economics.

  16. I would argue, and in fact, already have argued, that we have already been through one singularity already.

    The point of using the term ‘singularity’ is that we cannot know what is on the other side, no prediction is of any value. Well, been there, done that.

  17. This is a very interesting subject. Personally, I think that a “singularity” is very close. Not close in the terms of what I am having for breakfast next Wednesday, but in terms of human technological development.

    On Computer power, our processors are still essentially two dimensional devices, and largely based on electron flow. Yet to be exploited is the three dimensional construction and the photon operating medium. There are a number of orders of magnitude yet available. On the street computers may well not be keeping pace with what is possible. I suspect that Moore’s Law is intact technologically, but not fully exploited. Why? because there is insufficient need to for everyday processors to be operating on the boundaries of the technology. Processor development these days is more about fully exploiting the current level of electronics performance and spreading that out into the full field of market opportunities rather than make one shallow field of consumption unnecessarily over powerful.

    As for enhancing human performance I think that the guiding principle would be that there has to be a serviceable gain for there to be the reason to do this. We already have a massive range of operational performance with humans now, without enhancement from the most underdeveloped peoples to the most exotic scientific and commercial performers, but has this in fact improved overall human outcomes as we run headlong into resource depletion and Climate Instability.? The answer is that the outcome is different, not necessarily better.

    Personally, I believe that once ultra dense photon based crystal processor blocks are developed it will be possible to transfer ones consciousness to a crystal at death so that ones presence can be preserved. To understand the limitations of this though it is important to realize that a presence without sensation would be a very hollow experience. When I was musing over this prospect I finally understood that our memories are not in “movie” form, they are in the form of patterns, edges, attributes and sequence, and made tangible with the saturation of our real time sensory connection. When we construct a memory we actually rebuild that memory in our mind very much the same way that modern Computer Aided Design packages work. It is actually pretty cool.

    Ant way singularity certainly possible. Probable? As Syndrome said “once everyone is super, no-one will be”. Then there is the threat of domination. So maybe, just in case.

  18. 1) Oops, I posted on AI and checkers/chess @ Crooked Timber.
    [I worked at Bell Labs, saw Belle, I still have the very-limited-edition T-shirt (had a blue Bell symbol tweaked to look like the chess queen symbol. I got Ken Thompson to do the oral history on Belle for the Computer History Museum. The world-champ checkers program was done on MIPS and then SGI gear.]

    2) BIG DATA: people are scurrying around right now trying figure out where the term came from. See Origins in this.

  19. Moore’s “law” was never about clock speed (ie: the Mhz/Ghz number), it was about transistor density. Moore observed that the number of transistors in a chip doubles approximately every four years. As can be seen from this graph:
    his observation (/prediction) has been remarkably consistent for forty years.

    Further, transistor count is roughly proportional to *performance*. Not the clock speed, but how much “work” a CPU can do. In that regard, processors are still on track – the fastest CPUs today are easily twice as fast as those from ~4 years ago (even if they have the same clock speed).

  20. @Diewert

    Well we know people don’t use ‘Erwin’ for you first name, because that guy is a real gentleman, and would never say that even if it were merited (which, of course, it is not). I wonder if this sort of comment, from unknown quarters, is, on the internet, called a ‘drive by’?

  21. @drsmithy

    Yes it was and that is the most important thing that it about!

    Let me explain it too you. Thinking, for the most part, and computer programs on single CPU s operate sequentially. Having a single processor that operates 100 times faster is much better than having 100 processors at the initial speed. The increase in processor speed is what was important. Now the wall has been hit. New parallel processing ideas will be useful. But really Moore’s law, the important version of it, the one that people in the know used is history. That version was speed doubles every eighteen months and although also called Moore’s law, was based on the same sort of miniturisation reasons, was first articulated by someone else at Intelligence.

  22. @drsmithy

    Were you in “Lost in Space”? First name Zachary?

    Just wondering where you acquired your computer expertise. Conversations with the robot?

  23. I wrote Intell! How the h ell did it become intelligence. Google, you haven’t been hirering programmers from MS have you?

    Shame on you!

    You know there all cargo-cult!

  24. Oh my apologies. I didn’t see we were dealing with a Wikipedia scholar.

    I bow to your superior erudition.

  25. @John Mashey

    I’m very impressed. What a great institution! And all the great stuff that came out of the place! I’m referring, of course, to the intellectual output. The ideas that came out of the place. And those big names in computer chess programming and an the progression of that work, with some baton passes through eventually to deep blue.

  26. John Mashey,

    If you are into thought games, I was musing as to how one might create a approximation of a cubic processor with essentially 2D architecture. What I came up with was a standard processor wafer capped off both sides with a doublet (as they do with opal wafers) to produce processor “marbles”. The doublet caps would contain supporting devices. The surface of the “marble” would be a rubber coating with radiating conductive polymer fibrils (similar to that used to connect lcd display panels. The surface would also have a number of catalyst sites of two types for power energy collect from a chemical soup. To make the processor active you place a bunch of them in a container, add the chemical soup which reacts at the catalyst sites to supply or take away electrons, and away it goes. The processors would first look at their companions, establish a hierachy, then look for a purpose. IO would be by a number of marbles with optic fibre tails (sperm) through which information (sensory or command) would be transmitted. The processor cluster could be retasked easily with the addition or subtraction of specific purpose processor marbles, memory marbles, or IO marbles. In this way one could build a huge array of performance functions in just minutes. The chemical circulating chemical soup provides both power and temperature stability. The users manual would need to advise “for best results avoid losing any marbles”.

    With such a structure it would be possible to explore neural networks in a more meaning full way.

  27. Thinking, for the most part, and computer programs on single CPU s operate sequentially. Having a single processor that operates 100 times faster is much better than having 100 processors at the initial speed.

    Well, that depends a great deal on what you’re trying to do.

    The increase in processor speed is what was important. Now the wall has been hit. New parallel processing ideas will be useful.

    Firstly, how are you measuring “speed” ?
    Secondly, I’m pretty sure parallel processing will be a bit more than “useful”. You mentioned “big data”, which is a field rather intimately intertwined with parallel processing. Further, it’s been nearly fifteen years since I did my CS degree, and I’ve never taken a huge interest in the field anyway, but it was my understanding that highly efficient parallel processing was considered essential to emulating the brain’s parallelism and therefore workable AI.

    But really Moore’s law, the important version of it, the one that people in the know used is history. That version was speed doubles every eighteen months and although also called Moore’s law, was based on the same sort of miniturisation reasons, was first articulated by someone else at Intelligence.
    I have never heard “Moore’s Law” used in academic and formal contexts to mean anything other than transistor density, though it is often (heck, pretty much always) (mis-)used by people writing for consumer-level computing publications to refer to “speed” (ie: performance), largely because the correlation has been so strong. Who are these “people in the know” you’re talking about ?

  28. When it appears fruitful to do so, chips with dedicated neural net structures will be manufactured. Given that there would be a non trivial cost and the ordinary chip can be programed to get the same output some worthwhile benefit is needed. Dedicated chips would be faster.So when going faster is justifie, given the cost, they’ll happen.

  29. When you can make the transistors and associated things smaller on a chip the really important gain is that everything is closer together so a higher clock speed can be used. (Didn’t say this for Zach’s benefit) Just for those who don’t require what being able to do sequential operations faster, increasing speed , might mean.

  30. @Donald Oats

    Second your thoughts on the NBN! That infrastructure should provide some great opportunities. Even in its absence we have had some great entrepreneurial activity like Kaggle. Kaggle is now in silicon valley and even with the NBN that would have been the thing to have do. But they would probably have retained more of an Australian link after the move. (Or that’s my guess.)

  31. When you can make the transistors and associated things smaller on a chip the really important gain is that everything is closer together so a higher clock speed can be used.

    I guess that’s why clock speeds have been basically unchanged for a decade even though transistor densities have consistently doubled as predicted.

    Still wondering what your measure of “speed” is, as well.

  32. Where is a singularity when you need one? (That is, read to swallow up those everyday mundane annoyances) Why is it that so many feel entitled to demand me to do their thinking for them? Surely minimial manners and they would make an offer of moolah? But no. Apparently I owe them some sort of moral debt that I ought to work off. Gee! Enough to have me convert to Ayn Rand’s Godless religion.

  33. Is it interesting that you can put twice as many ‘things’ on chip while making those things halve the size or the same size. Does size matter? Well Zach, if you were told it doesn’t, she was trying to spare your feelings.

  34. JQ’s blog covers many topics, but I am afraid it is not a good venue for discussions of chip technology and architecture. If people *really* want that, I invite them to come to the 25th anniversary Hot Chips conference next year. We just finished Hot Chips #24, so people can see the sorts of topics discussed by people who actually do this for a living.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s