Tag: Stephen Hawking

Gates’ Law: How Progress Compounds and Why It Matters


It’s unclear exactly who first made that statement, when they said it, or how it was phrased. The most probable source is Roy Amara, a Stanford computer scientist. In the 1960s, Amara told colleagues that he believed that “we overestimate the impact of technology in the short-term and underestimate the effect in the long run.” For this reason, variations on that phrase are often known as Amara’s Law. However, Bill Gates made a similar statement (possibly paraphrasing Amara), so it’s also known as Gates’s Law.

You may have seen the same phrase attributed to Arthur C. Clarke, Tony Robbins, or Peter Drucker. There’s a good reason why Amara’s words have been appropriated by so many thinkers—they apply to so much more than technology. Almost universally, we tend to overestimate what can happen in the short term and underestimate what can happen in the long term.

Thinking about the future does not require endless hyperbole or even forecasting, which is usually pointless anyway. Instead, there are patterns we can identify if we take a long-term perspective.

Let’s look at what Bill Gates meant and why it matters.

Moore’s Law

Gates’s Law is often mentioned in conjunction with Moore’s Law. This is generally quoted as some variant of “the number of transistors on an inch of silicon doubles every eighteen months.” However, calling it Moore’s Law is misleading—at least if you think of laws as invariant. It’s more of an observation of a historical trend.

When Gordon Moore, co-founder of Fairchild Semiconductor and Intel, noticed in 1965 that the number of semiconductors on a chip doubled every year, he was not predicting that would continue in perpetuity. Indeed, Moore revised the doubling time to two years a decade later. But the world latched onto his words. Moore’s Law has been variously treated as a target, a limit, a self-fulfilling prophecy and a physical law as certain as the laws of thermodynamics.

Moore’s Law is now considered to be outdated, after holding true for several decades. That doesn’t mean the concept has gone anywhere. Moore’s Law is often regarded as a general principle in technological development. Certain performance metrics have a defined doubling time, the opposite of a half-life.

Why is Moore’s Law related to Amara’s Law?

Exponential growth is a concept we struggle to conceptualize. As University of Colorado physics professor Albert Allen Bartlett famously put it, “The greatest shortcoming of the human race is our inability to understand the exponential function.”

When we talk about Moore’s Law, we easily underestimate what happens when a value keeps doubling. Sure, it’s not that hard to imagine your laptop getting twice as fast in a year, for instance. Where it gets tricky is when we try to imagine what that means on a longer timescale. What does that mean for your laptop in 10 years? There is a reason your iPhone has more processing power than the first space shuttle.

One of the best illustrations of exponential growth is the legend about a peasant and the emperor of China. In the story, the peasant (sometimes said to be the inventor of chess), visits the emperor with a seemingly modest request: a chessboard with one grain of rice on the first square, then two on the second, four on the third and so on, doubling each time. The emperor agreed to this idiosyncratic request and ordered his men to start counting out rice grains.


— Robert Anton Wilson

If you haven’t heard this story before, it might seem like the peasant would end up with, at best, enough rice to feed their family that evening. In reality, the request was impossible to fulfil. Doubling one grain 63 times (the number of squares on a chessboard, minus the first one that only held one grain) would mean the emperor had to give the peasant over 18 million trillion grains of rice. To grow just half of that amount, he would have needed to drain the oceans and convert every bit of land on this planet into rice fields. And that’s for half.

In his essay “The Law of Accelerating Returns,” author and inventor Ray Kurzweil uses this story to show how we misunderstand the meaning of exponential growth in technology. For the first few squares, the growth was inconsequential, especially in the eyes of an emperor. It was only once they reached the halfway point that the rate began to snowball dramatically. (It’s no coincidence that Warren Buffett’s authorized biography is called The Snowball, and few people understand exponential growth better then Warren Buffett). It just so happens that by Kurzweil’s estimation, we’re at that inflection point in computing. Since the creation of the first computers, computation power has doubled roughly 32 times. We may underestimate the long-term impact because the idea of this continued doubling is so tricky to imagine.

The Technology Hype Cycle

To understand how this plays out, let’s take a look at the cycle innovations go through after their invention. Known as the Gartner hype cycle, it primarily concerns our perception of technology—not its actual value in our lives.

Hype cycles are obvious in hindsight, but fiendishly difficult to spot while they are happening. It’s important to bear in mind that this model is one way of looking at reality and is not a prediction or a template. Sometimes a step gets missed, sometimes there is a substantial gap between steps, sometimes a step is deceptive.

The hype cycle happens like this:

  • New technology: The media picks up on the existence of a new technology which may not exist in a usable form yet. Nonetheless, the publicity leads to significant interest. At this point, people working on research and development are probably not making any money from it. Lots of mistakes are made. In Everett Rogers’s diffusion of innovations theory, this is known as the innovation stage. If it seems like something new will have a dramatic payoff, it probably won’t last. If it seems we have found the perfect use for a brand-new technology, we may be wrong.
  • The peak of inflated expectations: A few well-publicized success stories lead to inflated expectations. Hype builds and new companies pop up to anticipate the demand. There may be a burst of funding for research and development. Scammers looking to make a quick buck may move into the area. Rogers calls this the syndication stage. It’s here that we overestimate the future applications and impact of the technology.
  • The trough of disillusionment: Prominent failures or a lack of progress break through the hype and lead to disillusionment. People become pessimistic about the technology’s potential and mostly lose interest. Reports of scams may contribute to this, as the media uses this as a reason to describe the technology as a fraud. If it seems like a new technology is dying, it may just be that its public perception has changed and the technology itself is still developing. Hype does not correlate directly with functionality.
  • The slope of enlightenment: As time passes, people continue to improve the technology and find better uses for it. Eventually, it’s clear how it can improve our lives and mainstream adoption begins. Mechanisms for preventing scams or lawbreaking emerge.
  • The plateau of productivity: The technology becomes mainstream. Development slows. It becomes part of our lives and ceases to seem novel. Those who move into the now saturated market tend to struggle, as a few dominant players take the lion’s share of the available profits. Rogers calls this the diffusion stage.

When we are cresting the peak of inflated expectations, we imagine that the new development will transform our lives within months. In the depths of the trough of disillusionment, we don’t expect it to get anywhere, even allowing years for it to improve. We typically fail to anticipate the significance of the plateau of productivity, even if it exceeds our initial expectations.

Smart people can usually see through the initial hype. But only a handful of people can—through foresight, stubbornness or perhaps pure luck—see through the trough of disillusionment. Most of the initial sceptics feel vindicated by the dramatic drop in interest and expect the innovation to disappear. It takes far greater expertise to support an unpopular technology than to deride a popular one.

Correctly spotting the cycle as it unfolds can be immensely profitable. Misreading it can be devastating. First movers in a new area often struggle to survive the trough, even if they are the ones who do the essential research and development. We tend to assume current trends will continue, so we expect sustained growth during the peak and expect linear decline during the trough.

If we are trying to assess the future impact of a new technology, we need to separate its true value from its public perception. When something is new, the mainstream hype is likely to be more noise than signal. After all, the peak of inflated expectations often happens before the technology is available in a usable form. It’s almost always before the public has access to it. Hype serves a real purpose in the early days: it draws interest, secures funding, attracts people with the right talents to move things forward and generates new ideas. Not all hype is equally important, because not all opinions are equally important. If there’s intense interest within a niche group with relevant expertise, that’s more telling than a general enthusiasm.

The hype cycle doesn’t just happen with technology. It plays out all over the place, and we’re usually fooled by it. Discrepancies between our short- and long-term estimates of achievement are everywhere. Consider the following situations. They’re hypothetical, but similar situations are common.

  • A musician releases an acclaimed debut album which creates enormous interest in their work. When their second album proves disappointing (or never materializes), most people lose interest. Over time, the performer develops a loyal, sustained following of people who accurately assess the merits of their music, not the hype.
  • A promising new pharmaceutical receives considerable attention—until it becomes apparent that there are unexpected side effects, or it isn’t as powerful as expected. With time, clinical trials find alternate uses which may prove even more beneficial. For example, a side effect could be helpful for another use. It’s estimated that over 20% of pharmaceuticals are prescribed for a different purpose than they were initially approved for, with that figure rising as high as 60% in some areas.
  • A propitious start-up receives an inflated valuation after a run of positive media attention. Its founders are lauded and extensively profiled and investors race to get involved. Then there’s an obvious failure—perhaps due to the overconfidence caused by hype—or early products fall flat or take too long to create. Interest wanes. The media gleefully dissects the company’s apparent demise. But the product continues to improve and ultimately becomes a part of our everyday lives.

In the short run the world is a voting machine affected by whims and marketing. In the long run it’s a weighing machine were quality and product matter.

The Adjacent Possible

Now that we know how Amara’s Law plays out in real life, the next question is: why does this happen? Why does technology grow in complexity at an exponential rate? And why don’t we see it coming?

One explanation is what Stuart Kauffman describes as “the adjacent possible.” Each new innovation adds to the number of achievable possible (future) innovations. It opens up adjacent possibilities which didn’t exist before, because better tools can be used to make even better tools.

Humanity is about expanding the realm of the possible. Discovering fire meant our ancestors could use the heat to soften or harden materials and make better tools. Inventing the wheel meant the ability to move resources around, which meant new possibilities such as the construction of more advanced buildings using materials from other areas. Domesticating animals meant a way to pull wheeled vehicles with less effort, meaning heavier loads, greater distances and more advanced construction. The invention of writing led to new ways of recording, sharing and developing knowledge which could then foster further innovation. The internet continues to give us countless new opportunities for innovation. Anyone with a new idea can access endless free information, find supporters, discuss their ideas and obtain resources. New doors to the adjacent possible open every day as we find different uses for technology.

“We like to think of our ideas as $40,000 incubators shipped direct from the factory, but in reality, they’ve been cobbled together with spare parts that happened to be sitting in the garage.”

— Steven Johnson, Where Good Ideas Come From

Take the case of GPS, an invention that was itself built out of the debris of its predecessors. In recent years, GPS has opened up new possibilities that didn’t exist before. The system was developed by the US government for military usage. In the 1980s, they decided to start allowing other organizations and individuals to use it. Civilian access to GPS gave us new options. Since then, it has led to numerous innovations that incorporate the system into old ideas: self-driving cars, mobile phone tracking (very useful for solving crime or finding people in emergency situations), tectonic plate trackers that help predict earthquakes, personal navigation systems, self-navigating robots and many others. None of these would have been possible without some sort of global positioning system. With the invention of GPS, human innovation sped up a little more.

Steven Johnson gives one example of how this happens in Where Good Ideas Come From. In 2008, MIT professor Timothy Presto visited a hospital in Indonesia and found that all eight of the incubators for newborn babies were broken. The incubators had been donated to the hospital by relief organizations, but the staff didn’t know how to fix them. Plus, the incubators were poorly suited to the humid climate and the repair instructions only came in English. Presto realized that donating medical equipment was pointless if local people couldn’t fix it. He and his team began working to design an incubator that could save the lives of babies for a lot longer than a couple of months.

Instead of continuing to tweak existing designs, Presto and his team devised a completely new incubator that used car parts. While the local people didn’t know how to fix an incubator, they were extremely adept at keeping their cars working no matter what. Named the NeoNurture, it used headlights for warmth, dashboard fans for ventilation, and a motorcycle battery for power. Hospital staff just needed to find someone who was good with cars to fix it—the principles were the same.

Even more telling is the origin of the incubators Presto and his team reconceptualized. The first incubator for newborn babies was designed by Stephane Tarnier in the late 19th century. While visiting a zoo on his day off, Tarnier noted that newborn chicks were kept in heated boxes. It’s not a big leap to imagine that the issue of infant mortality was permanently on his mind. Tarnier was an obstetrician, working at a time when the infant mortality rate for premature babies was about 66%. He must have been eager to try anything that could reduce that figure and its emotional toll. Tarnier’s rudimentary incubator immediately halved that mortality rate. The technology was right there, in the zoo. It just took someone to connect the dots and realize human babies aren’t that different from chicken babies.

Johnson explains the significance of this: “Good ideas are like the NeoNurture device. They are, inevitably, constrained by the parts and skills that surround them…ideas are works of bricolage; they’re built out of that detritus.” Tarnier could invent the incubator only because someone else had already invented a similar device. Presto and his team could only invent the NeoNurture because Tarnier had come up with the incubator in the first place.

This happens in our lives as well. If you learn a new skill, the number of skills you could potentially learn increases because some elements may be transferable. If you are introduced to a new person, the number of people you could meet grows, because they may introduce you to others. If you start learning a language, native speakers may be more willing to have conversations with you in it, meaning you can get a broader understanding. If you read a new book, you may find it easier to read other books by linking together the information in them. The list is endless. We can’t imagine what we’re capable of achieving in ten years because we forget about the adjacent possibilities that will emerge.

Accelerating Change

The adjacent possible has been expanding ever since the first person picked up a stone and started shaping it into a tool. Just look at what written and oral forms of communication made possible—no longer did each generation have to learn everything from scratch. Suddenly we could build upon what had come before us.

Some (annoying) people claim that there’s nothing new left. There are no new ideas to be had, no new creations to invent, no new options to explore. In fact, the opposite is true. Innovation is a non-zero-sum game. A crowded market actually means more opportunities to create something new than a barren one. Technology is a feedback loop. The creation of something new begets the creation of something even newer and so on.

Progress is exponential, not linear. So we overestimate the impact of a new technology during the early days when it is just finding its feet, then underestimate its impact in a decade or so when its full uses are emerging. As old limits and constraints melt away, our options explode. The exponential growth of technology is known as accelerating change. It’s a common belief among experts that the rate of change is speeding up and society will change dramatically alongside it.

“Ideas borrow, blend, subvert, develop and bounce off other ideas.”

— John Hegarty, Hegarty On Creativity

In 1999, author and inventor Ray Kurzweil posited the Law of Accelerating Change — that evolutionary systems develop at an exponential rate. While this is most obvious for technology, Kurzweil hypothesized that the principle is relevant in numerous other areas. Moore’s Law, initially referring only to semiconductors, has wider implications.

In an essay on the topic, he writes:


Progress is tricky to predict or even to notice as it happens. It’s hard to notice things in a system that we are part of. And it’s hard to notice incremental change because it lacks stark contrast. The current pace of change is our norm and we adjust to it. In hindsight, we can see how Amara’s Law plays out.

Look at where the internet was just twenty years ago. A report from the Pew Research Center shows us how change compounds. In 1998, a mere 41% of Americans used the internet at all—and the report expresses surprise that the users were beginning to include “people without college training, those with modest incomes, and women.” Less than a third of users had bought something online, email was predominantly just for work, and only a third of users looked at online news at least once per week. That’s a third of the 41% using the internet by the way, not of the general population. Wikipedia and Gmail didn’t exist. Internet users in the late nineties reported that their main problem was finding what they needed online.

That is perhaps the biggest change and one we may not have anticipated: the move towards personalization. Finding what we need is no longer a problem. Most of us have the opposite problem and struggle with information overwhelm. Twenty years ago, filter bubbles were barely a problem (at least, not online.) Now, almost everything we encounter online is personalized to ensure it’s ridiculously easy to find what we want. Newsletters, websites, and apps greet us by name. Newsfeeds are organized by our interests. Shopping sites recommend other products we might like. This has increased the amount the internet does for us to a level that would have been hard to imagine in the late 90s. Kevin Kelly, writing in The Inevitable,  describes filtering as one of the key forces that will shape the future.

History reveals an extraordinary acceleration of technological progress. Establishing the precise history of technology is problematic as some inventions occurred in several places at varying times, archaeological records are inevitably incomplete and dating methods are imperfect. However, accelerating change is a clear pattern. To truly understand the principle of accelerating change, we need to take a quick look at a simple overview of the history of technology.

Early innovations happened slowly. It took us about 30,000 years to invent clothing and about 120,000 years to invent jewelry. It took us about 130,000 years to invent art and about 136,000 years to come up with the bow and arrow. But things began to speed up in the Upper Paleolithic period. Between 50,000 and 10,000 years ago, we developed more sophisticated tools with specialized uses—think harpoons, darts, fishing tools and needles—early musical instruments, pottery, and the first domesticated animals. Between roughly 11,000 years and the 18th century, the pace truly accelerated. That time period essentially led to the creation of civilization, with the foundations of our current world.

More recently, the Industrial Revolution changed everything because it moved us significantly further away from relying on the strength of people and domesticated animals to power means of production. Steam engines and machinery replaced backbreaking labor, meaning more production at a lower cost. The number of adjacent possibilities began to snowball. Machinery enabled mass production and interchangeable parts. Steam powered trains meant people could move around far more easily, allowing people from different areas to mix together and share ideas. Improved communications did the same. It’s pointless to even try listing the ways technology has changed since then. Regardless of age, we’ve all lived through it and seen the acceleration. Few people dispute that the change is snowballing. The only question is how far that will go.

As Stephen Hawking put it in 1993:


But, as we saw with Moore’s Law, exponential growth cannot continue forever. Eventually we run into fundamental constraints. Hours in the day, people on the planet, availability of a resource, smallest possible size of a semiconductor, attention—there’s always a bottleneck we can’t eliminate.  We reach the point of diminishing returns. Growth slows or stops altogether. We must then either look at alternative routes to improvement, or leave things as they are. In Everett Rogers’s diffusion of innovation theory, this is known as the substitution stage, when usage declines and we start looking for substitutes.

This process is not linear. We can’t predict the future because there’s no way to take into account the tiny factors that will have a disproportionate impact in the long-run.

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    Image credit: tec_estromberg



Occam’s razor can be summarized as such:


The Basics





The History of Occam’s Razor





Mental Model Occam's Razor

Examples of the Use of Occam’s Razor





The Development of Scientific Theories










Prison Abolition and Fair Punishment




Crime solving and forensic work



Exceptions and Issues









Stephen Hawking Explains The Origin of the Universe


The Origin of the Universe, a lecture, by Stephen Hawking

According to the Boshongo people of central Africa, in the beginning, there was only darkness, water, and the great god Bumba. One day Bumba, in pain from a stomach ache, vomited up the sun. The sun dried up some of the water, leaving land. Still in pain, Bumba vomited up the moon, the stars, and then some animals. The leopard, the crocodile, the turtle, and finally, man.

This creation myth, like many others, tries to answer the questions we all ask. Why are we here? Where did we come from? The answer generally given was that humans were of comparatively recent origin, because it must have been obvious, even at early times, that the human race was improving in knowledge and technology. So it can’t have been around that long, or it would have progressed even more. For example, according to Bishop Usher, the Book of Genesis placed the creation of the world at 9 in the morning on October the 27th, 4,004 BC. On the other hand, the physical surroundings, like mountains and rivers, change very little in a human lifetime. They were therefore thought to be a constant background, and either to have existed forever as an empty landscape, or to have been created at the same time as the humans. Not everyone, however, was happy with the idea that the universe had a beginning.

For example, Aristotle, the most famous of the Greek philosophers, believed the universe had existed forever. Something eternal is more perfect than something created. He suggested the reason we see progress was that floods, or other natural disasters, had repeatedly set civilization back to the beginning. The motivation for believing in an eternal universe was the desire to avoid invoking divine intervention to create the universe and set it going. Conversely, those who believed the universe had a beginning, used it as an argument for the existence of God as the first cause, or prime mover, of the universe.

If one believed that the universe had a beginning, the obvious question was what happened before the beginning? What was God doing before He made the world? Was He preparing Hell for people who asked such questions? The problem of whether or not the universe had a beginning was a great concern to the German philosopher, Immanuel Kant. He felt there were logical contradictions, or antimonies, either way. If the universe had a beginning, why did it wait an infinite time before it began? He called that the thesis. On the other hand, if the universe had existed for ever, why did it take an infinite time to reach the present stage? He called that the antithesis. Both the thesis and the antithesis depended on Kant’s assumption, along with almost everyone else, that time was Absolute. That is to say, it went from the infinite past to the infinite future, independently of any universe that might or might not exist in this background. This is still the picture in the mind of many scientists today.

However in 1915, Einstein introduced his revolutionary General Theory of Relativity. In this, space and time were no longer Absolute, no longer a fixed background to events. Instead, they were dynamical quantities that were shaped by the matter and energy in the universe. They were defined only within the universe, so it made no sense to talk of a time before the universe began. It would be like asking for a point south of the South Pole. It is not defined. If the universe was essentially unchanging in time, as was generally assumed before the 1920s, there would be no reason that time should not be defined arbitrarily far back. Any so-called beginning of the universe would be artificial, in the sense that one could extend the history back to earlier times. Thus it might be that the universe was created last year, but with all the memories and physical evidence, to look like it was much older. This raises deep philosophical questions about the meaning of existence. I shall deal with these by adopting what is called, the positivist approach. In this, the idea is that we interpret the input from our senses in terms of a model we make of the world. One can not ask whether the model represents reality, only whether it works. A model is a good model if first it interprets a wide range of observations, in terms of a simple and elegant model. And second, if the model makes definite predictions that can be tested and possibly falsified by observation.

In terms of the positivist approach, one can compare two models of the universe. One in which the universe was created last year and one in which the universe existed much longer. The Model in which the universe existed for longer than a year can explain things like identical twins that have a common cause more than a year ago. On the other hand, the model in which the universe was created last year cannot explain such events. So the first model is better. One can not ask whether the universe really existed before a year ago or just appeared to. In the positivist approach, they are the same. In an unchanging universe, there would be no natural starting point. The situation changed radically however, when Edwin Hubble began to make observations with the hundred inch telescope on Mount Wilson, in the 1920s.


The expansion of the universe was one of the most important intellectual discoveries of the 20th century, or of any century. It transformed the debate about whether the universe had a beginning. If galaxies are moving apart now, they must have been closer together in the past. If their speed had been constant, they would all have been on top of one another about 15 billion years ago. Was this the beginning of the universe? Many scientists were still unhappy with the universe having a beginning because it seemed to imply that physics broke down. One would have to invoke an outside agency, which for convenience, one can call God, to determine how the universe began. They therefore advanced theories in which the universe was expanding at the present time, but didn’t have a beginning. One was the Steady State theory, proposed by Bondi, Gold, and Hoyle in 1948.

In the Steady State theory, as galaxies moved apart, the idea was that new galaxies would form from matter that was supposed to be continually being created throughout space. The universe would have existed for ever and would have looked the same at all times. This last property had the great virtue, from a positivist point of view, of being a definite prediction that could be tested by observation. The Cambridge radio astronomy group, under Martin Ryle, did a survey of weak radio sources in the early 1960s. These were distributed fairly uniformly across the sky, indicating that most of the sources lay outside our galaxy. The weaker sources would be further away, on average. The Steady State theory predicted the shape of the graph of the number of sources against source strength. But the observations showed more faint sources than predicted, indicating that the density sources were higher in the past. This was contrary to the basic assumption of the Steady State theory, that everything was constant in time. For this, and other reasons, the Steady State theory was abandoned.



Although the singularity theorems of Penrose and myself, predicted that the universe had a beginning, they didn’t say how it had begun. The equations of General Relativity would break down at the singularity. Thus Einstein’s theory cannot predict how the universe will begin, but only how it will evolve once it has begun. There are two attitudes one can take to the results of Penrose and myself. One is to that God chose how the universe began for reasons we could not understand. This was the view of Pope John Paul. At a conference on cosmology in the Vatican, the Pope told the delegates that it was OK to study the universe after it began, but they should not inquire into the beginning itself, because that was the moment of creation, and the work of God. I was glad he didn’t realize I had presented a paper at the conference suggesting how the universe began. I didn’t fancy the thought of being handed over to the Inquisition, like Galileo.

The other interpretation of our results, which is favored by most scientists, is that it indicates that the General Theory of Relativity breaks down in the very strong gravitational fields in the early universe. It has to be replaced by a more complete theory. One would expect this anyway, because General Relativity does not take account of the small scale structure of matter, which is governed by quantum theory. This does not matter normally, because the scale of the universe is enormous compared to the microscopic scales of quantum theory. But when the universe is the Planck size, a billion trillion trillionth of a centimeter, the two scales are the same, and quantum theory has to be taken into account.

In order to understand the Origin of the universe, we need to combine the General Theory of Relativity with quantum theory. The best way of doing so seems to be to use Feynman’s idea of a sum over histories. Richard Feynman was a colorful character, who played the bongo drums in a strip joint in Pasadena, and was a brilliant physicist at the California Institute of Technology. He proposed that a system got from a state A, to a state B, by every possible path or history. Each path or history has a certain amplitude or intensity, and the probability of the system going from A- to B, is given by adding up the amplitudes for each path. There will be a history in which the moon is made of blue cheese, but the amplitude is low, which is bad news for mice.

The probability for a state of the universe at the present time is given by adding up the amplitudes for all the histories that end with that state. But how did the histories start? This is the Origin question in another guise. Does it require a Creator to decree how the universe began? Or is the initial state of the universe, determined by a law of science? In fact, this question would arise even if the histories of the universe went back to the infinite past. But it is more immediate if the universe began only 15 billion years ago. The problem of what happens at the beginning of time is a bit like the question of what happened at the edge of the world, when people thought the world was flat. Is the world a flat plate with the sea pouring over the edge? I have tested this experimentally. I have been round the world, and I have not fallen off. As we all know, the problem of what happens at the edge of the world was solved when people realized that the world was not a flat plate, but a curved surface. Time however, seemed to be different. It appeared to be separate from space, and to be like a model railway track. If it had a beginning, there would have to be someone to set the trains going. Einstein’s General Theory of Relativity unified time and space as spacetime, but time was still different from space and was like a corridor, which either had a beginning and end, or went on forever. However, when one combines General Relativity with Quantum Theory, Jim Hartle and I realized that time can behave like another direction in space under extreme conditions. This means one can get rid of the problem of time having a beginning, in a similar way in which we got rid of the edge of the world. Suppose the beginning of the universe was like the South Pole of the earth, with degrees of latitude playing the role of time. The universe would start as a point at the South Pole. As one moves north, the circles of constant latitude, representing the size of the universe, would expand. To ask what happened before the beginning of the universe would become a meaningless question, because there is nothing south of the South Pole.

Time, as measured in degrees of latitude, would have a beginning at the South Pole, but the South Pole is much like any other point, at least so I have been told. I have been to Antarctica, but not to the South Pole. The same laws of Nature hold at the South Pole as in other places. This would remove the age-old objection to the universe having a beginning; that it would be a place where the normal laws broke down. The beginning of the universe would be governed by the laws of science. The picture Jim Hartle and I developed of the spontaneous quantum creation of the universe would be a bit like the formation of bubbles of steam in boiling water.

The idea is that the most probable histories of the universe would be like the surfaces of the bubbles. Many small bubbles would appear, and then disappear again. These would correspond to mini universes that would expand but would collapse again while still of microscopic size. They are possible alternative universes but they are not of much interest since they do not last long enough to develop galaxies and stars, let alone intelligent life. A few of the little bubbles, however, grow to a certain size at which they are safe from recollapse. They will continue to expand at an ever increasing rate, and will form the bubbles we see. They will correspond to universes that would start off expanding at an ever increasing rate. This is called inflation, like the way prices go up every year.

The world record for inflation was in Germany after the First World War. Prices rose by a factor of ten million in a period of 18 months. But that was nothing compared to inflation in the early universe. The universe expanded by a factor of million trillion trillion in a tiny fraction of a second. Unlike inflation in prices, inflation in the early universe was a very good thing. It produced a very large and uniform universe, just as we observe. However, it would not be completely uniform. In the sum over histories, histories that are very slightly irregular will have almost as high probabilities as the completely uniform and regular history. The theory therefore predicts that the early universe is likely to be slightly non-uniform. These irregularities would produce small variations in the intensity of the microwave background from different directions. The microwave background has been observed by the Map satellite, and was found to have exactly the kind of variations predicted. So we know we are on the right lines.

The irregularities in the early universe will mean that some regions will have slightly higher density than others. The gravitational attraction of the extra density will slow the expansion of the region, and can eventually cause the region to collapse to form galaxies and stars. So look well at the map of the microwave sky. It is the blue print for all the structure in the universe. We are the product of quantum fluctuations in the very early universe. God really does play dice.

Follow your curiosity to Nassim Taleb on the Notion of Alternative Histories.

In Pursuit of the Unknown: 17 Equations That Changed the World




Equations are a vital part of this world. And you don’t need to be a rocket scientist to appreciate them.

There are two kinds of equations in mathematics. Stewart writes:



An example of the first kind of equation is Pythagoras’s theorem, which is “an equation expressed in the language of geometry.” If you accept Euclid’s basic geometric assumptions, then Pythagoras’s theorem must be true.






A good example of the second kind of equation is Newton’s law of gravity. “It tells us how the attractive force between two bodies depends on their masses,” Stewart writes, “and how far apart they are. Solving the resulting equations tells us how the planets orbit the Sun, or how to design a trajectory for a space probe.” This isn’t a mathematical theorem but rather it’s true for physical reasons in that it fits the observations.




Pythagoras’s theorem


The first equation presented in the book.

What does it tell us?
Why is that important?
What did it lead to?






Triangulation is an indirect way of determining distance by employing angles.



Surveys routinely employed triangulation by the late 18th century. And while we don’t explicitly use it today, it is still there in how we deduce locations from satellite data.

