Wow. Somewhat in balance to yesterday’s reblog of part one of Coel Hellier’s post defending multiverse theories as scientific, here’s Amir Aczel skewering many of the proponents of multiverses and other untestable cosmological theories. He takes aim at Brian Greene, Max Tegmark, Lawrence Krauss, and others, for presenting metaphysical assertions as science.

The universe is a marvelous place to live in. Well, it’s the only place we know — and we know a very tiny part of it. Telescopes reveal to us farther parts of this wondrous cosmos, and through them we learn about fascinating objects and phenomena such as neutron stars, black holes, supernovas, and exoplanets. Equally, high-energy particle accelerators and other experiments reveal to us the workings of the very small — which always have a strong bearing on the nature of the universe as a whole. And of course theories are equally important. There is absolutely nothing wrong with speculation in physics — and the correct theories are eventually confirmed by experiment and observation. But it is definitely wrong — misleading and dishonest — to preach to an unsuspecting public, mostly uninitiated in science, mere hypotheses as if they were confirmed facts. This isn’t science, and it isn’t honest scientific reporting. Physicists should be the purveyors of facts, not dreams.

As I commented on Hellier’s piece, I actually have no problem with speculation on things like multiverses or on how the universe began, as long as it’s labeled as speculation. In truth, in Greene’s and Termark’s books, they are usually careful to delineate and label the more speculative aspects of their subject matter (I haven’t read Krauss’s book), but it doesn’t always seem to happen in their articles and commentary.

I think this speculation lies on the border between science and metaphysics. That doesn’t bother me. I mean it as no insult to those engaged in it. A lot of what I post about here is metaphysics. But I do find it ironic that many of the same people who dump on philosophy are effectively engaging in philosophy, and are so touchy about having that pointed out to them.

This is a subject that we’ve discussed repeatedly here, so I thought you might find Tegmark’s narration of this video interesting.

I’m pretty sure that mathematics are not only something humans created, that they are based on relations we observe in the world, but beyond that, I remain largely agnostic on the Mathematical Universe Hypothesis or overall on the precise ontological status of mathematical entities.

This essay by three physicists: Stephen Hawking, Max Tegmark, Frank Wilczek, along with Stuart Russell (the one computer scientist), seems to be getting a lot of attention. It keeps popping up in my various feeds, showing up in various venues.

With the Hollywood blockbuster Transcendence playing in cinemas, with Johnny Depp and Morgan Freeman showcasing clashing visions for the future of humanity, it’s tempting to dismiss the notion of highly intelligent machines as mere science fiction. But this would be a mistake, and potentially our worst mistake in history.

Artificial-intelligence (AI) research is now progressing rapidly. Recent landmarks such as self-driving cars, a computer winning at Jeopardy! and the digital personal assistants Siri, Google Now and Cortana are merely symptoms of an IT arms race fuelled by unprecedented investments and building on an increasingly mature theoretical foundation. Such achievements will probably pale against what the coming decades will bring.

As I indicated in one of the comment threads a few weeks ago, when physicists start talking about consciousness or artificial intelligence, I often find it cringeworthy, and a good example of the fact that when brilliant scientists speak about matters outside of their specialty, they often have little more insight than the rest of us. I’d feel a little bit better about this essay if I got the impression that these guys had talked with a lot of working AI researchers and perhaps some neuroscientists.

I’ve already written my own essay about why I’m not particularly worried about an AI revolt. We tend to equate intelligence with a self valuing agenda, a survival instinct. But that doesn’t come automatically. It only happened for us through billions of years of evolutionary programming. We’re as unlikely to accidentally develop malevolent AIs as your local garbage dump is to spontaneously evolve into Godzilla.

No, an AI’s instincts will have to be painstakingly programmed by us. Any accidents will be more likely to make it nonfunctional than malevolent. I do think there is a decent danger of unintended consequences from machines ardently trying to follow their programming, but that exists already with today’s computers and machines, and we haven’t destroyed ourselves yet. Actually AIs could arguably lessen that risk since it would give machines better judgment in the implementation of their directives.

My concern about essays like this, aside from the anxiety they cause, is that it might lead to politicians deciding they need to legislate AI research, to put limits and restrictions on it. All that would do is cause the US and Europe to cede such research to other countries without those restrictions. Legislation might eventually be needed to protect artificial life forms, but we’re still a ways off from that right now.

Am I completely sanguine on the dangers of AI? No, but I’m not completely sanguine on the dangers of any new technology. I’m personally a lot more worried about what we’re doing to the environment and our runaway population growth than I am about AIs turning on us.

If you enjoyed my write up on Tegmark’s Level II multiverse, you might enjoy this guest post that he makes on Sean Carroll’s blog, which includes a link to the chapter on inflation from his book, including those visual aids I referenced!

Since the BICEP2 breakthrough is generating such huge interest in inflation, I’ve decided to post my entire book chapter on inflation here so that you can get an up-to-date and self-contained account of what it’s all about. Here are some of the questions answered:

What does the theory of inflation really predict?

What physics does it assume?

Doesn’t creation of the matter around us from almost nothing violate energy conservation?

How could an infinite space get created in a finite time?

I recently read Max Tegmark’s latest book, ‘Our Mathematical Universe‘, about his views on multiverses and the ultimate nature of reality. This is the fourth and final post in a series on the concepts and views he covers in the book.

This final post in the series is a commentary on the overall book. Tegmark spends the early parts reviewing the current state of cosmology and physics. As described in the previous entries, he covers three increasingly diverse and grander definitions of the multiverse. These are fairly standard multiverse conceptions, and they aren’t all the one in currently circulation, but they are the ones most relevant to his main thesis.

The Mathematical Universe Hypothesis

Philosophy is written in this grand book, the universe … It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures –Galileo

Galileo wasn’t the first to say this of course. The ancient Greeks were also well aware of it. Mathematics is at the heart of science. Isaac Newton is credited with explaining the universal role of gravity, not because he was the first to come up with the idea (others had already contemplated it), but because he was the first to demonstrate the mathematics that described its dynamics.

The uncanny usefulness of mathematics in describing the world has often been a source of puzzlement for many philosophers. Indeed, there is a philosophy of mathematics field where a number of theories about this are discussed and debated, such as empiricism, platonism, nominalism, and many others.

So, the idea that mathematics describes the universe is well accepted. Tegmark, however, goes further by asserting that the universe is not just described by mathematics, but that it is mathematics, characterizing this as a radical form of platonism.

Now, immediately we have to do an important semantic clarification. When Tegmark refers to mathematics, he isn’t referring to the notation, the nomenclature, or the techniques that we use to express or explore mathematics. The ancient Greeks worked in math with a different notation than we use today, and no doubt an alien from Andromeda would have a radically different notation and process than anything humans have conceived of. But all these notations and processes should refer to the same underlying structures, the same underlying realities.

Tegmark points out that these mathematical structures are often identical to the underlying structures in nature. We have a tendency to view mathematical structures as abstract and separate from physical reality. But if those abstract structures match the physical ones, if we have two descriptions that are equivalent, then it makes sense to regard them as describing the same thing.

Many properties in science, such as empty space, the quantum wave function, or the spin property of elementary particles, are really only known only by their numeric properties. (“Spin” was originally thought to be descriptive of particles rotating in some classical manner. Subsequent developments showed that to be naive, but the name stuck.)

Most scientific theories are mathematical at their core, but require a qualitative explanation of one or more of the variables. In physics, this is often referred to as “baggage”. For example, the equation E=mc^{2} is fairly meaningless if you don’t know that E is energy, m is mass, and c is the speed of light.

Tegmark speculates that, if the Mathematical Universe Hypothesis is true, then the much sought after Theory of Everything should be a purely mathematical theory. It shouldn’t need any baggage. It’s entities should merely serve as points in relationships that should be enough to explain all of reality.

Addressing commons criticisms of the MUH, Tegmark spends a chapter on time. Mathematical structures are timeless structures, so how does that relate to a universe that evolves with time? Thinking in terms of spacetime, with time as one of the dimensions, the universe, including all of its history, could be viewed as a static structure. Tegmark uses the example of a DVD movie that appears to change when watching it, but is actually a static unchanging construct. He describes this concept in fascinating detail, in a manner that I can’t do justice to here.

Tegmark has an interesting discussion on time, infinity, and strange predictions that may call into question whether infinity is a valid concept. I found this section interesting because infinity seems to be an important assumption for the Level I and II multiverses. This discussion also included an excellent description of problems such as Boltzmann brains.

Finally, Tegmark addresses the most glaring criticism, that many mathematical structures do indeed match real world patterns, but not all of them. Many, such as the Mandelbrot set, exist only abstractly. Here is where all the earlier discussion of multiverses come to fruition. Tegmark’s answer is that all mathematical structures correspond with actual physical patterns, just not all in this universe.

The Level IV multiverse is one of mathematical structures. If our universe is a mathematical structure, then it is only one of an infinite variety of structures. All mathematical structures have physical reality in this multiverse. Exploring this multiverse is a matter of computation and ideas.

My take

Before reading this book, I was agnostic about the MUH, and I’m forced to say that I remain largely agnostic, albeit now in a much more informed fashion. Tegmark does an excellent job of describing the concept, along with the many required supporting ideas. But I often found him to exude a level of certainty that felt unwarranted.

His certitude is often related to what he sees as the inevitable mathematical consequences of well accepted theories. I don’t understand the mathematics of most of those theories well enough to judge first hand whether or not that certitude is warranted. But I’m aware that manyphysicists, who do understand those theories at the mathematical level, don’t necessarily concur.

I’m also aware that just because the mathematics lead to a certain conclusion doesn’t make that conclusion inevitable. The mathematical consequences of Newtonian mechanics allowed astronomers to predict the existence of Neptune because of Uranus’s orbit, but it also led them to predict the existence of Vulcan because of Mercury’s orbit. One was right, but the other was wrong, and a new theory (general relativity) was necessary to understand why.

I do strongly believe that mathematics rest on empirical foundations, foundations found in the patterns of nature. As a result, many mathematical constructs have real world correlates, and many others approximate real world patterns. This, to me, is sufficient to explain the powerful utility of mathematics in science, without necessarily having to adopt an absolutist position about all mathematical structures having physical existence.

Of course, many abstract mathematical structures have no known physical correlates. Here Tegmark’s extensive descriptions of multiverses serve an important purpose, since multiverses are necessary to explain how these abstract structures could exist physically. Interestingly, Tegmark himself does speculate that some mathematical structures might actually not exist. His focus is on infinite ones, but it doesn’t seem like much of a cognitive leap to conclude that many other types might not as well.

But if those abstract structures don’t have a physical existence, then where do they come from? I’m tempted to say that they come from the same place as Vulcan, that is a tautological conclusion with no real world correlate. But this implies that they’re not valid, and I don’t think that, particularly since abstract structures sometimes turn out to correspond to something physical that we just weren’t aware of when they were formulated.

To be clear, I do think the MUH is a valid candidate for reality. It might be true. In the first post on this blog, I discussed the possibility that reality might be structure all the way down, and the MUH is definitely compatible with that. Even if reality does have a brute physical layer, everything above it are patterns, most of which, if not all, are describable in mathematical terms.

I tend to think that whether or not the MUH is true is a philosophical matter. Tegmark asserts that the idea is falsifiable since if it isn’t true, physics will eventually hit a brick wall where mathematics is no longer useful. The problem is that if we hit such a wall, MUH proponents can always claim that we simply don’t know enough yet to apply mathematics to that wall.

Indeed, a case could be made that this is exactly what the indeterminancy of a single quantum particle is, and that quantum interpretations that rescue determinism are just saving appearances. Now, I’m agnostic on the major quantum interpretations, and I certainly don’t think it’s productive to assume we’ll never know more than we do about it, but it does seem that the MUH needs one of the deterministic interpretations of quantum mechanics to be true.

All that said, Tegmark is an excellent writer, and if you’ve found the ideas in this series interesting, then I highly recommend his book. It’s an excellent introduction to many ideas and I’ve only lightly scratched the surface in this and the previous posts.

Additional reading

Fellow blogger, Disagreeable Me, a advocate of the MUH, has written an excellent blog post on it, which I know some of you have already read. DM approaches the issue from a philosophical angle, and I found myself returning to his post after I had completed the book. A highly recommended read.

I recently finished reading Max Tegmark’s latest book, ‘Our Mathematical Universe‘, about his views on multiverses and the ultimate nature of reality. This is the third in a series of posts on the concepts and views he covers in the book.

In the early twentieth century, one of the mysteries of science was the constant speed of light. The speed of light was constant no matter how it was measured. This was in contrast to the speed of sound, or the speed of just about anything else, which varied depending on the speed of the observer.

Albert Einstein accepted the experimental evidence of the constancy of the speed of light, and explored its implications. If the speed of light was always constant, then something else had to give. Something that factored into that speed had to vary, something like mass, length, and time. Exploring those implications led to the special theory of relativity.

For several decades now, one of the mysteries of science has been wave / particle duality. We have strong evidence that light behaves like a wave, and strong evidence that it behaves like a particle. We have similar evidence for electrons and just about any other subatomic particle, as well as atoms themselves and even large molecules under isolated conditions.

The shape of a wave is modeled in a mathematical concept called the wave function. The particle will appear somewhere in that wave. There is no known way to predict where in the wave any individual particle will be found. All that can be known are probabilities of it appearing at various locations within the wave. Bizarrely, once the position of the particle is observed, once it is measured, all trace of the overall wave instantly disappears, with only the particle remaining.

Just to be clear, this is freaky strange, and no one is certain why it is so. Reality at the quantum level appears to be wavelike, to the degree that the wave can physically interfere with itself when split, but suddenly, instantly, becomes particle like when we look at it. As strange as it is, this has been confirmed for decades by extensive experimental data. It is reality.

There are a number of interpretations of what is happening. The oldest, and for a long time the most popular, is called the Copenhagen Interpretation. It is basically is a minimalist interpretation that says that this is simply reality, and that when a particle’s position is measured, the wave function “collapses”. Prior to the collapse, the particle exists in what’s called a superposition. It exists in multiple locations at the same time, but once the position is known, the existence of the particle in all but one of those locations disappears.

There are several other interpretations. All of them must throw one or more aspects of common sense reality under the bus in order to make sense of the data.

In the 1950s, Hugh Everett came up with a new interpretation. Everett accepted the mathematics of the wave function, but was troubled by the lack of anything in those mathematics that predicted a wave function collapse. The only reason that the wave function collapse is thought to exist is the fact that we only observe the particle in one location once it is measured.

Everett asked, what happens if the wave function, in fact, never collapses? If the wave function predicts two locations for the particle, then the mathematics say the particle is in both locations. Of course, we don’t observe it to be in both locations. So then, what’s going on? Similar to when Einstein was contemplating the constant speed of light, something else has to give.

According to the mathematics and our sensory data, we should see the particle in only one of the locations and we should see it only in the other one. No, the second “only” in the previous sentence is not a typo. We appear to have two realities in which we observe the particle. Prior to the measure, there was only one reality. After the measure, there are two.

In multiverse parlance, the many worlds interpretation asserts that our universe is cloned every time what appears to be a wave function collapse happens. Given that this happens an uncountable number of times per second throughout the universe, and given the large range of possibilities for each particle’s position, the number of universes being created every second is sublime.

The randomness of the particles location then is an illusion, created by the fact that we only observe the location particular to our universe. But the wave function unfolds unabated with the particle existing in each location in a different universe.

This means that there are an uncountable numbers of you in these alternate universes, where each quantum result is manifested. In other words, every random event that could happen, happens in some universe, and there are an uncountable versions of you living every conceivable version of your life.

In Tegmark’s framework, this is the Level III Multiverse. It is a superset of the Level I and II multiverses, although as formulated, there’s no particular reason that its existence, or non-existence, is dependent on the other ones. If all three levels exist, then Level III includes all the multiverses in the lower levels and reality continues to expand at an astounding rate.

Tegmark does note some similarities between the Level I and Level III multiverse. In both, there are an infinite number of you living every possible variation of your life. The result of every quantum possibility should be manifest in one of the Level I universes. Of course, if they were one and the same, it would mean that remote regions in the Level I multiverse were in some way quantum entangled with each other.

Tegmark also speculates about reconciling the Level II and III multiverse, but doesn’t currently see a way to do it.

Over time, support for the many worlds interpretation has grown in the particle physics community, although Copenhagen continues to hold a plurality in most polls. The question is, is there any way to test this idea? Brian Greene in ‘The Hidden Reality’ identified the possibility of the uncollapsed wave interfering with itself across universes, although he notes that observing this would be extremely difficult.

Tegmark proposes another one, although it’s not one that anyone is liable to volunteer for. The quantum suicide or subjective immortality thought experiment involves setting up a gun with a trigger set to fire if a random quantum event takes place, with a 50% chance of taking place in the first second. The experimenter then puts their head in front of the gun.

In 50% of the universes, the experimenter dies within the first second, but in the other 50%, they live. For each second, the probability of the experimenter being alive goes down. After a couple of minutes, the probability of the experimenter still being alive is infinitesimal. However, in at least some portion of the alternate universes, the experimenter lives on.

From the subjective point of view of the experimenter, the longer they live, the higher the probability of the many worlds interpretation being true. After a few hours, increasingly unlikely events (misfire, power outage, meteor strike, etc) begin to happen to prevent their death. If an experimenter subjectively survived this ordeal for several hours, they could have a high degree of confidence in the many worlds interpretation. (Of course, in virtually all universes, they would leave behind grieving friends and family who would be less convinced.)

Tegmark then points out that, if either the many worlds interpretation or infinite space scenario is true, then a version of each of us will, despite its improbability, live long enough to outlast all of humanity. In other words if is true, subjectively, you will live long enough to know it is true, at least assuming you recall reading this. Each of us may live to be the last human in our own improbable universe, knowing the truth of the multiverse.

In the next post, we will get into the main idea of Tegmark’s book, the mathematical universe hypothesis.

I recently finished reading Max Tegmark’s latest book, ‘Our Mathematical Universe‘, about his views on multiverses and the ultimate nature of reality. This is the second in a series of posts on the concepts and views he covers in the book. Tegmark postulates four levels of multiverse.

In the first post of this series, I described Termark’s Level I multiverse, which arises from space being infinite and the patterns of atoms in our local observable universe eventually being duplicated, with and without variations, in far off regions of space.

This post is on his Level II multiverse, which is a description of the bubble universes from eternal inflation.

There is strong and abundant evidence that our known universe started in a hot dense state about 14 billion years ago, commonly called the big bang. It has been expanding and cooling ever since.

However, there were unexplained aspects of the visible universe that complicated the picture. The main one was that the furthest teaches of the visible universe had the same temperature in all directions, even though those remote regions had never had a chance to interact and come into thermal equilibrium.

In order to explain the universe as we see it, cosmologists realized that a rapid exponential expansion must have happened in the earliest instances of the observable universe. During this period, called cosmic inflation, the universe would have expanded by a factor of at least 10^{26}. Once inflation was complete, the universe would have proceeded following the classic big bang timeline, going through the hot dense state and later expanding and cooling enough for normal matter and eventually galaxies to form.

What caused inflation? No one is sure, but quantum fluctuations are usually thought to be involved. Inflation expanded those quantum fluctuations into the universe wide variances that later led to the formation of galaxies. Some models actually posit that inflation is space’s natural state, and that the real question is what caused it to end?

If inflation is the natural state, and an event of some type such as a random quantum fluctuation caused it to end, and if space is infinite, then this event may not have caused inflation to end throughout all of space. There may be regions of space where inflation never ended, where it is eternal. Indeed, it may not have ended in most regions of space. Our universe may be a low probability bubble where inflation has ended, or perhaps slowed down to the more stately expansion we see today.

But a low probability event is not a zero probability event. So there may be other regions where inflation stopped, where other bubble universes formed. If so, we would be separated from these other bubble universes by a perpetually hyper inflating space.

If we could somehow leave our bubble (we can’t, more on that later), it would be fruitless to try to reach another bubble, because inflationary space would be expanding at an unimaginable rate. We’d never be able to catch up. For that matter, we’d never be able to return to our own bubble since the space we traveled would also have continued to expand, isolating us beyond all hope.

Although the fundamental laws of physics would be the same in each bubble universe, it’s thought that different random factors in formation of each bubble may cause the effective laws of physics to vary, notably in many of the physical constants that, in this universe, appear to be random.

The idea is that this would provide an explanation for what appears to be fine tuning in our universe. In most universes, the physical laws and constants would not allow for life. The fact that we are here contemplating this shows that we’re in one of the few universes that do have conditions for life to form. Of course, this assumes that those constants aren’t what they are in our universe due to some unknown physical necessity.

Is the Level II multiverse compatible with the Level I multiverse? After all, the Level I multiverse assumed infinite space. The Level II does as well, but doesn’t being in a bubble inherently mean that our universe is not infinite, at least at the Level I level? Furthermore, wouldn’t it be possible, in principle, for us to travel to the boundary of our bubble and enter inflationary space?

The answer is that the Level I and Level II multiverses are compatible, because within our bubble, space is infinite. How can this be? How can we have an infinite universe within a finite bubble? Warning: the answer is weird, and hard to explain. I’m probably going to fail. Tegmark attempts an explanation in the book. He’s an excellent writer, but I found Brian Greene’s explanation of this in ‘The Hidden Reality’ to be a bit more understandable.

It has to do with the relativity of time. We’ve known since Einstein that time flows differently depending on how fast you’re moving, or how intense gravity is. In short, time flows slower when the density of energy is high, and faster when the density of energy is lower. Not only is time relative, but so is simultaneity, that is, whether two events happened at the same time or not, is relative.

The bubble universe is constantly growing into inflationary space, bringing in new regions of space where inflation is coming to an end. From outside the bubble, the new outer regions are just beginning their big bang, their expansion and cooling, while the inner regions are well along into forming galaxies.

The bubble will expand forever, bring in new regions, ending inflation in those regions, infinitely far into the future, meaning that infinitely into the future, the bubble will be infinitely large. Of course, inflationary space is expanding much faster, so there’s no danger of the bubble eating everything.

But time flows differently within the bubble. Remember the relativity of simultaneity?

Ok, here’s the punchline. Within the bubble, all regions came out of inflation simultaneously. So, infinite time and expansion in inflationary space translates into infinite space, right from the beginning, within the bubble. Furthermore, we can’t travel to the boundary of the bubble, because the boundary is the big bang.

Yep, I warned you it was weird and hard to understand. It’s easier to understand with visual aids, which both Tegmark and Greene use in their books. I tried to find something on the web, but came up empty handed. Sorry.

Ok, so this multiverse is much more complicated than the Level I concept, which it ostensibly contains. It also makes a lot more assumptions. The first is inflation, which is actually turning out to be a good one with the new BICEP2 data. But is it the right inflationary model? Is inflation eternal? Is space infinite, or at least large enough for these bubbles to form? If they form, do the physical laws and/or constants vary among them?

There’s also the question of where all this continually hyper inflating space and energy is coming from. Tegmark, while affirming the evidence in our observable universe that it does happen at least on some scale, admits to being a bit apprehensive about this, as though maybe reality is the ultimate Ponzi scheme, where a helluva bill may come due someday.

Only time will tell, maybe.

The next post in this series will be on Tegmark’s Level III multiverse, where we’ll learn that we’re really just getting started pulling space and energy out of our hat.

I’ve just finished reading Max Tegmark’s latest book, ‘Our Mathematical Universe‘, about his views on multiverses and the ultimate nature of reality. This is the first in a series of posts that I plan to do on it. Tegmark postulates four levels of multiverse. This post is about the first, and simplest version, the Level I Multiverse.

No one knows for sure how large the universe is, but the size of the observable universe is actually pretty well known. The universe as we know it started expanding from a hot dense state about 14 billion years ago. Given the fast but finite speed of light, the further away we look, the further back in time we’re looking.

The furthest and oldest thing we can now see is the cosmic microwave background (CMB), which has been traveling for almost the entire history of the known universe, 14 billion years. However, due to the expansion of space, the CMB we are now seeing originated 46 billions light years away, making that the radius of the observable universe.

Cosmologists often consider the word “universe” to be synonymous with the observable universe. The reason for this is it’s all we can observe, and given that the speed of light is the fastest speed that any interaction or effect can travel, the edge of the observable universe is the limit of our causal influence, or of things that could influence us. In other words, we are currently causally disconnected from the regions beyond this point, beyond or cosmological horizon.

Of course, there’s no real reason to think the universe ends at the limits of our observation. Indeed, attempts to measure the curvature of space seem to indicate that the whole universe is at least hundreds of times the size of the observable universe.

Considering the observable universe to be our universe, Tegmark refers to the regions beyond as other universes. Semantically, this seems like a questionable move. These regions seem like more of just the same universe. Initially I suspected this might be an attempt to redefine the multiverse into something non-controversial so the controversial versions wouldn’t be as much of a leap.

However, if the universe is infinite, it leads to strange conclusions, and with enough distance, the phrase other universes starts to make sense. Within each local observable universe, there are a finite number of ways that the atoms can be arranged. (Note: a very large number of combinations, but a finite number.) This means that in an infinite universe, every possible configuration of matter will eventually be realized. Furthermore, if you could look far enough in this infinite space, eventually every configuration will repeat itself, infinitely.

In other words, in an infinite universe, somewhere there would be another observable universe, also with a 46 billion light year radius, that is identical to ours. If so, it would contain a duplicate version of you reading a duplicate of this blog entry. Actually, throughout infinite space, there would be infinite copies of you reading infinite copies of this blog entry. And there would be an infinite number of you in every possible variation of you leading every possible variation of your life.

These duplicate universes would exist within the same space that we inhabit, have the same laws of physics, and the same or varying histories. But even the closest would be unimaginably far way. Tegmark states that current calculations show that it would be 10^{1029} meters from here, which of course is an indescribable distance.

Do these duplicate regions, these parallel universes, exist? If space is infinite, or at least 10^{1029} meters in extent, it seems hard to argue that they don’t. If space is that large. But we don’t know yet whether or not it is. It might be, or it might not be.

Current measurements show space to be flat, but as I said above, the margin of error on these measurements leaves room for space to still eventually loop back onto itself, to make it that if you could travel in a certain direction long enough, you’d end up back at your starting point, similar to what happens on Earth, but in three dimensions.

But if space is flat and infinite, and these duplicate regions do exist, it’s hard to imagine how their existence could ever have an effect on us (unless maybe if someone invented an intergalactic warp drive). This also seems to make the idea unfalsifiable. Still, the concept, while speculative, is a fascinating one.

This is the simplest of the multiverses that Tegmark discusses in his book. In the next post in this series, I’ll discuss his Level II Multiverse.

James Franklin has an interesting piece today at Aeon, asking what exactly mathematics is. He looks at Nominalism and Platonism, but discounts both in favor of Aristotelian Realism, which is something I’d not heard of before but seems equivalent to the idea that mathematics is empirical.

What is mathematics about? We know what biology is about; it’s about living things. Or more exactly, the living aspects of living things – the motion of a cat thrown out of a window is a matter for physics, but its physiology is a topic for biology. Oceanography is about oceans; sociology is about human behaviour in the mass, long-term; and so on. When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.

I’m in the final chapters of Max Tegmark’s ‘Our Mathematical Universe’ and will have a lot of thoughts about this soon. Right now, I’ll just note that I do happen to think we learn mathematics initially through perception (some of which is instinctual) but that many derived mathematical structures built on top of those empirical foundations are non-physical.

The question is are those non-physical concepts “real”, meaning Platonism, or are they simply structures built on an incomplete or simplified understanding of our reality, that is built on quantitative relationships and patterns we have observed, but ignoring or in ignorance of constraints that prevent the physical embodiment of those derived structures?

Corey Powell has an interesting post up on what he calls the Four Great Eras of Exploration. The first era was Galileo’s discovery of the vastness of the universe, the second that stars were composed of chemical elements, and the third was Hubble’s discovery of other galaxies. The fourth, and main topic of his post, is the current age of discovery of exoplanets.

But for this post, I’m focusing on the third:

Overnight, the Andromeda Nebula became the Andromeda Galaxy, and our home galaxy became just one of a multitude. A scant 6 years later, Hubble measured the motions of those other galaxies and discovered that they were systematically moving away from us, with their speed directly proportional to their distance. This was the discovery of the expanding universe, which led to the idea of the Big Bang, galaxy evolution, dark energy, and all the other wild concepts of modern cosmology.

I still find it mind-boggling that less than a century ago nobody even knew whether other galaxies existed. The pace of astronomical discovery is truly shocking when you step back and look at it.

I very much agree with that last paragraph. Less than a century ago, we thought our galaxy was all that was, the entire universe. Reading this post reminded me of an old archaic term for galaxies: ‘island universes’, which I recall seeing in very old science fiction stories. The term didn’t stick because ‘universe’ was then understood to mean all of reality.

Thinking about our multiverse discussions, and Tegmark’s levels of multiverse, it occurs to me that galaxies could have been considered other universes at one time, and the space they’re all in, the multiverse. This is a counter-factual of course, since that’s not where the terminology went.

But it makes me wonder what might happen if we ever actually did discover other universes as described in one or more of the multiverse theories. Would we end up calling them universes, or something else like bubble, region, brane, or whatever ended up being descriptive? Maybe the term ‘universe’ might continue to encompass all of reality.

What if our currently observable universe was part of a large structure that was just one of several such structures separated from each other by vast voids. Would those other remote structures count as other universes, or just new regions of the current one? At what point is another aspect of reality another universe instead of just a previously unknown aspect of the universe?

Depending how we spliced it, the word ‘universe’ might become like ‘world’, where it would figuratively be used to refer to all of reality, but technically mean a distinct subset of it, similar to how ‘world’ now effectively means ‘planet’.

This line of thought (admittedly largely semantic) reminds me of something cultural anthropologists are always warning us about, that much of how we perceive reality is essentially convention. Conventions which are often ultimately the result of historical accidents.