Suppose, for example, that instead of the collected works of Shakespeare, we set our sights on something a bit less daunting, say, a 500-word essay I wrote the other day – about two, double-spaced pages in length. Further, suppose we perform the experiment with a computer instead of a monkey. Computers can generate thousands of random strings of characters per second and check their own work. Finally, instead of using a single computer, let’s run many in parallel, and we’ll stop when any one of the computers first completes the essay.

Suppose we had one computer for every atom in the universe (this is a thought experiment, so let’s think big). We’ll let the computers run from the beginning of the Big Bang to one hypothesized end of the universe known as the Big Freeze. Now we can ask: what is the chance one of the computers will type the essay in that time frame?

The answer? The chance is so small there’s nothing we can meaningfully compare it with. For more details, see http://www.uh.edu/engines/epi2924.htm.

In fairness, this result isn’t beyond comprehension. It’s surprising to most people, but the math is elementary. Where we’re led into the world of incomprehensibility is when we look at the complexity of, for example, the human body, where we can reasonably ask if the universe has been around long enough to generate the twenty thousand or so human genes found in DNA, each metaphorically equivalent to an essay. The alphabet of life consists of only four letters (nucleotide bases), but the twenty thousand essays are typically much longer than 500 words. Looking at purely random variations, the situation is stupefyingly worse than trying to randomly type a 500-word essay.

So what are we to make of this? First of all, random typing isn’t a good comparison with the way evolution operates. While evolution uses random mutations to create life, it doesn’t do so by trying everything until it finds something that works. Evolution functions in a more orderly fashion. We don’t fully understand all the steps along the way – for example, why molecules bind together to form life in the first place – but evolution most certainly

Yet having said this, I do think there’s a cautionary tale here. We tacitly assume that the earth’s been around long enough for randomness to do its magic. But when we work the numbers, it calls into question whether “long enough” has really been “long enough” to create something as immensely complex as human life. Remember, we’re not just talking recipes but the way the recipes work together to keep everything functioning. While I argue that the very existence of human consciousness is beyond comprehension, the fact that human life could come about randomly in the amount of time since the Big Bang only adds to the incomprehensibility.]]>

Volume and area are two different measures. Consider a box 10 feet on a side. It’s volume is 10 x 10 x 10 = 1000 cubic feet, while its surface area is the sum of the area of each of the six sides, 6 x 10 x 10 = 60 square feet. The point is that volume is a measure of

Location in a single dimension can be expressed with a single number (x); in two dimensions with a pair of numbers (x, y), and in three dimensions with a triple of numbers (x, y, z). If we look at our box that’s 10 feet on a side, we can think of the box corresponding to all coordinates (x, y, z) where x, y, and z take on all values between 0 and 10. The side on which the box rests is a square that corresponds to all coordinates (x, y, z) where x and y take on all values between 0 and 10 but z is limited to the single value 0. Analytically, the box can be thought of as made up of infinitely many squares stacked on top of one another, each square corresponding to a different value of z. Looked at in this way, we see that volume is very much a three-dimensional construct that appears infinitely larger than the two-dimensional surfaces. Is it reasonable to even make a comparison of the size of these two entities?

It all depends on what we mean by size. Size, in and of itself, isn’t a mathematically defined concept. That doesn’t mean the concept of size isn’t important. It just means that we have different definitions of size for different purposes. Two frequently used definitions are that of

Measure, in its various forms, is typically what we think of when dealing with areas, volumes and the like. It requires a deeper understanding of the foundations of math than can be covered here. However, I note that the definition of a measure is intimately related with the notion of dimension. The three-dimensional measure, or volume, of the

Even so, the fundamental conundrum remains. If we set out to measure the volume of the trumpet with tape measure and pencil (hypothetically, of course), we’d find it was finite, while we’d find the surface area was infinite using the same set of tools. We’re still left to ask “does this make intuitive sense”?

We can further complicate the issue by considering another approach to size, cardinality. Unlike measure, the concept of cardinality doesn’t depend on dimension. Instead, it focuses on the number of things in each set: how many points are in the surface when compared with the number of points in the box? While it may seem obvious that the number of points in the box is larger than the number of points in one side, Georg Cantor showed that the number was the same in that he could find a one-to-one correspondence between the two sets; that is, he could find a way to match one point in the one side of the box (or it’s entire surface) with exactly one point in the entire box. If this seems preposterous, it is. But the construction is reasonably straightforward (I cover it in detail in my book). So from the perspective of cardinality – the ability to show a one-to-one correspondence between the sets – the box and its surface are the same “size.” The same is true for Torricelli’s Trumpet and its surface.

Thus, at this more mathematical level, we might ask if it makes sense that two sets of the same cardinality –Torricelli’s Trumpet and its surface – have such different measures – one finite, the other infinite. The measure theorist might object that cardinality and measure are fundamentally different beasts. But the question raised by Torricelli’s Trumpet isn’t one of formalism. No matter how deeply we dive, we’re left with one, simple, basic question: does a figure with finite volume and infinite surface area make intuitive sense?]]>

So why have so few people heard or thought of it?

I suspect people don’t think about it because it’s just not the way we perceive the world. We innately feel that we can know everything if we just work hard enough. It took me decades of reflection before the BC conjecture hit me over the head and started to sink in.

The reason most people have never heard of the conjecture is that it’s a non-starter. It’s an interesting observation, but then what do you do with it? Various philosophers have touched on the idea: John Locke, Immanuel Kant, Noam Chomsky, Colin McGinn, Thomas Nagel, and John Searle are among those I’m aware of. If anyone is familiar with other scholars I’d love to hear of them. But once the point is made, there’s no way to develop it.

In my book I don’t develop the logical argument more than anyone else has, although I do offer my own perspectives. But the BC conjecture helped provide a framework for many of the perplexing conundrums we face in mathematics and the sciences. And the conundrums provide evidentiary support for the conjecture.]]>

As one very, very basic example, consider the following question: What’s the next number larger than 0? Suppose I claimed I had such a number. Call it x. If we take x and divide it by 2, we have a number that’s even closer to 0. From this, we conclude there’s no next larger number than 0.

Now consider the following experiment. Take a piece of paper and put two points on it. Label one point 0 and the other point 1. Now imagine a line segment connecting the two points. We can associate each point on the line segment with a number between 0 and 1. One-half would correspond to the point halfway between 0 and 1, one-tenth would correspond to the point one-tenth of the way from 0 to 1, and so on.

Suppose we now draw the line segment. As we start to draw, we must leave the point 0, and in doing so the pencil must pass over a next point on the line segment – the next number larger than 0. But it doesn’t, because there is no next number larger than 0. How can that be?]]>

The same is true of some truly remarkable facts about our world. Let me take Zeno’s paradoxes as an example, since I find they’re so familiar that many people simply roll their eyes when you bring them up.

I remember with clarity the day my father introduced me to a version of one. We were driving home from an A&W drive-through with a quart of root beer back when you couldn’t get A&W root beer in the store. He pointed out that in order to get home, we’d need to go half way, then half of the remaining distance, then half of the remaining distance, and so on. “Therefore,” he concluded for me, “we’ll never get home.”

As wonderfully surprising as his argument was, by the end of the day I was no longer puzzling over it. As I ran into it again and again over the years, it ceased to evoke any wonder. And when I was introduced to limits in high school I thought I had the explanation I needed (see my blog of 19 October 2017,

It was only decades later that I again came to appreciate Zeno’s paradoxes as I realized that some things are likely beyond comprehension.

I can’t recreate the wonder I once felt – that’s one of the consequences of familiarity. But I can appreciate that something’s askew. In traveling home with my father, not only did we have to cover an infinite number of distances, but individually each one had finite length. When would we actually cross the threshold and arrive at our destination?

Hopefully many of the examples in my book are new to the reader and elicit a sense of wonder. Others, like Zeno’s paradoxes, may not. But that doesn’t mean we shouldn’t stop and consider them one more time. The world’s filled with surprises. We’ve often just become too familiar with them to appreciate what’s right in front of us. ]]>

Recognizing that there likely exist things beyond human comprehension—beyond our ability to even know what we’re missing—doesn’t provide free license to believe in anything whatsoever. I have a chance to address this in detail in the book, but the length and format of spoken presentations limits what I can cover.

As humans, we have the ability to reason, and the use of reason is at the core of what we’ve achieved as a species. We can’t throw it out just because there may exist things we don’t have the ability to reason about. When a listener suggests that she now feels better able to defend her belief in astrology, I politely tell her no. Jumping to such conclusions is the bad side of the Beyond Comprehension conjecture.

The good side—apart from the joy of contemplating the many paradoxes we’re confronted with—is recognizing where we may be overstepping our cognitive limitations in the name of science. For example, it’s routinely argued by philosophers and neurobiologists that human free will doesn’t exist—the universe is deterministic and that’s it. Yet, we don’t understand why a relatively small collection of molecules—more than 85 percent of which are water and carbon—can give rise to human consciousness in the first place. That should be a clear warning sign that we need to be careful when making sweeping statements about free will.]]>

The number 1 needs no explanation. The number .999…, however, requires more thought. Starting down the road of formalization, it represents 9/10 + 9/100 + 9/1000 + … on to infinity. But formally we can’t carry out an infinite summation. Recognizing this, mathematicians have long relied on the notion of a limit. We can take enough terms in the summation 9/10 + 9/100 + 9/1000 + … to make the partial sum as close as we want to 1. In mathematical parlance, “in the limit as we take evermore terms in the summation 9/10 + 9/100 + 9/1000 + … we can get as close as we want to 1.”

The genius of this approach is that we sidestep the notion of performing an infinite summation. We’re always working in the land of the finite. The drawback is that we can’t say that 1 = .999…

It’s important to realize this isn’t a mere technicality. Admittedly, mathematicians will frequently write “1 = .999…” as shorthand, but infinite summations can lead to fundamental problems, which is why the notion of a limit was adopted in the first place.

With the expression .999… looked at through the lens of the limit, we start to see a version of one of Zeno’s famous paradoxes that goes as follows. A man reaches out to a wall. To do so, his hand must reach 9/10 of the way to the wall, then 9/10 of the remaining distance, then 9/10 of the remaining distance, and so on. Looked at in this way, his hand must cover an infinite number of distances, each of which individually has finite length. How, then, can his hand actually reach the wall?

The common response is “That’s what the notion of a limit was introduced for. If you understand limits the paradox disappears.” But that’s absolutely not the case. Translating the notion of a limit to the case of a man reaching toward a wall, a mathematician would say “yes, in reaching toward the wall in the way described, the hand can get as close as desired by passing through a large, though finite, number of distances.” As a mathematician, however, the individual would have to remain silent about actually reaching the wall. Zeno’s paradoxes are alive and well.

So does 1 = .999…? Working with the formal definitions provided by limits, we know the expressions can be used interchangeably. But are they actually equal? If we could carry out an infinite summation we’d find out.]]>

However, I believe it’s completely possible to encounter shadows of things that are beyond comprehension. I’ve touched upon some examples in my book, but there are many more. I maintain an ever-growing list, and I’d be happy to hear from readers with their thoughts, ideas and insights.

So the more appropriate question is "when are we peering at shadows of the incomprehensible?" To a large extent the answer is personal. That which is beyond comprehension is that which strikes

Let me take up one example today and come back to others in future posts.

It’s possible to construct any number of containers that have finite volume but stretch to infinity and have no bottom. Torricelli’s trumpet, covered in my book, is but one example. The question that presents itself is the following. How do you fill a container that has a hole in it? It would seem that any liquid you filled it with would simply continue forever downward, so that you could continue filling it without stop. Thus, how could the container have finite volume?

It’s true that if we attempted to fill any such container with a real liquid such as water, the water molecules would eventually get jammed in the neck of the container when it became too thin. But the bottomless containers we’re talking about are mathematical constructions, so we should feel free to fill them with another mathematical construction: dimensionless water drops.

It’s at this point that you must ponder the alternatives for yourself. What’s going on? For the mathematically inclined it’s possible to get into all sorts of detailed arguments about this and that. None are fully satisfying. Many miss the point. My personal perspective is that the problem stems from the way in which different mathematical dimensions relate to one another. There’s no "problem" about dimensions that one can point to, but the type of situation encountered here pops up in various scenarios. And even in all their mathematical glory, these scenarios remain head scratching.

But the simple point we can all agree upon, whether mathematically inclined or not, is the following: there exist infinite, bottomless containers that have finite volume. "This bottomless container contains six gallons" is a perfectly valid statement. But does it make any sense?

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humans can't comprehend -- isn't altogether new. It's been raised by various philosophers

over the centuries. One of the earliest references I'm familiar with since the time of the

Enlightenment can be found in John Locke's

entirety of the brief

It's definitely worth taking the time to read.

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The paper and responses are welcomely understandable, at least to gather the gist of the controversy. I don't intend a summary here. But I did want to bring up one issue in particular.

The authors of the paper call into question inflationary theories of the origin of the universe, whereby the universe underwent super-rapid expansion in the fractions of a second after the big bang. This, of course, doesn't sit well with proponents of inflationary theories, which are apparently the dominant collection of theorists in the field.

One major objection the authors put forth is that inflationary theories, as they presently stand, lead to a multiverse -- an infinite collection of universes each with its own properties. As is so commonly the case with multiverse theories, our own universe is then explained as one very random fluctuation. The authors refer to this state of affairs as both a "multimess" and unscientific, being something that isn't scientifically verifiable.

I hope the term multimess catches on. As I make a case for in my book, infinity gives rise to all sorts of truly remarkable paradoxes that remind us we need to be extremely careful when working with it. Multiverse theories have never sat well with me because they dive headlong into the depths of infinity to derive their thus-far scientifically unverifiable conclusions.

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