Of course, it's just a photograph of the jar, so you don't have enough information to answer the question. But if I were to give you a jar just like this, containing sand and rocks of various sizes, how would you count? What, exactly, is a rock? We don't ordinarily think of sand as being rocks but that's precisely what sand is, a whole lot of really small rocks. With enough patience, you could actually work through all the grains of sand in this jar, and give some definite answer. "There are XYZ rocks in this jar."
The point of this question is to think about relative scale and how this connects to the question of what is discrete and what is continuous. After all, we could grind the sand in this jar down into ever finer grit, so that the process of counting would become so laborious and time-consuming that you would give up. If we continue on grinding the sand down in this way, we know that we will eventually reach the point that we have broken the sand down into its molecular components, at which point, it will become impossible to keep it from dispersing into the air as an aerosol. Air, being continuous, is a gas and is not discrete -- it is continuous.
Let's consider for a moment the known mechanics of how the brain tells apart distinct objects. When we look at a photo containing two distinct objects, we know immediately whether they are different or the same:
The point of the rocks in the jar is that rocks, pebbles and sand show how our naive confidence in our ability to number things breaks down when we encounter objects at the limits of our perception. We can easily tell a dog from a tree under virtually any condition except the complete absence of light or being so far away that we cannot distinguish any objects at all. But defining the difference between a rock and sand is hard. Unless we are very close to it, we perceive sand in a similar way to how we perceive water. It is virtually fluid and it might as well be a continuum, even though we can see, upon close inspection, that it is not. Sand stands at the cross-roads between the discrete (rocks) and the continuous (air, water, etc.) For anyone who has studied the calculus, this will seem very natural. If you cut something up finely enough (infinitely fine, in the limit), you will be able to find its precise length, area or volume, no matter how complicated that object may originally have been.
Some organisms have much more limited senses and pattern-recognition ability than we do. For example, there are simple underwater life forms that can move towards warmth and away from cold by detecting a temperature difference. There are some animals with only the ability to sense the presence of light (and its direction) but are otherwise completely blind. And so on. Such limited sense abilities are almost binary, like "ON" and "OFF". Even though our brains are much fancier neuronal systems, we are really utilizing the same physical mechanisms, just refined and extended to a much greater degree. In short, pattern-recognition -- whether simple or complex -- is the same phenomenon in all living things.
Let's consider the set of natural numbers, or counting numbers. They are 1, 2, 3, ... The ellipsis can be read "on and on" or "so on, and so forth." It's a stand-in for the simple idea that we can always extend this sequence by incrementing the last element. If the last element were 173, we could extend it by adding 1, giving 174. The same is true for any other number. And we all know who the winner is in the Name the Biggest Number game: infinity!
But what, exactly, is infinity? Nobody has ever seen, heard or thought anything that is infinite. Every particular fact of our world and ourselves is finite. While infinity is common in modern mathematics, some mathematicians reject the idea of infinity altogether - this view is called finitism. Some of them have a conceptual objection to the idea of infinity. Some of them have formal objections to the idea of infinity. The view I will espouse here is that infinity is as real and useful as any other sort of mathematical object.
Let me begin by pointing out that number and enumeration are themselves really a metaphorical construct, built on our capacity to recognize patterns. Consider the image above depicting a tree and a dog. We say "there is one tree and one dog in this image," as though this is a fundamental fact of reality. But look again at the image of the jar of rocks -- how many rocks are in that jar? Is there any fundamental fact of reality that would serve as a reasonable answer to the question "How many rocks are in the jar?" We might object that the question is ill-posed in the case of the jar of rocks but this is just pedantry. The reason there is no good answer to the question is because what we mean by "the number of things" is itself vague in many cases, and this happens to be one of them!
At the risk of belaboring the point, let us spell out what we mean by the question, "How many dogs or trees or rocks are there in this image?" What we really mean is what answer would an ordinary human give when given the image and the question[1]. So, we can reduce the question of numeration itself to an empirical test by simply giving images to people and querying them as to how many objects of such-and-such type are in the image. (In fact, we could just as well use physical objects or images.) So, when we say, "there is one tree in this image," what we really mean is that any ordinary person would agree that they see one tree in the image (and we ourselves, being an ordinary person, also see this).
But as I pointed out above, pattern-recognition is not magic, nor is it unique to humans. In other words, there is no inherent "number-ness" in the one tree or one dog in the image. For any practitioner of Machine Learning, this is embarrassingly obvious -- an image is a large, two-dimensional array of numerical values called "pixels" [2] and there is no obvious answer to the question, "how many objects are there in this array of pixels?" Our brains are so accustomed to visual recognition of objects that it seems to us automatic and so fundamental as to be beyond analysis. How many trees in the image? There just is one tree, it needs no further elucidation. But imagine trying to write a program that would answer the question "How many objects are there in this image?" for any image I choose to give you. It turns out that this problem is so hard that it is unlikely that any human could ever write a program to do it. Instead, we have turned to Machine Learning algorithms to solve the problem.
So far, I have not mentioned the continuum. The continuum is an important concept in mathematics. It has been heavily studied but there are still important, unresolved foundational questions surrounding it. What is the continuum? Well, there are several ways to think about it but perhaps the easiest intuition is to think of an ideal geometric line, in the same way that the Greeks thought about it. An ideal geometric line is an analytical object that extends an ordinary line we might draw on a sheet of paper by imagining a line that is perfectly straight, having zero width, having some non-zero length, and being "dense" in some sense that, no matter how far you zoom in, you can't find any "gaps" in the line. Such a line corresponds to the mathematical idea of continuity.
In non-mathematical language, what the Continuum Hypothesis (CH) is really asking is how many points are there on the line? A naive answer would be "infinitely many." One reason this question is so interesting to mathematicians is that it turns out that the naive answer is wrong! Suppose you attempt to map the natural numbers 1, 2, 3, ... to the geometric line in such a way as to leave no gaps. 20th-century mathematics has proved that it is impossible to pack the natural numbers on to the line so densely that it is impossible to find a gap between any two points, even though there are infinitely many of them! Even if we map the rational numbers (a/b for any natural numbers a and b) to the number line, there will not be enough of them to make sure there are no gaps on the line. Therefore, in some very fundamental sense, there are "more than infinitely many" points on the line. This is a surprising result.
This isn't quite the whole story on the CH. There is a set of numbers which is "infinitely larger" than the set of all natural numbers. This set can be formed by taking the power set of the set of all natural numbers. This set (called omega-1) is so dense that no gaps can be found between any two distinct points, which gives it a suspicious resemblance to the continuum. While no one has proved that omega-1 is coextensive with the continuum, the CH is the hypothetical position that this the case (even though it is commonly stated in different terms, as it is on Wikipedia). The other alternative is that the continuum maps to some higher infinity[3].
One of the reasons I find this topic interesting is that it has indirect implications for the material world. In quantum theory, we can recognize some loose correspondence between the continuum and waves (continuous phenomena), and between discrete sets (such as the natural numbers) and particles (discrete phenomena).
To pursue this connection further, let's return to the problem of trying to write a program that can recognize objects. Recent developments in Machine Learning have led to a technology called super-resolution. Super-resolution works a lot like the fictional depictions in movies where spy satellite images can "freeze and enhance!" It allows more detail to be added to a low-resolution image by training a neural net to interpolate or "hallucinate" the missing details. Obviously, this won't give you license-plate numbers from a grainy image of a vehicle on a distant horizon, but it will allow a significant -- perhaps massive -- reduction in the amount of bandwidth required to transmit images, movies, audio and other forms of data.
What does this have to do with either infinity or the continuum? Well, if we suppose that Nature is as frugal with information as she is with all other material resources, it seems impossible to believe that the world constructed by our perception actually corresponds to what we imagine it to be. Let me explain. Suppose you are sitting on a mountain-top on a clear day, surveying a wide valley beneath you. As your eyes scan across the horizon, there are countless details that they can perceive. In fact, I have experienced a slight feeling of vertigo when looking across a large landscape, not from fear of heights or anything like that, but from the sheer immensity of how much detail is visible. It's almost like the brain is struggling to take it all in, there's just so much. But the idea of super-resolution tells us that almost all of the photons striking the eye are wasted information! If I look at a tree, my brain can (and does) interpolate the details of the tree, to one extent or another[4]. When we look very closely at something (with the "high-resolution" part of the eye), there is obviously much less interpolation occurring but we do not know that it is zero. In any case, the fact remains that physical theories based on the extrapolation from subjective experience assert that all the photons reflected off every leaf of every tree and every pebble or grain of sand in your field of view... every single one of those photons is striking your retina and most of them simply cannot be resolved because you only have so many rods and cones in your retina (resolution limit).
In computer systems, we use a "query-and-response" system to manage large information stores. This is precisely how any search engine works. You do not download the entire search engine's index (this index would be many terabytes or even petabytes in size!) Rather, you send a query to the search engine saying, "This is the information I'm looking for", and the search engine responds with the information it has, relevant to your query. If we take the observer effect seriously, it is not difficult to imagine that our observation of the material world has the same effect upon it.
This might seem a little strange because it feels like we're saying, "There are no trees until you open your eyes and look at them." But I think that's an over-simplification that misses the mark. It's not that there are no trees until we look. It's that there is no visual information about the trees that are there, until we open our eyes and look. An astute reader will object again, "Ah, but there is visual information since the photons produce enough light by reflection that some of the light is going through the eyelids, even when closed." Once again, this misses the point - we could put on black-out goggles or go indoors and close the door, or whatever. The point is that no information is sent but that information which would make a difference. If there is no effect upon the observer, there is no information transmitted at all. In short, when the proverbial tree falls with no one around to hear, it really makes no sound.
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[1] Note that I am not trying to establish the supremacy of empirical methods over analytical methods. As I see it, both have a place in any well-developed system of investigation into truth, as such.
[2] Actually, there are three such arrays in the case of color images, one each for the red, green and blue channels.
[3] Georg Cantor discovered that there is actually a hierarchy of infinite sets and that this hierarchy is infinitely large. The most basic hierarchy of infinite sets are called the aleph numbers and are formed by taking successive power sets of the set of natural numbers, N. Aleph-0 is the cardinality (magnitude/size) of N. Aleph-1 is the cardinality of the power set of N which is denoted 2aleph-0. Aleph-2 is the cardinality of the power set of aleph-1. And so on. This hierarchy is infinitely high but it does not exhaust infinite cardinals. There is a branch of mathematics that studies infinite cardinals beyond the alephs, it is called large cardinal theory.
[4] We know that it interpolates because of the way the peripheral vision works, especially the blind-spot.
