8. Atom
This is the eighth article in the series From Particles to Angels. If you are interested in this article you should read the previous articles in the series in order, beginning with the first (On Happiness).
In science and philosophy it can seem that the closer we get to things, the more their substantiality slips through our fingers. "Matter" is the very archetype and essence of reality and solidity. In essence, "materialism" says that there is only matter, and anything that isn't made of matter doesn't exist. With Einstein's formula E = mc2, that reality has been extended to include "energy"; that is, effectively, photons; although they do not have mass. In the previous two articles: Mysteries of Light - Part 1 and Mysteries of Light - Part 2 we examined the nature of photons (light). In this article we take a closer look at what is this stuff "matter" that is the basis of materialism.
When Conan the barbarian looked at the iron head of his axe, he likely considered it to be made of solid, smooth iron. And when he thought the word "solid" he likely imagined a substance that was continuous, solid in the sense of "not containing empty spaces". He likely considered the volume of space occupied by his axe-head to be completely filled with iron, through and through. Today of course, we know better. We know the iron of his axe-head is composed of tiny little particles called "atoms", and that it is mostly empty space. It does not merely contain holes like a honeycomb, but the particles that compose it are separated at a distance, like stars in the night sky, held in place only by invisible force fields. That is, they are held in place by the exchange of photons telling each particle where it should be.
"Ever since Ernest Rutherford discovered the atomic nucleus in 1911, physicists have known that the radius of an atom is about 100,000 times larger than the radius of an atomic nucleus. Here is a comparison to dramatize the factor 100,000. If a nucleus were the size of the pencil point in a mechanical pencil (or the ball in a ball-point pen), then the outermost electrons of the atom would be about one football field away."
"Leucippus, however, thought he had a theory which harmonized with sense-perception and would not abolish either coming-to-be and passing-away or motion and the multiplicity of things. He made these concessions to the facts of perception: on the other hand, he conceded to the Monists that there could be no motion without a void. The result is a theory which he states as follows: ‘The void is a "not being", and no part of "what is" is a "not-being"; for what "is" in the strict sense of the term is an absolute plenum. This plenum, however, is not "one": on the contrary, it is a "many” infinite in number and invisible owing to the minuteness of their bulk. The "many" move in the void (for there is a void): and by coming together they produce "coming to-be", while by separating they produce "passing-away". Moreover, they act and suffer action wherever they chance to be in contact (for there they are not "one"), and they generate by being put together and becoming intertwined.'"
Our knowledge of Leucippus' views comes to us from later philosophers writing about him. Leucippus and Democritus are now referred to as "Pre-Socratic philosophers", two of the group of Greek philosophers who preceded Socrates (470 – 399 BC), the teacher of Plato. Plato's writings are mostly writing about what Socrates had to say, although it is commonly assumed that he also included some of his own views in these writings, so that it is not known how much is Socrates and how much Plato. Plato is the first philosopher of whose writings we have more than small fragments.
Plato himself had his own brand of atomic theory, sometimes referred to as "geometric atomism". Atoms for Leucippus and Democritus were like tiny pebbles, an endless variety of irregular shapes all swirling and colliding. Plato suggested that the fundamental elements were triangles that formed special geometric solids now referred to as "Platonic Solids": specifically the tetrahedron, cube, octahedron, dodecahedron and icosahedron. These solids are special in that each one has all its faces identical. Plato believed in the four elements: fire, air, water and earth, associating each with a particular one of the Platonic Solids. The tetrahedron was fire, the cube was earth, the octahedron was air and the icosahedron was water. The dodecahedron was used somehow "in the delineation of the universe" ("Timaeus"). Instead of the aether (Aristotle's fifth element), Plato had something he called the "receptacle", which was apparently space, but which had the ability to shake like a sieve and sort and agitate the particles that occupied it. It was this property of the receptacle that facilitated earth to fall and fire to rise according to their natural attraction for their own kind.
Although Plato's ideas seem naïve, if we interpret his triangles to represent three particles held together by a force, his neat geometric model has many similarities to the very geometric form of real atoms and molecules. But of course, fire isn't really composed of little tetrahedrons, or water of little icosahedrons. If we consider fire to be composed of photons, then it is composed of point particles. Water molecules: H2O (that is: two hydrogen atoms and one oxygen atom) are little isosceles triangles. Methane molecules however are little tetrahedrons, and sulphur hexafluoride is little octahedrons, so Plato's geometric intuitions were not so very far off.
Aristotle was strongly against the idea of a vacuum, imagining space to be filled with continuous substance like a fluid. Movement could occur because the fluid background could slide around objects that moved through it. On the contrary he believed motion was impossible in a vacuum because he did not have the concept of momentum, and there would be nothing to push objects in this direction or that. The Dark Ages officially began around 529 AD, when the emperor Justinian closed all the philosophical schools in Athens, and after the destruction of what was left of the great Alexandrian Library in 391 AD in response to an order from the emperor Theodosius. The collapsing secular Roman Empire had transformed into the invigorated Holy Roman Empire. The state had failed to suppress the growing grass roots Christian religion and now instead adopted it, forming the Catholic Church and making it identical with the state, ruthlessly enforcing a narrow orthodoxy of belief. Like the other Greek philosophers, Aristotle was suppressed for several centuries in the West, but a slightly Christianised interpretation of his ideas eventually emerged and was adopted as official doctrine on all things philosophical and scientific. Through no fault of Aristotle himself, there followed several centuries of adoring commentaries on Aristotle's writings by obedient Scholastic academics. Throughout all of this period atomic theory lay in a long disfavour, still in place as we have seen in the 19th-century. The Dark Ages ended as ancient Greek knowledge continued trickling slowly back into Europe, or as the mindset changed sufficiently for it to be accepted, as the Renaissance and Enlightenment took shape, but this incredible intellectual hiatus meant that as science ("natural philosophy") came back to life, its proponents found themselves arguing against the entrenched doctrines of a canonised authority who had died two-millennia before, during which interval virtually no scientific work had been done in the West. A time when cutting edge philosophers believed the Earth was the centre of the universe. The contest between "religion and science" that Galileo was a party to was a contest between the old Aristotelian science and the new science, between new ideas and an existing institutionalised authority.
Atomic theory (and its cousin "corpuscularianism") began to re-emerge with proponents such as Francis Bacon (1561 – 1626), Galileo Galilei (1564–1642), Rene Descartes (1596–1650), Pierre Gassendi (1592–1655), Robert Boyle (1627 – 1692) and Roger Boscovich (1711 – 1787). The word "atom" was revived in science in 1805 by the British chemist John Dalton (1766-1844). We learned of the subsequent history of atomic theory in Mysteries of Light - Part 1 and Mysteries of Light - Part 2.
"It is found too that the same body changes its capacity for heat, or apparently assumes a new affinity, with a change of form. This no doubt arises from a new arrangement or disposition of its ultimate particles, by which their atmospheres of heat are influenced: Thus a solid body, as ice, on becoming liquid, acquires a larger capacity for heat, even though its bulk is diminished; and a liquid, as water, acquires a larger capacity for heat on being converted into an elastic fluid; this last increase is occasioned, we may conceive, solely by its being increased in bulk, in consequence of which every atom of liquid possesses a larger sphere than before." (emphasis added)
Gluons (that bind quarks into protons and neutrons and bind protons and neutrons together in the atomic nucleus) are also massless, and if gravitons exist, they are presumed to be massless as well. The photon, W boson, Z boson, gluon and graviton are all referred to as "force carriers" because they transmit the known forces. These particles are also all referred to as "bosons". Bosons have the peculiar property that more than one of them can be in the same place at the same time. In fact you can have as many of them as you like in the same place. The quarks, electrons and neutrinos (that each have mass) are all referred to as "fermions", and these obey what is called the "Pauli exclusion principle", which means you cannot have two of them in exactly the same place at exactly the same time. This property of matter is kind of what makes matter matter. Two objects are solid because they cannot be in the same place at the same time.
Because bosons can occupy the same place at the same time, they make up what we think of as fields and waves. Fields and waves have nice neat mathematical properties as a result of the fact that they pass right through each other when they collide. If you shine one torch beam through another they won't interfere with each other. Imagine if photons behaved like billiard balls on a pool table bouncing off each other chaotically. It would be a mess. Then we wouldn't have nice neat equations in electromagnetics, or the elegant properties of wave addition.
The three kinds of matter: electrons, up quarks and down quarks; are all like products rolled off of a factory production line. Every electron is exactly like every other electron in the universe. Every up quark or down quark is exactly like every other up quark or down quark, respectively, in the universe. And thus, every proton or neutron is exactly like every other proton or neutron, respectively, in the universe. Every electron has a mass of 0.5 and an electric charge of -1. Every up quark has a mass of 2.4 and an electric charge of 2/3. Every down quark has a mass of 4.8 and an electric charge of -1/3. Recall that a proton is composed of two up quarks and one down quark. Every proton has a mass of 938.3 and an electric charge of 1. Notice that the masses of the constituent quarks don't add up, but the charges of the constituent quarks do. Most of the mass of the proton is in the form of what is called "binding energy". Here we see the massless force carriers, "energy" (in this case gluons) being spoken of as if they have mass, or are somehow able to manifest mass. The vast bulk of the mass of protons and neutrons does not come from the mass of the quarks composing them, but from the "energy" associated with them. The difference between "mass" and "not mass" is not a difference between "something real that exists" and "something not real that does not exist"; rather, "mass" is energy behaving one way, and "not mass" is energy behaving another way. The bottom line then, is that matter as mass cannot serve as the fundamental basis of "real", because it is a derived behaviour of something else which does not possess mass. The neutron is composed of two down quarks and one up quark. It has a mass of 939.6 and no electric charge (notice that the charges of the constituent quarks add up to zero). Fundamental particles are like Lego blocks, extraordinarily simple and neat objects.
When I gave the masses of the particles above I did not say what were the units of measure I was using. That is, is mass being measured in pounds, kilograms, or what? The actual unit of measure is MeV/c2. The "V" in all of that is the Volt. It might seem strange to measure mass in Volts, but physicists typically do when working at this scale because energy is the more fundamental quantity. So mass is measured in terms of the quantity of energy it is convertible into via E = mc2. Mass is made out of energy. Matter is made out of something which itself is not matter.
In the macroscopic world, the everyday world, we are accustomed to force requiring work. If I want to hold a heavy suitcase over my head I have to strain my arms, which after a while will get tired and I will have to put the case down. To be able to do this work, even for a while, I need fuel: food, water and air. It is the same with machines that do work, whether a steam engine burning wood or coal, an electric power station burning coal or driven by steam heated by a nuclear reaction, or driven by wind or falling water, or a machine that runs on a battery, or gasoline and natural gas. When the fuel runs out, the machine stops working, the plant or animal dies. Even suns (stars) die when they eventually run out of fuel to burn. Eventually all the stars in all the galaxies will go out unless new stars are born to replace them.
But the fundamental particles are not like that. The force of gravity associated with a particle's mass does not run down. It always stays exactly the same. The electric charge of a particle always stays exactly the same. Two particles will stick together or repel each other with the same force for eternity, or until the universe ends, whichever comes first. A photon will always travel at exactly the speed of light. It won't ever get tired and slow down. The speed of light is usually given "in a vacuum", but this is not exactly because light travels slower when not in a vacuum, but because when not in a vacuum it is interacting with other particles it meets along the way, and those interactions take time.
Fundamental particles also have an innate behaviour called (quantum) "spin". The fermions (electron, up quark, down quark) always have a spin of 1/2. The bosons (photon, gluon, Z boson, W boson) always have a spin of 1 (except for the graviton if it exists).
What we call an atom today is not "indivisible" in the Greek sense. Soon after the atom was admitted into physics it was found to have parts. The electron was discovered by the English physicist Sir Joseph John Thomson (1856-1940) in 1897. The word "electron" was coined earlier in 1891 by the Irish physicist George J. Stoney (1826-1911). The New Zealand physicist, Ernest Rutherford (1871-1937) (1st Baron Rutherford of Nelson) discovered the atomic nucleus in 1911. Since electrons had a negative electric charge, Rutherford suggested the atomic nucleus contained particles with a positive charge, and he proposed the name "proton". Neutrons were discovered by the English physicist Sir James Chadwick (1891-1974) in 1932. The word "neutron" was coined in 1921 by the United States chemist William Draper Harkins (1873-1951) and alludes to the fact that these particles are electrically neutral (having neither a positive nor a negative charge). Later, protons and neutrons were found in turn to have parts, the quarks. Quarks are theorised to exist but no individual quark has ever actually been detected. In fact quarks cannot exist in isolation, by a rule called "confinement". So in that sense we could say that the hadrons (protons and neutrons) are indivisible. Our atoms are however the smallest units of the chemical elements. For example, an atom of oxygen (8 electrons around a nucleus of 8 protons and 8 neutrons) is the smallest unit of oxygen. If you break the atom apart into smaller pieces, it ceases to be oxygen and becomes some other element (such as some combination of nitrogen, carbon, boron, beryllium, lithium, helium or hydrogen), or free subatomic particles (electrons, protons and neutrons).
But the question of divisibility was more to the Greek philosophers than just determining the basic constituents of matter. The question held a philosophical dilemma. The pre-Socratic philosopher Zeno of Elea (ca. 490–430 BC) was a student of Parmenides (late sixth or early fifth century BC), and he is known for four paradoxes he posed to illustrate the teachings of Parmenides and which are still a cause for some mystification.
"The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. The result of the argument is that the slower is not overtaken"
This paradox is usually referred to as "Achilles and the Tortoise". It appears that tradition introduced the tortoise to represent what was originally just a slow runner. Achilles was a legendary hero of the Trojan War, apparently with a reputation for swiftness. He is imagined in a footrace with a tortoise, but the tortoise is given a head start of some distance. Let us say the distance is d. The two of them start racing, but really the paradox still works if the tortoise does not move at all. The question is posed: "When/where does Achilles overtake the tortoise?" The exact point and time will presumably depend on their respective speeds and the amount of the tortoise's head start. But now the presenter of the paradox raises the following difficulty. For simplicity we will imagine the tortoise to be motionless.
Before Achilles traverses the full distance (d) between him and the tortoise, he must first traverse half the distance (d/2). This means that he has the other half of the distance remaining (d/2). But before he traverses the whole remaining distance (d/2), he must first traverse half that distance (d/4). So that he now has one quarter of the total distance remaining (d/4). But before he traverses the whole remaining distance (d/4), he must first traverse half that distance (d/8). So that he now has one eighth of the total distance remaining (d/8), and so on. So Achilles is getting closer and closer to the tortoise, but when does he overtake it? If we continue: the distances he travels are: d/2 + d/4 + d/8 + d/16 + d/32 + d/64 ... and so on. This is an infinite series, because it is always possible to halve whatever distance remains. Although each new component of distance is smaller than the last, so that the additions to the distance travelled get smaller and smaller, still, however minute they become, they never become zero. What we have then is an infinite series containing elements all of which have non-zero magnitude. Any infinite number of non-zero magnitudes, however small the non-zero magnitudes, will add up to an infinite magnitude. This means that in order for Achilles to reach and overtake the tortoise, across the distance d, he must first traverse an infinite distance. Because it takes a non-zero amount of time to traverse each non-zero distance, it will take Achilles an infinite amount of time to traverse the infinity of non-zero distances. We know that the distance Achilles has to run is only d, and the time that it will take him depends simply on the speed he is running. If he is running at a speed of 1 d per minute, it will take him 1 minute to reach and overtake the tortoise. So how do we end up with an infinite distance and time? Aristotle had a clear understanding of the cause of the paradox.
"For there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number."
That is, things can be infinite in extent; for instance, we can surmise that the universe is infinite in extent, and that time is infinite in extent. We can also surmise that any continuous substance or magnitude is "infinitely divisible". That is, between any two points, will fit an infinite number, of infinitely small intervals. In mathematics, the Real Number Line is treated in this way. Between any two numbers, say 1.00002 and 1.00003 we can always give any number of other numbers; for example: 1.000021, 1.000022, 1,000029, 1.0000290001, etc. Aristotle offers the following solution to the paradox.
"So in the straight line in question any one of the points lying between the two extremes is potentially a middle-point: but it is not actually so unless that which is in motion divides the line by coming to a stand at that point and beginning its motion again.... Therefore to the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible: if they are potential, it is possible."
He suggests that the interval is only potentially infinitely divisible, and not actually infinitely divisible, because we will never actually arrive at having infinitely divided it. Rather than immersing ourselves deeply in this argument, I will say only the following. Aristotle's solution was accepted by many in Aristotle's own time, and down to the present day.
"Aristotle's solution to Zeno's paradoxes is genuinely a solution, and it is reasonable to suppose that it is not an invention of Aristotle's but represents the best mathematical thinking of his time"
Many others reject the solution.
"This is scarcely comprehensible, since it seems obvious that x is potentially ϕ only if it is possible that x should be actually ϕ.... For example, one might very well hold that the parts of a finite line do constitute an infinite totality, whether or not there has been any process of dividing the line into parts."
Calculus is the modern science that grapples with the infinite divisibility of physical quantities, and anyone who has learned calculus knows that they are only ever allowed to say that "n (the number of terms in the series) approaches infinity" or "n tends to infinity", but are absolutely not allowed to say that "n equals infinity". By avoiding infinity we avoid the problems posed by this paradox. Calculus effectively implements the Aristotelian solution. For all practical purposes the solution of the Calculus works perfectly well, but it does not resolve or even address the philosophical question. It only avoids and ignores it.
The issue also arises for what are known as "irrational numbers", such as the quantity π ("pi") which represents the ratio between the circumference of a circle and its diameter. We cannot exactly represent this magnitude, but can only approximate it. Pi is larger than 3.1415 and smaller than 3.1416. We can approximate π to any number of decimal places. Using computers it has been approximated to a billion decimal places. But to represent it precisely would require a decimal of infinite length. So instead we just say "pi". But presumably, despite its irrationality, π is still an actual physical magnitude. If you look at your ruler, somewhere between the mark for 31mm and the mark for 32mm will be the magnitude π centimetres. This location on the ruler does not just exist potentially (as the number of decimal places tends to infinity). It exists actually, presumably.
Aristotle's solution seems like a piece of slight of hand, pushing the paradox away to where it can no longer be seen, at the end of a task that cannot be completed in a finite time. Imagine if you will that the universe is infinite in extent. I get in my spaceship and set off for the end of the universe. I will never actually reach the end of the universe, because this will take eternity. But my ability to reach the end of the universe has no bearing on whether the universe is infinite or not. It either is or it is not, regardless of what I am doing. In the same way, an interval is either infinitely divisible or not, regardless of whether I have divided it. In fact, it is the very fact that I can never reach the end of the universe that defines it as infinite. If I could reach the end of it in a finite time, then it would not be infinite. Similarly, if I could complete the division of the interval in a finite time, then it would not be infinitely divisible; it would only be divisible. It is by definition that an infinite time or distance can never be traversed.
We know that the distance d between Achilles and the tortoise is finite, and that it can be traversed in, not only a finite time, but actually in quite a short time. So any solution that is offered to the paradox that arrives us at the conclusion: "... therefore we can traverse the interval in a finite time," is going to have a ring of plausibility to it, whether it is right or wrong. We know that the distance is finite, but if we reject Aristotle's solution we are stuck. Our mind struggles to reconcile the deduction of an infinite number of infinitely small distances, with the notion of a finite distance. What do we mean by an "infinitely small distance"? It cannot mean a distance of zero, because an infinite number of zero distances add up to zero. That is the problem with imagining a line as being composed of an infinite number of mathematical points, that is, points of zero dimension. An infinite number of points will not fill any interval, however small. An infinite number of points within any interval will still leave it completely empty. A point has no existence, it is what we call a "limit". It marks the limit or extremity or boundary of something else, the limit of an interval, where one line ends and another begins. It marks a location only.
Our reasoning mind is suspended between two irreconcilable conclusions, neither of which can be correct: either (1) infinitely small intervals have a non-zero magnitude, or (2) infinitely small magnitudes have a zero magnitude. Option (1) will give us a sum of infinity, while option (2) will give us a sum of zero. We search in vain for an option (3). This is another example of an antinomy. (We first encountered antinomies in The Scientific Creation Myth). The ancient Greek philosophers posed the "atom" as a solution, as the ultimate limit of divisibility. We see then that we meet antinomies looking forward and back to the beginning and the end of time, and therefore the beginning and the end of causality; and we meet an antinomy looking out to the edge of the universe. Now we also meet an antinomy looking in to the smallest parts of reality. So in a way we seem to be hemmed in on all four sides (forward and back, outward and inward) by antinomies where reasoning breaks down altogether. These boundaries mark the limits of reason.
The dilemma does not only apply to the question of what are the smallest constituents of matter, or more generally to the subdivisions of space itself. It also applies to time and, as stated by Aristotle, any continuous quantity. Another of Zeno's paradoxes was about an arrow in flight. If we consider the arrow at any instant in time, it is motionless. Therefore it is motionless for its entire journey. Therefore, when does it move?
Consider an object that is motionless and then starts to move. At the time t=0 it is stationary. Then say at the time t=1 second it is moving at a speed of 1cm per hour. At what point between t=0 and t=1 did its speed change from 0 to some nonzero value? Again we fall into the trap of infinite divisibility. An instantaneous change in velocity, however small, represents an infinite acceleration, which is also a problem. Some Greek philosophers, such as Diodorus, argued for time atoms as well.
Therefore this problem affects all change in the universe and all causation. How did you and I get from a moment ago to this moment? There is another way we can think about the divisibility problem that is perhaps a little more intuitive.
"Long ago, some of the greatest minds in theoretical physics such as Pauli, Heisenberg, Dirac, and Feynman, did suggest that nature's constituents might not actually be points but rather small undulating 'blobs' or 'nuggets.' They and others found, however, that it is very hard to construct such a theory, whose fundamental constituent is not a point particle, that is nonetheless consistent with the most basic of physical principles such as conservation of quantum-mechanical probability (so that physical objects do not suddenly vanish from the universe, without a trace) and the impossibility of faster-than-light-speed transmission of information. From a variety of perspectives, their research showed time and again that one or both of these principles were violated when the point-particle paradigm was discarded."
We know about particles via the force fields that surround them. We surmise there is something at the centre of the force field, responsible for the field (the source of the force carrier particles). A force field such as gravity or an electric field works according to an "inverse square law", which means the strength of the force field varies according to the distance from the particle. Mathematically we have f/(d2), where f is some factor. Let us imagine a simple force field where the factor f=1 so that the field strength is 1/(d2). At a distance d=1 the field strength will be 1. At a distance of d=10, the field strength will be 1/100. At a distance of d=100 the field strength will be 1/10000. So the field strength becomes gradually less with distance, presumably becoming (approaching) zero at a distance (approaching that) of infinity.
But look at what happens as we get closer to the particle for distances less than 1. At a distance of 1/10, the field strength becomes 100. At a distance of 1/100 the field strength becomes 10,000. So as we get closer and closer to the particle the field strength increases exponentially. At a distance d=0 the field strength will be infinite. Which is not allowed. That is one of the problems in quantum physics, especially in efforts to form a theory of quantum gravity. String Theory promotes itself as offering a solution to this problem by offering a workable non-zero dimension form for fundamental particles, but at present there is no way to test or verify String Theory.
If a fundamental particle is a point particle, it only marks a position in space around which certain events are occurring. Let us assume that fundamental particles are not point particles. Where does that get us? Let us ask the question: "What does a fundamental particle look like?"
The first problem with this question is that the particles are too small to resolve an image of them with visible light, because they are smaller than the wavelength of visible light. We use electron microscopes to see very small objects instead of using visible light microscopes because electrons (remember these have wave properties as well) have shorter wavelengths than visible light. But we cannot use an electron microscope to take pictures of something the size as an electron. There is a process called "scanning tunnelling microscopy" that can generate cool but fuzzy pictures of atomic structure. But let's imagine we can somehow shrink ourselves down and look at a fundamental particle. What would it look like. Would it look like a smooth shiny sphere like in illustrations in Physics and Chemistry text books? Or would it be irregular like one of the nuggets of Leucippus, Democritus, Pauli, Heisenberg, Dirac, and Feynman?
If we imagine that it is a smooth sphere, how did it get that way? Is there some mould that stamps out these little spheres, some process that forms them? They are too small to be affected by any constructive process we are aware of. Are they maybe little bubbles in space-time where space-time curves in on itself? But this still requires a cause and something to be acted upon by the cause. Perhaps they have no surface at all but really are fuzzy like in the scanning tunnelling microscope images. Somehow it seems easier to believe that they are fuzzy than that they have some surface. Perhaps this is why we cannot get a clear photo of UFOs or bigfoot. They may be clear photos of fuzzy objects. But whatever form they have we have the problem of how they came to be that way. We also have the problem of how they have the ability to do anything. How do they possess mass and gravity and charge and spin etc. How do they emit or absorb force carrier particles? How do they convert to and from energy or other particles. In short, if they are simple in the sense that they have no constituent parts, how do they do anything? If they are not simple, then they are not fundamental. But if they are not fundamental then the same problem merely arises again for their constituent parts. There is no way around this problem, so it is again an antinomy.
"A truly elementary particle-completely devoid of internal structure-could not be subject to any forces that would allow us to detect its existence. The mere knowledge of a particle’s existence, that is to say, implies that the particle possesses internal structure!"
String theory does not resolve the philosophical problem even if it is true. We are just looking at an infinitely thin loop of string instead of a point or a little sphere or blob.
The hope seems to be that "laws" can be discovered to explain all of this, and that once we have all the necessary laws, we will have a self contained system of explanation. The question of the origin of laws gives rise to the same problems as the origin of anything else. But we feel less compelled to answer questions about the origins of physical laws. When it comes to these we are more likely to be content to say: "It's just the way it is." We also have the problem of: What is a law? A law is not a cause. A law merely describes what happens, what is. It is not what makes it happen. But more and more laws (like probability and gravitation) are treated as sufficient causes of events. Why does a negatively charged electron not fall into the positively charged nucleus? Because a law prevents it.
We have arrived at last at the end of our review of the nature of matter. I hope at this point to have convinced you that our knowledge of matter does not resolve the mysteries of the universe because matter is one of the mysteries of the universe. In the next article: Like a Seed from a Tree, we will look at some of the implications and possibilities of evolution.
Any comments welcome.
In science and philosophy it can seem that the closer we get to things, the more their substantiality slips through our fingers. "Matter" is the very archetype and essence of reality and solidity. In essence, "materialism" says that there is only matter, and anything that isn't made of matter doesn't exist. With Einstein's formula E = mc2, that reality has been extended to include "energy"; that is, effectively, photons; although they do not have mass. In the previous two articles: Mysteries of Light - Part 1 and Mysteries of Light - Part 2 we examined the nature of photons (light). In this article we take a closer look at what is this stuff "matter" that is the basis of materialism.
When Conan the barbarian looked at the iron head of his axe, he likely considered it to be made of solid, smooth iron. And when he thought the word "solid" he likely imagined a substance that was continuous, solid in the sense of "not containing empty spaces". He likely considered the volume of space occupied by his axe-head to be completely filled with iron, through and through. Today of course, we know better. We know the iron of his axe-head is composed of tiny little particles called "atoms", and that it is mostly empty space. It does not merely contain holes like a honeycomb, but the particles that compose it are separated at a distance, like stars in the night sky, held in place only by invisible force fields. That is, they are held in place by the exchange of photons telling each particle where it should be.
"Ever since Ernest Rutherford discovered the atomic nucleus in 1911, physicists have known that the radius of an atom is about 100,000 times larger than the radius of an atomic nucleus. Here is a comparison to dramatize the factor 100,000. If a nucleus were the size of the pencil point in a mechanical pencil (or the ball in a ball-point pen), then the outermost electrons of the atom would be about one football field away."
("Newton to Einstein: the trail of light", by Ralph Baierlein, p.258.)
The Early History of Atomic Theory
Our word "atom" comes from the Greek word άτομο (atomo), the literal translation of which is "indivisible" or "un-cuttable". The word is constructed from the Greek word τομή (tomi) meaning "cut", and the prefix ά (a) indicating negation, that is "uncut". English uses the prefix "a" in a similar way in cases such as "asymmetric" to mean "not symmetric", or "apolitical" to mean "not political". Atomic theory, like the theory of the aether, began with those remarkable ancient Greek philosophers. What we now call science was once called "natural philosophy" and was considered a branch of philosophy, specifically philosophising about the natural world. The philosopher Leucippus (5th century BC) is credited with originating atomic theory. His pupil Democritus (c.460 – c.370 BC) also promoted the theory. The atomic theory of the ancient Greek philosophers was motivated by philosophical considerations. They theorised that if space contained no voids, there could be no motion, everything would be locked immovably in the solid mass in which it was in."Leucippus, however, thought he had a theory which harmonized with sense-perception and would not abolish either coming-to-be and passing-away or motion and the multiplicity of things. He made these concessions to the facts of perception: on the other hand, he conceded to the Monists that there could be no motion without a void. The result is a theory which he states as follows: ‘The void is a "not being", and no part of "what is" is a "not-being"; for what "is" in the strict sense of the term is an absolute plenum. This plenum, however, is not "one": on the contrary, it is a "many” infinite in number and invisible owing to the minuteness of their bulk. The "many" move in the void (for there is a void): and by coming together they produce "coming to-be", while by separating they produce "passing-away". Moreover, they act and suffer action wherever they chance to be in contact (for there they are not "one"), and they generate by being put together and becoming intertwined.'"
(Aristotle: On Generation and Corruption: Book I: Chapter 8.)
Our knowledge of Leucippus' views comes to us from later philosophers writing about him. Leucippus and Democritus are now referred to as "Pre-Socratic philosophers", two of the group of Greek philosophers who preceded Socrates (470 – 399 BC), the teacher of Plato. Plato's writings are mostly writing about what Socrates had to say, although it is commonly assumed that he also included some of his own views in these writings, so that it is not known how much is Socrates and how much Plato. Plato is the first philosopher of whose writings we have more than small fragments.
Plato himself had his own brand of atomic theory, sometimes referred to as "geometric atomism". Atoms for Leucippus and Democritus were like tiny pebbles, an endless variety of irregular shapes all swirling and colliding. Plato suggested that the fundamental elements were triangles that formed special geometric solids now referred to as "Platonic Solids": specifically the tetrahedron, cube, octahedron, dodecahedron and icosahedron. These solids are special in that each one has all its faces identical. Plato believed in the four elements: fire, air, water and earth, associating each with a particular one of the Platonic Solids. The tetrahedron was fire, the cube was earth, the octahedron was air and the icosahedron was water. The dodecahedron was used somehow "in the delineation of the universe" ("Timaeus"). Instead of the aether (Aristotle's fifth element), Plato had something he called the "receptacle", which was apparently space, but which had the ability to shake like a sieve and sort and agitate the particles that occupied it. It was this property of the receptacle that facilitated earth to fall and fire to rise according to their natural attraction for their own kind.
Although Plato's ideas seem naïve, if we interpret his triangles to represent three particles held together by a force, his neat geometric model has many similarities to the very geometric form of real atoms and molecules. But of course, fire isn't really composed of little tetrahedrons, or water of little icosahedrons. If we consider fire to be composed of photons, then it is composed of point particles. Water molecules: H2O (that is: two hydrogen atoms and one oxygen atom) are little isosceles triangles. Methane molecules however are little tetrahedrons, and sulphur hexafluoride is little octahedrons, so Plato's geometric intuitions were not so very far off.
Aristotle was strongly against the idea of a vacuum, imagining space to be filled with continuous substance like a fluid. Movement could occur because the fluid background could slide around objects that moved through it. On the contrary he believed motion was impossible in a vacuum because he did not have the concept of momentum, and there would be nothing to push objects in this direction or that. The Dark Ages officially began around 529 AD, when the emperor Justinian closed all the philosophical schools in Athens, and after the destruction of what was left of the great Alexandrian Library in 391 AD in response to an order from the emperor Theodosius. The collapsing secular Roman Empire had transformed into the invigorated Holy Roman Empire. The state had failed to suppress the growing grass roots Christian religion and now instead adopted it, forming the Catholic Church and making it identical with the state, ruthlessly enforcing a narrow orthodoxy of belief. Like the other Greek philosophers, Aristotle was suppressed for several centuries in the West, but a slightly Christianised interpretation of his ideas eventually emerged and was adopted as official doctrine on all things philosophical and scientific. Through no fault of Aristotle himself, there followed several centuries of adoring commentaries on Aristotle's writings by obedient Scholastic academics. Throughout all of this period atomic theory lay in a long disfavour, still in place as we have seen in the 19th-century. The Dark Ages ended as ancient Greek knowledge continued trickling slowly back into Europe, or as the mindset changed sufficiently for it to be accepted, as the Renaissance and Enlightenment took shape, but this incredible intellectual hiatus meant that as science ("natural philosophy") came back to life, its proponents found themselves arguing against the entrenched doctrines of a canonised authority who had died two-millennia before, during which interval virtually no scientific work had been done in the West. A time when cutting edge philosophers believed the Earth was the centre of the universe. The contest between "religion and science" that Galileo was a party to was a contest between the old Aristotelian science and the new science, between new ideas and an existing institutionalised authority.
Atomic theory (and its cousin "corpuscularianism") began to re-emerge with proponents such as Francis Bacon (1561 – 1626), Galileo Galilei (1564–1642), Rene Descartes (1596–1650), Pierre Gassendi (1592–1655), Robert Boyle (1627 – 1692) and Roger Boscovich (1711 – 1787). The word "atom" was revived in science in 1805 by the British chemist John Dalton (1766-1844). We learned of the subsequent history of atomic theory in Mysteries of Light - Part 1 and Mysteries of Light - Part 2.
"It is found too that the same body changes its capacity for heat, or apparently assumes a new affinity, with a change of form. This no doubt arises from a new arrangement or disposition of its ultimate particles, by which their atmospheres of heat are influenced: Thus a solid body, as ice, on becoming liquid, acquires a larger capacity for heat, even though its bulk is diminished; and a liquid, as water, acquires a larger capacity for heat on being converted into an elastic fluid; this last increase is occasioned, we may conceive, solely by its being increased in bulk, in consequence of which every atom of liquid possesses a larger sphere than before." (emphasis added)
("A New System of Chemical Philosophy" (1808) by John Dalton, pp.48-9.)
The Building Blocks of Reality
We saw also in Mysteries of Light - Part 2 that all matter is composed of three kinds of particles: electrons, up quarks and down quarks. The two bosons (W boson and Z boson) that take part in the rarely occurring "weak interaction" also have mass, and are therefore matter, but they wink in and out of existence in a tiny fraction of a second, and so are not counted among normal matter. Photons (light) do not have mass, and so are not matter. Photons are described as having energy rather than being energy, but photons are such puzzling phenomena that we do not know quite what to make of them. There are more exotic particles, some with mass, some without. Neutrinos have a tiny mass. The earth is constantly bathed in neutrinos from the sun, but these particles hardly ever interact with normal matter, so they pass right through us, and right through the Earth without touching anything.Gluons (that bind quarks into protons and neutrons and bind protons and neutrons together in the atomic nucleus) are also massless, and if gravitons exist, they are presumed to be massless as well. The photon, W boson, Z boson, gluon and graviton are all referred to as "force carriers" because they transmit the known forces. These particles are also all referred to as "bosons". Bosons have the peculiar property that more than one of them can be in the same place at the same time. In fact you can have as many of them as you like in the same place. The quarks, electrons and neutrinos (that each have mass) are all referred to as "fermions", and these obey what is called the "Pauli exclusion principle", which means you cannot have two of them in exactly the same place at exactly the same time. This property of matter is kind of what makes matter matter. Two objects are solid because they cannot be in the same place at the same time.
Because bosons can occupy the same place at the same time, they make up what we think of as fields and waves. Fields and waves have nice neat mathematical properties as a result of the fact that they pass right through each other when they collide. If you shine one torch beam through another they won't interfere with each other. Imagine if photons behaved like billiard balls on a pool table bouncing off each other chaotically. It would be a mess. Then we wouldn't have nice neat equations in electromagnetics, or the elegant properties of wave addition.
The three kinds of matter: electrons, up quarks and down quarks; are all like products rolled off of a factory production line. Every electron is exactly like every other electron in the universe. Every up quark or down quark is exactly like every other up quark or down quark, respectively, in the universe. And thus, every proton or neutron is exactly like every other proton or neutron, respectively, in the universe. Every electron has a mass of 0.5 and an electric charge of -1. Every up quark has a mass of 2.4 and an electric charge of 2/3. Every down quark has a mass of 4.8 and an electric charge of -1/3. Recall that a proton is composed of two up quarks and one down quark. Every proton has a mass of 938.3 and an electric charge of 1. Notice that the masses of the constituent quarks don't add up, but the charges of the constituent quarks do. Most of the mass of the proton is in the form of what is called "binding energy". Here we see the massless force carriers, "energy" (in this case gluons) being spoken of as if they have mass, or are somehow able to manifest mass. The vast bulk of the mass of protons and neutrons does not come from the mass of the quarks composing them, but from the "energy" associated with them. The difference between "mass" and "not mass" is not a difference between "something real that exists" and "something not real that does not exist"; rather, "mass" is energy behaving one way, and "not mass" is energy behaving another way. The bottom line then, is that matter as mass cannot serve as the fundamental basis of "real", because it is a derived behaviour of something else which does not possess mass. The neutron is composed of two down quarks and one up quark. It has a mass of 939.6 and no electric charge (notice that the charges of the constituent quarks add up to zero). Fundamental particles are like Lego blocks, extraordinarily simple and neat objects.
When I gave the masses of the particles above I did not say what were the units of measure I was using. That is, is mass being measured in pounds, kilograms, or what? The actual unit of measure is MeV/c2. The "V" in all of that is the Volt. It might seem strange to measure mass in Volts, but physicists typically do when working at this scale because energy is the more fundamental quantity. So mass is measured in terms of the quantity of energy it is convertible into via E = mc2. Mass is made out of energy. Matter is made out of something which itself is not matter.
In the macroscopic world, the everyday world, we are accustomed to force requiring work. If I want to hold a heavy suitcase over my head I have to strain my arms, which after a while will get tired and I will have to put the case down. To be able to do this work, even for a while, I need fuel: food, water and air. It is the same with machines that do work, whether a steam engine burning wood or coal, an electric power station burning coal or driven by steam heated by a nuclear reaction, or driven by wind or falling water, or a machine that runs on a battery, or gasoline and natural gas. When the fuel runs out, the machine stops working, the plant or animal dies. Even suns (stars) die when they eventually run out of fuel to burn. Eventually all the stars in all the galaxies will go out unless new stars are born to replace them.
But the fundamental particles are not like that. The force of gravity associated with a particle's mass does not run down. It always stays exactly the same. The electric charge of a particle always stays exactly the same. Two particles will stick together or repel each other with the same force for eternity, or until the universe ends, whichever comes first. A photon will always travel at exactly the speed of light. It won't ever get tired and slow down. The speed of light is usually given "in a vacuum", but this is not exactly because light travels slower when not in a vacuum, but because when not in a vacuum it is interacting with other particles it meets along the way, and those interactions take time.
Fundamental particles also have an innate behaviour called (quantum) "spin". The fermions (electron, up quark, down quark) always have a spin of 1/2. The bosons (photon, gluon, Z boson, W boson) always have a spin of 1 (except for the graviton if it exists).
Divisibility
So why did the ancient Greek philosophers call the fundamental particles "atoms" (that is "indivisible")? The name is meant to communicate the idea that they are the smallest units of matter, the fundamental building blocks of matter. If they were divisible, then whatever they are divisible into would be the fundamental building blocks of matter. Unless those in turn were divisible, in which case whatever they are divisible into would be the fundamental building blocks of matter, and so on.What we call an atom today is not "indivisible" in the Greek sense. Soon after the atom was admitted into physics it was found to have parts. The electron was discovered by the English physicist Sir Joseph John Thomson (1856-1940) in 1897. The word "electron" was coined earlier in 1891 by the Irish physicist George J. Stoney (1826-1911). The New Zealand physicist, Ernest Rutherford (1871-1937) (1st Baron Rutherford of Nelson) discovered the atomic nucleus in 1911. Since electrons had a negative electric charge, Rutherford suggested the atomic nucleus contained particles with a positive charge, and he proposed the name "proton". Neutrons were discovered by the English physicist Sir James Chadwick (1891-1974) in 1932. The word "neutron" was coined in 1921 by the United States chemist William Draper Harkins (1873-1951) and alludes to the fact that these particles are electrically neutral (having neither a positive nor a negative charge). Later, protons and neutrons were found in turn to have parts, the quarks. Quarks are theorised to exist but no individual quark has ever actually been detected. In fact quarks cannot exist in isolation, by a rule called "confinement". So in that sense we could say that the hadrons (protons and neutrons) are indivisible. Our atoms are however the smallest units of the chemical elements. For example, an atom of oxygen (8 electrons around a nucleus of 8 protons and 8 neutrons) is the smallest unit of oxygen. If you break the atom apart into smaller pieces, it ceases to be oxygen and becomes some other element (such as some combination of nitrogen, carbon, boron, beryllium, lithium, helium or hydrogen), or free subatomic particles (electrons, protons and neutrons).
But the question of divisibility was more to the Greek philosophers than just determining the basic constituents of matter. The question held a philosophical dilemma. The pre-Socratic philosopher Zeno of Elea (ca. 490–430 BC) was a student of Parmenides (late sixth or early fifth century BC), and he is known for four paradoxes he posed to illustrate the teachings of Parmenides and which are still a cause for some mystification.
"The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. The result of the argument is that the slower is not overtaken"
(Aristotle: Physics: Book VI: Chapter 9.)
This paradox is usually referred to as "Achilles and the Tortoise". It appears that tradition introduced the tortoise to represent what was originally just a slow runner. Achilles was a legendary hero of the Trojan War, apparently with a reputation for swiftness. He is imagined in a footrace with a tortoise, but the tortoise is given a head start of some distance. Let us say the distance is d. The two of them start racing, but really the paradox still works if the tortoise does not move at all. The question is posed: "When/where does Achilles overtake the tortoise?" The exact point and time will presumably depend on their respective speeds and the amount of the tortoise's head start. But now the presenter of the paradox raises the following difficulty. For simplicity we will imagine the tortoise to be motionless.
Before Achilles traverses the full distance (d) between him and the tortoise, he must first traverse half the distance (d/2). This means that he has the other half of the distance remaining (d/2). But before he traverses the whole remaining distance (d/2), he must first traverse half that distance (d/4). So that he now has one quarter of the total distance remaining (d/4). But before he traverses the whole remaining distance (d/4), he must first traverse half that distance (d/8). So that he now has one eighth of the total distance remaining (d/8), and so on. So Achilles is getting closer and closer to the tortoise, but when does he overtake it? If we continue: the distances he travels are: d/2 + d/4 + d/8 + d/16 + d/32 + d/64 ... and so on. This is an infinite series, because it is always possible to halve whatever distance remains. Although each new component of distance is smaller than the last, so that the additions to the distance travelled get smaller and smaller, still, however minute they become, they never become zero. What we have then is an infinite series containing elements all of which have non-zero magnitude. Any infinite number of non-zero magnitudes, however small the non-zero magnitudes, will add up to an infinite magnitude. This means that in order for Achilles to reach and overtake the tortoise, across the distance d, he must first traverse an infinite distance. Because it takes a non-zero amount of time to traverse each non-zero distance, it will take Achilles an infinite amount of time to traverse the infinity of non-zero distances. We know that the distance Achilles has to run is only d, and the time that it will take him depends simply on the speed he is running. If he is running at a speed of 1 d per minute, it will take him 1 minute to reach and overtake the tortoise. So how do we end up with an infinite distance and time? Aristotle had a clear understanding of the cause of the paradox.
"For there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number."
(Aristotle: Physics: Book VI: Chapter 2.)
That is, things can be infinite in extent; for instance, we can surmise that the universe is infinite in extent, and that time is infinite in extent. We can also surmise that any continuous substance or magnitude is "infinitely divisible". That is, between any two points, will fit an infinite number, of infinitely small intervals. In mathematics, the Real Number Line is treated in this way. Between any two numbers, say 1.00002 and 1.00003 we can always give any number of other numbers; for example: 1.000021, 1.000022, 1,000029, 1.0000290001, etc. Aristotle offers the following solution to the paradox.
"So in the straight line in question any one of the points lying between the two extremes is potentially a middle-point: but it is not actually so unless that which is in motion divides the line by coming to a stand at that point and beginning its motion again.... Therefore to the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible: if they are potential, it is possible."
(Aristotle: Physics: Book VIII: Chapter 8.)
He suggests that the interval is only potentially infinitely divisible, and not actually infinitely divisible, because we will never actually arrive at having infinitely divided it. Rather than immersing ourselves deeply in this argument, I will say only the following. Aristotle's solution was accepted by many in Aristotle's own time, and down to the present day.
"Aristotle's solution to Zeno's paradoxes is genuinely a solution, and it is reasonable to suppose that it is not an invention of Aristotle's but represents the best mathematical thinking of his time"
(Plato's Parmenides (Revised Edition), translated with comment by R. E. Allen, pp.253-4.)
Many others reject the solution.
"This is scarcely comprehensible, since it seems obvious that x is potentially ϕ only if it is possible that x should be actually ϕ.... For example, one might very well hold that the parts of a finite line do constitute an infinite totality, whether or not there has been any process of dividing the line into parts."
(Aristotle: Physics, A new translation by Robin Waterfield, pp. XXXIV-XXXV.)
Calculus is the modern science that grapples with the infinite divisibility of physical quantities, and anyone who has learned calculus knows that they are only ever allowed to say that "n (the number of terms in the series) approaches infinity" or "n tends to infinity", but are absolutely not allowed to say that "n equals infinity". By avoiding infinity we avoid the problems posed by this paradox. Calculus effectively implements the Aristotelian solution. For all practical purposes the solution of the Calculus works perfectly well, but it does not resolve or even address the philosophical question. It only avoids and ignores it.
The issue also arises for what are known as "irrational numbers", such as the quantity π ("pi") which represents the ratio between the circumference of a circle and its diameter. We cannot exactly represent this magnitude, but can only approximate it. Pi is larger than 3.1415 and smaller than 3.1416. We can approximate π to any number of decimal places. Using computers it has been approximated to a billion decimal places. But to represent it precisely would require a decimal of infinite length. So instead we just say "pi". But presumably, despite its irrationality, π is still an actual physical magnitude. If you look at your ruler, somewhere between the mark for 31mm and the mark for 32mm will be the magnitude π centimetres. This location on the ruler does not just exist potentially (as the number of decimal places tends to infinity). It exists actually, presumably.
Aristotle's solution seems like a piece of slight of hand, pushing the paradox away to where it can no longer be seen, at the end of a task that cannot be completed in a finite time. Imagine if you will that the universe is infinite in extent. I get in my spaceship and set off for the end of the universe. I will never actually reach the end of the universe, because this will take eternity. But my ability to reach the end of the universe has no bearing on whether the universe is infinite or not. It either is or it is not, regardless of what I am doing. In the same way, an interval is either infinitely divisible or not, regardless of whether I have divided it. In fact, it is the very fact that I can never reach the end of the universe that defines it as infinite. If I could reach the end of it in a finite time, then it would not be infinite. Similarly, if I could complete the division of the interval in a finite time, then it would not be infinitely divisible; it would only be divisible. It is by definition that an infinite time or distance can never be traversed.
We know that the distance d between Achilles and the tortoise is finite, and that it can be traversed in, not only a finite time, but actually in quite a short time. So any solution that is offered to the paradox that arrives us at the conclusion: "... therefore we can traverse the interval in a finite time," is going to have a ring of plausibility to it, whether it is right or wrong. We know that the distance is finite, but if we reject Aristotle's solution we are stuck. Our mind struggles to reconcile the deduction of an infinite number of infinitely small distances, with the notion of a finite distance. What do we mean by an "infinitely small distance"? It cannot mean a distance of zero, because an infinite number of zero distances add up to zero. That is the problem with imagining a line as being composed of an infinite number of mathematical points, that is, points of zero dimension. An infinite number of points will not fill any interval, however small. An infinite number of points within any interval will still leave it completely empty. A point has no existence, it is what we call a "limit". It marks the limit or extremity or boundary of something else, the limit of an interval, where one line ends and another begins. It marks a location only.
Our reasoning mind is suspended between two irreconcilable conclusions, neither of which can be correct: either (1) infinitely small intervals have a non-zero magnitude, or (2) infinitely small magnitudes have a zero magnitude. Option (1) will give us a sum of infinity, while option (2) will give us a sum of zero. We search in vain for an option (3). This is another example of an antinomy. (We first encountered antinomies in The Scientific Creation Myth). The ancient Greek philosophers posed the "atom" as a solution, as the ultimate limit of divisibility. We see then that we meet antinomies looking forward and back to the beginning and the end of time, and therefore the beginning and the end of causality; and we meet an antinomy looking out to the edge of the universe. Now we also meet an antinomy looking in to the smallest parts of reality. So in a way we seem to be hemmed in on all four sides (forward and back, outward and inward) by antinomies where reasoning breaks down altogether. These boundaries mark the limits of reason.
The dilemma does not only apply to the question of what are the smallest constituents of matter, or more generally to the subdivisions of space itself. It also applies to time and, as stated by Aristotle, any continuous quantity. Another of Zeno's paradoxes was about an arrow in flight. If we consider the arrow at any instant in time, it is motionless. Therefore it is motionless for its entire journey. Therefore, when does it move?
Consider an object that is motionless and then starts to move. At the time t=0 it is stationary. Then say at the time t=1 second it is moving at a speed of 1cm per hour. At what point between t=0 and t=1 did its speed change from 0 to some nonzero value? Again we fall into the trap of infinite divisibility. An instantaneous change in velocity, however small, represents an infinite acceleration, which is also a problem. Some Greek philosophers, such as Diodorus, argued for time atoms as well.
Therefore this problem affects all change in the universe and all causation. How did you and I get from a moment ago to this moment? There is another way we can think about the divisibility problem that is perhaps a little more intuitive.
What does a Fundamental Particle Look Like?
Our previous quote about: "If a nucleus were the size of the pencil point in a mechanical pencil (or the ball in a ball-point pen), then the outermost electrons of the atom would be about one football field away" gives a ratio between the physical dimensions of an atom, and an atomic nucleus. But a problem arises when we ask about the size of fundamental particles because physics treats them as point particles. That is, as having no physical size, zero dimension. If fundamental particles actually have zero dimension, in what sense are they something as opposed to nothing? It is generally assumed that they don't really have zero dimension, but that this is just a simplification, a model, like the way stars and planets can be treated as points when calculating their gravitational influences on each other. But it has proved difficult to depart from the point particle model of fundamental particles."Long ago, some of the greatest minds in theoretical physics such as Pauli, Heisenberg, Dirac, and Feynman, did suggest that nature's constituents might not actually be points but rather small undulating 'blobs' or 'nuggets.' They and others found, however, that it is very hard to construct such a theory, whose fundamental constituent is not a point particle, that is nonetheless consistent with the most basic of physical principles such as conservation of quantum-mechanical probability (so that physical objects do not suddenly vanish from the universe, without a trace) and the impossibility of faster-than-light-speed transmission of information. From a variety of perspectives, their research showed time and again that one or both of these principles were violated when the point-particle paradigm was discarded."
(The Elegant Universe, by Brian Greene, pp.157-8.)
We know about particles via the force fields that surround them. We surmise there is something at the centre of the force field, responsible for the field (the source of the force carrier particles). A force field such as gravity or an electric field works according to an "inverse square law", which means the strength of the force field varies according to the distance from the particle. Mathematically we have f/(d2), where f is some factor. Let us imagine a simple force field where the factor f=1 so that the field strength is 1/(d2). At a distance d=1 the field strength will be 1. At a distance of d=10, the field strength will be 1/100. At a distance of d=100 the field strength will be 1/10000. So the field strength becomes gradually less with distance, presumably becoming (approaching) zero at a distance (approaching that) of infinity.
But look at what happens as we get closer to the particle for distances less than 1. At a distance of 1/10, the field strength becomes 100. At a distance of 1/100 the field strength becomes 10,000. So as we get closer and closer to the particle the field strength increases exponentially. At a distance d=0 the field strength will be infinite. Which is not allowed. That is one of the problems in quantum physics, especially in efforts to form a theory of quantum gravity. String Theory promotes itself as offering a solution to this problem by offering a workable non-zero dimension form for fundamental particles, but at present there is no way to test or verify String Theory.
If a fundamental particle is a point particle, it only marks a position in space around which certain events are occurring. Let us assume that fundamental particles are not point particles. Where does that get us? Let us ask the question: "What does a fundamental particle look like?"
The first problem with this question is that the particles are too small to resolve an image of them with visible light, because they are smaller than the wavelength of visible light. We use electron microscopes to see very small objects instead of using visible light microscopes because electrons (remember these have wave properties as well) have shorter wavelengths than visible light. But we cannot use an electron microscope to take pictures of something the size as an electron. There is a process called "scanning tunnelling microscopy" that can generate cool but fuzzy pictures of atomic structure. But let's imagine we can somehow shrink ourselves down and look at a fundamental particle. What would it look like. Would it look like a smooth shiny sphere like in illustrations in Physics and Chemistry text books? Or would it be irregular like one of the nuggets of Leucippus, Democritus, Pauli, Heisenberg, Dirac, and Feynman?
If we imagine that it is a smooth sphere, how did it get that way? Is there some mould that stamps out these little spheres, some process that forms them? They are too small to be affected by any constructive process we are aware of. Are they maybe little bubbles in space-time where space-time curves in on itself? But this still requires a cause and something to be acted upon by the cause. Perhaps they have no surface at all but really are fuzzy like in the scanning tunnelling microscope images. Somehow it seems easier to believe that they are fuzzy than that they have some surface. Perhaps this is why we cannot get a clear photo of UFOs or bigfoot. They may be clear photos of fuzzy objects. But whatever form they have we have the problem of how they came to be that way. We also have the problem of how they have the ability to do anything. How do they possess mass and gravity and charge and spin etc. How do they emit or absorb force carrier particles? How do they convert to and from energy or other particles. In short, if they are simple in the sense that they have no constituent parts, how do they do anything? If they are not simple, then they are not fundamental. But if they are not fundamental then the same problem merely arises again for their constituent parts. There is no way around this problem, so it is again an antinomy.
"A truly elementary particle-completely devoid of internal structure-could not be subject to any forces that would allow us to detect its existence. The mere knowledge of a particle’s existence, that is to say, implies that the particle possesses internal structure!"
(Geoffrey Chew, theoretical physicist in "Impasse for the Elementary Particle Concept: The Great ideas Today",
(William Benton, Chicago, 1974), p. 99.)
(William Benton, Chicago, 1974), p. 99.)
String theory does not resolve the philosophical problem even if it is true. We are just looking at an infinitely thin loop of string instead of a point or a little sphere or blob.
The hope seems to be that "laws" can be discovered to explain all of this, and that once we have all the necessary laws, we will have a self contained system of explanation. The question of the origin of laws gives rise to the same problems as the origin of anything else. But we feel less compelled to answer questions about the origins of physical laws. When it comes to these we are more likely to be content to say: "It's just the way it is." We also have the problem of: What is a law? A law is not a cause. A law merely describes what happens, what is. It is not what makes it happen. But more and more laws (like probability and gravitation) are treated as sufficient causes of events. Why does a negatively charged electron not fall into the positively charged nucleus? Because a law prevents it.
We have arrived at last at the end of our review of the nature of matter. I hope at this point to have convinced you that our knowledge of matter does not resolve the mysteries of the universe because matter is one of the mysteries of the universe. In the next article: Like a Seed from a Tree, we will look at some of the implications and possibilities of evolution.
Any comments welcome.
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