Celtic knot rings are composed of similar interweaving lines that are inspired by the infinity symbol. These Celtic knots pay homage to Irish, Scottish and Welsh culture and behold a similar meaning to the infinity symbol. The endless loops of the trinity knot similarly represent an everlasting love.
Throughout the ages and various cultures, the infinity symbol has represented a variety of concept and beliefs.
As a mathematical device, the very first person to have written about the infinity symbol was Mathematician and Scientist, John Wallis in However, its symbolic significance has been around for much longer.
There is a strong belief that the meaning of infinity ritualised from ancient India and Tibet where the symbol represented dualism, perfection and equilibrium between different genders. Despite the figurative nature of this symbol differing, most notions affirm that it represents eternity and everlasting love.
Proposing with an Infinity Engagement Rings is the perfect symbolic gift for the person you love. Infinitas, meaning unbounded or limitless, makes infinity diamond engagement rings an apt symbol.
This representation of everlasting or endless devotion is the perfect embodiment of eternal love and equilibrium. The situation where our three-dimensional space is finite and unbounded, and the reality of four-dimensional space denied, falls under this case.
In this case we either have an infinite number of universes, duoverses, etc. So is space infinite? It seems that we can insist that at some dimensional level it is infinite; adopt the Aristotelian stance that space is finite at some level beyond which nothing lies; or accept the view that there is an infinite sequence of dimensional levels.
In this last case we already have a qualitative infinity in the dimensionality of space, and we may or may not have a quantitative infinity in terms, say, of the total volume of all the 3-D spaces involved. In this subsection I will discuss the existence of the infinity in the small, as opposed to the infinity in the large, which has just been discussed.
Since a point has no length, no finite number of points could ever constitute a line segment, which does have length. So it seems evident that every line segment, or, for that matter, every continuous plane segment or region of space, must consist of an infinite number of points.
By the same token, any interval of time should consist of an infinite number of instants; and any continuous region of space-time would consist of an infinite number of events event being the technical term for a space-time location, i. It is undeniable that a continuous region of mathematical space has an infinite number of mathematical points. Right now, however, we are concerned with physical space.
We should not be too hasty in assuming that every property of the abstract mathematical space we use to organize our experiences is an actual property of the concrete physical space we live in. But what is "the space we live in"? If it is not the space of mathematical physics, is it the space of material objects? Is it the space of our perceptions?
In terms of material objects or of perceptions, points do not really exist; for any material or perceptual phenomenon is spread over a certain finite region of space-time. So when we look for the infinity in the small in matter, we do not ask whether matter consists of an infinity of unobservable mass-points, but, rather, whether matter is infinitely divisible.
A commitment to avoiding the formless made it natural for Greek atomists such as Democritus to adopt a theory of matter under which the seemingly irregular bodies of the world are in fact collections of indivisible, perfectly formed atoms. The four kinds of atoms were shaped, according to Plato, like four of the regular polyhedra. There is one other polyhedron, the twelve-sided dodecahedron, and this was thought somehow to represent the Universe with its twelve signs of the zodiac.
For the atomists, it was as if the world were an immense Lego set, with four kinds of blocks. The diverse substances of the world -- oil, wood, stone, metal, flesh, wine, and so on -- were regarded as being mixtures of the four elemental substances: Earth, Air, Fire, and Water.
Thus, gold was regarded by Plato as being a very dense sort of Water, and copper was viewed as gold with a small amount of Earth mixed in. The alchemists and early chemists adopted a similar system, only the number of elemental substances became vastly enlarged to include all homogeneous substances, such as the various ores, salts, and essences.
The fundamental unit here was the molecule. A new stage in man's conception of matter came when it was discovered that if an electric current is passed through water, it can be decomposed into hydrogen and oxygen. Eventually, the vast diversity of existing molecules was brought under control by regarding molecules as collections of atoms. Soon some ninety different types of atoms or chemical elements were known.
A new simplification occurred when it was discovered, by bombarding a sheet of foil with alpha rays, that an atom consists of a positive nucleus surrounded by electrons. Shortly after this the neutron was discovered, and the physical properties of the various atoms were accounted for by regarding them as collections of protons, neutrons, and electrons.
Over the last half century it has been learned, by using particle accelerators, that there are actually many types of "elementary particles" other than the neutron, electron, and proton.
The situation in high-energy physics today is as follows. A few particles -- electrons, neutrinos, and muons -- seem to be absolutely indivisible. These particles are called leptons.
All others -- protons, neutrons, mesons, lambdas, etc. The historical pattern in the investigation of matter has been the explanation of diverse substances as combinations of a few simpler substances. Diversity of form replaces diversity of substance. So it is no surprise that it has been proposed that the great variety of divisible particles that exist can be accounted for by assuming that these particles are all built up out of quarks.
A second element in the historical pattern is that as more powerful tools of investigation are used, it becomes evident that there are more types of new building blocks than had been suspected initially. This is the phase that high-energy physics is currently moving into. First there were three kinds of quark: up, down, and strange. Now, the charmed quark has been admitted, and there are two new possible quarks: the top quark and the bottom quark.
It seems likely that the many diverse types of quark will eventually be accounted for by assuming that each quark is a combination of a few, let us say, darks. The cycle will then repeat, with more and more different sorts of dark being indirectly observed, the new diversity being accounted for by viewing each dark as a collection of a few smaller particles of which there are a limited variety, this limited variety beginning to proliferate, and so on. If this sort of development can indeed continue indefinitely, then we are left with the fact that a stone is a collection of collections of collections of.
The stone thus consists of an infinite number of particles, no one of which is indivisible. There is, finally, no matter -- only form. For a stone is mostly empty space with a few molecules in it, a molecule is a cloud of atoms, an atom is a few electrons circling a tiny nucleus.
What if any seemingly solid bit of matter proves on closer inspection to be a cloud of smaller bits of matter, which are in turn clouds, and so on? Note that the branching matter tree that I began to draw for the stone has only a finite number of forks or nodes at each level, but that since there are infinitely many levels, there are in all an infinite number of nodes or component particles.
There are various objections to this sort of physical infinity. One is the Aristotelian argument that unless one is actually smashing the stone down to the quark level, the quarks are only potentially as opposed to actually there. The point would be that the stone may be indefinitely divisible, but that since no one will ever carry out infinitely many divisions, there are not really infinite numbers of particles in the stone right now.
There is a more practical objection as well. This is that no quark has ever been observed in isolation; the existence of quarks is deduced only indirectly as a way of explaining the symmetries of structure that occur in tables of the elementary particles. This argument is not very strong, however. For one thing, a great number of the things we believe in can be observed only indirectly; and, more practically, if we can continue to increase the energy of our measuring tools, there is no reason to think that quarks cannot be more convincingly detected.
A more fundamental objection to the whole idea of particles, subparticles, etc. By splitting particles indefinitely we arrived at the conclusion that there is only form, and no content; many physicists prefer to start with this viewpoint.
For these physicists, the various features of the world are to be explained in terms of the geometry of space-time. To get a feeling for this viewpoint, one should look carefully at the surface of a river or small brook. There are circular ripples, flow bulges, whirlpools and eddies, bubbles that form, drops that fly up and fall back, waves that crest into foam. The geometrodynamic worldview regards space-time as a substance like the surface of a brook; the various fields and particles that seem to exist are explained as features of the flow.
Does the space-time of geometrodynamics allow an infinity in the small? There is really no answer to this question at present. According to one viewpoint there should be a sort of graininess to space-time, and the grain size would represent a sort of indivisible atom; a different viewpoint suggests that space-time should be as infinitely continuous as mathematical space. What if there really is nothing smaller than electrons and quarks?
Is there then any hope of an infinity in the small? One can argue that a given electron can have infinitely many locations along a given meter stick, so that our space really does have infinitely many points. It is sometimes asserted that the uncertainty principle of quantum mechanics nullifies this argument, but this is not the case.
Quantum mechanics puts no upper limit on the precision with which one can, in principle, determine the position of an electron. It is just that the more precisely the electron's position is known, the less precisely are its speed and direction of motion known.
Infinite precision is basically a nonphysical notion, but any desired finite degree of precision is, in principle, obtainable.
The precision with which something can be measured is thus a good example of something that is potentially infinite, but never actually infinite. But this still gives us an actual infinity in the world. For if our electron is located somewhere between zero and one, then each member of the following infinite collection is a possible outcome of a possible measurement:.
Although infinite precision is impossible, an electron can be found to occupy any of the infinitely many points between zero and one whose distance from zero is a terminating decimal. There are, however, some modern physical speculations that regard "space" and "time" as being abstractions which apply to our size level, but which become utterly meaningless out past the thirtieth decimal place. What would be there instead? Our old friend the apeiron. But even if we cannot really speak of infinitely many space locations, we might hope to find infinitely many sorts of particle.
It is sometimes thought that quantum mechanics proves that there is a smallest size of particle that could exist. This is not true. Quantum mechanics insists only that in order to "see" very small particles, we must use very energetic processes to look for them. It is illuminating, after all this, to learn how the high-energy physicists actually go about finding new particles.
The process is a little like finding stations on the radio by inching the dial back and forth until you hear music instead of static. One uses a particle accelerator in which collisions between electrons and positrons are continually taking place. The energy of the collision processes is varied by turning the voltage on the accelerator up and down. There is number R that measures the "particleness" of the reaction taking place.
R can be thought of as being a little like the information parameter that enables you to tell whether you have found a station, even though the sound of music is no louder than the sound of the static.
When an energy is found at which the graph of R versus energy has a sudden peak, then it is assumed that the energy in question is characteristic of the rest-mass of a new particle. This process is called "bump-hunting. The question of whether or not matter is infinitely divisible may never be decided. For whenever an allegedly minimal particle is exhibited, there will be those who claim that if a high enough energy were available, the particle could be decomposed; and whenever someone wishes to claim that matter is infinitely divisible, there will be some smallest known particle which cannot be split.
One is almost tempted to doubt if the question of the infinite divisibility of matter has any real meaning at all, particularly in view of the fact that such concepts as "matter" and "space" have no real meaning in the micro-world of quantum mechanics. To return to something a little more concrete, let us consider the divisibility of our perceptual field. There is a limit to the subdivisions that this field can undergo. If two clicks happen close enough together in time, they cannot be distinguished; if a spot of ink is small enough, we can no longer see it.
Hume makes much of this fact in his Treatise of Human Nature of Put a spot of ink upon paper, fix your eye upon that spot, and retire to such a distance, that at last you lose sight of it; 'tis plain, that the moment before it vanish'd the image or impression was perfectly indivisible. The best way to understand Hume's view of the world is to regard our space-time as being supplemented by an additional dimension of scale.
To represent what I have in mind, let us forget about time and drop all the space dimensions but one. In Figure 24 I have drawn the space-scale continuum for a one-dimensional world. An individual's perceptual field has a certain fixed size, as drawn; the field is made up of a certain finite number of slots or tiles -- minimal perceptual units.
In this model, the one-dimensional creature has two dimensions in which he can move his perceptual field. He can move to the left and right in space, and he can enlarge and contract his perceptual field. Rather than thinking of the field as enlarging and contracting, we think of the field moving up and down on the scale axis.
If the labelled objects mountain, stone, speck of rock dust occupy the appropriate regions of the space-scale continuum, then we can think of the ordinary perceptual level as being when the field is placed somewhere in the middle of the picture. At this perceptual level stones are visible, but one has neither enlarged one's field of vision enough to see the mountain as a single object, nor contracted one's attention enough to see the specks of dust on the rock.
Notice that changing the size of one's perceptual field amounts just to moving this field about in the space-scale continuum. Hume takes perceptions as primary. Although he is often thought of as an empiricist, his is actually an extremely idealistic viewpoint.
The perceptions are "out there"; one's consciousness seems to move among them like a butterfly flitting from flower to flower. One's perceptual field has minimal elements, yet these minimal elements can be resolved into smaller elements by altering one's field by paying closer attention, using a telescope, or moving closer to the object in question.
The only way to reconcile these two apparently contradictory aspects of our perceptual world is to view the world as a five-dimensional, space-time-scale continuum. The question of the existence of an infinity in the small now becomes the question of whether or not the space-scale continuum drawn in Figure 24 extends downward indefinitely; similarly, the question of the existence of infinity in the large is the question of whether or not the continuum extends upward indefinitely.
I have long been interested in a curious trick that eliminates the infinity in the large and the infinity in the small without introducing any absolute perceptual minimum or maximum. This is simply the trick of bending the space-scale diagram into a tube, by turning the scale axis into a circle.
Here the universe could consist of many galaxies, which consist of many star systems, which consist of many planets, which consist of many rocks, which consist of many molecules, which consist of many atoms, which consist of many elementary particles, which consist of many quarks and leptons, which consist of many darks, which could consist of many universes.
A problem with the circular scale model is that if our universe is broken down far enough, one gets many universes, each of which will break down into many more universes. Are all of these universes the same? Perhaps, but then it would be hard to see how there could really be more than one object in the world.
Another difficulty is that if there are many universes, each of which breaks up into many more universes, how can each of the component universes be one of the starting universes?
There is no problem if we have infinitely many universes. To illustrate this, I have drawn a picture of the simplest case: the case in which each universe is made up of two universes. We can see that 1 splits into 1 and 2, 2 splits into 3 and 4, 3 splits into 5 and 6, and in general n splits into 2 n - 1 and 2 n.
We can continue splitting any given universe indefinitely, thus obtaining an infinite number of components in any bit of matter. What is gained here is freedom from the belief that any size scale is intrinsically more basic or important or complex than any other size scale.
Why waste time on the six o'clock news when you are no more nor less important than a galaxy or an atom? The point of this question is that one is often pressured to feel that the concerns of society or the world are more significant than one's own immediate personal concerns.
But this is based on the assumption that some sizes are in an absolute sense bigger than others, and it is this assumption that circular scale undermines. In conclusion, note that it is entirely possible that our universe is in every sense finite. A toroidal space-time of the sort mentioned in the section on temporal infinities eliminates all infinities in the large; and if circular scale is introduced as in the section on infinities in the small, then there are no discrete infinities in the small.
These finitizations can be accomplished smoothly: there need be no end of time, edge of space, or smallest particle. But it is hard to believe that there would be only one of these totally finite universes. First, it is difficult to see how to apply circular scale unproblematically unless there are infinitely many universes; second, the principle of sufficient reason is violated if only this particular finite universe exists; and third, there is the feeling that the "space" in which our space-time is curved should be real.
In the section on spatial infinities it was pointed out that if, on the one hand, one repeatedly finitizes by replacing lines with circles, and if, on the other hand, one never accepts some particular finite n -verse as the end of the line -- if, in other words, one thinks along the lines sketched in the last two paragraphs --then one is forced to conclude that space is infinite dimensional and that there are infinitely many objects in this cosmic space.
In the last section I discussed some of the ways in which an actual infinity could physically arise. But there are things that are not physical. There are minds, thoughts, ideas, and forms. In this section we will see if any of these familiar nonphysical entities are actually infinite. In order to appreciate the section at hand, it is necessary to keep an open mind on the question of whether or not mind equals brain, for if one assumes a priori that a thought is nothing more than a certain biochemical configuration in a certain finite region of matter, then unless one has infinite divisibility of matter it seems to follow automatically that infinite thoughts are impossible.
To cast a few preliminary doubts on the hypothesis that brain equals mind, let me quickly raise a few questions. Is what you thought yesterday still part of your mind? If you own and use an encyclopedia, are the facts in that encyclopedia part of your mind? Does a dream which you never remember really exist?
How can you grasp a book as a whole, even though you only read it a word at a time? Would the truths of mathematics still exist if the universe disappeared? Did the Pythagorean theorem exist before Pythagoras? If three people see the same animal, we say the animal is real; what if three people see the same idea? I think of consciousness as a point, an "eye," that moves about in a sort of mental space. All thoughts are already there in this multi-dimensional space, which we might as well call the Mindscape.
Our bodies move about in the physical space called the Universe; our consciousnesses move about in the mental space called the Mindscape. Just as we all share the same Universe, we all share the same Mindscape.
For just as you can physically occupy the same position in the Universe that anyone else does, you can, in principle, mentally occupy the same state of mind or position in the Mindscape that anyone else does. It is, of course, difficult to show someone exactly how to see things your way, but all of mankind's cultural heritage attests that this is not impossible. Just as a rock is already in the Universe, whether or not someone is handling it, an idea is already in the Mindscape, whether or not someone is thinking it.
A person who does mathematical research, writes stories, or meditates is an explorer of the Mindscape in much the same way that Armstrong, Livingstone, or Cousteau are explorers of the physical features of our Universe. The rocks on the Moon were there before the lunar module landed; and all the possible thoughts are already out there in the Mindscape.
The mind of an individual would seem to be analogous to the room or to the neighborhood in which that person lives. One is never in touch with the whole Universe through one's physical perceptions, and it is doubtful whether one's mind is ever able to fill the entire Mindscape. One last analogy. Note that there is always a certain region of physical space that only I can ordinarily know of -- barring surgery, no one but me is in a position to assess the physical conditions obtaining within my stomach.
In the same way, there is a certain part of the Mindscape that only I can ordinarily know of -- unless I am to be greatly favored by the Muse, the feelings that pass over me when I think of my childhood will always remain private and inexpressible. Nevertheless, these almost ineffable feelings are part of the common Mindscape -- they are simply difficult for anyone else to get to. The point of all this is that just as the finiteness of our physical bodies does not imply that every physical object is finite, the finiteness of the number of cells in our brains does not mean that every mental object is finite.
The most familiar candidate is the set N of all natural numbers. What the ". The idea, of course, is that all of the natural numbers are to be collected together into a whole. Each of them would seem to exist individually in the Mindscape, and one would suppose that the set consisting of exactly the natural numbers would be in the Mindscape as well -- one almost feels as if one can see it.
We might try to avoid the use of the ". We might try to get around this difficulty by saying that N is the smallest set in the Mindscape that has one in it, and that has x plus one whenever it has x. But, for reasons that I will begin to explain in the next section, the word "Mindscape" cannot be meaningfully used in a definition. The concept of "Mindscape" is too vast to be represented by any word or symbol.
If we try to avoid this difficulty by substituting some sort of finite description of the mental universe for the word "Mindscape," then we get the same problem as before. So it is quite literally true that what is really meant by the ". Some thinkers have taken this to mean that there is, after all, no unique N in the Mindscape. This could be true. But one need not take this to mean that there are no infinite sets in the Mindscape: if there are many, many versions of the set of natural numbers, then there are many, many infinite sets.
However, it is normally more desirable to assume that there is a simple unique N in the Mindscape, just as it is simpler to assume that there is only one universe instead of a whole slew of "parallel worlds. I might note here that if time is indeed infinite, then just as we can indicate Earth by saying, "this planet," we could indicate our N by saying, "the number of seconds left in this time. If infinite forms are actually out there in the Mindscape, then maybe we can, by some strange trick of mental perspective, see some of these forms.
The philosopher Josiah Royce maintained that a person's mental image of his own mind must be infinite. So the image includes an image that includes an image, and so on. This infinite regress can be nicely visualized by imagining a United States in which a vast and fanatically accurate scale model of the country occupies most of the Midwest.
The scale model, being absolutely accurate, includes a copy of the scale model, etc. This regress is occasionally used to make a striking label for a commercial product. The old can of Pet Milk, for instance, bore a picture of a can of Pet Milk, which bore a picture of a can of Pet Milk, etc.
In a physical situation we would probably never actually be able to finish making such a label in all its infinite detail. But this is not to say that no such label or country-plus-scale-model could exist. There would be no problem, if matter were infinitely divisible. If scale is indeed circular, then everything is, in a sense, already an object of this nature!
There is certainly no reason why a nonphysical mind should not be infinite; and Royce's point is that if you believe that one of the things present in your mind is a perfect image of this mind and its contents, then your mind is infinite. I would like to discuss this a bit more, but first let me formally introduce some of the apparatus of set theory.
That is Ouroboros itself is a sign of the infinite and drawn in the shape of 8. In the 17th century infinity symbol got its mathematical meaning. In it was first used by John Wallis but he never said that why he used 8 on its side as a symbol of infinity.
In fact, this type of similar symbol was used by Romans to express large numbers. Top 10 most intelligent people on Earth. In mathematics, Calculus, Leibniz speculated infinite numbers and their use in mathematics. In Real analysis also the symbol infinity is used to denote an unbounded limit. Even in Complex analysis the symbol infinity denotes an unsigned infinite limit etc.
The infinity symbol appears in the Tarot , as a part of the Magician card. Like in the Pamela Colman rider Waite version has a lemniscate floating boldly infinitive symbol above his head and in other decks, the brim of his hat conceals the shape.
Are you worried or stressed? Click here for Expert Advice.
0コメント