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Thursday, January 23, 2014

Does infinity really exist?



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Our universe appears to be bound by a finite set of laws, yet we often talk about things that go on for an eternity. "Infinity" is a strange idea. But it's crucial if you want to understand anything from philosophy to mathematics. Here’s why.

There are three broad domains where infinity can be applied. It’s used as a conceptual tool to help us describe the properties and values of objects and processes, it’s an important notion in philosophy, cosmology and metaphysics, and, of course, it’s crucial to mathematics. Let’s take a look at each of these in detail.

The Totality of Infinity

We often use the word ‘infinity’ when describing something that goes on forever. But it can also be used to describe something that doesn’t go on forever, or for which its value is absolute.

Take, for example, the use of infinity in sports and gaming. As every chess player knows, each piece is assigned a numerical value according to its tactical importance and strength. These values range from one (pawns) to nine (the queen), and are often used to keep a kind of score as the game progresses. But the king is assigned infinite value — and for very good reason. Losing the king is fatal. It’s instant game over, regardless of whatever else might be happening in the match. The king’s worth, therefore, cannot be bound within a finite set of values.
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Image: Shutterstock/18percentgrey.

Similarly, an overtime goal in hockey, or a ‘golden goal’ in soccer, can likewise be ascribed infinite value. Any situation in which a single goal causes the match to come to an immediate end, and along with it both instant victory and defeat, hinges on an absolutist event.

More importantly, philosophers, religious scholars, legal experts, and ethicists often assign an infinite value to human life. And indeed, as an enlightened society, we’re appalled by the idea of attaching a price to such a thing. We simply cannot buy, sell, or trade each other. Of course, this doesn’t always work in practice. During war, human lives are sacrificed to protect the larger body of citizens and to uphold certain values and institutions. And illicit human trafficking is an ongoing problem. But at a conceptual level, and as many religious thinkers argue (especially Christians), there is no price high enough to allow for the ‘buying’ of human life, and there’s no situation grave enough to warrant killing (i.e. Thou shalt not kill); only God has the power — and the right — to take away the gift of life.

And therein lies another kind of infinity: death. Assuming that nothing awaits us in the afterlife, the termination of our lives represents a kind of eternity. It’s an eternity of nothingness, but an eternity nonetheless.

Infinite In All Directions

But while infinity can be used as a helpful conceptual tool, it’s also something that may be physically, tangibly real — especially when considering some of the freakier aspects of philosophy, cosmology and metaphysics.

Take for example the eerie findings of string theory. Though controversial, the theory suggests the presence of 10 or 11 spacetime dimensions. The extra six or seven dimensions could be compacted at an insanely small scale, or our universe could be located on a D-brane — a dynamic (3+1)-dimensional object. Columbia University physicist Brian Greene argues that these branes could support parallel universe, giving rise to the multiverse hypothesis and the potential for an infinite set of braneworlds; these worlds aren’t always parallel and out of reach, resulting in constant collisions that cause an infinite succession of Big Bangs.


But there are other examples as well. Cosmologist Lee Smolin, in his book The Life of the Cosmos, proposed that universes are nothing more than black hole generators. According to his theory of cosmological natural selection, each black hole spawns a daughter universe. The process is then repeated — likely for an infinity.

And indeed, we’re not certain if the universe (or multiverse) has an end — or a beginning for that matter. Perhaps it’s always been here and it always will.

We’re also not sure about the shape of space-time. If it’s flat, then it could stretch out for an infinity — but it must start repeating at some point owing to the finite number of ways particles can be arranged in space and time. If this is the case, then there are an infinite number of universes.

Quantum mechanics also suggests an infinite universe. Everett’s Many Worlds Interpretation (MWI) states that the universe branches off into distinct worlds to accommodate every single possible outcome. We may live in an infinite web of alternate timelines.
This raises some interesting — if not deeply troubling — issues.

If the MWI is true, for example, every possible outcome that can be observed will be observed — no matter how ludicrous or extreme.

And given an infinite amount of time, freak disembodied minds, called Boltzman Brains, could eventually pop into existence owing to the right configurations of matter and energy. Thankfully, a new interpretation of string theory suggests that humanity will never be overrun by these spontaneously forming disembodied space brains.

The Math of the Matter

Lastly, infinity is used in math — though not without controversy. It’s often mistaken for a number or a singular entity, but it’s more a label that can be used to describe a variety of mathematical objects and concepts that are larger than anything that can be physically or conceptually expressed in the real world. More simply, it’s a term that can be given to any non-finite number or groups of numbers. And indeed, the whole point of it is that it can’t be characteristized as there is no end to it.

There are two distinct areas in which infinity is used in mathematics, namely set theory and topologies (there are others as well, like limits and algebra, but we’re not going to discuss those at this time).

Fun With Sets

Set theory, in which numbers with corresponding items can be grouped into sets, shows that there are multiple types of infinities and that some are bigger than others — a surprising result to say the least. Back in the 19th century, Georg Cantor used this insight to describe two different kind of infinities, countable and uncountable.

Countability describes anything that can be lined up such that it can be numbered; all sets that can be put into a one-to-one correspondence with natural numbers can be considered countable.
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Image: Rasmus Holmboe Dahl/Shutterstock.

In set theory, this would include a group like {1,2,3,4} and {banana, cake, fork, napkin) — a set with a cardinality of 4 (i.e. the size of a set). Some of these countable sets are infinite (called enumerable infinities), like the set of all integers (yes, members of a countably infinite set can be counted, even though it would take an infinity to do so). But because a single number cannot describe the size of an infinite set, Cantor used the term aleph-0, or aleph-null (aleph being the first letter of the Hebrew alphabet) to signify the cardinality of an infinite set.

Indeed, aleph-0 is a cardinal number despite the fact that infinity should not be considered a number; but numeration can in fact be extended to infinite quantities so long as consistent definitions are maintained. Needless to say, infinite cardinal numbers don’t follow the same mathematical rules as finite number — you can’t just throw them into an equation and hope for the best.

Aleph-0 can include such sets as the set of all prime numbers, all rational numbers, all algebraic numbers, the set of binary strings of all finite lengths, and the set of all finite subsets of any countably infinite set.

Interestingly, the number 2 is a countable infinity. It’s a number that can be physically realized, but it consists of an infinite number of fractions.

Uncountable infinities, on the other hand, happens when you get into irrational numbers, like the square root of 2, or transcendental numbers like pi and e. The reason why these sets are considered uncountable is because they cannot be numbered.

Cantor proved this by offering a thought experiment. If we had a numbering scheme that was supposed to count all the real numbers, we could go down that list, and for the nth term, read off the nth digit. But we could then change it to something different and use it on our new number. But since this “newly constructed number” is different from every other number on the list (at the nth digit at the very least), then it must have been missed in the enumeration. The list cannot be enumerated in any way. It’s simply too big, and thus uncountable.

Numbering Nonsense

As an aside, number systems cannot accommodate the concept of infinity. No “infinity” concept exists in the context of any number system, if by number system we mean any collection of concepts that have operations like addition and multiplication — operations which obey the usual properties of arithmetic.

For example, what would infinity minus 1 be? It couldn’t be a finite number, since no finite number plus 1 equals infinity. This would violate the rules of arithmetic, leading to such absurd equations like -1=0, which isn’t true. So infinity doesn’t exist, if by “exist” we’re talking about the context of a number system.

Topological Spaces

Topological spaces describe the properties of surfaces outside of angles and distances. So, if two surfaces can be mapped together (i.e. made the same) by stretching and pulling them, rather than cutting and pasting them, they’re considered identical from a topological perspective.

Interestingly, the real number system is considered a topological space. We can come up with a definition of what it means for a sequences of numbers, rather than surfaces, to converge. For example, the sequence {1.1, 1.01, 1.001, 1.0001...} converges to the number 1, while the sequence {1, 2, 1, 2, 1, 2, 1, 2...} doesn’t converge to anything.

In calculus, it’s said that sequences like {1,2,3,4...} converge to infinity. But can it truly be said that there’s an actual object called “infinity” that this sequence can be converged to? Is there some topological space — i.e. a set of objects plus a definition of what convergence means — that includes real numbers and also an infinity concept to which some sequences of real numbers converge?

Perhaps surprisingly, the answer is yes.

Ask a Mathematician provides an explanation:

When topologists work with the real number line (i.e. the set of real numbers together with the usual notion of distance which induces topological structure), they sometimes introduce a “point at infinity”. This point, denoted \infty can be thought of as the point that you would always be heading towards if you started at 0 and traveled in either direction at any speed for as long as you liked. Strangely, when this infinite point is added to the real number line, it makes it topologically equivalent to a circle (think about the two ends of the number line both joining up to this single infinite point, which closes a loop of sorts). This same procedure can also be carried out for the plane (which is the two dimensional surface consisting of points (x,y) where x and y are any real numbers). By adding a point at infinity we compactify the plane, turning it into something topologically equivalent to a sphere (imagine, if you can, the edges of the infinite plane being folded up until they all join together at a single infinity point).
Does infinity really exist?
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Image: AAM.

As a concluding note, and as Ask a Mathematician points out, it’s important to remember that the question of infinity in mathematics cannot be answered indisputably. We can ask, “how do infinite things arise in math”, but we can only answer that they arise in many, very important ways.

Top image: Vadim Sadovski/Shutterstock.
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Top image: Vadim Sadovski/Shutterstock.

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SUBTITLES IN ENGLISH, ESPAÑOL, PORTUGUÊS

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MUFON

How to Digitally Record/Video a UFO sighting:


Como registar digitalmente ou gravar um vídeo de um avistamento de um UFO:




Stabilize the camera on a tripod. If there is no tripod, then set it on top of a stable, flat surface. If that is not possible lean against a wall to stabilize your body and prevent the camera from filming in a shaky, unsteady manner.

Estabilize a camera com um tripé. Se não tiver um tripé, então coloque-a em cima de uma superfície estável. Se não for possível, então encoste-se a uma parede para estabilizar o corpo e evitar que a camera registe de maneira tremida e instável.

Provide visual reference points for comparison. This includes the horizon, treetops, lampposts, houses, and geographical landmarks (i.e., Horsetooth Reservoir, Mt. Adams, etc.) Provide this in the video whenever is appropriate and doesn’t detract from what your focus is, the UFO.

Forneça pontos visuais de referência para comparação. Isso inclui o horizonte, cimo das árvores, postes de iluminação, pontos de referência geográficos (como o Reservatório de Horsetooth, Mone Adams, etc) Forneça esses pontos no vídeo sempre que for apropriado e não se distraia do que é o seu foco, o UFO/a Nave.

Narrate your videotape. Provide details of the date, time, location, and direction (N,S,E,W) you are looking in. Provide your observations on the weather, including approximate temperature, windspeed, any visible cloud cover or noticeable weather anomalies or events. Narrate on the shape, size, color, movements, approximate altitude of the UFO, etc and what it appears to be doing. Also include any unusual physical, psychological or emotional sensations you might have. Narrate any visual reference points on camera so they correlate with what the viewer will see, and thereby will be better able to understand.

Faça a narração do vídeo. Forneça pormenores sobre a data, hora, local e direcção (Norte, Sul, Este, Oeste) que está a observar. Faça observações sobre as condições atmosféricas, incluindo a temperatura aproximada, velocidade do vento, quantidade de nuvens, anomalias ou acontecimentos meteorológicos evidentes. Descreva a forma, o tamanho, a cor, os movimentos, a altitude aproximada onde se encontra o UFO/nave, etc e o que aparenta estar a fazer. Inclua também quaisquer aspectos pouco habituais de sensações físicas, psicológicas ou emocionais que possa ter. Faça a narração de todos os pontos de referência visual que o espectador irá ver e que, deste modo, será capaz de compreender melhor.

Be persistent and consistent. Return to the scene to videotape and record at this same location. If you have been successful once, the UFO sightings may be occurring in this region regularly, perhaps for specific reasons unknown, and you may be successful again. You may also wish to return to the same location at a different time of day (daylight hours) for better orientation and reference. Film just a minute or two under “normal” circumstances for comparison. Write down what you remember immediately after. As soon as you are done recording the experience/event, immediately write down your impressions, memories, thoughts, emotions, etc. so it is on the record in writing. If there were other witnesses, have them independently record their own impressions, thoughts, etc. Include in this exercise any drawings, sketches, or diagrams. Make sure you date and sign your documentation.

Seja persistente e não contraditório. Volte ao local da cena e registe o mesmo local. Se foi bem sucedido uma vez, pode ser que nessa região ocorram avistamentos de UFOs/naves com regularidade, talvez por razões específicas desconhecidas, e talvez possa ser novamente bem sucedido. Pode também desejar voltar ao mesmo lugar a horas diferentes do dia (durante as horas de luz)para ter uma orientação e referência melhor. Filme apenas um ,inuto ou dois em circunstâncias “normais” para ter um termo de comparação. Escreva tudo o que viu imediatamente após o acontecimento. Logo após ter feito o registo da experiência/acontecimento, escreva imediatamente as impressões, memórias, pensamentos, emoções, etc para que fiquem registadas por escrito. Se houver outras testemunhas, peça-lhes para registar independentemente as suas próprias impressões, pensamentos, etc. Inclua quaisquer desenhos, esbolos, diagramas. Certifique-se que data e assina o seu documento/testemunho.

Always be prepared. Have a digital camera or better yet a video camera with you, charged and ready to go, at all times. Make sure you know how to use your camera (and your cell phone video/photo camera) quickly and properly. These events can occur suddenly, unexpectedly, and often quite randomly, so you will need to be prepared.

Esteja sempre preparado, Tenha sempre uma camera digital, melhor ainda, uma camera vídeo consigo, carregada e pronta a usar sempre que necessário. Certifique-se que sabe como lidar com a sua camera (ou com o seu celular/camera fotográfica) rápida e adequadamente. Esses acontecimentos podem acontecer súbita e inesperadamente e, por vezes, acidentalmente, por isso, necessita estar preparado.

Look up. Be prepared. Report. Share.

Olhe para cima, Esteja preparado, Relate, Partilhe.

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