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Tuesday, November 27, 2012

Fitting an Infinite Universe Into a Finite Head





To see a world in a grain of sand
and a heaven in a wildflower,
hold infinity in the palm of your hand
and eternity in an hour.”


-
William Blake, from “Auguries of Innocence”


Believe me, I am a novice about all the knowledge necessary to even logically contemplate the Universe. I know very little about planets, stars, galaxies, and the contents of intergalactic space. I am content to be amazed by the totality of existence, and I find a deep spiritual connection to the cosmos. However, I find myself feeling like a grade schooler without the mental capability to understand most discussions about creation, space, and time. Theories and models pertaining to the universe tend to speed "over my head,” and I find myself soon losing interest.

Many renowned scientists claim the Universe is believed to be at least 93 billion light years (A light year is about 6 trillion miles.) in diameter and has existed for about 13.7 billion years, since it was created by the Big Bang.

In addition, physicists studying various multiverse hypotheses have suggested that the Universe might be one among many universes that likewise exist.

What part of the Universe can humans actually see? The farthest distance that it is theoretically possible for humans to see is described as the observable universe. Of course, more lies beyond the limits of human sight.

In fact, physicists say observations have shown that the universe appears to be expanding at an accelerating rate (One model claims the time it takes for the linear size scale of the universe to expand to double its size is approximately 11.4 billion years.), and a number of models have arisen to predict its ultimate fate by meeting the Big Freeze, the Heat Death, the Big Crunch, the Big Rip, or the Big Bounce.



A Finite or Infinite Universe

See what I mean. It doesn't take long to get lost. I have always believed the Universe is infinite, yet some scientists suggest it is, indeed, finite. Infinite may be defined as “unmeasurable in extent of space or duration of time.” A mathematician may add that infinite refers to that “being without an upper or lower numerical bound.”

Let's try to understand the finite/infinite nature of natural numbers. Then, we can apply it to determining the bounds of the Universe. Go slowly, now, at least if you are as mathematically deprived as I.

“The natural numbers are the integers we use daily to count like 1, 2, 3, ... etc. The aggregate of all the natural numbers is usually represented by the symbol N although sometimes it is also symbolized by the symbol Z+ to clearly state that we are leaving out the integer zero (0).

“The series of the natural numbers never ends because to any given number n it is always possible to add 1 to find its successor n+1. Because this series never ends is why we call it "infinite". Sometimes, this is also denoted as 1, 2, 3 ... ∞, but we must keep in mind that the infinite cannot be properly represented by any symbol, because the symbol ∞ is not an arithmetic entity that can be manipulated under the common mathematical operations.”

“Another way of defining the infinitude of the natural numbers is as follows: 2 is the successor of 1, 3 is the successor of 2, 4 is the successor of 3, etc. Thus, 3 is the successor of the successor of 1, 4 is the successor of the successor of the successor of 1. Hence, N is the collection —or set, or aggregate— of all the successors of 1

N = {1, 1 + 1, 1 + 1 + 1, ...}.

“Therefore, the infinitude of the natural numbers is nothing more than the collection of the possible successors of the unit element one.”

We cannot prove that the natural numbers are infinite; we take them as the definition of infinitude. The aggregate of the natural numbers are the "standard yardstick" to measure other infinitudes.

“Giving a symbol —like N— for the aggregate of all natural numbers does not make this set more comprehensible than not naming it at all. Every infinitude is intrinsically incomprehensible, the reason for this nobody knows, but the role of the language may be a relevant barrier because our languages —all of them, be it spoken, written, symbolic or not— are tools for the manipulation of finite and bounded everyday objects, and therefore, the abstract relations among them is forcibly limited.”
("Some Myths About the Infinite," Android Mind, 4dlab.info)

Watch Video:
 
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Wow, infinite but finite? Math always says you can add one more number. If you can count that number, I guess it would be finite, yet an infinite set (nothing more than the collection of the possible successors of the unit element one) has to prove infinitude. And, maybe our language contributes to the finite nature of numbers since we use words and symbols to express natural numbers. Ouch! My head hurts already. Could math help us define the Universe?
 
 
Sand and Stars

Now, let's try to understand the finite nature of a grain of sand. If it were possible to separate all the grains of sand on all the beaches of the world, do you think someone could count them? Remember, the sand grains are real; therefore, they do exist in a physical form that could be (But, how?) measured and counted. Boy, I wouldn't want that job. It seems impossible.

How about the stars in the Universe? Can they be counted like grains of sand? If they could be numbered, what number – sand grains or stars – would actually total the most?

To give an analogy to the infinite nature of the universe,
Carl Sagan is quoted as saying,
"The total number of stars in the Universe
is larger than all the grains of sand
on all the beaches of the planet Earth."

Sagan goes on to say:

“The beach reminds us of space. Fine sand grains, all more or less uniform in size, have been produced from larger rocks through ages of jostling and rubbing, abrasion and erosion, again driven through waves and weather by the distant Moon and Sun...

“A handful of sand contains about 10,000 grains, more than the number of stars we can see with the naked eye on a clear night. But the number of stars we can see is only the tiniest fraction of the number of stars that are. What we see at night is the merest smattering of the nearest stars. Meanwhile the Cosmos is rich beyond measure: the total number of stars in the universe is greater than all the grains of sand on all the beaches of the planet Earth.”

(Carl Sagan, Cosmos, Chapters VIII and XII, 1980)

Watch Video:

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And, wouldn't you know, one guy thinks he should test Sagan's theory. Glen Mackie, lecturer of the Centre for Astrophysics and Supercomputing, Swinburne University of Technology in Melbourne, Australia, claims he can figure out how a number for sand grains (really an estimate) and a number for stars (likewise, an estimate) might be made. Mackie says the following:

“What are the dimensions of an average, sandy beach on Earth? I was lucky. I know a Coastal Geomorphologist (doesn't everyone?) who provided some estimates. Take 360,000 kilometres of coast (the total coastline of the world is about 1 million kilometres, of which about 36 percent is sandy), mix with an average beach width of 50 metres (high to low tide lines), and add a dash of average beach depth of 4 metres.

“The diameters of sand grains range from about 0.1 to 2 millimetres. We'll adopt an average diameter of 0.5 millimetres. Lets stack them simply, one on top of another (not very likely in a beach I admit), so that 8000 fit into 1 cubic centimetre. Drum roll please. The total number of sand grains on our Earth beach is then 600 billion billion. But wait!, I hear some of you remarking about sand dunes and sand below the low tide mark. To placate you worriers I'm happy to increase my initial estimate by a factor of 3. That gives me a grand total of about 2000 billion billion grains of sand on Earth.

“The stars win! Carl was right, ... maybe. The excess factor (the number of stars divided by the number of grains) of 50,000/2000 or 25 is greater than an order of magnitude, but only just. My individual assumptions must be wrong at some level. Maybe the average galaxy has only 40 billion stars (faint surveys do detect many small, ``dwarf'' galaxies). As well, maybe the number of sand grains is three times greater. (In fact my Coastal Geomorphologist suggested that the average grain diameter could be smaller than 0.5 millimetres, probably allowing us to stack a factor of 3 more!). Combine these two changes and the grains (barely) win!

“Not so fast! Any (conspired) increase in the number of sand grains is probably offset by the recent discovery of many more galaxies by HST in the near-infrared region of the electromagnetic spectrum, not previously detected in optical surveys. Observing in the infrared minimizes the absorbing effects of dust allowing us to see galaxies enshrouded in large amounts of dust present because of vigorous star formation. The odds are that there are many more than 130 billion galaxies in the universe. I'll cast my final vote in favor of the stars, and I'll admit a slight astronomical bias.”

(Glen Mackie, To See the Universe in a Grain of Taranaki Sand,
North and South Magazine -- New Zealand, May 1999)

Another expert adds this:

If a grain of sand represented an entire galaxy; so each grain of sand, or galaxy, contains 100’s of billions of stars, you would need to fill six rooms full of sand to contain all the galaxies in the known universe. If you drilled a tiny whole in one of the grains of sand, 'our Milky Way universe,' that would be the area that we have been capable of searching for planets so far. 534 planets have been discovered so far.”
(Dan Goods, “Visual Strategist” for NASA’s Jet Propulsion Laboratory, directedplay.com)
 
 

My Take

So, numbers and measurements and calculations and theories are just not enough to make me appreciate the enormous size of our Universe, even our known Universe whose definition is being greatly expanded by the use of space telescopes and observation devices. I think we must admit that science is as yet ignorant to defining any limitations.

A finite or infinite Universe? I see the concepts of both as totally unimaginable. My human brain simply cannot comprehend a model of either. How can it distinguish a space that has beginnings and ends or a space that has no end? I can't even comprehend what would lie beyond the boundaries of the Universe if it had bounds. It is certain infinity is just a word symbol, not a concept I understand.

I love to think about the great mysteries of our existence, yet I realize these things exist as part of a creation that will continue to baffle me. I am just grateful I have a few more years to marvel at the wonders created by the Master plan. Humans are very fortunate to be able to think and imagine.

I hope I never lose that sense of wonder, and I fully expect to remain “in the dark” about the nature of the Universe. To me, that is the best way to think about it – under a star-covered sky on a dark night, alone and silently staring at the vast cosmos, and realizing my little human life is of microscopic importance. Then, a heavenly realization humbles me as it should: “I am just a minuscule grain of sand on one tiny planet in a beautiful, endless Universe.”

A Man Said to the Universe
A man said to the universe:
"Sir I exist!"
"However," replied the universe,
"The fact has not created in me
A sense of obligation."
-Stephen Crane

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