The Infinite. Part 4. Or What Makes the Federal Debt Look Tiny.

Journalists and pop science writers have a little fun trying to make large numbers “real” to us.[1] For example, a billion dollars in one hundred dollar denomination bills would require about ten standard storage pallets in your garage, stacked 5 feet high or so. A trillion dollars in one hundred dollar bills would occupy a warehouse with 10,000 such pallets. I don’t know what you’d spend it on, but if you spent a million dollars every day since the time of Moses, until now, you may have used up the trillion dollars. Spendthrift! Stacking up a hundred million trillion one dollar bills will get you out a bit more than one light year from earth. (Making that many dollar bills would require more than the mass of the earth – and it would take too long – not to mention the ink.)

In the previous post I considered some issues surrounding the use of “infinite” in a religious context. For the most part, it appears that “infinite” (and cognates like eternity, eternal, forever) as found in scripture and classical LDS discourse is rarely used to refer to some actualized infinity. It seems to be employed to emphasize the distance between God and creation (including man) or to otherwise indicate the grandeur of the Almighty. There are threads in Mormon thought that narrow, even erase the ontological distance between God and man, while at the same time they place an existential gulf between one person and another (and perhaps that is what sealing is all about – a bridging of existence) but even so, no one postulates that God and man are much alike in any cosmological sense. I’m going to come back to this in a later post (don’t worry, this series is not infinitely long).

A God-like view (ht:BCC) of the universe is difficult to comprehend even for, perhaps especially for, moderns. Let me illustrate. Imagine a tiny bit of the universe: a ball, one light-year in radius. Such a ball would contain roughly 900,000,000,000,000,000,000,000,000,000,000,000,000 cubic miles of space. It would take about 30,000,000,000,000,000,000,000,000,000 copies of earth to fill up such a ball of empty space. Astronomical, right? And relatively speaking, our stellar neighborhood is rather bare. There is no other star within a light-year of us. No other stellar system within that 9 x 1038 cubic miles around us. (It’s possible that there is some burned out star coming our way because of the rather chaotic gravitational conditions in more central regions of our galaxy. Let’s hope not.)

These numbers seem large and they are difficult to process. On a more mundane level, our brains store information (memory). We do this in a rather fancy way using somewhat mysterious compression techniques. Even so, aside from reconstructive smoothing (“I lost Jenny at the park,” “What was she wearing?” “I can’t recall but maybe something green.”) the brain probably has a raw storage capacity of something like 1300 gigabytes. In any case, there is certainly an upper bound for storable data.[2] Hence our ability to artfully reflect and forget, encode systematic behaviors in formulae, etc. It’s this ability that allows some logical boundaries for the infinite.

There is no universal time reference in the physical universe. But as earth-bound humans, our in-house sensors don’t perceive this. Indeed, we’re pretty limited in the way we can observe things with our unaided “five senses.” But imagine this: suppose we could construct a space vehicle capable of safely achieving something very close to the speed of light, say 99.9999% of that speed. It would then be possible for you to hop on board and circumnavigate the big-bang universe in less than 40 years (onboard time). Time for you would pass at a normal rate in your surroundings on board. But to someone on earth (assume they could get a signal from you) it would seem that you were stone. No perceivable movement on board (your return journey would be a disappointing one – this physical earth would no longer exist).

Among the cosmologies proposed in the past, some saw the universe as either bounded by the “spheres” or infinite in extent. (The Hebrew scriptures picture the universe as a flat earth bounded above by a dome – the windows of heaven.) It’s not clear in modern cosmology whether space is “infinite.” The issue is complicated by a lack of knowledge about the actual shape of things. Time however, provides a useful backdrop for thinking about large things. We measure time with the common unit of one second. Once upon a time, a second was defined as 1⁄86,400 of an earth day. But astronomical observations showed this period was gradually getting longer. So at the present time a second is defined by (theoretical) “vibrations” of caesium 133. Given that, we can talk about things like the age of the universe (~14,000,000,000 years).[3]

What’s the largest named number around, aside from exponential numeral representation? A googolplex [ed: but see comments for larger ones]. That’s 10 to the googol, or 1010100. It’s well nigh impossible to consider writing this number down in all its glory. 1 followed by a googol of zeros. You would die long before you wrote this number. Even if you used every elementary particle in the universe to stand for a zero, you would run out long before being able to write out a googolplex. Indeed, there is not enough space in the observable universe to actually write down a googolplex. It’s big, kind of. But clearly not anything like a “large” number. Now ω. That’s starting to get big, relatively speaking. After the next few installments, whenever you read or hear “Alpha and Omega,” you’ll get a little shiver down your spine. Also, I may discuss time in more detail at some point. Meanwhile, I encourage you to get hold of a copy of Steven Peck’s little book, A Short Stay in Hell. I was lucky enough to get a kindle version. It’s a lovely little story about time and what it might mean to live forever. More on this coming up.

(Part five, here.
———
[1] I use the term large to mean “out of normal” experience. There is really no such thing as a large number. All numbers you can think of are nearly equal to zero, comparatively.

[2] There are examples of people who seem to retain everything, even huge lists of digits by simply reading them through once – including randomly described subsets of lists, etc. Unfortunately, this doesn’t seem to correlate with great intuitive insight. See, A. P. Лурия, МАЛЕНЬКАЯ КНИЖКА О БОЛЬШОЙ ПАМЯТИ. English: Mind of a Mnemonist, (Harvard UP, 1987).

[3] Standard years (orbital time for earth around sun) require the insertion of one official second per year so that the sun is directly overhead at the Greenwich Meridian at noon. This keeps caesium time and solar time in sync more or less. I feel unsettled about abandoning solar time, but it isn’t uniform.

Comments

  1. I love your footnotes in particular, WVS. For example, this:

    There is really no such thing as a large number. All numbers you can think of are nearly equal to zero, comparatively.

    I can’t tell you how many times I’ve told my oldest son this, in discussions of large numbers. Seriously!

    And regarding #3, have you seen this Slate article on the possibility of letting the cesium-based time drift away from astronomical time?

  2. I knew people were thinking about letting things diverge. But had not seen the article. Thanks for the link.

  3. “It would then be possible for you to hop on board and circumnavigate the big-bang universe in less than 40 years.”

    I was convinced that this couldn’t be right, but I finally realized that you were speaking of time as experienced by the traveler. The universe is a strange place, and it’s easy to forget that.

  4. Is God greater than any pile, including an infinity, of exponents? Or any nest of recursive factorials?

    n!!!!!!!!!!!!!!!!!!!!!!!….!!!! where the n > infinity and the number of factorials > infinity?

    Mathematics is a modeling language. It is not reality. I do not believe it has anything much to do with God.

  5. Alright WVS, you keep leading me on with good posts. There better be some awesome punchlines coming up.

    Again, just rattling off what comes to mind, but so far the physical universe would indicate that relativity is the norm, that there are no absolutes. That is, there is no inertial frame, everything is rotating relative to something else. The earth is indeed the center of the universe (in a sense). Large numbers don’t exist, small numbers don’t exist, only their relative size compared to another number. If relativity seems to be the case, is it not reasonable to extrapolate this to other spheres of influence? Why is moral relativity so unpopular (especially in the church) (this is a rhetorical question because I know what the standard Mormon view on this is)? Why is philosophical relativism considered silly these days? Indeed, when I look at the moral landscape of the history of the earth I have to hypothesize that morality and relativism are in fact relative to the social constructs of the group. It feels to me like the label of “absolute,” “law,” etc. are really codified terms for perceived “consistency,” or “contract,” etc. Even God himself, if we’re to accept the scriptures, consistently commands his people to take exception to previously ordained laws.

    For me, coming full circle, just like infinity, the idea of an absolute ANYTHING requires a leap of faith that isn’t supported by the evidence we have available in the physical world. It’s like we insist on a platonic form of absoluteness to assuage our fears of uncertainty, randomness, and other messy aspects of the physical world.

  6. Re RW-

    Mathematics is a modeling language. It is not reality.

    It depends on who you ask. The question as to whether or not math is created or discovered is not an easy one to answer. It seems to me the question can be divided along the lines of engineers vs. physicist/mathematicians. Pragmatic individuals tend to believe math is created, a modeling language (this is what I think so I actually agree with you), but theoretical physicists hold out for one equation for everything as if it’s only a matter of time before such an equation is discovered. I think both sides have some compelling arguments.

  7. I don’t know about awesome punchlines, but there is some interesting interface with Mormonism coming up.

  8. jmb275,

    It is a conceit that mathematics is real. It cannot exist without an intelligence to draw the relationships between the symbol and the thing. In this regard it is like a real language. Is language created or discovered?

    Real language has rules which make it intelligible, so does mathematics. The rules of real language seem to carry over, albeit loosely, into all languages. Mathematics’ rules, more strictly, are used to describe the world. The world would not cease to exist if there were no mathematics or intelligence to use it. It is a mapping system by which intelligence maps the internal world to the external.

    I could be wrong.

    Bring on the punch line, WVS

  9. Love this post, and I for one hope the series goes on forever.

  10. There are much bigger named numbers than googolplex. Graham’s Number makes a googolplex seem paltry by comparison, which, of course, is right in line with your point. Great series by the way.

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