Puzzles for Pleasure #2- Hotel Infinity

A long, long time ago in a galaxy far far away, there was a hotel with an infinite number of rooms. The room numbers start at 1 and go on forever...

One day, when every room was occupied, a space pilot on his way to deep space nine (or something) dropped by to spend the night. Even though there was no vacancy, the hotel manager simply notified every guest to move the the room number that was one higher than their current room.

This left room 1 open for the pilot.

The next day, 10 couples on their honeymoon showed up. The hotel manager did the same, only notifying everyone to move to a room number ten higher than their current room, leaving 1 through 10 available to the honeymooners.

The next day, an INFINITE number of guests seek lodging. Is it possible for the hotel manager to accept an infinite number of guests when the hotel already is lodging an infinite number?

TAKE A MOMENT TO THINK ABOUT THAT BEFORE READING ON...THE ANSWER FOLLOWS:

In set theory, no finite set can be put into one to one correspondence with any of its subsets. This means that, if you have a set of 100 colored objects, and within those is a subset of 20 blue objects, you cannot match one blue object to each of the 100 colored objects. Common sense, right?

Well, this is not true for infinite sets. They violate the rule that a whole is greater than any of its individual parts, and an infinite set can be defined as any set which can be put in one to one correspondence with one of it's subsets.

In this case, what the manager should do is move every guest to the room number TWICE as large as their current room number. This moves every one of the infinite guests into an even numbered room, leaving the entire infinite set of odd numbered rooms vacant. Thus, he is able to accept the infinite number of new guests, adding infinity to infinity.

This demonstrates how the infinite set of individuals currently lodging can be placed into one to one correspondence with the subset of all even numbered rooms.

I hope you've enjoyed this installment of puzzles for pleasure. As a bonus, see below:

Can the rational number line (an infinite set consisting of all numbers which are not irrational- numbers like 1.001, 3.11111111111111777773, 4, and 5/3 are rational. Pi and phi are not, because they cannot be expressed as the ratio of two whole numbers) be placed in one to one correspondence with the natural numbers(1, 2, 3, 4, 5)?

Remember, one to one correspondence is matching each member of a set one for one with a member of another set...

To answer this question, you may want to research the concept of Cardinality and Aleph Numbers.

Puzzles for Pleasure #1- The Prediction

I'm going to try something new for a while, I'm going to regularly post paradoxes, puzzles, and logical oddities just to entertain and stimulate! I hope you enjoy...

Can a fortune teller see the future in his crystal ball? 

Let's say a fortune teller, Genie, has a teenage daughter, call her Mary, whom he has promised to buy a car for when she graduates school. 

One day, Mary tells her father that he's a fake who can't tell the future, and challenges him to a test:

She writes something on a piece of paper and locks it into a box. She says, "I've written down an event that will either happen or not happen before sundown." She hands him a blank piece of paper, and says "If the event will happen, write "yes," if it will not, write "no." If you are wrong, buy me a car now, if you are right, don't buy me one at all."

Genie says it's a deal.

He writes something on the card, and at sundown Mary unlocks the box. Genie reads what she wrote:

"Before sundown, you will write NO on the card."

"You've tricked me!" declares Genie. "I wrote YES, so I was wrong, but if I'd written NO I'd be wrong too!"

Mary used her new sports car to drive across country and never went back to her crazy Dad's house.

The original version of this story is about a computer that can only respond "true" or "false" (which is exactly how computers work). We ask it to tell us whether it's next response will be "true," or "false." Since it is not possible for the prediction to be correct, the computer spends ten thousand years thinking and spits out "42," or something like that.

We can reduce this paradox to the question "Will the next word you speak be "no?" Please say "yes" or "no.""

This is a disguised version of the well known "liar" paradox. The liar paradox arises when one states "this sentence is false." It is what is known as a semantic, or truth-value, paradox.

The fun thing about semantic paradoxes is that they can all be rephrased as set theory paradoxes. The sentence, "this statement is false," for example, can be rephrased as "this assertion is a member of the set of all false assertions."

Every semantic paradox has a set theoretical analog and vice versa.

I hope you have enjoyed this first installment of "Puzzles for Pleasure."

Thanks for reading!

Atheism 101: The Thomistic Cosmological Argument

It’s a very common argument for god, and it’s a topic atheists should understand.

Have you ever heard a theist say, “Well, everything has to come from something”?

That’s the Thomistic Cosmological Argument. Formally, it can be put as such:

1.    What we observe in this universe is contingent (i.e. dependent, or conditional)

2.    A sequence of causally related contingent things cannot be infinite

3.    The sequence of causally dependent contingent things must be finite

4.    Conclusion: There must be a first cause in the sequence of contingent causes

Technically, there are a few versions of the cosmological argument, but Thomas Aquinas’ version is the most commonly used. In general, a cosmological argument is any argument made from the position of first cause. In lay terms, it asserts that:

1.       Everything must have some cause

2.       The universe exists

3.       There must be a cause for the existence of the universe

4.       THEREFOR GOD(see illustration to right)

Does this mean that you are wrong, that God exists and is going to burn your ass forever? Not really. Where do we begin with our discussion of this fallacious argument?

In logic, there is an informal fallacy known as “special pleading.” I pulled this definition from the first paragraph of the Wikipedia page on the subject, adding the italics for my own emphasis:

“Special pleading (also known as stacking the deck, ignoring the counterevidence, slanting, and one-sided assessment[1]) is a form of spurious argument where a position in a dispute introduces favourable details or excludes unfavourable details by alleging a need to apply additional considerations without proper criticism of these considerations. Essentially, this involves someone attempting to cite something as an exception to a generally accepted rule, principle, etc. without justifying the exception.”

You may recall that a few moments ago you read that, according to the cosmological argument, everything must be caused by something else. The argument is free to make this assumption, but then it proceeds to deduce the existence of god without applying the same rule to his existence. In essence, it argues that everything must come from something else EXCEPT FOR GOD and makes no justification for this caveat. Recall also that the definition of special pleading is citing something as an exception to a rule without justifying that exception. The god that is introduced in this argument, the one which exists in a system in which everything requires a cause, and yet has no cause for himself, is a god which is deduced by logical fallacy.

Theist logic

The easiest way to point this out is to evoke the greatest tool in all of science: Occam’s Razor(WARNING: THAT WAS A VERY OPINIONATED STATEMENT).

When this subject comes up, simply point out that their god does not require a creator, and yet is more complex than the universe. If they are willing to accept that such a great and complex and powerful being can exist of its own accord to create the universe, why not just cut the middle man(that’s where the razor comes in) because he is an entirely unnecessary assumption, and allow the universe to be the one which exists without a creator?

After all, the universe isn’t sentient, all powerful, or prone to irrational behavior. God is all of that and more. Which one is more likely to simply happen all by itself?

A God Atheists Can Believe In

The Omega Point Theory

I’m going to stretch your brain a little bit today. First, I’m going to warn you that this is a little long and difficult to read at times. Apologies in advance, but I believe that in the end the reading will be worth it.

In this day and age, we are all very familiar with pseudoscience. Internet wackos with a half cooked, laughable understanding of quantum mechanics make bold and ridiculous statements regarding the consequences of their supposed knowledge.

Things being as they are, it’s easy to brush off profound, grandiose statements about gods or souls or morality based off of so called “physics.” I wonder though- what if there were a physically contrived theological position that weren’t so easy to brush aside? Well, there is.

Frank Tipler is a professor of mathematical physics who earned his bachelor’s degree in physics at MIT, proceeded to complete his doctorate at the University of Maryland and worked as a research scientist at three major universities before settling down as a permanent staff member at Tulane University. He knows his stuff, and as a part of his research he has introduced something which is now called the “Omega Point Theory.” His theory includes its own axioms, such as the in-retrospect-ill-named “Final Anthropic Principle” (FAP).  It builds somewhat upon the work of the Jesuit philosopher Pierre Teilhard de Chardin, who developed an evolutionary theory called the “Law of Complexity/Consciousness.”

My point is that there is an awful lot of background which goes into the Omega Point Theory, including advanced mathematics and physics and some evolutionary work which was unfortunately performed by a priest. I won’t be able to relay all of that information in its entirety, but I would like to introduce you to this theory and would be interested as to your thoughts regarding it.

Before introducing it, I am compelled to tell you that this is only one interpretation in a long line of attempts to understand the concepts that Tipler addresses. John Bennett, the British mathematician, has an alternative view which he calls the hyparchic future, which rejects the two most common propositions for the universe(that it was created and designed, or that it came into existence by accident and is governed only by physical law) in favor of a third alternative which invokes the existence of the universe based on a physically inherent destiny as well as a patterned behavior governed by some sort of potential energy field. Most of this work is presented in bookstores and websites as spiritual material, because that is what it seems to address. What sparked my interest was that none of this work was originally created for that purpose, but instead for introduction to the peer review community as physical concepts. (It isn’t always well received.) The other thing which I find interesting is that I’ve often wondered(my own thinking here) whether the concepts they address might be the ultimate consequences of a successful transhumanist campaign…you will see what I mean.

On to business:

Tipler borrows the term “Omega Point” from our Jesuit priest and uses it to describe what he deduces to be the ultimate fate of the universe.  We can summarize his theory as the following points, these are not the arguments themselves but their conclusions, presented with a little commentary to explain:

·         The universe has finite size, and the topology of a 4 dimensional sphere

This is based on his physical calculations, which are detailed in his “can’t judge a book by its cover” titled book: The Physics of Immortality. It’s not an unreasonable conclusion, as the universe is four dimensional and a 3 sphere(as it is called) is the simplest shape to describe a four dimensional space.

·         There are no event horizons. The future c-boundary will therefore be a point rather than a singularity

If you follow recent updates in physics, it turns out that the hypothesis about event horizons is probably true. The c-boundary is a difficult concept to explain, but basically it is similar to an asymptote- ya know, that thing you get on either side of the point where you approach division by zero in a rational equation. The C-Boundary is the result of an intelligence explosion, a situation where the universe tends toward more and more complexity and intelligence. In a universe with event horizons, that tendency toward infinite complexity would result in an information loss. If event horizons don’t exist, the c-boundary cannot be a singularity but rather a defined point. I’m struggling to explain it…perhaps you can ponder the consequences of any variable tending towards infinity. It’s not essential that this be completely understood to address the point further down

·         Sentient life must eventually engulf the universe

This is a conclusion reached based on the complexity theory

·         The amount of information processed between now and the Omega Point is infinite

Skipping for now

·         The amount of information stored in the universe asymptotically tends to infinity as the Omega Point is approached

Also skipping for now

Essentially, Tipler’s predictions require two things: That the universe is closed and that it is going to end. If those two things are true, he arrives at the conclusion that the increasing complexity of life will continue to iterate upon itself as the universe approaches its end. As the end grows nearer and nearer, the processing power available to the combined intelligent beings living in the universe tends closer and closer to infinity, until a point is reached such that processing power is growing at such a rate that an infinite amount of processing can be done before the end of the universe.

Transhumanists may be familiar with the concept of escape velocity with regards to lifespans, but Tipler introduced the concept a couple decades ago with regards to intelligence. To illustrate this point intuitively, imagine if you could think twice as fast as you can today. If, to me, you live for a hundred years, you will experience the equivalent of two hundred years of life in that same amount of time. If you thought ten times as fast you would experience a thousand years of thoughts during the hundred years that you live. The Omega Point is the point when intelligent beings think so fast that they experience an infinite amount of life between whatever point they are currently at on a timeline and the eventual death of the universe.

At this point I must step back from the exact work of Tipler, whose proofs and theorems are beyond my ability to understand, and discuss this from my own perspective. Could you imagine an existence where thought is so fast as to create an infinite time line of thought and experience? A society which has achieved this level of existence would essentially have created exactly what our religious counterparts believe in: god. They, collectively, would have so much processing power available(due simply to the fact that their bound on computation time is infinite) that they could simulate every possible timeline. They could resurrect dead individuals by recreating the exact timeline that caused them to exist. They would live forever, have every thought, know everything, and they would be what our counterparts call god, and would live in the state which our counterparts call heaven.

To be clear, I’m not saying that I believe this. Tipler’s theory is based upon some laws I’m not sure exist. I consider it to be more of “The Omega Point Possibility.” It is certainly mathematically possible, even if his notion of an infinite lifespan through infinite processing doesn’t seem intuitively possible. The Omega Point is more likely at this point than any theistic god, because we can demonstrate mathematically that the Omega Point can exist. The beings living there would likely be able to master all the secrets of the universe and would be the closest thing imagineable to a god and an afterlife.

I actually did write a science fiction story about this concept a while back, told in unique timeline fashion, and it is essentially about the creation of the Omega Point. If you are interested, check it out the story, Destiny, here.

Now, I’d like to hear your thoughts. If you’ve made it this far, take a moment to tell me how you feel about the Omega Point, about life eventually evolving into this being that the theists call god, and about it’s supposed inevitability. I don’t introduce this to persuade you or anything along those lines, simply to discuss something other than the usual “OMG CHRISTIANS ARE SOO STUPID” that you here everywhere else.

May the dark lord watch over you ;)

Atheism 101: The Anthropic Principle

Maybe you’ve heard of it, maybe you haven’t, but I guarantee you that you’ve heard theists arguing for god because they don’t understand it.

You could just read the Wikipedia article, here, and I encourage you to do so, but that page doesn’t address the Anthropic Principle from the perspective of atheism.

So, what is it? In short:

It’s the philosophical position that the universe couldn’t be any other way than the way it is right now.

It sounds like a presumptuous statement, until you understand why we make it.

From a theist’s point of view, the universe has been fine tuned by some god in order to support the existence of life. It is true that a universe which deviated in the slightest from this one’s fundamental constants would not support life as we know it. Were the strengths of the fundamental forces any different, stars would not form, matter may not even exist, and the universe would be nothing more than a vast expanse of space and energy.

A theist, confronted with the precise nature of this universe’s tendency toward creating life concludes that the “fine tuning” of this universe is evidence of god’s hand.

The more reasonable way of looking at this, however, is to simply observe that, were it any different, we would not be here to notice. That’s it. It’s that simple.

Those who reject the anthropic principle assume that a universe has to be fine tuned to support life because they suspect that precision is improbable. There is no reason to assume that.

We know nothing about universe creation. We don’t know how to do it, we don’t know how these constants are decided and we don’t know how many universes there are. For all we know there are a trillion trillion universes in existence which don’t meet the requirements to support the development of life, and just this one which does.

What are the odds, in that case, that we would find ourselves in the needle in the haystack, that we would be so fortunate as to come to existence in the one, improbable, chance universe that can support life amongst the infinitude which can’t? Why, 100 percent of course. After all, there is no other way it could be. In order for life to exist, it must exist in a universe which is habitable and caters to its own existence. To that particular life form, the universe may appear “fine tuned.”

Perhaps those individuals might take a moment to consider that maybe they are fine tuned to the universe.

After you allow the self evident nature of the anthropic principle to sink in, I suggest that you investigate it beyond the scope of this article. Some of the philosophical implications of it are fascinating, including the idea that life is necessary for a universe to exist.

Before moving on to the practical applications of the anthropic principle in discourse with theists, I’ll close my section on what it is with a quote from Michael Frayn,

“ It's this simple paradox. The Universe is very old and very large. Humankind, by comparison, is only a tiny disturbance in one small corner of it - and a very recent one. Yet the universe is only very large and very old because we are here to say it is... And yet, of course, we all know perfectly well that it is what it is whether we are here or not.”

When will this come up?

It’s a common refrain from theists to suggest that the fine tuned universe is evidence of god. Someone who hasn’t studied the anthropic principle may find themselves disarmed when confronted by detailed explanation from a theist as to how precisely the fundamental constants of the universe are “tuned” to enable human life. Understanding this position allows retort.

How can I use it in a discussion about god?

Honestly, it’s an advanced concept. I don’t know why I chose it as my first Atheism 101 post(I do- I brought it up because everyone should know about it), but it’s definitely not the best go-to for general discussion. It’s difficult to explain, and I’m having a hard time explaining it here, so I recommend being aware of it and bringing it up when you are talking with the sort of theist who wants to discuss fundamental constants as an aspect of their defense of god.

Are there downsides to this argument?

Unfortunately, yes. For starters, it is not testable. If a theist asks you to “prove it,” you won’t be able to. It is a logical inference that we make when considering the nature of life and the prerequisites for its existence. It is also a dangerous cop out for scientists, as saying “it can’t be any other way” might discourage one from finding physical explanations for observed phenomena.

Just remember: The burden of proof is not on you. Don’t lean on the anthropic principle as the crux of your argument, just be aware of it as a retort and keep the burden where it needs to be- on the believer. While you may not be able to prove the anthropic principle, it is a reasonable observation and a suitable retort to theists who took Physics 101 and are amazed at how purposefully god designed our universe.

How should I word it if I bring it up?

However suits you, but you could always quote me:

“Were it any different, we would not be here to notice.”