The Freedom of Not Knowing

Religions claim to have the answers. Go to a local Rabbi or Priest and ask them what is the meaning of life. They will have a ready answer. How was the world created? They'll have an answer. Why is X, Y? They'll have an answer. They are actually, in a way, required to have these answers. This is especially true for the more import metaphysical questions such as the meaning of life.  A religion that couldn't provide an answer for that, would be a sorry religion indeed. If you asked a Rabbi what the meaning of life was, and he said, "I don't know" you probably wouldn't feel very impressed with his particular wealth of knowledge.

Because religions are required to weigh in on many of these topics they often offer up solutions that have absolutely no backing. As our frontiers of knowledge expand, lots of these answers wind up being shown false. An example of one is "how was the world created." The Bible offers up an answer to this, which fits in very nicely to the numerous creation myths found across the globe. Pretty much every culture has one, and they're all similar in that absolutely none of them have any basis in reality. Though, I must admit that some people go through valiant efforts to reinterpret the Biblical creation myth to align with what we now know. Still others claim that the story is obviously allegorical and no one ever took it literally anyway, completely ignoring the numerous fundamentalists who adhere to a very strict literal interpretation of those words even today.

When I moved from a religious framework to a non-religious framework, one of the freedoms I gained was the freedom of saying "I don't know." Additionally, I could give probable answers if I wasn't certain, but thought something was true. This also allowed me to update my knowledge with new information as it came in. Being proven wrong wasn't an assault on the entire edifice my life was built around, but rather just another opportunity to update and improve my current beliefs.

The rest of this post (and the next) will be a fairly long-winded description of the various kinds of not-knowing.  It turns out there are many different ways to not know something!

I Don't Know

If someone asked you, "what is the population of Sri Lanka," chances are you'd say "I don't know." If you had internet access, you could get the answer for them without too much trouble. This is the simplest form of not knowing. There is an answer but you just may not have access to it right now.

Consider the question, "Why is the sky blue?" If you have some background in physics you might be able to give a basic explanation of refracting light in the atmosphere. If you have a lot of physics background, and the person you're speaking with is also similarly inclined, you could derive the Rayleigh scattering formula and show that the scattering goes like the frequency of light to the fourth power, so that blue light (higher frequency) is scattered far more than red light. "Why is the sky blue" is a question which has an answer, but it takes some effort to understand and explain. It is this kind of question that tripped up religious advocates in the past. The Bible has its own answer to why the sky is blue, which has to do with the Ancient Near East cosmological structure. The bible explains that the earth is surrounded by chaotic water both above and below. The water is prevented from coming up through the ground by the earth, and is held back in the sky by a solid barrier. This, is of course very very wrong. They would have been far better off with "I don't know." Nevertheless, religions all over the world are filled with explanations like these.

What about the question, "What is dark energy?" If you're an astrophysicist, you might have some theories. But in the end the best answer you are likely to offer is "I don't know" or better yet, "we don't know."  There is probably an answer there, but we don't have sufficient data to resolve it. There are people who think this is a major weakness of scientific knowledge. Gaps exist in our understanding. Some religious advocates like to shove their preferred deity into these gaps. But this is unsatisfying, the gaps are constantly getting smaller and with them so does the deity.

Finally let's consider the question "Why are we here?" This is an interesting question to look at because science offers no answer to it. Every religion offers at least one. From my point of view the question is faulty. It assumes that there is some reason, some teleological purpose for humanity, but there very well may not be. The best answer I could give is, "I don't know that this question makes any sense." For some people this is very unsatisfying. We'll definitely touch on topics like this in the future.

Now let's look at some probabilistic uncertainties that arise all the time in our life.

Fundamental Errors

Let's say you want to weigh something.  You put it on the scale in your house and it reads 1.5 kg.  But you're not satisfied, so you take it to many other scales and weigh several more times. You get several readings of 1.4 kg, 1.5 kg, 1.6 kg and even one 1.3 kg. What is the weight of your object? If you assume that all the errors in the scales are random instead of all the scales being biased in the same manner, you can state the most probably weight, and a region of confidence. This depends a bit on how many weighings you made, but let's say this is calculated at 1.45 kg with a 95% probability of being between 1.35 and 1.55 kg. Now let's ask, what is the probability that the real weight is over 2 kg. None of your measurements came close to that value, but still you can ask the question. For this case the answer is somewhat less than 1 in 1 quintillion.  But it's not zero. (The above calculation assumes a normal distribution.)

Here's another game. You are having a coin flipping contest with your friend. Heads you win, tails you lose. You flip and it comes up tails. The second time: tails. Third?  tails again. Fourth, fifth, sixth, seventh are all tails. At what point do you argue that the coin is faulty?

The probability of a fair coin coming up tails on one flip is 1 in 2, 50%. Twice is 25%, you can continue calculating in this manner until you get to the desired amount of flips. For seven flips, the probability of a fair coin coming up tails seven times in a row is 1/2^7 or  0.7%. Is this enough evidence to accuse your friend of cheating? What if you did 10 flips and it was 1 heads and 9 tails?  Well you can calculate that also. The probability of getting 9 or 10 tails in 10 flips is 11  / 2^10 or  about 1%.

The main point of this section is that every measurement has some error associated with it.  These errors can be purely mathematical or due to real world limits on precision, such as the inaccuracies in the scales. We scientists spend a great deal of time quantifying our errors. However, for everyone, there is a threshold where they don't care about the probability of a specific result because it is so very low. Given our scale weightings, there's no reason to extend your error bars to encompass the extremely low probability that your item weighs more than 2 kg. 

But there's also a counter-side. Humans are really bad at very big or very small numbers. It's very hard for us to envision the difference between the 0.7% and the 1 in 1 quintillion above. But they are very very different. Let's say instead of flipping the coin once 7 times, you flip it 100 sets of 7. What is the probability that at least one of those sets of 10 is all tails?  To calculate this, first calculate the probability of not getting seven tails, this is just 1-0.7% or 99.3%. Then calculate the probability of getting that result 100 times in a row. This is 0.993^100 or just below 50%.  If you try 100 sets of 7 flips you have a better than 50% probability of getting at least one set of all seven tails.  If you did 1000 sets of 7, you are all but guaranteed to get at least one set of all 7 tails (over 99.9%).

This is informally called the law of large numbers, which isn't really a mathematical law. But the basic idea is that if your sample size is large enough compared to the probability of the event, then it's very likely that the event will occur at least once. To put it in other terms, let's say you assign everyone in the world a number and then pick one person at random to win a prize. The probability of you winning is less than 1 in 7 billion. But the probability of any person winning is exactly 100%.

Originally I planned to discuss a lot more things in this post, but I think it will have to wait for the next one, otherwise it would be too long. The next post will talk about some of the more exotic ways of not knowing. We'll talk about quantum uncertainty, chaos theory, which are real world examples of not being able to know something. We'll also talk about some surprising pure mathematical results regarding the inability to know everything.

Please let me know in the comments if these topics are interesting and if there are specific areas you'd be most intrigued about.