Londoner wrote:
Just looking at the word, I would ask what the opposite of 'random' would be? I would think it would involve 'selection' or 'purpose'.
Suppose I was asked to 'give a selection of numbers' and I picked: 2,4,6,8. I would say that as a group they are not random in that all those symbols were examples of numbers; I have purposely not included any letters or punctuation marks.
Technically the opposite of random would be "compressible." Let me explain.
Before explaining I should put this in context. The mathematical ideas I'm talking about are not necessarily related to the various philosophical or linguistic meanings of the word random. We might ask, are the things that happen to us determined? Or are they random? That's a perfectly good use of the word; even though it has nothing at all to do with the mathematical meaning.
Even within mathematics there are various different concepts of randomness. Statistical randomness, etc. Everything I'm saying is in the context of one particular meaning of randomness, namely incompressibility.
Ok, so instead of the set {2, 4, 6, 8}, which is too small to be interesting to us, suppose we had the set {2, 4, 6, 8, 10, 12 ..., 100,000,000}. In other words,
All the even numbers between 2 and 100 million, inclusive.
Now I have just described a set of 50 million numbers with the 57 characters in italics above. That's very efficient compression. I could send those 57 characters to you over the Internet, and you would receive those 57 characters and be able to perfectly regenerate a copy of the original set of 50,000,000 even numbers.
So we are not just talking about abstractions. This is about data compression on the Internet. Figuring out how to send 57 characters instead of 50,000,000 numbers.
I hope this is clear. Sets that can described using far fewer characters than the size of the set, are compressible. Sets that are not compressible are random. And there are degrees in between. The fewer characters it takes to describe a set, the less random it is.
Londoner wrote:
Now it is also true that we might look at those numbers and see a pattern of relationships between them, but since that pattern has no meaning in the context of 'give a selection of numbers' then I don't think it makes them non-random.
If thre's any pattern at all, and if you can describe the pattern using fewer characters than there are numbers in the set, then you can achieve communications efficiency by transmitting the description of the pattern, rather than the set, across the Internet.
And what is another word for a description of a pattern? How about algorithm? Sets that are generated by short algorithms are not random. I don't need to send you all the infinitely many digits of pi. I can just send you one of the many known algorithms and you can generated the digits for yourself. The digits are highly compressible. They are not random.
[However do note that the digits of pi are highly
statistically random. That's a completely different mathematical meaning of the word random].
Londoner wrote:
To put it another way, if somebody saw significance in the fact that each number was 2 more than the last, they would be misunderstanding the meaning of the sample.
If I understand you correctly than I disagree. "Start with 2. Keep adding 2. Stop at 100,000,000" is only 48 characters long. You compressed my data better than I did. You understand this perfectly.
The meaning isn't the algorithm. The meaning is the set they generate. We have two different algorithms that uniquely describe the same set of numbers. How the set is generated is not where the meaning lies.
It's interesting that you see the meaning in the
method. "Start with 2 and keep adding 2" is not the same as "Even numbers starting with 2." You see these as having different meanings.
And perhaps you are right in some circumstances. For example, I disagree that the ends justify the means. In other words even if two methods achieve the same goal, one method might be moral and the other immoral. So in the real world, the meaning is often in the method as much as in the result.