During a discussion on a internet board, I espoused a pretty weird theory. I like it so much I’m going to try to explain it here.
The gist of the theory is that there is no such thing as integers or rational numbers, at least not in reality. Sounds crazy, I know, and it’s probably just a bunch of bunk, but I like the “feel” of the argument so I’d like to capture it in words.
When we started inventing the thing we call math, it was all about counting things. One sheep, two sheep, three sheep, four. We made words to associate with these concepts and started constructing a set of words and symbols to work with adding and subtracting sheep.
We soon reached a point where we had to deal with the idea of taking away more sheep than we had — meaning we had to have negative numbers. I’m sure it didn’t make sense to a lot of folks at the time (how can you have -4 sheep?) but it worked well enough so that the math became a lot more useful.
When we added multiplication and division into the mix, a couple of interesting things happened. First of all, we ended up trying to divide 5 sheep among 3 people. How could you split a sheep? Never fear, we invented rational numbers, which are any numbers that can be repesented as p/q (for those of you sleeping in math class), where p and q are integers. Once again, it didn’t make much sense to sheep herders, but the math became even more useful.
We also had the interesting question of “what is something divided by zero?”. As we all know from elementary school, you cannot divide a number by zero, because how many times do you need to multiply zero together to get any other number? You can’t, so it didn’t make sense.
Here we did something very interesting. Instead of re-examining where we were with our math system, we just ignored the problem and said “don’t worry about it, it just never can happen”. We didn’t invent a new type of number, we didn’t make up new symbols. We just kind of grimmaced and continued our counting, adding, multiplying, and dividing.
Some of you know more of the story. Later on we wondered what the square root of -1 was. For this, we invented (discovered?) imaginary numbers. Then we got quaternions, vectors, matrices — we’ve made all sorts of extensions to our beloved system for enumerating sheep.
At each step, we learned more about reality. For instance, when we had to represent those 5 sheep split among 3 people, that was a real-world situation. We would make up words and symbols, see if it mapped to reality, and then make up new words and symbols if it didn’t work exactly right.
Which leads me to physics today. As I understand it (and I am a total idiot but I do stay at Holiday Inn from time to time) now that we’re getting to very small pieces of things we are finding that the “pieces” aren’t really there. Take an electron, for instance. There’s no such thing as “an electron”, rather there is a potentiality field where an electron _might_ be at any one time. The universe, in effect, consists of probability waves that intersect. Nothing is really somewhere — everything is everywhere. If you know it’s exact momentum, you have no idea where it is, and if you know it’s exact position, you have no idea how fast it is going. We really never know either of those totally, everything is “fuzzy”.
This is the craziness of quantum theory. If you don’t get it, don’t worry — it looks nothing at all like counting sheep, that’s for sure. But it describes what we know about reality currently. For any given amount of space, there is some probability that some particle may exist in it.
This means that probability and statistics play a very important role in how we define reality today.
Now you can take this two ways. You can think in terms of particles and the probability that they are somewhere. That’s the way most people think about it. Or — you can think that the particles really don’t exist at all but only the probability waves.
I like to think of it that way.
I think there is no such thing as zero. Sure, there might be zero sheep, but the things that make up the universe are not sheep. A sheep is either there or not, but a quark exists in some other reality where it is kind of everywhere but most likely in a certain area. To say there is no quark in a room has no meaning — there is always some probability, no matter how small, that a quark is there. There is no zero.
If we had to start over again with math, and we were very small (or had great eyesight), our system would look much different than it does today! First, there would be no zero. Once we accepted that reality, it logically follows that there is no 1, 2, or 3 (because how do you really know there isn’t one more or less?). You can’t really count anything. Counting, the verb, is meaningless.
So how would we express states of reality? How does one say, compare a room full of electrons with a room that looks pretty empty? My response is that from our vantage point these concepts are expressed as ratios of truly irrational numbers. Although perhaps the term “ratio” is a little misleading. We might need to invent new operators to describe how these true irrationals relate to one another.
There is probably an underlying computer system for the universe — a “calculable substrate” on which the rest of reality runs. This is the reason so many small things are currently expressed as probabilities — we’re really just measuring a whole bunch of something else. That “something else” may not be observable from our perspective. In which case, it follows that the concept of zero, integers, and rational numbers do not help us understand reality at that level. Instead we may need a set of symbols that describe computer relations, operations, and the interaction of computing fields.
It was a fun mental exercise.If you've read this far and you're interested in Agile, you should take my No-frills Agile Tune-up Email Course, and follow me on Twitter.