Counting day. How many ways can you skin a cat if the number of sharp versus dull knives is this and such. Simple stuff, easy to get the hang off. There were no cats present—I saw a chipmunk—thus no practicum. Maybe next year.
Speaking of counting, the discrete solution to Zeno’s so-called paradox came up. That conundrum goes: if you start a journey from A to B by going half way, then going half way from there, and so on, you’ll never get to B because no matter how long a distance is left, you’ll only get half way to the destination on the next step.
This has an analytic solution, which requires calculus to understand, and which is fine. But it requires the premise that the universe can be infinitely divided in space. And there’s no indication that that’s true.
But if we abandon that premise and embrace a quantum, which is to say discrete, universe, then Zeno is easily answered. Between A and B are a discrete or quantum number of steps. You can either go halfway to B if n is odd, or something other than halfway if n is even. For instance, if you are at A and there is only one step to B (n = 2) it is not possible to go half way. You go to B or you go nowhere. Movement cannot be infinitely graduated. You must move at least one step at a time.
This is a reminder that no measurement can be infinitely graduated, or have infinite precision. Everything we can do for real that can be quantified is necessarily discrete. But, of course, we can approximate the discrete with the continuous. Fine. As long as we remember it is always an approximation, lest the deadly sin of reification appear. And, even more of course, is that not everything is quantifiable, nor should we seek for quantification where it is not necessary to do so.
The number 10 is just as far from it as 10100, a googol, is. And a googol is just as far from infinity as a googolplex is. A googolplex is 10googol. And if you were to take that creature to the power of 10, it too would be just as far from infinity.
Infinity is big. BIG. It is really far away. It is infinitely far away. Unimaginably far away. So far away you’ll never get there. Even if you really want to.
Normal distributions are ubiquitous and always an approximation, usually a crude or awful one. The probability of any observable given a normal is 0, forever 0. Normals say nothing can happen, because why? Because normals, and other continuous distributions, attempt to give probability to infinite numbers of events. And we’ve just seen this is an awful lot of events.
Imagine a bag with an infinite number of white marbles and one black one. What’s the probability (given these premises) you pull out the black? Zero.
But math says normals are pretty, and many statisticians think statistics is a branch of math, instead of philosophy, and so off we go down the bell-shaped road, memorizing equations and not meanings.
Enter the most Deadly Sin Of Reification!
Raise your hands. Who’s heard a sentence like, “Temperature is normally distributed”? Everybody! That’s the sin right there, in all its ugliness. Temperature is not normally distributed. Nothing is, and nothing can be.
We can say, “Our uncertainty in temperature is quantified by a normal with this and such parameters.” But there is nothing causing temperatures to line up in the shape of a normal. It is impossible that temperature can, since it is finite.
Reification happens when the temperature is forgotten and the normal representing it becomes as real or realer than the temperature. Of course, it isn’t just normals and it isn’t just temperature. It happens every time the model—and the normal is a model—becomes realer than reality. This is magic: and not the only time we meet magical thinking in probability.
Example? Time series are a perpetual source of magical thinking. How many times have we seen plots of the actual real this-is-it hey-it’s-me data overlaid with “running averages” or other smoothers (this includes straight-line regressions)? Directly this is done, the original data is all but forgotten and the smoother becomes the real thing.
So powerful is the spell of this smoothing miracle that you cannot convince an observer that the non-data is the non-data. The non-data, so pretty so pure and smooth and free of defect, must be real!
I haven’t yet found anything to counter this evil charm. There isn’t enough holy water to go around and every time I try to stuff garlic into somebody’s computer, the d***ed thing freezes up. Suggestions welcomed.