My work on the philosophy of mathematics has sparked a recognition of a new epistemological-ethical principle: establishing thresholds is essential to success in thought and action.
A “threshold” is a lower bound of significance–a degree below which something has too little cognitive or existential impact to be entertained.
For instance, your chance of buying a winning lottery ticket or of getting hit by falling space debris is sub-threshold, so you should take no action based on that and give it no thought (beyond the judgment that these events are sub-threshold).
This supplements the Objectivist understanding of the arbitrary. The arbitrary has no evidence and is asserted on the premise of “evidence—who needs it?!” Entertaining the arbitrary is treating imagination as if it were cognition.
But the sub-threshold is different. “You have a 1 in 12 million chance of winning this lottery” is put forward on the basis of mathematics, not emotion. One could argue that by implication acting on this mathematics is emotionalist, but that presupposes the point about thresholds that I’m going to make: to grant significance to things with too little evidence or too little value is to engage in context dropping.
What context is dropped? The context of all the other things that have, in the mathematical sense, a 1 in 12 million chance of happening. There’s a a much better chance (1 in 2.6 million) chance of being dealt a royal straight flush in poker, so before one buys that lottery ticket one would have to consider betting the limit, sight unseen, on the next poker hand. There’s no doubt at least a 1 in 12 million chance that while you are in the store to buy the lottery ticket, an armed robber will enter and you will get shot. There’s a 1 in 12 million chance that you will receive a fortune in the near future in some other way. There’s a 1 in 12 million chance you will be struck by lightning, that building you are in will collapse, that a talent scout will decide you are have just the right look for a certain role in the next blockbuster movie. Etc.
There are far too many things to think about and do if all sub-threshold chances are to be taken as significant. And in practice what happens is that whatever fantastically unlikely scenario is talked up becomes the one that is treated as if it were the only one.
Many people are impressed, for instance, with the thousands of cases of, alleged or real, bad reactions to the Covid vaccines. But they have forget that one thousand bad outcomes out of one billion vaccinations is one in a million. And they drop the context of the greater than one in a million chance of experiencing severe medical problems from being unvaccinated and contracting Covid.
Thresholds are contextual. A very slight chance of getting a cold is, for most people, negligible. The same chance of dying is quite significant. At the end of We the Living, we see Kira taking a high degree of risk trying to escape Soviet Russia, because the alternative was unthinkable.
It is important, therefore, to determine whether a given degree of evidence (and when to count statistics as evidence) or a given degree of value or disvalue is entitled to one’s attention.
In mathematics, I promote the concept of “nil,” which is a magnitude that isn’t zero but is too small to detect or too small to matter—to matter in application.
For instance, in measuring rugs for your home, the threshold length may be a half foot, it may be an inch, it could even be an eighth of an inch. But it cannot be a millionth of an inch. But in measuring the size of molecules that can cross a given cell membrane, a millionth of an inch may make all the difference.
There’s the flip side of “too small too matter”: so big that increases don’t matter. This is the rational meaning of “infinity” in one sense of that term. Something is infinitely big if additions to it make no difference. (In effect, for any n, ∞+n — ∞ = nil.)
It is good to apply thresholds to establish what’s “enough.” Perfectionism is precisely the error of dropping the context and thinking any improvement, no matter how small, is significant. For the perfectionist, infinity is never reached; his work is never good enough, because there’s always more polishing of it that can be done. (As an advocate of contextual perfection, I must add that the charge of “perfectionism” is often hurled at those who simply demand more of themselves and others than the accuser wishes they did.)
Perfectionism is the same issue as thinking one “just might” win the lottery: it’s failing to consider what the operative threshold of significance implies as to other things. Yes, by spending more time editing this post, I could add some to its value; but what are the other things I could be using that time for? Yes, by working an extra hour longer a billionaire probably can make an extra $1,000 dollars. But what else could he be expending that hour on? Are the selfish rewards of increasing his wealth from $1,000,000,000 to $1,000,001,000 greater than the selfish rewards of dinner with friends? Improving his tennis game? Thinking about a philosophic topic, such as the need to set thresholds?
The point is not what answer one gives to those judgments; the point is the need to face these issues consciously and deal with them rationally.
Note that overconcern with the threshold being “exactly” right is itself a violation of the threshold principle. In the region of the threshold, small differences are way sub-threshold. Suppose our billionaire is trying to set a monetary threshold for what amount of money begins to matter to him. Suppose his estimate is: $1,000. Anything below that is, to him, what less than a penny is to us. But then he wonders: “Maybe my threshold should be $1200 per hour. But $200 has to be nil to him. The difference for him between a threshold of $1,000 and $1200 isn’t, for him, worth worrying about. It would be like, for us, worrying about the difference between a penny and 1.2 pennies.
So, thresholds are normally approximate, because differences close to the threshold make no difference.
Between “nothing’s there” and “something’s there, what do I do about it?” there is a third condition: “something is there, but it’s too little to devote any of my scarcest resource—time—to thinking about or dealing with.”