I’m one of those aging folks who still remember the original run of Star Trek (no colon, no The Original Series or any other kind of elaboration — just Star Trek). It was a groundbreaking show, and whether you like it or not (there are plenty of reasons to do both), it held out a positive vision for the future, and sketched a societal ethos that was not entirely acquisitive, and not even as secular and materialistic as later outings in the Star Trek franchise. The officers of the Enterprise were not latter-day conquistadors. They were genuine explorers, with a Prime Directive to help them avoid destroying too many other nascent cultures. (Yes, I know: they violated it very frequently, but that was part of the point of the story. Sometimes there was even a good reason for doing so.)
It also offered the nerds among us a point of contact. Sure, Captain Kirk was kind of a cowboy hero, galloping into situations with fists swinging and phasers blazing, and, more often than not, reducing complex situations to polar binaries and then referring them either to fisticuffs or an outpouring of excruciatingly impassioned rhetoric. Dr. McCoy, on the other hand, was the splenetic physician, constantly kvetching about everything he couldn’t fix, and blaming people who were trying to work the problem for not being sensitive enough to be as ineffectual as he was. But Mr. Spock (usually the object of McCoy’s invective) was different. He was consummately cool, and he relied upon what he called Logic (I’m sure it had a capital “L” in his lexicon) for all his decision-making. He was the science officer on the Enterprise, and also the first officer in the command structure. Most of the more technically savvy kids aspired to be like him.
It was an article of faith that whatever conclusions Spock reached were, because he was relying on Logic, logical. They were the right answer, too, unless this week’s episode was explicitly making a concession to the value of feelings over logic (which happened occasionally, but not often enough to be really off-putting), and they could be validated by science and reason. You can’t argue with facts. People who try are doomed to failure, and their attempt is at best a distraction, and often worse.
Up to that point, I am more or less on board, though I was always kind of on the periphery of the nerd cluster, myself. I suspected then (as I still do) that there are things that logic (with an upper-case or a lower-case L) or mathematics cannot really address. Certainly not everything is even quantifiable. But it was the concept of significant digits that ultimately demolished, for me, Mr. Spock’s credibility as a science officer. When faced with command decisions, he usually did reasonably well, but when pontificating on mathematics, he really did rather badly. (Arguably he was exactly as bad at it as some of the writers of the series. Small wonder: see the Sherlock Holmes Law, which I’ve discussed here previously.)
The concept of significant digits (or figures) is really a simple one, though its exact specifications involve some fussy details. Basically it means that you can’t make your information more accurate merely by performing arithmetic on it. (It’s more formally explained here on Wikipedia.) By combining a number of things that you know only approximately and doing some calculations on them, you’re not going to get a more accurate answer: you’re going to get a less accurate one. The uncertainty of each of those terms or factors will increase the uncertainty of the whole.
So how does Spock, for all his putative scientific and logical prowess, lose track of this notion, essential to any kind of genuine scientific thinking? In the first-season episode “Errand of Mercy”, he has a memorable exchange with Kirk:
Kirk: What would you say the odds are on our getting out of here?
Spock: Difficult to be precise, Captain. I should say approximately 7,824.7 to 1.
Kirk: Difficult to be precise? 7,824 to 1?
Spock: 7,824.7 to 1.
Kirk: That’s pretty close approximation.
Spock: I endeavor to be accurate.
Kirk: You do quite well.
No, he doesn’t do quite well. He does miserably: he has assumed in his runaway calculations that the input values on which he bases this fantastically precise number are known to levels of precision that could not possibly be ascertained in the real world, especially in the middle of a military operation — even a skirmish in which all the participants and tactical elements are known in detail (as they are not here). The concept of the “fog of war” has something to say about how even apparent certainties can quickly degrade, in the midst of battle, into fatal ignorance. Most of the statistical odds for this kind of thing couldn’t be discovered by any rational means whatever.
Precision and accuracy are not at all the same thing. Yes, you can calculate arbitrarily precise answers based on any data, however precise or imprecise the data may be. Beyond the range of its significant digits, however, this manufactured precision is worse than meaningless: it conveys fuzzy knowledge as if it were better understood than it really is. It certainly adds nothing to the accuracy of the result, and only a terrible scientist would assume that it did. Spock’s answer is more precise, therefore, than “about 8000 to one”, but it’s less accurate, because it suggests that the value is known to a much higher degree of precision than it possibly could be. Even “about 8000 to one” is probably not justifiable, given what the characters actually know. (It’s also kind of stupid, in the middle of a firefight, to give your commanding officer gratuitously complex answers to simple questions: “Exceedingly poor,” would be more accurate and more useful.
This has not entirely escaped the fan community, of course: “How many Vulcans does it take to change a lightbulb?” is answered with, “1.000000”. This is funny, because it is, for all its pointless precision, no more accurate than “one”, and in no situations would fractional persons form a meaningful category when it comes to changing light bulbs. (Fractional persons might be valid measurements in other contexts — for example, in a cannibalistic society. Don’t think about it too hard.)
Elsewhere in the series, too, logic is invoked as a kind of deus ex machina — something to which the writer of the episode could appeal to justify any decision Mr. Spock might come up with, irrespective of whether it was reasonable or not. Seldom (I’m inclined to say never, but I’m not going to bother to watch the whole series over again just to verify the fact) are we shown the operation of even one actual logical operation.
The structures of deductive reasoning (logic’s home turf) seldom have a great deal to do with science, in any case. Mathematical procedures are typically deductive. Some philosophical disciplines, including traditional logic, are too. Physical science, however, is almost entirely inductive. In induction, one generalizes tentatively from an accumulation of data; such collections of data are seldom either definitive or complete. Refining hypotheses as new information comes to light is integral to the scientific process as it’s generally understood. The concept of significant digits is only one of those things that helps optimize our induction.
Odds are a measure of ignorance, not knowledge. They do not submit to purely deductive analysis. For determinate events, there are no odds. Something either happens or it doesn’t, Mr. Spock notwithstanding. However impossibly remote it might have seemed yesterday, the meteorite that actually landed in your back yard containing a message from the Great Pumpkin written in Old Church Slavonic now has a probability of 100% if it actually happened. If it didn’t, its probability is zero. There are no valid degrees between the two.
Am I bashing Star Trek at this point? Well, maybe a little. I think they had an opportunity to teach an important concept, and they blew it. It would have been really refreshing (and arguably much more realistic) to have Spock occasionally say, “Captain, why are you asking me this? You know as well as I do that we can’t really know that, because we have almost no data,” or “Well, I can compute an answer of 28.63725, but it has a margin of error in the thousands, so it’s not worth relying upon.” Obviously quiet data-gathering is not the stuff of edge-of-the-seat television. I get that. But it’s what the situation really would require. (Spock, to his credit, often says, “It’s like nothing we’ve ever seen before,” but that’s usually just prior to his reaching another unsubstantiated conclusion about it.)
I do think, however, that the Star Trek promotion of science as an oracular fount of uncontested truth — a myth that few real scientists believe, but a whole lot of others (including certain scientistic pundits one could name) do believe — is actively pernicious. It oversells and undercuts the legitimate prerogatives of science, and in the long run undermines our confidence in what it actually can do well. There are many things in this world that we don’t know. Some of the things we do know are even pretty improbable. Some very plausible constructs, on the other hand, are in fact false. I’m all in favor of doing our best to find out, and of relying on logical inference where it’s valid, but it’s not life’s deus ex machina. At best, it’s a machina ex Deo: the exercise of one — but only one — of our God-given capacities. Like most of them, it should be used responsibly, and in concert with the rest.
One thing obscured in Spock’s appeal to “logic” is the distinction between deductive and inductive reasoning. The former is what we usually think of as “logic.” Indeed, deduction plays a role in science but only in tandem with induction. Induction is the process of gathering data and sorting it out into general principles (Hmm, when these cats have babies, they are always kittens, whereas these geese always have goslings, I hypothesize that all animals have babies that are like themselves). Deduction can then be used to make predictions from the general principles (Hmm, I predict that these kangaroos will give birth to baby kangaroos). Seeing the two processes together is important. Another fictional character who is misleading in this respect is Sherlock Holmes, supposed to be an expert at the “science of deduction,” when in fact nearly all of his successes are inductive. These issues are among those explored in my course on Reasoning.