Proving a negative and the reindeer in your house

# Proving a negative and the reindeer in your house

TL;DR – We cannot always validate all answers to a scientific question with the same process. For some yes/no questions, only one side can be actually experimentally verified.

Some people write me things like “Is it true that absent of evidence is not evidence of absence?” “Is it true that there my be a China teapot orbiting between Earth and Mars?” “Is it tru that scince caan’t proved a negatitive?” They clearly give evidence that some things are lacking.

There is some kind of fascination on whether scientific investigation can prove negatives. For example: can reindeer fly? Well, we could bring a bunch of them on top of the Empire State building, push them off one by one and see what happens. The best we could conclude is that those particular reindeer, on that particular day, in those particular weather conditions, weren’t able to fly. Or, if they were, they chose not to (which may still tell us something about reindeer). This was pointed out by James Randi a long time ago, and it is an example of what he meant by “you can’t prove a negative.”

Then some people told him, “Of course you can prove a negative! What you say is poppycock. Therefore it is not non-poppycock, thus proving a negative!” This is interesting because it provides an example of how you can be logically pedantic and yet miss the point.

In the same vein, there is the old adage “absence of evidence is not evidence of absence.” Which is catchy because it reverses the word order and not the order of the words. This, again, tells us that if we didn’t see a flying reindeer, it doesn’t necessarily mean it doesn’t exist. In fact, there may be tiny ones flying inside your brain. I know! That’s kind of freaky! Just to be sure, I did an MRI and there were none. Well, at least last time I checked.

This frustrates some people, who stress the foolishness of sticking to experimental evidence. Would you generalize already? Use induction, for Pete’s sake! Nobody has seen flying reindeer, therefore they don’t exist! Ah, so impatient… Plus, I don’t think Saint Peter cares about flying reindeer.

My takeaway is somewhat different. The problem is not that we can’t prove a negative. The problem is that the ability to experimentally verify a claim may or may not correspond to the ability to experimentally verify its negation. But it’s something better discussed after one of those ludicrous and baffling thought experiments.

# The reindeer in your house

Suppose you are sitting comfortably at home by yourself. It’s a quiet summer afternoon and you are in your favorite chair reading a book. Or watching TV… whatever you do in your favorite chair on a summer afternoon. Someone knocks at your door. You open the door. You don’t recognize the stranger… though somehow he feels familiar. From underneath his big white beard he says, “I am so very sorry to disturb you. I seem to have misplaced my reindeer. His name is Rudolph. Could you check if, by any chance, he happens to be in your house?”

You now find yourself in a state of “What?!?”, the one where the situation is so bizarre you do not know how to react, and you wonder whether you are sleeping and it’s all a dream… Or whether you should just go to bed and end the day right there. You mumble something incoherent but the stranger insists “Could you just check? It will only take a minute… Just go around the house and see if you see him. He’s fairly big, you can’t miss him. I’ll wait here.”

You go with the flow. Since all reason already went out of the window, you don’t see any point in giving this more rational thought. This is a typical reaction when in a state of “What?!?” In the end, you just have to go around and look for a reindeer. If you do not see it, that’s evidence Rudolph is not here. If you see it, that’s evidence that Rudolph is one sneaky reindeer.

No reindeer in the living room. No reindeer in the kitchen. No reindeer in the bathroom. No reindeer… wait! There is one in your bedroom… on your bed. And he appears to be sleeping. How did he…? No matter… You go back to the door, and say “Yes, he is actually here… in my bedroom.” “Oh, thank goodness!” the stranger says, and you hurry his chubby figure inside.

You stand in from of your bedroom door. “Oh!” utters the stranger “That’s not my reindeer. Rudolph has a rash on his nose… Well, thanks anyway. Bye!” And he goes away. You start wondering how you are going to explain this to animal control.

# Discussion

While the fact that you couldn’t see a reindeer was enough evidence to conclude that Rudolph was not in your house, the fact that you could see a reindeer was indeed not enough evidence to conclude that Rudolph was in your house. Presence of evidence was not evidence of presence.

You see, the problem is better understood in a different way. Suppose you have a yes or no question. Will the experimental evidence gathered by a single procedure lead to a yes or no answer with equal certainty? Or do we need one procedure to answer “yes” and another one to answer “no”? In the reindeer case we needed an extra step for the “yes” answer: make sure the reindeer is the right one.

In many cases one procedure is enough. If I ask you, “Do you have twenty dollars in your pockets?” you can dump everything on a table, see that there are twenty dollars and say, “Yes!”  Only to realize you are wrong because now you have zero dollars in your pockets. The point is: after you count how many dollars you have, and you put them back, you can conclusively say either yes or no to the original question. But that’s not always the case.

Originally neutrinos were thought to have exactly zero mass. In 1957 Bruno Pontecorvo (which roughly translates as Brown Crowbridge) predicted that if neutrinos had mass they would oscillate. Experiments were made and their results were consistent with neutrino oscillation, and therefore the neutrino mass could not be zero. Note that, while we still don’t have a precise measurement of the actual value, we do know conclusively that it’s not zero. That is: we can measure “mass is not zero” very very well.

On the other hand, the photons are still thought to have exactly zero mass. Do they? Well, if they had mass you would see some effects. It would modify Coulomb’s law, which experiments didn’t detect. It would affect the galactic plasma, which experiments didn’t detect. In the end, we just know that the mass of the photon has to be less than 1×10−18 eV/c2 because if it were greater, we would have detected those effects. That is: we can’t measure “mass is zero” very well. Absence of mass is not mass of absence.

The question “Is the mass zero?” has an exact no answer or non-exact less-than-this yes answer. But this has nothing to do with the value zero itself. Any value on a continuous scale (such as length, volume, weight, energy, temperature, stupidity, and so on) can only be measured within a finite precision. But we can still rule out a particular exact number. For example, I may not know the precise length of my right arm, but I do know that it’s not 3 meters precisely. I know because I can’t get a soda from the kitchen fridge when I am watching TV in the living room.

Consider another question: “Do you have biological children?” If you are female, that is easily verifiable: if you didn’t notice, you didn’t have any. If you are male, and you think you have children, you can do a paternity test which may lead you to a conclusive yes. But to answer no conclusively, performing a paternity test on the whole population of earth is not a good strategy: you may get a lot of false positives and may get stuck with a son 40 years older than you. Also note that if the first million people didn’t test positive, that would not be enough evidence to conclude you don’t have any children. So, for the no answer, you need an altogether different strategy. You should retrace your steps and maybe make a few phone calls. Good luck!

Once you realize that different answers may warrant different procedures, you may also realize that, for some questions, only a procedure for one side may exist. Will this complicated calculation I just started on my computer terminate? If it eventually does, then yes. Otherwise, beats me! Turing says you can’t know that in general.

So, if we wanted to be precise, we should say, “The negation of an experimentally verifiable statement is not necessarily an experimentally verifiable statement.” See the problem now? It should be really really clear: that’s not at all catchy! It takes more than a minute to wrap your head around it! How can this idea be communicated in these modern times, where the attention span is… Wait! Cat video! Sooooo cute! What was I saying? Oh, whatever…