Why isn’t it rational to believe in unicorns? Why shouldn’t we waste our time investigating whether the (actual) tabloid headlines “Noah’s ark found on Mars!” or “Hillary Clinton dating a space alien” are true? How do we draw some sort of line between what is reasonable to believe true and what clearly isn’t?
If you are anything like me, you may marvel at the achievements of science. I don’t mean things like cars and DVD players, as impressive as they are. Rather, it is the ability of science to force wider our conceptual horizons and show us ideas we never would have dreamt possible. I think, for example, about Einstein’s wondrous theories of relativity (which show that a baseball hurled towards a batter ages more slowly than the pitcher does and actually gets heavier in the process). Or the modern view of evolutionary biology, where evolution’s battles are played out at the level of the gene, and bodies are most properly viewed as the gene’s survival machines – programs enabling them to be replicated. And if you seek mystery, there is nothing so bizarre and confounding as the world of quantum physics, with particles appearing and disappearing for no apparent reason, all the while simultaneously appearing a mind-wrenching duality of both wave and particle.
These ideas are as counter-intuitive as they come; in fact I find them even more bizarre (and far more interesting) than the tabloid headlines I quoted. And yet, scientists believe them. Now at this point you may well be asking what the difference is. How is it that a scientist is entitled to his ghostly particles, whereas, following the headline, anyone hopefully swivelling his telescope towards Mars should be treated with derision?
The difference is proof. Science starts with its default position at scepticism[1] – that is, with disbelief. If there were no proof, a scientist would know nothing. But from this position of avowed cluelessness, science (ideally, at least) accepts under its wings those claims for which there is sufficient evidence. If science had a spokesman, she would no doubt be endlessly repeating to us recalcitrant humans that disbelieving a claim is the correct starting point on the road to truth. If anything is to be believed, there should be a reason for believing it.
To my ears though, to stop here would leave us with little more than unjustified rhetoric. After all, why should I only believe something if there is good evidence for it? Why not shun the scientific method and adopt a perversely opposite approach, believing something unless it is disproven? Incidentally, this is more than an interesting aside, for we get ourselves into quite a philosophical quagmire if we say that everything that is believed should be justifiable – and then refuse to justify this statement.
But the riddle does have a solution, and it is fiercely interesting. What we are really looking for is a method for limiting inaccuracy. In other words, if I have a statement – we could call it ‘X’ – does it have the same odds of being wrong as if does of being right? If it does, whether we choose to start with acceptance or start with scepticism would not be a logical matter, for neither would have any advantage. But if the odds are stacked in a particular direction, things start to get interesting. If any statement is more likely to be correct than incorrect, then (in the absence of any other data that might sway us) the best bet would be to believe it and change our minds only if it is proven to be untrue. We would be right more times than we would be wrong this way. On the other hand, if an isolated statement is more likely to be wrong than right, our default position should then be disbelief, again because it would guarantee correctness on a greater number of occasions.
The situation is really a bit of logical gambling. Provided that our given statement is isolated – that is, that there is no predisposing external reason for believing in it or spurning it[2] – then we have no way of knowing whether is will turn out to be true or false. But the shrewd gambler works the odds. Given a statement, the question confronting him is really whether he should put money ‘true’ or ‘false’. And the answer to that will depend on whether, all things being equal, the odds are skewed towards statements being either true or false.
So which is it? One method might be to count the number of true statements and compare this number with the number of false statements. Clearly, if there are, say, only a hundred true statements about the world, and a hundred million false ones, the odds of any particular statement being true would be… well, the proverbial one in a million. Placing my money on any particular statement being true could well be considered diagnostic of insanity – I would most likely stay richer longer if I piled all my money in the middle of your room and set fire to it. (Not a sensible bet then.)
Unfortunately such a theoretically easy solution is unavailable to us. The logic only works if there are a finite number of either true or false statements, even if this number is too great to actually count. Alas though, it turns out that there are an infinite number of both statements either way. The easiest way to illustrate this is to use numbers, which are a readily understandable built-in source of infinity.
First take the false statements. ‘1 + 1 = 2’ may be correct, but ‘1 + 1 = 3’ isn’t. Nor is ‘1 + 1 = 4’ or ‘1 + 1 = 5’. And since the number line goes onward infinitely, one can simply keep adding 1 to whatever ‘1 + 1’ isn’t for a new and equally spectacularly false answer!
With a slight modification, the same trick can prove that there are an infinite number of true statements. Instead of continually adding 1 to the answer column, just add it to one of the other columns, and adjust the answer accordingly:
‘1 + 1’ may only equal ‘2’,
but ‘1 + 2 = 3’ is just as true,
and so is ‘1 + 3 = 4’…
You don’t have to use numbers to prove the infinities, although it is easier. To prove false infinities in another manner, you could say, for example, “That [pointing to a computer] is spelt ‘COMPUTER’”. The statement would obviously be true, but there are an infinite number of wrong attempts. One could simply continue to add a letter: COMPUTERA, then COMPUTERAA, then COMPUTERAAA, and so on.
So our first attempt is destined for failure. Should we abandon the attempt here? Should we accept that our scepticism when confronted with fairies, unicorns, trolls and the like is just a matter of temperament, devoid of any foundation? No! Help is at hand! Something may just have caught your eye in all this – a loophole through which to squeeze our way out of our predicament. Even though there are definitely an infinite number of both true and false statements, the two proofs are not simply mirror images of each other. For the subject ‘1 + 1 = …’ there is only one possible right answer and an infinite number of false answers. But to prove that there are an infinite number of true answers, we had to change the subject. This slight of hand obscured a weakness – a weakness that provides an opening for our next assault.
What happens if we ‘constrain’ the subject? By this I mean to take the subject of the phrase (which in the above example corresponds to ‘1 + 1 = ’) and lock it down, so that, unlike the answer, it is not free to vary. Now what is the ratio of true statements to false statements when the subject is constrained? Clearly, there is only one true statement but an infinite number of false ones! So the ratio is set nicely at 1:infinity (1:µ), which is – by definition – as poor odds as you are ever capable of laying eyes on. And from this insight it is only a small jump to understand why one’s initial position should be disbelief if one is to be at all rational about it. No gambler could be so stupid as to open with credulity.
It isn’t always quite this cut and dry, naturally. Take the example, “Bob is …” which we shall take to be the constrained subject. Now there appear to be many true answers, for example:
… a living organism
… a mammal
… a vertebrate
… stupid
However, if this appears to be a problem, then one often simply hasn’t been specific enough about the subject. Language has had a strained relationship with logic in the last century or so. In this case, the problem is that language can be a kind of shorthand conglomeration of many subjects, and which particular subject is implied may only reveal itself when the rest of the sentence plays itself out. Starting with “Bob is …” I have no idea of what the right answer is, whereas if we unpack the enmeshment, things start getting clearer, if much less aesthetically pleasing. “Regarding Bob’s status as a living or non-living thing, Bob is … a living organism” corrects this somewhat (although it sounds truly awful). In this reformulated subject, the answer “stupid” would be wrong. In a similar vein, “As regards Bob’s intelligence, with stupid being defined as an IQ of less than 80, Bob is … a living organism” sounds vaguely like an insult, but nonetheless joins the infinite list of incorrect answers.
Language’s ability potential to confuse can reveal itself later in the phrase too. For example, to answer “a living thing” instead of “a living organism” is really a similar same trick played over again – this time in reverse. This time, language is being too expansive, and toying with different words for the same answer altogether. I wasn’t asking whether he was a “thing” or an “organism” (which might be dead), I was asking whether he was living or non-living. The fact that other words can be used to describe the same concept is neither here nor there, or an answer in another language would have to count as a different answer. The point is that, disregarding language’s red herrings, there is still only one correct answer to each of these questions. If you are getting more than one plausible answer, then it is the question that isn’t tight enough. For instance, if “living thing” versus “living organism” bothers you, just phrase the question as pedantically as, “Which of the two phrases ‘living organism’ or ‘non-living thing’ best describes Bob?”
A slightly more fundamental limitation is that the ratio of true:false statements can be effortlessly reversed with a minus sign. To show this, let us start with another example “I am a South African citizen” (correct) and “I am an Italian citizen”, “I am an American Citizen”, “I am an Iraqi citizen”, etc. Again, there is an infinite number of wrong answers (192 plus answers like ‘stone’ and ‘no’) - enough, one would hope, to reassure us that the best default position would again be scepticism, unless attended to by proof. What if I were to introduce a negative here, though? Now, the statements become “I am not a South African citizen” (incorrect), and “I am not an Italian citizen”, “I am not an American citizen”, “I am not an Iraqi citizen”, etc. In this (negative) case, the odds are reversed, and the odds of a phrase being false are now 1:µ. Is this a serious challenge to our progress so far? Not really, we just need to rephrase the central theorem as being “Any positive statement is considered untrue unless proven otherwise”. It is worth pointing out that, far from being a challenge our original idea, this is really the same proof in reverse – once again, it shows that the best starting point is scepticism, whether it is disbelieving a positive point, or believing a negative one.
The only time that there genuinely is more than one correct answer is when the question deals in relativity, exemplified by the phrase ‘X is greater/smaller/ more than/less than/etc. Y’. “Lucy is shorter than 2 metres tall” is true, and you will hopefully easily see that she would also be shorter than 3 metres, and 4 metres, and 5… There are, of course, still an infinite number of wrong answers too, like “Lucy is shorter than 1 metre tall” and “Lucy is shorter than 1.01 metres tall”, and “Lucy is shorter than 1.001 metres tall”, etc. These questions require a range of answers, so we have to give it to them. Alas, with this range comes the infinity problem again, and so the theorem can’t be applied in either direction.
Lastly, it should be obvious that the theorem applies only to statements where there is a right and wrong answer. Questions of subjective taste and evaluation (“The play was good”, “Apples are nice”) can never be adjudicated on objectively.
Minor objections aside, so far we have nonetheless showed that if the subject can be ‘constrained’, then it is best (usually by the biggest possible margin) to bet on a statement being false, rather than true. Furthermore, by our original logic, we have showed that this most rationally translates into the strategy of requiring proof for statements if they are to be believed. There must be good reasons for believing something, or else the statement is all but guaranteed to be wrong. But when in real life is the subject constrained? The answer is almost all of the time! Whenever we say, “A dog has four feet” we don’t permit any accusers to retort that we are wrong because a millipede (a different subject) doesn’t. No, if a statement is to be opposed, the subject must be constrained. Similarly, to condemn the statement, “South Africa’s first democratically elected president was Abraham Lincoln”, we can’t respond that Europe is small by continental standards. We would (accurately) be accused of lunacy, for the essence of an argument is to oppose the commentary made on a fixed subject – to talk simultaneously of two different things (the equivalent of not constraining the subject) is to argue like a madman.
Incidentally, this also illuminates in a slightly mathematical way just how important definitions are. If what we mean by the word “love” is different, then our argument will almost certainly be fruitless. We would revert to the problem of unconstrained subjects, where there are an infinite number of potentially correct answers, causing debate to be futile. Not having the same definition of the subject in mind is really just a less dramatic way of talking in two unrelated languages – and getting upset when the other man doesn’t talk yours!
The ‘divergent definitions’ problem aside, the understanding that disbelief is the only rational stance for proofless claims generates some massive implications. For one, this settles the irritating little debate that goes something like this: “You can’t prove me wrong, therefore I’m entitled to believe it”. No – not anymore. We have just shown that your position is next to worthless unless you can back it up. If this still seems a bit theoretical, then let me hand over to the philosopher who wrote the best prose of all, Bertrand Russell. In 1952, he famously noted:
If I were to suggest that between the Earth and Mars there is a china teapot revolving around the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be observed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presumption on the part of human reason to doubt it, I should rightly be though to be talking nonsense.
It sounds so obvious, doesn’t it – so rational. But carry this logic forward into that territory traditionally given a free pass from logic – religion – and you may get quite a jolt. Continuing, Russell notes, not without a trace of bitterness:
If, however, the existence of such a teapot were affirmed in ancient books, taught as truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.
Russell’s ‘celestial teapot’ has become a cult classic amongst atheists, and has recently been joined by other sarcastic inventions along the same line, including like the Flying Spaghetti Monster (“may you be touched by His noodly appendage”) and the Invisible Pink Unicorn. All are deliberately made both as ridiculous and as un-disprovable as possible, and clearly illuminate the madness of treating unjustified claims with credulity. The point, again, is this: without justification, any concept – including that of a god – should be treated with the same scepticism and scorn as Russell’s celestial teapot. And if there is any reason to estimate God (or anything else) as being more likely than this, then let’s hear it, for those reasons would fall quite comfortably into the realm of either logic or science. Needless to say, neither field has proved at all accommodating.
Naturally, the ambit is not limited to religion. Along other targets I would push towards the firing line are homeopathy (scientifically proven not to work), the ‘power’ of prayer (likewise the subject of several withering analyses), ‘magic’ crystals (often ‘blessed’ by the local shaman), most ‘alternative’ health schemes (a few may work, but virtually none bother testing their extravagant claims), all religious texts, John Edward and anyone else who claims they can talk to dead people, and many and much more.
Perhaps I should step down off my soap box now, lest this turn into a moan. Let us instead return to our original observation – that science has progressed an amazingly far amount despite it’s eternally cocked “Spockian eyebrow of doubt” (to use Natalie Angier’s wonderful phrase). But that isn’t quite the right way to phrase it – science’s success is not ‘despite’ such initial scepticism, it’s because of it. Instead of guessing, intuiting, divining or accepting the voice of tradition or authority, science has chosen the most gruelling method of all. And the much harder task eventually pays off, with rich rewards indeed.
[1] ‘Scepticism’ (or ‘skepticism’) is technically a branch of philosophy centred upon the idea that absolute knowledge is impossible. For the purposes of this essay, I will use the term as it is meant in conventional English today – as the opposite of credulousness.
[2] Perhaps this needs a little clarification. If I face the statement “lions have whiskers” armed with the knowledge that lions belong to the cat family, members of which generally have whiskers, then this knowledge represents what I mean by a ‘predisposing external reason’. These are really just clues that would (and should) sway us in our assessment of the truth of the original statement. An isolated statement has no such clues to help us, as in “The Andromedean species of Xweuirts have blue eyes”…
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