I struggle to keep up with my many interests. I adore pure mathematics (functional analysis, PDEs, for a time even number theory) and I need to contribute to the scientific effort to combat ageing. Of course, if you're on this blog, you must know that I love fiction too, reading and writing. Fiction, science, and mathematics each hold a special place in my heart and each competes for the attention I have left after attending to life's necessities such as food, socialising, sleep and exercise. I fear that I have no time to devote to even one of these, never mind all.
This is one of the reasons I want to live forever: it feels like it would take a lifetime to become an expert in just one subfield of each of these areas and I cannot abandon any one of them because to do so would be to abandon myself. But recently I came across an idea that these things are not actually dissimilar and that what one experiences in one can in fact be applied to the others, especially if one is aware of exactly how they relate to each other.
"Event A implies Events B and C implies event D but only if event E." This is a verbal causal diagram (as popularised by Judea Pearl). This applies to fictional stories but also to science as mentioned in Judea's book "The Book of Why?". I say that it also applies to mathematics in a sense, even though nothing 'causes' another thing, only that one arrives at truths by connecting them using logic.
Our brains are still causal machines and we apply causal and probabilistic reasoning even when a statement is certainly true or false. For instance, many mathematicians say it is 'likely' that the Riemann Hypothesis is true, but this is not really the same thing one means when they say it is 'likely' alien life exists. In the former case the mathematicians probably mean that based on their experience of similar problems it seems reasonable to them that it's true. It is probability in the Bayesian sense, a degree of belief. It is not the case that there are many possible words that could lead to the hypothesis being either false or true. It is either true in all possible worlds, or false in every one.
Conversely, when it comes to alien life, one can still interpret this probability as frequentist. A sequence of effectively random events could lead to life being present on any given planet. The probability that there is no life is 1 - (probability there is life on no planet).
Of course this is not criticism of either approach; doing the best we can with our brains we have achieved remarkable progress in both science and mathematics. Some say the success of mathematics in physics is surprising, but I think that's more surprising is that our brains wired for causal inference can also be adept at mathematics which is not causal in the same sense. But this difference should be noted because it changes how we approach each subject.
In science one eliminates conditions systematically, and hopes that the hypothesized connections are really the connections that are there (or are sufficiently close so that the theory has the same utility). One does this by comparing examples where one believes the general theory applies: in physics, we can apply the theory of gravity to humans and Earth, Earth and the sun, blackholes and stars, ect. This is not in fact too dissimilar to what one does in mathematics; one creates a general theory (for instance of weak solutions to elliptic partial differential questions) and is only satisfied when the theory can deal with all examples which we believe are similar. Of course, our beliefs in what examples are similar or different may change according to the theory.
The difference is that with mathematics, one knows exactly when statements and ideas connect. Connections can be verified or falsified to 100% certainty (at least in elementary proofs). The difficultly comes in deriving these connections but once they are known they are certain. For example, I can use the idea that a (product of primes)+1 is not divisible by any primes in that product to show that there are infinitely many primes. If I, as I once did, believed that (product of primes)+1 is itself prime, I could falsify this idea quickly by noticing that 3*5+1=16.
In science we do not eliminate, only make less plausible. That said, a lot of the same methods can be used to at least find plausible connections: inspired guessing, determining counterexamples, evaluating why those counterexamples fail and updating one's perceived connections. To find a couterexample, one needs to do an experiment (or find the work of someone else who has done an experiment) which is generally much harder to do.
But what of fiction? As I said at the start, stories are also built up of plausible connections. The difference is that one cannot make these more or less plausible, only conform more or less to one's intuition. By this I mean a story is not beholden on the rules of real life even if it is meant to be realistic; if you write a character whose only ambition in life is to own an ice-cream store, and that they stop at nothing, not even murder, to do this, I might say this character is unrealistic but what I really mean is that I don't believe real life people behave this way.
Similarly, this is the difference between good and bad plot twists. A good plot twist conforms to the readers intuition as it is developed, so that when it is revealed the reader is prepared to accept it. Conversely, a plot twist may be bad because it does not conform to the reader's intuition. They do not accept the premises or logic that lead to it, so it feels unsatisfying, as if someone is telling you a falsehood.
The magic of stories lies in their ability to both conform to their reader's intuitions and simultaneously draw unexpected conclusions from them. In this way, they expand the reader's world view. Consider Watchmen; it reaches the surprising conclusion "Superheros wouldn't actually be able to cope with their power and remain good people" but I don't find it surprising by the end of the film because it matches my intuition of what real people can be like.
While mathematics reveals definite truths, science reveals plausible truths, fiction reveals possible truths; not of the world as such, but about our own intuitions. This has to be the case because stories cannot be bound by the rules of our physical world or even mathematical logic.
Each of these domains has key differences but they are part of a greater whole. By noticing these differences, I believe we (especially myself) can promote our understanding of the other domains using our understanding of one. Indeed by devoting time to all three, one can develop insights that might not otherwise be possible. That said, I still fear that I am destined to fail by spreading myself too thin by attempting to write novels, do pure mathematics, and solve the scientific problem of ageing, but now that I see all of these things as part of a greater whole I know the pursuit is worth it, and will not feel guilty for focusing on just one of these domains for a while.
