Approximation to the Truth
Will the sun rise tomorrow morning? Granted this not a question, most people peer into too much. However, this question serves as the crux of all human endeavour into the nature of truth through science. One might say that the answer to this question is obvious; although upon further inspection one comes to a stark realisation. This problem was posed by a French astronomer and mathematician called Pierre-Simon Laplace (1749–1827). He frames it as such; imagine that you have never seen a sunrise, this would then lead you to answer this question with a very low probability that the sun will rise tomorrow morning. You then wake up the next morning and you observe that the sun has indeed risen; thus when you are then framed the question once again, you would then answer yes with a higher probability. After a succession of events of the sun rising in the morning, you sequentially increase that probability higher and higher approximating closer and closer to a probability of a 100% yet never quite reaching it. Now, why is this so? If you have seen the sun rising for all of your life, wouldn’t the probability be absolute …. this simply cannot be the case as at any given moment the sun might be destroyed and further than that you simply wouldn’t be able to answer the question through an infinite span of time. This leads to a realisation that even a supposedly true and obvious statement that sun will rise tomorrow morning is not absolute; despite this notion, you are still able to continuously come to a closer approximation of the truth by obtaining more and more evidence.
These set of axioms form the central tenet of scientific investigation and the scientific method; in that, you form falsifiable hypotheses and you update it through the influx of new pieces of evidence. This can be formulated in mathematical terms by forming a hypothesis (H) which is supported by a piece of evidence (E), you then update that hypothesis when given a new set of evidence (E). Once this notion is extrapolated it forms a mathematical theorem called Bayes’ Theorem (can be seen on the image below). The theorem is named after Reverend Thomas Bayes (1701–1761) and its main application is in Bayesian inference. Bayesian inference is an interpretation of probability that uses the theorem to update the probability for a hypothesis as more evidence becomes available².
Let’s try to dissect this theorem by giving an example. Suppose that a zoologist has observed from a distance what appears to be a Bonobo (Pan paniscus) because of the noticeable feature that distinguishes them from Chimpanzees (Pan troglodytes) which are pink lips. Suppose that this feature is present in 95% of the Bonobo population. In the Chimpanzee population, this feature is present 5% of the time. However, the Chimpanzee population vastly outweighs the Bonobo population as Bonobos only account for 20% of the total population of the two species. What is the probability that the ape the zoologist sees is, in fact, a Bonobo?
- Hypothesis (H): The ape that was seen was a Bonobo
- Evidence (E): It is a Bonobo because of their distinguishing feature (pink lips)
- P(H) = Probability of hypothesis → Bonobo population.
- P(E) = Probability of evidence → Features present in Chimpanzee and Bonobo population. This can be found using an extended form of the theorem: P(E | H) P(H) + P(E | X) P(X) → as in order to determine the occurrence of the features we have to find the sum of both the Chimpanzee and the Bonobo population.
- P(H | E)= Conditional probability of a hypothesis (H) being true given evidence (E) → the ape that the zoologist observes is a Bonobo because of the pink lips feature that is present.
- P(E | H)= Conditional probability of the evidence (E) occurring given the hypothesis (H) → feature present in Bonobo population
- P(X)=Probability of Chimpanzee population
- P(E | X) = Probability of evidence (E) occurring in (X) → feature present in Chimpanzee population.
By using the theorem, you retrieve a seemingly counter-intuitive result. Despite the seemingly high chance of the ape spotted being a Bonobo as the feature present is in 95% of the Bonobo population; Bayes’ Theorem tells us not to neglect a significant variable that is the Bonobo population compared to the Chimpanzee population which is significantly lower. The Bonobo is rarer to find than a Chimpanzee thus it is more likely that the ape spotted is in fact not a Bonobo which is highlighted by the answer: 32%. The zoologist is understandably disappointed by this discovery.
Well, now we know the mathematical significance of the theorem, but how does it relate to science; why’s the theorem so significant in scientific investigation? Let’s start by making some presumptions: the first presumption is that at large scientific development occurs through empirical observations; we also know that our senses do not give us a true, one-to-one perception of the world, at large our perceptions are reconstructions of reality (this notion is explained in a previous article: The Machiavellian Mind — https://medium.com/absurd-existence/the-machiavellian-mind-737859f3bc36). The second presumption is that science does not find truth rather it attempts to gain closer approximations of the truth. In a nutshell, this is what Bayesian reasoning and inference is; it attempts to gain closer approximations of the truth as new evidence comes in, however, it will never reach absolute truth as it simply cannot take in all the variables and evidences that would a comprise a truth. This is the same in science, we will never find absolute truths in science as it inherently impossible within the system of science because firstly our senses do not give an objective perception of the world and secondly we cannot fathom all the variables that would comprise a truth within the system of science. There is a reason as to why the Big Bang is called a theory and not a fact despite the compelling evidence for it; it is called a theory because firstly a theory is an explanation that comprises a host of facts and secondly a theory must be falsifiable within the system of science and the scientific method. A theory implies that it must be falsifiable through new empirical or mathematical evidence, thus, according to science, no truth can be absolute as this would imply that it is unfalsifiable. In the realm of science absolute truth can never be found, only closer approximations to the truth through Bayesian inference.
“Bayes’s Theorem is one of those insights that can change the way we go through life. Each of us comes equipped with a rich variety of beliefs, for or against all sorts of propositions. Bayes teaches us (1) never to assign perfect certainty to any such belief; (2) always to be prepared to update our credences when new evidence comes along; and (3) how exactly such evidence alters the credences we assign. It’s a road map for coming closer and closer to the truth.”
Excerpt From: Sean Carroll. “The Big Picture”
[1]: Wikimedia Commons. (n.d) Olive Trees with Yellow Sky and Sun. https://upload.wikimedia.org/wikipedia/commons/f/f2/Vincent_van_Gogh_-_Olive_Trees_with_Yellow_Sky_and_Sun.jpg
[2]: Wikipedia. (n.d) Bayesian Inference. https://en.wikipedia.org/wiki/Bayesian_inference
[3]: Wikimedia Commons. (n.d) Apeldoorn Apenheul zoo Bonobo. https://upload.wikimedia.org/wikipedia/commons/e/e2/Apeldoorn_Apenheul_zoo_Bonobo.jpg
[4]: Wikimedia Commons. (n.d) CMB Timeline300 no WMAP. https://upload.wikimedia.org/wikipedia/commons/6/6f/CMB_Timeline300_no_WMAP.jpg
