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The scientific method rests on faith in God and Man.

The so-called "scientific method" is, I think, rather poorly understood. For example, let us consider one of the best-known laws of nature, often simply referred to as the Law of Gravity:

Newton's Law of Universal Gravitation: Every object in the universe attracts every other object toward it with a force proportional to the product of their masses, divided by the square of the distance between their centers of mass.

Now here is a series of questions for you, which I often ask audiences when I give lectures on the philosophy of science:

  1. Do you believe Newton's Law of Universal Gravitation is true?
  2. If so, how sure are you that it is true?
  3. Why do you believe it, with that degree of certainty?

The most common answers to these questions are "yes", "very sure", and "because it has been extensively experimentally verified." Those answers sound reasonable to any child of the Enlightenment -- but I submit, on the contrary, that this set of answers has no objective basis whatsoever. To begin with, let us ask, how many confirming experiments do you think would have been done, to qualify as "extensive experimental verification." I would ask that you, the reader, actually pick a number as a rough, round guess.

Whatever number N you picked, I now challenge you state the rule of inference that allows you to conclude, from N uniform observations, that a given effect is always about from a given alleged cause. If you dust off your stats book and thumb through it, you will find no such rule of inference rule there. What you will find are principles that allow you to conclude from a certain number N of observations that with confidence c, the proportion of positive cases is z, where c < 1 and z < 1. But there is no finite number of observations that would justify, with any nonzero confidence, that any law held universally, without exception (that is, z can never be 1 for any finite number of observations, no matter how small the desired confidence c is, unless c = 0). . And isn't that exactly what laws of nature are supposed to do? For Pete's sake it is called the law of universal gravitation, and it begins with the universal quantifier every (both of which may have seemed pretty innocuous up until now).

Let me repeat myself for clarity: I am not saying that there is no statistical law that would allow you to conclude the law with absolute certainty; absolute certainty is not even on the table. I am saying that there is no statistical law that would justify belief in the law of universal gravitation with even one tenth of one percent of one percent confidence, based on any finite number of observations. My point is that the laws of the physical sciences -- laws like the Ideal gas laws, the laws of gravity, Ohm's law, etc. -- are not based on statistical reasoning and could never be based on statistical reasoning, if they are supposed, with any confidence whatsoever, to hold universally.

So, if the scientific method is not based on the laws of statistics, what is it based on? In fact it is based on the

Principle of Abductive Inference: Given general principle as a hypothesis, if we have tried to experimentally disprove the hypothesis, with no disconfirming experiments, then we may infer that it is likely to be true -- with confidence justified by the ingenuity and diligence that has been exercised in attempting to disprove it.

In layman's terms, if we have tried to find and/or manufacture counterexamples to a hypothesis, extensively and cleverly, and found none, then we should be surprised if we then find a counterexample by accident. That is the essence of the scientific method that underpins most of the corpus of the physical sciences. Note that it is not statistical in nature. The methods of statistics are very different, in that they rest on theorems that justify confidence in those methods, under assumptions corresponding to the premises of the theorems. There is no such theorem for the Principle of Abductive Inference -- nor will there ever be, because, in fact, for reasons I will explain below, it is a miracle that the scientific method works (if it works).

Why would it take a miracle for the scientific method to work? Remember that the confidence with which we are entitled to infer a natural law is a function of the capability and diligence we have exercised in trying to disprove it. Thus, to conclude a general law with some moderate degree of confidence (say, 75%), we must have done due diligence in trying to disprove it, to the degree necessary to justify that level confidence, given the complexity of the system under study. But what in the world entitles us to think that the source code of the universe is so neat and simple, and its human denizens so smart, that we are capable of the diligence that is due?

For an illuminating analogy, consider that software testing is a process of experimentation that is closely analogous to scientific experimentation. In the case of software testing, the hypothesis being tested -- the general law that we are attempting to disconfirm -- is that a given program satisfies its specification for all inputs. Now do you suppose that we could effectively debug Microsoft Office, or gain justified confidence in its correctness with respect to on item of its specification, by letting a weasel crawl around on the keyboard while the software is running, and observing the results? Of course not: the program is far too complex, its behavior too nuanced, and the weasel too dimwitted (no offense to weasels) for that. Now, do you expect the source code of the Universe itself to be simpler and friendlier to the human brain than the source code of MS Office is to the brain of a weasel? That would be a miraculous thing to expect, for the following reason: a priori, if the complexity of that source code could be arbitrarily large. It could be a googleplex lines of spaghetti code -- and that would be a infinitesimally small level of complexity, given the realm of possible complexities -- namely the right-hand side of the number line.

In this light, if the human brain is better equipped to discover the laws of nature than a weasel is to confidently establish the correctness an item in the spec of MS Office, it would be a stunning coincidence. That is looking at it from the side of the a priori expected complexity of the problem, compared to any finite being's ability to solve it. But there is another side to look from, which is the side of the distribution of intelligence levels of the potential problem-solvers themselves. Obviously, a paramecium, for example, is not equipped to discover the laws of physics. Nor is an octopus, nor a turtle, nor a panther, nor an orangutan. In the spectrum of natural intelligences we know of, it just so happens that there is exactly one kind of creature that just barely has the capacity to uncover the laws of nature. It is as if some cosmic Dungeon Master was optimizing the problem from both sides, by making the source code of the universe just simple enough that the smartest beings within it (that we know of) were just barely capable of solving the puzzle. That is just the goldilocks situation that good DM's try to achieve with their puzzles: not so hard they can't be solved, not so easy that the players can't take pride in solving them

There is a salient counterargument I must respond to. It might be argued that, while it is a priori unlikely that any finite being would be capable of profitably employing the scientific method in a randomly constructed universe, it might be claimed that in hindsight of the scientific method having worked for us in this particular universe, we are now entitled, a posteriori, to embrace the Principle of Abductive Inference as a reliable method. My response is that we have no objective reason whatsoever to believe the scientific method has worked in hindsight -- at least not for the purpose of discovering universal laws of nature! I will grant that we have had pretty good luck with science-based engineering in the tiny little spec of the universe observable to us. I will even grant that this justifies the continued use of engineering for practical purposes with relative confidence -- under the laws of statistics, so long as, say, one anomaly per hundred thousand hours of use is an acceptable risk. But this gives no objective reason whatsoever (again under the laws of statistics) to believe that any of the alleged "laws of nature" we talk about is actually a universal law. That is to say, if you believe, with even one percent confidence, that we ever have, or ever will, uncover a single line of the source code of the universe -- a single law of Nature that holds without exception -- then you, my friend, believe in miracles. There is no reason to expect the scientific method to work, and good reason to expect it not to work -- unless human mind was designed to be able to uncover and understand the laws of nature, by Someone who knew exactly how complex they are.

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Do you believe Newton's Law of Universal Gravitation is true?

This strikes me as a Socratic question. Socrates used to ask Greeks questions that were slightly off. Being polite, the Greeks would refrain from nit-picking the questions, and try to answer. Then Socrates, being an arse-hole, would nit pick the answers. He would entangle his victims with his verbal dexterity, and skillfully obscure that bad answers were down-stream from bad questions.

There are many stories to tell about gravity. Kepler discovered that the planets moved in ellipses. Newton invented a theory of mechanics and new mathematics. Then he was able to respond to speculation that the ellipses were due to an inverse square law of attraction by filling in the details of what that actually meant, and solving the mathematical problems to demonstrate it.

Newton went further, spotting that gravity was "universal". By "universal" Newton meant that the attraction was not a specific property of the Sun (which would leave gravity on the surface of the Earth as a separate mystery) but was about all matter attracting all matter. So a cannon ball fired by an artillery man follows an elliptical trajectory with one focus at the center of the Earth. Obviously an artillery man uses the parabolic approximation (until the Paris gun in 1918. But (unless my memory is playing tricks on me) Newton had the idea that a cannon fired horizontally with sufficient force would cause the cannon ball to orbit the Earth, just as the moon orbits the Earth.

"Universal" creates a loose end. Jupiter is attracting Saturn and Saturn is attracting Jupiter. The Sun is not the only player in the solar system. That loose thread went unpulled until it was noticed that Jupiter was spiraling in. Jupiter's orbit was decaying and it would in time destroy the Earth. Then a French mathematician (LaPlace?) got stuck into the details. Jupiter and Saturn are nearly in a five to two orbital resonance. The difference frequency is about 800 years. Four hundred years of Jupiter spiraling in and Saturn spiraling out get followed by four hundred years of Jupiter spiraling out and Saturn spiraling in. Theory and accurate astronomical observation agreed; panic over.

Other stories include Halley working out the orbital parameters of a comet and predicting its return. That was a big deal at the time, because comets were traditionally seen as bad omens. If they simply moved in obedience to Kepler's Laws, they stopped being frightening. The comet returned as predicted and is now called Halley's Comet.

After Hershel discovered Uranus, both John Couch Adams and Urbain Le Verrier puzzled over anomalies in the orbit of Uranus. Could there be another planet. Verrier go Johann Galle and Heinrich D'Arrest to look, and there was Neptune, discovered in 1846 by mathematics and Newton's Law of Gravitation. Verrier tried to repeat his success with anomalies in the orbit of Mercury, and inferred the existence of the planet Vulcan. Which wasn't there, leading eventually, by a circuitous route to Einstein's General Theory of Relativity.

For me, this raises questions about the word believe. I'm comfortable with three interpersonal meanings. Do I believe a person's testimony: did the things he tells me actually happen? Do I believe a person's promises: will he keep them? Do I believe a person's predictions: will they actually happen? But how does one extend the word believe to cover scientific theories? The tale that I've told goes well beyond my personal experience. The largest telescope that I have looked through is a twelve inch reflector. Maybe the story about Neptune is made up; I've seen Saturn, but neither Uranus nor Neptune. Interpersonal belief is at issue. Yet when we talk of belief in Newton's Law of Gravity, we assume the honesty of astronomers and are talking about something else. I'm not clear what. Contemplating the long narrative that I have sketched is valuable because it gives a concrete example of what successful science looks like. Trying to abstract a high level concept of "belief"? That is the kind of unmotivated abstraction that confuses things.

I'm comfortable with two meanings of the word true. One is person testimony (again). Did that actually happen? The other is in my books on mathematical logic. When is (A and B) true? When A is true, but not just A, B must also true. Add in first order logic, sets, and model theory and there is lots to read about. But neither notion of truth fits well with generalisations arising from empirical investigations.

The most promising notion of truth, appropriate to empirical investigation, that I have encountered is Probably Approximately Correct learning theory and Vapnik-Chervonenkis Dimension. Those are crap links. To get the basic idea, image rolling a d6 six hundred times to estimate the probabilities of each face. You get numbers like 100, 118, 95, 88, 114, 85 or like 112, 103, 93, 104, 99, 89. Empirical work always has a certain about of random slop and your empirical estimate will never be true in the sense of being exact. But what about being approximately true? Fix an unambitious goal for accuracy and ponder the probability of being approximately correct. Things can still go horribly wrong; an unlucky sequence of rolls could give you 600, 0, 0, 0, 0, 0 and your empirical work is not even approximately correct. But something interesting happens when the Vapnik-Chervonenkis dimension is finite. Fix your desired level of approximation and keep rolling the d6. The probability of not meeting your approximation goal eventually starts to decline exponentially with the number of rolls. Exponentially! You are on the route to the practical man's version of certain knowledge. Well, that is nice, but God is it complicated.

Asking "Do you believe Newton's Law of Universal Gravitation is true?" is doing the 20000 foot overview thing. It can only lead to vague waffle. On the other hand, waffling vaguely is rather fun; what am I actually proposing as a rival ideal? I think that interesting gap is between social science and "hard" science. There is a gap between "hard" science and ideal certainty, but it seems unimportant compared to the gap between social science and "hard" science. Let me give a concrete example of how little we know in social science so that you can see how well Newton's Law of Gravity compares.

Think about Laffer Curve effects. Here are four theories.

  1. The Laffer Curve is bunk. If the government increases income tax from 40 pence in the pound to 83 pence in the pound, that will increase revenue. Revenue will probably double.

  2. Rich businessmen are trapped by their commitments. If their take home pay falls, they won't be willing to give up their yacht or their mistress. They will draw more salary from their business, to maintain their take home pay. Rather than pay themselves $1,600,000 to take home $1,000,000 they will pay themselves $5,882,353 to take home $1,000,000. Tax revenue will rise from $600,000 to $4,882,353 Eight fold, not two fold.

  3. Don't ask where we are on the Laffer Curve, ask when we are. The government is taxing fifty year old businessmen, expecting revenues to hold up indefinitely. But in thirty years time they will all have retired. Will today's twenty year olds replace them? No, once Boxer goes to the knacker's yard, no-one is taking his place.

  4. Laffer Curve effects are prompt. When taxes are low the rich businessman pays his mistress from after tax income. When tax rates soar, he cuts the money that he withdraws from his business as personal income, and preserves his lifestyle by having his company employ his mistress as a secretary. Tax revenue falls.

What would it be like to have a theory of taxation with the accuracy of Newton's Law of Gravity? The very idea is mind boggling. A good philosophy of science would help us construct a scientific theory of tax revenue. A good discussion of the philosophy of science would look at areas of science where we are doing badly and wonder how to do a little better. Perhaps a good discussion of the philosophy of science would also look at successes, such as Newton's Law of Gravity and try to extract lessons, about how to do science, that we could apply to where we are failing. That is very different from looking at Newton's Law of Gravity and worrying about miracles or something.