The facts of life

"Facts are sacred"

Well, no - sacred things are sacred: facts are just facts.

But I think the quote is trying to say that facts are true, which is different from sacred - except in the sense that one can use 'sacred' to mean 'inviolable', or 'unquestionable' - or 'matters of faith".

A short talk by James at Woking's Cafe Scientific:

https://www.facebook.com/events/1559113751055062/

and an animated discussion that preceded it, reminded me of a statement that is commonly made but that I suspect is not so.

In a discussion, one participant may say that something they are asserting is a 'fact' - and that should end the argument, because facts are unquestionable. Except they are not. And I don't mean just in the sense that people sometimes use, where they imply that the person questioning their fact is doing so only in some deluded counter-factual sense.

In fact (sic) I would suggest that facts are inherently questionable, and mutable. In fact (sorry, I can't resist..) we can't get to the facts, we can only get to the observations that lead us to suggest what the facts might be. And worse, in science and mathematics we (or at least some of us) have learnt that facts are just not available - so even the apparent 'facts' that we infer from our observations, turn out to be only mathematical or logical models - simplifications - whose aim it to allow us to predict what the result of other observations will yield. In fact (OK, I'll stop now..) that is all that science, maths and logic do - they let us predict observations, based on models. If our predictions match observation, we continue to use the model - if not, then we discard that model and use another one. No tears, no impassioned arguments (well, OK, scientists are human, so quite a few impassioned arguments, tears and rage) - if the model doesn't work, we throw it out and adopt a new one.

So at the moment we often say we live in a 'post-fact' world - one where facts are dependent on opinion. And we are right, in the sense that if my opinion is fixed firmly enough that I will ignore the observations and stick to my own model, then I am indeed counter-factual - or at least, counter-observational. But here I am not talking about the current political 'elite', who know things that are not so just by being so clever that they don't need to doubt - I am talking about facts as ordinary people - and ordinary scientists - see them. Even our sensible, logical, clear-thinking, scientific facts are questionable, mutable, evanescent - they shift and change and evaporate in the face of new evidence - so they cannot ever have been sacred, can they?

Why do I say this?

First, let's assume that at least in some discussions, opposing parties to the discussion may both be sane and logical, clear thinking, 'scientific' people. So if one advances a 'fact' and the other questions its factuality, and agreement cannot be reached through reasoned and informed debate, then I would say the fact is perhaps not so much as fact as an assertion - perhaps even an opinion.

Second, let's take James's talk. He was talking about Information Warfare and Cyber Security, and he introduced the CIA acronym - which says that information should be Confidential, Available - and crucially should have Integrity: this last meaning, the information that was received should be the same as that which was sent. And Integrity of information, we all know in this 'post-truth' age, is incredibly hard to verify.

But we can verify the integrity of information, if we follow sound practices - mainly, keeping an audit trail. Suppose I make a measurement - say, of the distance between two magnets in a particle accelerator. As it happens, designing a computer-controlled instrument to do just this was one of my first projects as a Fellow at CERN in the 1980s - its outcome, the Distinvar, measures half a metre to an accuracy of 10 microns, as quickly as it can. So I make my measurement and it is passed on through a chain of processing and coding and transmission. The eventual result is perhaps read by some engineer whose job is to adjust the positions of the magnets. That engineer might justifiably ask if the measurement is correct - which means, really, if it matches the actual distance between the magnets. He might ask me - if I am still available - and I might say yes, I was careful, and yes, the measurement is correct. If he accepted that, then he was basing that acceptance on trust - trusting me, perhaps because he knows me, or knows my reputation, or qualification. But before he could do that, he would need to know who made the measurement, so he could know it was me, and so he could then ask me. That is one element in an audit trail.

That type of measurement, which is critical to the functioning of a very expensive system (in fact, it was the LEP, the particle accelerator that preceded the LHC that produced the Higgs Boson) - that type of measurement is in fact accompanied by a rigorous audit trail, that goes like this:

• The Distinvar works by pulling taut a half metre length of Invar wire - a platinum-iridum alloy that has the strange property of not changing in length as it warms up. That piece of wire is a ruler - so the first thing we record is which ruler we used - and who used it, in which Distinvar instrument, when, and where.

• The invar wire is calibrated - checked, made accurate - by comparing it to a better ruler - in fact (sorry, I really can't help it..) a bar of platinum-iridium alloy that is kept as a standard reference in controlled conditions in a tunnel at CERN for that purpose. So the next thing we record is the calibration - when it was done, what was its result, who made it, when and where, against what.

• The CERN platinum-iridium bar is itself calibrated, by being taken to Paris each year, where it is calibrated against the bar of platinum-iridium that is (or was, before we changed to wavelengths of krypton lasers) the standard definition of the standard metre. This is the end of the audit trail - at the defined standard reference.

There will be other entries in the audit trail too - the computer that recorded the measurement, the software processes, and their version numbers, that processed it and transmitted it.

This audit trail is what makes the measurement a measurement. In conversation with a head of metrology at the National Physical Laboratory a few years ago, I was explaining a software processing chain that did not record such an audit trail, and this eminent metrologist had difficulty understanding that anyone could possibly be so daft as to not record an audit trail. (Sadly for him, software engineers don't know squat about metrology so that sort of daft mistake is common as muck.) And he said this:

"If it lacks an audit trail, then it isn't a measurement"

which is insightful - and typical of metrologists.

If you don't know who made the observation, you don't know if you can trust them. If you don't know what they did, you don't know if they did it right. If you don't know what they did it with, you don't know if that was right. This is a chain of trust - a way of establishing the integrity of the observation - the integrity of the information, the I in James's CIA acronym. To establish the integrity of the information you need its audit trail, its chain of trust.

This applies to news too, and to scams, and to hacked emails offered to newspapers. If I read in the Sunday Sport that a statue of Elvis was discovered on Mars (yes, it is true - that the Sunday Sport said it, not that there is a statue of Elvis on Mars) then I could check it by asking who published it (the Sunday Sport, now closed, was not considered by some to be a trustworthy source of news - its famous headline 'Aliens changed our son into a fish finger' saw to that) - and where they got the story from, how that person knew, and so on. If that audit trail is unavailable, or suspiciously short, than we might accord less trust to the information, and doubt its integrity.

So a chain of trust can establish the integrity of the information, insofar as it maintains the same thing at the end of the chain of communication as it did at the beginning. In science this is de rigeur: you always publish your sources, you litter your publications with references to the source that provided that information, you show your working. In fact it gets in the way of reading science, this constant referencing, even though it is essential. And it doesn't establish as much as it perhaps should: the referenced sources might not be as trustworthy as you would hope: the weakest link in the chain determines the strength of the chain. With news, when it is published on the internet, we can sometimes establish a chain of trust: you may trust the news source, or the news source may provide links to their own sources. It isn't obvious you can trust a source even if you think you can: I well remember in the Falklands War the BBC insisting that we (the British) had not lost a capital ship: only to discover after several days that the BBC - bastion of honest truth, as I then thought - had lied, parroting the Government line for propaganda purposes and to stop us (the ignorant populace) from despairing and perhaps demanding a stop to the casualties. So the chain of references - the audit trail - becomes more significant as an indicator of truth. Traditional media don't do audit trails as much as I would like: so you have to search - Googling, or verifying on a site such as Snopes that is dedicated to checking the provenance of stories. (But do I trust Snopes?)

So when the President-elect of the United States says the CIA are making stuff up, and the BBC say they probably aren't, the chain of trust can come down to opinion - to who you do trust.

But the really interesting question is, what is a fact?

In fact, we shouldn't really talk about facts, but about observations, and inferences.

An observation is something that was observed - an inference is something you… infer … to explain your observations.

Let me try to clarify.

If someone sees a saber-tooth tiger enter a cave, hears screams, and after the tiger has left, enters the cave and finds dead cavemen, then those are observations: the saber-tooth tiger was seen to enter the cave, the screams were heard, the dead cavemen were seen in the cave after the tiger left. Those observations could become facts, if I trusted their provenance enough. That chain of observations leads to an obvious conclusion: that the saber-tooth tiger killed the cavemen. That conclusion is an inference: the observations (which can be verified and audit trailed) lead to the inference of what connected, or led to, those observations. But the inference is not a fact: it is a model, an explanation, an inference. If we were doing science, we would put more cavemen in caves and send more saber-tooth tigers into those caves, and then go and see if we found dead cavemen. If we found dead cavemen in all cases, we would feel safe in inferring that saber-tooth tigers kill cavemen - and by further inference, that in the first case, the saber-tooth tiger did indeed kill the cavemen because now we have established that is what saber-tooth tigers do. But this inference is not a fact in the same way as the observations could be: because the inference might be wrong, and I would change my belief in it according to new observations. Imagine for instance that I observe more carefully - I follow the saber-tooth tigers into the caves - and I see that when I do, the cavemen are so freaked at seeing a saber-tooth tiger that they scream so much they die from hyperventilation. Now I have a new inference - that seeing a saber-tooth tiger freaks cavemen out so much that they die of hyperventilation. I am just as happy with my new inference as I was with my old one. Just as happy in fact as physicists were with Einstein's new-fangled Relativity, which replaced Newton's old Mechanics.

So I would say that we need to distinguish between observations - which can at least in principle be checked for integrity - and inferences, which are contingent upon observations. An observation is a measurement, an inference is an explanation for - or strictly speaking a predictor of - measurements.

This of course leads us into the science that most prizes observation - Quantum Mechanics.

Everybody knows about quantum mechanics, and most people know that quantum mechanics is fact - it must be, since no-one understands it, so we have to believe it in and trust it as fact.

Quantum Mechanics is the most proven of sciences - in the sense of the most tested. It predicts observations to an amazing degree. For example the finding of the Higgs Boson at CERN has a statistical significance of one in 3.5 million: whereas in fields such as cancer research a significance of 1 in 20 is considered good. Basically, that means that no measurement ever (no trusted measurement, anyway..) has ever, ever, disagreed with the predictions of Quantum Mechanics.

So one might think that Quantum Mechanics, of all sciences, deals in facts.

Well, yes and no. There is a nice equation in quantum mechanics called the Schroedinger Equation. Yes, Schroedinger of the Cat. It used to be called Schroedinger's Wave Equation, but then people started thinking everything was waves and so people left out the wave bit - which is significant. Schroedinger's Equation describes the probability of an observation yielding a particular result. That is, it predicts observations. Actually, that is all it does. Schroedninger's Equation doesn't in itself say anything - it does not infer any fact, any basis for what it predicts - it just predicts.

Which is interesting - because there is a basis, Jim, but not as we know it.

When Quantum Mechanics was first proposed, furious arguments broke out. Einstein - a trusted source if ever there was one - ridiculed it, even though he played his part in inventing it, by saying he 'refused to believe that God plays dice'. That statement in itself is intriguing - containing the word 'belief' - opinion, you might say. Those arguments turned on the oddity of Quantum Mechanics - an uncertainty intrinsic to it, encapsulated in Heisenberg's Principle of Uncertainty. The Principle of Uncertainty basically says that you can't ever know everything about a system - because to know about it you have to measure it, and to measure it you have to perturb  it - to change it. That uncertainty is intrinsic - inevitable - it arises from fundamental properties of the formulae of Quantum Mechanics. And the argument was, whether the fact that you could not know everything, in fact meant that there was nothing to know: if what you could not measure in fact existed at all, or - being unknowable - could not be considered to be real in any way. Einstein argued no: that there must be an underlying reality - unknowable, yes, but existing and following its own rules, if only we could know them. Niels Bohr - the Father of Quantum Mechanics - on the other hand argued that those things that could not be observed, that could not be measured, could not be considered as being real: could not be considered as facts. Einstein and Bohr, and many others - including a grand total of 19 scientists who either then or later won Nobel Prizes - argued this out at Copenhagen, and arrived at the Copenhagen Interpretation: agreeing ultimately with Niels Bohr, that what you can't observe, cannot be considered real.

It is this aspect of Quantum Mechanics that non Quantum Mechanicians find hardest: just because you can't see something doesn't mean it is isn't there, does it? Well, no and yes. If a tree falls in the forest, and no-one sees it fall, did the tree fall? Answer: don't know. If you looked later and saw the tree fallen you could say it fell: or maybe it was pushed? You could draw up a rule of observation: that if you see a tree standing in the forest, and later you see the same tree fallen, then the tree fell. But if someone showed you that what in fact happens is that trees sort of flip between standing and fallen, without actually falling, then you would have to give up your 'tree falling' inference and accept another, weirder, explanation.

But even this isn't my point. (I will get to my point, in the end, with luck..).

Schroedinger's Equation is an inverse equation. It is an equation that encapsulates the probability that an observation will yield a particular result, but the equation does not directly yield the result: you have to invert it - to 'solve' it. And like many such inverse equations, Schroedinger's lacks a solution.

Think of an inverse equation as something like:

distance = speed * time

(which is an equation of motion).

You can invert this to find the time as a function of speed and distance:

time = distance / speed

But you can only do this because we already know the solution: someone, in ancient times, discovered the solution to this inverse equation and it was passed down to us through generations, so we know it and can use it. But most equations aren't like that: they don't have simple known inverse solutions: we can't solve them. Schroedinger's Equation is like that: it hasn't been solved.

Well, that is not quite true. Schroedinger's Equation has been solved for a very few, very simple, cases: like the very simple system of a hydrogen atom, or a hydrogen nucleus.

And if we solved an equation for some simple cases, then we can use a trick: known as a Transformation. A transform transforms a problem into a different problem. Specifically, if we know a number of simple solutions to an inverse equation, we can transform a more complex, unknown problem into a sum of those simpler problems: and so its solution becomes a sum of the solutions to those simpler solutions. This is the magic of a transform: it is the basis (a pun, as we will soon see..) for much of western science. It is sometimes called 'the whole is the sum of its parts' - and the matching phrase 'the whole is more than the sum of its parts' shows it weakness, too. It is an approximation: but often a good one. What we do is, to break down a very complex problem into a sum of lots of simple ones: then we solve each of those simple problems, add those solutions together, and hope the result is the solution to the complex problem (it isn't, but it is often so close as to be satisfying).

The simple problems are called the 'basis' functions. We break down the complex problem into a sum - a 'linear superposition' - of basis functions, solve those, and say the sum of the solutions is the solution to the complex problem.

Neat, huh?

Except that the our desire for facts leads us to think the basis functions are real.

Take what are called 'atomic orbitals'. These are the 'clouds' within which electrons orbit the nuclei of atoms. They are lovely - beautiful spheres, figure-eights - and they define the shape of molecules (which are assemblages of atoms, linked by their atomic orbitals). Each orbital contains a pair of electrons (electrons always go round in pairs, but that is another story…). Except that they don't. Electrons don't follow the atomic orbitals at all: the orbitals are just basis functions - simplifications, mathematical abstractions to help us to solve a more complex problem. The electrons don't follow orbits - but even more, they don't even follow the orbitals as probability distributions. The orbitals are basis functions - actually, the solutions to Schroedinger's Equation for the simplest of systems, a hydrogen atom. In fact the electron's probability distribution for any mildly complex atomic system is incredibly complex - a weird Spirograph cloud - but it can be approximated reasonably (we think) as the sum of the solutions to the simple hydrogen atom case.

This wouldn't matter - in a way it is a philosophical question, whether the electron is in the orbitals or not, because its behaviour can be predicted using them. But text books are full of such cute pictures of these orbitals. You can buy plastic models of them, with which you can build models of atoms and molecules. People talk of them as if they are real. But the Copenhagen Interpretation specifically, clearly, says they are not: the electron state can be modelled by a linear superposition of basis functions, but that does not mean the basis functions are real.

How easy, how seductive, how compelling, is the slippery slide from inference to fact: from model to reality.

One last example to illustrate my point.

We all learn at school how naïve were the ancient astronomers, who thought the sun and all the planets went round the Earth. We learnt to smile at poor old Ptolemy, who modelled the observed planetary motions with the crystal spheres - and then, because the planets move in strange ways, had to add spheres within spheres, crystal spheres like cogs meshing, circles on circles, in ever-increasing complexity, to model the motions. They thought of the circles within circles as showing the harmony of the universe - they talked of the harmony of the crystal spheres. How simplistic, to model planetary motions with perfect circles.

It is one of the best known facts we learn: that the Earth goes round the Sun, not the Sun round the Earth. In fact, knowing this fact is almost a badge of being Scientific: of living in the age of the triumph of fact over belief. If anyone suggested otherwise we would think them ignorant or possibly deluded. The Earth going round the Sun is a central fact, it anchors our understanding of our solar system, we all know it to be a fact.

There is a funny story, about a man whose son asks him why the ancients thought the Sun went round the Earth. The father says: "because it looked like it did" - and the son, after a pause, asks innocently: "what would it have looked like if the Earth had gone round the Sun". To which of course the answer is: "the same". Silly Ancients, thinking otherwise...

But the same applies the other way. It did indeed look like the Sun went round the Earth - it still does.

The difference is, you need less circles within circles if you make the Sun the centre of the circles. The model isn't better, except in this sense that it is simpler - less circles. If you let the circles be ellipses it gets even simpler - less ellipses than circles - so we now choose to model with ellipsoidal functions (which are, by the way, a damn sight more awkward mathematically than circles)

There is a transformation called the Fourier Transform, that is very popular. It models all sorts of problems as sums of circular functions - harmonics. People use the Fourier Transform to model the incredible complexity of speech, of music, of images and moving pictures and of radar and sonar and medical imaging. The Fourier Transform is one of the most widely used transforms, used in almost every consumer device in all sorts of ways. All of these uses of the Fourier Transform user harmonic functions - circular functions - as basis functions to model the complex behaviours. In speech and music we even talk of the 'harmonics' - the circles within circles that make up the richness and harmoniousness of those sounds.

You can predict the observed motions of the planets by a Fourier Transform - the circles within circles are the harmonic functions of the transform. It works perfectly well: easily as well as the ellipse model - you can't tell the difference. So then, are the Ptolemaic spheres like the atomic orbitals - real things? Or are they like the atomic orbitals - abstractions whose only purpose is to make a simple mathematical model that predicts observations?

I think, and here I am following the Copenhagen Interpretation, that the problem lies in saying what is a fact. An observation - a measurement - can be a fact. It can be verified, checked, audited. Observations of planetary motions are facts in this sense. An inference Is not a fact. Ellipsoidal, or circular, orbits of planets are inferences, so in the same sense they are not facts.

So we do need to be sure of our facts: but we also need to be sure what a fact is.