Recent trends have begun to drive a confluence between Mathematics and aspects of the data arena, notably to do with Data Science and Artificial Intelligence approaches like Machine Learning. For this reason, I will periodically share selected articles from the Maths & Science Section here on the main site. I hope that they are of interest to at least some regular readers. I’ll kick this off with three, interconnecting articles about one of the most important numbers in Mathematics, , one of the most misunderstood, , and finally their beautiful relationship to each other .
The number π is surrounded by a fog of misunderstanding and even mysticism. This article seeks to address some common misconceptions about π, to show that in many ways it is just like any other number, but also to demonstrate some of its less common properties.
I have some form when it comes to getting irritated by quasi-mathematical social media memes (see Facebook squares “puzzle” for example). Facebook, which I find myself using less and less frequently these days, has always been plagued by clickbait articles. Some of these can be rather unsavoury. One that does not have this particular issue, but which more than makes up for this in terms of general annoyance, is the many variants of:
Only a math[s] genius can solve [insert some dumb problem here] – can u?
Life is too short to complain about Facebook content, but this particular virus now seems to have infected LinkedIn (aka MicrosoftedIn) as well. Indeed as LinkedIn’s current “strategy” seems to be to ape what Facebook was doing a few years ago, perhaps this is not too surprising. Nevertheless, back in the day, LinkedIn used to be a reasonably serious site dedicated to networking and exchanging points of view with fellow professionals.
Those days appear to be fading fast, something I find sad. It seems that a number of people agree with me as – at the time of writing – over 9,000 people have viewed a LinkedIn article I briefly penned bemoaning this development. While some of the focus inevitably turned to general scorn being heaped on the new LinekdIn user experience (UX), it seemed that most people are of the same opinion as I am.
However, I suspect that there is little to be done and the folks at LinkedIn probably have their hands full trying to figure out how to address their UX catastrophe. Given this, I thought that if you can’t beat them, join them. So above appears my very own Mathematical meme, maybe it will catch on.
It should be noted that in this case “Less than 1% can do it!!!” is true, in the strictest sense. Unlike the original meme, so is the first piece of text!
After 100s of views on my blog, 1,000s of views on LinkedIn and 10,000s of views on Twitter, it took Neil Raden (@NeilRaden) to point out that in the original image I had the sum running from n=0 as opposed to n=1. The former makes no sense whatsoever. I guess his company is called Hired Brains for a reason! This was meant to be a humorous post, but at least part of the joke is now on me.
Note: In the following I have used the abridgement Maths when referring to Mathematics, I appreciate that this may be jarring to US readers, omitting the ‘s’ is jarring to me, so please accept my apologies in advance.
Regular readers of this blog will be aware of my penchant for analogies. Dominant amongst these have been sporting ones, which have formed a major part of articles such as:
I have also used other types of analogy from time to time, notably scientific ones such as in the middle sections of Recipes for Success?, or A Single Version of the Truth? – I was clearly feeling quizzical when I wrote both of those pieces! Sometimes these analogies have been buried in illustrations rather than the text as in:
Jim Harris (@ocdqblog) frequently employs analogies on his excellent Obsessive Compulsive Data Quality blog. If there is a way to form a title “The X of Data Quality”, and relate this in a meaningful way back to his area of expertise, Jim’s creative brain will find it. So it is encouraging to feel that I am not alone in adopting this approach. Indeed I see analogies employed increasingly frequently in business and technology blogs, to say nothing of in day-to-day business life.
However, recently two things have given me pause for thought. The first was the edition of Randall Munroe’s highly addictive webcomic, xkcd.com, that appeared on 6th May 2011, entitled “Teaching Physics”. The second was a blog article I read which likened a highly abstract research topic in one branch of Theoretical Physics to what BI practitioners do in their day job.
An homage to xkcd.com
Let’s consider xkcd.com first. Anyone who finds some nuggets of interest in the type of – generally rather oblique – references to matters Mathematical or Scientific that I mention above is likely to fall in love with xkcd.com. Indeed anyone who did a numerate degree, works in a technical role, or is simply interested in Mathematics, Science or Engineering would as well – as Randall says in a footnote:
“this comic occasionally contains […] advanced mathematics (which may be unsuitable for liberal-arts majors)”
Although Randall’s main aim is to entertain – something he manages to excel at – his posts can also be thought-provoking, bitter-sweet and even resonate with quite profound experiences and emotions. Who would have thought that some stick figures could achieve all that? It is perhaps indicative of the range of topics dealt with on xkcd.com that I have used it to illustrate no fewer than seven of my articles (including this one, a full list appears at the end of the article). It is encouraging that Randall’s team of corporate lawyers has generally viewed my requests to republish his work favourably.
The example of Randall’s work that I wanted to focus on is as follows.
It is worth noting that often the funniest / most challenging xkcd.com observations appear in the mouse-over text of comic strips (alt or title text for any HTML heads out there – assuming that there are any of us left). I’ll reproduce this below as it is pertinent to the discussion:
Space-time is like some simple and familiar system which is both intuitively understandable and precisely analogous, and if I were Richard Feynman I’d be able to come up with it.
If anyone needs some background on the science referred to then have a skim of this article if you need some background on the scientist mentioned (who has also made an appearance on peterjamesthomas.com in Presenting in Public) then glance through this second one.
Here comes the Science…
Randall points out the dangers of over-extending an analogy. While it has always helped me to employ the rubber-sheet analogy of warped space-time when thinking about the area, it is rather tough (for most people) to extrapolate a 2D surface being warped to a 4D hyperspace experiencing the same thing. As an erstwhile Mathematician, I find it easy enough to cope with the following generalisation:
The set of all points defined by one variable (x1)
– i.e. a straight line
The set of all points defined by two variables (x1, x2)
– i.e. a plane
The set of all points defined by three variables (x1, x2, x3)
– i.e. “normal” 3-space
The set of all points defined by four variables (x1, x2, x3, x4)
– i.e. 4-space
” ” ” “
The set of all points defined by n variables (x1, x2, … , xn)
– i.e. n-space
As we increase the dimensions, the Maths continues to work and you can do calculations in n-space (e.g. to determine the distance between two points) just as easily (OK with some more arithmetic) as in 3-space; Pythagoras still holds true. However, actually visualising say 7-space might be rather taxing for even a Field’s Medallist or Nobel-winning Physicist.
… and the Maths
More importantly while you can – for example – use 3-space as an analogue for some aspects of 4-space, there are also major differences. To pick on just one area, some pieces of string that are irretrievably knotted in 3-space can be untangled with ease in 4-space.
To briefly reference a probably familiar example, starting with 2-space we can look at what is clearly a family of related objects:
A square has 4 vertexes, 4 edges joining them and 4 “faces” (each consisting of a line – so the same as edges in this case)
A cube has 8 vertexes, 12 edges and 6 “faces” (each consisting of a square)
A tesseract (or 4-hypercube) has 16 vertexes, 32 edges and 8 “faces” (each consisting of a cube)
Note: The reason that faces appears in inverted commas is that the physical meaning changes, only in 3-space does this have the normal connotation of a surface with two dimensions. Instead of faces, one would normally talk about the bounding cubes of a tesseract forming its cells.
Even without any particular insight into multidimensional geometry, it is not hard to see from the way that the numbers stack up that:
An n-hypercube has 2n vertexes, 2n-1n edges and 2n “faces” (each consisting of an (n-1)-hypercube)
Again, while the Maths is compelling, it is pretty hard to visualise a tesseract. If you think that a drawing of a cube, is an attempt to render a 3D object on a 2D surface, then a picture of a tesseract would be a projection of a projection. The French (with a proud history of Mathematics) came up with a solution, just do one projection by building a 3D “picture” of a tesseract.
As aside it could be noted that the above photograph is of course a 2D projection of a 3D building, which is in turn a projection of a 4D shape; however recursion can sometimes be pushed too far!
Drawing multidimensional objects in 2D, or even building them in 3D, is perhaps a bit like employing an analogy (this sentence being of course a meta-analogy). You may get some shadowy sense of what the true object is like in n-space, but the projection can also mask essential features, or even mislead. For some things, this shadowy sense may be more than good enough and even allow you to better understand the more complex reality. However, a 2D projection will not be good enough (indeed cannot be good enough) to help you understand all properties of the 3D, let alone the 4D. Hopefully, I have used one element of the very subject matter that Randall raises in his webcomic to further bolster what I believe are a few of the general points that he is making, namely:
Analogies only work to a degree and you over-extend them at your peril
Sometimes the wholly understandable desire to make a complex subject accessible by comparing it to something simpler can confuse rather than illuminate
There are subject areas that very manfully resist any attempts to approach them in a manner other than doing the hard yards – not everything is like something less complex
Why BI is not [always] like Theoretical Physics
Having hopefully supported these points, I’ll move on to the second thing that I mentioned reading; a BI-related blog also referencing Theoretical Physics. I am not going to name the author, mention where I read their piece, state what the title was, or even cite the precise area of Physics they referred to. If you are really that interested, I’m sure that the nice people at Google can help to assuage your curiosity. With that out of the way, what were the concerns that reading this piece raised in my mind?
Well first of all, from the above discussion (and indeed the general tone of this blog), you might think that such an article would be right up my street. Sadly I came away feeling that the connection made was, tenuous at best, rather unhelpful (it didn’t really tell you anything about Business Intelligence) and also exhibited a lack of anything bar a superficial understanding of the scientific theory involved.
The analogy had been drawn based on a single word which is used in both some emerging (but as yet unvalidated) hypotheses in Theoretical Physics and in Business Intelligence. While, just like the 2D projection of a 4D shape, there are some elements in common between the two, there are some fundamental differences. This is a general problem in Science and Mathematics, everyday words are used because they have some connection with the concept in hand, but this does not always imply as close a relationship as the casual reader might infer. Some examples:
In Pure Mathematics, the members of a group may be associative, but this doesn’t mean that they tend to hang out together.
In Particle Physics, an object may have spin, but this does not mean that it has been bowled by Murali
In Structural Biology, a residue is not precisely what a Chemist might mean by one, let alone a lay-person
Part of the blame for what was, in my opinion, an erroneous connection between things that are not actually that similar lies with something that, in general, I view more positively; the popular science book. The author of the BI/Physics blog post referred to just such a tome in making his argument. I have consumed many of these books myself and I find them an interesting window into areas in which I do not have a background. The danger with them lies when – in an attempt to convey meaning that is only truly embodied (if that is the word) in Mathematical equations – our good friend the analogy is employed again. When done well, this can be very powerful and provide real insight for the non-expert reader (often the writers of pop-science books are better at this kind of thing than the scientists themselves). When done less well, this can do more than fail to illuminate, it can confuse, or even in some circumstances leave people with the wrong impression.
During my MSc, I spent a year studying the Riemann Hypothesis and the myriad of results that are built on the (unproven) assumption that it is true. Before this I had spent three years obtaining a Mathematics BSc. Before this I had taken two Maths A-levels (national exams taken in the UK during and at the end of what would equate to High School in the US), plus (less relevantly perhaps) Physics and Chemistry. One way or another I had been studying Maths for probably 15 plus years before I encountered this most famous and important of ideas.
So what is the Riemann Hypotheis? A statement of it is as follows:
The real part of all non-trivial zeros of the Riemann Zeta function is equal to one half
There! Are you any the wiser? If I wanted to explain this statement to those who have not studied Pure Mathematics at a graduate level, how would I go about it? Maybe my abilities to think laterally and be creative are not well-developed, but I struggle to think of an easily accessible way to rephrase the proposal. I could say something gnomic such as, “it is to do with the distribution of prime numbers” (while trying to avoid the heresy of adding that prime numbers are important because of cryptography – I believe that they are important because they are prime numbers!).
I spent a humble year studying this area, after years of preparation. Some of the finest Mathematical minds of the last century (sadly not a set of which I am a member) have spent vast chunks of their careers trying to inch towards a proof. The Riemann Hypothesis is not like something from normal experience; it is complicated. Some things are complicated and not easily susceptible to analogy.
Equally – despite how interesting, stimulating, rewarding and even important Business Intelligence can be – it is not Theoretical Physics and n’er the twain shall meet.
And so what?
So after this typically elliptical journey through various parts of Science and Mathematics, what have I learnt? Mainly that analogies must be treated with care and not over-extended lest they collapse in a heap. Will I therefore stop filling these pages with BI-related analogies, both textual and visual? Probably not, but maybe I’ll think twice before hitting the publish key in future!
Chronological list of articles using xkcd.com illustrations:
A lot of human scientific and technological progress over the span of recorded history has been related to discerning patterns. People noticed that the Sun and Moon both had regular periodicity to their movements, leading to models that ultimately changed our view of our place in the Universe. The apparently wandering trails swept out by the planets were later regularised by the work of Johannes Kepler and Tycho Brahe; an outstanding example of a simple idea explaining more complex observations.
In general Mathematics has provided a framework for understanding the world around us; perhaps most elegantly (at least in work that is generally accessible to the non-professional) in Newton’s Laws of Motion (which explained why Kepler and Brahe’s models for planetary movement worked). The simple formulae employed by Newton seemed to offer a precise set of rules governing everything from the trajectory of an arrow to the orbits of the planets and indeed galaxies; a triumph for the application of Mathematics to the natural world and surely one of humankind’s greatest achievements.
For centuries it appeared that natural phenomena seemed to have simple principles underlying them, which were susceptible to description in the language of Mathematics. Sometimes (actually much more often than you might think) the Mathematics became complicated and precision was dropped in favour of – generally more than good enough – estimation; but philosophically Mathematics and the nature of things appeared to be inextricably interlinked. The Physicist and Nobel Laureate E.P. Wigner put this rather more eloquently:
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
In my youth I studied Group Theory, a branch of mathematics concerned with patterns and symmetry. The historical roots (no pun intended) of Group Theory are in the solvability of polynomial equations, but the relation with symmetry emerged over time; revealing an important linkage between geometry and algebra. While Group Theory is a part of Pure Mathematics (supposedly studied for its own intrinsic worth, rather than any real-world applications), its applications are actually manifold. Just one example is that groups lie (again no pun intended) at the heart of the Standard Model of Particle Physics.
However, two major challenges to this happy symbiosis between Mathematics and the Natural Sciences arose. One was an abrupt earthquake caused by Kurt Gödel in 1931. The other was more of a slowly rising flood, beginning in the 1880s with Henri Poincaré and (arguably) culminating with Ruelle, May and Yorke in 1977 (though with many other notables contributing both before and after 1977). The linkage between Mathematics and Science persists, but maybe some of the chains that form it have been weakened.
Potentially fallacious patterns
However, rather than this article becoming a dissertation on incompleteness theorems or (the rather misleadingly named) chaos theory, I wanted to return to something more visceral that probably underpins at least the beginnings of the long association of Mathematics and Science. Here I refer to people’s general view that things tend to behave the same way as they have in the past. As mentioned at the beginning of this article, the sun comes up each morning, the moon waxes and wanes each month, summer becomes autumn (fall) becomes winter becomes spring and so on. When you knock your coffee cup over it reliably falls to the ground and the contents spill everywhere. These observations about genuine patterns have served us well over the centuries.
It seems a very common human trait to look for patterns. Given the ubiquity of this, it is likely to have had some evolutionary benefit. Indeed patterns are often there and are often useful – there is indeed normally more traffic on the roads at 5pm on Fridays than on other days of the week. Government spending does (with the possible exception of current circumstances) generally go up in advance of an election. However such patterns may be less useful in other areas. While winter is generally colder than summer (in the Northern hemisphere), the average temperature and average rainfall in any given month varies a lot year-on-year. Nevertheless, even within this variability, we try to discern patterns to changes that occur in the weather.
We may come to the conclusion that winters are less severe than when we were younger and thus impute a trend in gradually moderating winters; perhaps punctuated by some years that don’t fit what we assume is an underlying curve. We may take rolling averages to try to iron out local “noise” in various phenomena such as stock prices. This technique relies on the assumption that things change gradually. If the average July temperature has increased by 2°C in the last 100 years, then it maybe makes sense to assume that it will increase by the same 2°C ±0.2°C in the next 100 years. Some of the work I described earlier has rigorously proved that a lot of these human precepts are untrue in many important fields, not least weather prediction. The phrase long-term forecast has been 100% shown to be an oxymoron. Many systems – even the simplest, even those which are apparently stable – can change rapidly and unpredictably and weather is one of them.
For the avoidance of doubt I am not leaping into the general Climate Change debate here – except in the most general sense. Instead I am highlighting the often erroneous human tendency to believe that when things change they do so smoothly and predictably. That when a pattern shifts, it does so to something quite like the previous pattern. While this assumed smoothness is at the foundation of many of our most powerful models and techniques (for example the grand edifice of The Calculus), in many circumstances it is not a good fit for the choppiness seen in nature.
Obligatory topical section on volcanoes
The above observations about the occasionally illusory nature of patterns lead us to more current matters. I was recently reading an article about the Eyjafjallajokull eruption in The Economist. This is suffused with a search for patterns in the history of volcanic eruptions. Here are just a few examples:
Last time Eyjafjallajokull erupted, from late 1821 to early 1823, it also had quite viscous lava. But that does not mean it produced fine ash continuously all the time. The activity settled into a pattern of flaring up every now and then before dying back down to a grumble. If this eruption continues for a similar length of time, it would seem fair to expect something similar.
Previous eruptions of Eyjafjallajokull seem to have acted as harbingers of a subsequent Katla [a nearby volcano] eruptions.
[However] Only two or three […] of the 23 eruptions of Katla over historical times (which in Iceland means the past 1,200 years or so) have been preceded by eruptions of Eyjafjallajokull.
Katla does seem to erupt on a semi-regular basis, with typical periods between eruptions of between 30 and 80 years. The last eruption was in 1918, which makes the next overdue.
To be fair, The Economist did lace their piece with various caveats, for example the above-quoted “it would seem fair to expect”, but not all publications are so scrupulous. There is perhaps something comforting in all this numerology, maybe it gives us the illusion that we can make meaningful predictions about what a volcano will do next. Modern geologists have used a number of techniques to warn of imminent eruptions and these approaches have been successful and saved lives. However this is not the same thing as predicting that an eruption is likely in the next ten years solely because they normally occur every century and it is 90 years since the last one. Long-term forecasts of volcanic activity are as chimerical as long-term weather forecasts.
A little light analysis
Looking at another famous volcano, Vesuvius, I have put together the following simple chart.
The average period between eruptions is just shy of 14 years, but the pattern is anything but regular. If we expand our range a bit, we might ask how many eruptions occurred between 10 and 20 years after the previous one. The answer is just 9 of the 26, or about 35%. Even if we expand our range to periods of calm lasting between 5 and 25 years (so 10 years of leeway on either side), we only capture 77% of eruptions. The standard deviation of the periods between recorded eruptions is a whopping 12.5; eruptions of Vesuvius are not regular events.
One aspect of truly random distributions at first seems counterfactual, this is their lumpiness. It might seem reasonable to assume that a random set of events would lead to a nicely spaced out distribution; maybe not a set of evenly-spaced points, but a close approximation to one. In fact the opposite is generally true; random distributions will have clusters of events close to each other and large gaps between them.
The above exhibit (a non-wrapped version of which may be viewed by clicking on it) illustrates this point. It compares a set of pseudo-random numbers (the upper points) with a set of truly random numbers (the lower points). There are some gaps in the upper distribution, but none are large and the spread is pretty even. By contrast in the lower set there are many large gaps (some of the more major ones being tagged a, … ,h) and significant clumping. Which of these two distributions more closely matches the eruptions of Vesuvius? What does this tell us about the predictability of its eruptions?
The predictive analytics angle
As always in closing I will bring these discussions back to a business focus. The above observations should give people involved in applying statistical techniques to make predictions about the future some pause for thought. Here I am not targeting the professional statistician; I assume such people will be more than aware of potential pitfalls and possess much greater depth of knowledge than myself about how to avoid them. However many users of numbers will not have this background and we are all genetically programmed to seek patterns, even where none may exist. Predictive analytics is a very useful tool when applied correctly and when its findings are presented as a potential range of outcomes, complete with associated probabilities. Unfortunately this is not always the case.
It is worth noting that many business events can be just as unpredictable as volcanic eruptions. Trying to foresee the future with too much precision is going to lead to disappointment; to say nothing of being engulfed by lava flows.
The solvability of polynomials is of course equivalent to whether or not roots of them exist.
Lie groups lie at the heart of quantum field theory – a interesting lexicographical symmetry in itself
Indeed it has been argued that non-linear systems are more robust in response to external stimuli than classical ones. The latter tend to respond to “jolts” in a smooth manner leading to a change in state. The former often will revert to their previous strange attractor. It has been postulated that evolution has taken advantage of this fact in demonstrably chaotic systems such as the human heart.
Here I include the – to date – 66 years since Vesuvius’ last eruption in 1944 and exclude the eruption in 1631 as there is no record of the preceding one.
For anyone interested, the upper set of numbers were generated using Excel’s RAND() function and the lower are successive triplets of the decimal expansion of pi, e.g. 141, 592, 653 etc.
Again for those interested the average gap in the upper set is 10.1 with a standard deviation of 4.3; the figures for the lower set are 9.7 and 9.6 respectively.
By way of [very necessary] explanation, this post is a response to an idea started on the blog of Curt Monash (@CurtMonash), doyen of software industry analysts. You can read the full article here. This is intended as an early April Fools celebration.
[…] the Rules of the No-Fooling Meme are:
Rule 1: Post on your blog 1 or more surprisingly true things about you,* plus their explanations. I’m starting off with 10, but it’s OK to be a lot less wordy than I’m being. I suggest the following format:
A noteworthy capsule sentence. (Example: “I was not of mortal woman born.”)
A perfectly reasonable explanation. (Example: “I was untimely ripped from my mother’s womb. In modern parlance, she had a C-section.”)
Rule 2: Link back to this post. That explains what you’re doing.
Rule 3: Drop a link to your post into the comment thread. That will let people who check here know that you’ve contributed too.
Rule 4: Ping 1 or more other people encouraging them to join in the meme with posts of their own.
*If you want to relax the “about you” part, that’s fine too.
I won’t be as dramatic as Curt, nor will I drop any names (they have been changed to protect the guilty). I also think that my list is closer to a “things you didn’t know about me” than Curt’s original intention, but hopefully it is in the spirit of his original post. I have relaxed the “about me” part for one fact as well, but claim extenuating circumstances.
My “no-fooling” facts are, in (broadly) reverse chronological order:
I have recently corrected a Physics paper in Science – and please bear in mind that I was a Mathematician not a Physicist; I’m not linking to the paper as the error was Science’s fault not the scientists’ and the lead author was very nice about it.
My partner is shortly going to be working with one of last year’s Nobel Laureates at one of the world’s premier research institues – I’m proud, so sue me!
My partner, my eldest son and I have all attended (or are attending) the same University – though separated by over 20 years.
The same University awarded me 120% in my MSc. Number Theory exam – the irony of this appeals to me to this day; I was taught Number Theory by a Fields Medalist; by way of contrast, I got a gamma minus in second year Applied Mathematics.
Not only did I used to own a fan-site for a computer game character, I co-administered a universal bulletin board (yes I am that old) dedicated to the same character – even more amazingly, there were female members!
As far as I can tell, my code is still part of the core of software that is used rather widely in the UK and elsewhere – though I suspect that a high percentage of it has succumbed to evolutionary pressures.
I have recorded an eagle playing golf – despite not being very good at it and not playing at all now.
I have played cricket against the national teams of both Zimbabwe (in less traumatic times) and the Netherlands – Under 15s and Under 19s respectively; I have also played both with and against an England cricketer and against a West Indies cricketer (who also got me out), but I said that I wasn’t going to name drop.
[Unlike Curt] I only competed in one chess tournament – I came fourth, but only after being threatened with expulsion over an argument to do with whether I had let go of a bishop for a nanosecond; I think I was 11 at the time.
At least allegedly, one of my antecedents was one of the last hangmen in England – I’m not sure how you would go about substantiating this fact as they were meant to be sworn to secrecy; equally I’m not sure that I would want to substantiate it.
And a bonus fact (which could also be seen as oneupmanship vis à vis Curt):
One of the articles that I wrote for the UK climbing press has had substantially more unique views than any of my business-related articles on here (save for the home page itself) – sad, but true, if you don’t believe me, the proof is here.
Standard note: You need to be a member of both LinkedIn.com and the group mentioned to view the discussions.
Here are a couple of sections from the original poster’s starting comments:
I’ve been thinking: is one version of the truth attainable or is it a bit of snake oil? Is it a helpful concept that powerfully communicates a way out of spreadmart purgatory? Or does the idea of one version of the truth gloss over the fact that context or point of view are an inherent part of any statement about data, which effectively makes truth relative? I’m leaning toward the latter position.
There can only be one version of the truth if everyone speaks the same language and has a common point of view. I’m not sure this is attainable. To the extent that it is, it’s definitely not a technology exercise. It’s organizational change management. It’s about changing the culture of an organization and potentially breaking down longstanding barriers.
Please join the group if you would like to read the whole post and the subsequent discussions, which were very lively. Here I am only going to refer to these tangentially and instead focus on the concept of a single version of the truth itself.
Readers who are not interested in the ellipitcal section of this article and who would instead like to cut to the chase are invited to click here (warning there are still some ellipses in the latter sections).
A [very] brief and occasionally accurate history of truth
I have discovered a truly marvellous proof of the nature of truth, which this column is too narrow to contain.
— Pierre de Tomas (1637)
Instead of trying to rediscover M. Tomas’ proof, I’ll simply catalogue some of the disciplines that have been associated (rightly or wrongly) with trying to grapple with the area:
Various branches of Philosophy, including:
Religion (or more perhaps more generally spirituality)
and of course Polygraphism
Given my background in Pure Mathematics the reader might expect me to trumpet the claims of this discipline to be the sole arbiter of truth; I would reply yes and no. Mathematics does indeed deal in absolute truth, but only of the type: if we assume A and B, it then follows that C is true. This is known as the axiomatic approach. Mathematics makes no claim for the veracity of axioms themselves (though clearly many axioms would be regarded as self-evidently true to the non-professional). I will also manfully resist the temptation to refer to the wrecking ball that Kurt Gödel’s took to axiomatic systems in 1931.
I have also made reference (admittedly often rather obliquely) to various branches of science on this blog, so perhaps this is another place to search for truth. However the Physical sciences do not really deal in anything as absolute as truth. Instead they develop models that approximate observations, these are called scientific theories. A good theory will both explain aspects of currently observed phenomena and offer predictions for yet-to-be-observed behaviour (what use is a model if it doesn’t tell us things that we don’t already know?). In this way scientific theories are rather like Business Analytics.
Unlike mathematical theories, the scientific versions are rather resistant to proof. Somewhat unfairly, while a mountain of experiments that are consistent with a scientific theory do not prove it, it takes only one incompatible data point to disprove it. When such an inconvenient fact rears its head, the theory will need to be revised to accommodate the new data, or entirely discarded and replaced by a new theory. This is of course an iterative process and precisely how our scientific learning increases. Warning bells generally start to ring when a scientist starts to talk about their theory being true, as opposed to a useful tool. The same observation could be made of those who begin to view their Business Analytics models as being true, but that is perhaps a story for another time.
I am going to come back to Physical science (or more specifically Physics) a little later, but for now let’s agree that this area is not going to result in defining truth either. Some people would argue that truth is the preserve of one of the other subjects listed above, either Philosophy or Religion. I’m not going to get into a debate on the merits of either of these views, but I will state that perhaps the latter is more concerned with personal truth than supra-individual truth (otherwise why do so many religious people disagree with each other?).
Discussing religion on a blog is also a certain way to start a fire, so I’ll move quickly on. I’m a little more relaxed about criticising some aspects of Philosophy; to me this can all too easily descend into solipism (sometimes even quicker than artificial intelligence and cognitive science do). Although Philosophy could be described as the search for truth, I’m not convinced that this is the same as finding it. Maybe truth itself doesn’t really exist, so attempting to create a single version of it is doomed to failure. However, perhaps there is hope.
Trusting your GUT feeling
After the preceding divertimento, it is time to return to the more prosaic world of Business Intelligence. However there is first room for the promised reference to Physics. For me, the phrase “a single version of the truth” always has echoes of the search for a Grand Unified Theory (GUT). Analogous to our discussions about truth, there are some (minor) definitional issues with GUT as well.
Some hold that GUT applies to a unification of the electromagnetic, weak nuclear and strong nuclear forces at very high energy levels (the first two having already been paired in the electroweak force). Others that GUT refers to a merging of the particles and forces covered by the Standard Model of Quantum Mechanics (which works well for the very small) with General Relativity (which works well for the very big). People in the first camp might refer to this second unification as a ToE (Theory of Everything), but there is sometimes a limit to how much Douglas Adams’ esteemed work applies to reality.
For the purposes of this article, I’ll perform the standard scientific trick of a simplifying assumption and use GUT in the grander sense of the term.
Scientists have striven to find a GUT for decades, if not centuries, and several candidates have been proposed. GUT has proved to be something of a Holy Grail for Physicists. Work in this area, while not as yet having been successful (at least at the time of writing), has undeniably helped to shed a light on many other areas where our understanding was previously rather dim.
This is where the connection with a single version of the truth comes in. Not so much that either concept is guaranteed to be achievable, but that a lot of good and useful things can be accomplished on a journey towards both of them. If, in a given organisation, the journey to a single version of the truth reaches its ultimate destination, then great. However if, in an another company, a single version of the truth remains eternally just over the next hill, or round the next corner, then this is hardly disastrous and maybe it is the journey itself (and the aspirations with which it is commenced on) that matters more than the destination.
Before I begin to sound too philosophical (cf. above) let me try to make this more concrete by going back to our starting point with some Mathematics and considering some Venn diagrams.
Ordo ab chao
In my experience the following is the type of situation that a good Business Intelligence programme should address:
The problems here are manifold:
Although the various report systems are shown as separate, the real situation is probably much worse. Each of the reporting and analysis systems will overlap, perhaps substantially, with one or more or the other ones. Indeed the overlapping may be so convoluted that it would be difficult to represent this in two dimensions and I am not going to try. This means that you can invariably ask the same question (how much have we sold this month) of different systems and get different answers. It may be difficult to tell which of these is correct, indeed none of them may be a true reflection of business reality.
There are a whole set of things that may be treated differently in the different ellipses. I’ll mention just two for now: date and currency. In one system a transaction may be recorded in a month when it is entered into the system. In another it may be allocated to the month when the event actually occurred (sometimes quite a while before it is entered). In a third perhaps the transaction is only dated once it has been authorised by a supervisor.
In a multi-currency environment reports may be in the transactional currency, rolled-up to the currency of the country in which they occurred, or perhaps aggregated across many countries in a number of “corporate” currencies. Which rate to use (rate on the day, average for the month, rolling average for the last year, a rate tied to some earlier business transaction etc.) may be different in different systems, equally the rate may well vary according to the date of the transaction (making the last set of comments about which date is used even more pertinent).
A whole set of other issues arise when you begin to consider things such as taxation (are figures nett or gross), discounts, commissions to other parties, phased transactions and financial estimates. Some reports may totally ignore these, others my take account of some but not others. A mist of misunderstanding is likely to arise.
Something that is not drawn on the above diagram is the flow of data between systems. Typically there will be a spaghetti-like flow of bits and bytes between the different areas. What is also not that uncommon is that there is both bifurcation and merging in these flows. For example, some sorts of transactions from Business Unit A may end up in the Marketing database, whereas others do not. Perhaps transactions carried out on behalf of another company in the group appear in Business Unit B’s reports, but must be excluded from the local P&L. The combinations are almost limitless.
Interfaces can also do interesting things to data, re-labelling it, correcting (or so their authors hope) errors in source data and generally twisting the input to form output that may be radically different. Also, when interfaces are anything other than real-time, they introduce a whole new arena in which dates can get muddled. For instance, what if a business transaction occurred in a front-end system on the last day of a year, but was not interfaced to a corporate database until the first day of the next one – which year does it get allocated to in the two places?
Finally, the above says nothing about the costs (staff and software) of maintaining a heterogeneous reporting landscape; or indeed the costs of wasted time arguing about which numbers are right, or attempting to perform tortuous (and ultimately fruitless) reconciliations.
Now the ideal situation is that we move to the following diagram:
This looks all very nice and tidy, but there are still two major problems.
A full realisation of this transformation may be prohibitively expensive, or time-consuming.
Having brought everything together into one place offers an opportunity to standardise terminology and to eliminate the confusion caused by redundancy. However, it doesn’t per se address the other points made from 2. onwards above.
The need to focus on what is possible in a reasonable time-frame and at a reasonable cost may lead to a more pragmatic approach where the number of reporting and analysis systems is reduced, but to a number greater than one. Good project management may indeed dictate a rolling programme of consolidation, with opportunities to review what has worked and what has not and to ascertain whether business value is indeed being generated by the programme.
Nevertheless, I would argue that it is beneficial to envisage a final state for the information architecture, even if there is a tacit acceptance that this may not be realised for years, if at all. Such a framework helps to guide work in a way that making it up as we go along does not. I cover this area in more detail in both Holistic vs Incremental approaches to BI and Tactical Meandering for those who are interested.
It is also inevitable that even in a single BI system data will need to be presented in different ways for different purposes. To take just one example, if you goal is to see how the make up of a book of business has varied over time, then it is eminently sensible to use a current exchange rate for all transactions; thereby removing any skewing of the figures caused by forex fluctuations. This is particularly the case when trying to assess the profitability of business where revenue occurs at a discrete point in the past, but costs may be spread out over time.
However, if it is necessary to look at how the organisation’s cash-flow is changing over time, then the impact of fluctuations in foreign exchange rates must be taken into account. Sadly if an American company wants to report how much revenue it has from its French subsidiary then the figures must reflect real-life euro / dollar rates (unrealised and realised foreign currency gains and losses notwithstanding).
What is important here is labelling. Ideally each report should show the assumptions under which it has been compiled at the top. This would include the exchange rate strategy used, the method by which transactions are allocated to dates, whether figures are nett or gross and which transactions (if any) have been excluded. Under this approach, while it is inevitable that the totals on some reports will not agree, at least the reports themselves will explain why this is the case.
So this is my take on a single version of the truth. It is both a) an aspirational description of the ideal situation and something that is worth striving for and b) a convenient marketing term – a sound-bite if you will – that presents a palatable way of describing a complex set of concepts. I tried to capture this essence in my reply to the LinkedIn.com thread, which was as follows:
To me, the (extremely hackneyed) phrase “a single version of the truth” means a few things:
One place to go to run reports and perform analysis (as opposed to several different, unreconciled, overlapping systems and local spreadsheets / Access DBs)
When something, say “growth” appears on a report, cube, or dashboard, it is always calculated the same way and means the same thing (e.g. if you have growth in dollar terms and growth excluding the impact of currency fluctuations, then these are two measures and should be clearly tagged as such).
More importantly, that the organisation buys into there being just one set of figures that will be used and self-polices attempts to subvert this with roll-your-own data.
Of course none of this equates to anything to do with truth in the normal sense of the word. However life is full of imprecise terminology, which nevertheless manages to convey meaning better than overly precise alternatives.
More’s Utopia was never intended to depict a realistic place or system of government. These facts have not stopped generations of thinkers and doers from aspiring to make the world a better place, while realising that the ultimate goal may remain out of reach. In my opinion neither should the unlikelihood of achieving a perfect single version of the truth deter Business Intelligence professionals from aspiring to this Utopian vision.
I have come pretty close to achieving a single version of the truth in a large, complex organisation. Pretty close is not 100%, but in Business Intelligence anything above 80% is certainly more than worth the effort.