Analogies

Disaster Area's chief research accountant has recently been appointed Professor of Neomathematics at the University of Maximegalon, in recognition of both his General and his Special Theories of Disaster Area Tax Returns, in which he proves that the whole fabric of the space- time continuum is not merely curved, it is in fact totally bent.

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.

Introduction

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:

Rock climbing: Perseverance
A bad workman blames his [BI] tools
Running before you can walk
Feasibility studies continued…
Incremental Progress and Rock Climbing
Cricket: Accuracy
The Big Picture
Mountain Biking: Mountain Biking and Systems Integration
Football (Soccer): “Big vs. Small BI” by Ann All at IT Business Edge

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:

Synthesis RNA Polymerase transcribing DNA to produce RNA in the first step of protein synthesis
The Business Intelligence / Data Quality symbiosis A mitochondria, the possible product of endosymbiosis of proteobacteria and eukaryots
New Adventures in Wi-Fi – Track 2: Twitter Paul Dirac, the greatest British Physicist since Newton

On other occasions I have posted overtly Mathematical articles such as Patterns, patterns everywhere, The triangle paradox and the final segment of my recently posted trilogy Using historical data to justify BI investments.

Jim Harris' OCDQ Blog

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.

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.
© xkcd.com (adapted from the original to fit the dimensions of this page)

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:

S(1) = The set of all points defined by one variable (x1)
– i.e. a straight line
S(2) = The set of all points defined by two variables (x1, x2)
– i.e. a plane
S(3) = The set of all points defined by three variables (x1, x2, x3)
– i.e. “normal” 3-space
S(4) = The set of all points defined by four variables (x1, x2, x3, x4)
– i.e. 4-space
” ” ” “
S(n) = 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:

2-space: 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)
3-space: A cube has 8 vertexes, 12 edges and 6 “faces” (each consisting of a square)
4-space: 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:

n-space: 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.

La Grande Arche de la Défense

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:

  1. Analogies only work to a degree and you over-extend them at your peril
  2. Sometimes the wholly understandable desire to make a complex subject accessible by comparing it to something simpler can confuse rather than illuminate
  3. 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

Hand with reflecting sphere - Maurits Cornelis Escher (1935). This is your only clue.

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:

  1. In Pure Mathematics, the members of a group may be associative, but this doesn’t mean that they tend to hang out together.
  2. In Particle Physics, an object may have spin, but this does not mean that it has been bowled by Murali
  3. 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.

Tridimensional realisation of the Riemann Zeta function
© Jean-François Colonna

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!

Euler's product formula for the Riemann Zeta function


Chronological list of articles using xkcd.com illustrations:

  1. A single version of the truth?
  2. Especially for all Business Analytics professionals out there
  3. New Adventures in Wi-Fi – Track 1: Blogging
  4. Business logic [My adaptation]
  5. New Adventures in Wi-Fi – Track 2: Twitter
  6. Using historical data to justify BI investments – Part III

 

13 thoughts on “Analogies

  1. Brilliant blog post, Peter.

    Thanks for the very kind words about my blog and my affinity for analogies, otherwise known as metaphorically blogging.

    Although I lack both a robust mathematics education and a robust interest in mathematics, I have always been fascinated by Theoretical Physics. One very accessible book that I recently read on the subject is Why Does E=mc2?, which attempts to explain one of the most famous, easily remembered, but difficult to truly understand formulas.

    Cheers,

    Jim

    • Hi Jim,

      The kind words are deserved :-).

      Brian Cox is becoming a worldwide phenomenon – not bad for a lad from Oldham who got a D in A-level (High School) Mathematics!

      Peter

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