Fact and Fiction

Thoughts about a funny old world, and what is real, and what is not. Comments are welcome, but please keep them on topic.

Monday, December 25, 2006

Demystifying hard subjects

Some subjects are called "hard" because you need years of training to understand and practice them. Examples of hard subjects are mathematics, physics, chemistry, etc, and examples of soft subjects are sociology, media studies, etc. Essentially, the difference between these two categories is that hard subjects are far from our everyday common sense so they are not intuitively accessible, whereas soft subjects are close to our everyday common sense so they are intuitively accessible.

It seems to me that the "hardness" of a subject is strongly correlated with the following two properties to do with reasoning about and sensing the world:

  1. How mathematical/logical its formulation is; this property is to do with reasoning about the world. Anyone who does not have the appropriate mathematical/logical training is automatically excluded from the subject, which thus falls into the "hard" category.
  2. How accessible it is to observation and verification by our senses; this property is to do with sensing the world. Any subject that relates to phenomena that are accessible only by using specially designed sensors falls into the "hard" category.

Either way, our common sense is limited by what our bodies can do unaided, whether it is sensing (e.g. using our eyes) or reasoning with our brains. To develop an enhanced common sense we need to enhance our sensors (e.g. use a microscope) and/or our enhance our ability to reason (e.g. attend further education classes).

Here I want to focus on the improvements to common sense that can be gained by enhancing our ability to reason, which are gained (often at great expense in time and money) by further education. Of course, I have a vested interest in this aspect because my own further education didn't finish until I was around 27 years old (or 1/3 of a typical lifetime), and it was heavily mathematical towards the end. However, I have a very ambivalent attitude to the type of enhancements to my common sense that I gained as a result of my lengthy education. I can do fancy maths to calculate results that are both correct and useful, and which thus can serve as enhancements to my prior common sense. But I can't do these calculations in real-time, because I have to do each calculation laboriously off-line rather than on-line, so they are rather slow enhancements to my common sense. It would be much better if these enhancements could be "compiled down" to become fast response (i.e. on-line rather than off-line) enhancements, which could then more justifiably (i.e. they now work in real-time) be called enhancements to my common sense reasoning ability.

All of this "compiling down" amounts to finding ways of short-circuiting mathematical calculations so that results can be seen directly without having to go through all the intermediate steps of the mathematical derivation. This possibility is called computational reducibility, and it is not guaranteed to even be possible, e.g. many simulations in NKS are strictly computationally irreducible. Where a calculation is computationally reducible it is because there is a higher level principle at work (whose inner details are implemented by the individual steps of the mathematical derivation) which allows us to take large strides forwards through a derivation rather than achieve the same thing by a large number of time consuming little steps. Generally speaking, the higher the level of the principles the greater the speed-up that can be achieved. Note (for physicists) that another way of viewing these high-level principles is that they bear the same relationship to the low-level mathematics as effective theories bear to underlying theories.

Here is a simple example. When I was about 10 years old I realised that the formula p=ρ.g.h for pressure p at a depth h in a medium of density ρ could be understood directly and intuitively; in effect I never memorised that formula (even when I was 10 years old) because I always rederive it in real-time every time that I need it. This early insight was a complete revelation to me, and it has influenced the way that I do science and mathematics ever since. I carry very little memorised information in my head, but I do carry a small number of principles that I use to quickly rederive whatever I need. Naturally, I have now gone far beyond p=ρ.g.h but the same general approach still applies. Very conveniently, this approach (i.e. rederive rather than memorise) served me very well in various school/university examinations!

So, how are computational reducibility, higher level principles, and enhanced common sense related? How is it possible to see results directly and intuitively? Here I will have to assume that what works for me also works for other people, so I will focus on the use of visual intuition where low-level mathematics become elementary operations for manipulating a visualised world, and then high-level principles emerge as composite operations on this visualisation. It should be clear that the amalgamation of many low-level operations into fewer high-level operations amounts to a form of visual intuition about the behaviour of this visualised world.

Let's discuss a concrete example to illustrate the strengths and weaknesses of visualisation. How about the simple p=ρ.g.h example given above? The medium can be visualised as a cloud of small blobs (atoms), where the average number of blobs per unit volume (number density) is n, the mass of each blob is m, so the average mass of blobs per unit volume (density) is ρ=n.m. Each unit area of whatever this cloud rests upon must supply an (upward) force that is equal to the (downward) weight of the blobs in the cloud resting on the unit area, otherwise the upward and downward forces would not be in balance. The thickness of the cloud (depth) is h, so the average mass of blobs per unit area is ρ.h. Finally, the (downward) acceleration due to gravity g causes this mass per unit area ρ.h to have a (downward) weight g.ρ.h. Gathering it all together, the (downward) weight per unit area is ρ.g.h, which must be equal to the (upward) force per unit area, which is one and the same thing as the pressure p at the base of the cloud of blobs, so p=ρ.g.h.

Whew! Written out like that the derivation of p=ρ.g.h is incredibly verbose. However, I am not suggesting that you actually write things out in this way, but that you create a visualisation whose properties are described by the various sentences in the above paragraph. With exercise this process of visualisation becomes semi-automatic, and the whole of the above paragraph can be quickly "simulated" in your head. This simulation then becomes a virtual derivation (written in visual symbols) of the formula p=ρ.g.h, which can be converted into a real derivation (written in standard algebraic notation) as necessary. This example illustrates the relationship between the visual and the algebraic approaches to doing mathematics.

Of course, p=ρ.g.h is a very simple example where the visual representation is so easy to create from the algebraic representation that you might assume that you could always start from the algebra and derive the visualisation from it, rather than the other way around. However, almost everything else is more complicated than p=ρ.g.h, and although the algebraic approach always works correctly, it rapidly becomes so complicated that its results can no longer be obtained directly and intuitively, which is where the alternative visual approach allows you to take intuitive visual shortcuts to quickly derive qualitative results.

For an example of a more substantial use of a visualisation type of approach have a look at my Spooky action at a distance? posting where I discuss the so-called EPR paradox in quantum mechanics. In fact there is no paradox there, but only confused thinking caused by a failure to use the mathematics of QM consistently, i.e. assuming that "observers" somehow exist outside the QM system that they are "observing". I was taught QM using this inconsistent approach, which was fine for passing physics exams, but started me off with a hopelessly confused point of view when I eventually had time to sit down and really think about QM. It took me a while, and several false starts, before I finally realised that "observers" have no special status, and should therefore be lumped in with the system that they are observing, which led inevitably to me rediscovering what I later learnt was the Everett interpretation of QM (see the The Everett FAQ) which I wrote about in State vector collapse? Anyway, I now have a visual language (e.g. Spooky action at a distance?) that enhances my everyday common sense so that I can intuitively and directly obtain qualitative QM results. Why can't this type of approach be taught in QM courses?

Of course, I can't prove that it will always be possible to find intuitive visual representations of complicated mathematics, but I certainly feel that I understand the mathematics much better if I can find such a representation, and use it to intuitively and directly derive qualitative results. I have heard it claimed that some fields of work lie outside the realm of direct intuitive comprehension (e.g. when a simple result seems to magically pop out of an enormously complicated piece of mathematics), but I hope that this claim is the result of a lack of imagination rather than there being a fundamental limitation to what we can visualise using appropriate techniques.

The visual approach described above can be used to enhance your everyday common sense, so that it becomes able to intuitively reason about what are normally viewed as "hard" subjects. In effect, "hard" subjects become "soft" subjects, when addressed using a common sense that has been enhanced by use of visualisation techniques.


At 26 December 2006 at 12:29, Anonymous Anonymous said...

Dear Steve,

You use very nice computer science metaphors, like "compiling down", to describe some natural functions of human brain. I take the liberty to develop these metaphors further.

Mathematics is difficult (and, for many people - impossibly difficult) because it requires some neural circuits in the brain beeing developed exactly at the time when growing and developing brain of a child is susceptive for precisely that kind of development. You were lucky to have sufficient intellectual stimuli to derive the pressure formula at age of 10. Your early learning environment made it possible for you to become a mathematician and computer scientist: your growing brain developed sufficient circuitry for "compiling down" and other forms of "intuitive" thinking about things "far from our everyday common sense".

I have to admit that this is a topic I avoid to discuss in public. Indeed, I believe that persons for whom mathematics is "hard" were, in effect, braindamaged at early stages of their education: the necessary neural circuitry had not been supported at the time when it had a chance to develop, and it is highly likely that this chance is lost forever.

At 26 December 2006 at 14:53, Blogger Steve said...

I wonder whether we are using the phrase "compiled down" in the same sense.

As I understand it, you are referring to the development of low level neural circuitry that directly implements whatever detailed operations are needed. Through appropriate training when you are young enough to be susceptible, you can implement a wide range of brain functionality. As you say, once the brain gets too old your chance to rewire it at a very low level is gone forever. The development of the visual cortex is an extreme example of this, where you "use it or lose it" very early on in your life.

However, what I was thinking about was a different sort of "compiling down", which is the replacement of a difficult sequence of operations by a different and easier sequence of operations. I apologise for not making this explicit in my original posting. In this case the difficult version of these operations correspond to the doing of a "hard" subject (such as mathematics, etc), whereas the easy version of the operations achieves the same qualitative effect using methods that are "soft" and accessible to a large number of people.

Note that the "soft" version does not in general rival the "hard" version in the level of detail and precision that it can compute. However, it does allow you to directly and intuitively see qualitative solutions to problems which would otherwise be "hard" to arrive at.

As for the "brain damage" viewpoint, I think we are each brain damaged (as you put it) by our own unique environmental experiences. I'll bet the children of musicians have highly developed musical neural circuits, and others who are less exposed to music are relatively "brain damaged" in that respect. For instance, musical perfect pitch (i.e. having an absolute pitch standard wired into your brain) is an interesting example where early exposure to music is really important for its development.

Anyway, I am less pessimistic than you are about whether people can do "hard" subjects, because I think that they can learn how to do a "soft" and qualitative version of the "hard" version of the subject. What is not clear yet is how early one has to develop this way of thinking in order to become adept at it.

At 26 December 2006 at 15:08, Blogger Steve said...

I have now downloaded your book "MATHEMATICS UNDER THE MICROSCOPE", and glancing through the contents list I can see many topics of interest to me.


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