Cargo Cults, Causality, Capitalism (and Mathematics) (Part II)

And now the next, and quite possibly last chapter.

Teaching mathematics, especially in Africa, is a bit thankless. The foremost reason for this is that, more often than not, no one really cares. People take a course and they say "I cant wait to be done with this. All I need is a pass and then I will never have to take another mathematics course again." And I ask "so why are you in this course then?" And more often than not, the answer is something to the effect of "because I have to"; most of these individuals are in economics or math finance or actuarial or something like this. In other words, mathematics has gone from a search for truth to a means to perpetuate greed. Hurrah!

But there is a reason for this, and it points to a much larger problem. Basically, I think there are two elements at play here. One is that the lack of real conflict in the last century on North American soil has meant that it has been able to maintain a great deal of uniformity in its educational approach, and hence it has been allowed to develop in a way that maintains a certain amount of continuity with much more historical treatments of education, namely in terms of a more meritocratic system of developing ideas, experimentation, discovery, and notions of 'truth'. Because of this, the strong roots that it was allowed to maintain with mathematics in terms of major ideological debates that strike at its very core, such as the one that surrounded the Hilbert program, etc. This has allowed it to able to maintain a certain level of 'ivory tower' status against the recent push for the commodification and corporatization of education. On the other hand, South Africa in particular and Africa in general does not have: a) the historical continuity that allows for this, b) the economic history to that allows it to put 'theory' before 'practice' (since so many people have so little, it is incredibly difficult to conceive of people having time to spend 'contemplating their navels' as it were), and c) the current economic feasibility to try to instill any sort of change in ideology, etc.

But the bigger problem is the state of education in general. A recent article that notes that there are only one or two universities in Africa that feature in the top echelons of university rankings systems provides some insight:

“Africa inherited a higher education system that was a carbon copy of [that of] the powers that colonised it. Right from the beginning, Africa started on a wrong footing – well behind the starting line, so to speak. Despite all the political and economic turmoil it has gone through since independence – often of its own making – it is now expected to compete on a completely non-level playing field. Not only is this unfair, it is also inappropriate,” says Mohamedbhai, who has also served as vice-chancellor of the University of Mauritius. “One could argue that other regions that were also colonised – South Asia, Latin America – are doing reasonably well. However, none of these regions suffered from the sort of exploitation that Africa underwent and continues to experience.”

Latin America cannot be put in this category simply because their colonization occurred centuries before Africa and Asia: they have been able to generate a reasonable amount of uniformity and, moreover, when the Europeans relinquished control of these countries there was still a fairly long period within which they could develop out of the limelight of the 'modern world' that allows for individuals, groups, corporations, etc. to jump from one continent to the next at the drop of a hat. Brazil's independence from Portugal was 'recognized' in 1825, Argentina from Spain in 1816, Chile in 1844, etc. In fact, other than the three 'Guyanas' that were claimed by the British, Dutch, and French respectively, and are largely seen politically as part of the West Indies, only one country achieved full independence after Canada did in 1867. That country was Peru: although it was declared long before in 1821, it was only officially 'recognized' in 1879. Contrast that to Africa where, aside from Egypt which has basically been able to maintain its independence throughout its history because of the very strong historical legacy of Egypt, all countries only emerged from European colonization in the mid-20th century, with Tunisia and Morocco the first to gain independence in 1956, and Ghana the first sub-Saharan country to gain their independence in 1957.

Moreover, South Asian universities also tend to be quite low on the scale, and aside from Thailand, which has maintained its monarchical rule throughout its history, all of these other South Asian nations also emerged from colonial rule only post-WWII. Those countries that have done exceptionally well in Asia, namely Japan, China, and South Korea have largely been spared the Western colonial cosh, though they have had their differences throughout history (i.e. the Japanese invasion of Manchuria, and their occupation of the Korea peninsula until the division of Korea occurred post-WWII).

The two exceptions are Hong Kong (1841 to Britain, 1997 back to China) and Singapore (1963 from UK, 1965 from Malaysia). But it should be no surprise that these are exceptions: they have few natural resources, and a high population density, meaning that the people were forced to develop their know-how since they couldn't very well live off the land. On the other hand, most colonial 'land-grabs' were for the purposes of owning land and the corresponding resources. There was no need to educate the people, since they basically were enslaved to serve as free labour for the production of goods. When they were allowed to declare their independence, globalization had already had its say: transport of people, goods, and information across great distances was now fairly routine, so they were given no period of 'privacy' to establish what was to be done. For example, Indonesia were able to declare independence from Japan in 1945, but were besieged for the next four years by the British and the Dutch trying to retake control of the archipelago. After this, they had to deal with the fact that most of the industry was ethnically controlled, namely by the Dutch and ethnic Chinese.

Thus, the problem is not that such nations are incapable due to their people, it is rather that history has dealt them a very cruel hand, and they have only emerged with some difficulty from colonial rule during a period when they are being constantly scrutinized by powerful countries with superior military and economic might. Emerging, as it were, with these very same countries still owning the brunt of the economic resources (e.g. the situation as it has played out in Zimbabwe). It is difficult to decide what is to be done in one's own country when no one will leave a people alone to make this decision, and no one will leave a people alone to take stock of THEIR OWN RESOURCES to decide how best to allocate them for THEIR OWN INTERESTS. I already dealt with the other side of this problem in Part I.

So let me get back to the mathematical side of things. Although frustrating, it seems that one cannot but accept that mathematics as a tool to business is, in some sense, a necessity. It is unfathomable why an individual would study Banach algebras over finance if there is little money available in mathematical research and one is likely living like the much of the rest of the population: day-by-day hand-to-mouth.

It is not difficult to see why the educational systems in these countries are poorly conceived of and poorly managed; it is because they have not had any time to themselves since independence: they have been thrown into the deep end of a world that is constantly spinning about them both economically and politically, with their former colonial masters standing at the poolside constantly taunting them and pushing them back into the middle should they get to close to the edge where they might gain some sort of respite from their perpetual treading of water. Continuing the above article:

In light of the continent’s urgent problems, Mohamedbhai thinks that African universities should absent themselves from the race to rise up the rankings and focus their efforts on immediate needs. “Do African universities need to be ranked globally? I don’t think so. Their mission should be to produce the appropriate manpower required for Africa’s development, to undertake research that is of direct relevance to Africa – which may not be acceptable for publication in the best scientific journals – and to reach out to assist the communities in the many challenges they are facing, especially poverty reduction. None of these fits the criteria used for global ranking. African universities have a duty to serve their countries and region first before seeking global glory. The tragedy is that many African governments, blinded by the prestige of global rankings, are challenging their universities to be ranked without understanding the consequences of the grossly inappropriate use of resources that that would entail. At the end of the day, this brings us back to the very purpose of higher education in a country. Not all universities in the world can have the same mission. Priorities are different in different countries, and universities must not be forced to conform to a single model of a world-class university.”

I personally believe that there is a different type of education that Africa (and many countries in Asia) requires: a push towards a much more critical pedagogy that provides its people with an understanding and a genuine belief that it is not them that is the problem; it is rather their colonial wardens that give them no peace that are the problem. They must learn and understand who they are and what is at stake. They must realize that Africa and its people are ontologically no different than anywhere else. That it has the potential for greatness if only enough people see and genuinely believe in this greatness and are willing to come together to enact real change towards a real Africa that is Africa through and through rather than an poor interpretation of Western society.

And, most importantly, they must understand that this change can only come from them.

Probability, Philosophy, and Monty Hall

I'm not sure that it is possible to come up with something original every weekday, and today is one of those days where my mind is elsewhere. So here is something I've plagiarized word-for-word from a photocopy tacked to a wall in the mathematics department at UCT (the original author is Dr. J Ritchie, a senior lecturer on the Philosophy of Science here at UCT). I thought the ending fit nicely with some of the things I said yesterday with regard to consequentialism and finitism.

The Monty Hall Problem

Here's a puzzle you might have heard before. Imagine you are taking part in a game show. The host, Monty Hall, has three doors in front of him. Behind one there is a car and behind the other two a goat. At the end of the game you will have chosen one of the doors and you'll win whatever is inside. You want to win the car.

You start by choosing one door at random. Monty looks behind the other two doors and opens one of them to reveal a goat. He now offers you the chance to swap your door for the one he didn't open. What should you do?

Many people argue like this. There are only two possibilities: the car is either behind your door or Monty's, each is equally likely, so it doesn't matter which door you choose. But that's a mistake. There are three possibilities at the beginning of the game, all of which we assume are equally likely. Either you have chosen the door with the car or you've chosen the door with goat 1 or you've chosen the door with goat 2. If you swap in the first case, you'll win a goat. If you swap in the other two cases, then you'll win the car. Hence you're twice as likely to win if you swap, so you should swap.

That's a simple problem in probability theory but let's now think about it in a different way. Imagine instead of winning a car that Monty promises to shower you in riches, if you win. But if you lose, Monty will shoot you. Of course, you might not want to play that game. But tough, Monty has kidnapped all your family and threatened to kill them all and you unless you play. The game proceeds as before. You choose a door, Monty opens one of his doors to reveal a goat (which now represents your imminent death) and asks if you want to swap. What should you do?

We've ratcheted up the drama a bit, you might think, but the logic of the case remains the same. You are more likely to win if you swap. So you should swap. Let's say you swap and it turns out the door you are left with contains a goat; so Monty shoots you. In what sense, then, was it the right decision to swap?

The obvious response is that it is the right thing to do because, if you play the game a lot, you will win twice as often (on average) as you lose. Probabilities in other words tell us about long-run frequencies. But this kind of game you can't play a lot. Once you lose, you are in a very serious way out of the game for good. In fact since relative frequencies only converge on probabilities in the infinite long run and in the long-run we're all dead, is there any good reason ever to choose the more probable option? Now we've moved from a simple problem in probability theory to a hard problem in philosophy... But I've run out of space to offer you any solutions.

(By the way, you are invited to send possible solutions to jack.ritchie@uct.ac.za)

'Luki'

I felt there was only one topic for my first post. By request: Wittgenstein.

My interest in philosophy began when I returned to Canada after spending a bit of time off and on living in England. I had heard of this thing called ‘philosophy’ that had ‘all questions and no answers’ and, cocky as I was, I thought ‘how is this possible? Let me at it and I’ll put these issues to bed.” Not knowing where to start, and worrying that a turgid slog through a random philosophical treatise might put me off (books had not really appealed to me much since I had got out of school), I went to the local library and began collecting all of the possible books in the ‘Great Philosophers’ series that I could find. Basically, these were pocketbooks of between 50 and 70 pages giving a quick summary of their major philosophical points. Since I was new to philosophy at the time, I had only heard a few names here and there, and of books I found with names I knew: Plato, Aristotle, Rousseau, I found it a bit difficult to get a good picture of just what was going on. I had not really done much critical thinking or testing of my understanding of the world at the time; it simply was what it was. Two philosophers in the series changed everything for me and put me on the path to having a genuine interest in philosophy and all that it deals with. One of those was Wittgenstein (the other was Schopenhauer, but more on that some other time).

Many historians and ‘Wittgensteinians’ refer to ‘Wittgenstein I’ and ‘Wittgenstein II’. Wittgenstein I was the man who had claimed to have ‘solved all philosophical problems’ in the Tractatus Logico-Philosophicus in which the final and most important point is

‘Whereof one cannot speak, thereof one must remain silent.’

The meaning of which was simply that for Wittgenstein (at that time), there was no such thing as a philosophical ‘problem’. What were actually being debated in philosophical circles were ‘puzzles’ that arose because our use of language and our attempts to understand each other’s meaning are imperfect. And so we disagree about things not because there are ‘realities’ to disagree about, but rather because we are not able to properly define terms and communicate ideas. It is only language and meaning that could properly be debated. After a hiatus that he spent teaching schoolchildren in the mountains somewhere in Scandinavia (if I remember correctly) Wittgenstein II then went back on this rather bold summary of philosophy, and wrote his ‘Philosophical Investigations’, which, in my opinion is both a brilliantly ‘playful’ and a brilliantly profound exploration of language (in contrast to the rigorous formality of the Tractatus). The Investigations is famous for, amongst other things, the ‘private language argument’ (he argued that one cannot have a ‘private’ language, that is, a language limited entirely to oneself).

If anyone reading this is interested, I recommend ‘Wittgenstein’s Poker’ for a brilliantly entertaining synopsis of Wittgenstein the man and Wittgenstein the philosopher based around a ‘legendary’ meeting (and the only one ever) between Wittgenstein and another very ‘self-assured’ (i.e. cocky) philosopher, Karl Popper.

At the time of reading this 60-odd-page intro to Wittgenstein, I knew none of this, and a lot of the finer details were lost on me. What struck me as so radical (and still does today) is that all of a sudden language went from a tool to communicate to a thing that can be analyzed in and of itself (and it is especially radical when it is presented in such a way as to lead one to believe it could possibly be a metaphysical basis of reality). I had never really had a reason to ponder over the fact that we call the white liquid that comes from cows ‘milk’ on no other pretense than that a bunch of people once upon a time agreed that it should be so (of course, etymology enters the picture, but one can apply the same argument to the etymological roots of any word). It is nothing more than custom that we use the term ‘milk’ as opposed to, say, ‘rabagooba’ to denote this item in English.

And this idea can be useful to try to make mathematics less intimidating. Of course, we usually take mathematics as the ‘everyday’ mathematics that is grounded in arithmetic and such, and this is entirely justified. We have utilized both language and mathematics for thousands of years, and it has only been within the past 150 years with Frege and Cantor that each of language and mathematics have been seen as an ‘item’ to be analyzed in and of itself. However, if one looks at mathematics itself, one sees that as opposed to all of the other ‘sciences’, which are based on understanding and interpreting reality to some extent, mathematics is a self-contained axiomatic system. Thus it is, in a sense, a ‘language’ as well (and therefore, from a certain point of view, it is entirely a ‘human invention’ not wholly grounded in reality), though one based on the rules of logic. Gödel’s Incompleteness Proof established that there is no such thing as a logical system that can prove itself. Basically, if one begins with different axioms, one can get an entirely different system of mathematics, so long as the axioms are not, in some way, self-contradictory. Hence, there is no ‘true’ mathematics, like one may say there is a ‘true’ chemistry, physics, or biology that accurately describes the secrets of Mother Nature, just like there is no objective reason why one should call milk ‘milk’ rather than ‘rabagooba’. The only difference is that the ‘rightness’ or ‘wrongness’ of mathematics is based on rules of logic in contrast to language, which is based on custom and the systematization of grammar. That is, in terms of mathematics as a ‘system’; the manner in which mathematics is actually applied and 'done' is based on custom. Once students understand that ‘x’ is simply a 'placeholder' and its use to designate an unknown is based entirely on custom (and ‘ease’, since mathematicians are lazy)—there is nothing to stop one designating said unknown by drawing a map of Indonesia, making a chicken noise, or dancing a jig—they stop making silly mistakes, like using x to designate different unknowns that are not equal. The main problem with the other three options is one’s ability to duplicate it when needed: dancing can get pretty tiring, and making a chicken noise? That's just stupid.

Gauss once called mathematics ‘the queen of the sciences’. I forget his justification for this statement, but I always think that maybe his intention was to say that mathematics is such that although it doesn’t rule, you cannot do anything without it. Maybe this is a somewhat sexist (and, along the same vein, overly ‘biological’) interpretation, but one must remember that it was said about 300 years ago, and the human race and its approach to gender equality and human rights has come a long way since. Supposedly.