On the nature of mathematical knowledge

l by Carlo Fonseka

(February 05, Colombo, Sri Lanka Guardian) For most of my long life, I believed that mathematics is the subject that gave us absolutely certain knowledge. I thought that nothing could be more certain than that two plus two are four. Then one day I read an essay titled ‘Mathematics and the Metaphysicians’ by Bertrand Russell written in 1901. In it he says that "pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true". To tell the truth, I found this very puzzling and disturbing. So I asked my mathematician friend Prof. Douglas Amarasekara to explain it to me. He told me that pure mathematics being indistinguishable from symbolic logic, Russell’s assertion made sense.

Without quite understanding what he meant, I felt reassured and was able to accept tentatively Russell’s conclusion that mathematics is "the subject in which we never know what we are talking about, nor whether what we are saying is true". I should say, however, that Russell’s definition of mathematics shook my faith in the certainty of mathematical knowledge. Matters became much worse when I subsequently read somewhere Cambridge mathematician G. H. Hardy’s nonchalant remark that "a mathematician is someone who not only does not know what he is talking about but also does not care".

If the Z-score on the basis of which students are selected to our universities is founded on mathematical concepts, and mathematics is what Russell and Hardy say it is, what can we possibly tell students (and their parents) who are struggling to come to terms with the so-called Z-score ‘imbroglio’?

Reassure

The first thing we should tell them is that unlike the ivory tower intellectual Prof. G. H. Hardy, we deeply care about their current plight and anxiety and are determined to do all that can be done to ensure justice and fair-play. Secondly, we should tell them that it is not through anybody’s fault or negligence that an absolutely reliable non-controversial method is still not available to us for selecting students to our universities. Uncertainty is a part of life we have to learn to live with. But things are not as uncertain as they appear to be at first sight.

Although humankind may not have absolutely certain knowledge, science based on mathematics has proved itself to be an enormously successful problem-solving activity. Galileo famously said that "the book of nature is written in the language of mathematics". Precisely because modern science has been so successful in changing the world by means of computers and television and jet-planes and so on, the mathematics itself underlying science has acquired great credibility. So the concept of the Z-score is not without validity.

The first thing we should tell them is that unlike the ivory tower intellectual Prof. G. H. Hardy, we deeply care about their current plight and anxiety and are determined to do all that can be done to ensure justice and fair-play. Secondly, we should tell them that it is not through anybody’s fault or negligence that an absolutely reliable non-controversial method is still not available to us for selecting students to our universities. Uncertainty is a part of life we have to learn to live with. But things are not as uncertain as they appear to be at first sight.

Least unsatisfactory

The selection of students for university admission is of course not a scientific process and mathematics itself is not a branch of natural science. Whether mathematical truths are empirical in nature or are true by definition or are capable of giving us a priori insight into the structure of reality (Platonism) are unsettled issues. However that may be, the selection of students for university admission by a statistical procedure which is based on a tried and tested mathematical concept is basically a rational process. Because in fields like medicine and engineering, we are basing selection of students for university entrance entirely on written exams (no practicals, no aptitude tests, no interviews, no reports from school principals about the extra-curricular activities and attitudes of students) the Z-score is the least unsatisfactory mechanism we have for the purpose.

What the current imbroglio implies is that the use of the Z-score involves not simply an application of a formula to the raw marks obtained by students but some judgments about appropriate sampling procedures to use when applying the formula. Here, there can be legitimate differences of opinion about the sampling techniques to adopt. Predictably therefore, students who are after all, members of the ‘species’ Homo-economicus will be hell-bent on maximizing their chances of achieving their desire to enter the faculties of their choice. Therefore, they will figure out which statistical procedure maximizes their chances of gaining entry and move heaven and earth to have it accepted by the relevant authorities. In such a situation where two legitimate ways are available for processing the results to arrive at a Z-score what should be done? It seems to me that the only course open to us is to use the utilitarian principle of the greatest happiness of the greatest number.