I was browsing the internet a few weeks ago, and came across an opinion piece lamenting the poor state of mathematical education and the detrimental effect it has had on science. The provocative piece starts as follows:
It is a subject of wonder and regret to many, that this island, after having astonished Europe by the most glorious display of talents in mathematics and the sciences dependent upon them, should suddenly suffer its ardour to cool, and almost entirely to neglect those studies in which it infinitely excelled all other nations. After having made the most wonderful and unhoped-for discoveries, and pointed out the road to more; suddenly to desist, and leave these to be cultivated, and the road to more to be explored, by other nations, is very remarkable. It seems as strange as the conduct of a conqueror would be, was he to conquer all the countries around him, and then tamely to suffer his own and the subjugated ones to be possessed, governed, and cultivated, by those whom he had conquered.
It is a very great disgrace for a nation like this, which can proudly boast of a superiority over all others in arts, arms and commerce, to suffer the sublimest sciences, which once were its greatest pride and glory, to be neglected. Surely a much more solid fame accrues to a people from their superiority in talents than in arms. Athens is as celebrated for its learning as its commerce or its victories. It cannot be owing to any want of importance in the sciences themselves that they are neglected; the discoveries made in them are of the most astonishing nature, and such as seemed absolutely beyond the reach of human intellect. By the marvellous assistance of the mathematics from the simple law of gravity are deduced the orbits of the planets and satellites, their distances, the times of their revolutions, their densities, quantities of matter, and many other remarkable properties too well known to be enumerated. Were it not for them, mechanics, optics, hydrostatics, geography, and other branches of natural philosophy, would hardly have been known as science. It is possible that discoveries more wonderful and of greater utility than those already made by the help of mathematics, may some time or other be effected, should some great genius once point out the way. It is the opinion of many philosophers, that the various forms and diversified properties of bodies are owing to the various laws of attraction and repulsion which their constituent particles exercise upon each other. Should these laws ever be discovered, we shall become as well acquainted with the structure, affinities, and mutual operations of boides, as we are with the revolutions and actions of the planets upon each other.
As you probably have noted already from the style of writing, this is not a particularly recent article. In fact, this call-to-arms in favor of mathematics education was written in 1804!
The article is by the Reverend John Toplis, A.M., “On the decline of mathematical studies, and the sciences dependent upon them,” Philosophical Magazine 20 (1805), 25-31. There seems to be relatively little information available about Toplis online, though he was apparently a relatively influential mathematician in his time. He had earned himself a master of arts degree (A.M.), as well as a bachelor of divinity (B.D.) degree from Queen’s College, Cambridge. Circa 1813, he published a translation of some of the works of Laplace on mechanics; he also seems to have guided the great mathematician George Green in his career, possibly informing Green on the current developments in mathematics. Toplis was also the headmaster at Nottingham High School from 1806 to 1819, an indicator that he was broadly interested in educational issues.
Toplis is concerned with maintaining the advantage that England had gained in mathematics starting with the great Isaac Newton and his development of calculus around 1700, as well as the slightly earlier founding of the world’s first modern academic journal, the Philosophical Transactions. Toplis laments,
We seem, as a nation, for this last half century, to be sunk into a great degree of supineness with respect to the sciences, regardless of our former fame. The generality of the papers in the Philosophical Transactions are no longer of that importance they were formerly. We have long ceased to study those sciences in which we took the lead and excelled, and are content to follow, at a very humble distance, the steps of the philosophers of the continent, in those which they have in a manner discovered and made plain by their glorious exertions. We, after having discovered and conquered regions in science, suddenly quit them to be possessed and cultivated by other nations, that we may pick up a few gleanings in the countries found out and cultivated by their exertions.
This argument is reminiscent of those used today by those warning of the decline of science in the U.S.: we risk losing our competitive edge in science to other countries.
To what strange infatuation can it be owing that we tamely give up what was once our greatest boast? Is it because a century back our philosophers made such advances in science that all other nations were left at an immeasurable distance, that we are contented with their glory, and think our country sufficiently immortalized? Can we tamely sit down with what they have done, and see other nations gaining fame where our ancestors immortalized themselves? By such conduct, the fame they have acquired reflects double disgrace upon ourselves; we show that we are degenerate, and unworthy of such fathers.
The explanations that Toplis gives for the decline in mathematics also sound very familiar, though some are more valid than others:
The mathematics, and the sciences dependent upon them, cannot be neglected from their want of importance and utility: they are a much nobler study than the present favourite one of natural history, the various branches of which seem to require more the efforts of memory than judgment; in the pursuit of which, the highest object to be attained is the discovery of some nondescript insect or plant, in which chance more than judgment is concerned.
Oh, no he didn’t! Toplis is basically denouncing biology as an inferior field of study that requires no more than memorization! Here we have a religious person taking cheap shots at the biological sciences: if P.Z. Myers lived in Toplis’ era, the good Reverend would be able to insulate his house with all the angry letters he would be getting from Pharyngulites.
Though one can appreciate Toplis’ concern, it was undoubtedly short-sighted: the biological sciences are perhaps the most important areas of research today, and require far more mathematics and statistics than Toplis could have ever imagined.
Other concerns of Toplis have sadly changed little through the present day:
Perhaps one reason to be assigned for the deficiency of mathematicians and natural philosophers is the want of patronage. These sciences are so abstruse, that, to excel in them, a student must give up his whole time, and that without any prospect of recompense; and should his talents and application enable him to compose a work of the highest merit, he must never expect, by publishing it, to clear one-half of the expense of printing. All those men, therefore, who have not fortune sufficient to enable them to give up their time in the study, and part of their property to the publication, of works in these sciences, are in a manner excluded from advancing them. In France and most other nations of Europe it is different: in them the student may look forward to a place in the National Institute or Academy of Sciences, where he will have an allowance sufficient to enable him to comfortably pursue his studies; and should he produce works worthy of publishing, they will be printed at the expense of the nation.
To this day, brilliant people going into the sciences can expect that they will be making far less money than they could have made in other fields, and will have spent the better part of their life making almost no pay at all as a graduate student. We like to think that the love of the research makes up for this financial shortcoming, but it is undeniable that the sacrifice can be painful enough to drive away a lot of talented people.
Toplis also alludes to another still-extant problem: the lack of jobs! Even if a graduate student excels in research and distinguishes his or herself, there may not be jobs available upon graduation. This is a concern that has been much-debated in science blogs and the greater academic community for a long, long, time.
Another interesting concern that Toplis raises is the misguided emphasis amongst those who in fact do study the mathematical arts:
It is remarkable, that amongst the very few men who still pursue mathematical studies in this country, a considerable part, instead of being dazzled and delighted by the wonderful and matchless powers of modern analysis, still obstinately attach themselves to geometry. It is a science, perhaps, of all others, from the clearness and accuracy of its proofs, the most proper to be taught young men, that from the study of it their reasoning faculties may be improved; but at the same time, as a science, it is confined in its application, feeble, tedious, and almost impracticable in its powers of discovery in natural philosophy. But what is called analysis possesses boundless and almost supernatural powers in its application to science; and the discoveries made by it in natural philosophy are of so surprising a nature, that to pretend to despise it, and obstinately to grovel amongst a few properties of surfaces and solid bodies, part of which were discovered by means of analysis, denotes a very narrow and prejudiced mind.
It is worth providing some explanation here: mathematics in Toplis’ time could be broken into four major branches, namely number theory, algebra, geometry, and analysis. Number theory and geometry were instituted by the ancient Greeks, while algebra was founded relatively more recently by the Indians and Arabs circa 700-800 C.E. Analysis, which began with calculus, was a relative newcomer in mathematical research and was initiated by Newton.
Analysis opened many doors in mathematics and allowed the solution of formerly unsolvable problems, including those in the other branches of study. It seems that many mathematicians regarded the notions of infinity and infinitesimal quantities introduced by calculus with suspicion, however, and it is perhaps not surprising that many aspiring mathematicians would be drawn to the “established” arts such as geometry.
So what solutions does Toplis suggest to correct the sorry state of mathematics in his time? First, he notes what needs to be changed:
I cannot here forbear making a few remarks upon the method of study made use of in the university of Oxford and the principal seminaries of this kingdom, as I look upon it as a very great interruption to the progress of science. Regardless of the wonderful advances made in the sciences and arts, they treat their learners with contempt, and, obstinately shutting their eyes against their present most enlightened state, seem determined that nothing but the study of words and ridiculous attempts at elegant composition in the Greek and Latin languages shall employ their scholars.
Toplis is quick to point out that he does not want to eradicate the study of the classics, but educators should not put so much effort into it “for the sake of reading half a dozen poets, historians, and orators.” The time currently wasted in providing a standardized education of dead languages should be replaced with a study of mathematics:
Mathematics, and the sciences dependent upon them, ought to make the principal part of a good education. The strictness and accuracy of their reasonings would contribute in the highest degree to improve the mind of the student. By them he would learn to become patient in investigation, and severe in judgment. It would serve to check in him all conceited and arrogant pretensions to knowledge, and render him more diffident, by showing how careful and laborious it is necessary to be to acquire a few truths. They tend likewise to improve the morals, and give a steady serenity to the mind. In studying them, we seem to leave the jarring world, convulsed and rendered turbulent by the prejudices and frantic passions of men, to lead a life of pure enjoyment. In the pursuit of them we proceed by incontestable truths; every thing is certain, and the laws which take place throughout nature invariable. No prejudices, passions, or wrong bias of education, can involve us in errors and perplexities; any defect in the chain of reasoning can always be detected, and the mind may rest satisfied with the assured discovery of truth… The mathematical sciences are not only of the greatest importance to us from the beneficial effects attending their study, but at the same time most sublime in their application. Through their assistance we become acquainted with some of the laws by which the omniscient and eternal Creator governs the universe, and are enabled to predict their effects in distant ages. Our condition becomes superior to the common lot of humanity, and we may be held out to the world as an example of the perfection to which it is possible for the human species to arrive.
Here, in somewhat flowery prose, we see another familiar, and valid, argument for the teaching of math and science: it helps people to think better! I point this out to my students and friends as often as I can. At the very least, some study of mathematical proof and the scientific method serves as a “bullshit detector” in a world where con men, quacks and politicians are happy to feed you a steady supply of crap.
Toplis’ argument is a bit out of date today, however, because of his reference to mathematical “truths”. The work on non-Euclidean geometry by Nicholai Lobatchevsky (1793-1856) in the early 1800s and on the consistency of mathematics by Kurt Gödel (1906-1978) in the early 1900s destroyed the notion of mathematics as “absolute truth”, and showed that there are limits to what we can know for certain in the mathematical world.
So what lessons can we learn from Toplis’ 1805 paper? On the one hand, it seems to show that arguments about the decline of mathematical education are somewhat overblown — if mathematics has been in decline for 200 years, it is a wonder we can still add and subtract! On the other hand, I think most math and science educators in the U.S. would agree that there is a very real problem with the scientific literacy of today’s students.
One lesson to be learned, I suppose, is that education should not be a monolithic, unalterable program. Toplis’ bashing of biological science seems quite silly in modern times, and any shift in emphasis towards biology probably helped England become the originator of the next great scientific revolution, namely Charles Darwin‘s theory of evolution. As educators, we have to be careful that a perceived loss of scientific literacy is not in fact our own curmudgeonly resistance to change.
Toplis’ paper can also be used as a “call to arms” for modern educators. The fact that, 200 years ago, a mathematician was faced with the same problems of science and math literacy that we are facing today suggests that we must always be vigilant in pushing such topics in the classroom and finding new ways to make the material accessible to new generations of students.