TOK discussion

Mathematics: Invented or Discovered? Consider logarithms

The question is whether mathematics is an invention (a creation of the human mind) or a discovery (something that exists independent of us).

Intuitionists say that mathematics is a creation of the human mind; therefore, it is invented by humans. Any mathematical idea or object exists only in our mind and does not have an existence independent of us.

Alternatively, Platonists claim that any mathematical idea or object exists and we can only "see" them through our mind. This means that any mathematical concept or construction exists independent of humans and can, therefore, be discovered.

In 1614, the Scottish nobleman, mathematician, scientist and theologian John Napier (1550-1617) published Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Law of Logarithms). This Latin text explained (although it consisted mostly of tables of numbers) Napier's invention of logarithms. His motivation is made clear in the book's preface when he wrote (taken from 1616 English translation):

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.

The logarithms that Napier invented are not the same as common logarithms (base 10) and natural logarithms (base e) with which all Maths HL & SL students are very familiar. However, Napier certainly introduced the central concept of logarithms where one manipulates the powers (exponents) of numbers to carry out a calculation (multiplication, division, roots, etc) instead of performing the tedious calculation directly - thus, turning difficult multiplications and divisions into simple additions and subtractions. The invention was incredibly beneficial to scientists - especially astronomers - who routinely had to perform operations with very large numbers. The French mathematician and astronomer Pierre-Simon Laplace (1749-1827) is reputed to have said that logarithms, "by shortening the labours, doubled the life of the astronomer."

The English mathematician Henry Briggs (1561-1630) met with Napier and made improvements to Napier's invention - the first was changing logarithms to have a base of 10 (common logarithms). After the invention of logarithms came the invention of the logarithmic scale by Edmund Gunter (1581-1626). Gunter was able to perform calculations by using dividers (a device similar to a compass) to add and subtract distances along the logarithmic scale. A few years later, the English mathematician William Oughtred (1574-1660) - who invented the notation × for multiplication - put two logarithmic scales together (so that one could slide along the other) and inventing the first linear slide rule. In 1632 he published Circles of Proportion and the Horizontal Instrument which described slide rules and sundials. For over 300 years, the slide rule was the most effective handheld computing device.

Up to this point in this blog entry some form of the word invent is used eight times. Virtually all references to John Napier state that he "invented" logarithms. Was it an 'invention' or a 'discovery'? Certainly, all would agree that Oughtred's introduction of the × symbol for multiplication is an invention because the symbol was new and had not been used for this purpose before, but Oughtred did not invent multiplication. Early in the 18th century the greek letter π (pi) started to be used to represent the ratio of a circle's circumference to its diameter. Again, most would agree that the use of this symbol was an invention but what about the number (i.e. ratio of circumference to diameter) that the symbol represents - was that invented? Most would say 'no'. Even if humans did not exist, wouldn't the ratio of the circumference of any circle to its diameter still be the same constant that we decided to represent with the greek letter π ?

No doubt Napier invented the name logarithm (or, in Latin, logarithmus) which combined the Greek words logos (reckoning or ratio) and arithmos (number). But did he invent the idea behind logarithms, or did he simply uncover and utilize a concept that already existed and was waiting to be discovered by someone who was sufficiently astute and motivated? A discussion of this question can be a very useful and interesting way to integrate some TOK thinking/activity into a maths lesson. It can also serve as the starting point of an activity that could be used in the TOK classroom itself.

Consider the following passage from the University of Stirling (Scotland) announcing a lecture to commemorate what it refers to as the "400th anniversary of Napier's discovery of logarithms." (bold highlights added here and below)

One of the most important discoveries in the history of mathematics was the invention in 1614 of logarithms as a calculating aid by the famous Scottish mathematician John Napier. This came at an opportune time for the great astronomers of the day who were making massive strides in their understanding of the movement of the planets. Those of a certain age will also remember using logarithms in school before the days of electronic calculators! To celebrate this quatercentenary we shall have a look at Napier's life and some of his other inventions as well as seeing why logarithms proved such an important breakthrough.