Scientific American has a guest blog post with the title: *Mathematicians Are Overselling the Idea That “Math Is Everywhere*, which argues in its subtitle: *The mathematics that is most important to society is the province of the exceptional few—and that’s always been true*. Now I’m not really interested in the substantial argument of the article but the author, Michael J. Barany, opens his piece with some historical comments that I find to be substantially wrong; a situation made worse by the fact that the author is a historian of mathematics.

Barany’s third paragraph starts as follows:

*In the first agricultural societies in the cradle of civilization, math connected the heavens and the earth. Priests used astronomical calculations to mark the seasons and interpret divine will, and their special command of mathematics gave them power and privilege in their societies.*

We are taking about the area loosely known as Babylon, although the names and culture changed over the millennia, and it is largely a myth, not only for this culture, that astronomical calculations were used to mark the seasons. The Babylonian astrologers certainly interpreted the divine will but they were civil servants who whilst certainly belonging to the upper echelons of society did not have much in the way of power or privilege. They were trained experts who did a job for which they got paid. If they did it well they lived a peaceful life and if they did it badly they risked an awful lot, including their lives.

Barany continues as follows:

*As early economies grew larger and more complex, merchants and craftsmen incorporated more and more basic mathematics into their work, but for them mathematics was a trick of the trade rather than a public good. For millennia, advanced math remained the concern of the well-off, as either a philosophical pastime or a means to assert special authority.*

It is certainly true that merchants and craftsmen in advanced societies – Babylon, Greece, Rome – used basic mathematics in their work but as these people provide the bedrock of their societies – food, housing etc. – I think it is safe to say that their maths based activities were in general for the public good. As for advanced maths, and here I restrict myself to European history, it appeared no earlier than 1500 BCE in Babylon and had disappeared again by the fourth century CE with the collapse of the Roman Empire, so we are talking about two millennia at the most. Also for a large part of that time the Romans, who were the dominant power of the period, didn’t really have much interest in advance maths at all.

With the rebirth of European learned culture in the High Middle ages we have a society that founded the European universities but, like the Romans, didn’t really care for advanced maths, which only really began to reappear in the fifteenth century. Barany’s next paragraph contains an inherent contradiction:

*The first relatively widespread suggestions that anything beyond simple practical math ought to have a wider reach date to what historians call the Early Modern period, beginning around five centuries ago, when many of our modern social structures and institutions started to take shape. Just as Martin Luther and other early Protestants began to insist that Scripture should be available to the masses in their own languages, scientific writers like Welsh polymath Robert Recorde used the relatively new technology of the printing press to promote math for the people. Recorde’s 1543 English arithmetic textbook began with an argument that “no man can do any thing alone, and much less talk or bargain with another, but he shall still have to do with number” and that numbers’ uses were “unnumerable” (pun intended).*

Barany says, “that anything beyond simple practical math ought to have a wider reach…” and then goes on to suggest that this was typified by Robert Recorde with his *The Grounde of Artes *from 1543. Recorde’s book is very basic arithmetic; it is an abbacus or reckoning book for teaching basic arithmetic and book keeping to apprentices. In other words it is a book of simple practical maths. Historically what makes Recorde’s book interesting is that it is the first such book written in English, whereas on the continent such books had been being produced in the vernacular as manuscripts and then later as printed books since the thirteenth century when Leonardo of Pisa produced his *Libre Abbaci*, the book that gave the genre its name. Abbaci comes from the Italian verb to calculate or to reckon.

What however led me to write this post is the beginning of Barany’s next paragraph:

*Far more influential and representative of this period, however, was Recorde’s contemporary John Dee, who used his mathematical reputation to gain a powerful position advising Queen Elizabeth I. Dee hewed so closely to the idea of math as a secret and privileged kind of knowledge that his detractors accused him of conjuring and other occult practices.*

Barany is contrasting Recorde, man of the people bringing mathematic to the masses in his opinion with Dee an elitist defender of mathematics as secret and privileged knowledge. This would be quite funny if it wasn’t contained in an essay in *Scientific American*. Let us examine the two founders of the so-called English School of Mathematics a little more closely.

Robert Recorde who obtained a doctorate in medicine from Cambridge University was in fact personal physician to both Edward VI and Queen Mary. He served as comptroller of the Bristol Mint and supervisor of the Dublin Mint both important high level government appointments. Dee acquired a BA at St John’s College Cambridge and became a fellow of Trinity College. He then travelled extensively on the continent studying in Leuven under Gemma Frisius. Shortly after his return to England he was thrown into to prison on suspicion of sedition against Queen Mary; a charge of which he was eventually cleared. Although consulted oft by Queen Elizabeth he never, as opposed to Recorde, managed to obtain an official court appointment.

On the mathematical side Recorde did indeed write and publish, in English, a series of four introductory mathematics textbooks establishing the so-called English School of Mathematics. Following Recorde’s death it was Dee who edited and published further editions of Recorde’s mathematics books. Dee, having studied under Gemma Frisius and Gerard Mercator, introduced modern cartography and globe making into Britain. He also taught navigation and cartography to the captains of the Muscovy Trading Company. In his home in Mortlake, Dee assembled the largest mathematics library in Europe, which functioned as a sort of open university for all who wished to come and study with him. His most important pupil was his foster son Thomas Digges who went on to become the most important English mathematical practitioner of the next generation. Dee also wrote the preface to the first English translation of Euclid’s Elements by Henry Billingsley. The preface is a brilliant tour de force surveying, in English, all the existing branches of mathematics. Somehow this is not the picture of a man, who *hewed so closely to the idea of math as a secret and privileged kind of knowledge*. Dee was an evangelising populariser and propagator of mathematics for everyman.

It is however Barany’s next segment that should leave any historian of science or mathematics totally gobsmacked and gasping for words. He writes:

*In the seventeenth century’s Scientific Revolution, the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty.*

What can I say? I hardly know where to begin. Let us just list the major seventeenth-century contributors to the so-called Scientific Revolution, which itself has been characterised as the *mathematization* *of nature* (my emphasis). Simon Stevin, Johannes Kepler, Galileo Galilei, René Descartes, Blaise Pascal, Christiaan Huygens and last but by no means least Isaac Newton. Every single one of them a mathematician, whose very substantial contributions to the so-called Scientific Revolution were all mathematical. I could also add an even longer list of not quite so well known mathematicians who contributed. The seventeenth century has also been characterised, by more than one historian of mathematics as the golden age of mathematics, producing as it did modern algebra, analytical geometry and calculus along with a whole raft full of other mathematical developments.

The only thing I can say in Barany’s defence is that he in apparently a history of modern, i.e. twentieth-century, mathematics. I would politely suggest that should he again venture somewhat deeper into the past that he first does a little more research.

Thanks for your reactions, Thony! Here’s my reply: http://mbarany.com/ReplyToThonyC.html

I found Barany’s argument rather confused. For example in the last paragraph of his Scientific American blog he says:

“We have to recognize that elite mathematics today, while much more inclusive than it was one or five or fifty centuries ago, remains a discipline that vests special authority in those who, by virtue of gender, race, and class, are often already among our society’s most powerful.”

I could replace that word ‘mathematics’ with any number of others: sport, classical music, literature etc., without making the paragraph less true. Olympic athletes, concert pianists, prize-winning authors are all members of elites, but this does not make sport, classical music or literature any less all-pervasive in our society.

What Barany does not consider is that everyone needs a certain level of mathematical understanding just to operate in modern society, otherwise the elite-level mathematics that underlies it, becomes no more than ‘magic’. Jordan Ellenberg, whom he criticises in his blog, at least is trying to educate the public. We need to remember that what Ellenberg is popularising now, would have been elite mathematics when Fermat and Pascal were responding to the Chevalier de Mere’s question.

Hi Laurence, thanks for chiming in! Just to set the record straight on Ellenberg: I think what he’s doing is great and I’m a big fan of his. I quoted from his HNTBW publicity to make a broader point about current math-policy rhetoric. I wonder, though, per your point, just what mathematical understanding everyone does need (and I’m not alone in wondering about this!). I think numeracy is extremely important, particularly the kind of numeracy required to understand budgets and public policies. That’s an argument for a certain kind of math education that does not boil down to saying math is all around us, but instead focuses on a few key areas where certain kinds of math (in this case some fairly elementary kinds mixed with some much more advanced points) have a large effect on one’s life.

Your comparisons to sports and music show something important, too. Neither really make their case for public support by claiming to be “everywhere” in the way math does, but it’s true that they cultivate elites of their own. And their responses to inequalities of access show what a response to historical elitism can look like. Consider the move in many orchestras to do blind auditions, which dramatically increased those orchestras’ diversity. Or, in the United States, the long legacy of Title IX regulations on women’s access to sports. By understanding that their fields are built around elitism with numerous gateways and hurdles, those connected to music and sports are better able to take steps to make their fields more fair and just than they would otherwise be.

Michael,

Unfortunately, the system here does not allow me to thread this comment as a reply to your comment of August 19th, but I would like to make one point. Firstly, I was not thinking of basic numeracy but of an understanding of probability (which I think that Ellenburg also majors on – I will have to get his book to see). Now even though you may say this is just one area of maths, it really is all around us in the sense that it affects us in so many different ways.

For example, when the weather forecaster says there is a 30% chance of rain today, do you take your raincoat, or just an umbrella, or do you take a chance on avoiding it? if you go to a new web site and it requires you create a password to use it, do you know how to create a sufficiently strong password that you can also remember? Do you know the difference between Bayesian and frequentist statistics and how it can affect the significance of a result?

That last case may seem esoteric, but there have been miscarriges of justice when expert witnesses have got their statistics wrong. In one such case in the UK, an expert witness gave a statistical argument that an outcome that had happened was highly improbable. One cannot be surprised that the jury were taken in, but so were the judge and both prosecution and defence barristers who should have known better, or even known that they did not know and had the wit to ask the right questions.

Now, as I said above, all that would have been elite mathematics in the 17-18th Century (Pierre Fermat’s & Thomas Bayes’ lifetimes). Who is to say that, for example, vector or matrix algebra, or even number theory will not be as important in the future as understanding probability is now. A child being educated now, may (in the West at least) still be alive in 2100, Are we really confident that basic numeracy will still be sufficient then?.

Reblogged this on dohalu and commented:

Very interesting,

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I’m afraid you have committed a vicious error by omitting the greatest mind of the seventeenth century.

“Libre Abbaci”? Italian?

Leonardo of Pisa’s Liber Abbaci is in Latin, but there are texts in Italian based on it in the 13th century — not sure what the date of the earliest is. They are libri d’abaco.

Hi David.

I have a digital copy of Leonardo Pisano’s Libro d’abbaco MS 2404. The dialect is northern Italian. It’s the original ‘mathematics for dummies’ that popularized ‘modern’ math in Europe.

See

http://devlinsangle.blogspot.com.au/2011/09/first-arithmetic-textbook-in-western.html

and

https://www.maa.org/external_archive/devlin/Fibonacci.pdf

Best wishes

Jonathan

Jonathan, The first link takes me to an image of a MS written in 1290 which may or (more probably) may not be a copy of a text by Leonardo Pisano. Its title is Livero de l’abbecho. The second takes me to images of the Latin text of Leonardo’s Liber Abbaci. The claim that MS2404 is by Leonardo is rather at odds with its first sentence which you quote in translation as “This is the book of abacus according to the opinion of master Leonardo of the house of sons of Bonacie from Pisa.” “According to the opinion of” would seem more likely to mean “based on the work of” than “written by”. But you know more about all this than I do, David

“I would politely suggest that should he again venture somewhat deeper into the past that he first does a little more research.”

I would suggest some comparative reading. Lot of interesting recent research on literacy and power in the ancient world.

A comparative approach on maths and literacy as forms of symbolic media would be quite fruitfull and interesting I suspect.

Reading on the growth of ancient literacy and its relationship with power would also help frame the questions better and understand the issues, which seem to be shared.

Some comparative reading here would have resolved some of the issues made in the article and help frame the subject better.

It is an interesting subject. I would encourage Micheal to get reading and learning and start again.

Sure their is an interesting and fruitfull paper waiting to emerge from the ashes of this one.

I know that Newton was at least a little suspicious of some forms of math, he did not condone the use of elementary algebra, electing to write his Principia entirely using geometry and diagrams. Newton never saw anything resembling an F=ma.

Read this, in particular the last paragraphs

Hi Thony

Re: “Historically what makes Recorde’s (1543) book interesting is that it is the first such book written in English…”. This is incorrect. I provide two examples that predate Recorde.

An introduccion for to lerne to rekyn, Anon, printed by John Herford, St. Albans, 1537. (NOTE: I am grateful to the British Library for making images of this book available to me.)

An Introductiō for to Lerne to Recken with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym, Anon, printed by Nicholas Bourman, London, 1539.

The oldest known English text on arithmetic was printed and published in England in 1526.

Best wishes

Jonathan Crabtree

Thank you! How come I’ve never come across your wonderful website before?

I think one of the most significant flaws made with history is to suggest that social status is derived from having a particular skill and to assume that this repeats through time and across cultures.

Its not the case and you do not find a consistent pattern across the ancient world.

No hard and fast rule you can make other than not to make generlized assumptions as the situation alters from one society to the next.

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European maths took off only after it came in contact with India. All the early maths and science were directly copied from Indian Mathematics Schools and Indian ancient text books.

Al-Khwārizmī’s algebra was not from either India or Greece.

Quite simply not true, Mr Singh