Question

 

• What do you know about how math and historical developments are related? Share any information you already have about the topic.
• Are there any questions you still have?
• Do you believe that the teaching and learning of math has changed?
• What are some of your first conjectures of how math affected the growth of civilization?

Answer 1

The history of math goes all the way back to pre-historic times when hunter-gatherers created words to identify one, two, or more than two animals or objects. As you can imagine, it was critical for them to know how many saber-tooth tigers were outside the cave! Archaeologists have found artifacts showing early math development in Africa that are more than 20,000 years old. Other archaeologists have found multiplication tables on clay tablets that they date to the Babylonians in 2500 BCE; some of these tablets may have even been geometry homework!

Pre-historic Africans started using numbers to track time about 20,000 years ago. The Rhind Papyrus (1650 BCE) shows how ancient Egyptians worked out arithmetic and geometry problems in the first math textbook.

Babylonian mathematicians were the first known to create a character for zero. Hypatia worked with her father Theon to translate math texts into Greek. The Greeks expanded the math developed by the ancient Egyptians and Babylonians to promote a systematic study of math. Pythagorasdeveloped his famous theorem about right triangles around 530 BCE and even inspired a religion and school that worshipped numbers.

Today's geometry textbooks are direct descendants of Euclid's Elementswritten by the famous Greek mathematician. Meanwhile, thousands of miles away in what would one day be Mexico, Mayan mathematicians independently developed a number system sophisticated enough to predict astronomy-related events.

The development of math in Europe almost came to a stop during the medieval centuries but continued to progress quickly in China. Liu Hui used a 192-sided polygon to calculate the value of pi to five decimal places around 263 CE. The first abacus was probably created in China to help civil servants calculate taxes, wages, and engineering solutions. During the 13th century, Qin Jiushao used approximations to solve quadratic and cubic equations. These solutions were unknown in the West until the 17th century.

Mathematicians in India were also centuries ahead of those in Europe. Many historians believe that Pythagoras learned his geometry from an Indian textbook, the Shulba Sutras, which may have been written as early as 700 BCE. Jain mathematicians recognized different types of infinities a couple of hundred years later. Indian mathematicians were also the first to see zero as a number rather than just a placeholder and develop trigonometric concepts like sine and tangent to calculate distances. By the 12th century, Indian mathematicians developed the early foundations of calculus.

The study of history traditionally brings to mind images of dark, winding archives and the smell of dust; visions of academics poring through piles of documents to uncover secrets and find the missing piece, piles that are invariably too small to cover the subject and too high to be worked through.

But thanks to the internet, an almost inconceivable amount of sources are now available to the historian. The basis of historical research – manuscripts, rare books, images, and documents of a private and administrative nature like letters and financial plans – can now be accessed from almost everywhere. And this increased quantity of available historical sources doesn’t just mean that we now know it better. It means that now, we can know it differently. This quantity has affected the nature of our research. It has not only changed the kinds of answers historical study can provide, but also what questions we ask.

History comes in two flavours. There’s what I call micro history, and then there’s long-durée historical reconstruction. The first is characterised by detailed but temporally and spatially limited case studies; the second is rather a second-order reflection oriented by a historical hypothesis. This sort embraces a long spatial and temporal span but is informed by a limited number of selected case studies.

This has long restricted the kind of history that can be studied. But by mathematically analysing large historical data sets, it becomes possible to integrate the two approaches, conducting deep source analysis systematically while covering long spatial and temporal distances. In the field of history of science, which I work in, this is allowing us to investigate how the scientific knowledge systems that now dictate our lives formed.

Why is this possible? First, because the selection of sources against which historical hypotheses are proved, modified and sometimes rejected has increased. But also because such an increased number of sources allows for the consideration of more perspectives simultaneously.

For instance, historians of knowledge can now not only consider a much larger corpus of sources, such as a large quantity of scientific treatises from the past, but also the sources concerned with the institutional, economic and social context in which such treaties were produced. Historians have long called for a contextualised history of knowledge, but until now, long-durée historical reconstructions could only connect a few well studied examples by means of specific hypotheses of an economic or conceptual nature.

But if a much larger corpus of sources is able to be considered and analysed in detail, we can reflect more broadly about mechanisms of knowledge evolution. This allows us to move toward a more abstract understanding of our past. We can speak about the mechanisms of history – and other humanities – in a totally new, informed way.

An entire new discipline – the digital humanities – emerged in order to allow scholars to manage this wealth of information. Historical sources, their electronic copies and bibliographic metadata are increasingly immersed in a frame of annotations, ideas and relations electronically produced by historians while studying our material and intellectual heritage. Appropriate repositories have been created for all this data and a standard format for its preservation and reuse independently from these platforms and tools is being developed.

Open access to data, even more than to publications, is therefore becoming imperative. History writing is leading the humanities to contribute to that new frontier of science called big data.

Historical data is being explored by means of graph visualisations and network parameters. In particular, some models and packages allow historians to simulate how networks are changed by the decisions of those involved. These simulations are based on hypotheses that are formulated by the historians and hard-coded in the scripts. One example hypotheses could be “religious differences do not represent any obstacle for communication in the scientific society”.

For example, the historian Ingeborg van Vugt has used this multi-layered approach to explore the different ways in which information circulated in the Republic of Letters, the long-distance intellectual community of the late 17th and 18th centuries in Europe and America. Such research allows us to better visualise how the Age of Enlightenment, driven forward by these intellectuals, developed. The next step could be to statistically model this network, and so be able to pursue her research question by integrating an even broader wealth of data.

A network model for studies in history of knowledge has to consider an unusually varied set of data. There is the data of a social nature concerned with people and organisations; that concerned with material aspects of history, such as the conservation life of a book; and the data that represent the actual knowledge, the content of the sources. These are three different levels of one and the same evolving network for which explanatory mathematical models have been rarely conceived and even less realised. From this perspective, history writing is even about to challenge applied statistics.

Although mathematical modelling in the frame of history is clearly at its first steps, its introduction already appears unstoppable. This is creating the conditions for the emergence of a new vision, according to which we might be able to develop general mathematical models to explain how ideas and knowledge changed from a social and historical perspective. Perhaps we could even use these models in different areas of scientific research dedicated to the present and the future. And in such a future, humanities and exact science will begin to use the same mathematical language.

Pick any number. If that number is even, divide it by 2. If it's odd, multiply it by 3 and add 1. Now repeat the process with your new number. If you keep going, you'll eventually end up at 1. Every time.

Mathematicians have tried millions of numbers and they've never found a single one that didn't end up at 1 eventually. The thing is, they've never been able to prove that there isn't a special number out there that never leads to 1. It's possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1. But no one has ever been able to prove that for certain.

So you're moving into your new apartment, and you're trying to bring your sofa. The problem is, the hallway turns and you have to fit your sofa around a corner. If it's a small sofa, that might not be a problem, but a really big sofa is sure to get stuck. If you're a mathematician, you ask yourself: What's the largest sofa you could possibly fit around the corner? It doesn't have to be a rectangular sofa either, it can be any shape.

This is the essence of the moving sofa problem. Here are the specifics: the whole problem is in two dimensions, the corner is a 90-degree angle, and the width of the corridor is 1. What is the largest two-dimensional area that can fit around the corner?

The largest area that can fit around a corner is called—I kid you not—the sofa constant. Nobody knows for sure how big it is, but we have some pretty big sofas that do work, so we know it has to be at least as big as them. We also have some sofas that don't work, so it has to be smaller than those. All together, we know the sofa constant has to be between 2.2195 and 2.8284

Remember the pythagorean theorem, A2 + B2 = C2? The three letters correspond to the three sides of a right triangle. In a Pythagorean triangle, and all three sides are whole numbers. Let's extend this idea to three dimensions. In three dimensions, there are four numbers. In the image above, they are A, B, C, and G. The first three are the dimensions of a box, and G is the diagonal running from one of the top corners to the opposite bottom corner.

Just as there are some triangles where all three sides are whole numbers, there are also some boxes where the three sides and the spatial diagonal (A, B, C, and G) are whole numbers. But there are also three more diagonals on the three surfaces (D, E, and F) and that raises an interesting question: can there be a box where all seven of these lengths are integers?

The goal is to find a box where A2 + B2+ C2 = G2, and where all seven numbers are integers. This is called a perfect cuboid. Mathematicians have tried many different possibilities and have yet to find a single one that works. But they also haven't been able to prove that such a box doesn't exist, so the hunt is on for a perfect cuboid

Draw a closed loop. The loop doesn't have to be a circle, it can be any shape you want, but the beginning and the end have to meet and the loop can't cross itself. It should be possible to draw a square inside the loop so that all four corners of the square are touching the loop. According to the inscribed square hypothesis, every closed loop (specifically every plane simple closed curve) should have an inscribed square, a square where all four corners lie somewhere on the loop.

This has already been solved for a number of other shapes, such as triangles and rectangles. But squares are tricky, and so far a formal proof has eluded mathematicians

The happy ending problem is so named because it led to the marriage of two mathematicians who worked on it, George Szekeres and Esther Klein. Essentially, the problem works like this:

Make five dots at random places on a piece of paper. Assuming the dots aren't deliberately arranged—say, in a line—you should always be able to connect four of them to create a convex quadrilateral, which is a shape with four sides where all of the corners are less than 180 degrees. The gist of this theorem is that you'll always be able to create a convex quadrilateral with five random dots, regardless of where those dots are positioned.

So that's how it works for four sides. But for a pentagon, a five-sided shape, it turns out you need nine dots. For a hexagon, it's 17 dots. But beyond that, we don't know. It's a mystery how many dots is required to create a heptagon or any larger shapes. More importantly, there should be a formula to tell us how many dots are required for any shape. Mathematicians suspect the equation is M=1+2^N-2 where M is the number of dots and N is the number of sides in the shape. But as yet, they've only been able to prove that the answer is at least as big as the answer you get that way.