"Numbers are everywhere, hidden servants that scurry around behind the scenes – carrying messages, correcting our spelling when we type, scheduling our holiday flights to the Caribbean, keeping track of our goods, ensuring that our medicines are safe and effective."
Title: Taming the Infinite; The story of mathematics from the first numbers to chaos theory
Author: Ian Stewart
Category: Expository – Mathematics/History
What is the book about as a whole?
The book is a brief but concise story about the evolution of mathematics through time. It talks not only about the theoretical and abstract dimension of it but as well about how it has shaped the human life.
What is being said in detail, and how?
In the preface Ian Stewart tell us how Mathematics is the working force behind the apparent magic of our contemporary technology. In a sense it has been one of the most precious discoveries of humanity and for that deserves a book telling its story. However this book doesn’t aim to be “The ultimate history” of math. Instead seeks to be an introduction for the reader to explore some of the fascinating evolution of mathematics from the first numbers to the chaos theory.
Although mathematics is no longer limited just to numbers, they were part of the primeval forest where this discovery took its first steps. In a very literal sense it started with numbers and from it has set the mood for huge leaps in the evolution of math.
But what are numbers? Stewart mentions “Numbers are some kind of mental construct, yet we feel that they would continue to have meaning even if humanity were wiped out by a global catastrophe and there were no minds left to contemplate them” This brought me back to one of the main questions I’ve been struggling for a long time: “Are numbers implicit in nature and we discover them? Or are rational impositions from humans on nature?
But, aside from these interesting philosophical questions the mere history of the evolution of numbers is super interesting. From rather humble beginnings in tally marks about 37,000 years ago; they evolve to the highly sophisticated system of marks in tokens during Babylon’s times to more complex and full of symbolism during the Egyptian times. Finally arriving to our time and facilitating our lives.
As Stewart states “Mathematics and culture co-evolve” and in great part our current society is the result among many other things of mathematical progress. But Math does his thing on the back and is not all obvious its progress and relationship with our life. In days were a pocket calculator is common and even trivial thing in the world, we tend to underestimate or completely ignore the importance of mathematics. It helps us from tiny tasks as calculating the tip in a restaurant to locate the next coffee shop via GPS.
Besides symbolic reasoning in mathematic we also have visual reasoning. The second then being intrinsic related with shape and giving place to the birth of a way of thinking. This system called Geometry was one of the greatest discoveries in math since it uses figure to derive conclusions. It has often being called less formal than symbols. But the work of Euclid of Alexandria (who constructed it on the work of other mathematicians as Pythagoras) linked the use of pictures and the rigidness structure of logical proofs.
Pythagoras deserves a note on this story for he was the founder of a cult that placed mathematics in the core of the universe. Although “they understood that mathematics is about abstract concepts, not reality”; “they also believed that these abstractions were somehow embodied in ideal concepts”. One of the most influential beliefs of Pythagoreans was the belief that the universe is founded on numbers. They assign an extra symbolism to each number, for example 1= Unity; 2= Female principle, 3= Male principle and 4= the four elements. Then 10, being the sum of them, is deeply significant.
Then continue talking about Euclid we can say that he although not the most original mathematician he was a great synthetize. Indeed his books have become some of the most read and sold in the history of ideas. From the 5 books that survive of him, The Elements it’s his magna opus. In a 13 volume treatise he explains and works geometric propositions that will set the ground for further developments not only in geometry but in mathematical and logical theory. But you his book cannot be learn by repetition or memorization the only way to learn about Euclid understands what he is explaining.
In his theorems for example, he proves that the angles at the base of an isosceles triangle are equal (I.5), or that the angles inside a triangle are equal to two right angles (180°) (I.32) or the Pythagoras theorem (I.47).
But what of Euclid and later Archimedes work have to do with our lives, or to be fair ancient Greeks lives? In fact a lot because this way of thinking and logic help them to give sense to the world they live and shape it to their needs. To construct tunnels, make huge and beautiful buildings, make weapons and even calculate with impressing accuracy the size of the earth! And geometry is still working for us…
Greek mathematic (although without a great number system) contributed two crucial ideas to the human development. First a systematic understanding of geometry, that allows us to enhance our understanding. And second, the systematic use of logical deduction to make sure that what was being asserted could also be justified. Both remain vital today.
We does our numbers come from? Why “10” is so special as to be the basis of our number system? We are very used to use them but us rarely if not never wonder about its origin or if there are other systems of numbers. In this chapter the author explores the evolution of number systems and how we manage to get to our familiar decimal system.
We are reminded that most westerners (Is Latin-America part of the west?) know about the roman numbers but this is often the single other system that we know. We talked a little bit before about the primeval systems such as the Babylonic (based on the 60 and with marks) or the highly symbolic Egyptian. But then after their cultures decayed the greeks continued mathematic in the west. Although they took a big step backwards from Babylonians since its number system, with unrelated symbols or later complex and finally based on letters do not suit for easy mathematical calculation.
Although the romans didn’t actually succeed on overcoming that messy system their numbers where a little better for calculation and lasted in the west until the middle ages. On the other hand Indian number system was working from some time ago with 10 digits. They used positional value for 9 symbols, moving to the left would mean multiply by 10. But in the beginning they only left a blank space and this created the chance of mistake. And this is the birth of 0 to note that there was no value on that space. This invention of Zero lead to the 10 digits that we are so familiar with.
This Hindu system spread into the Arabic world. There are accounts that in 776 this method was shown to the Caliph court. Being a commercial civilization they were particularly interested in its applications for commerce. By 830 their use was popular and there were an increased awareness “of the possibility of performing all numerical calculations using only the ten digits.”
From this point this new system jumped to middle ages Western Europe that was tight to the Arab world by links of commerce. A very influential figure was Leonardo of Pisa, better known as Fibonacci, who in his book Liber Abbaci (1202) introduced the system to Europe. They were not easily accepted as well as negative numbers. But since the arithmetic worked perfectly well they were so useful that “it would have been silly not to use them.”
Different notations are still in use and alive. And even our computers work in the binary system not in the decimal that we are so used to. Now, have you ever wonder what is the X in an equation? Well is the Unknown. The next big step in mathematics was the elaboration of a way to devise the unknown from the known. Now we get to the field of equations, linear, quadratic and trigonometric. We might don’t recognize but our world is highly functional as we know it today given the learned ability of humanity to work with equations. One of the most exciting times for mathematicians was the renaissance where duels of mathematics were held in public plazas in Italy as a way to show mastery of mathematics. I wonder how many of these “masters” of their time could pass a 9th grade math exam. Well, it actually doesn’t matter they opened the field.
Now let’s talk about trigonometry. Yes we are going to talk about triangles and their magnificent properties. During the first centuries of the first millennia astronomers needed desperately a way to calculate in reliable fashion the movements of the heaven. Realizing the huge distances that they needed to operate they find out about the properties of triangles. Ptolemy the great astronomer proponent of the geocentric model of the universe, was one of the users of trigonometry in those times. But what most people don’t know is that triangles are still essential in our life, from calculating an extension of land in India to deliver our text message in or friend’s cellphones. Our current world still benefits of trigonometry, and you if you have look for a coffee shop in the last few days.
So now, your GPS tells you it found a coffee shop near to you and gives you the coordinates. Well that which the GPS gives you is the topic of the next chapter. Coordinates and planes. But they not only help us to find the coffee, they help us to do visual representations of mathematical functions. It is really helpful to have this since our mind likes the visual dimension of the problems and might help us to ge the idea faster. Although Descartes is famous for his chart, Piere de Fermat was also another great thinker that helped to develop the usage of planes and coordinates.
Our mind likes patterns. Actually I would say it loves them. Well this lifelong love of humanity has a great expression in numbers. We realized that numbers might create wonderful patterns that might allow us to predict and describe isomorphically our reality. That is the birth of number theory an our friend Euclid is the first example that we have of such an effort to systematize numbers. Of particular and even esoteric interest are Prime numbers. Their properties seem so reachable and even though they have managed to evade any kind of predictably. So what is next in the amazing story of mathematics? Well, is the invention of Calculus that we associate to the great name of Sir Isaac Newton, is somehow like the invention of a new grammar that have empowered us to overcome difficulties and master in better way problems we face in our day to day world. But is there any mathematic implicit in our world or rather is our mind the one that impress a mathematical bias in reality? That question is really important one and many authors have dealt with that for a long time, on philosophy, science and in random talks on coffee shops. The truth is that the tools that calculus in its different forms give us somehow make that our world otherwise frightening and unpredictable become a cozy and often reliable place to live. We see applications on music, finances, industry, all over the place!
Let’s talk about impossible things now. First Impossible numbers… are they possible? Well at least not in the abstract world of mathematics, in the imaginary realm that the concepts can create by themselves. Think about imaginary numbers like the square root of 1. Or rational numbers. Or decimals. Well we can imagine them and they are certainly useful. What stroke me the most is how we are so used to them now and even just some centuries ago it would have been just crazy talk. As well as it happen with the Zero. But how useful they are! Now connected to that we have the impossible numbers we can talk about impossible triangles. Euclid probably would have gone nuts just trying to conceive it, but never the less they are useful to think if we are dealing with surfaces different to the plane for example a triangle in a sphere. But let us give a quick thought… our world is not flat is roughly a sphere. So after all this non-Euclidean geometry is essential if we are dealing with certain kind of problems in reality.
The uncertain has always been a problem for humans, but how mathematics can help us to tame that gigantic problem. Well, the arising of probability theory allowed us to make calculations about the future and about the way things might go. Not that we are now able to predict for sure what is going to happen but now we have an often reliable approximation about what is likely to happen. And this is not only useful in throwing the dice it is also crucial in so many areas of our life today from insurance to aircraft design. And knowing that probability and in fact mathematic have become a major part of the human endeavors, we have been making continuous efforts to improve our calculating power. Those efforts have ended in what now we have in front of our eyes, this computer that is running thanks a hidden mathematic. Our technology now the highest link so far achieved in a chain with humble beginnings. Finally, as a cherry on the cake we might talk about Chaos and Complexity being the current dialogue in modern mathematics. They represent the verge of our ever expanding knowledge of mathematics.
What are the author’s questions and problems?
The author is concerned about present a brief and exciting story about the evolution of mathematics. He is faced with the problem of how to synthetize such much history. He I is also concerned about how to relate our everyday activities to this fascinating story, in order to do that he present us with little charts relating each chapter with how mathematics solved practical problems both for the people in ancient time but also for us today.
What of it?
Without doubt this has been one of my favorite books since it present in such epic terms a topic that usually one would disregard as boring. I particularly loved that he made me realized how fascinating and fantastic mathematics are and how much we ought to be in debt with all the efforts that people invested in the evolution of mathematics. Honestly it changed my view to the everyday objects, it made me realize how much math is implied in our everyday life and how different (for the worst) our life would be without math.
What quotes did I like of the book?
The rise of human civilization and the rise of mathematics have gone hand in hand.
Without numbers civilization as we now it could not exist.
Numbers are everywhere, hidden servants that scurry around behind the scenes – carrying messages, correcting our spelling when we type, scheduling our holiday flights to the Caribbean, keeping track of our goods, ensuring that our medicines are safe and effective.
And, for balance, making nuclear weapons possible, and guiding bombs and missiles to their targets. Not every application of mathematics has improved the human condition.
Whether you like arithmetic or not, it is hard to deny the profound effects that numbers have had on the development of human civilization.
…mathematics and culture co-evolve.
Perhaps the best way to think of Euclid’s work is as an examination of the logic of spatial relationships.
To modern mathematicians, what is most interesting about Euclid’s geometry is not its content, but its logical structure. […] Euclid does not merely assert that some theorem is true. He provides a proof.
Book I Proposition 5 proves that the angles at the base of an isosceles triangle (one with two equal sides) are equal. This theorem was known to generations of Victorian schoolboys as the pons asinorum or bridge of asses: the diagram looks like a bridge, and it was the first serious stumbling block for students who tried to learn the subject by rote instead of understanding it.
In 1906 the Danish scholar Heiberg was studying a 13th century parchment, with prayers written on it. He noticed faint lines from an earlier inscription, which had been erased to make room for the prayers. He discovered that the original document was a copy of several works by Archimedes, some of them previously unknown.
Our world is too complex, potentially dangerous, for us to base decisions on what we want to believe, rather than on what is actually the case.
In science, emphasis is placed on trying to prove that what you deeply believe to be the case is wrong. Ideas that survive stringent attempts to disprove them are more likely to be correct.
How can “0” be a number when a number is a quantity of things? Is nothing a quantity?
Bhaskara (known as ‘the teacher’) wrote three important works: Lilavati, Bijanganita and Siddahanta Siromani. According to Fyzi, court poet of the Mogul emperor Akbar, Lilavati was the name of Bhaskara´s daughter. Her father cast his daughter’s horoscope, and determined the most auspicious time for her wedding. To dramatize his forecast, he put a cup with a hole in it inside a bowl of water, constructed so that it would sink when the propitious moment arrived. But Lilavati leaned over the bowl and a pearl from her clothing fell into the cup and blocked the hole. The cup did not sink, which meant that Lilavati could never get married. To cheer her up, Bhaskara wrote a mathematics textbook for her. The legend does not record what she thought of this.
What books are connected with it?
Gödel, Escher, Bach by Douglas Hofstadter
The Copernican Revolution by Thomas Kuhn
The Elements by Euclid of Alexandria
Against the Gods. The remarkable story of risk by Bernstein
