A Brief History of Mathematical Thought: Key concepts and where they come from

by Luke Heaton

Mathematics is a product of human culture which has developed along with our attempts to comprehend the world around us. In A Brief History of Mathematical Thought, Luke Heaton explores how the language of mathematics has evolved over time, enabling new technologies and shaping the way people think. From stone-age rituals to algebra, calculus, and the concept of computation, Heaton shows the enormous influence of mathematics on science, philosophy and the broader human story.

The book traces the fascinating history of mathematical practice, focusing on the impact of key conceptual innovations. Its structure of thirteen chapters split between four sections is dictated by a combination of historical and thematic considerations.

In the first section, Heaton illuminates the fundamental concept of number. He begins with a speculative and rhetorical account of prehistoric rituals, before describing the practice of mathematics in Ancient Egypt, Babylon and Greece. He then examines the relationship between counting and the continuum of measurement, and explains how the rise of algebra has dramatically transformed our world. In the second section, he explores the origins of calculus and the conceptual shift that accompanied the birth of non-Euclidean geometries. In the third section, he examines the concept of the infinite and the fundamentals of formal logic. Finally, in section four, he considers the limits of formal proof, and the critical role of mathematics in our ongoing attempts to comprehend the world around us.

The story of mathematics is fascinating in its own right, but Heaton does more than simply outline a history of mathematical ideas. More importantly, he shows clearly how the history and philosophy of maths provides an invaluable perspective on human nature.

Genres: Mathematics, Science, Nonfiction, History, Collections, Stem, Academic

336 pages, Paperback

First published April 2, 2015

Reviews

[Brian Clegg · Rating 3 out of 5]

I had great hopes at the start of this book that we'd get a meaty but approachable history of the development of mathematics. When describing the origins of number, arithmetic and mathematical processes the opening section is pitched well, but things go downhill when we get to detailed mathematical exposition.

The problem may be that Luke Heaton is a plant scientist, which may have prepared him better for using maths than for explaining it. If we take, for instance, his explanation of the demonstration that the square root of two is irrational, I was lost after about two lines. As soon as Heaton gets into mathematical detail, his fluency and readability are lost.

There's a central chunk of the book where the mathematical content seems too heavy for the way the contextual text is written. We then get back onto more effective ground when dealing with logic and Turing's work, before diving back into rather more impenetrable territory.

One slight concern is that, in talking about Turing, Heaton presents the largely discredited idea that Turing committed suicide by eating a poisoned apple, laying the blame for this on his arrest and 'treatment' for homosexuality, a narrative that doesn't stand up (it's a shame the author didn't have a chance to read The Turing Guide). I suspect this is a one-off and much of the mathematical history is fine.

This was, then, a curate's egg book for me. I really enjoyed it when it was dealing with history and philosophy of maths, but found the technical explanations, even of mathematics I understand perfectly well, hard to follow.

[Brian Clegg · Rating 3 out of 5]

I had great hopes at the start of this book that we'd get a meaty but approachable history of the development of mathematics. When describing the origins of number, arithmetic and mathematical processes the opening section is pitched well, but things go downhill when we get to detailed mathematical exposition.

The problem may be that Luke Heaton is a plant scientist, which may have prepared him better for using maths than for explaining it. If we take, for instance, his explanation of the demonstration that the square root of two is irrational, I was lost after about two lines. As soon as Heaton gets into mathematical detail, his fluency and readability are lost.

There's a central chunk of the book where the mathematical content seems too heavy for the way the contextual text is written. We then get back onto more effective ground when dealing with logic and Turing's work, before diving back into rather more impenetrable territory.

One slight concern is that, in talking about Turing, Heaton presents the largely discredited idea that Turing committed suicide by eating a poisoned apple, laying the blame for this on his arrest and 'treatment' for homosexuality, a narrative that doesn't stand up (it's a shame the author didn't have a chance to read The Turing Guide). I suspect this is a one-off and much of the mathematical history is fine.

This was, then, a curate's egg book for me. I really enjoyed it when it was dealing with history and philosophy of maths, but found the technical explanations, even of mathematics I understand perfectly well, hard to follow.

[Al Bità · Rating 3 out of 5]

I am a bit of a sucker for books which promise to elucidate things mathematical. This is not because I am a mathematician, but because I am fascinated by the subject itself. I have been “caught” by promises in books that the the mathematics involved are “not difficult”, only to find that for my poor brain, they are! So their detailed maths contents end up not being read… None of this has stopped me being interested. Now, however, I have learned to read what I can, skim-read (even skip past) the rest, and hopefully thus managing to glean something from the book. Sometimes this works, but sometimes it doesn’t.

Heaton’s book is one of those that sits somewhere in the middle (for me) and I found it helpful (if not always easy to read). The intention here is to provide key concepts in mathematics, and Heaton provides the reader with thirteen chapters on just that. It is written in a way that should be basically accessible to the ordinary reader (though a slight tendency to prolixity can sometimes make this a little harder to grasp) and in general I think he does this pretty well.

Unfortunately, there are still situations when the mathematics “gets in the way” as it were. I would argue that maths is essentially about abstractions, and then about abstractions about these abstractions, and again, and again… For the ordinary reader, the jumps into higher abstractions can be difficult to comprehend. I would imagine that such readers would prefer that the writer just simply says something like “trust me, it is possible to do such abstractions and arrive at certain conclusions, such as…” I know I would. Instead the writers seem, at a certain point in their argument, to be unable to resist actually diving into the actual mathematical process, and the reader’s brain switches off. I feel that in many cases this is not necessary (but, of course, I may be wrong). So my default position takes over, and I skim-read through the esoteric symbols and complex logistics, then bravely read on, desperately trying to convince myself that I will not drown in my own ignorance…

My technique of ploughing on regardless seems to have worked for me with Heaton’s book. One may not always fully comprehend exactly what is going on, but one does get an idea about what is being talked about within each particular branch of mathematics, and I feel that this is better than not knowing anything at all!

[Danika Christensen · Rating 3 out of 5]

It has a pretty thorough description of math history. You can definitely tell Heaton is a biologist though, the whole first chapter is about evolution (which I hate), and there are stretches to include biology sometimes. You can also tell that he is passionate about math, but that he struggles with formulating proofs into easier to understand versions. (He gets cred for trying tho its not easy) He kind of loses me when he starts talking about Turing machines/cards. Also, at the very end, Heaton makes some very definitive claims about the existence of mathematics, instead of just keeping it as a recount of historical positions like the rest of the book.

[Ibrahim Zaidan · Rating 4 out of 5]

Quite interesting from the beginning to the end. However I found that when the writer explains math he does it a bit in a weird way, where reasoning is kind of unclear and vague.

[Mari · Rating 3 out of 5]

This was certainly an interesting book. It is exactly what the title says: A brief history of mathematical thought. So you get what you have been promised.

I especially liked that there were a lot of concepts that I never learned in school. Especially geometries were presented in an interesting way and throughout the reading process I was googling the concepts to learn some more. So it was also inspiring. Sometimes though, I was a bit lost on how the things presented connected to each other and why the author chose to explain some concepts and not others. So because of this, I would minus one point from the maximum five.

The second minus point I will have to give because of the errors. This is something that really makes me question the quality of the book. I read the paperback version so the page numbers correspond to that. Let me know if I have been mistaken on some of these. I'm not an expert in math, so I might also be wrong.

- Page 69: Both in the top half of the page and in the very end shouldn't there be q1 and not twice q2 in the function?
- Page 77: Somayaji's equation describes pi/4, not pi. Just like it was explained on the previous page.
- Page 79: 1000x != 2345.23452345... It should be 10,000x.
- Page 111: Halfway through the page there is A=a (mod M) and A=b (mod M). Shouldn't the second A be a B?
- Page 254: The last Peano's axiom should be n x (m+1) = (n x m) + n.

I found these with quite a superficial look into the book. It leads me to believe that there could be more which is a shame.

[Benjamin Manning · Rating 3 out of 5]

I saw it on a shelf for 2$ at a book store and I had to buy it. I'd say the first 3rd of this book was fascinating - outlining how mathematics has integrated into human evolution (we inherently understand certain mathematical concepts) and the kind of pre-2000 BC history of math; along with many examples of the very math being done. The remaining two thirds became FAR more technical (Heaton tried to explain Godel's incompleteness theorem...) and while this is cool stuff, it's hard to follow in quasi words/symbols that are trying to be readable. Just bit off a little more than can be chewed.

3 things I learned/will remember.

1. "Compared to other artifacts, the world of mathematics is strangely timeless: you don't need to know how the ancient Greeks lived to understand Euclidean Geometry." I'm currently in a quasi-philosophy of technological science course and this quote resonates with me as we discuss the state of the social sciences and how any social scientific result really depends on a time and place - mathematical knowledge, maybe it's different?

2. 2,000 year old chinese civilizations were solving systems of equations using elimination, and the west wasn't doing this until Gauss! (note, we were using substitution, which worked as well; still cool though).

3. Why are some people just SO good at eveything??? Alan Turing made essential discoveries in understanding how twins versus single babies develop in embryos. Come on Alan, leave some parts of science for the rest of us!!

[Diocletian · Rating 3 out of 5]

I liked the bits about Euclid's algorithm to find ratios and continued fractions solving quadratic equations. Revising the proof of why 0.9999...==1 was cool. The author makes a logical fallacy by using the phrase "everybody knows" quite often. Liberal use of "in other words" was much appreciated. I didn't enjoy the final third of the book too much.

[Bagja Hidayat · Rating 4 out of 5]

Luke Heaton investigated why and how human found the mathematical concept. Same like mammalian, human ‘s brain was mathematical thought. We are thinking on mathematical. Bird always remember how many their eggs like us always measure everything in daily life.

[Chris · Rating 4 out of 5]

This little book was quite readable even for a student who never went beyond the first semester of calculus. Some sections remained somewhat beyond my grasp, but there was enough there that I could follow, at least to a degree, and there was a lot there to think about. I think I gained a broader understanding of what the study of math is about and how it has developed over centuries. There is just a touch of the philosophical here, and that is what sustained my interest when my own grasp of the math faltered.

[Tom · Rating 2 out of 5]

This is a book that I think deserves a higher rating than I gave it, yet I personally had a 2/5 experience with it. In fairness to him, I think that this is largely because it is titled poorly rather than because it is not a good representation of his field.

With a background in engineering, much of this book felt like revision of old concepts rather than a history of the thoughts themselves. He brings too much mathematical specificity to the titular question - what do math and how? I was more interested in the philosophical and historical contexts of mathematics than the maths itself, and in fairness to him he covers those points as well, but not often enough. He often lapses into mathematical proofs where a believe the reader of this would prefer a broader "take it as it is" answer.

Rating:
2/5 - Heaton loves his field too much to write about it for someone not immersed in it, and I say that as someone with a relatively strong mathematical background.


Who I'd recommend this to:
Someone with an interest in actually learning mathematics rather than its history - the book does the former much better than the latter.

[Alberto Croquevielle · Rating 4 out of 5]

Interesting book, with a nice summary of key mathematical ideas originating both in Antiquity and in modern Europe. Some of the specific examples are hard to follow, and would have benefited from more pictures and longer explanations, but the main ideas are generally easy to grasp. I especially appreciate the exposition of how mathematics was done and thought about in old cultures (especially Ancient Greece), and some very nice insights into the issue of mathematical Platonism and whether math is discovered or invented.

[Matt · Rating 4 out of 5]

I think another editing/rewriting pass could have made this an excellent book. As it is, I'm really glad I read it, but I'd put it in the 'ambitious but uneven' basket. A lot of ground is covered, most of it is interesting (even the more philosophical strands that run through the book and take over in the final chapter), and I think I learned a few things. You can tell that Heaton is really into this stuff and wants to share genuine understanding with the reader, not just show off his own knowledge. But I found some of the explanations unclear, and although that's partly down to my own weaknesses, I think it's also a flaw in the book: Heaton's way of explaining Turing machines, for example, was quite confusing to me, even though I've encountered the concept many times before and happily followed explanations that went at least as deep, if not deeper. There are quite a lot of typos (not so much misspellings as missing or misplaced words, wrong parts of speech, that sort of thing), which isn't a big deal in itself but suggests a lack of careful editing, which would also explain why the clarity and flow are rather uneven. Still, I enjoyed this and will definitely have a look at anything else Heaton publishes. Do note that although the title is accurate, there's a bit more emphasis on the 'mathematical thought', and a bit less on the 'history', than you might expect. That suited me well, though, as I'm squarely in the target audience for books pitched at mathematically ignorant (and only moderately intelligent) readers, but willing to make them think.

[Alexandr Iscenco · Rating 2 out of 5]

Quite an interesting overview of the development of mathematics and its domains throughout history. The book has a number of things to satisfy a curious reader's mind: for example, the origins of zero, the foundations of computer programs, the concept of infinity, etc. However, the book is rather difficult to read, as it is full of mathematical concepts, which are not always explained clearly. The last chapters are the hardest due to combinations of hard-core mathematics and philosophical language.

[Lotte · Rating 3 out of 5]

The book started out great and exactly what I was expecting from it: easy to read and understand whilst explaining the value, history and marvel of mathematics. However as soon as actual mathematical theories were introduced the book quickly lost its legibility for me; too much knowledge was assumed for what is supposed to be a book to draw people into the amazing world of mathematical thought. I understood the overal point that Heaton was trying to make with the book, but it's a shame the explanations weren't more accessible.

[Arun Babji · Rating 4 out of 5]

Gives a good historical background, motivation and philosophical questions that led to the development of number theory, set theory. I appreciated why specific problems like string theory and need to handle infinity consistently are good representations and starting points for several other real world and pure mathematical problems. These mathematicians are pure geniuses to abstract out such difficult concepts.

[Yves · Rating 4 out of 5]

I'm not a mathematician, in fact I suffered from not really caring too much about my schooling when I was there... however in recent years I've gotten a thirst for learning and this book was one that I picked up to help quench that thirst.

Although I found some parts a bit challenging to follow, simply because I lacked some knowledge on the subject matter, I found several chapters quite interesting (start to finish). I especially enjoyed the few closing chapters.

[Sambasivan · Rating 5 out of 5]

These kind of books are rare to come by. In the simplest possible language the author elucidates difficult mathematical concepts and inculcates a passion into the reader through his impeccable logic and perseverance. Great read.

[Antti Rasinen · Rating 5 out of 5]

This book contained several new things to me, even though I am a confessed math-head. I particularly enjoyed the segments about history and non-Euclidean geometry.

[T.R. · Rating 2 out of 5]

It was interesting before it promised to start becoming interesting.

[Dirk Hennebel]

Not my cup of tea.

[Ultan · Rating 4 out of 5]

It is very rare to read such a fantastic book which stimulates an interest in a subject such as mathematics. The book is very well written and accomplishes its task well. The book gives a great explanation of the concepts that it discusses and how they were developed with time. In short Dr.Luke Heaton did a fantastic job in writing this book and I would recommend it to anyone with an interest in mathematics and discovering more about the origins of mathematical concepts.

[Brett T · Rating 3 out of 5]

Most of the time we use math we do just that: use math. We rarely think about things like why numbers come together the way they do. Or why certain mathematical functions and relationships seem to matter in the world of actual things as well as within the realm of pure equation and solution. But almost every mathematical advance throughout human history has often stemmed from and also sparked some serious thought.

Luke Heaton's A Brief History of Mathematical Thought skims through history and sketches some of the thinking that accompanied the ciphering. He begins as close to the beginning as possible, offering some ideas on how our stone-age ancestors may have begun to progress beyond the simple counting of objects into understanding the numbers behind the counting had relationships that could be regularized and predicted. At what point, for example, did some forgotten genius figure out that two of anything added to three of that same thing would always make five of that thing? If you had two rocks and were handed three, you did not need to count all five of them over again -- you could add the three to your two and know you had five whether you counted them or not. And once people had developed this understanding, how did it change their civilization and culture?

History is better in the earlier sections, such as the one mentioned above and others that deal with numerical development among the ancient Greeks, ancient Indians and other civilizations. It's also a good overview of how the switch to Arabic numerals and the use of the zero as a place-keeper propelled scientific thought far beyond what had been possible with cumbersome systems like Roman numerals. Later sections, though, deal with more esoteric subjects within math and their impact seems less obvious. Heaton offers reasons to spend some time pondering non-Euclidian geometry, for example, but has fewer explanations about how this particular wrinkle affects the way we live and work. Still, History is a good primer on what kind of thought can come from dwelling on even the most mundane of numerical tasks, as well as how that thought has shaped who we are today.

Original available here.

[Ilib4kids]

p 46 composite numbers
p80 0.999..9 = 1,
p81 continued fraction
p306 "mathematical arguments can be deeply striking.....

p71
The connection between the geometric facts of right-angles triangles and purely arithmetic facts such as 32+ 42=52 has greatly impressed hundreds of generations. On first inspection there appears to be little connection between arithmetic and geometric knowledge, for as John Stillwell writes in his excellent Mathematics and its History:
Arithmetic is based on counting, the epitome of a discrete process, The facts of arithmetic can be clearly understood as outcomes of certain counting processes, and one does not expect them to have any meaning beyond this. Geometry, on the other hand, involves continuous rather than discrete objects, such as lines, curves, and surfaces. Continuous objects cannot be built from simple elements by discrete processes, and one expects to see geometric facts rather than arrive at them through calculation.
Despite the many differences between arithmetic and geometry, Pythagoras Theorem hints at a depth of inner connection

====
p107
The notion that techniques for handling equations might constitute an independent branch of mathematics was an extremely important development, which the Arabs were uniquely placed to make. As John Stillwell wrote in his classic Mathematics and its History:
In Indian mathematics, algebra was inseparable from number theory and elementary arithmetic. In Greek mathematics, algebra was hidden by geometry. Other possible sources of algebra, Babylonia and China, were lost or cut off from the West until it was too late for them to be influential. Arabic mathematics developed at the right time and place to absorb both geometry of the West and the algebra of the East and to recognize algebra as a separate field with its own methods. The concept of algebra that emerged - the theory of polynomial equations - proved its worth by holding firm for 1000 years. Only in the nineteenth century did algebra grow beyond the bounds of the theory of equations, and this was at a time when most fields of mathematics were outgrowing their established habitats."

[Ricardo · Rating 3 out of 5]

What kind of philosopher can you really be nowadays if you don't know the empiric sciences that established the modern world, am i right?

Through this continuous journey, i started now with Mathematics simply, because.. i have sucked at it since a kid, i sucked at it as a teenager, and still kinda suck at it now as an adult. No shame, right? So, what better way to start learning it with that metaphysical/ontological side i like so much!

This is an amazing book that tries it's best simplyfing super-hard concepts answers those questions of "Where did it all came from?" I say try because, well, try to explain what is quantum physics and advanced algebra to a man that has 10 IQ on maths. So, it tries it's best to stay humble, and i would like to thank Heaton because of that, simultaneously not treating us as dumbasses, but, at the same time, knowing our limitations. A Brief History of Mathematical Thought is linear, kind-of-simple, funny sometimes, and one of the best all-around books that teaches you some maths in an somewhat quasi-serious way.

Who knew that one of the most charming books i've read this year, came from the area of Mathematics. Life, huh?

[Koen Crolla · Rating 4 out of 5]

All histories of mathematics are broadly the same. Heaton retreads the ground in the usual way, bringing no surprises or particular new insights, but also not doing anything especially wrong. If you haven't yet read a hundred of these, there are worse ones to pick up.

[Juan Burolleau · Rating 4 out of 5]

I really enjoyed reading it, some parts are harder than others but nothing a fully functional brain can't understand.