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| 014198628X
| 9780141986289
| 014198628X
| 4.27
| 4,319
| May 02, 2019
| Mar 05, 2020
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it was amazing
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Hair is so personal to ourselves, yet in many ways it is also political. Hairstyles can signal status—gender, affluence, class, or cultures. As Emma D
Hair is so personal to ourselves, yet in many ways it is also political. Hairstyles can signal status—gender, affluence, class, or cultures. As Emma Dabiri explores in Don’t Touch My Hair, this is particularly true for Black women. This book goes far deeper than I expected given its length; Dabiri fuses her personal experience growing up Black in Ireland and the United States with meticulous research. The latter takes us from enslaved people in the Americas to Yoruban culture and mathematics to the sprawling, technologically sophisticated cities of African empires. This book is about far more than hair; it is a story of culture and history as it is written on people’s bodies. As a white person who grew up in a city with a very small number of Black people, Black hair has never been something I have had much familiarity with or call to think about. Somewhere along the way, I learned about the controversial idea of Black people rocking “natural hair” instead of relaxed or straightened hair—but again, my racialization and upbringing meant that I really didn’t understand the internal politics of such decisions. Dabiri is really doing white people a favour when she discusses the history and weight of Black hairstyles, for she helps us understand how colonialism extended its control over Black bodies long past the formal end of slavery. In so doing, we go far beyond the simple eponymous admonishment and actually get to the root (pun intended) of our society’s misogynoir attitude towards Black women’s natural hair. Dabiri’s grounds her connections between colonialism and hairstyling in her Yoruba heritage, but I suspect similar stories exist from other African cultures. She relates what she has discovered about the role that hairstyling played in Yoruban life, from the status of travelling hairstylists to the way that one’s hairstyle could signal one’s social position, such as a messenger. Dabiri admits that taking care of natural Black hair is time-consuming, then goes on to say:
Truly in this paragraph Dabiri demonstrates why it is so necessary for white people like myself to continue, always, to read books about racism by Black people. It isn’t enough to stop at “ok, don’t touch Black people’s hair without asking, and don’t ask to touch a Black person’s hair.” That’s merely being not racist. If one wants to be antiracist, one needs to go deeper than mere behaviours and actually understand the connection between Black hair and the forces that maintain racist oppression. That’s what Dabiri does in the above paragraph and throughout this book, and it is why no white person will ever be an expert on anti-Black racism no matter how many of these books we read. But one of the benefits of reading to learn more about being antiracist is that it also encourages me to think about how white supremacy, while not oppressing me, also forces me into certain patterns of behaviour. Reading this book inspired me to reflect on how my relationship with my own hair has changed over the past few years, mostly as a result of my transition. Since that isn’t relevant to my thoughts on this book, I turned that reflection into a companion blog post that you can read if you are interested in my thoughts. Beyond the antiracist education made available in Don’t Touch My Hair, there is just a wealth of cultural and personal knowledge that Dabiri shares. I was not expecting the final chapter to be all about mathematics! It was with such delight that I read Dabiri’s account of research done by white ethnomathematician Ron Eglash. In this way she summarizes how, historically, Yoruban and other African cultures have used hair as a way to describe mathematical knowledge, such as fractals, long before these concepts were laid out in writing by European mathematicians. We often give a tip of our hat in mathematics to the contributions of “Islamic mathematicians” while forgetting that a large portion of Muslims were, indeed, Black Africans. This erasure is, in and of itself, a form of racist revisionist history wherein even as we re-admit Islamic contributions into science and mathematics, we whitewash Islamic scientists and mathematicians just enough that they become palatable to Eurocentric stories of these disciplines. Dabiri’s point? African people have always participated in scientific and mathematical discovery and innovation long before Europeans showed up in Africa, looked around, and promised to deliver “civilization” by railway at gunpoint. Moreover, African people did this through complex, three-dimensional ways of storytelling, from the construction of their cities and weaving of their clothes to the styling of their hair. Highly recommend this book to a wide audience, particularly for white people like myself. It isn’t too long, it is rigorously cited, and it is packed full of important ideas and information. Dabiri’s writing challenges you, pushes you to consider your own complicity in these systems, and exposes wide without recourse the ways in which white supremacy continue to oppress Black bodies despite supposedly centuries of freedom. It’s time to change that. Originally posted on Kara.Reviews, where you can easily browse all my reviews and subscribe to my newsletter. [image] ...more |
Notes are private!
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1
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Feb 05, 2022
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Feb 06, 2022
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Feb 18, 2022
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| 1137279842
| 9781137279842
| 1137279842
| 3.57
| 883
| Jan 06, 2015
| Jan 06, 2015
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it was ok
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The origins of our numbers, of our decimal place value system, of our numerals, is certainly an interesting topic! After all, we take for granted that
The origins of our numbers, of our decimal place value system, of our numerals, is certainly an interesting topic! After all, we take for granted that we write numbers the way we do today—most of us learned Roman numerals as kids and quickly realize they are clunky and formidable as we try to write the year we were born (although anyone born after 2000 has a much easier time of it now!). But Amir Aczel was curious about the origins of our number system, and in particular its linchpin of zero. Finding Zero is his very personal story of searching for evidence that the earliest known use of what became our zero symbol was in what is now Cambodia. Aczel opens the book by describing his childhood aboard the cruise ship his father captained across the Mediterranean. Here, his father’s steward fostered a love of mathematics. Now, as a professor of mathematics in the United States, Aczel still dreamed of the origins of our numbers. Eventually he took a trip to India, which was basically the birthplace of the Arabic numerals we use today, to visit some of the oldest known examples of zero. Finally, he discovered the work of Georges Cœdès, an anthropologist who had previously noted the presence of a 0-like symbol in a Khmer inscription on a stele. The actual artifact, however, went missing during the Khmer Rouge’s destruction of Cambodia’s cultural history. Aczel’s story climaxes with his trip to Cambodia to find this artifact—if it still exists. Often when a writer includes personal anecdotes, it’s relatable and interesting. I can’t say the same for this book. I was so interested in hearing Aczel talk about the properties of zero and why it’s important, but I could have done without the discussion of his childhood, etc. While it’s ultimately his choice how he decides to tell this story, it isn’t satisfying to me, and it’s quite self-aggrandizing. Aczel seems to see himself as a mathematical Indiana Jones on an epic quest to find the first 0. This is less about his discovery and more about his discovery. I would be much more tolerant of that if the writing were better—to be clear, I don’t think Aczel is doing anything wrong by writing this in a memoir form. I applaud him for trying to make the history of mathematics into an intense, exciting quest. Similarly, this book sheds light on the bias of Western mathematicians, the way we have shunned or dismissed the contributions of Asian—particularly south Asian—mathematics. Aczel does his best to explain how the inscription fits into what was a vibrant, advanced culture; similarly, he asserts the importance of making sure that the inscription survives and remains in Cambodia. These are laudable attitudes. But honestly, there are better books about zero. Although 20 years old now, Charles Seife’s Zero: The Biography of a Dangerous Idea remains my favourite book about this number. Seife certainly doesn’t go into the same level of detail that Aczel devotes to tracking the origin of the 0 symbol, that’s true. He basically attributes it to India and leaves it at that. Nevertheless, Seife’s book is so rich in history and ideas—and very well-written. Moreover, it’s worth noting that in the years since this book was published, the Bahkshali manuscript has been carbon dated. Aczel mentions this manuscript in his book—it contains some of the suspected earliest examples of a 0 symbol in India. At the time he wrote the book, no one had been allowed to extract samples from the manuscript to date it for fear of irreparably damaging the fragile artifact. I guess that changed, and the results are in: pars of the manuscript pre-date, by several centuries, the inscription Aczel rediscovered in Cambodia. So Finding Zero is also somewhat out of date in this respect. This is not a bad book, but it also isn’t one I would recommend. The mathematics are explored elsewhere in more detailed and interesting ways. And as much as I applaud Aczel’s adventurous spirit, I didn’t enjoy the way he told the story of his quest for the 0 symbol. I had hoped for a lot more here. Originally posted on Kara.Reviews, where you can easily browse all my reviews and subscribe to my newsletter. [image] ...more |
Notes are private!
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1
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Sep 04, 2020
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Sep 06, 2020
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Sep 04, 2020
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Hardcover
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| 1541646509
| 9781541646506
| 1541646509
| 3.73
| 842
| Jul 16, 2020
| Aug 25, 2020
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liked it
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At first I admit to some scepticism about the idea that we could use mathematics to rethink our conversations around gender. I was apprehensive becaus
At first I admit to some scepticism about the idea that we could use mathematics to rethink our conversations around gender. I was apprehensive because science, and even to some extent mathematics (or at least more applied subsets of its, like statistics) have been misused and abused in service of gender stereotype fallacies. Indeed, Eugenia Cheng points this out herself, and this, along with her careful and patient exposition of her topic, eventually won me over. X + y: A Mathematician’s Manifesto for Rethinking Gender is a good example of how an interdisciplinary approach to gender issues can often yield interesting new ideas. Cheng has clearly taken a lot of time to consider how to model and talk about disparities in our society when viewed through the lens of gender. Her conclusion? Sometimes when we think we’re talking about gender, we aren’t, and that creates too much confusion for us to make effective change. Cheng’s central argument goes like this. We spend a lot of time observing differences between men and women in various aspects of society (professional life comes up a lot as an example). Some people hold that these differences are innate. Otherwise believe the differences are caused by environment—that is, structural inequities. And the truth, probably, is somewhere in between. But as Cheng points out, researching innate gender or sex-linked differences is very hard and every time someone purports to have sorted it, people come along and very easily poke holes in the findings. Similarly, we have this tendency to refer to certain behaviours as masculine or feminine, yet that association is not as useful as we think: there are plenty of women who behave in so-called masculine ways, and likewise there are many men who exhibit so-called feminine attributes. As a side note, I have struggled with these terms myself lately as I transition. Technically anything I do, as a woman, is feminine by definition. Yet in everyday language, when I discuss how I dress, wearing makeup, etc., I talk about “expressing my femininity” and “being feminine.” I do this because I have an idea in my mind about how to express myself as a woman, but that idea is wrapped up in what we have been socialized to believe is feminine as a result of our society. For me, as a woman, wearing makeup is a feminizing act—but if a man wears makeup, is that feminine or feminizing? I would argue that context matters greatly here: some men put on makeup to feminize themselves (e.g., drag); others do it merely to hide a blemish or look better, just as many women do, and in that context I would argue that wearing makeup is in fact a masculine behaviour, if we are defining masculine as something done by men. Hopefully you can see how this quickly becomes confusing! Cheng points this out and then tries to help us make sense of it by falling back on her experience as a mathematician. If you were hoping to escape any mathematics in this book, you’ll be disappointed, but you also don’t need to understand the mathematics Cheng references to understand her point. Basically, math is good at definitions. Math is also very good at contextual definitions: infinity means something different depending on which mathematical world you’ve chosen to play in. Finally, Cheng argues that her particular field, category theory, is of supreme usefulness in this discussion because it tries to discuss different items in terms of relationships rather than membership/attributes. Now, in this particular case I don’t think Cheng is on to anything new. Plenty of people before and after Foucault have written about social justice from the point of view of power dynamics. If all she brought to x + y was some category theory, I don’t think this book would be very useful or successful. However, the discussion of category theory merely lays the ground work for Cheng’s main thesis. This goes back to what I discussed above about the equivocating around the terms masculine and feminine. Cheng proposes two new terms: ingressive and congressive. I’m not going to explain it as well as she does, but the gist goes like this: ingressive actions look inwards, centreing the individual; congressive actions work to bring the community together. When I first heard these terms, I immediately thought, “is this just a rehashing of individualism versus collectivism? In some ways, perhaps, but I will credit Cheng with building atop such concepts. My next thought was, “this is a nice attempt, but won’t people just use ‘ingressive’ as a synonym for ‘masculine’ and ‘congressive’ as a synonym for ‘feminine’?” I didn’t think Cheng was intending it, but I can see how someone who isn’t being careful might view this as a one-to-one mapping. Cheng makes it clear that this isn’t the case, going so far as to outline her journey from acting ingressively to keep up as a research mathematician to realizing that she truly preferred to foster a congressive environment while teaching mathematics. Lest you think that this is merely semantic sophistry to chronicle her journey from trying to act like one of the guys to reclaiming her femininity, Cheng tries to help us understand that this is not, in her opinion, a matter of gender. What these terms allow us to do, she argues, is discuss our ways of relating to one another without making stereotypical statements about gender. When someone jumps to ask a question that highlights their own expertise, that’s not “typical masculine behaviour”; it’s ingressive. People of all genders can do this. Likewise, if someone is trying to build consensus and help everyone get on to the same page, that’s not the empathetic behaviour of a woman—it’s congressive, and again, people of all genders can do this. So we can challenge the dominance of ingressivity in areas like academia in a way that removes the complication of talking about gender. Great, right? I’m not sure. I do like the new terminology, and I see the value in what Cheng proposes. I agree that sometimes our focus on gender can obscure the true power dynamics at work. Cheng demonstrates this aptly by referring to critiques of “lean in” feminism as trumpeted by Sheryl Sandberg. Cheng understands, and I agree with her, that merely putting women in positions of power within the current system is insufficient. It ignores intersectionality and the idea that there may be other marginalizations at work (race, class, etc.) that contribute to oppression or unequal power dynamics. Her solution is to restructure parts of our society to encourage congressivity, presumably because a congressive social order would allow people to participate more equitably regardless of their identities. It’s a nice vision. I want to acknowledge that it’s not entirely pie in the sky, that Cheng takes her time to lay out how we can build a congressive future from the ground up. That’s more than some dreamers do in their books where they try to explain why their one neat trick for saving society is the one we should enact. I hesitate to endorse this fully, however. Cheng tries hard to be congressive here, to encourage us to rethink our discussions around gender because she doesn’t want us to be “divisive.” She offers up competing definitions of feminism and slogans like “smash the patriarchy” as examples of how current thinking on gender polarizes the conversation and prevents true progress. I am sympathetic to this view. Yet I think there is an appropriate time and place for polarizing or divisive messages. Let’s take transgender people, for example. (And I note that Cheng makes every effort to be inclusive here, using cis and trans appropriately and acknowledging that, for example, some trans men are capable of becoming pregnant.) We trans people are, just by existing in current society, polarizing. TERFs or gender-critical feminists or whatever you want to call them (I prefer the simple transphobe label) would really rather prefer we don’t exist at all. No amount of re-labeling or rethinking the gender conversation will change this fact, because at the end of the day, this is not about how trans people behave or even about how transphobes behave: it is, ultimately, an ideological divide. It is not one that can be argued away. For trans people to be safe and able to participate fully in society, we and our allies must fight, passionately and aggressively, against discrimination. I, personally, hope that some transphobic people, if they are exposed to more trans people and come to know us and understand that we are not a threat, will change their tune. In that respect I do not think this is an “us vs. them” situation. Nevertheless, this is an example of how some aspects of gender-linked discrimination cannot be rectified through new labels. If you come to x + y expecting a totally revolutionary blueprint for how to think about gender, you might be disappointed. I came to this book with sceptical expectations, however, and I was pleasantly surprised. This book reminds me of Beyond Trans: Does Gender Matter? , in which trans man Heath Fogg Davis argues that there are many areas of society where gender doesn’t matter even though, at the moment, we insist it does. Cheng and Davis would probably agree on a lot of points, I think, as do I with both of them. I see value in critiquing the epistemology of gender, and I like that Cheng tries to apply the rigor and flexibility of mathematics. However, her arguments and ideas here can only take us so far. This is a great contribution to the ongoing meta-discussion. Originally posted on Kara.Reviews, where you can easily browse all my reviews and subscribe to my newsletter. [image] ...more |
Notes are private!
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1
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Aug 16, 2020
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Aug 18, 2020
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Aug 16, 2020
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Hardcover
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| 1579550304
| 9781579550301
| B088TTFTDR
| 3.68
| 92
| Jun 10, 2020
| Jun 10, 2020
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really liked it
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The Math(s) Fix wants you to believe that computers are coming for your math. Scary, isn’t it? You should find it scary. Computers are way better at ca The Math(s) Fix wants you to believe that computers are coming for your math. Scary, isn’t it? You should find it scary. Computers are way better at calculating than we are, yet we insist that “real math” means learning how to do long division by hand! Wolfram Media kindly provided me an eARC of this book via NetGalley in exchange for this review. I was definitely very interested in this. Some positionality, because even though this review is not about me, my perspective informed my reaction to the book. I am a math and English teacher. I have taught high school in the UK, and I currently teach high school to adult students in Ontario who need their diploma. I have 7 years of intimate experience with how the math curriculum and our wider system of educating and assessing students fails them. My current position allows me a lot of latitude that I wouldn’t necessarily have if I had to answer to parents, so I’ve had the enjoyment of doing things like going gradeless. A lot of what Wolfram suggests in The Math(s) Fix aligns with what I am already doing or planning to do in my classroom—however, as he points out, teachers alone cannot implement this fix. For this reason, I am a proponent of radical change to all levels of our education system. But what if you’re not? What if you’re someone who doesn’t know much about our current education system? You’ve been out of school for years. Maybe you’re a parent, maybe not. Will this book convince you that Wolfram is right, that there is a problem and he has the right solution? I hope so. I really hope so. Here’s what you need to know about this book. First, it’s not a math book. It’s not an education book. It’s not a policy book. There are no advanced equations in here. You don’t need anything beyond your basic education to read this. Wolfram also doesn’t delve too deeply into theory of pedagogy here (he brushes up against it, at times, but nothing that’s too hard to follow). Similarly, Wolfram keeps the aim of the book general enough, in terms of policy changes, to apply to any jurisdiction and any scale—local, regional, national. If you’re wondering, “Does this book apply to me, to my children, to my school, to my board or authority?” the answer is “Yes.” Second, this is a book about the necessity of unity mathematics as a school subject with computational thinking, but it is not about how we should replace educators with computers, and if that’s the reading you take away from this, go back and read it again. I’ll admit I was skeptical as all-get-out when I saw who had written this. Indubitably Conrad Wolfram is qualified to speak on this subject, but would The Math(s) Fix just be a thinly-veiled advertisement for Wolfram products in schools? It’s unavoidable that Wolfram’s companies would benefit from the shift he outlines here, and he acknowledges this. Yet the arguments he makes for the necessity of this shift are persuasive and have nothing to do with the Wolfram bottom line. Moreover, Wolfram recognizes—indeed, is intimately familiar with—the limitations of computer-based math. At one point he condemns people who interpret calls for CBM to mean “computer-based assessments.” He argues that computers can help with the organization and presentation of material, that computers can help with computation, but that at the end of the day, both qualitative and quantitative assessments are best left to human educators. This is true even for quantitative assessments, because it is hard to quantify problem-solving. Which brings me to … Third, this book clearly defines what math is and is not—or rather, what math has become in schools versus what it should be. One of the first things I say to my new math class full of anxious adult learners traumatized by their years of failing to do math in high school? “This,” I hold up my phone calculator, “is not math.” I proceed to explain how math is not “doing calculations,” because we have computers for that. I explain to them that real math is about creatively solving problems. And then I try, in eight weeks, 110 hours, to somehow undo as much of the years and years of abuse they’ve endured at the hands of our industrialized education system. Don’t get me wrong: Ontario probably has one of the best math curricula out there. Yet I still want to tear it up and start fresh, because I think our whole approach is fundamentally backwards and obsolete in the world of computation. Wolfram is very passionate about this change. He explains why this is not something we teachers can tackle alone. We need politicians, parents, and basically everyone else on board too—after all, this affects everyone. The Math(s) Fix is impeccably organized in such a way to lead you through the problem, the solution, and counterarguments to those who think this is unnecessary or unworkable. What’s missing from The Math(s) Fix is probably a patina of prosaic writing. Wolfram admits he has shortcomings in this area. The arguments are logical, and the rhetoric itself isn’t bad. Yet despite his frequent references to experiences with his daughter, not to mention his own days learning defunct subjects like Latin, Wolfram is not great with the emotional appeals. As a reader, I definitely value these elements of a manifesto. There are others who have made similar arguments in more accessible, emotionally-intelligent ways. And there will hopefully be more to come. Wolfram himself acknowledges that this book cannot be the beginning nor the end of this movement for a new “core computational subject,” as he calls it. So here’s my evaluation and my recommendation: The Math(s) Fix should be read by anyone with a strong interest in education policy, reform, or decision-making at any level. If you are a school board trustee, an educator, a politician … this book is for you. If you are a member of the general public and you feel like you have the stamina to wade through a book that is not at all math-heavy but definitely logic-encumbered then I’d recommend this book to you as well. If you want a book that makes a plea based more on anecdotes or broader social data, then you won’t find that here (and that’s ok). The Math(s) Fix is an important, well-presented addition to what is one of the most crucial conversations of our age. We are either going to get ahead of the computational revolution or we are going to do our children a disservice. Will you contribute to the fix? Other good education reform reads: Hacking Assessment , For White Folks Who Teach in the Hood , The Curiosity of School Hook me up with more education reform reads in the comments! [image] ...more |
Notes are private!
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1
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May 19, 2020
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Jun 10, 2020
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May 19, 2020
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ebook
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| 039363499X
| 9780393634990
| 039363499X
| 4.12
| 10,816
| Sep 2018
| Sep 18, 2018
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really liked it
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Algorithms are increasingly an important part of our lives, yet even as more of us become aware of this, how much do we actually stop to consider what
Algorithms are increasingly an important part of our lives, yet even as more of us become aware of this, how much do we actually stop to consider what that means? How much do we stop to consider who is designing these algorithms and how they actually work? And why are we willing to give up so much control to them in the first place? Hello World is a short tour through the various ways in which algorithms intersect with human decision-making. It is neither comprehensive nor particularly in-depth. Nevertheless, through a few choice examples, Hannah Fry illustrates why this is an important topic and an issue we should think about more often and more deeply. Fry has organized the book into 7 discrete chapters: Power, Data, Justice, Medicine, Cars, Crime, and Art. Each chapter explores the role of algorithms in these parts of our society. Fry intersperses explanations of various algorithms with anecdotes, many of which, as she notes about the now-infamous story of Target outing a teenager’s pregnancy, the reader might already have heard. I believe that one of Fry’s goals is to demonstrate for the reader how algorithms aren’t exotic animals confined to the zoo of a computer science lab. They have real-world applications and real-world effects. It’s probably inevitable that I compare this to Weapons of Math Destruction , given the similarity of these two books. (Cathy O’Neil blurbed Hello World as well—very good job, marketing team!) Honestly, the subject matter of these two books is very similar, yet I’m not willing to say that one is better than the other. O’Neil writes from the perspective of a mathematician who spent a significant part of her career embedded in the financial sector; Fry is a mathematician who studies Big Data from as an academic career. Many of the concepts elucidated in Weapons of Math Destruction make an appearance here, and Fry often draws similar conclusions as O’Neil. Whereas O’Neil is mostly concerned with the negative effects of algorithms, however, I’d argue that Fry is more interested in raising our awareness about the complexity of these algorithms. This is a math book at its finest—by which I mean it’s a math book with very few equations in it. Lay people often assume a good mathematics book needs a lot of formulas and numbers, and that’s not true. Math isn’t formulas (that’s engineering—sorry not sorry). Math is about developing a system for solving problems creatively. Fry breaks down what an algorithm is in simple terms, and I loved the chapters on Medicine and Cars, because Fry uses these to explain some great statistical concepts: false positives and negatives, in the former; and Bayesian inference, in the latter. So even though a lot of the anecdotes, specific algorithm examples, etc., were already familiar to me, I still enjoyed how Fry tackles these fundamental but often overlooked mathematical ideas. (As a fan of graph theory and decision math, I also liked the discussion of random forests.) Fry spends a lot of time discussing how algorithms can get thing wrong. She points out (perhaps obviously) that algorithms will never have “human” judgment—algorithms can’t be empathetic or sympathetic. She illustrates how an algorithm is always going to be biased, so we should be less concerned with chasing after “objective” algorithms but instead focus on building algorithms that are more honest about their biases. The problem with machine learning is two-fold: it’s the data sets we feed in, but it’s also the fact that the decision-making that leads to the output is often opaque. For all that Fry paints a dire picture, though, she presents a balanced viewpoint that also endorses algorithms as potentially beneficial and necessary. In the Medicine chapter, she points out that algorithmic recognition of diseases like breast cancer is going to make the healthcare system more efficient—as long as these tools are used in conjunction with human judgment, not as a replacement for it. These sentiments are echoed in every chapter, from her exploration of the justice system to her explication of driverless cars, repeated once more at the end of the book where she mentions Kasparov’s centaur chess. If Fry is correct, then perhaps our optimal future is a cyborg future: one in which algorithms enhance our decision-making and help defuse the fallibility of our human judgment, but where humans remain in control of the ultimate decision process and can audit the algorithm. Hello World is a clear, easy to follow discussion of an extremely relevant topic in today’s society. If, like me, you’re well-read on this subject already, there isn’t a lot of new stuff in here—but I suspect you’ll probably find something. Even so, you’ll hopefully appreciate Fry’s talent for writing and explaining these ideas. As for anyone who has only recently become interested in this subject, you’ll not find many books that explain these ideas so well. Like I said above, this pairs nicely with Weapons of Math Destruction—read both! [image] ...more |
Notes are private!
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1
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Mar 12, 2019
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Mar 14, 2019
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Sep 03, 2018
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Hardcover
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| 0465094252
| 9780465094257
| 0465094252
| 4.06
| 3,122
| Jun 2018
| Jun 12, 2018
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liked it
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Is truth beauty and beauty, truth? It can be hard to tell. In Lost in Math: How Beauty Leads Physics Astray, Sabine Hossenfelder argues that these two Is truth beauty and beauty, truth? It can be hard to tell. In Lost in Math: How Beauty Leads Physics Astray, Sabine Hossenfelder argues that these two concepts are not equivalent. As the subtitle implies, Hossenfelder feels that theoretical physicists are too obsessed with creating “beautiful” theories, in the sense that the mathematics that underpins the theories (because these days, theories are basically math, even though, as Hossenfelder stresses, physics isn’t math) must be beautiful and use “natural” numbers (by which she means numbers close to 1). Theories that don’t conform to these criteria tend to be unpopular, to receive less funding for experiments and less attention in papers. This, Hossenfelder contends, is a mistake. She fears it reinforces an orthodoxy that threatens theoretical physics with stagnation and, worse, undermines the scientific method. In her view, since theoretical physics is often regarded as the “hardest” of hard sciences, if faith in the foundations of physics goes, so too goes trust in science—right when we need it more than ever. I guess this book kind of hits the sweet spot for me, because I’m big into the intersections of philosophy and mathematics and science. Thanks to NetGalley and Basic Books for the free eARC! I love thinking about the limits not just of what we know but also what is knowable. This, to me, is why theoretical physics fascinates me—not just because it explains the foundations of our physical existence, but because it knocks up against literally the limits of our ability to measure and quantify. Each time we want to go looking for heavier and heavier particles, for example, we need bigger and bigger particle accelerators. That’s why we built the LHC, which—in Hossenfelder’s opinion—has been a bit of a bust in terms of new physics. But we’re running up against the limits of what we can do on Earth, or even in orbit … what’s next? Particle accelerators the size of our solar system? I get chills. Lost in Math is not really about physics in the popular science sense. This is more accurately a philosophy book. Hossenfelder discusses a lot of physics concepts (most of which, to be honest, go way over my head), but ultimately she is more interested in looking at why her colleagues in theoretical physics chase the theories that they do. Unlike some of them, she confesses that she doesn’t seem to have a nose for beauty, that she doesn’t recognize a beautiful theory when she sees it—and she is uneasy about this reliance on ideas of beauty. So while the book follows the fairly standard approach in popular science texts of doing a brief overview of the history of physics, Hossenfelder is looking at the philosophies that were at work rather than the merits of the actual theories. As familiar names—Bohr and Einstein and Schrodinger and Heisenberg and Dirac et al—move across the page, we learn more about their thinking—insofar as we can know it—than their specific contributions. Hossenfelder isn’t looking to teach physics here. She’s asking us to think critically about why we have the physics we do. I think this is a really interesting and important point. To laypeople, like myself, it might seem inevitable that we’ve ended up here. After all, there is only one true science, right? We might have made a bunch of false starts, but along the way, as we uncover more and more “facts” and tinker with our theories and run better experiments, we’re narrowing it down and getting closer to “the truth”, right? Well … it’s complicated. As Hossenfelder explains, it isn’t so much that the physics we have now is The One True Physics as it is Something That Mostly Works. And in the case of quantum physics, there are actually a whole bunch of competing interpretations that explain the same phenomena, just differently, and at the moment they all tend to be valid because no one has figured out a way to test between them. So as much as both philosophers and physicists would like the other camp to stay out of their business, when you get right down to it, the two are entwined at the moment. Hossenfelder tours the landscape of theoretical physics, interviewing researchers in different fields to help her understand the obsession with naturalness and beauty. Along the way, you will pick up on her clear sense of exasperation with what’s happening in her profession. It isn’t just the naturalness argument: it’s the whole system, the fighting over short-term grants and positions, the tendency to reward people who publish more often, on more accepted topics, over people who spend their time tinkering with more heterodox approaches. And maybe how surprised I am by Hossenfelder’s tone and voice, or even the fact that this book got written, further supports this idea, since we are so used to “gee whiz” pop physics books that emphasize the beauty of the universe and of the theories that explain it. Physicists who write books for popular consumption are generally trying to build a following, and I get the impression Hossenfelder really doesn’t care about that. While I find Hossenfelder’s writing, in general, to be mediocre, her forthright and honest tone is refreshing and interesting. There’s a fair bit of mathematical concepts in this book too. That probably shouldn’t be surprising, given its title. There aren’t actual equations, but Hossenfelder throws around terms like “groups” fairly generously without really going into what they are (and maybe that’s for the best). As with the physics shop talk, if you don’t have much of a grounding in abstract algebra, you’re going to feel a little out of the loop. This is not a light read. It is, however, enjoyable in the sense that it tickles the part of your brain that really wants to think hard about things. Lost in Math succeeds, largely, in what it sets out to do. It demonstrates that certain elements of how theoretical physicists theorize right now aren’t the most conducive or productive. It pulls back the curtain for a wider audience, exposing us to some of the philosophical debates and issues that have long been happening within the physics community, which laypeople might wrongly perceive as monolithic in approach, if not in interpretations. Hossenfelder’s writing is a little dry, and the book is full of challenging concepts … but I think it’s worth a try if you want some philosophy in with your science. [image] ...more |
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| Jan 07, 2014
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Who doesn’t like a good controversy in their popular science books? What’s a philosophical theory about the nature of the universe if it doesn’t ruffl
Who doesn’t like a good controversy in their popular science books? What’s a philosophical theory about the nature of the universe if it doesn’t ruffle some feathers? No one wants to write a book and then have everyone turn around and shrug at you. That doesn’t sell! So it’s not really surprising that Our Mathematical Universe: My Quest for the Ultimate Nature of Reality is a controversial book by a somewhat controversial physicist. I received this as a Christmas gift a few years ago, and that was the first I’ve heard of Max Tegmark. Since then he has popped up a few times here or there, and now I’ve finally made time to read this long and detailed treatise on the current state of physics and Tegmark’s personal conception of, well, reality. I don’t actually find it all that controversial, per se—though I should clarify that I’m a mathematician by training, and not a physicist, so maybe the way Tegmark presents these ideas is more insulting or seems more radical when one is a physicist. That being said, I’m also not saying I agree with Tegmark’s Mathematical Universe Hypothesis (MUH), because, despite probably being a mathematical realist, Platonism itself strangely makes me uncomfortable…. Oh boy, I think I’ve already used too many strange terms! This review is probably going to get pretty heady and philosophical at some point, much like Our Mathematical Universe does. So let me spend the first part here just discussing the book, its structure and writing, etc., in a more general way, to give you an idea of whether or not it is of interest to you before you read my whole review. I’ll get to my thoughts about Tegmark’s specific claims later. Firstly, regardless of any reservations I might have, I still recommend this book. This is a really well-written and approachable popular science work. Tegmark’s style is really accessible—despite going heavy on scientific and mathematical terminology, he is careful to proceed in a systematic way. This is not a book you want to be reading just before bed, maybe, or during a busy commute—it took me pretty much a week, albeit a busy week, to work my way through it. Nevertheless, I think it is a worthwhile use of one’s time. Tegmark first impressed me with a table at the end of Chapter 1 called “How to read this book”. He lists every chapter of the book, along with three columns: Science-curious reader, hard-core reader of popular science, and physicist. Each column lists the chapters that reader would be best to read/skip—i.e., the science-curious reader should read the entire book; the hard-core reader can skip several of the earlier chapters because they presumably will have seen these explanations before; and the physicist can skip all but the controversial chapters (Tegmark also labels each chapter as “mainstream”, “controversial”, or “extremely controversial”). I love this approach and hope more popular science authors use it. Now, I, of course, ignored these suggestions and read the whole book anyway, because I wanted to see how Tegmark explained the Big Bang, inflation, etc. Yet I confess I skimmed some parts and felt better about it because I knew it was sanctioned. One reason I’ll recommend this book is simply because Tegmark’s explanations for the origins of our universe, as currently understood by “mainstream” cosmology, are really lucid. He clarified several aspects of the Big Bang and inflation that, until now, I not only did not understand but didn’t realize I didn’t understand. He didn’t just improve my comprehension: he actually showed me parts of my comprehension of these theories that were inaccurate. I am not a physicist by training by any stretch of the imagination (I only took physics up to Grade 12 in high school, and they don’t even get into relativity by then, let alone QM); all of this knowledge is entirely autodidactic, and hence it isn’t surprising a lot if it is inaccurately understood. But I think I’ve plateaued a lot lately because I was having trouble finding explanations that were calibrated for my knowledge level: either the explanations get too technical and lose me, or else I just end up reading the same ground-floor “hey have you heard of this thing called the double-slit experiment?” stories over and over again, which isn’t fun either. In particular, I really enjoyed Chapter 5, in which Tegmark explains inflation and why it is necessary to account for problems with the Big Bang theory. The idea of the Big Bang itself is now probably within the realm of general public knowledge, assuming a half-decent education (and regardless of whether one “accepts” the theory or prefers creationist nonsense). Yet there are probably as many misconceptions about this theory as there are explanations of it in popular science books, and once any two non-cosmologists start talking about it, we inevitably run into quasi-philosophical walls. Tegmark very clearly presents what the theory actually says; why it is compelling given the evidence; the problems with the theory without inflation and why inflation itself solves those problems. Tegmark refers a lot to data gathered by several satellites and ground-based microwave telescopes that have observed the Cosmic Background Microwave Radiation (CBMR). He himself worked quite a bit on many of these projects, or with the data from these projects, to help sharpen and analyze this evidence. And this is another reason I enjoyed and recommend Our Mathematical Universe: Tegmark provides a great perspective on how science is done. From conferences to international projects poring over satellite data to writing and publishing papers, Tegmark shows us the act of physics research as much as the end result. He shows us how individual physicists’ opinions of theories will evolve over time. He shows us how people have different specializations, which in turn lead to different predilections and levels of knowledge about parts of physics. It’s really fascinating, and it’s an aspect to the discourse around science that I wish more media would cover. So the first 6 or 7 chapters of this book are excellent, and I recommend reading at least those. After Chapter 8, Tegmark introduces the more “controversial” content. As I said above, I don’t see it as controversial so much as a bundle of claims that are either uninteresting because they are obvious or unappealing because they are largely unintelligible. Now we arrive at the part of the review that gets technical. Let me refer you to Scott Aaronson’s review. He is a computer scientist and much more well-versed in this stuff than I am, so his review goes into more depth behind the mathematical/physics claims that Tegmark makes. I found myself largely nodding along and agreeing with most of Aaronson’s opinions there. You might think that I, as a mathematically-inclined person, might seize upon the idea presented here. Tegmark’s MUH says not only that we can describe the universe using mathematics (a notion almost axiomatic to our physics) but that all of our physical reality itself is literally mathematical. That is, our entire subjective human experiences are simply the consequence of certain facets of a certain mathematical structure within a superset of structures, the entirety of which comprise the Level IV multiverse, i.e., the sum total of all existence and anything that could ever possibly exist. It’s tempting. And yet…. Years ago I read The Grand Design . This was back in my university days, mind, when I was high on philosophy classes of all kinds and armed much more to purpose for these kinds of throw-downs. Nowadays, my memory of the differences between ontological and epistemological arguments requiring jogging from Wikipedia, I’m not so sure I’m up to the task. Yet one idea has stayed with me from Hawking and Mlodinow’s book: that of model-dependent realism. They proposed that the reason we are having so much trouble finding a “theory of everything” to unify the physics of the big (relativity) and the physics of the small (QM) is because no such theory exists. Rather, different theories are required depending on the situation one is trying to model. It is an intriguing idea, one I hadn’t really encountered in a science book before. And I really liked how it short-circuited many anti-realist objections to scientific realism. Tegmark appears to move in the opposite direction. He backs the ToE horse (which is fine) by insisting that the ToE is reality. And then he kind of dodges the question of whether that means we will ever actually find a ToE (because if we did, wouldn’t that mean we just have … reality?). That’s what I mean about the MUH being uninteresting and unintelligible. He starts off by talking about how the movement of time is an illusion, all very much standard stuff depending on how you define spacetime, etc. Yawn. When we get into the more “controversial” material, his argument just sort of breaks down. He starts making a whole bunch of probabilistic paradox arguments, like quantum suicide, the doomsday argument, etc.—the kind of thought experiments that are fun to put into a first-year philosophy textbook but that have little connection to, you know, reality. These thought experiments rely explicitly on making assumptions to make up for our near-total lack of knowledge about a situation. The whole point is that, as we acquire more certain knowledge, we are in a better position to see if we are indeed a representative sample or if, perhaps however improbably, we are not. Tegmark’s MUH is also, despite his claims to the contrary, completely untestable/unfalsifiable. He insists that we will uncover evidence and create theories which logically imply the MUH, and that’s just silly. The MUH is untestable because we currently have no alternative to mathematics as a way of describing physical theories of reality. It is unfalsifiable, because even if we can get past the testing problem, how will we know if we’ve discovered a physical law or property that violates the MUH? Almost by definition, the MUH can take nearly any observational evidence and somehow fit into its framework. Tegmark claims that if the MUH is false, then we will one day run up against an insurmountable “wall” in physics beyond which our knowledge of reality can progress no further, since our mathematics will no longer be able to express reality. I disagree. I think model-dependent realism would be an effective way to counteract such a wall: maybe to progress, all we need do is abandon the search for a ToE and instead create theories of everything. The last half of Our Mathematical Universe is a wild ride of philosophy of mathematics and science. I loved reading it. I found parts of it very convincing, but I don’t think those parts (combined with the other parts) necessarily add up to the whole that Tegmark calls the Level IV multiverse, the Mathematical Universe Hypothesis. I think he is incredibly enthusiastic about this idea and has clearly spent a lot of time thinking on it—which is great. I loved that I got a chance to read it. But I don’t think his arguments are as sound as he thinks they are. I say this not from a physicist’s position (because I’m not one) nor even a mathematician/logician (because, let’s face it, my memory of higher math dims with each passing day) but as the target demographic for this book, the hard-core popular science reader who is looking for a new hit to bring on that theoretical physics high. It’s a nice try, Tegmark, and you almost had me going. [image] ...more |
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| 3.88
| 27,906
| Sep 06, 2016
| Sep 06, 2016
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The profile of the term “Big Data” has risen recently. Yet, like so many buzzwords, people often don’t fully grasp the significance of the term. “Big
The profile of the term “Big Data” has risen recently. Yet, like so many buzzwords, people often don’t fully grasp the significance of the term. “Big Data” is more than the nebulous connotation of corporations collecting our information, and perhaps packaging and selling it—although it is that. It is, in fact, about how corporations quantify everything we do, even the information we don’t realize we’re leaking out into the world, and then use that data to make decisions on our behalf, or for us, or about us, without even informing us of how they reached those decisions. It’s creepy. And it is already everywhere in our lives. Weapons of Math Destruction is Cathy O’Neil’s impassioned plea not to let this spread further, and indeed, for us to take a hard look at how we are using Big Data algorithms already. I was drawn to this book because it’s at the intersection of two things I love: mathematics and social justice. As a mathematician, I love learning about how mathematics interacts with our society. People scoff at the utility of math, particularly the higher-level, “pure” math, yet ultimately that is what powers the digital devices we use every day and allows us to do things like make video calls across continents. You don’t have to understand the math to appreciate its power. Similarly, as someone passionate about social justice, I am sympathetic to O’Neil’s argument here. To be clear, you don’t have to have either of these qualities in order to find Weapons of Math Destruction informative or valuable; that’s just to indicate where I’m coming from. Some of the topics O’Neil covers will be familiar, in part or all, to readers. You might already have heard about predictive policing using models of criminal activity, and the way in which it leads to a self-fulfilling prophecy (the more heavily you police an area, the more crime you uncover, the more you police the area…). You’ve probably heard about how credit scores are notoriously far-flung now yet frustratingly opaque. I also appreciate how O’Neil mentions lower-tech WMDs, like the college ranking system, which started off as something not based on computer algorithms at all. The various and sundry examples given here reinforce the fact that WMDs exist throughout our society rather than in any particular field. More broadly, O’Neil’s overall thesis is that we can’t fix this with more technology. This is not a problem of not having “enough data” or not having “good enough models” or programs or whatever. Fundamentally this is a social problem. I appreciate that she outright states that mathematics is not the solution here, nor is it being used in a neutral way. Often people like to pretend that math and science, because they are so-called “hard” disciplines (as in their rigour, not their difficulty) are objective or neutral. That’s not the case, as O’Neil demonstrates here. O’Neil makes the companion point, though, that models are not in and of themselves negative social forces. A model is not a WMD until it is deployed improperly or its flaws are ignored. To some extent, we are stuck, now that we have the technological capability to do this. Although, on the surface, this might seem like a contradiction of her thesis, it’s actually just the logical conclusion: O’Neil reminds us that the only way we can fix these mathematical tools is through social pressure, i.e., as a society we have to decide how we want these data-crunching algorithms to operate. So, Weapons of Math Destruction admirably fulfills its purpose: it will educate you in more detail about algorithms, big data, and the decisions that they make about our lives. It could have been fuller and longer, sure. O’Neil’s writing style is pretty basic and leaves something to be desired (or developed further). Yet that just means it’s a quick read. It has the word “math” in the title, and sure, there is some discussion of math (mostly statistics)—but I promise you there is nary an equation to be found in these pages, and while that might be disappointing to me, I suspect it won’t be to many other readers. [image] ...more |
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it was amazing
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I first heard about this on Quirks & Quarks from CBC Radio. Then Josie, one of my Canadian friends still teaching in England, was filling me in on how
I first heard about this on Quirks & Quarks from CBC Radio. Then Josie, one of my Canadian friends still teaching in England, was filling me in on how she went to one of Matt Parker’s stand-up events and how awesome it was. When I informed her I had purchased a signed copy of Things to Make and Do in the Fourth Dimension on the Internets, she was suitably envious. Not, however, as envious as I was for her singular stand-up experience—I don’t like stand-up, but I’d probably watch math stand-up. Here’s my secret when reviewing math books: don’t focus on the math. Because, you know, anyone with a math degree can write about math. Writing about math is not hard. It’s making math accessible that’s hard. Now, that’s not because math is somehow more difficult for the average person to comprehend than any other highly-specialized field. We only have this perception as an unfortunate side-effect of our industrialized education system, which has traditionally insisted that we should learn math through rote memorization of rules. Matt Parker rightly embraces a much more flexible idea about how we can learn math. Specifically, he champions recreational mathematics. That’s right, people: doing math for fun! If you’re sceptical, I don’t blame you—see my point above about school systems. It’s really unfortunate we break people and squash their love of math so early like this. If I were better with young children I might consider becoming a primary school teacher to rectify this. As it is, my head stuck up here in the calculus clouds, I can only evangelize recreational math from afar. See, we mathematicians know what people with a warped idea of math do not: mathematics is a creative discipline. Someone had to find the Fibonacci sequence, and they didn’t do it by looking at nature. Someone had to devise and name different dimensions of shapes. And mathematicians do this by investigating, by looking at what we already know and finding the gaps. Yes, they do this is a systematic way, and they have to do it rigorously before other mathematicians will agree with them. But a lot of mathematical discoveries have literally come about because of mathematicians just playing with numbers and shapes and ideas. This idea pervades Things to Make and Do in the Fourth Dimension, which is organized in such a way to progress from basic ideas about numbers to very abstract ideas about functions, dimensions, and infinity. You’re not going to understand all of it, and that’s OK. Understanding everything is not the goal of reading a popular math or popular science book—getting a glimpse behind the curtain, understanding why it’s important, piquing your interest to learn more; these are the goals. (I’m trying to pump you up and help you be more resilient here, because I won’t lie to you and pretend it’s easy to follow everything, either in this book or in others like it.) Don’t worry though, because the author will always be around to help you out. Parker writes with a sense of humour that’s only to be expected considering his comedic career. (Britain really does seem to have cornered the market on funny mathematicians….) There are also lots of practical exercises too. And I don’t mean questions you need to calculate and answer. I mean activities, templates for you to cut out and puzzles for you to consider. Parker is very proactive in demonstrating some of the practical ramifications of even the most esoteric ideas, from calculating digital roots to knitting 3D projections of 4D shapes. I could easily see some of this stuff working in a classroom setting if, you know, you’re not the kind of math teacher that thinks we should just memorize it all. Really, when it gets down to it, this is how we need to be teaching and learning math. Reading a book about math is all well and good—I love doing it. But you need to learn by doing math. You need to try these things yourself, to investigate a problem until you hit upon interesting and sometimes unexpected results. This is one of the greatest things about mathematics: you can, in theory, verify every math result ever discovered by someone else. And you don’t even need specialized equipment: most of the time you just need a ruler, some scissors, and some paper. (And maybe a calculator or a computer for the recent discoveries!) This is DIY math at its finest. I learned some neat things in the chapters that Parker devotes to higher-dimensional shapes. This is not an area of math I’ve studied in much detail, and conceptualizing higher-dimensional shapes is, of course, very difficult! Yet he explains it clearly. I also appreciate how much he uses computer programs to help him investigate relationships and ideas. As someone who also enjoys writing Python scripts, I’m always happy to see my interest in math and computers come together. On the flip side, I know a lot about graph theory and enjoyed his section on that. He doesn’t really do anything new when it comes to talking about old chestnuts like the Four Colour Theorem and its infamous proof. Nevertheless, this is one of those areas of math that people never hear about unless they go into university, despite it being so interesting and widely applicable. Things to Make and Do in the Fourth Dimension is a lovely and informative book. It’s a great example of how to write well about doing math for fun. Parker is ever-encouraging, ever-understanding, ready to make fun of math, mathematicians, school, and himself—and yes, my dear reader, you as well. This is a safe book in that sense: you’re not going to be judged for not liking math or not having much luck, so far, with it. But thanks to Matt Parker, you can roll your own math and enjoy doing it. We need more books like this! Until then, read this one. [image] ...more |
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Oct 26, 2015
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Really, it’s my fault that mathematics gets such a bad rap. And by me, I mean math teachers in general. And by math teachers, I actually mean the pedag Really, it’s my fault that mathematics gets such a bad rap. And by me, I mean math teachers in general. And by math teachers, I actually mean the pedagogical paradigm in which most of us are embedded, and the questionable premises of the educational system that encourages such pedagogy. Math anxiety is often caused by general test anxiety, combined with a lingering sensation that there is “one right answer,” as well as a misunderstanding what math is and how we use it. Other factors: parents communicating anxiety/resisting innovative ways of teaching, and a generalized anti-intellectual snobbery in our society in which those who are interested in how the world works are “geeks” and “nerds.” (This is independent of the fact that, in recent years, geekdom and nerdery has become trendy. Capitalist structures might be co-opting the symbols and fashions of geek culture, but that doesn’t translate into broader tolerance or embracing of geek interests.) With How to Solve the Da Vinci Code and 34 Other Really Interesting Uses of Mathematics (I hate the title), Richard Elwes sets out to make some of the most important fields or problems in math more accessible to the layperson. This is a worthy goal. From the titles of his other books, it looks like this is Elwes’ pet cause: he likes to break mathematics into small but fascinating facts, problems, or ideas that he can explore in five-minute chunks. As a result, this is the sort of book you can dip in and out of, say at bedtime, for a number of evenings. You don’t have to remember a lot or pay attention to a plot. Nor does Elwes demand much in the way of memory or understanding. He covers some of the basics of algebra in the earlier chapters, but even understanding those is not a requirement. This book doesn’t so much teach you mathematics as it describes the different types and fields of mathematics and some of the most interesting results or problems from them. Perhaps the most complicated concept you really want to understand is prime numbers: if you know what those are, then you’re good. Some of Elwes’ explanations are great. Within this are many “standard” explanations that I’ve seen before and skimmed—that being said, I am a mathematician and a math educator and a math enthusiast, so what’s familiar to me is not necessarily familiar to you, and this might be someone’s first exposure to Russell’s paradox or the theory of sets or graph theory. So that’s not a negative in my book, just an observation that the more mathematically-inclined have likely come across most of the content here, in one place or another. On a related note, I want to stress that this really is a survey of mathematical results. Some chapters are longer than others, but none go into the type of depth one wants for a truly comprehensible explanation of what’s going on. To reiterate: you won’t learn a lot of math here; you’ll learn about math. Also valuable and important, but it’s a keen distinction. For me, some of the highlights were: Chapter 7, “How to unleash chaos” (chaotic systems and strange attractors); Chapter 15, “How to arrange the perfect dinner party” (Ramsey’s theorem); Chapter 18, “How to draw an impossible triangle” (non-Euclidean geometry); Chapter 19, “How to unknot your DNA” (knot theory); and Chapter 23, “How to build the perfect beehive” (2D/3D tesselation and packing). I like these chapters because they taught me something or reminded me of something I had forgotten, or Elwes’ explanations are particularly thoughtful and useful. For example, the knot theory chapter doesn’t just talk about knots—as the title implies, he mentions DNA, enzymes, proteins, etc. It’s a reminder that mathematical discoveries end up having applications in places you wouldn’t suspect. That’s another thing that this book does well. In chapters such as the one on the four-colour theorem, or Benford’s law, Elwes emphasizes two important and related things about mathematics. Firstly, mathematical discoveries don’t always happen in isolation or as strokes of genius. We tend to tell those stories, because they are exciting. But for something like Benford’s law or the four-colour theorem, the discoveries build on decades (or centuries) of work. Several mathematicians independently notice something cool, make a conjecture, fail to prove it, and discard it—only for another generation to succeed where they didn’t. Math is a progressive, ongoing effort. And something we don’t make clear often enough in the classroom is that new mathematical research is still ongoing at a furious pace. We present math as an accomplished, finished product: here’s how you find the missing side of a triangle; the Babylonians knew how to do it, and now you do too! But like science, mathematics isn’t a stable set of knowledge. It behoves us to raise awareness among the general public of how people research math and what we still research. Elwes points to the Clay Institute’s Millennium Prizes as one example. He also mentions a few other questions that remain open problems. While it’s true that genuine mathematics research is not for the faint of heart or the interested amateur, that tends to be true of any specialized discipline. Math is not necessarily more difficult or special in this regard. How to Solve the Da Vinci Code is not the warmest of math books I’ve read. Elwes’ tone is conversational, yes, and has a hint of humour to it. However, the broad strokes of his descriptions necessarily make them less personal than they might otherwise be. He tells a story in most of the chapters, but it’s not with the same level of vivacity that other authors often employ. Instead, his style is one step up from an encyclopedia article. Again, this isn’t really a positive or negative in and of itself—it depends on what you want from a book like this. I, personally, want to know more about the author. I want to know where they’re coming from, what interests them, and hear them tell the story of mathematics from their perspective. We don’t get that here—Elwes never inserts himself into the text—and I feel like that’s unfortunate. But others might find it more objective and informative. Would I recommend? Not for someone like me, who has read a lot of math books and studied math. For neophytes and laypeople? Maybe, depending on the person. I’d rather find a book that gets them more excited about one specific thing, rather than throw everything at them like Elwes does here. Maybe this book is best for someone who already likes math, has a passing interest or understanding of it, and wants to sort of survey the field and see what kinds of things are out there. In that case, there’s definitely 35 good ideas here. [image] ...more |
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Mar 23, 2015
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Mar 25, 2015
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Aug 09, 2014
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| 0465050743
| 9780465050741
| 0465050743
| 3.69
| 3,668
| Oct 01, 2013
| Oct 01, 2013
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liked it
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I’m disappointed that so many people seem underwhelmed by the autobiographical parts of this book and feel that they are ancillary to Frenkel’s purpos
I’m disappointed that so many people seem underwhelmed by the autobiographical parts of this book and feel that they are ancillary to Frenkel’s purpose. I disagree: they are, in fact, the heart and soul of Love & Math. Without them, this would be a fairly intense treatise on deep connections between abstract algebra, algebraic geometry, and quantum physics. With them, Frenkel demonstrates how the study of mathematics and a devotion to thought for thought’s sake, to fulfil human curiosity helped him personally through anti-Semitism and Soviet persecution. In some ways I was reminded of remarks Neil Turok makes in
The Universe Within
(if I am remembering correctly) about the state of education in many African countries depriving us of staggering potential intellects. How many people, poor or Jewish or otherwise unprivileged, were not as lucky as Frenkel happened to be? Frenkel’s personal recollections are also interesting because they provide a glimpse into the lifestyle and community of professional mathematicians. This is not something most people think about, even people who are scientifically-minded. There are a few famously reclusive or otherwise lone-wolf mathematicians out there (though I think that most of them at least maintain some kind of correspondence with a few respected colleagues), but for the most part, twentieth and twenty-first century mathematics is very much a group endeavour. Frenkel describes how he helped to organize new research in the Langlands Program by gathering together mathematicians from various institutions to hear their input. Belying the stereotypes, mathematics is a very social world. Ultimately, of course, the personal parts of the story are essential to Frenkel’s explanation of why he loves math. Again, I must disagree with those reviewers who pan this book because it doesn’t inspire them to love math … that was never the aim. Neither the book nor Frenkel are naive enough to believe that, I think. But I suspect one reason many people react the way they do when one reveals one’s mathematical inclinations is genuine bewilderment over the idea that a “normal” person could actually love math. As Frenkel points out, even when mathematical achievements are depicted in popular culture, the subject is always a social outsider. (In a way, it’s similar to this whole idea of left brain/right brain people. “Oh, you’re a left brain person!” and, when people find out I teach both math and English, “You’ve got a weird left and right brain thing going on!” But the truth is, a lot of people in “left brain” positions that require logical reasoning are also very creative and passionate and linguistic—and a lot of “right brain” thinkers are also organized and calculating. Humans are diverse, and the stereotypes and categories we create are not that good at classifying us.) The autobiographical elements also humanize what might otherwise be a fairly involved book. When Frenkel talks about loving math, he isn’t pulling a Cabinet of Curiosities here. Don’t get me wrong: I’m all for books explaining elementary math. But I’m pleased that Frenkel tackles much higher-concept, abstract mathematics in a nonetheless accessible and approachable way. I’ve forgotten a lot of my undergraduate math, I am sorry to say. One day I’ll delve back into ring theory and group theory for some fun. I’m pleased by how much I do remember, however. I recognized a great deal of what Frenkel explained, even though some of it still managed to escape me. So when I say Love & Math is accessible, I’m not claiming Frenkel is going to help you comprehend abstract algebra. Rather, he demonstrates some of the concepts that power abstract algebra through some clever diagrams and explanations, and he connects abstract algebra to quantum physics. I particularly enjoyed this latter endeavour. I knew that symmetry was one of the most significant aspects of group theory, but I didn’t understand the specific ways in which group theory actually underlies a good deal of the interactions between subatomic particles. So that was cool. There are many points where Frenkel basically explains the math behind the physics, then says, “Oh, and mathematicians figured this out long before physicists came along and discovered the math was useful.” That’s not to say math is more important than physics (that’s just, like, self-evident), but I love that we can build these models in math without any reference to the physical world … and then somehow, these models become useful in explaining the physical world. That is just mind-boggling. As an educator, I also sympathized with another remark Frenkel makes, rather early in the book. He compares the teaching of math in high schools now to the prospect of teaching art by having students paint fences. That is, we barely get to scratch the surface of what mathematics is in high school. Frenkel speaks of quadratics with the disdain only a pure mathematician could muster. But it’s true: I don’t blame students for thinking that math is boring, because the topics we drill into them and the way we do it tends to communicate that fact. You really don’t need to know the quadratic formula—not in the days of Wolfram Alpha—but symmetry? That’s not only important but beautiful as well. Honestly, Love & Math is not going to make you love math, and it was never supposed to. It’s not going to teach you group theory or representation theory, and you probably won’t have any clue what a Riemannian Surface or a Kac–Moody Algebra is after reading the book. (Maybe you’ll understand what a group is, in some way.) If you’re really interested in learning those things, there are books and videos and courses and wikis to help you out. Instead, Love & Math is one mathematician’s story of how he fell in love with math, how it saved and defined his life, and how he feels honoured and awed that he has had the chance to give back to the mathematical community. Frenkel goes so far as to make a weird surrealist movie about loving math … and that is not my thing, but it’s clearly his thing, and I’m all for people doing their thing. So you go, Frenkel. And while you do that, hopefully some of the people who read this book come away with a better understanding of what it might mean to love math, even if they don’t quite share that feeling themselves. [image] ...more |
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May 29, 2015
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Jun 03, 2015
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Aug 09, 2014
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Hardcover
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| 184614678X
| 9781846146787
| 184614678X
| 3.95
| 20,048
| May 29, 2014
| 2014
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it was amazing
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I math for a living. I mathed, both amateurly and professionally, at school. I math quite a bit. And as a math teacher, I like reading "pop math" book
I math for a living. I mathed, both amateurly and professionally, at school. I math quite a bit. And as a math teacher, I like reading "pop math" books that try to do for math what many science writers have done for science. So picking up How Not to Be Wrong was a no-brainer when I saw it on that bookstore shelf. I’ve read and enjoyed some of Jordan Ellenberg’s columns on Slate and elsewhere (some of them appear or are adapted as chapters of this book). And he doesn’t disappoint. I should make one thing clear: I mainlined this book like it was the finest heroin. Partly that’s because I just love reading about math, but in this case I was also days away from moving back to Canada from the UK when I started this, and luggage space was at a premium, so I was on a deadline to finish this book. I injected chapters at a time into my veins, revelling in that rush as Ellenberg charismatically and entertainingly explores the math behind a lot of everyday concepts and ideas. Unlike similar attempts, however, Ellenberg doesn’t pull the punches. He’s more than willing to go into the higher-concept ideas behind the math, and when it starts getting too esoteric or academic even for this venue, he’s always ready with a book recommendation for those interested in some further reading. Early in my reading, I tweeted I had already decided to give this book five stars because Ellenberg alludes to Mean Girls in a footnote. (Specifically, he says, “As Lindsay Lohan would put it, ’the limit does not exist!’”) That’s really all you need to know about Ellenberg’s writing style and sense of humour. Actually, I’m not all that enamoured with the footnotes in general; they interrupted the flow of my reading and the symbols used to mark them were slightly too small, so I kept missing them in the text—but that’s a design issue. The content of the footnotes themselves is often informative or, as in the case above, humorous. Ellenberg might be a university math professor, but he also has a sense of humour and an awareness of pop culture that helps to make his writing accessible. I’m impressed by the way Ellenberg effortlessly straddles pure and applied mathematics. The child of two statisticians, he clearly has a good grasp and appreciation of the way applied math drives so many areas of society. From economics to gambling, he makes passionate appeals for informed perspectives over simplistic analogies or fallacies. His first chapter criticizes analogies that promote linear thinking about taxation when the very same economists writing these analogies know that taxation probably isn’t linear. He doesn’t argue for or against an increase in taxes, but rather he points out that it’s wrong to oversimplify the concept when trying to sell it to the public. Is a curve really all that much harder to understand than a line? There’s also some great chapters on odds and the lottery, in which Ellenberg recounts how a group of MIT students set up a legitimate operation to bulk buy lottery tickets from a certain game that actually gave them good odds of winning. They made a profit, because they used math to turn a game of chance into a predictable investment strategy (which is more than we can say for the stock market). So, you know, stay in school kids. But actually, the parts about the lottery that impressed me were more towards the purer end of the math spectrum. Ellenberg started discussing, for example, how best to pick the numbers on one’s tickets so that one could maximize the chance of winning at each tier of prizes. It turns out that it’s possible to represent the way of picking these numbers geometrically (yes, as in pictures) and that it’s related to the way we create error-correcting codes (which allow us to send instructions to spacecraft, and compress data in JPEGs, MP3s, and on discs). He goes into quite a bit of detail about the more advanced concepts behind these ideas. Later, he points out how correlation on scatter plots corresponds to an ellipse—and we know how to deal with ellipses algebraically, which gives us a good toolset for talking about correlation algebraically too. So, How Not to Be Wrong makes an effort time and again to belie the impression that we often get in school that math consists of a series of discrete topics: arithmetic, geometry, statistics, and the dreaded algebra. We teach it that way because it’s easier to lay out as a curriculum and focus on the essential skills of each discipline. And also because we are boring. If you’re lucky, like me, then as a student you’ll start to see the connections yourself. Circles and pi start showing up everywhere, to the point where suddenly you feel like you’re being stalked, and no amount of infinite series or integration is going to save you. But really, good teachers start showing these connections as soon as possible. We fail students and leave them behind because, in our rush to equip them with the skills we’ve been told they need, we rob them of the idea that math is a creative process, instead fostering this false impression that math is a sterile, difficult, procedural slog. If it is, then you might be a computer. Ellenberg never demands a knowledge of integral calculus, of set theory, or of transfinite numbers. What he does demand is an open mind, a willingness to be convinced that not only does math have a useful place in life (it’s pretty obvious to most people that someone needs to know how to math; they just don’t see why it should be them) but that a deeper understanding of the roles and uses of math can enrich anyone’s life. One can be a believer in the power of mathematics without necessarily worshipping at its altar, and it’s this quest for adherents rather than acolytes that makes this popular math book successful. It helps that Ellenberg’s style is witty. It helps that he is passionate without sounding too evangelical. He weaves in enough history, anecdotes, and allusions to demonstrate that mathematicians’ journeys and the development of mathematics as a discipline has been just like everything else in life: alternately dramatic and dull, intense, occasionally acrimonious. We don’t like to admit it, but we mathematicians are people too. And occasionally we’re wrong, very wrong (like those nineteenth-century French eugenicists…). The title here is tongue-in-cheek, and How Not to Be Wrong can’t guarantee your future correctness with great certitude. All it can do is help you think more critically, more logically, but more creatively about the problems and questions that you’ll face in the future. Because mathematics is a tool for helping us to do amazing things. You can be a novice, or you can be a proficient user of this tool, but either way you’ll need to pick it up at some point to do a little handiwork. Don’t fear it: embrace it. Oh, and read this book. [image] ...more |
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![A Brief Guide to the Great Equations: The Hunt for Cosmic Beauty in Numbers (Brief Histories) [Sep 20, 2002] Crease, Robert](https://cdn.statically.io/img/i.gr-assets.com/images/S/compressed.photo.goodreads.com/books/1347789709l/6455281._SY75_.jpg)
| 1845292812
| 9781845292812
| 1845292812
| 3.82
| 585
| 2008
| Jan 01, 2009
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really liked it
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To paraphrase Mr T, I pity the fool who doesn’t see the beauty of mathematics inherent in the world around us. As a teacher, I feel rather complicit a
To paraphrase Mr T, I pity the fool who doesn’t see the beauty of mathematics inherent in the world around us. As a teacher, I feel rather complicit at times in robbing children of the joy of mathematics. The systemic, industrial tone of education does not often lend itself well to the investigation and discovery that should be the cornerstone of maths; I find this particularly true in the UK, where standardized tests and levels are the order of the day. There are times when I am conflicted about how to cover subject matter. I have to find a balance between a breakneck schedule and a desire to achieve the comprehension that only comes with time and careful practice, strive to find the equilibrium between exploring interesting lines of inquiry and curtailing those lines in order to teach what’s on the test. I hope that as I become more experienced finding this balance becomes smoother. For now, though, it’s a struggle. Because the secret that everyone learns as a child and then has beaten out of them by the endless grind of daily mathematics lessons is this: mathematics is not numbers. It is not arithmetic. There, I said it. I gave my students a test today on our statistics unit, which involved data collection: designing surveys, selecting sampling methods and sample sizes, etc. As they worked through the test, a few questioned its connection to mathematics. "This is words!" they protested, as if I were somehow an imposter trying to sneak extra English content into their day. Somewhere along the line—I don’t know precisely where—they developed this notion that mathematics is solely about manipulating numbers. Really, though, mathematics is about relationships between things. Mathematics is a process for understanding the world, as well as understanding theoretical constructs that, while not directly observable in the real world, can still have useful and fascinating properties. Math can be numbers, but it’s also truth, in one of the most fundamental ways possible. This is what Robert P. Crease attempts to communicate in A Brief Guide to the Great Equations. He foregrounds each equation and carefully explains how it became a part of the great canon of mathematics. He also explains why the result is so exciting, not just to mathematicians but to the population at large. I’m pretty enthusiastic about all this crazy math stuff, but Crease manages to stoke even my considerable flames of fanaticism and set my heart racing. The way he breathlessly extols the beauty and utility of Maxwell’s equations or Einstein’s relativity … it’s like a BBC Four documentary in paper form. When it comes to books on popular mathematics, I always try to anticipate how a layperson would receive the book. As a mathematician, I don’t have a problem following the equations and explanations; it comes naturally. It still staggers me how some people are able to understand the intense nuances of some of the higher-level mathematics involved in quantum mechanics and relativity; I’m somewhat reassured by Crease’s claims that physicists often rejected new developments that required them to learn a lot of complicated new math. Yet I still know what Crease means when he carelessly bandies around certain terminology, expecting his reader to keep up to speed based on a high school education alone. As far as pop math goes, A Brief Guide to the Great Equations is not the most friendly book. I’d probably hesitate to recommend it to casual readers, preferring maybe Zero: The Biography of a Dangerous Idea . For someone very interested in the history and philosophy of science, however, this book would appeal even if one’s math knowledge isn’t quite up to snuff. Crease recounts without fail some of the more interesting scuffles and disagreements among famous mathematicians and scientists; he also carefully lays out his own views on what constitutes a scientific revolution, and the role that developments of equations can have in revolutions. It’s easy enough to follow the history and soak up the spectacle without following the math. I don’t mean to say that you shouldn’t read this unless you’ve studied math in university. If anything, Crease hopefully sheds light on how and why people can find math such an interesting occupation. By reading these stories of how Maxwell and Einstein and Schrödinger dedicated years of their lives to these problems, one gets the sense that the problems are more interesting and worthwhile than the equations themselves indicate. Crease explains how the problems consumed and intrigued these brilliant minds in such a way that, even if one doesn’t understand the nature of the problem—or its resolution—itself, one can still appreciate the passion and dedication involved. Such passion and dedication are more universal than even the mathematics that unites the great thinkers featured in this book. One need not like math to be good at it or to succeed at it in school or in life. One need only appreciate its versatility, utility, and beauty. Crease tries and succeeds admirably in showcasing such attributes through the equations and history that he includes here. Math is beautiful. You just need to open your mind, cast aside the "but I just don’t have the brain for it", and embrace the wonderful freedom of being able to figure out how the world works. [image] ...more |
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Nov 27, 2013
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Nov 30, 2013
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Nov 27, 2013
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| 1408822385
| 9781408822388
| 1408822385
| 3.67
| 2,065
| 2011
| Jan 01, 2012
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really liked it
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I came across this book while browsing the science section in Waterstones, because that’s where they hide all the good mathematics books as well, and
I came across this book while browsing the science section in Waterstones, because that’s where they hide all the good mathematics books as well, and I was looking for an appropriate math book to give to a fellow math friend for her birthday. (I opted for Ian Stewart’s Hoard of Mathematical Treasures.) Having read Dava Sobel’s explication of John Harrison and the marine chronometer in
Longitude
, I snapped this up without a second thought. Later, I discovered it was already on my to-read list. Serendipity! A More Perfect Heaven is a biography of Nicolas Copernicus. As such, it reveals so much more about him than his importance to the adoption of heliocentric theory. I knew that Copernicus was a Polish mathematician who lived in the early 1500s, and that his work was largely adopted on a mathematical basis rather than a physical one. That was about it. I had no idea of his extensive involvement in the Church, including his canonry and relationships with local bishops. I didn’t know that he developed most of his theory early in his life but held off on publishing until a Lutheran mathematician showed up out of the blue to persuade him to share his theory. So in this respect, Sobel fills in some very large gaps. She brings Copernicus to life, giving names to his parents and friends, setting up the relationships and geography that would define him and influence him as he considered the movements of the heavens. As I mentioned in my review of Longitude, I’m sceptical of the “Great Man” theory of history. It’s undeniable, however, that Copernicus’ book influenced a great many astronomers and mathematicians, a case Sobel makes in the last chapters of the book, with Brahe, Kepler, Galileo, and others. Copernicus was neither the first nor the only great proponent of helocentric theory, but he was in the right place at the right time, and he had the right help, to put it forth. While most of the world wasn’t quite ready to accept it, the idea was now there, ripening in the collective unconsciousness of a generation of scientists. Speaking of which, I felt smug during much of this book. As I read about the Church’s attempts to stifle suggestions that the Earth revolves about the Sun, I mentally giggled at the amount of power religion could wield in the face of scientific discovery. But I laughed much too soon, because while it’s true that heliocentric theory has won the day, there are plenty of contemporary issues that have inherited its political controversy. The sad truth is that not much has changed in the past five centuries. Though the Catholic Church itself is much more friendly towards scientific discoveries than it once was, other elements of religion continue to push against science they see as inimical to their worldview. These deniers rail against everything from global warming to evolution to vaccines. These positions aren’t just quaint throwbacks; they’re actively dangerous. Human nature and human society has not changed all that much since Copernicus’ time, and we should not be fooled into thinking so simply because our scientific understanding has changed since then. Fortunately, A More Perfect Heaven also tells us that, eventually, science will prevail. Copernicus’ calculations were just so accurate that they became the gold standard. That wasn’t quite enough for astronomers to accept his model as fact (cognitive dissonance is a really awesome phenomenon). But it kept the Copernican ideas alive long enough to reach the ears of people like Galileo and Kepler. The former’s discovery of Jupiter’s four largest satellites was a philosophical blow to the idea that everything in the skies must orbit the Earth. The latter’s obsession with finding a beautiful mathematical explanation for certain types of orbital problems led him to expand the Copernican model based on all the data he could obtain from Tycho Brahe’s careful observations. A few centuries on, Copernicus was vindicated, and opinions began to shift. This is probably the exciting part of the story, the part that seem most relevant today. But most of the book is about Copernicus himself and his involvement in Varmia, the Prussian province of his canonry. Sobel recounts Nicolas’ various administrative duties throughout his life as a Varmian canon. I was amazed to learn of his wide interests in everything from medicine to economics, though I shouldn’t be have been so surprised. Copernicus even wrote extensively on money reform! He might not have been a Great Man, depending on your point of view, but he was a great man. Sobel departs from the typical biographical style by presenting the middle of the book as a two-act play, “And the Sun Stood Still”. She dramatizes the interactions between Rheticus and Copernicus that persuade the latter to finish and publish his overall theory. Since little in the way of documentation survives, Sobel has to take certain artistic license with this interpretation. It’s an interesting way to do it, and I was a little sceptical I would enjoy the sudden arrival of a play in the midst of a non-fiction experience. Much to my relief, the play is interesting, easy to follow, and actually rather entertaining. Sobel does it again. Like Longitude, A More Perfect Heaven is the perfect type of popular science history. It’s not too long, yet it’s amazing in its wealth of information. Sobel communicates with a passion for her subject that can’t help but be contagious. She takes the time to lay out exactly why these giants are indeed giants, people who made such a significant and lasting contribution to the way we think and operate in this world. These are the types of books that get me excited and thinking about science even as I marvel at the history of such discoveries. [image] ...more |
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Sep 28, 2013
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Sep 30, 2013
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Sep 30, 2013
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| 1444737449
| 9781444737448
| 1444737449
| 3.53
| 1,853
| Aug 01, 2012
| Jan 01, 2013
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liked it
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I can’t resist picking up mathy books when I’m in a bookstore. As a mathematician, I love broadening my knowledge about the field—and seeing what pass
I can’t resist picking up mathy books when I’m in a bookstore. As a mathematician, I love broadening my knowledge about the field—and seeing what passes for “popular mathematics” these days. Thinking in Numbers is a slim volume that promises to “change the way you think about maths and fire your imagination to see the world with fresh eyes”. It didn’t do that for me—but maybe that’s because I already think about maths that way. Daniel Tammet is an exceptionally talented voice when it comes to presenting the inspirational elements of mathematics, so I hope that for people who don’t quite understand why I get so excited about maths, the book does make a difference. I last wrote about why I love math in 2011. Since then, I’ve graduated from university. I’ve completed research in mathematics and had a paper published. I’ve begun teaching math and English at a high school level. All of these changes have deepened, broadened, and otherwise changed my love for math. As a student, math can be a mystery, a puzzle that demands both ruthless logic and amazing creativity, something that can tickle both the left and right hemispheres of the brain. As a teacher, I’ve tried to make my math classroom as “safe zone” where students can learn, and indeed where they can express a dislike for math, if that’s their opinion. Of course, I’m always out on a little bit of an evangelical mission to change people’s minds. But I’m not asking people to love math; I’m just asking them to reconsider whether they actually hate it, whether they are wrong when they say, “I just can’t do math”. Everyone can do math; everyone does math every day. Math is an integral (no pun intended) part of our society. And it’s just wonderful. Tammet captures a lot of these sentiments in Thinking in Numbers. This is a very unusual math book, in that it isn’t really about math. It’s a collection of 25 very short essays on topics that relate to math tangentially. There are precious few equations or formulae in this book. Instead, Tammet takes a what I might even call an intersectional approach to math. In one of my favourite essays, “Counting to Four in Icelandic,” he explores how different languages form words for numbers. Some languages, Icelandic included, have completely different words for the same numeral depending on whether what it describes is abstract or concrete (whereas, in English, we just say four regardless). In another essay, he ponders the recurrence of the motif of nothingness and synonyms for zero in Shakespeare’s works. He connects this to the spread of zero, from the Arabic world through Italy to the rest of Europe, during Shakespeare’s time. The essays are bite-sized. This is a book easy to devour over the course of a few evenings: read a few essays, then put it down and mull over them before going to bed. There is a preface but no conclusion, and there is no overarching connection or theme, beyond Tammet’s obvious love for the relationship between life and math. On a related note, the topics are quite varied. There is little to suggest a pattern beyond different connections between math and life that have occurred to Tammet over the years. This might prove frustrating for people who are used to more forthright or even argumentative non-fiction. Tammet isn’t so much presenting an argument as opening the door to another perspective on the topic. It’s an invitation, not one side of a debate. Tammet’s writing style always verges on the intimate and philosophical, and he always leans on anecdotes or autobiographical details to furnish his asides. This can work well—I wasn’t familiar with his name, so his account of memorizing and reciting 22,514 decimal places of pi for a new record was fascinating. His essay expounding upon mathematical models using his mother as an example, less so. The book is at its best when Tammet takes a concrete piece of mathematics—pi, calculus, primes—and links to another field, whether it’s the literature of Tolstoy or the possibilities in a chess game. In this way, he demonstrates how math is more than just a series of problems in a textbook, and it’s not just something mathematicians, physicists, and engineers need in their daily lives. It’s this pervasiveness of mathematics that comes to the fore in this book. The dearth of equations, proofs, and even diagrams attests to this: Tammet is not out to explain mathematics. Instead, he finds and traces the connections between math and life. He talks about how an Amazonian tribe that lacks names for numbers conceptualizes the world. He examines Tolstoy’s use of calculus as an analogy for analyzing history. Having recently read War and Peace, I really enjoyed those little allusions to math. For people who only see the epic as this massive work of literature, however, it might seem strange to think that Tolstoy owes his view of history to math. Tammet teases out the cool, unsuspected ways that math can pop up and connect to parts of our lives, and it’s wonderful. Not every essay in this collection is amazing. I’d probably recommend this to most of my friends, with the caveat that they shouldn't read the book all the way through. Instead, this is a collection where it's appropriate to leaf through the chapters and read those that pique one's interest. Tammet covers enough topics that there is probably at least one essay in here for everyone. I was sceptical, when I saw the title of the book and read the brief description, that Thinking in Numbers could impress me. It looked so thin, so insubstantial, that I expected it would be too light, too far on the popular side of popular mathematics. Instead, Tammet delivers something that I wasn't anticipating at all—and it works. [image] ...more |
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Jun 2013
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Jun 02, 2013
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Jun 01, 2013
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| 0771016964
| 9780771016967
| 0771016964
| 3.09
| 47
| 2009
| Apr 28, 2009
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liked it
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I read math books for fun. I realize that, right away, this puts me in an unusual portion of the population. It’s not just my fancy math degree that m
I read math books for fun. I realize that, right away, this puts me in an unusual portion of the population. It’s not just my fancy math degree that makes these books attractive. However, I do think that there are some math books written for people interested in math (whether professionally or amateurly), and then there are math books written for people who, usually thanks to a bad experience in school, have sworn off math like they said they would swear off cheap booze. Our Days Are Numbered is one of the latter. In a passionate and personal exploration of shape, algebra, geometry, and number, Jason I. Brown illuminates the fundamental mathematics behind some everyday tasks. While some people will still run away screaming, others will hopefully begin to see math in a new way. Among the topics Brown explores are: converting between units, using graphs to display data, the meaning behind averages, the role of chance in decision-making, networks and coincidences, prime numbers in cryptography, fractals in art, and the math behind the mystery of the Beatles’ Chord. Each chapter is bookended by a short, two- or three-paragraph anecdote related to its given topic. For the main body of the chapter, Brown gradually develops some of the math behind common tasks. For example, he shows how an understanding of ratio and conversion factors makes converting between units a breeze without any memorization (aside from the factor itself, of course). Later, he explains why the Web and social networking has guaranteed that graph theory will remain a practical and important field of math for a long time. This is not really my kind of math book, and that isn’t even because of the audience or the way Brown presents the math. Rather, I read math books for the story. I’m interested in math books that take a specific topic and explore its history, its present state, and the different ways to interpret it using mathematics. Our Days Are Numbered instead covers a variety of topics. There isn’t anything wrong with this approach. However, each of these topics can be (and has been) the subject of entire, weighty tombs. It’s difficult for Brown to do them justice. Sometimes, such as with the chapter on conversion factors, he does a very thorough job. Other times, such as with his explanation of prime numbers and Internet security, he leaves something to be desired. Also, much of one’s enjoyment will hinge on how one much one likes or dislikes Brown’s writing style. As the chapter titles and subheadings demonstrate, he is a man of corny humour, easy puns, and deprecating remarks towards himself and fellow mathematicians. I can get behind the first and third attribute, and I can ignore the second. Although I think a book any longer might have begun pushing its luck, as it is, I enjoyed Brown’s conversational and easygoing style. Others will find it overbearing and intrusive, however, and there is no escape from it here. So, Our Days Are Numbered isn’t my mathematical cup of tea, but could it be anyone’s? Well, one way in which this book excels is Brown’s unrelenting insistence that math is useful, relevant, and not at all scary. As a math enthusiast and math teacher, the opposites of these sentiments besiege me constantly. I love how Brown comments on the somewhat unique reception math receives at parties: When I tell people what I do for a living, the most common response is a look of dismay, followed by “I always hated mathematics!” This statement is made with relish and without a hint of embarrassment. I don’t think there is another profession out there that gets the same response. Do people state they’ve always hated English? Music? Lawn care? I think not. Tongue-in-cheek, Brown touches on a very crucial and deplorable fact: hating math is socially acceptable. It’s cool to disparage math and one’s ability to do math. To some extent, the aura of nerdery surrounds all of the STEM fields, but scientists and engineers get a little more recognition—people’s eyes might glaze over if one announces oneself as a theoretical physicist, but there is a little gleam of gruding respect. Mathematicians, however … what do they even do? The social acceptability of disparaging mathematics troubles me. Math is the foundation of the other three STEM fields. Science, technlogy, and engineering are all fields that require creative, passionate thinkers. Yet from an early age we send children signals that math is a dull, uncreative subject and it’s OK to hate it for being boring and irrelevant. This is nothing short of educational sabotage. It’s certainly fine for people not to like math, and I understand how parts of the educational system foster that feeling. But we should do everything we can to avoid reinforcing that notion, especially among our children. Hence the power of this book. Brown takes it as a given that math is a useful, powerful tool in the everyday world. He isn’t out to convert everyone to a science or engineering job. He isn’t trying to shoehorn calculus into a discussion of changing a car tire. (As a teacher, the incessant call to include real-world applications and contexts in my lessons wearies me at times.) He is careful not to insist that everyone uses or needs all of this math all the time—you don’t need to know how to use prime numbers in order to keep your online banking secure. But isn’t it nice to know why it is secure? Brown’s non-evangelical stance is refreshing, though it can also be a little frustrating. Our Days Are Numbered lacks a true, cohesive message, aside from the idea in the title. With no introduction and no conclusion, Brown relies on the title and the chapters to come together to create that singular idea. While not essential, some kind of introduction or meta-narrative would lend additional structure to this otherwise scattered text. With brilliant mathematics, hardcore mystery-solving, and no small amount of humour, Our Days Are Numbered is a well-written and very successful math book. It isn’t anywhere close to my Platonic ideal of what a math book should be—but that’s me being picky. Nor do I think, in the long run, that people convinced math is uninteresting or “not for me” will find their convictions toppled by anything in here. But for anyone who is open to learning about the role of math in everyday life, there is definitely something here, waiting to be read. [image] ...more |
Notes are private!
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Mar 05, 2013
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Mar 11, 2013
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Dec 31, 2012
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Hardcover
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| 1402785844
| 9781402785849
| 1402785844
| 3.65
| 147
| Oct 04, 2011
| Oct 04, 2011
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liked it
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On my last official day with my Grade 8 class, I did not want to teach them more about fractions. Instead I had asked them to submit a question they h
On my last official day with my Grade 8 class, I did not want to teach them more about fractions. Instead I had asked them to submit a question they had about mathematics—anything, from something they’d learned earlier in the year but didn’t understand to a question that had been simmering since sixth grade. The cards I got back were all across the scale, from earnest to uninterested. Quite a few were about pi. I decided to take the questions and weave them into a broader narrative about the use, purpose, and history of mathematics. I wanted to talk about how we figured out math and discuss some of the milestones in mathematical discovery. Prezi has become a pretty big deal at my university’s faculty of education, but until now I had avoided it. I decided that I should probably make at least one before I was finished my undergraduate degree. Plus, my partner student teacher had the Grade 8s make their own prezis for a history project. So I made my first prezi to talk to Grade 8s about math. Part of my goal as a teacher is to expose my students to the wider world of mathematics, to impress upon them that math is more than just skills and concepts they learn out of a textbook in the fulfilment of curriculum expectations. I want to make the usefulness and purpose of all that math explicit—and I want to go even further and show that math can be beautiful. Finally, it’s important to provide a sense of history and context to all this math. Because the history of mathematics—and the lives of those caught up in it—is intensely fascinating. Or at least I find it so. Stories of love, betrayal, comedy, and tragedy pervade story of math. Because doing math is ultimately an act of discovery and of creativity—and those acts are what make us human. Amir D. Aczel recognizes this in A Strange Wilderness, which is a history of mathematics disguised as a biography of mathematicians. He makes it his mission to relate the stories behind the math, such as Pythagoras’ travels and interesting diet to Archimedes’ famous bathtub epiphany. (Lucky for me, my Grade 8s had not heard the Eureka! story, despite having just concluded their unit on fluids. So I got to tell it to them for the first time!) This is a laudable goal, and one that coincides with my own. Owing to the way it’s taught in school, we often treat mathematics like received wisdom, far more than we do even science. Mathematical concepts just exist, passed down to us by the teacher and the textbook. It’s difficult, if you don’t actually go out and look for it, to realize that someone had to ask the questions and make the leaps that gave us these concepts. These people were all living, breathing individuals at some point in history, with the same mundane concerns as any human being. For reason, though, through a combination of genius and effort and luck, they made a lasting contribution to our wealth of knowledge as a species. Aczel brings a wealth of knowledge and enthusiasm to this endeavour. I discovered a lot of cool things about names I already knew, and I met a few fresh faces as well. I marvelled at the chain of events that led to people like Isaac Newton becoming the juggernauts of their day. Newton’s mother, after abandoning him for a new husband, apparently pulled him out of grammar school to live on a farm. It was only through the intervention of his uncle that he returned to finish his education and end up at Cambridge. I shudder to imagine how history would have played out differently if Newton had stayed on a farm! Of course, a book this size can’t do justice to the history of mathematics or all the mathematicians involved in it. Aczel seems to do his best to hit the high notes. That being said, he makes some curious decisions about who to leave out. In particular, the book seems to start off strong but lose steam, and by the time we reach the twentieth century, great minds like Lebesgue, Zermelo, Russell, Hilbert, and Gödel get cameos if they’re mentioned at all. I don’t know if this is just a consequence of the rather dense nature of twentieth-century mathematics compared to the previous centuries or if Aczel was worried about the complicated nature of the math. Certainly he focuses less on the math itself and more on the mathematicians, as is the case with the final mathematician, the reclusive Alexander Grothendieck. I guess you can’t please everyone, of course, and Aczel does his best while trying to keep the book to a manageable length. As you might be able to tell, I’m passionate about the history of mathematics. While I’m sure Aczel is too, I have to confess that the stories in this book come across much drier than they should. Maybe it’s a result of reading so many short biographies back to back—it’s just a steady diet of mathematical dessert. Whatever the reason, as much as I enjoyed A Strange Wilderness in small doses, it took me longer to read than I expected. There’s something to be said for books with narrower scopes and their ability to take a detailed look at the lives of a select few. In combination with other resources, for it is certainly not exhaustive, A Strange Wilderness is a fine book on the history of mathematics. People who aren’t that familiar with (or comfortable) with math shouldn’t have a problem reading this book. Aczel will often discuss the details of the mathematics that his featured geniuses discovered. However, he characterizes the most esoteric items (like group theory) in very general terms, and even when he gets a little more specific (such as with his discussion of Leibniz and Newton’s calculus), it’s never too technical. The math in this math book consists mostly of shout-outs, an understanding of which is far from essential for enjoying this book. As usual, it comes down to what you want out of your mathematics book. If, like me, your interest in the history of mathematics burns bright and you’re familiar with quite a few of these lives already, then there are probably better books dealing with more specific topics. You can certainly discover new things in this book, but it won’t blow you away. This is definitely a good starting point, however, for those who know that mathematics has some interesting stories to tell but just aren’t sure where to find them. [image] ...more |
Notes are private!
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1
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Apr 12, 2012
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Apr 24, 2012
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Jan 02, 2012
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Hardcover
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| 0767908163
| 9780767908160
| 0767908163
| 3.79
| 6,134
| 2002
| Sep 23, 2003
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liked it
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This is one of the oldest (perhaps the oldest?) physical books I own and have yet to read. Goodreads suggests I’ve had it for nearly a decade. Oops. T
This is one of the oldest (perhaps the oldest?) physical books I own and have yet to read. Goodreads suggests I’ve had it for nearly a decade. Oops. The truth is, I was never excited to read this. I love reading math books! But I am not particularly enamoured of books that explore one or two “special numbers,” and phi is perhaps my least favourite special number. The blurb from Dan Brown on the cover didn’t help. See, phi has been egregiously sexed up and romanticized by people, turned into a mystical number that recurs exactly throughout art and nature, and ascribed aesthetic properties it doesn’t deserve. I was nervous this book would repeat these claims. Well, I owe Mario Livio an apology. Not only does he critically challenge those claims and debunk a lot of the hogwash surrounding the golden ratio, but he also takes it upon himself to tell a broader and more complete story than focusing solely on this number. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number is a good story of the intersection of mathematics and society, and it provided one key insight that, as a math and English teacher, I find very valuable. You would be forgiven for, having begun the book, thinking that Livio has entirely forgotten about phi for the first couple of chapters. Rather, he explores the history of numbers and counting in general, eventually ended up in ancient Babylon and Greece and making some connections with geometry. This creates a much richer backdrop for Livio’s later exposition of the golden ratio, and it also broadens the reader’s awareness for how various cultures developed and practised mathematics at different points in history. For example, Livio discusses the Rhind/Ahmes papyrus, which famously provides insight into Egyptian mathematics about 3500 years ago. He emphasizes the papyrus’ purpose as a teaching/reference tool—it specifically explains how to do practical calculations. Fast-forward a couple of millennia, and Fibonacci was doing the same thing—writing tutorials, essentially, for accountants. See, I appreciate this, because most approaches to discussing the golden ratio focus on the idea that its use in architecture, art, etc., invokes certain ingrained aesthetic ideals in us. These approaches further seek to ground the golden ratio in the idea that its proponents and adherents throughout history have sought it out as a result of being fascinated with mathematical beauty. Livio, on the other hand, reminds us that a great deal of mathematics was (and remains) practical. It’s true that the Pythagoreans were a semi-mystical cult that believed their discoveries reflected the beauty of nature—but the problems they solved were motivated by questions of geometry and arithmetic that were relevant to life in Greece at the time. This has remained true throughout history: our development of mathematical approaches is driven by our needs as a society. The adoption of Hindu-Arabic numerals, for example, didn’t happen because they are “more beautiful” than Roman numerals—the accounts liked them better for arithmetic! It might seem strange for a mathematician, especially one who loves pure math, to be arguing against the idea that beauty should be a foundational concept of mathematics. And I’m not, not really. But I agree with Livio that viewing mathematics in the past through a lens of beauty/aesthetics is ultimately an ahistorical reading that confuses more than it illuminates. Understanding the emphasis on practical applications for math helps us understand its place in our society. And this is where The Golden Ratio really got me. Several chapters examine whether well-known artists used the golden ratio in their work. Livio discusses the works themselves, as well as numerous scholarly intrepretations both for and against the idea that the golden ratio played a part. I appreciate his extensive use of references and the way he engages with the topic as objectively as possible. Most importantly, Livio suggests that our desire to spot the golden ratio in this artwork undermines and devalues the artists’ general mathematical brilliance. If the aesthetic quality of a work of art were simply the matter of using the right shape of rectangle everywhere, what does that say about art and artists? Why wouldn’t we have made a computer program that can generate “the perfect work of art” by now? No, Livio concludes, the brilliance of these works of art is independent of their use, or lack of use, of the golden ratio. It comes from a far deeper grounding in mathematics than we care to credit—from the use of perspective to plane geometry, math is everywhere in art. He points out how some artists, like Durer, studied mathematics purposefully to improve and influence their artistic output. I teach math. I also teach English. People treat me like a unicorn because of this, but I really don’t see them as all that different. Neither did Charles Dodgson, who wrote Alice in Wonderland. Livio cites numerous other poems and literary works that use math, as subject matter or metrical inspiration or both. He reminds us that this siloing of STEM is a recent and very artificial phenomenon, that throughout the majority of history, STEAM indeed was the rule of the day. The idea that if you have an artistic sensibility you must somehow be allergic to mathematics is ahistorical and untrue, for as Livio points out here, many of the most celebrated and famous artists studied, understood, and used math in their work. In this way, The Golden Ratio provides a far more valuable story than simply “the world’s most astonishing number” (which phi is not). Livio’s tangents into philosophy, history, art, and music remind the layreader that mathematics is not this alien construct that only super-intelligent people can appreciate or do. It is fundamental to our lives, to our praxis, and to our pleasure—not for any innate beauty it possesses, but for the way its practice can help us create what we consider beautiful. The golden ratio does not play as big a role in this process as some want you to believe. Rather, as is usually the case, the truth is far more wonderful and broader in scope than the simple idea that one number can rule them all. Originally posted on Kara.Reviews, where you can easily browse all my reviews and subscribe to my newsletter. [image] ...more |
Notes are private!
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1
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Dec 23, 2020
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Dec 26, 2020
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May 20, 2011
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Paperback
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| 1926851099
| 9781926851099
| 1926851099
| 3.17
| 266
| Jan 01, 2011
| Mar 31, 2011
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it was ok
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Do you ever feel like you have let down a book, not the other way around? That if you had been smarter, funnier, prettier, then the book wouldn’t have
Do you ever feel like you have let down a book, not the other way around? That if you had been smarter, funnier, prettier, then the book wouldn’t have broken up with you by text message and started dating your friend, who really isn’t all that much prettier than you and has terrible taste in clothing and music and restaurants anyway? No? Just me? OK. I kind of feel that way about Napier’s Bones. I first heard of this book from a “Big Idea” piece on John Scalzi’s blog. It sounded amazing: mathematics as magic! People, called numerates, who can see and manipulate numbers. I had a coworker this summer who has a form of synesthesia where numbers move around for her, and I was really curious about this phenomenon. Derryl Murphy’s concept reminds me of that, and of course as a mathematician myself, I’m fascinated by the idea of being able to manipulate numbers in a very real way. So I was excited for this novel and bought it new a few months later. Alas, this is one of those times when the premise is far superior to the execution. For the first few chapters, this is an OK book. Indeed, my mathematical interests had me positively tingling as I read about Dom’s acquisition of an adjunct, the shade Billy, and his newly-minted status as a fugitive from a shadowy opponent. It was an in media res opening that promised Murphy would keep the action going right until the last page. For the most part, he delivers on this promise, which is one reason I decided to go with two stars instead of one. I have many criticisms of Napier’s Bones, but “dull” is not one of them. The cracks are tiny but appear early. Murphy loves his exposition, and although Jenna is by no means a minor character, her primary role for the first half of the novel is as a listener to Dom’s Mr. Exposition-pants (TVTropes). The action/travel sequences are really just what happens in between the lengthy conversations in which Dom explains how numbers behave, how numerates manipulate them, how mojo enters into the equation, etc. Jenna nods and smiles. It’s the most unsatisfactory way to explore a mythology and a magical system; I wish Murphy had put as much effort into unfolding his universe as he did constructing it in the first place. The real trouble begins about halfway through, when Jenna and Dom get rescued from near-certain death by a mysterious, defrocked priest who introduces himself as Father Thomas. It turns out that John Napier is back from the dead, has possessed someone important to Dom or Jenna, and is after some of his old artifacts in a quest for ultimate power. Fair enough. I mean, his name is in the title, so I was expecting Napier to show up—in body or in spirit—at some point, and I was pretty sure Napier’s actual bones would be an important part of the story. I have no problems with this. Once again, however, I take issue with how Murphy communicates all this information. (And I could have done without being told, almost every time his name comes up, that Napier invented logarithms. I get it.) Father Thomas explains why it is so important for Dom and Jenna to hop on a flight he has booked them to Scotland. And then we never see Father Thomas again. When you introduce a character for a single chapter whose only purpose is to provide major exposition and a plane ticket, you are doing something wrong. So Dom and Jenna hop the pond to Scotland, where they gallivant across the countryside, searching for several important artifacts. They meet another ally in their quest, in the form of intelligent numbers who collectively choose to call themselves “Arithmos”. I wish I were making this up. When the talking numbers enter the story is where I draw the line and where Napier’s Bones goes from slightly flawed to outright bad. Murphy’s interesting idea about numbers being a form of magic degenerates into a messy equivocation of magic and quantum mechanics. With each chapter, he introduces new rules—and exposition to go along with those rules—and more conditions for victory (or failure) on the part of Dom and Jenna. I dislike it when magical systems don’t feel consistent but instead appear to change based on the needs of plot. Speaking of which, the ending itself is somewhat of a literal deus ex machina, at least as far as I can tell. By the time we got that point, I was not so interested in the story any more. The plot had become hard to follow, and my emotional connection to Dom and Jenna was tenuous. This is probably the dealbreaker, in my book, even though it is the most subjective part of the relationship between reader and story. I can handle oppressive levels of exposition and poorly-constructed systems of magic. But ultimately a story is about the reader connecting to a character (or characters), and that did not happen here. Napier’s Bones is just a mess. Its narrative is jumbled, chaotic, and confusing. Its themes are feeble and spread thinly across a book that is longer than it needs to be and still feels far too short. The “magical system” that underlies the story is unsatisfactory and, worse, feels completely arbitrary. The characters start off as interesting and actually become less well-defined as the story progresses. For all of these reasons, I had a difficult time feeling anything more than ambivalence toward this book. Is it me? Did I do something wrong, Napier’s Bones? Am I too mathy for you—was that voice in my head going, “This isn’t why mathematics is magical!” too loud? Did I say the wrong thing in front of your parents on that one Saturday night when I was tired from a long day at work and they dropped by, despite the fact I told you to tell them Saturdays weren’t good for me, and I would prefer that they give us advanced warning so I could at least tidy up the place, because it’s not like you ever bother to do it? Just … give me a sign, please. I can change. Maybe it’s better if I just see some other books for a while, you some other readers. We can get some perspective. A lot of perspective. [image] ...more |
Notes are private!
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1
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Feb 07, 2012
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Feb 10, 2012
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Mar 29, 2011
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Paperback
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| 1841196509
| 9781841196503
| 1841196509
| 3.58
| 797
| Sep 12, 2003
| 2003
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liked it
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My two teachables, the subjects which I will be qualified to teach when I graduate from my education program in May, are mathematics and English. When
My two teachables, the subjects which I will be qualified to teach when I graduate from my education program in May, are mathematics and English. When I tell people this, they usually express surprise, saying something like, “Well, aren’t those very different subjects!” And it irks me so. They’re not, not really. Firstly, mathematics and English are both forms of communication. Both rely on the manipulation of symbols to tell a tale. As with writers of English, writers of mathematics have styles: some are elegant yet terse, seemingly expending little effort while getting their point across with an admirable economy of symbols; others are expansive and eloquent, elaborating at some length in order to furnish the reader with an adequate explanation. Secondly, as with English, mathematics is very much grounded in philosophy and history, and it is a subject that is open to deep, almost spiritual interpretation. If you balk at that last idea, don’t worry. You’ve probably had it drilled into your head since elementary school that in mathematics there is only one correct answer! How could such a reassuringly logical subject be open to interpretation? Despite its apparent objectivity, mathematics is just another human endeavour, and like all our mortal works, it is vulnerable to our flaws, foibles, and fits of passion. Mathematicians can be just as stubborn and argumentative, if not more, than other people. There are many famous follies and feuds in the history of mathematics, and that is one of the reasons I enjoy learning about it so much. Infinity is one of the mathematical concepts most central to those feuds. It’s one of the areas where math rubs up against the spiritual realm—for, as some mathematicians and philosophers have wondered, what is infinity if not God or some kind of greater being? So it seems natural to look at our shifting views on the infinite along the continuum of the history of maths. In A Brief History of Infinity, Brian Clegg does just that, following the classical, somewhat Eurocentric development of math from Greece to Rome, then zig-zagging down to the Middle East and India before flying back to Britain, France, and Germany. As with most tricky math concepts, the trouble with infinity begins with its definition. One must be very careful with definitions in math—for example, it is not enough merely to say that infinity means “goes on without end”. After all, the surface of the Earth has no “end”, but that does not mean the Earth has infinite surface area! Rather, the surface of the Earth is unbounded. Grasping the idea of infinity as “not finite” is easy enough, though: there is no “last” counting number, because you can always add one to the largest number you can conceive, and suddenly you have a new largest number. So infinity is a quicksilver of a concept: intuitive and easy to grasp, yet also elusive and far too fluid for some mathematicians to handle. The Greeks, with their mathematics strictly confined to the geometric figure, would have no dealings with the infinite. Infinity confused Galileo, who nevertheless bravely meditated upon it in his final days. And the shadow of infinity hangs over the controversy of the calculus that caused the divide between Newton and Leibniz, and correspondingly, between Britain and the Continent. The story of infinity gets even more interesting after that. In general, I love the history of mathematics during the 1700s and 1800s. So many brilliant minds pop up during that time: as Newton and Leibniz exit, Euler and Gauss enter. Later, Cauchy and Weierstrass formalize the concept of the limit, which does away with any need for infinity in calculus at all! There are plenty of names and plenty of stories—and this is where A Brief History of Infinity starts to lose its edge. The first few chapters of this book are fascinating. Clegg devotes a lot more space to the Greek philosophers than others might, going so far as to mention some of the more obscure ones, like Anaxagoras. He provides a considerably detailed development of Zeno’s paradox (well, paradoxes) and a nice, if basic, grounding in the idea of an infinite series. Clegg lays the ground well for what will come in later chapters, all the while emphasizing the reluctance of the Greek philosophers to abandon the solidity of numbers found in the real world. But as we get closer to those magical two centuries following the great Newton–Leibniz schism, the story of infinity gets more complicated as more people get involved. This book is very similar to Zero: The Biography of a Dangerous Idea . In my review of Zero, I praised the author’s ability to stay focused: The story intersects with the lives of many famous mathematicians, but the obvious slimness of this book testifies that Seife managed to distill only what was necessary about their lives in his quest to explain the mystery of zero. To be fair to Clegg, this book is almost as slim as Zero. And although he happens to go off on many a tangent, he at least has the ability to find his way back on track quickly enough—that is, his tangents are interesting and informative. He sometimes seems to go into more detail than is strictly necessary to get the point across, and once in a while he waxes melodramatic—as is the case when he links Cantor’s madness to his study of infinity. Overall, however, Clegg’s writing is crisp and clear. I’m also impressed by the detail and depth of Clegg’s explanation of the math. He goes so far as to list and briefly elaborate upon each of the axioms of Zermelo-Fraenkel set theory! I was half expecting him to mention the Banach–Tarski paradox after that—he doesn’t quite get there, but he does explain the difference between ordinals and cardinals, develop the continuum hypothesis, and even mention Gödel’s Incompleteness Theorem. He tackles whether imaginary numbers are truly all they’re cracked up to be. And he even discusses nonstandard analysis—we didn’t even learn about that in university. Don’t let my awe scare you away, though. Rather, think of it like this: if you are not particularly mathematical and read this book, you will gain a wealth of knowledge. You will be fun at parties! If you are particularly mathematical, then depending on how much you like the history of math, you might already be familiar with most of these anecdotes. But the book will still be fun to read, and chances are you will learn at least one or two new things. So I would recommend A Brief History of Infinity to most people—perhaps not with the same zeal that I do Charles Seife’s Zero, but with a similar hope in mind. I hope this book, or at least my review of this book, demonstrates why I find math, as well as the history of math, so fascinating. It’s not just all about numbers, solving for x, and finding the One True Solution. Mathematics is a subject with a long and storied past, one that is fun to explore by looking at the humans who progressed—or regressed—throughout the centuries. A Brief History of Infinity is a book in this mould. While its organization and its focus leaves something to be desired, its scope and ambition do not. [image] ...more |
Notes are private!
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Dec 19, 2011
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Dec 23, 2011
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Jan 08, 2011
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| 4.27
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it was amazing
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Feb 06, 2022
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Feb 18, 2022
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| 3.57
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it was ok
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Sep 06, 2020
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Sep 04, 2020
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| 3.73
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liked it
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Aug 18, 2020
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Aug 16, 2020
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| 3.68
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really liked it
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Jun 10, 2020
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May 19, 2020
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| 4.12
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really liked it
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Mar 14, 2019
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Sep 03, 2018
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Jun 24, 2018
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Jun 22, 2018
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| 4.19
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Jan 27, 2017
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Jan 20, 2017
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| 3.88
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Apr 30, 2018
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Oct 07, 2016
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| 4.17
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it was amazing
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Nov 04, 2015
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Dec 30, 2014
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| 3.71
|
liked it
|
Mar 25, 2015
|
Aug 09, 2014
|
|||||
|
| 3.69
|
liked it
|
Jun 03, 2015
|
Aug 09, 2014
|
|||||
|
| 3.95
|
it was amazing
|
Aug 02, 2014
|
Jul 31, 2014
|
|||||
|
| 3.82
|
really liked it
|
Nov 30, 2013
|
Nov 27, 2013
|
|||||
|
| 3.67
|
really liked it
|
Sep 30, 2013
|
Sep 30, 2013
|
|||||
|
| 3.53
|
liked it
|
Jun 02, 2013
|
Jun 01, 2013
|
|||||
|
| 3.09
|
liked it
|
Mar 11, 2013
|
Dec 31, 2012
|
|||||
|
| 3.65
|
liked it
|
Apr 24, 2012
|
Jan 02, 2012
|
|||||
|
| 3.79
|
liked it
|
Dec 26, 2020
|
May 20, 2011
|
|||||
|
| 3.17
|
it was ok
|
Feb 10, 2012
|
Mar 29, 2011
|
|||||
|
| 3.58
|
liked it
|
Dec 23, 2011
|
Jan 08, 2011
|
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