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What interested me most immediately was the cast: the people involved and their intellectual and personal relationships. Hobbes' first public involvement with squaring the circle, was his intervention in the dispute between the Danish mathematician Longoburg (Longomontanus) and John Pell. Interestingly, Hobbes demonstrated that Pell was correct in rejecting Longomontanus's claim to having squared the circle. Pell was author of an essay regarding establishment of a library of mathematical books and instruments. That essay was published as an appendix to his friend John Dury's The Reformed Librarian. Pell and Dury were part of the Hartlib circle, as was Theodor Haak, who reported Hobbes' role in the Pell/Longomontanus dispute to John Aubrey. After that we see Henry Oldenburg, secretary of the Royal Society, asking Hobbes to make up a list of mathematical texts. Presumably, the list was for Robert Boyle. In 1658 we find Thomas White and Kenelm Digby carrying communications between Fermat and Fenicle de Bessey in France and Lord Brouncker and Wallis in England. White, a Jesuit, had headed the English College at Douai; he had engaged Hobbes in debate over Galileo's Dialogues; and, at the end of a long life of intellectual distinction, he was caught up in the Popish Plot fabricated by Simon Oates. Etc.
It's worth describing the problem of squaring the circle, because it accounts for so much. Simply put: can a square be constructed that has the same area as a given circle, or vice versa? The key to the answer is in the word "constructed." By means of certain curves, such a square or circle can be defined. But those curves cannot be constructed using the classical instruments: ruler and compasses. Essentially, we're dealing with the transcendental character of pi, something that wasn't proved mathematically until the early 19th century.
Jesseph asks the question: why did Hobbes insist so long, to the end of his life, that he had squared the circle? Wallis refuted him again and again, but Hobbes hung on like a snapping turtle. Answering the question opens a window on the political, philosophical, social, and mathematical developments of the time. To summarize:
Hobbes's materialist philosophy brought him to support Isaac Barrow in Barrow's contention with Wallis over the primacy of geometry over arithmetic. These were the two components of classical mathematics, geometry dealing with continuous quantities and arithmetic with discrete. Barrow held that geometry was primary because it dealt with real physical reality, like distances and area, whereas numbers were abstractions: 1, 2, 3, etc. are concepts drawn from one egg, two brothers, three ships, and the like.
Hobbes' Erastianism--his contention in Leviathan that the king should be ruler over the religion of his subjects--placed him on the side of the Independents (congregationalists) against the Presbyterians in Protectorate religious politics. Hobbes, himself an atheist, felt that the Independents could be more easily be brought to heel by the King than the Presbyterians. Wallis was a Presbyterian, had even been Secretary of the Westminster Assembly.
Hobbes shared with the eduicational reformer John Webster and the Independent preacher William Dell a mistrust of the universities as temples to the scholasticism deriving from Aquinas. Wallis and his friend Seth Ward, Savilian Professor of Astronomy, were both actively involved in defending the universities from these attacks.
Of particular interest is the first of these issues. The debate between geometry and arithmetic came to the fore at this time because the distinction between the two was breaking down. Descartes (1637) and then Viete (1646) had defined analytical geometry. Napier (1620s?) had derived the logarithm from his work on the geometrical problem of the parallax. Cavalieri (1635) developed a "method of indivisibles": parallel slices of a geometric figure which, taken together, define the figure. Where this all is going, it seems to us in Whiggish hindsight, is toward notions of infinity, infinite series, and Newton's calculus.
I'll close by noting that Jesseph responds to Shapin and Schaffer, who, in their Leviathan and the Air-Pump, make "the success or failure of a scientific research program entirely independent of the truth or falsehood of the program's central claims." Jesseph calls their approach Wittgensteinian, but it was recognized when the book came out as post-modernist radicalism: science as pure social construct, without necessary basis in either the physical world or agreed canons of reason and method. The fact that Hobbes persisted so long in his claims is certainly less important than that after 1670 he was no longer taken seriously as a mathematician, except insofar as Wallis took the time and effort to refute him in the Transactions of the Royal Society.
Regardless, I think this a great book. Once I got comfortable with the terms, I realized this might be construed as something much more interesting than the traditional 'discovery' of mathematical truths. I'm still not exactly sure how to characterize it, but I'm having fun thinking it a history of science fiction. After all, 'squaring the circle' is the 17th century equivalent to predicting the winner of the Kentucky Derby or tomorrow's change in the Dow Jones Industrials. I may be stretching a bit here, but there is clearly more on the plate than justifying 17th century mathematic revolutions with apples falling on the head of a reclining Newton.
Rather than placing the subject matter in purely mathematical terms, Jesseph considers his material in a wider context, one that makes room for Restoration style science fiction. Keep in mind that Newton's alchemy was an early form of teleportation and the monads of Leibnitz took advantage of an early warp drive. Despite overtly humiliating Hobbs for his mathematical errors, its clear that Jesseph finds Hobbs the ultimate winner. Hobbs suffers no more from his lapse of academic rigor than any contemporary science fiction author. And like Jules Verne, the spirit of his ideas has won if not his details. The political science advocated by Hobbs in Leviathan is hard to distinguish from contemporary standards. Hobbs advocated severely limiting the legal authority of church bishops, scientific materialism and the notion that good laws could produce a good society. Meanwhile, Wallis would be known as a dogmatic right-wing Christian fundamentalist. Further, most of us are convinced that science has 'solved' the problem of 'squaring the circle' which is all Hobbs was advocating, anyway.
And so, Jesseph does a subtle job of indicting my modern sensibility. Painting Hobbs to be the fool, he is actually pointing a finger at my happy secular humanism. Bravo!
So, why is squaring the circle so tricky?
In short, an attempt to find the circle which is exactly 1 square foot in area forces us to confront conflicting intuitions about how we prove the existence of 'real' objects. Try it out for yourself. It is something you can attempt with pencil and paper, or better... try it with a home computer. One of the great things about this book is that it lays out the mathematical issues clearly enough that anyone with high school algebra and maybe an ability to write an excel spreadsheet, can play the 17th century geometer and mathematician. I had a great time doing a 'quadrature' in Excel. If you are interested, I'll email you the spreadsheet (see users.htcomp.net/markmills).
In summary, I think Jessup's book fits into a broad, ongoing reappraisal of mathematical history. I can identify 3 trends, 'Squaring the circle belonging to the 3rd and most important of them:
1. Bringing non-western mathematics to western readers. The best of this is the ongoing research into ancient Chinese mathematics. See 'Chinese Mathematics: A Concise History', Li Yan, Du Shiran, John N. Crossley, Anthony W.-C. Lun, Shih-Jan Tu or 'Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing', Christopher Cullen. In short, most of what the Europeans called 'new' math in the 1500s had been around for 500 years in China.
2. Finding a physiological basis for mathematic intuitions. Check out the cognitive research described by 'Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being', George Lakoff, Rafael E. Nunez, Rafael Nuñez. All you really need to read is the first chapter. The rest is rather speculative.
3. Reappraising the conventional myths about heroic ancient European mathematicians. Unless your ambition is a tenured job teaching the history of math, you will have a great time reading sensible inquiries into pre-modern western math. Take a look at 'Biographies of Scientific Objects , Lorraine Daston (Editor), or Pappus of Alexandria and the Mathematics of Late Antiquity, Serafina Cuomo.
A first fundamental issue concerned Hobbes' materialistic foundation of mathematics where Wallis reasoned from the traditional account that mathematical facts do not depend on the structure of the material world. A second source of dispute was the completely different concept of ratios: Wallis defended that those ratios could only be applied to homogoneous quantities. Thirdly, the angle of contact between a circle and its tangent was a subject of wilful misunderstanding from the part of Wallis. Finally, the important 17th-century debate on infinitesimal small quantities was heavily criticised by Hobbes and although he did not develop an alternative, his objections to some of the obscurities of Wallis' arithmetic of infinities were well justified. These disputed foundations on the philosophy of mathematics were not the only sources of the irreconcilable conflict. Wallis and Hobbes also held opposing views on methodological issues such as the nature of demonstration and the centuries-old discussion on analytic and synthetic methods. For Hobbes all demonstration must arise from causes and as such he rejected techniques from algebra and analytic geometry in which one starts by assuming the truth of the proposition that is ultimately sought and deduces consequences from that assumption. In doing so he tossed aside the tools that might have helped him in his desperate attempt to make his mark as mathematician. Apart from diverging views on the fundaments of mathematics and methodological issues, religious and political positions play part in explaining the controversy. Wallis fitted neatly into the reformed tradition while Hobbes' religious opinions stood far apart. Hobbes was excluded from the newly established Royal Society for ideological and personal reasons and as such was deprived from a forum to respond to his critics. His political opinions and his vitriolic polemics at universities brought him into conflict with many important people and explain at least some of the vehemence with which the dispute was conducted. However, Jesseph refrains from pursueing sociological reductionist account and spends part of the last chapter convincingly demonstrating the inadequacy of a purely sociological explanation of the dispute. By uncovering the conceptual gulf dividing Hobbes and Wallis, Jesseph succeeds in demonstrating important differences in the philosophy of mathematics in the 17th century and explains why these two men engaged in such ferocious fight. Objectively, Hobbes miserably failed attempt at the solution of classic geometric problems makes him the loser of this dispute. But one can feel in this book also some sympathy for the consequent way in which Hobbes rigorously applied principles of his philosophy to mathematics and ultimately rejected classical geometry to avoid an even worse fate: the refutation of his own philosophy.
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I love this book, this children's book...
Adults read children's books to children, but seldom pick one up for their own pleasure. They miss a lot by the omission. THE WEIGHTY WORD BOOK is an example of a book written for children that is delightful for adults, at least adults who love words.
It's written by three University of Colorado English professors and consists of a series of whimsical and enchanting stories about words -- selected words, useful and syllabic.
In the first story, for example, Benjamin Van Der Bellows, a bear "who did not know the things he was supposed to know" is moved each time he makes a mistake to an office on a lower floor. Predictably, he winds up in the basement. "So, whenever a person has been lowered in position or rank or office, we say that person has suffered an abasement."
There's an imaginative and articulate story about each letter of the alphabet. Don't miss this one -- it's for all ages and it's loads of fun.
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Enter Pierrette, an intelligent young girl who sees an apocalyptic vision of the future. She can only save the world by training to become a sorceress, and _The Sacred Pool_ is the story of her education. Pierrette must experiment with long-forgotten spells and newfangled science in order to defeat a demon that plagues her sister, and in the long run, to save magic itself from being destroyed.
This book starts out slow, but gets interesting once Pierrette begins her studies. It is thought-provoking and intelligent, and one of the few novels dealing with paganism and Christianity that says anything more profound than that one is "good" and one is "bad" (take your pick which is which; there are plenty of books taking each side). If you like fantasy that makes you think, check out this tale of magic, belief, science, and philosophy.
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"There are lots of things/ that regular people can do/ but dads can't."
" . . . can't cross the street without holding hands."
"Dads can push, but they can't swing."
"Dads can't pitch a baseball very hard or hit one very far."
"When dads play hide-and-seek they always get found, but they have a hard time finding you."
"They aren't very good wrestlers."
"Dads lose at checkers/ and cards/ and almost every other game."
"Dads aren't good at sleeping late. They can't comb their hair or shave by themselves."
"Dads like to go camping but they need lots of help setting up the tent." "And cooking."
As you can see, the manly virtues being praised here are inclusiveness and helpfulness. When dad operates like mom, then he "can't" do something . . . but that's really all right.
My favorites in the book include:
"Dads seem to have trouble holding on to their money." This is combined with an illustration of a dad buying a child a toy in a store.
"Dads like to go fishing, but they don't like to go alone. And they need extra practice baiting the hook."
"Dads can't read a book by themselves."
"Sometimes they leave a night-light on because they're a little scared of the dark."
"And most of all, whatever happens, a dad never ever stops loving you."
The book is written in such a way that this book can be seen as being about a dad who's married to a wonderful mom, one who's divorced and whose children are visiting, or to a single dad who's raising children by himself. I liked that feature very much.
The book is also appealing because it will make a dad feel good whenever he does one of these things. He knows that his son or daughter will appreciate his consideration from having read this book. So having this book available in your house is a little bit like making every day Father's Day!
After you finish the book, dad, you can develop even more closeness by asking your wee ones what else dad can't do. I'm sure they'll come up with some dillies to amuse one and all.
Find more ways to help . . . because you never stand higher in the world's estimation than when you stoop to aid another.
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