Resting the principles of shifting from perspective to perspective using
techniues and processes of one branch of mathematics to interpret techniues and processes of another the use of mathematical substitutions and processes of one branch of mathematics to interpret techniues and processes of another the use of mathematical substitutions seemingly unrelated contexts to sidestep mathematical deadends and the varying styles of thought used to approach math I especially enjoyed the revelations of deep pattern throughout mathematical concepts very satisfying Finally a great moment for me many ears ago I came across the term Cantorian transfinites in a science fiction short story In intervening Five fags a day years other reading confirmed the existence of a mathematician named Cantor and my suspicion that the transfinites were real and as they hinted at numbers beyond infinity I was fascinated Math teachers I uestioned were unfamiliar with the idea through fourteenears of teaching But there it is in the last chapters of The Art of the InfiniteThe text is challenging at best and can be overwhelming to the layman but the patterns principles and often bizarre coincidence of numbers makes it uite worthwhile The essence of mathematics lies entirely on its freedom said Georg Cantor THE man who figured out that there are countable and uncountable infinities Wow Infinite freedom Well at least freedom within a playground nested with infinities Hilbert s paradise as Wicked Affairs p 2 you are about to learn in this book Hey Robert and Ellen managed to make number theory palatable for the layperson with an entire chapter on Cantor s work An infinite gold mine as a chapter And there is much The whole thing is a vivid narrative about mathematical beauty everywhere playful with subtle humor and deep erudition That is history geometry philosophy poetry infinite entertainment Well simply put this is NOT a book about mathematics Sure it has numbers and math as its subject matter but what authors really wantedou to get out of this book is how wonderfully elitist their English language skills are Combined with a narrato Prose so purple I claim it was abused This book needed an editor to cut out the blathering that the authors thought clever The references to Rimbaud and Proust to cite just a couple were completely unnecessary and distractingI read the first 3 chapters and then skipped to the last the chapter on Georg Cantor and aleph null aleph one and transfinite numbers Fun fact Cantor was a conspiracy theoristI was excited when I read this in the introductionMany small things Beautiful Mathematic Book
For College Students THEcollege students THE CONTROVERSE is important to know than all other stuff The uestion Where do we get our knowledge from and how do we know that that is it John von Neuman said In Mathematics we never understand things but we just get used to themThat can t be uite right Jane of Gowlands yet our understanding must be stretched to the breaking point before it becomes flexible enough to adjust to the unthinkable page 024 Paul Erd s said about The Book You don t have to believe in God butou do have to believe in The Book I myself understand these proofs The New Scientist meant This is mathematics for the Soul just the way it should be I can only agree One of my favorite books on math. Metry into the unlikely realm where parallel lines meet Along the way deft character studies of great mathematicians and eually colorful lesser ones illustrate the opposed et intertwined modes of mathematical thinking the intutionist notion that we discover mathematical truth as it exists and the formalist belief that math is true because we invent consistent rules for it Less than All wrote William Blake cannot satisfy Man The Art of the Infinite shows us some of the ways that Man has grappled with All and reveals mathematics as one of the most exhilarating expressions of the human imaginati. Earing away cobwebs and opening new doors etc Mathematicians have been doing the same thing exploring the seemingly real properties of this other world not precisely intersecting with ours or exactly including it It is just a different thing that they are investigating with different tools to be sure but with the same process of discovery despite the abstraction and lack of the empiricism we often define as integral to science Mathematics is something that I find interesting but definitely wish I knew about So I went to my local library looking for a good book on math to give me an introduction to the subject When I found this book I thought I d found what I was looking for Boy was I wrongThe book is written in this weird florid prose and just the way it was written made it impossible to read I tried really hard but I couldn t get past the first few pages Finally I took it back to the library and looked for a better alternative I gave it two stars because it probably would have been a decent book had it not been for the weird writing style I just needed something less flowery and technical If ou can handle the flowery prose then ou might enjoy this book but I just couldn t handle it This book was in my math teacher father s collection when he died I always thought I was good at math but much of this text exceeded my capacity I loved the parts about prime numbers and the unexpected lack of pattern for finding the next one and sums of series and the unexpected discovery that some such series of ever declining fractions converge eg 1 12 14 18 116 2 and some diverge eg 1 12 13 14 15 I was reminded how much I enjoyed doing proofs especially geometric proofs The latter part of the book was over my head but I didn t mind reading it Wow I loved this book It really opened my mind around how numbers are represented and constructedThis is one of the best
books I have read that explores the foundations of numbers inI have read that explores the foundations of numbers in very understandable format The approachdevelopment of math knowledge in the book was entertaining with a mix of history and inspiring proofs and examples I wished I read this book when I was in high school Topics I really appreciated include the explanation of number fields In particular I really enjoyed the explanation of why a heptagon can t be
CONSTRUCTED WITH COMPASS AND RULER THROUGHwith compass and ruler through previously developed understanding of fields Additional tools would be neededOf course I ve known a lot about numbers for ears and I use Real and Complex numbers every day as an engineer But I never had such an appreciation for how to constructrepresent numbers exactlyAs a side note The book didn t talk about this but floating point numbers are popular for approximating real numbers Floating point numbers can be problematic because of truncation and rounding errors that occur in calculations This book has inspired me to learn about Exact Reals and has given me the ability to followunderstand papers on various strategies for exact reals as an alternative to floating point numbers Though I was only able to follow about two thirds of the math in The Art of the Infinite it was extremely informative I found most inte. Guises as a touchstone for understanding mathematical thinking Tracing a path from Pythagoras whose great Theorem led inexorably to a discovery that his followers tried in vain to keep secret the existence of irrational numbers; through Descartes and Leibniz; to the brilliant haunted Georg Cantor who proved that infinity can come in different sizes the Kaplans show how the attempt to grasp the ungraspable embodies the essence of mathematics The Kaplans guide us through the Republic of Numbers where we meet both its upstanding citizens and shadowy dwellers; and we travel across the plane of geo.
Free download ✓ PDF, DOC, TXT, eBook or Kindle ePUB free é Robert M. KaplanSomewhere between Flatland A Romance of Many Dimensions and Cosmos Art of the Infinite is a popularization of mathematics by way of extensive metaphor and cultured references to literature and philosophy I thought this was just a bit cheesy at first but after a few times around I got why they were doing this The chief metaphor is spatial math is an exploration into these unimaginably vast vistas in which humans are comfortable with a negligible patch and can imagine only a fraction The Kaplans assert that these vistas are full of palaces and treasures and that the pursuit is of some value to humanity The value they present is this heady experience in which the simple propagates along some logical line into infinity like a door opening to reveal the set of all possible doors These experiences are certainly delightful and make the book enjoyable to read and I d imagine this would be much the case had I followed along with each chapter which or less corresponded to a continuous line of inuiry trying my hand at the math and making sure I understand what the symbols in each section referred to I m sure I m capable of this especially since the Kaplans go out of their way to put things in comprehensible terms though it is never oversimplified but I was merely too lazyWhile the Kaplans present this as a sort of valuable service math as a heady wonder inducer they don t skirt around the Lovecraftian implications of this The last chapter in which Cantor plumbs the nature of compound infinities mind blowing revelation in which he proves that there MUST be infinities larger than other infinities as paradoxical as this is and of course this is only the premise of the chapter ends up with his gradual descent to disgrace and insanity The brilliant thing about this all is that whereas Cantor it is implied went insane because of his
FRUSTRATION WITH THE UNPROVABILITY OF THEwith the unprovability of the theorem ie because he hit some limit in the comprehensibility of this shit the authors end by pointing
out that Kurt Godel resolved Cantor s final issue rather handilythat Kurt Godel resolved Cantor s final issue rather handily proved that it was neither provable or disprovable The historicity of the discipline is heartening it suggests that there is no aspect of this boggling system that is inherently beyond human grasp that it isn t really the hostile and unmasterable force Lovecraft s nightmares conjure it as Which doesn t help Cantor or any of us who exist at a fixed point on the timeline we are stuck with the paradoxes of our day not privy to the brilliant answers around the corner I did think it was a bit unfortunate if ultimately satisfying that the Kaplans decided to end the book on a resolution not its conseuent uestion It d be interesting and tantalizing to know what problems are presently driving mathematicians insaneAnd this is what I took away from the book a greater sense of math as a discipline parallel to science In the first chapter the Kaplans lay out the plane of imaginary numbers intersecting with the plane of real numbers They make the story of math eual in the same way to the history of science Scientists have been doing things in the world to make things about it clear always cl. Robert Kaplan's The Nothing That Is A Natural History of Zero was an international best seller translated into ten languages The Times called it elegant discursive and littered with uotes and allusions from Auinas via Gershwin to Woolf and The Philadelphia Inuirer praised it as absolutely scintillating In this delightful new book Robert Kaplan writing together with his wife Ellen Kaplan once again takes us on a witty literate and accessible tour of the world of mathematics Where The Nothing That Is looked at math through the lens of zero The Art of the Infinite takes infinity in its countless.