Number: The Language of Science
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- Author：Tobias Dantzig
- ISBN：0452288118
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Summary
Number is an eloquent， accessible tour de force that reveals how the concept of number evolved from prehistoric times through the twentieth century。 Tobias Dantzig shows that the development of math—from the invention of counting to the discovery of infinity—is a profoundly human story that progressed by “trying and erring， by groping and stumbling。” He shows how commerce， war， and religion led to advances in math， and he recounts the stories of individuals whose breakthroughs expanded the concept of number and created the mathematics that we know today。
Reviews
Prakash Yadav
，
This book describes Maths like no other。 Well maths is a big subject， this focuses on Numbers entirely。 Its mythology， mysticism， secrecy， tragedies， developments， spiritualism， all rolled into a narrative from history to the modern day， or at least 1950s when the book was written。I am at a loss of words to describe my second most favourite book I have ever read。 Believe me， it's a good book。 Around 70 years old yet as relevant as ever。 I bought a tattered copy at a flea market， best of luck fin This book describes Maths like no other。 Well maths is a big subject， this focuses on Numbers entirely。 Its mythology， mysticism， secrecy， tragedies， developments， spiritualism， all rolled into a narrative from history to the modern day， or at least 1950s when the book was written。I am at a loss of words to describe my second most favourite book I have ever read。 Believe me， it's a good book。 Around 70 years old yet as relevant as ever。 I bought a tattered copy at a flea market， best of luck finding it in print。 。。。more
Ivan Izo
，
This was an interesting review of how number systems entered human society and developed over time。 Each chapter gets a bit more complex， almost like it's a test of how well you know your mathematics。 I was mostly lost for the last 80 or so pages， but I'm going to read it again - after studying more math。 Dantzig's writing creates an enthusiasm for the subject。 At least it did for me。 This was an interesting review of how number systems entered human society and developed over time。 Each chapter gets a bit more complex， almost like it's a test of how well you know your mathematics。 I was mostly lost for the last 80 or so pages， but I'm going to read it again - after studying more math。 Dantzig's writing creates an enthusiasm for the subject。 At least it did for me。 。。。more
John
，
I nearly abandoned this in Chapter 3， disliking it a lot， and apparently for quite different reasons from others （who found it dry and difficult; I didn't）。 Dantzig is much stronger when it comes to summarizing actual math than he is at historical/cultural analysis。 I understand the affection this book evokes in most reviewers here; the style is chummy and charming in a way， although to me it just comes off as loose rambling。 Part of my distaste derives from my being more used to tight， rigorous I nearly abandoned this in Chapter 3， disliking it a lot， and apparently for quite different reasons from others （who found it dry and difficult; I didn't）。 Dantzig is much stronger when it comes to summarizing actual math than he is at historical/cultural analysis。 I understand the affection this book evokes in most reviewers here; the style is chummy and charming in a way， although to me it just comes off as loose rambling。 Part of my distaste derives from my being more used to tight， rigorous histories of ideas; compared to those， Dantzig's facile analysis of Pythagoreanism as "Bad because superstitious， good because anticipating modern quantitative research methods" （p。 44） is laughable。 It compares very unfavorably to， say， the excellent historical syntheses in The Concepts of Space and Time: Their Structure and Their Development by Milic Capek。 I've just come off of reading The Passion of the Western Mind: Understanding the Ideas that Have Shaped Our World View， which （whatever you think of the author's other intellectual commitments） offers a much more insightful and balanced reading of topics like Pythagoreanism and Platonism， and without the silly， knee-jerk evaluations from a limited Enlightenment standpoint that Dantzig offers。In fact， that example typifies a tendency Dantzig has of inserting his own Enlightenment-era values （reason， progress， eradication of superstition， science rules， tradition is stupid and holds us back） into historical method， where they don't belong。 No， I'm not an out-and-out relativist， but the degree of intrusive， shoot-from-the-hip evaluation Dantzig indulges in is simply too much。 The history itself gets slighted in favor of this stupidly smug editorializing， as for example in Chapter 2 when the actual transition from "Abacism" to "Algorism" gets glossed over and mystified。 We are told that it "went through all the usual stages of obscurantism and reaction" （p。 33）。 One paragraph with scant detail on what is ostensibly one of the main points of the chapter。It seems that Dantzig's main interest in historical states and changes is the contemptuous little jokes he can wring from them， rather than the genuine insight they can offer。 Most annoying， after an incredibly brief drive-by of the fact that zero was invented in India， was the non sequitur that zero was "a gift from blind chance" （p。 35）。 That algebra was invented concurrently with positional numeration in India is "strange" （p。 30）， full stop。 Never mind actually doing the work of researching and explicating the state of Indian mathematics at the time， such that this apparent coincidence could happen; no， it's just strange， it's just chance。 There was a real missed opportunity here to actually learn something。 （And don't give me the insulting treatment of Hindu mathematicians， "Fools" Dantzig calls them， on p。 84， in which a single almost fact-free page tries to cover this multitude of sins。）These problems are noted in the otherwise complimentary review by Joseph Schrock here。 Another reviewer rightly points out an apparent inconsistency with Dantzig's insistence that "man is the measure of all things" and seemingly relativist conclusions in Appendix D。 I'm wondering if the Appendices were added by the author at a more mature， sober age。 But in any case， how does this jibe with the value judgments in the previous chapters？ Is Dantzig advocating some kind of Nietzschean willing of a set of values （reason， progress） without any extrahuman basis？ It doesn't add up。I could see enjoying this as an offering in the spirit of Montaigne's essays， whose errant nature is part of their charm。 The difference， though， is that Montaigne was a much better writer， much more erudite in his way， wittier， with a much more profound influence on the Western mind。 We have good reasons to be interested in his wandering thoughts， reasons that we lack in Dantzig's case。I suspect that many here who loved the book weren't bothered by this because they sympathize with or share the author's worldview。 Number: The Language of Science might be all right if you're looking for entertainment and little tidbits to share at cocktail parties for nerds， but I think this sort of fare carries a real danger of infecting the reader with its own intellectual sloppiness。 For me， keep the real academic treatments coming。 I'd much rather read a reliable， academic history of math or science than this smug， rambling thing。 。。。more
Maria
，
Fascinating read but definitely dated。 I am clearly on the discrete side of math because the most interesting things were seeing the evolution of arithmetic and algebra。
esplovago
，
Il breve commento di Einstein sul libro coglie perfettamente l'essenza del libro。 Attualissimo ancora cent'anni dopo per fare luce sulla storia della matematica。 Interessantissimo il capitolo sulle "due realtà" nel quale cerca di legare il concetto di continuità e di immutabililità a quelli della fisica （allora） contemporanea。 Il breve commento di Einstein sul libro coglie perfettamente l'essenza del libro。 Attualissimo ancora cent'anni dopo per fare luce sulla storia della matematica。 Interessantissimo il capitolo sulle "due realtà" nel quale cerca di legare il concetto di continuità e di immutabililità a quelli della fisica （allora） contemporanea。 。。。more
Mirek Kukla
，
This was a frustrating read。 Primarily for the annoyances detailed below， but doubly so because this could have been such a good book。 You have here all the makings of a winner: an ambitious topic， presented in a spunky tone， even spiced up with a bit of philosophy。 Unfortunately， "Number" is marred by so many editorial fuckups that it borders on conspiracy， and ultimately， the pain is not worth the gain。As the title suggests， this is a book about the historical evolution of the very notion of " This was a frustrating read。 Primarily for the annoyances detailed below， but doubly so because this could have been such a good book。 You have here all the makings of a winner: an ambitious topic， presented in a spunky tone， even spiced up with a bit of philosophy。 Unfortunately， "Number" is marred by so many editorial fuckups that it borders on conspiracy， and ultimately， the pain is not worth the gain。As the title suggests， this is a book about the historical evolution of the very notion of "number。" Danzig's tour starts with the crude number sense of animals， and by the time you reach the heady heights of Cantor's transcendentals， you'll have covered a healthy swath of math history。 While most of this is pretty standard fare， Danzig has a philosophical flavor to his investigations that kept me engaged （e。g。 the validity of infinity， the relation between physics and mathematics， intuitionism vs formalism）。On the whole， it's light and breezy stuff - though that's not necessarily a good thing。 As is often the case in books like this， everything even remotely resembling a proof has been relegated to a lengthy appendix， and as a result， the bulk of the discussion feel a bit dumbed down。 Again， hiding "the math" like this isn't terribly uncommon， but here it's taken to an extreme。 As a result， the bulk of the book suffers from being overly non-technical， whereas the dense and technical appendix suffers from a lack of context。But。But。[Takes a deep breath]Ladies and gentlemen of the jury: I do not pride myself on being a diligent proof reader。 Errors evade my eyes with silky， startling ease。 Typos sneak past my glazy gaze for breakfast。 But my God are there a lot of errors in this book。Note that we're not talking about a missing comma here and there。 Most of the typos show up in the proofs， and are seemingly chosen so as to maximize confusion: "<" instead of ">"， "*2" instead of "^2"， "^2" instead of "^3"， and in one egregious case， overloading a symbol to represent two different things （have fun with with the proof on page 300）。 I can't tell you how much time I burned trying to make sense of flat-out falseness。Look: I get it。 Editors aren't mathematicians， and some of these proofs are tricky。 But this book was published in 1930 and has seen over 12 different editions since。 You'd think someone would have bothered。 The preface， at least， is written by a mathematician， and the cover quotes Einstein as favorable reviewer。 Only conspiracy explains it。 I rest my case。Anyways。In many ways， Danzig's "Number" reminds me DFW's "Everything and More。" It's a different take on math-for-the-masses， written in a creative tone， infused with some high ambitions。 And like "Everything and More"， it's ultimately ruined by sloppiness。 Back in the day， the warts might have been worth it。 The same can no longer be said。 These days there is so much good math writing out there that there's no need to let in bathwater with the baby。Key takeawaysThere have been three pivotal ideas in the history of mathematics:1。 The principle of position （coupled with the notion of zero）: as soon as you interpret the leading digit of "20" as representing something different than the leading digit of "2"， you effectively unlock arithmetic。2。 Using abstract symbols to represent numbers: variables let you think in terms of general solutions， and ultimately unlock algebraic manipulation。3。 The Cartesian plane: this unified algebra and geometry and changed how people think about equations， imaginary numbers， etc。Reading notes here。Quotes"And so it was that the complex number， which has its origin in a symbol for fiction， ended up becoming an indispensable tool。。。 moral: fiction is a form in search of an interpretation。" （213）"The mathematician may be compared to the designer of garments， who is utterly oblivious of the creatures whom his garments may fit。。。 a shape will occasionally appear which will fit into the garment as if the garment has been made for it。 Then there is no end of surprise and delight！" （240-241） 。。。more
Dharmendra
，
A fantastic insight unto a language which doesn't know to lie & it's struggle to break chains of human intuition。 A fantastic insight unto a language which doesn't know to lie & it's struggle to break chains of human intuition。 。。。more
Mary
，
A bit dense but if you are mathematically inclined， it's worth the read。 It was an interesting look at how disjointed the history of math as a science really is。 So much intuition coupled with genius， that through lots of hard work， over time， eventually became locked down and relegated to solid scientific foundations。 A bit dense but if you are mathematically inclined， it's worth the read。 It was an interesting look at how disjointed the history of math as a science really is。 So much intuition coupled with genius， that through lots of hard work， over time， eventually became locked down and relegated to solid scientific foundations。 。。。more
Mark Whitehead
，
A nice summary of the history of math， starting with the ideas of correspondence and succession and proceeding through the natural sequence， the rational domain， algebraic numbers， the arithmetic continuum， and the surprising properties of infinity。 It is likely a shame that math education in this country is not organized and presented within the context of this history。 If it were， students could see that math is developed through intuition and logic and mistakes; they might learn that math was A nice summary of the history of math， starting with the ideas of correspondence and succession and proceeding through the natural sequence， the rational domain， algebraic numbers， the arithmetic continuum， and the surprising properties of infinity。 It is likely a shame that math education in this country is not organized and presented within the context of this history。 If it were， students could see that math is developed through intuition and logic and mistakes; they might learn that math wasn't handed down by God and isn't only accessible to the cognoscenti。 。。。more
George
，
This book may be old， but its a classic。 The book was written in the 30s， author made his last edits in the 50s and my edition is from the 90s。 This might tell you that the book is still quite relevant and a landmark for the history of the number from the humble integer to the mind boggling infinities and with it a short history of mathematics， phylosophy and science。 If you are researching these topics this is where you start。Considering that this book mostly deals with the history of the conce This book may be old， but its a classic。 The book was written in the 30s， author made his last edits in the 50s and my edition is from the 90s。 This might tell you that the book is still quite relevant and a landmark for the history of the number from the humble integer to the mind boggling infinities and with it a short history of mathematics， phylosophy and science。 If you are researching these topics this is where you start。Considering that this book mostly deals with the history of the concept of number， it avoids going too deep into mathematical proofs and instead tries to tell you a story， so that it could be enjoyed by everyone。 Of course there must be explanations for some of these proofs and the Author has done an admirable job explaining it， but knowing some math would help。 Everyone who has read a math textbook or other popular books on mathematics has seen how difficult it is to express mathematics in words。 I personally would've given up a long time ago and just left pages and pages of pure math and be done with it。 Now because the book is so old you could totally skip his chapter of "Future problems"， because they are all solved， but still interesting to take a look at just to see some of these math problems and research their solutions。 And finally the editors thankfully added some book recommendations for further reading， those are on my to read list now of course。 。。。more
Panagiotis Jones
，
I am not well-versed in the history of mathematics， so my opinion of this book comes from my exclusive experience with it， rather than a comparison with its kind。It has enough detail to satisfy you， while keeping away from excessiveness。 It is brilliantly written and with a touch of philosophy。 I have learned a great deal about the history of mathematics， and have come to realize and understand many fundamental ideas--how they came to be， why they came to be， and what they led to。 I am glad to a I am not well-versed in the history of mathematics， so my opinion of this book comes from my exclusive experience with it， rather than a comparison with its kind。It has enough detail to satisfy you， while keeping away from excessiveness。 It is brilliantly written and with a touch of philosophy。 I have learned a great deal about the history of mathematics， and have come to realize and understand many fundamental ideas--how they came to be， why they came to be， and what they led to。 I am glad to add this book to my collection！ 。。。more
Joseph Schrock
，
This book was quite interesting to me; however， the latter portions were more technical than I could well tackle， given my limited mathematical expertise。 I did not agree with the author's nominalist views on the ultimate nature of mathematics。 My views are more "Platonist"， given that I am convinced that mathematics has some form of objective existence， and in view of my belief that mathematics is discovered rather than created or invented。 Tobias Dantzig believed that logic is a branch of math This book was quite interesting to me; however， the latter portions were more technical than I could well tackle， given my limited mathematical expertise。 I did not agree with the author's nominalist views on the ultimate nature of mathematics。 My views are more "Platonist"， given that I am convinced that mathematics has some form of objective existence， and in view of my belief that mathematics is discovered rather than created or invented。 Tobias Dantzig believed that logic is a branch of mathematics。 I quote from page 245: "How then can we arrive at a criterion [for the reality of the number concept]？ Not by evidence， for the dice of evidence are loaded。 Not by logic， for logic has no existence independent of mathematics。 How then shall mathematical concepts be judged？ They shall not be judged！ Mathematics is the supreme judge; from its decisions there is no appeal。"I conclude that this is backward。 Mathematics is a branch of logic。 Logic is that set of rules of thought apart from which NO THINKING -- including mathematical thinking -- can possibly occur。 Since thinking that is non-mathematical can occur， it follows that logic is more fundamental than mathematics。 Dantzig placed great emphasis on mathematical intuition as the ultimate guide for mathematics。 I believe that intuition is the faculty of the intellect that enables discovery of mathematics， but it is logic that is the arbiter of valid mathematical thought。 Logic has the final word。 Furthermore， logic is the principle behind all thought， but mathematics consists of logic APPLIED to quantifiable entities and structures。 Thus， mathematics is subsumed in logic。 I was also not impressed favorably by Dantzig's ridicule of Christianity。 On page 129 he wrote: "When， after a thousand-year stupor， European thought shook off the effect of the sleeping powders so skilfully administered by the Christian Fathers， the problem of infinity was one of the first to be revived。" And on page 197 he writes: "I see in the work of Galileo， Fermat， Pascal， Descartes， and others the consummation of an historical process which could not reach its climax in a period of general decline。 Roman indifference and the long Dark Ages of religious obscurantism prevented a resumption of this process for fifteen hundred years。"If Christianity was such a dark period intellectually， WHY was it that modern science sprang up in Christian Europe？ It was not in India， China， or the Muslim countries where science blossomed， but in the heart of Christian Europe。 It is not at all obvious that this alleged "religious obscurantism" was， in fact， inhibiting mathematical and scientific advances。 Admittedly， Christianity had placed a great deal of emphasis on the spiritual life， but there were medieval scholars who were doing serious intellectual work， sowing the seeds for scientific and mathematical advances， even as universities sprang up in Western Europe in the twelfth and thirteenth centuries。 To relegate this time period to "Dark Ages" does not do justice to the intellectual ferments that were brewing during the middle ages in Christian Europe。 I found Dantzig's book well worth reading -- and quite challenging。 However， as a philosopher and thinker， this author does not get high marks from me。 Of course， his profession was mathematics -- not philosophy or religion。 However， being a mathematician should not vitiate sound philosophical thinking。 To any mathematical hobbyist （among which I am one） or mathematics student， or scientist or philosopher， I would recommend Dantzig's challenging and thought-provoking book。 。。。more
Niharika
，
The most amazing thing about numbers is they don’t actually exist。 It is all our imagination。 Mathematics is high art as it creates a whole new world like number system。 In this book author changes your perception about numbers。 The book goes from the history of creation of numbers in different societies to real， trancedental and complex numbers。Its a must read for anyone who is atall interested in mathematics。
BlackOxford
，
Literate MathematicsA classic in every sense: a model of style and erudition to rank with Oscar Wilde， as inspiring as Zadie Smith， as concise as a page from George Orwell， and as timeless as any of Dickens’s tales。 If you have an interest in mathematics， or if you have been scarred by the imposition of tedious calculating techniques in your school days， or if you simply want to understand an enormous part of intellectual history， this is the single most important book you could have at hand。 Th Literate MathematicsA classic in every sense: a model of style and erudition to rank with Oscar Wilde， as inspiring as Zadie Smith， as concise as a page from George Orwell， and as timeless as any of Dickens’s tales。 If you have an interest in mathematics， or if you have been scarred by the imposition of tedious calculating techniques in your school days， or if you simply want to understand an enormous part of intellectual history， this is the single most important book you could have at hand。 The first edition was published almost 90 years ago。 Yet it is fresh and witty and simply full of the most remarkable facts and astute observations about the development and use of numbers。 Apparently， for example， birds （particularly crows） have a relatively developed sense of number （at least up to five）。 Dogs， horses and other domestic animals appear to have none。 And the English trice has the double meaning of three times as well as simply many， plausibly echoing the Latin ‘tres’ and ‘trans’ - beyond - thus memorializing an ancient method of base 3 counting。Dantzig‘s factual anecdotes are similarly captivating: “Thus， to this day， the peasant of central France （Auvergne） uses a curious method for multiplying numbers above 5。 If he wishes to multiply 9 × 8， he bends down 4 fingers on his left hand （4 being the excess of 9 over 5）， and 3 fingers on his right hand （8 – 5 = 3）。 Then the number of the bent-down fingers gives him the tens of the result （4 + 3 = 7）， while the product of the unbent fingers gives him the units （1 × 2 = 2）。”The only misjudgment Dantzig makes is his underestimation of binary arithmetic。 “It is the mystic elegance of the binary system，” he says somewhat disapprovingly， “that made Leibnitz exclaim: Omnibus ex nihil ducendis sufficit unum。 （One suffices to derive all out of nothing。）” Little could Dantzig （much less Leibniz） have foreseen the rather non-mystical importance of the base-two counting in the age of the digital computer。Dantzig is acutely sensitive to the cultural matrix of mathematics。 That matrix， he points out， is neither commercial nor academic; it is largely religious。 “Religion is the mother of the sciences。” The Greeks of course had several mathematically based religious cults。 Even the most recent （and difficult） mathematical field， number theory “had its precursor in a sort of numerology” of biblical texts。 （See here for more on the religious inspiration in mathematics: https://www。goodreads。com/review/show。。。）。But he also recognises religion as a major impediment to the development of mathematical knowledge: “When， after a thousand-year stupor， European thought shook off the effect of the sleeping powders so skillfully administered by the Christian Fathers， the problem of infinity was one of the first to be revived。” Religion， thankfully， shot itself in the foot in interesting ways: “Now， the acquisition of culture was certainly not a part of the Crusader’s program。 Yet， this is exactly what the Crusades accomplished。 For three centuries the Christian powers tried by sword to impose their “culture” upon Moslem。 But the net result was that the superior culture of the Arabs slowly yet surely penetrated into Europe。”Perhaps most impressive is Dantzig’s intellectual humility。 He begs ignorance of the philosophical issue of whether or not numbers exist outside of human thought about them。 But he is not without an important philosophical view: “Herein I see the genesis of the conflict between geometrical intuition， from which our physical concepts derive， and the logic of arithmetic。 The harmony of the universe knows only one musical form—the legato; while the symphony of number knows only its opposite—the staccato。 All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as a legato。 Yet our intellect will always brand such attempts as deceptions and reject such theories as an insult， as a metaphysics that purports to explain away a concept by resolving it into its opposite。”To conclude with this sort of poetic image justifies entirely the description of this book as “an ode to the beauties of mathematics。” 。。。more
A
，
Loved the chapter: The Anatomy of the Infinite。
Munthir Mahir
，
Have to give it to the author for apparently trying to make the topic accessible to the general reader。 It is certainly not written for the serious mathematicians （maybe for the aspiring mathematician）。 It， also， contains equations in detailed explanation which I find unnecessary。 A lot of the discussion， although fits to the subject of number evolution， is dry and involves mathematical proofs。 This book is definitely not accessible。
Dennis Littrell
，
Einstein called this "the most interesting book on the evolution of mathematics which has ever fallen into my hands。"Number was first published in 1930 with the fourth edition coming out in 1954。 This is a republication of that fourth edition （Dantzig died in 1956） edited by Joseph Mazur with a foreword by Barry Mazur。 It is an eminently readable book like something from the pages of that fascinating four-volume work The World of Mathematics （1956） edited by James R。 Newman in that it is aimed a Einstein called this "the most interesting book on the evolution of mathematics which has ever fallen into my hands。"Number was first published in 1930 with the fourth edition coming out in 1954。 This is a republication of that fourth edition （Dantzig died in 1956） edited by Joseph Mazur with a foreword by Barry Mazur。 It is an eminently readable book like something from the pages of that fascinating four-volume work The World of Mathematics （1956） edited by James R。 Newman in that it is aimed at mathematicians and the educated lay public alike。Part history， part mathematics and part philosophy， Number is the story of how we humans got from "one， two。。。many" to various levels of infinity。 Strange to say it is also about reality。 Here is Dantzig's concluding statement from page 341 in Appendix D: "。。。modern science differs from its classical predecessor: it has recognized the anthropomorphic origin and nature of human knowledge。 Be it determinism or rationality， empiricism or the mathematical method， it has recognized that man is the measure of all things， and that there is no other measure。"Or more pointedly from a couple of pages earlier: "Man's confident belief in the absolute validity of the two methods [mathematics and experiment] has been found to be of an anthropomorphic origin; both have been found to rest on articles of faith。"These are inescapably the statements of a postmodernist。 I was surprised to read them in a book on the theory of numbers， and even more surprised to realize that if mathematics is a distinctly human language， it is entirely possible that beings from distant worlds may speak an entirely different language; and therefore our attempts to use what many consider the "universal" language of mathematics to communicate with them may be in vain。And this thought makes me wonder。 Is the concept "two，" for example， （as opposed to the number "2"） really just a human construction？ Would not intelligent life anywhere be able to make a distinction， just as we have， between， say， two things and three things？ And if so， would they not be able to count？ And would not then the entire edifice of mathematics （or at least most of it） follow？I wonder if Dantzig was not in contradiction with himself on this point because earlier he writes （p。 252） "。。。any measuring device， however simple and natural it may appear to us， implies the whole apparatus of the arithmetic of real numbers: behind any scientific instrument there is the master-instrument， arithmetic， without which the special device can neither be used nor even conceived。" Does this not imply that measurements （by any beings） and therefore numbers have an existence outside of the human mind and do not rest on "articles of faith"？As to the numbers themselves （putting philosophy aside） we learn that the two biggest bugaboos in the history of number are zero and infinity。 It took a long， long time for humans， as Dantzig relates， to accept the idea of zero as a number。 Today zero is also a place-holder。 But what does it mean to say that there are zero pink elephants dancing about my living room？ I can see one cow in the yard， or two or three， but I cannot see zero cows in the yard。Of course， today it is easy to see that zero is a number that is less than one and greater than minus one。 I have one cow and I sell that one cow。 Now I have zero cows。 （Curiously， note that the plural noun "cows" is grammatically required。） However， the imperfect fit within the entire structure of mathematics that zero has achieved may be appreciated by realizing that every other number can be a denominator; that is， three over one equals three， three over two equals 1。5， etc。， but what does three over zero equal？It is a convention of mathematics to say that division by zero is "undefined。" There is no other number about which the same can be said。I used to think when I was young that infinity was the proper answer to division by zero。 For Dantzig this is clearly not correct because to him infinity is not a number at all but a part of the process。 He writes， "the concept of infinity has been woven into the very fabric of our generalized number concept。" He adds， "The domain of natural numbers rested on the assumption that the operation of adding one can be repeated indefinitely， and it was expressly stipulated that never shall the ultra-ultimate step of this process be itself regarded as a number。" Of course he is talking about "natural" numbers。 He notes in the next sentence that in the generalization to "real" numbers， "the limits of these processes" were "admitted。。。as bona fide numbers。" （p。 245） In other words， part of the process became a number itself！The culmination of Dantzig's argument here is that infinity itself is a construction of the human mind and exists nowhere （that we can prove） outside of the human mind。 He believes that the basis for our belief in the existence of infinity comes from our （erroneous） conception of time as a continuum。 Dantzig notes that Planck time and indeed all aspects of the world are to be seen in terms of discrete quanta and not continuous streams。Ultimately， Dantzig gives this sweeping advice to the scientist: "。。。he will be wise to wonder what role his mind has played in。。。[a] discovery， and whether the beautiful image he sees in the pool of eternity reveals the nature of this eternity， or is but a reflection of his own mind。" （p。 242） --Dennis Littrell， author of “The World Is Not as We Think It Is” 。。。more
Maggie
，
Great book， profound but easy to understand， substantial but light。 Mr Dantzig writes:“Our intuition permits us， by an act of the mind， to sever all time into the two classes， the past and the future， which are mutually exclusive and yet together comprise all of time， eternity。The now is the partition which separates all the past from all the future; any instant of the past was once a now， any instant of the future will be a now anon， and so any instant may itself act as such a partition。 To be Great book， profound but easy to understand， substantial but light。 Mr Dantzig writes:“Our intuition permits us， by an act of the mind， to sever all time into the two classes， the past and the future， which are mutually exclusive and yet together comprise all of time， eternity。The now is the partition which separates all the past from all the future; any instant of the past was once a now， any instant of the future will be a now anon， and so any instant may itself act as such a partition。 To be sure， of the past we know only disparate instants， yet， by an act of the mind we fill out the gaps; we conceive that between any two instants - no matter how closely these may be associated in our memory - there were other instants， and we postulate the same compactness for the future。 This is what we mean by the flow of time。” 。。。more
Paul
，
Even though I loved this book， I put it down for a while since I was intimidated by the appendices。 While that fear turned out to be valid， I still enjoyed the book as a whole and the last appendix especially for the way in which it made me think harder about things I had already learned and for introducing me to interesting things I probably should have learned。
Alberto
，
A good read。 I have a BS in mathematics， and there were some things in here （particularly， some of the geometric interpretations of algebraic results） that I had not seen before。 The additional readings suggested at the end are mostly outstanding。
Jane
，
The anthropological survey about number systems in the first few chapters was pretty interesting， but the dryness of the writing really came into the forefront when the later chapters turned to increasingly technical mathematics。 While I appreciate the rigor， it ended up feeling like I was reading a math textbook， which is not my jam right now。
Daniel
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The first two chapters are slow， but after that it speeds up quickly。 Tells the history of mathematics （up to set theory） as a story of uncovering/intuiting new types of numbers and the process of proving that these new （and increasingly abstract） entities do follow the central axioms that numbers follow， and can be used as such。 It's a history of mathematics that I've never come across before。 I thought it was fucking brilliant。 The first two chapters are slow， but after that it speeds up quickly。 Tells the history of mathematics （up to set theory） as a story of uncovering/intuiting new types of numbers and the process of proving that these new （and increasingly abstract） entities do follow the central axioms that numbers follow， and can be used as such。 It's a history of mathematics that I've never come across before。 I thought it was fucking brilliant。 。。。more
Deepak Mathivanan
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Prolly the best classic on history of mathematics
Binit
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This is a rather old book （originally published in 1930's） and so the language is slightly dated and some of the concepts are explained rather confusingly。 But it does give a great explanation of some basic facts about Number theory came to be。 The book does not not shy away from Math when appropriate but most of denser math is in the appendices。 But overall this is a good read for anyone interested in the history of Math。 This is a rather old book （originally published in 1930's） and so the language is slightly dated and some of the concepts are explained rather confusingly。 But it does give a great explanation of some basic facts about Number theory came to be。 The book does not not shy away from Math when appropriate but most of denser math is in the appendices。 But overall this is a good read for anyone interested in the history of Math。 。。。more
Chris Cousins
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Starts off simply with a history of the development of numbers。 Gets very involved in high-end maths as you go on。
Francisco Rodríguez
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I was fascinated by the fact that the act of counting and the concept of number is not a self-evident truth inherent in human beings， but something that had to be learnt。Realizing that a pair of horses and the passage of two days are two different instances of the same concept of the number "two" required thousands of years。 Proof of this， given in the book， is that some primitive tribes have different names for the numbers when referring to people， days， objects， places， etc。The book is full of I was fascinated by the fact that the act of counting and the concept of number is not a self-evident truth inherent in human beings， but something that had to be learnt。Realizing that a pair of horses and the passage of two days are two different instances of the same concept of the number "two" required thousands of years。 Proof of this， given in the book， is that some primitive tribes have different names for the numbers when referring to people， days， objects， places， etc。The book is full of extremely interesting stories like the one above， and I loved it。I do not give the book 5 stars only because the writing style is （a little bit） too serious in some cases。 I later realized that the book was written in the early part of the twentieth century， which might explain this different style to what would be expected in a popular science book from today。 。。。more
Steve Gathje
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Who could have thought a history of numbers could be so interesting。 Okay， I could have thought that。 And it was very interesting！
Greg Talbot
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It's a lot to ask of a book to give a portrayal of what it means to be human。Near impossible to find so much reflected humanity in a book on mathematical principles。 Tobias Dantzig gives us an elegant history of the discovery of mathematical principles。 This discoveries are products of imagination， abstract reasoning， and the outposts of thought were erected before it's territory was marginally searched。From finger counting， to abacuses， to expanding the domain of numbers。。。it's a slow road。 One It's a lot to ask of a book to give a portrayal of what it means to be human。Near impossible to find so much reflected humanity in a book on mathematical principles。 Tobias Dantzig gives us an elegant history of the discovery of mathematical principles。 This discoveries are products of imagination， abstract reasoning， and the outposts of thought were erected before it's territory was marginally searched。From finger counting， to abacuses， to expanding the domain of numbers。。。it's a slow road。 One of the highlights of the book is Dantzig getitng into the gritty Dedekind-Cantor law。 Isn't it easy to to take it for granted that algebra and geometry can be used interchangeably。 The human imagination stretched from abasic domain of real numbers， to a model of the infite filled with compound， irrationals and never-ending prime numbersTobias re-worked this work multiple times， indeed it's his primary work， and legacy。 It's not hard to imagine many people having been turned on， and dramatically influenced by this passion。Oh， and Albert Einstein endorsed it too。 。。。more
Stephen Armstrong
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Number theory clearly explained in this classic。 Beautifully written。 2007 publication date， original was in 1930。 What book on number theory survives 77 years， unless it is extraordinary？
Bob
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