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Number: The Language of Science
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.
416 pages, Paperback
First published January 1, 1930
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About the author
Tobias Dantzig
30 books10 followersTobias Dantzig (February 19, 1884 – August 9, 1956) was a mathematician of Baltic German and Russian American heritage, the father of George Dantzig, and the author of Number: The Language of Science (A critical survey written for the cultured non-mathematician) (1930) and Aspects of Science (New York, Macmillan, 1937).
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December 25, 2019
Literate Mathematics
A 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.”
A 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.”
July 26, 2019
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.
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.
March 26, 2010
On the cover is an interesting mini-review: "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands." -- Albert Einstein.
With an endorsement like that, what more needs to be said? Indeed, this is a very interesting and very informative book, scarcely dimmed by the passage of years -- it was first published in 1930! In the interim, much has changed; one amusing example is the following statement on page 121: "Today [1930:] over 700 correct decimals of the number pi are known." By contrast, in 2010 nearly 3 *trillion* digits are known. And yet much has not changed, and this book remains one of the best exposition of how our system of mathematics arose.
One of the best chapters is the second chapter, where the author clearly describes how our modern Indo-Arabic numerical system, which is arguably the greatest mathematical discovery of all time, arose in India in the first few centuries of the common era, and from their percolated to the Arab world, and then to a kicking-and-screaming European world. Dantzig introduces Chapter 2, where this is discussed, with this interesting quote from Laplace:
"It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity."
With an endorsement like that, what more needs to be said? Indeed, this is a very interesting and very informative book, scarcely dimmed by the passage of years -- it was first published in 1930! In the interim, much has changed; one amusing example is the following statement on page 121: "Today [1930:] over 700 correct decimals of the number pi are known." By contrast, in 2010 nearly 3 *trillion* digits are known. And yet much has not changed, and this book remains one of the best exposition of how our system of mathematics arose.
One of the best chapters is the second chapter, where the author clearly describes how our modern Indo-Arabic numerical system, which is arguably the greatest mathematical discovery of all time, arose in India in the first few centuries of the common era, and from their percolated to the Arab world, and then to a kicking-and-screaming European world. Dantzig introduces Chapter 2, where this is discussed, with this interesting quote from Laplace:
"It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity."
April 7, 2018
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”
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”
December 16, 2020
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.
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.
October 19, 2014
Loved this (near) closing line - The reality of today was but an illusion yesterday.
October 7, 2018
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.
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.
June 24, 2016
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.
August 29, 2009
This is another one I pick up a lot. There is some really dense math that is really outside my understanding, but also some incredibly lucid analysis of the development of mathematics and how it has effected the way we perceive and cognate. Tremendous stuff, and humbling!
April 12, 2021
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.
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.
January 21, 2023
This book assumes a pretty intensive math background but goes into some interesting nuances.
October 24, 2025
Reading this book reminded me of this strip by the inimitable Randall Munroe
The first few chapters speaking about the history of counting etc is fascinating and offers some hope to the layperson, which is slowly and then geometrically dashed as we move into the middle topics and the difficulty levels just ramp up and before you know it you're just reading symbols and equations in a page.
The first few chapters speaking about the history of counting etc is fascinating and offers some hope to the layperson, which is slowly and then geometrically dashed as we move into the middle topics and the difficulty levels just ramp up and before you know it you're just reading symbols and equations in a page.
February 21, 2024
1. Elements of Geometry, we encounter Euclid’s definition of a line: “Definition 2. A line is breadthless length
2. Observation and experiments on dogs, horses and other domestic animals have failed to reveal any number sense
3. The Bushmen of South Africa have no number words beyond one, two and many, and these words are so inarticulate that it may be doubted whether the natives attach a clear meaning to them
4. There is a plausible connection between the Latin tres, three, and trans, beyond;
5. A rudimentary number sense, not greater in scope than that possessed by birds, was the nucleus from which the number concept grew
6. until one of the collections, or both, are exhausted. The number technique of many primitive peoples is confined to just such such a matching or tallying. They keep the record of their herds and armies by means of notches cut in a tree or pebbles gathered in a pile. That our own ancestors were adept in such methods is evidenced by the etymology of the words tally and calculate, of which the first comes from the Latin talea, cutting, and the second from the Latin calculus, pebble
7. of time the very connection between the two is lost to memory. As man learns to rely more and more on his language, the sounds supersede the images for which they stood, and the originally concrete models take the abstract form of number words
8. in many primitive languages the number-words up to four are identical with the names given to the four fingers. The more civilized languages underwent a process of attrition
9. in many primitive languages the number-words up to four are identical with the names given to the four fingers.
10. Compare the Sanskrit pantcha, five, with the related Persian pentcha, hand; the Russian “piat,” five, with “piast,” the outstretched hand
11. It is these fingers which have taught him to count and thus extend the scope of number indefinitely.
12. And it is reasonable to conjecture that without our fingers the development of number, and consequently that of the exact sciences, to which we owe our material and intellectual progress, would have been hopelessly dwarfed.
13. How old is our number language? It is impossible to indicate the exact period in which number words originated, yet there is unmistakable evidence that it preceded written history by many thousands of years
14. In all IndoEuropean languages, as well as Semitic, Mongolian, and most primitive languages, the base of numeration is ten, i.e., there are independent number words up to ten, beyond which some compounding principle is used until 100 is reached
15. These two other systems are the quinary, base 5, and the vigesimal,
16. These two other systems are the quinary, base 5, and the vigesimal, base 20
17. In the quinary system there are independent number words up to five, and the compounding begins thereafter. (See table at the end of chapter.) It evidently originated among people who had the habit of counting on one hand.
18. They point to the Greek word pempazein, to count by fives, and also to the unquestionably quinary character of the Roman numerals. However, there is no other evidence of this sort, and it is much more probable that our group of languages passed through a preliminary vigesimal stage
19. The day of the Aztecs was divided into 20 hours; a division of the army contained 8000 soldiers (8000 = 20 × 20 × 20)
20. We have the English score, two-score, and three-score; the French vingt (20) and quatre-vingt (4 × 20)
21. There exists among the most primitive tribes of Australia and Africa a system of numeration which has neither 5, 10, nor 20 for base. It is a binary system, i.e. , of base two. These savages have not yet reached finger counting. They have independent numbers for one and two, and composite numbers up to six. Beyond six everything is denoted by “heap
22. I mention this merely to show how the prejudices of childhood may cloud the vision even of the greatest men!
23. Almost any other base, with the possible exception of nine, would have done as well and probably better
24. if the choice of a base were left to a group of experts, we should probably witness a conflict between the practical man, who would insist on a base with the greatest number of divisors, such as twelve, and the mathematician, who would want a prime number, such as seven or eleven, for a base
25. Numeration is at least as old as written language, and there is evidence that it preceded it. Perhaps, even, the recording of numbers had suggested the recording of sounds
26. from the beginning of history until the advent of our modern positional numeration, so little progress was made in the art of reckoning.
27. The greatly increased facility with which the average man today manipulates number has been often taken as proof of the growth of the human intellect
28. The scheme can be typified by the curious method of counting an army which has been found in Madagascar
29. The soldiers are made to file through a narrow passage, and one pebble is dropped for each. When 10 pebbles are counted, a pebble is cast into another pile representing tens, and the counting continues. When 10 pebbles are amassed in the second pile, a pebble is cast into a third pile representing hundreds, and so on until all the soldiers have been accounted for
30. In order to avoid this ambiguity it is essential to have some method of representing the gaps, i.e., what is needed is a symbol for an empty column.
31. We see therefore that no progress was possible until a symbol was invented for an empty class, a symbol for nothing, our modern zero. The concrete mind of the ancient Greeks could not conceive the void as a number, let alone endow the void with a symbol
32. The Indian term for zero was sunya, which meant empty or blank, but had no connotation of “void” of “nothing.” And so, from all appearances, the discovery of zero was an accident brought about by an attempt to make an unambiguous permanent record of a counting board operation.
33. they translated the Indian sunya by their own, sifr, which meant empty in Arabic. When the Indo-Arabic numeration was first introduced into Italy, sifr was latinized into zephirum. This happened at the beginning of the thirteenth century, and in the course of the next hundred years the word underwent a series of changes which culminated in the Italian zero.
34. We now know that the word is merely a corruption of Al Kworesmi, the name of the Arabian mathematician of the ninth century whose book (in Latin translation) was the first work on this subject to reach Western Europe.
35. In fact, the numerals did not assume a stable form until the introduction of printing. It can be added parenthetically that so great was the stabilizing influence of printing that the numerals of today have essentially the same appearance as those of the fifteenth century.
36. the theory of numbers is the branch of mathematics which has found the least number of applications.
37. The theory of integers is one of the oldest branches of mathematics, while modern arithmetic is scarcely four hundred years old.
38. , seven and forty were the ominous numbers of the Hebrews, and Christian theology inherited the seven: the seven deadly sins, the seven virtues, the seven spirits of God, seven joys of the Virgin Mary, seven devils cast out of Magdalen.
39. Thus Abraham proceeding to the rescue of his brother Eliasar drives forth 318 slaves. Is it just a coincidence that the Hebrew word Eliasar adds up to 318
40. Christian theology made particular use of Gematria in interpreting the past as well as in forecasting the future. Of special significance was 666, the number of the Beast of Revelation
41. Strangely enough we find a striking correspondence in Chinese mythology. Here the odd numbers symbolized white, day, heat, sun, fire; the even numbers, on the other hand, black, night, cold, matter, water, earth
42. The amicable numbers were known to the Hindus even before the days of Pythagoras. Also certain passages of the Bible seem to indicate that the Hebrews attached a good omen to such numbers
43. The smallest perfect numbers are 6 and 28, and were known to the Hindus as well as to the Hebrews
44. If Nicomachus meant to imply that there was a perfect number in every decimal class, he was wrong, for the fifth perfect number is 33,550,336
45. However, we know enough to conclude that the primes do not become substantially rarer as we go on
46. Bertrand asserted that between any number and its double there exists at least one prime
47. In 1640 the great French mathematician Fermat announced that he had found a form which represented primes only
48. The failure to obtain a general form for generating prime numbers led to indirect criteria for testing primality.
49. One trouble with it is that while it is true, it is not a criterion, i.e., the condition is necessary but not sufficient
50. postulate of Goldbach, a contemporary of Euler. This postulate alleges that every even number is the sum of two primes
51. The concrete has ever preceded the abstract. That is why the theory of numbers preceded arithmetic
52. mathematics is the cement which holds this structure together. A problem, in fact, is not considered solved until the studied phenomenon has been formulated as a mathematical law
53. When we analyze these mathematical processes we find that they rest on the two concepts: Number and Function
54. the sum does not depend on the order of its terms. The mathematician says no more when he states: addition is a commutative operation
55. never attached much importance to these statements. Yet they are fundamental. On them is based the rule for adding larger numbers
56. In the history of mathematics, the “how” always preceded the “why,” the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The counting technique and the rules of reckoning were established facts at the end of the Renaissance period. But the philosophy of number did not come into its own until the last quarter of the nineteenth century
57. The strength of arithmetic lies in its absolute generality. Its rules admit of no exceptions: they apply to all numbers
58. There is no last number! The process of counting cannot conceivably be terminated. Every number has a successor. There is an infinity of numbers
59. psalmist
60. It is a plausible hypothesis that the early conception of infinity was not the uncountable, but the yet-uncounted
61. For, since the properties of whole numbers form the basis of mathematics, if these properties can be proved by the rules of formal logic, then all of mathematics is a logical discipline
62. is desired that each assumption should be independent of all the others, and that the whole system be exhaustive, i.e., completely cover the question under investigation. The branch of mathematics which deals with such problems is called axiomatics and has been cultivated by such men as Peano, Russell and Hilbert
63. It is the result of observation and experience. To discover a property of a certain class of objects we repeat the observation or tests as many times as feasible, and under circumstances as nearly similar as possible. Then it may happen that a certain definite tendency will manifest itself throughout our observation or experimentation. This tendency is then accepted as the property of the
64. It is the result of observation and experience. To discover a property of a certain class of objects we repeat the observation or tests as many times as feasible, and under circumstances as nearly similar as possible. Then it may happen that a certain definite tendency will manifest itself throughout our observation or experimentation. This tendency is then accepted as the property of the class.
65. For, in order to prove a mathematical proposition, the evidence of any number of cases would be insufficient, whereas to disprove a statement one example will suffice. A mathematical proposition is true, if it leads to no logical contradiction, false otherwise. The method of deduction is based on the principle of contradiction and on nothing else.
66. Mathematics is a deductive science, arithmetic is a branch of mathematics. Induction is inadmissible. The propositions of arithmetic, the associative, commutative and distributive properties of the operations, for instance, which play such a fundamental rôle even in the most simple calculations, must be demonstrated by deductive methods. What is the principle involved?
67. It is significant that we owe the first explicit formulation of the principle of recurrence to the genius of Blaise Pascal, a contemporary and friend of Fermat.
68. , if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the theorems which fill so many volumes are nothing but devious ways of saying that A is A
69. Induction, as applied in the physical sciences, is always uncertain, because it rests on the belief in a general order in the universe, an order outside of us
70. And yet the concept of infinity, though not imposed upon us either by logic or by experience, is a mathematical necessity. What is, then, behind this power of the mind to conceive the indefinite repetition of an act when this act is once possible?
71. The word is of Arabic origin.“Al” is the Arabic article the, and gebar is the verb to set, to restitute. To this day the word “Algebrista” is used in Spain to designate a bonesetter, a sort of chiropractor.
72. The full title of the book is “Algebar wal Muquabalah,” an exact translation of which would read “On Restitution and Adjustment.” Ben
73. Traces of a primitive algebra are found on the clay tablets of the Sumerians, and it probably reached quite a high degree of development among the ancient Egyptians
74. Be this as it may, there is no doubt that Egyptian algebra antedates the papyrus by many centuries
75. The Hindus may have inherited some of the bare facts of Greek science, but not the Greek critical acumen
76. They played with number and ratio, zero and infinity, as with so many words: the same sunya, for instance, which stood for the void and eventually became our zero, was also used to designate the unknown
77. We must remember in this connection that the richest library of Hellenic antiquity, that of Alexandria, was twice pillaged or destroyed: first by Christian vandals in the fourth century, then by Moslem fanatics in the seventh. As a result of this destruction, a great number of ancient manuscripts disappeared, and would have been completely lost to posterity if it were not for their Arabic translations.
78. Omar Khayyám. The author of the Rubaiyat was
79. he can be considered as the originator of graphical methods. Furthermore, there are indications that he anticipated Newton in the discovery of the binomial formulas.
80. Was it that the Arabs pushed their claim of being the intellectual heirs of the Hellenes to the point of refusing to acknowledge the debt they owed to the Brahmins?
81. The dawn of this day in history is known as the Renaissance or the Revival of Learning.
82. 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. The Arabs of Spain and the Arabs of the Levant were largely responsible for the revival of European learning
83. The systematic use of letters for undetermined but constant magnitudes, the “Logistica Speciosa” as he called it, which has played such a dominant rôle in the development
84. There is a striking analogy between the history of algebra and that of arithmetic. There, we saw, humanity struggled for thousands of years with an inadequate numeration for lack of symbol for naught. Here, the absence of a general notation reduced algebra to a collection of haphazard rules for the solution of numerical equations
85. Just as the discovery of zero created the arithmetic of today, so did the literal notation usher in a new era in the history of algebra.
86. the letter liberated algebra from the slavery of the word. And by this I do not mean merely that without the literal notation any general statement would become a mere flow of verbiage, subject to all the ambiguities and misinterpretations of human speech
87. the letter is free from the taboos which have attached to words through centuries of use. The arithmos of Diophantus, the res of Fibonacci, were preconceived notions: they meant a whole number, an integer. But the A of Vieta or our present x has an existence independent of the concrete object which it is assumed to represent. The symbol has a meaning which transcends the object symbolized: that is why it is not a mere formality
88. What distinguishes modern arithmetic from that of the preVieta period is the changed attitude towards the “impossible. ” Up to the seventeenth century the algebraists invested this term with an absolute sense. Committed to natural numbers as the exclusive field for all arithmetic operations, they regarded possibility, or restricted possibility, as an intrinsic property of these operations.
89. Thus, the direct operations of arithmetic—addition (a + b), multiplication (ab), potentiation (ab)—were omnipossible; whereas the inverse operations—subtraction (a – b), division (a/b), extraction of roots ba ,—were possible only under restricted conditions
90. neither is an intrinsic property of the operation but merely a restriction which human tradition has imposed on the field of the operand. Remove the barrier, extend the field, and the impossible becomes possible.
91. Or, let us assume that the field is restricted to odd numbers only. Multiplication is still omnipossible, for the product of any two odd numbers is odd. However, in such a restricted field addition is an altogether impossible operation, because the sum of any two odd numbers is never an odd number.
2. Observation and experiments on dogs, horses and other domestic animals have failed to reveal any number sense
3. The Bushmen of South Africa have no number words beyond one, two and many, and these words are so inarticulate that it may be doubted whether the natives attach a clear meaning to them
4. There is a plausible connection between the Latin tres, three, and trans, beyond;
5. A rudimentary number sense, not greater in scope than that possessed by birds, was the nucleus from which the number concept grew
6. until one of the collections, or both, are exhausted. The number technique of many primitive peoples is confined to just such such a matching or tallying. They keep the record of their herds and armies by means of notches cut in a tree or pebbles gathered in a pile. That our own ancestors were adept in such methods is evidenced by the etymology of the words tally and calculate, of which the first comes from the Latin talea, cutting, and the second from the Latin calculus, pebble
7. of time the very connection between the two is lost to memory. As man learns to rely more and more on his language, the sounds supersede the images for which they stood, and the originally concrete models take the abstract form of number words
8. in many primitive languages the number-words up to four are identical with the names given to the four fingers. The more civilized languages underwent a process of attrition
9. in many primitive languages the number-words up to four are identical with the names given to the four fingers.
10. Compare the Sanskrit pantcha, five, with the related Persian pentcha, hand; the Russian “piat,” five, with “piast,” the outstretched hand
11. It is these fingers which have taught him to count and thus extend the scope of number indefinitely.
12. And it is reasonable to conjecture that without our fingers the development of number, and consequently that of the exact sciences, to which we owe our material and intellectual progress, would have been hopelessly dwarfed.
13. How old is our number language? It is impossible to indicate the exact period in which number words originated, yet there is unmistakable evidence that it preceded written history by many thousands of years
14. In all IndoEuropean languages, as well as Semitic, Mongolian, and most primitive languages, the base of numeration is ten, i.e., there are independent number words up to ten, beyond which some compounding principle is used until 100 is reached
15. These two other systems are the quinary, base 5, and the vigesimal,
16. These two other systems are the quinary, base 5, and the vigesimal, base 20
17. In the quinary system there are independent number words up to five, and the compounding begins thereafter. (See table at the end of chapter.) It evidently originated among people who had the habit of counting on one hand.
18. They point to the Greek word pempazein, to count by fives, and also to the unquestionably quinary character of the Roman numerals. However, there is no other evidence of this sort, and it is much more probable that our group of languages passed through a preliminary vigesimal stage
19. The day of the Aztecs was divided into 20 hours; a division of the army contained 8000 soldiers (8000 = 20 × 20 × 20)
20. We have the English score, two-score, and three-score; the French vingt (20) and quatre-vingt (4 × 20)
21. There exists among the most primitive tribes of Australia and Africa a system of numeration which has neither 5, 10, nor 20 for base. It is a binary system, i.e. , of base two. These savages have not yet reached finger counting. They have independent numbers for one and two, and composite numbers up to six. Beyond six everything is denoted by “heap
22. I mention this merely to show how the prejudices of childhood may cloud the vision even of the greatest men!
23. Almost any other base, with the possible exception of nine, would have done as well and probably better
24. if the choice of a base were left to a group of experts, we should probably witness a conflict between the practical man, who would insist on a base with the greatest number of divisors, such as twelve, and the mathematician, who would want a prime number, such as seven or eleven, for a base
25. Numeration is at least as old as written language, and there is evidence that it preceded it. Perhaps, even, the recording of numbers had suggested the recording of sounds
26. from the beginning of history until the advent of our modern positional numeration, so little progress was made in the art of reckoning.
27. The greatly increased facility with which the average man today manipulates number has been often taken as proof of the growth of the human intellect
28. The scheme can be typified by the curious method of counting an army which has been found in Madagascar
29. The soldiers are made to file through a narrow passage, and one pebble is dropped for each. When 10 pebbles are counted, a pebble is cast into another pile representing tens, and the counting continues. When 10 pebbles are amassed in the second pile, a pebble is cast into a third pile representing hundreds, and so on until all the soldiers have been accounted for
30. In order to avoid this ambiguity it is essential to have some method of representing the gaps, i.e., what is needed is a symbol for an empty column.
31. We see therefore that no progress was possible until a symbol was invented for an empty class, a symbol for nothing, our modern zero. The concrete mind of the ancient Greeks could not conceive the void as a number, let alone endow the void with a symbol
32. The Indian term for zero was sunya, which meant empty or blank, but had no connotation of “void” of “nothing.” And so, from all appearances, the discovery of zero was an accident brought about by an attempt to make an unambiguous permanent record of a counting board operation.
33. they translated the Indian sunya by their own, sifr, which meant empty in Arabic. When the Indo-Arabic numeration was first introduced into Italy, sifr was latinized into zephirum. This happened at the beginning of the thirteenth century, and in the course of the next hundred years the word underwent a series of changes which culminated in the Italian zero.
34. We now know that the word is merely a corruption of Al Kworesmi, the name of the Arabian mathematician of the ninth century whose book (in Latin translation) was the first work on this subject to reach Western Europe.
35. In fact, the numerals did not assume a stable form until the introduction of printing. It can be added parenthetically that so great was the stabilizing influence of printing that the numerals of today have essentially the same appearance as those of the fifteenth century.
36. the theory of numbers is the branch of mathematics which has found the least number of applications.
37. The theory of integers is one of the oldest branches of mathematics, while modern arithmetic is scarcely four hundred years old.
38. , seven and forty were the ominous numbers of the Hebrews, and Christian theology inherited the seven: the seven deadly sins, the seven virtues, the seven spirits of God, seven joys of the Virgin Mary, seven devils cast out of Magdalen.
39. Thus Abraham proceeding to the rescue of his brother Eliasar drives forth 318 slaves. Is it just a coincidence that the Hebrew word Eliasar adds up to 318
40. Christian theology made particular use of Gematria in interpreting the past as well as in forecasting the future. Of special significance was 666, the number of the Beast of Revelation
41. Strangely enough we find a striking correspondence in Chinese mythology. Here the odd numbers symbolized white, day, heat, sun, fire; the even numbers, on the other hand, black, night, cold, matter, water, earth
42. The amicable numbers were known to the Hindus even before the days of Pythagoras. Also certain passages of the Bible seem to indicate that the Hebrews attached a good omen to such numbers
43. The smallest perfect numbers are 6 and 28, and were known to the Hindus as well as to the Hebrews
44. If Nicomachus meant to imply that there was a perfect number in every decimal class, he was wrong, for the fifth perfect number is 33,550,336
45. However, we know enough to conclude that the primes do not become substantially rarer as we go on
46. Bertrand asserted that between any number and its double there exists at least one prime
47. In 1640 the great French mathematician Fermat announced that he had found a form which represented primes only
48. The failure to obtain a general form for generating prime numbers led to indirect criteria for testing primality.
49. One trouble with it is that while it is true, it is not a criterion, i.e., the condition is necessary but not sufficient
50. postulate of Goldbach, a contemporary of Euler. This postulate alleges that every even number is the sum of two primes
51. The concrete has ever preceded the abstract. That is why the theory of numbers preceded arithmetic
52. mathematics is the cement which holds this structure together. A problem, in fact, is not considered solved until the studied phenomenon has been formulated as a mathematical law
53. When we analyze these mathematical processes we find that they rest on the two concepts: Number and Function
54. the sum does not depend on the order of its terms. The mathematician says no more when he states: addition is a commutative operation
55. never attached much importance to these statements. Yet they are fundamental. On them is based the rule for adding larger numbers
56. In the history of mathematics, the “how” always preceded the “why,” the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The counting technique and the rules of reckoning were established facts at the end of the Renaissance period. But the philosophy of number did not come into its own until the last quarter of the nineteenth century
57. The strength of arithmetic lies in its absolute generality. Its rules admit of no exceptions: they apply to all numbers
58. There is no last number! The process of counting cannot conceivably be terminated. Every number has a successor. There is an infinity of numbers
59. psalmist
60. It is a plausible hypothesis that the early conception of infinity was not the uncountable, but the yet-uncounted
61. For, since the properties of whole numbers form the basis of mathematics, if these properties can be proved by the rules of formal logic, then all of mathematics is a logical discipline
62. is desired that each assumption should be independent of all the others, and that the whole system be exhaustive, i.e., completely cover the question under investigation. The branch of mathematics which deals with such problems is called axiomatics and has been cultivated by such men as Peano, Russell and Hilbert
63. It is the result of observation and experience. To discover a property of a certain class of objects we repeat the observation or tests as many times as feasible, and under circumstances as nearly similar as possible. Then it may happen that a certain definite tendency will manifest itself throughout our observation or experimentation. This tendency is then accepted as the property of the
64. It is the result of observation and experience. To discover a property of a certain class of objects we repeat the observation or tests as many times as feasible, and under circumstances as nearly similar as possible. Then it may happen that a certain definite tendency will manifest itself throughout our observation or experimentation. This tendency is then accepted as the property of the class.
65. For, in order to prove a mathematical proposition, the evidence of any number of cases would be insufficient, whereas to disprove a statement one example will suffice. A mathematical proposition is true, if it leads to no logical contradiction, false otherwise. The method of deduction is based on the principle of contradiction and on nothing else.
66. Mathematics is a deductive science, arithmetic is a branch of mathematics. Induction is inadmissible. The propositions of arithmetic, the associative, commutative and distributive properties of the operations, for instance, which play such a fundamental rôle even in the most simple calculations, must be demonstrated by deductive methods. What is the principle involved?
67. It is significant that we owe the first explicit formulation of the principle of recurrence to the genius of Blaise Pascal, a contemporary and friend of Fermat.
68. , if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the theorems which fill so many volumes are nothing but devious ways of saying that A is A
69. Induction, as applied in the physical sciences, is always uncertain, because it rests on the belief in a general order in the universe, an order outside of us
70. And yet the concept of infinity, though not imposed upon us either by logic or by experience, is a mathematical necessity. What is, then, behind this power of the mind to conceive the indefinite repetition of an act when this act is once possible?
71. The word is of Arabic origin.“Al” is the Arabic article the, and gebar is the verb to set, to restitute. To this day the word “Algebrista” is used in Spain to designate a bonesetter, a sort of chiropractor.
72. The full title of the book is “Algebar wal Muquabalah,” an exact translation of which would read “On Restitution and Adjustment.” Ben
73. Traces of a primitive algebra are found on the clay tablets of the Sumerians, and it probably reached quite a high degree of development among the ancient Egyptians
74. Be this as it may, there is no doubt that Egyptian algebra antedates the papyrus by many centuries
75. The Hindus may have inherited some of the bare facts of Greek science, but not the Greek critical acumen
76. They played with number and ratio, zero and infinity, as with so many words: the same sunya, for instance, which stood for the void and eventually became our zero, was also used to designate the unknown
77. We must remember in this connection that the richest library of Hellenic antiquity, that of Alexandria, was twice pillaged or destroyed: first by Christian vandals in the fourth century, then by Moslem fanatics in the seventh. As a result of this destruction, a great number of ancient manuscripts disappeared, and would have been completely lost to posterity if it were not for their Arabic translations.
78. Omar Khayyám. The author of the Rubaiyat was
79. he can be considered as the originator of graphical methods. Furthermore, there are indications that he anticipated Newton in the discovery of the binomial formulas.
80. Was it that the Arabs pushed their claim of being the intellectual heirs of the Hellenes to the point of refusing to acknowledge the debt they owed to the Brahmins?
81. The dawn of this day in history is known as the Renaissance or the Revival of Learning.
82. 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. The Arabs of Spain and the Arabs of the Levant were largely responsible for the revival of European learning
83. The systematic use of letters for undetermined but constant magnitudes, the “Logistica Speciosa” as he called it, which has played such a dominant rôle in the development
84. There is a striking analogy between the history of algebra and that of arithmetic. There, we saw, humanity struggled for thousands of years with an inadequate numeration for lack of symbol for naught. Here, the absence of a general notation reduced algebra to a collection of haphazard rules for the solution of numerical equations
85. Just as the discovery of zero created the arithmetic of today, so did the literal notation usher in a new era in the history of algebra.
86. the letter liberated algebra from the slavery of the word. And by this I do not mean merely that without the literal notation any general statement would become a mere flow of verbiage, subject to all the ambiguities and misinterpretations of human speech
87. the letter is free from the taboos which have attached to words through centuries of use. The arithmos of Diophantus, the res of Fibonacci, were preconceived notions: they meant a whole number, an integer. But the A of Vieta or our present x has an existence independent of the concrete object which it is assumed to represent. The symbol has a meaning which transcends the object symbolized: that is why it is not a mere formality
88. What distinguishes modern arithmetic from that of the preVieta period is the changed attitude towards the “impossible. ” Up to the seventeenth century the algebraists invested this term with an absolute sense. Committed to natural numbers as the exclusive field for all arithmetic operations, they regarded possibility, or restricted possibility, as an intrinsic property of these operations.
89. Thus, the direct operations of arithmetic—addition (a + b), multiplication (ab), potentiation (ab)—were omnipossible; whereas the inverse operations—subtraction (a – b), division (a/b), extraction of roots ba ,—were possible only under restricted conditions
90. neither is an intrinsic property of the operation but merely a restriction which human tradition has imposed on the field of the operand. Remove the barrier, extend the field, and the impossible becomes possible.
91. Or, let us assume that the field is restricted to odd numbers only. Multiplication is still omnipossible, for the product of any two odd numbers is odd. However, in such a restricted field addition is an altogether impossible operation, because the sum of any two odd numbers is never an odd number.
March 19, 2024
Un ejemplo excelente del nivel que puede alcanzar la literatura de divulgación en las manos adecuadas. Dantzig toma el concepto de número y nos pasea por múltiples aristas: la intuición psicológica de número y cantidad, su representación gráfica, la evolución de la aritmética. Su estudio sobre el infinito y su necesidad para la matemática es disfrutable e intrigante, tocando el análisis matemático, algebra, teoría de números y otras areas. La exposición histórica sobre la escritura de los números en occidente y el desarrollo de la aritmética es quizás lo mas enriquecedor de la obra para el no especialista.
Su único defecto es profundizar en ocasiones mas de lo razonable para un público profano. Los apéndices son interesantes pero desentonan un poco por su excesivo tecnicismo. Nada de esto opaca sustancialmente la obra.
Su único defecto es profundizar en ocasiones mas de lo razonable para un público profano. Los apéndices son interesantes pero desentonan un poco por su excesivo tecnicismo. Nada de esto opaca sustancialmente la obra.
February 21, 2023
A very comprehensive history of numbers and number theory; the prose is organized into various chapters on continuity, irrational numbers, transcendental numbers, and imaginary numbers. Dantzig's classic still resonates today because the intellectual adventures that the ancient Greeks, Arabs, and modern Europeans are still adventurous for us to think through.
The text at times borders on the philosophical, with Dantzig always reminding us that particular properties and tenets of mathematics hold fast, and once those principles are considered immutable, then the proofs and theorems that come out of those principles must also be true.
To be honest I skimmed through this volume much too quickly, especially the Section II which delves into particular lemmas and proofs in number theory of historical importance.
But otherwise it's a very impressive volume and worth reading again with care.
The text at times borders on the philosophical, with Dantzig always reminding us that particular properties and tenets of mathematics hold fast, and once those principles are considered immutable, then the proofs and theorems that come out of those principles must also be true.
To be honest I skimmed through this volume much too quickly, especially the Section II which delves into particular lemmas and proofs in number theory of historical importance.
But otherwise it's a very impressive volume and worth reading again with care.
July 27, 2010
This was recommended to me by someone of much higher intelligence and more knowledge than I. He called it an "easy" book on the history, philosophy, and central concepts of mathematics. Well, I understood the history easily enough, and the philosophy was at times challenging but I think I understood it. The math, though, was simply beyond me. I think I should revisit this once I've studied set theory....
My mathematical ineptitude aside, I highly recommend this book to anyone interested in the topic. I really think that "philosophers" should become better acquainted with math and science - how can you claim to make any claims about "truth" without them?
My mathematical ineptitude aside, I highly recommend this book to anyone interested in the topic. I really think that "philosophers" should become better acquainted with math and science - how can you claim to make any claims about "truth" without them?
January 17, 2015
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.
June 24, 2019
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.
Its a must read for anyone who is atall interested in mathematics.
April 24, 2017
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.
August 18, 2008
Very informative book about number theory. At some points it is a bit technical, which can't be avoided, but if you are literate in advanced mathematics there should be no problem. If you are not, it is still a great read.
July 4, 2011
The first couple of chapters were interesting--about the evolution of counting and the development of language to describe abstract concepts like "how many", but after that, the book got extremely tedious and boring. Not one of my favorites on math.
March 29, 2017
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.
June 19, 2010
Loved this book. I didn't retain much but his wonder about numbers as well as the progression of complexity of calculations was fun reading.
August 4, 2013
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?
December 13, 2012
"Though the source be obscure, still the stream flows on"
June 10, 2020
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 takeaways
There 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)
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 takeaways
There 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)
July 1, 2025
An exciting read!
The book covers the breadth of evolution of Math in such a short span. Being somewhat trained in Math, there were questions that I have always pondered over; finding answers was easy but it was not illuminating enough as context was missing. This book does a great job in summarizing evolution of Math developing a new perspective in looking at things.
Interesting things I found in this book:
1) Connecting linguistics and Math to show how naturally Math could have evolved.
2) That some major development in Math happened much before rigor. That intuition and necessity played a great role, and the Math itself was solidified much later.
3) The book is ordered in increasing generality of the Math. However, at any point of time, there were some vague ideas from the past that are connected to the currently developing Math.
Loved how the book connects different eras in this respect.
Overall, an amazing book if you want to develop a historical/philosophical perspective of Math. This book may not be regarded as a book of history of Math. It's more about learning the forces and connections behind the ideas.
The book covers the breadth of evolution of Math in such a short span. Being somewhat trained in Math, there were questions that I have always pondered over; finding answers was easy but it was not illuminating enough as context was missing. This book does a great job in summarizing evolution of Math developing a new perspective in looking at things.
Interesting things I found in this book:
1) Connecting linguistics and Math to show how naturally Math could have evolved.
2) That some major development in Math happened much before rigor. That intuition and necessity played a great role, and the Math itself was solidified much later.
3) The book is ordered in increasing generality of the Math. However, at any point of time, there were some vague ideas from the past that are connected to the currently developing Math.
Loved how the book connects different eras in this respect.
Overall, an amazing book if you want to develop a historical/philosophical perspective of Math. This book may not be regarded as a book of history of Math. It's more about learning the forces and connections behind the ideas.
August 3, 2024
I don't believe I am the target audience for this book, and I feel a touch of discomfort with reviewing it (God-forbid rating it) at all. I understood the first two thirds well enough, but the latter third was generally beyond me. Still, it was an interesting mix of mathematics and philosophy, toned down for a well-educated, but not necessarily mathematician audience. Every page presented a new thought-provoking insight, with the conversational tone of the writing leading the reader through the thought process. I would recommend this to anyone interested in and possessing a foundational knowledge of a STEM field with the advice of being patient and taking it slow.
November 3, 2018
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!
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!
January 17, 2024
I like Math. It was my major in college, and I did a year of grad school work in Math.
This book was just not my cup of tea. There seemed to be some melodrama in it, although writing and reading Mathematics can be extremely dry, so some drama might liven things up.
Nevertheless, I skimmed it in the end. It wasn't bad, brought up a few topics I would like to explore further, reminded me that not all infinities are the same (different orders of infinities), and reminded me how much I enjoy Mathematics.
I may just go look up the Pythagoreans just now!
This book was just not my cup of tea. There seemed to be some melodrama in it, although writing and reading Mathematics can be extremely dry, so some drama might liven things up.
Nevertheless, I skimmed it in the end. It wasn't bad, brought up a few topics I would like to explore further, reminded me that not all infinities are the same (different orders of infinities), and reminded me how much I enjoy Mathematics.
I may just go look up the Pythagoreans just now!
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