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One, Two, Three: Absolutely Elementary Mathematics
From the acclaimed author of A Tour of the Calculus and The Advent of the Algorithm, here is a riveting look at mathematics that reveals a hidden world in some of its most fundamental concepts. In his latest foray into mathematics, David Berlinski takes on the simplest questions that can be What is a number? How do addition, subtraction, multiplication, and division actually work? What are geometry and logic? As he delves into these subjects, he discovers and lucidly describes the beauty and complexity behind their seemingly simple exteriors, making clear how and why these mercurial, often slippery concepts are essential to who we are. Filled with illuminating historical anecdotes and asides on some of the most fascinating mathematicians through the ages, One, Two, Three is a captivating exploration of the foundation of how it originated, who thought of it, and why it matters.
224 pages, Kindle Edition
First published January 1, 2011
142 people are currently reading
391 people want to read
About the author
David Berlinski
32 books263 followersDavid Berlinski is a senior fellow in the Discovery Institute’s Center for Science and Culture.
Recent articles by Berlinski have been prominently featured in Commentary, Forbes ASAP, and the Boston Review. Two of his articles, “On the Origins of the Mind” (November 2004) and “What Brings a World into Being” (March 2001), have been anthologized in The Best American Science Writing 2005, edited by Alan Lightman (Harper Perennial), and The Best American Science Writing 2002, edited by Jesse Cohen, respectively.
Berlinski received his Ph.D. in philosophy from Princeton University and was later a postdoctoral fellow in mathematics and molecular biology at Columbia University. He has authored works on systems analysis, differential topology, theoretical biology, analytic philosophy, and the philosophy of mathematics, as well as three novels. He has also taught philosophy, mathematics and English at Stanford, Rutgers, the City University of New York and the Université de Paris. In addition, he has held research fellowships at the International Institute for Applied Systems Analysis in Austria and the Institut des Hautes Études Scientifiques. He lives in Paris.
Recent articles by Berlinski have been prominently featured in Commentary, Forbes ASAP, and the Boston Review. Two of his articles, “On the Origins of the Mind” (November 2004) and “What Brings a World into Being” (March 2001), have been anthologized in The Best American Science Writing 2005, edited by Alan Lightman (Harper Perennial), and The Best American Science Writing 2002, edited by Jesse Cohen, respectively.
Berlinski received his Ph.D. in philosophy from Princeton University and was later a postdoctoral fellow in mathematics and molecular biology at Columbia University. He has authored works on systems analysis, differential topology, theoretical biology, analytic philosophy, and the philosophy of mathematics, as well as three novels. He has also taught philosophy, mathematics and English at Stanford, Rutgers, the City University of New York and the Université de Paris. In addition, he has held research fellowships at the International Institute for Applied Systems Analysis in Austria and the Institut des Hautes Études Scientifiques. He lives in Paris.
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Displaying 1 - 30 of 42 reviews
June 5, 2013
In the end, this book is about defining and proving elementary mathematics procedures. Some of the proofs are elegant, such as why two negatives equal a positive and the work with fractions. However, movement through the book is choppy and a background in math is necessary to enjoy this book's thesis.
November 15, 2011
Dr. Berlinski's little book was a pleasure to read. He managed to make book about elementary mathematics engaging and entertaining. Highly recommended.
March 7, 2024
One, Two, Three is an explanation of “absolutely elementary mathematics.” David Berlinski uses history and logic to show how simple math, like counting, addition, exponenents, fractions, etc. actually work. It addresses questions like: why is any number to the 0 power equal to 1? Why does multiplying two negatives result in a positive number? How should one practically think about a negative number?
I have always wondered how we know math corresponds to reality. It’s one thing to see addition or subtraction play out in normal life, but it’s harder to see a real-world justification for why quadratic equations are valid, for example.
I think the beauty of mathematics is that it’s all connected and works out predictably, ranging from the most simple to complex operations. Though the book is understandably dense, Berlinski clearly wants to communicate that beauty to a broad audience.
I have always wondered how we know math corresponds to reality. It’s one thing to see addition or subtraction play out in normal life, but it’s harder to see a real-world justification for why quadratic equations are valid, for example.
I think the beauty of mathematics is that it’s all connected and works out predictably, ranging from the most simple to complex operations. Though the book is understandably dense, Berlinski clearly wants to communicate that beauty to a broad audience.
May 17, 2012
"...what is primitive and thus given, and what is civilized and thus made, find one another in contentment."
December 10, 2019
Somewhat Interesting, but dry.
This book is the author's reflection on mathematics and consists of a historical overview, the foundation of mathematics in commerce, general observations and proofs. Berlinski tries to make it interesting and winsome, and sometimes succeeds, but hardly enough to win me over. I don't feel bettered by reading this, and I felt it a mercy to finish.
One word about the other reviewers who state that this work is "just the author's opinions." One wonders what universe these ignoramuses live in. Presumably the same as me, but they are oblivious to the fact. One's banker might take issue with one's "opinions, " but of the numbers there can be no doubt.
This book is the author's reflection on mathematics and consists of a historical overview, the foundation of mathematics in commerce, general observations and proofs. Berlinski tries to make it interesting and winsome, and sometimes succeeds, but hardly enough to win me over. I don't feel bettered by reading this, and I felt it a mercy to finish.
One word about the other reviewers who state that this work is "just the author's opinions." One wonders what universe these ignoramuses live in. Presumably the same as me, but they are oblivious to the fact. One's banker might take issue with one's "opinions, " but of the numbers there can be no doubt.
March 9, 2024
Felicitări traducătorilor, textul curge așa de lin încât nici n-ai ști că citești o traducere.
Cartea asta mi-a lăsat impresia unui eseu lingvistic despre semnificația numerelor și a conceptelor matematice de bază. De exemplu, știm acum că niciun copil nu e tabula rasa, ci că nevoia de structură (sintaxă) e înnăscută. La fel, ne naștem cu nevoia de diferențiere a numerelor (și nu suntem singurii care facem distincția între un fruct și două fructe, un bol de mâncare mai mic ori mai mare etc). Deși acum pot părea intuitive ideile astea, de fapt sunt revoluționare istoric vorbind.
Dacă ești pasionat de lingvistică dar te fascinează și matematica, atunci cartea asta e fix ce cauți.
Cartea asta mi-a lăsat impresia unui eseu lingvistic despre semnificația numerelor și a conceptelor matematice de bază. De exemplu, știm acum că niciun copil nu e tabula rasa, ci că nevoia de structură (sintaxă) e înnăscută. La fel, ne naștem cu nevoia de diferențiere a numerelor (și nu suntem singurii care facem distincția între un fruct și două fructe, un bol de mâncare mai mic ori mai mare etc). Deși acum pot părea intuitive ideile astea, de fapt sunt revoluționare istoric vorbind.
Dacă ești pasionat de lingvistică dar te fascinează și matematica, atunci cartea asta e fix ce cauți.
January 25, 2024
The subtitle of this book is 'Absolutely Elementary Mathematics". I am not sure that that is an accurate description of the book because, at times, it gets quite complex. The part of the book that I liked the most was the history of various parts of mathematics. The author has a sense of humor but at times I really did get the joke.
May 5, 2013
The author, a mathematics and philosophy professor, writes about the basic concepts of simple arithmetic (addition, subtraction, multiplication, division), starting with the premise that numbers exist outside of human endeavor, then on to the definition of addition (which is just adding by one), lingering at the problem of zero, then through some rather convoluted proofs of various theorems, to stop at the abstract algebraic concepts of rings (structures which include sets of integers and provide the definition of addition and multiplication) and fields (which define division through multiplicative inverses).
If the summary above makes it seem as though this is a jaunt through the math you learned in elementary school, think again: “The recursion theorem justifies definitional descent by drawing a connection between the recipe or algorithm embodied in definitional descent and the existence of a unique function, the one that definitional descent has presumably defined.” Berlinski is often this recursive; I often found myself wondering what was being proved or defined, and what was being simply assumed. But aside from tortuous mathematical definitions, the book is written in an airy, conversational, sometimes jocular (sometimes smug) tone, with many sentences given their own paragraphs in order to give them Weight. Berlinski is even quite funny, as when he discusses Guiseppe Peano (whose axioms provide the groundwork for what Berlinksi attempts to show) and his bizarre simplified Latin that no one used or understood, or when he imagines early mathematicians’ dialogue when encountering the apparent absurdity that is negative numbers (“Can I do that?” “Why not?” “I’m just asking.” “What next? I mean besides giving up. That always works”). As a philosophical treatise on the concept of mathematics itself, the book makes some trenchant points (“across the vast range of arguments [in psychology, logic, physics, etc.]… it is only within mathematics that arguments achieve the power to compel allegiance because they are seen to command assent”). But as a tour of elementary abstract principles, it’s a bit abstruse for the layman. I enjoyed his insights on sets and some of the simpler chapters, but finished the book feeling as though Berlinski was a bit too clever for his own good, and yet not quite clever enough to make it all clear.
If the summary above makes it seem as though this is a jaunt through the math you learned in elementary school, think again: “The recursion theorem justifies definitional descent by drawing a connection between the recipe or algorithm embodied in definitional descent and the existence of a unique function, the one that definitional descent has presumably defined.” Berlinski is often this recursive; I often found myself wondering what was being proved or defined, and what was being simply assumed. But aside from tortuous mathematical definitions, the book is written in an airy, conversational, sometimes jocular (sometimes smug) tone, with many sentences given their own paragraphs in order to give them Weight. Berlinski is even quite funny, as when he discusses Guiseppe Peano (whose axioms provide the groundwork for what Berlinksi attempts to show) and his bizarre simplified Latin that no one used or understood, or when he imagines early mathematicians’ dialogue when encountering the apparent absurdity that is negative numbers (“Can I do that?” “Why not?” “I’m just asking.” “What next? I mean besides giving up. That always works”). As a philosophical treatise on the concept of mathematics itself, the book makes some trenchant points (“across the vast range of arguments [in psychology, logic, physics, etc.]… it is only within mathematics that arguments achieve the power to compel allegiance because they are seen to command assent”). But as a tour of elementary abstract principles, it’s a bit abstruse for the layman. I enjoyed his insights on sets and some of the simpler chapters, but finished the book feeling as though Berlinski was a bit too clever for his own good, and yet not quite clever enough to make it all clear.
October 20, 2022
Happy with math as a black box thank you very much. It works; I don't care why.
December 25, 2015
Weird but enjoyable. Literature plus mathematics I guess?
"But, really, isn’t this how we all are, much impressed by things we do not understand and hoping that they represent something very wise and interesting?"
"To get the number from the fraction, it is necessary only to keep the fraction’s numerators while discarding its denominators, the decimal point serving to separate the integer from the fraction that follows. In place of 1314/1000—rather ungainly, let us be honest—there is 1.314, sleek as a seal, and as easy to train; all that is needed to use it is a willingness to keep track of the decimal point and the places that it commands."
"Mud we might leave to the philosophers. They love the stuff."
"But, really, isn’t this how we all are, much impressed by things we do not understand and hoping that they represent something very wise and interesting?"
"To get the number from the fraction, it is necessary only to keep the fraction’s numerators while discarding its denominators, the decimal point serving to separate the integer from the fraction that follows. In place of 1314/1000—rather ungainly, let us be honest—there is 1.314, sleek as a seal, and as easy to train; all that is needed to use it is a willingness to keep track of the decimal point and the places that it commands."
"Mud we might leave to the philosophers. They love the stuff."
August 7, 2012
An entertaining jaunt through some important points in the history and philosophy of the most basic foundations in mathematics.
October 5, 2014
This might have been a really good book if the author had stayed on topic, but two things made me stop reading: to many digressions and to many completely irrelevant anecdotes.
December 10, 2022
Absolutely elementary mathematics originates from counting by one. In math we want certainty. But certainty does not come cheap. A relatively obvious statement may involve long proof with lots of details.
Knowing that Russell wrote Introduction to Mathematical Philosophy while in prison (like Nehru's Glimpses), creates a renewed urge to read the work. The work (among other things) tries to explain natural numbers without invoking the concept of number. What makes three sheep, "three"? It's its similarity with other sets of three things. Each thing in a pair of three-sets can be matched elementwise, exactly. Four-sets can also have this matching, then how do we differentiate between three and four? Three-set is bigger than two-set and smaller than four-set. Eventually we end up recognizing a one-set or even an empty set? In any case, the numbers are then properties of not objects but sets of objects.
Membership, are you in or out, is the fundamental relationship in set theory.
Say 0 represents the empty set. Then we can build a sequence of sets like {0}, {0, {0}}, {0, {0}, {0, {0}}}, ... which corresponds to the natural numbers 1, 2, 3, ...; for counting we can try matching , say the apples, to the elements of each set in the sequence and whichever matches exactly the number of apples then is the same as the number of elements in that matching set.
Validity of a logical argument depends on form not content. First check premises and conclusion are in right form. Then comes checking the truth of the premises which is outside of logic.
Author seems to hold the opinion that proof is the most distinguishing characteristic of math because no other field has "proof". Given that a math proof, in the end, is a sociological construct (seems very successful though), I would say "proof" is what a field has established as a process to agree if a statement is true or false. In that sense, science has its own sort of proof.
The five Peano axioms are like five Euclidean ones, but for natural numbers. The central focus of the axioms is the idea of "succession". Dedekind took counting as fundamental, and from that numbers come forth.
To define x+y in terms of "succession" we need three axioms: (1) x+0=x (2) x+z=x+(y+1) (3) x+z=(x+y)+1; by the third we have actually defined x+z as the successor of x+y. As an example, 4+3 = 4 + (2+1) = (4+2) + 1. If we continue: 4+2 = (4+1) + 1, 4+1 = (4+0) + 1; so 4+3 = (4+0) + 1 + 1 + 1. Applying the same for 4, we have 4+3 = 0 + 1 + 1 + 1 + 1 + 1 + 1 + 1 or with S as successor 4+3 = SSSSSSS(0). This last one is what kids do, they collect the 4 and 3 objects in a pile and count.
To define multiplication we assume addition has already been defined and we have three more axioms: (1) x * 0 = 0 (2) x = y+1 (3) x * y = x * (y+1) = x*y + x. These operations on natural numbers are defined recursively. Kleene's recursive theorems are important in this respect.
Law of induction can be proved from the assumption (well ordering principle) that every non-empty set of natural numbers contain a smallest element. Well ordering principle can be used to demonstrate that there isn't a natural number between 0 and 1.
Noether established the link between symmetry and conservation laws. She also brought rings to the front. Using the axioms of rings we can demonstrate that (-x)(-y) = xy. Polynomials form ring. Fractions form a field.
Knowing that Russell wrote Introduction to Mathematical Philosophy while in prison (like Nehru's Glimpses), creates a renewed urge to read the work. The work (among other things) tries to explain natural numbers without invoking the concept of number. What makes three sheep, "three"? It's its similarity with other sets of three things. Each thing in a pair of three-sets can be matched elementwise, exactly. Four-sets can also have this matching, then how do we differentiate between three and four? Three-set is bigger than two-set and smaller than four-set. Eventually we end up recognizing a one-set or even an empty set? In any case, the numbers are then properties of not objects but sets of objects.
Membership, are you in or out, is the fundamental relationship in set theory.
Say 0 represents the empty set. Then we can build a sequence of sets like {0}, {0, {0}}, {0, {0}, {0, {0}}}, ... which corresponds to the natural numbers 1, 2, 3, ...; for counting we can try matching , say the apples, to the elements of each set in the sequence and whichever matches exactly the number of apples then is the same as the number of elements in that matching set.
Validity of a logical argument depends on form not content. First check premises and conclusion are in right form. Then comes checking the truth of the premises which is outside of logic.
Author seems to hold the opinion that proof is the most distinguishing characteristic of math because no other field has "proof". Given that a math proof, in the end, is a sociological construct (seems very successful though), I would say "proof" is what a field has established as a process to agree if a statement is true or false. In that sense, science has its own sort of proof.
The five Peano axioms are like five Euclidean ones, but for natural numbers. The central focus of the axioms is the idea of "succession". Dedekind took counting as fundamental, and from that numbers come forth.
To define x+y in terms of "succession" we need three axioms: (1) x+0=x (2) x+z=x+(y+1) (3) x+z=(x+y)+1; by the third we have actually defined x+z as the successor of x+y. As an example, 4+3 = 4 + (2+1) = (4+2) + 1. If we continue: 4+2 = (4+1) + 1, 4+1 = (4+0) + 1; so 4+3 = (4+0) + 1 + 1 + 1. Applying the same for 4, we have 4+3 = 0 + 1 + 1 + 1 + 1 + 1 + 1 + 1 or with S as successor 4+3 = SSSSSSS(0). This last one is what kids do, they collect the 4 and 3 objects in a pile and count.
To define multiplication we assume addition has already been defined and we have three more axioms: (1) x * 0 = 0 (2) x = y+1 (3) x * y = x * (y+1) = x*y + x. These operations on natural numbers are defined recursively. Kleene's recursive theorems are important in this respect.
Law of induction can be proved from the assumption (well ordering principle) that every non-empty set of natural numbers contain a smallest element. Well ordering principle can be used to demonstrate that there isn't a natural number between 0 and 1.
Noether established the link between symmetry and conservation laws. She also brought rings to the front. Using the axioms of rings we can demonstrate that (-x)(-y) = xy. Polynomials form ring. Fractions form a field.
December 26, 2018
David Berlinksi is among the leaders of writing so-called popular books about different aspects of math -- some that is highly advanced, as in The Advent of the Algorithm, and some that is very very basic, as in 2011's One, Two Three.
Berlinksi assigns the basic mathematical functions the group name "AEM" or Absolutely Elementary Mathematics. The four major functions of addition, subtraction, multiplication and division are outlined as the building blocks of far more complicated functions and equations. Berlinksi also digs even deeper, offering ways to think about even the idea of "number."
Berlinski holds doctorates in philosophy and mathematics, so he is a good choice to explain math concepts in terms that don't lean too heavily on equations. His purpose in One, Two Three is to suggest answers for these simplest and most basic questions about AEM and to show how such answers can be deduced via logic from some very simple assumptions.
One, Two Three is both aided by and labors under Berlinksi's habit of breezy and almost flippant writing. On the one hand he largely succeeds in getting complex ideas boiled down to terms that most people can understand, and presents his arguments in ways that can be followed without specialized knowledge. But on the other hand, his tone sometimes crosses over into flippancy in ways that can slow readers down while they finish rolling their eyes.
He too often sacrifices some clarity and direction in order to make a witty observation and in more than one place sticks in some jokes for their own sake rather than explanatory value. Whether or not Berlinksi is actually all that impressed with his own wit, he gives a good enough imitation of being so to make several parts of One, Two Three way more annoying and way less useful than they could have been.
Original available here.
Berlinksi assigns the basic mathematical functions the group name "AEM" or Absolutely Elementary Mathematics. The four major functions of addition, subtraction, multiplication and division are outlined as the building blocks of far more complicated functions and equations. Berlinksi also digs even deeper, offering ways to think about even the idea of "number."
Berlinski holds doctorates in philosophy and mathematics, so he is a good choice to explain math concepts in terms that don't lean too heavily on equations. His purpose in One, Two Three is to suggest answers for these simplest and most basic questions about AEM and to show how such answers can be deduced via logic from some very simple assumptions.
One, Two Three is both aided by and labors under Berlinksi's habit of breezy and almost flippant writing. On the one hand he largely succeeds in getting complex ideas boiled down to terms that most people can understand, and presents his arguments in ways that can be followed without specialized knowledge. But on the other hand, his tone sometimes crosses over into flippancy in ways that can slow readers down while they finish rolling their eyes.
He too often sacrifices some clarity and direction in order to make a witty observation and in more than one place sticks in some jokes for their own sake rather than explanatory value. Whether or not Berlinksi is actually all that impressed with his own wit, he gives a good enough imitation of being so to make several parts of One, Two Three way more annoying and way less useful than they could have been.
Original available here.
January 6, 2025
I hesitantly rounded up 3.5 to 4 stars for this one. The book could have used another round of editing. I have read a couple of the author's other books (The Advent of the Algorithm and The Devil's Delusion) and don't remember them wandering off-point as much as this one did.
I usually enjoy anecdotes about mathematicians that place their discoveries in a historical and personal/biographical context. However, the selection of these that Berlinski chose to include for this work came off as random asides. It was really difficult to see how some of the longer biographies (especially Chapter 5) connected to the overall story of the refinement and evolution of the understanding of elementary arithmetic operations.
The "humorous" descriptions of historical figures sometimes fell flat for me since his visions of what these historical figures were like as people often came off as unnecessarily speculative and at times insulting or in poor taste.
As for the mathematical content - it was excellent and basically accounted for all 4 stars I ended up assigning to this. Anyone who ever wondered how elementary arithmetic works at a technical level will benefit from the book. They just have to cut through the rest of the thicket - other reviewers seem to have had the same basic issue with this book.
This probably would have been better as two separate books but overall I don't regret reading it.
I usually enjoy anecdotes about mathematicians that place their discoveries in a historical and personal/biographical context. However, the selection of these that Berlinski chose to include for this work came off as random asides. It was really difficult to see how some of the longer biographies (especially Chapter 5) connected to the overall story of the refinement and evolution of the understanding of elementary arithmetic operations.
The "humorous" descriptions of historical figures sometimes fell flat for me since his visions of what these historical figures were like as people often came off as unnecessarily speculative and at times insulting or in poor taste.
As for the mathematical content - it was excellent and basically accounted for all 4 stars I ended up assigning to this. Anyone who ever wondered how elementary arithmetic works at a technical level will benefit from the book. They just have to cut through the rest of the thicket - other reviewers seem to have had the same basic issue with this book.
This probably would have been better as two separate books but overall I don't regret reading it.
February 4, 2019
This book made me think about how much we take for granted in basic mathematics. What is the abstract concept of a number, where did it come from? Berlinski touches on some important developments of absolutely elementary mathematics and presents gobs of proofs of fundamental concepts I hadn’t seen before. However this book was tedious in the detail and minutiae of proving every concept he presents. It felt like a text book it this way, but also the overly detailed stories of the mathematicians involved with the topics didn’t help the book flow much better. I think this book could have been 50-75 pages shorter and spent less time spelling out all the details. It’s great as a general overview.
(The illustration of the number line folding on itself at zero so that the positive and negative number cancel each other out into nothingness and then unfold again to reveal creation... excellent!)
(The illustration of the number line folding on itself at zero so that the positive and negative number cancel each other out into nothingness and then unfold again to reveal creation... excellent!)
February 19, 2020
Too wordy, by the multiplicative inverse of one's successor
Berlinski is far too enamored by his own words, to the point that even when he is making very simple arguments, it becomes easy to lose track of where he is heading. Don't get me wrong; I love math history, anecdotes and analogies. But when the telling is so ornate and circumlocutory, it obscures more than reveals. As evidenced by those fake proofs of 1=2 which depend on the reader not noticing that one has sneaked a division by zero into a long set of equations, mathematics is not well served by overcomplication for its own sake.
That is to say, too wordy by half.
Berlinski is far too enamored by his own words, to the point that even when he is making very simple arguments, it becomes easy to lose track of where he is heading. Don't get me wrong; I love math history, anecdotes and analogies. But when the telling is so ornate and circumlocutory, it obscures more than reveals. As evidenced by those fake proofs of 1=2 which depend on the reader not noticing that one has sneaked a division by zero into a long set of equations, mathematics is not well served by overcomplication for its own sake.
That is to say, too wordy by half.
November 8, 2017
An absolutely delightful work that was not only informative but also transformative and a bit humorous. Blessed with a very easy to read writing style, Berlinski punctuates his thoughts about Absolutely Elementary Mathematics with some light honor here and there. I think this book would be enjoyed by every math teacher in the world, by anyone who enjoyed math in high school or college and by anyone interested in learning about the development of mathematics. I never knew zero and one could be so interesting.
February 21, 2018
Everyone is familiar with numbers, counting and basic arithmetic. But mathematician Berlinski shows the theory behind why these exist and work and the relationship of the ineffable mathematical universe and concrete reality. Although many consider math among the most rigorous disciplines resting on solid proofs, Berlinski also makes clear that in an axiomatic system where induction provides the method of proof, a good deal of faith is involved before one can even begin counting.
February 26, 2021
I really enjoy David Berlinski's sarcasm and humor. This short book was full of historical and philosophical anecdotes about math (not my favorite subject) I did it on Audible, however, which was a mistake. The formula's, although basic, were very difficult for a mathaphobe like me to follow on audio. I would advise getting the book.
May 30, 2022
As a "writer of obscure books" I will always enjoy reading his writings regardless of his conviction to the truth. But remember, if take zero, fractional numbers and negative numbers out of mathematics you will be left with the Devine.
Read
July 1, 2019A overview of advanced mathematics and algebra. Includes topics like sets and rings. The basic building blocks of arithmetic and integers.
August 14, 2019
16/20ths of this book is a humblebrag. The other 1/5th is great. You'll have to read the book to prove (actually PROVE) that 16/20 = 4/5.
Read
December 21, 2019DNF
December 27, 2019
Didn't finish. Did not like his writing style. He seemed more interested in trying to show he was clever than actually providing information.
February 3, 2021
Read to prepare for tutoring grand children some great antidotes. Used many. Good book for picking and choosing AN backgrounds.
January 28, 2024
A brilliant mind. If you like math and philosophy, you'll enjoy it -- if you can get past the brilliant wittiness of the genius Dr David Berlinski, that is.
July 4, 2024
Because there are no FEW.
December 31, 2024
hated it!
A waste of my time. Not for people who are not in love with Math etc.
Wish I had not wasted my time or $$!
A waste of my time. Not for people who are not in love with Math etc.
Wish I had not wasted my time or $$!
Displaying 1 - 30 of 42 reviews