出版社: Oxford University Press
副标题: How the Mind Creates Mathematics
出版年: 1999129
页数: 288
定价: USD 39.95
装帧: Paperback
ISBN: 9780195132403
内容简介 · · · · · ·
The Number Sense is an enlightening exploration of the mathematical mind. Describing experiments that show that human infants have a rudimentary number sense, Stanislas Dehaene suggests that this sense is as basic as our perception of color, and that it is wired into the brain. Dehaene shows that it was the invention of symbolic systems of numerals that started us on the climb ...
The Number Sense is an enlightening exploration of the mathematical mind. Describing experiments that show that human infants have a rudimentary number sense, Stanislas Dehaene suggests that this sense is as basic as our perception of color, and that it is wired into the brain. Dehaene shows that it was the invention of symbolic systems of numerals that started us on the climb to higher mathematics. A fascinating look at the crossroads where numbers and neurons intersect, The Number Sense offers an intriguing tour of how the structure of the brain shapes our mathematical abilities, and how our mathematics opens up a window on the human mind.
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In all truth, matters are trifle more complex because only a certain version of Peano's axioms that mathematicians call "firstorder Peano arithmetic" suffers from this infinite expansion of nonstandard models. Yet this version is generally thought to be the best axiomatization of number theroy that we have.
20130612 18:21
In all truth, matters are trifle more complex because only a certain version of Peano's axioms that mathematicians call "firstorder Peano arithmetic" suffers from this infinite expansion of nonstandard models. Yet this version is generally thought to be the best axiomatization of number theroy that we have.
回应 20130612 18:21 
Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation...
20130414 23:48
Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation and subtract the top digit from the bottom one. Little does it matter that this operation is meaningless. Indeed, the result often exceeds the starting number, without disturbing the pupil in the least. Calculation appears to him as a pure manipulation of symbols, a surrealist game largely devoid of meaning.
Where do these bugs come from? Strange as it might seem, no textbook ever describes the correct subtraction recipe in its full generality. A computer scientist can vainly search his kid's arithmetic manual for instructions precise enough to program a general subtraction routine.
Only a refined understanding of the algorithm's design and purpose can help. Yet the very occurrence if such absurd errors suggests that the child's brain registers and executes most calculation algorithms without caring much about their meaning.
回应 20130414 23:48 
We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It we...
20130414 20:48
We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It weaves multiple links among disparate data. Associative links permit the reconstruction of memories on the basis of fragmented information. We invoke this reconstruction process, consciously or not, whenever we try to retrieve a past fact. Step by step, the perfume of Proust's madeleine evokes a universe of memories rich in sounds, visions, words, and past feelings. Associative memory is a strength as well as a weakness. It is a strength when it enables us, starting from a vague reminiscence, to unwind a whole ball of memories that once seemed lost. No computer program to date reproduces anything close to this "addressing by content." It is a strength again when it permits us to take advantage of analogies and allows us to apply knowledge acquired under other circumstances to a novel situation. Associative memory is a weakness, however, in domains such as the multiplication table where the various pieces of knowledge must be kept from interfering with each other at all costs. When faced with a tiger, we must quickly activate our related memories of lions. But when trying to retrieve the result of 7 * 6, we court disaster by activating our knowledge of 7 + 6 or 7 * 5. Unfortunately for mathematicians, our brain evolved for millions of years in an environment where the advantages of associative memory largely compensated for its drawbacks in domains like arithmetic. We are now condemned to live with inappropriate arithmetical associations that our memory recalls automatically, with little regard for our efforts to suppress them.
回应 20130414 20:48 
Placevalue coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an a...
20130414 20:40
Placevalue coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an abacus. By contrast, our Arabic numerals, based on the placevalue principle, make the magnitude relations between 5, 50, 500, and 5,000 completely transparent. Placevalue notations are the only ones that reduce the complexity of multiplication to the mere memorization of a table of products from 2 * 2 up to 9 * 9. Their invention revolutionized the art of numerical computation.
回应 20130414 20:40

A quantitative representation, inherited from our evolutionary past, underlies our intuitive understanding of numbers. If we did not already posses some internal nonverbal representation of the quantity "eight", we would probably be unable to attribute a meaning to the digit 8. We would then be reduced to purely formal manipulations of digital systems, in exactly the same way that a com...
20130414 20:36
A quantitative representation, inherited from our evolutionary past, underlies our intuitive understanding of numbers. If we did not already posses some internal nonverbal representation of the quantity "eight", we would probably be unable to attribute a meaning to the digit 8. We would then be reduced to purely formal manipulations of digital systems, in exactly the same way that a computer follows an algorithm without ever understanding its meaning.
I would like to suggest that these mathematical entities are so difficult for us to accept and so defy intuition because they do not correspond to any preexisting category in our brain. Positive integers naturally find an echo in the innate representation of numerosity; hence a fouryearold can understand them. Other sorts of numbers, however, do not have any direct analogue in the brain. To really understand them, one must piece together a novel mental model that provides for intuitive understanding. This is exactly what teachers do when they introduce negative numbers with such metaphors as temperatures below zero, money borrowed from the bank, or simply a leftward extension of the number line. This is also why the English mathematician John Wallis, in 1685, made a unique gift to the mathematical community when he introduced a concrete representation of complex numbershe first saw that they could be envisioned as a plane where the "real" numbers dwelled along a horizontal axis. To function in an intuitive mode, our brain needs imagesand as far as number theory is concerned, evolution has endowed us with an intuitive picture only of positive integers.
回应 20130414 20:36 
Placevalue coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an a...
20130414 20:40
Placevalue coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an abacus. By contrast, our Arabic numerals, based on the placevalue principle, make the magnitude relations between 5, 50, 500, and 5,000 completely transparent. Placevalue notations are the only ones that reduce the complexity of multiplication to the mere memorization of a table of products from 2 * 2 up to 9 * 9. Their invention revolutionized the art of numerical computation.
回应 20130414 20:40 
We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It we...
20130414 20:48
We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It weaves multiple links among disparate data. Associative links permit the reconstruction of memories on the basis of fragmented information. We invoke this reconstruction process, consciously or not, whenever we try to retrieve a past fact. Step by step, the perfume of Proust's madeleine evokes a universe of memories rich in sounds, visions, words, and past feelings. Associative memory is a strength as well as a weakness. It is a strength when it enables us, starting from a vague reminiscence, to unwind a whole ball of memories that once seemed lost. No computer program to date reproduces anything close to this "addressing by content." It is a strength again when it permits us to take advantage of analogies and allows us to apply knowledge acquired under other circumstances to a novel situation. Associative memory is a weakness, however, in domains such as the multiplication table where the various pieces of knowledge must be kept from interfering with each other at all costs. When faced with a tiger, we must quickly activate our related memories of lions. But when trying to retrieve the result of 7 * 6, we court disaster by activating our knowledge of 7 + 6 or 7 * 5. Unfortunately for mathematicians, our brain evolved for millions of years in an environment where the advantages of associative memory largely compensated for its drawbacks in domains like arithmetic. We are now condemned to live with inappropriate arithmetical associations that our memory recalls automatically, with little regard for our efforts to suppress them.
回应 20130414 20:48 
Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation...
20130414 23:48
Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation and subtract the top digit from the bottom one. Little does it matter that this operation is meaningless. Indeed, the result often exceeds the starting number, without disturbing the pupil in the least. Calculation appears to him as a pure manipulation of symbols, a surrealist game largely devoid of meaning.
Where do these bugs come from? Strange as it might seem, no textbook ever describes the correct subtraction recipe in its full generality. A computer scientist can vainly search his kid's arithmetic manual for instructions precise enough to program a general subtraction routine.
Only a refined understanding of the algorithm's design and purpose can help. Yet the very occurrence if such absurd errors suggests that the child's brain registers and executes most calculation algorithms without caring much about their meaning.
回应 20130414 23:48

In all truth, matters are trifle more complex because only a certain version of Peano's axioms that mathematicians call "firstorder Peano arithmetic" suffers from this infinite expansion of nonstandard models. Yet this version is generally thought to be the best axiomatization of number theroy that we have.
20130612 18:21
In all truth, matters are trifle more complex because only a certain version of Peano's axioms that mathematicians call "firstorder Peano arithmetic" suffers from this infinite expansion of nonstandard models. Yet this version is generally thought to be the best axiomatization of number theroy that we have.
回应 20130612 18:21 
Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation...
20130414 23:48
Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation and subtract the top digit from the bottom one. Little does it matter that this operation is meaningless. Indeed, the result often exceeds the starting number, without disturbing the pupil in the least. Calculation appears to him as a pure manipulation of symbols, a surrealist game largely devoid of meaning.
Where do these bugs come from? Strange as it might seem, no textbook ever describes the correct subtraction recipe in its full generality. A computer scientist can vainly search his kid's arithmetic manual for instructions precise enough to program a general subtraction routine.
Only a refined understanding of the algorithm's design and purpose can help. Yet the very occurrence if such absurd errors suggests that the child's brain registers and executes most calculation algorithms without caring much about their meaning.
回应 20130414 23:48 
We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It we...
20130414 20:48
We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It weaves multiple links among disparate data. Associative links permit the reconstruction of memories on the basis of fragmented information. We invoke this reconstruction process, consciously or not, whenever we try to retrieve a past fact. Step by step, the perfume of Proust's madeleine evokes a universe of memories rich in sounds, visions, words, and past feelings. Associative memory is a strength as well as a weakness. It is a strength when it enables us, starting from a vague reminiscence, to unwind a whole ball of memories that once seemed lost. No computer program to date reproduces anything close to this "addressing by content." It is a strength again when it permits us to take advantage of analogies and allows us to apply knowledge acquired under other circumstances to a novel situation. Associative memory is a weakness, however, in domains such as the multiplication table where the various pieces of knowledge must be kept from interfering with each other at all costs. When faced with a tiger, we must quickly activate our related memories of lions. But when trying to retrieve the result of 7 * 6, we court disaster by activating our knowledge of 7 + 6 or 7 * 5. Unfortunately for mathematicians, our brain evolved for millions of years in an environment where the advantages of associative memory largely compensated for its drawbacks in domains like arithmetic. We are now condemned to live with inappropriate arithmetical associations that our memory recalls automatically, with little regard for our efforts to suppress them.
回应 20130414 20:48 
Placevalue coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an a...
20130414 20:40
Placevalue coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an abacus. By contrast, our Arabic numerals, based on the placevalue principle, make the magnitude relations between 5, 50, 500, and 5,000 completely transparent. Placevalue notations are the only ones that reduce the complexity of multiplication to the mere memorization of a table of products from 2 * 2 up to 9 * 9. Their invention revolutionized the art of numerical computation.
回应 20130414 20:40
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2 有用 serendipitous 20150828
本书讲了动物和人类天生的数感、大脑处理数字和数量的区域和方式、数感与空间感的关系、计算能力（和数感不是一回事，数感关键是右脑叶中专门处理数量的区域，跟空间和位置有关；计算能力跟记忆力以及左脑主语言的区域有关）、数学天才和计算天才、合理的数学教学方法（应该发挥儿童天生的数感）、数学是客观存在还是人为创造（作为认为柏拉图派能想象出一个数学世界，不是因为那个世界存在，而是数字在大脑里是空间式呈现的）、为何数学能巧妙的解释世界（人类的数感是进化塑造的，进化的大脑必然有一套适用的能认知周围世界的机制；另外那些不能合理解释世界的数学都被淘汰了。不是上帝用数学创造了世界，而是人脑只能用数学认知世界）、以及后天可以培养数感等问题。没有谈到人类的逻辑本能是怎么回事，逻辑和数感有什么关系。
0 有用 春云 20130117
对我这种刚入门的小白来说，Dahaene的书就算是启蒙篇啦
0 有用 Don't Panic 20150802
Education is wiping out the treasure we're born with
2 有用 serendipitous 20150828
本书讲了动物和人类天生的数感、大脑处理数字和数量的区域和方式、数感与空间感的关系、计算能力（和数感不是一回事，数感关键是右脑叶中专门处理数量的区域，跟空间和位置有关；计算能力跟记忆力以及左脑主语言的区域有关）、数学天才和计算天才、合理的数学教学方法（应该发挥儿童天生的数感）、数学是客观存在还是人为创造（作为认为柏拉图派能想象出一个数学世界，不是因为那个世界存在，而是数字在大脑里是空间式呈现的）、为何数学能巧妙的解释世界（人类的数感是进化塑造的，进化的大脑必然有一套适用的能认知周围世界的机制；另外那些不能合理解释世界的数学都被淘汰了。不是上帝用数学创造了世界，而是人脑只能用数学认知世界）、以及后天可以培养数感等问题。没有谈到人类的逻辑本能是怎么回事，逻辑和数感有什么关系。
0 有用 Don't Panic 20150802
Education is wiping out the treasure we're born with
0 有用 春云 20130117
对我这种刚入门的小白来说，Dahaene的书就算是启蒙篇啦