出版社: Dover Publications
出版年: 196311
页数: 115
定价: USD 8.95
装帧: Paperback
ISBN: 9780486210100
内容简介 · · · · · ·
Two most important essays by the famous German mathematician: one provides an arithmetic, rigorous foundation for the irrational numbers, thereby a rigorous meaning of continuity in analysis. The other is an attempt to give logical basis for transfinite numbers and properties of the natural numbers.
Essays on the Theory of Numbers的书评 · · · · · · (全部 0 条)

小魚 (Be brave)
2016/Nov/17th Open paragraphs About hundreds of years after Leibniz and Newton, Dedekind thinks the geometric foundation is not solid enough to "infinitesimal analysis"(the term he used to indicate differential calculus). In this paper, he aims to provide a "purely arithmetic and perfectly rigorous" scientific foundation of differential calculus. In another word, Dedekind...20161118 03:27 1人喜欢
2016/Nov/17thOpen paragraphsAbout hundreds of years after Leibniz and Newton, Dedekind thinks the geometric foundation is not solid enough to "infinitesimal analysis"(the term he used to indicate differential calculus). In this paper, he aims to provide a "purely arithmetic and perfectly rigorous" scientific foundation of differential calculus. In another word, Dedekind is working on number's property to build his arithmetic foundation. The essential principle he tries to establish here is the continuity of number.About Dedekind's purpose, we need to question firstly why he thinks the geometric foundation of calculus is weak and why he calls the arithmetic foundation is the scientific one. A huge dispute might arise about whether algebra(number) or geometry is the primary beings and what is the relation between algebra and geometry.There could be at least three possibilities for their relation.1, Algebra are truly different from geometry. Thus, we cannot apply geometric inventions directly on numbers. We need a theory of number to develop its “legitimacy” of calculus. 2, Algebra and geometry are the same beings, but they show different ways of behaving and expressing themselves. Thus we need solid foundations for both them to reach the same goal. 3, Algebra and numbers are the same beings, however, the number is the primary existence, and geometry is a second derivative, numberlike thought. If so, geometry should be understood by numbers and thus the geometric foundation of calculus is not solid.From my previous schools' learning experience, algebra was treated as universal math, but really, it is a dispute that whether geometry or algebra appear first. Are they both primary existence or it has to be the case that one appears first and the other derivative from it? I was on the side that geometry appears first in the past. Because I understood numbers as the ratio of geometric units. But I am not sure now. I still incline to believe that geometry is more primary, however, could they both be primary source? It is hard for me to point out a number that existing alone without referring to any unit. I can not see how the number exist by itself naturally. Meanwhile, I think the units, which could be anything, like plants, trees, stones and etc., as an abstract conception, have been extracted from physical beings. (I try to list all natural beings first rather than artificial works like chairs or cups.) A unit could also be considered as not a part of geometric objects. Can we thus neglect geometric consideration and just leave numbers and physical beings here? But there are not any exactly similar natural object exist. By counting one tree or two trees, we still need to distinguish the tree and others, and we also need to make the similarities between two different trees. It seems an abstract understanding jumps in at this moment, and it has to go through a geometric process. Even though the physical beings seems to be more primary than geometric objects, by counting trees, there is an undeclared geometric reference.I also have another conjecture about numbers. For any vocabulary, its meaning changes with time. Could number mean differently in ancient and modern times? Maybe number comes from geometry but then with more property that human added to it, it becomes independent beings. The number that we are talking now is different from the one that Dedekind have learned in his past. Again, the order of our discoveries might not be the same as how the beings exist by nature. (I learned this from Conics.) This can not prove numbers has to be later produced than geometry.If geometry and numbers are both artificial intelligent presentations of nature, and they are not nature, what is the meaning for us to clarify their primary? I may leave this question remain open here and think about it later.Let's assume that we only know it is important for Dedekind to establish an arithmetic foundation for calculus. He thinks the first step would be proven that the numbers are continuous. In his own words, the theorem would be
Numbers are discontinuous before Dedekind's time. Dedekind also thinks the numbers are greater than infinity from reading his friend' paper. This motivates him to build his theory on numbers.every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value.
回应 20161118 03:27 
小魚 (Be brave)
2016/Dec/8th Before I discuss the new definition of addition set by Dedekind, I recall the definition of irrational numbers: they are defined by a negative way and they are numbers that are not rational. For a detail of the passage: c2 = (a 1/2p) + (B1/2p) As long as the p is wholly distributed into two parts, the equation holds. But p cannot be partly left because then it would be unabl...20161213 00:43
2016/Dec/8thBefore I discuss the new definition of addition set by Dedekind, I recall the definition of irrational numbers: they are defined by a negative way and they are numbers that are not rational.For a detail of the passage: c2 = (a 1/2p) + (B1/2p) As long as the p is wholly distributed into two parts, the equation holds. But p cannot be partly left because then it would be unable to prove the contradiction of definition.The new definition of arithmetic operations enlarges the application of the arithmetic operations from the realm of rational numbers to the realm of all real numbers. The arithmetic operations corresponding to a method of finding the exact cuts on the line. Following by the new definitions of arithmetic operations, Dedekind starts to discuss the meaning of an interval. Interval is defined by bounding limits and a good example would be 2ac < ac + bc < 2bc if a < b. 2ac and 2bc would be two extremities in the case to define the interval. I think the discussion of the interval is a preparation of the following topic: infinitesimal analysis.The infinitesimal analysis is an another name of Calculus. Dedekind is reapproaching the calculus with his new found arithmetic foundations. The system of symbol lim(xoo)F(x) = L follows Dedekind's theory.Does Dedekind successfully build a new arithmetic foundation of calculus? I may say yes. No one has proved or even thought about the question whether the points are continuous in a line. Dedekind proves that numbers are continuous by themselves. Calculus talks about the infinity which is continuous. Algebra now is more solid than geometry as the ground of calculus.Meanwhile, Dedekind proves the selfcompleteness and individuality of the numbers. In the past paper that involves complex calculations, we questioned the meaning of the equations, especially for the ones cannot refer to physics or the geometrical world. But now we know that the algebra equations stand for themselves and there is no need to keep finding the correlations of geometry.Dedekind's paper really resolves me from a longtime confusion about number and geometry. Now I may believe that numbers are ideal and intelligible beings and geometry is the most accessible way for us to approach it. Geometry is more perceivable and they are one of many representations of numbers. There are more secret hiding by numbers.Others: Is geometry purely about the space?Descartes thinks the intuition is an ability to see all the things at once.回应 20161213 00:43 
小魚 (Be brave)
2016/Dec/7th Now, we turn our sight toward the system of real numbers. Real number combines both rational numbers and irrational numbers. The rules of real number equally apply to rational numbers and irrational numbers. Why is there an incommensurability? Why cannot people cut things so small to measure all things? The second question, in fact, is the answer of the first one. The incomme...20161208 08:32
2016/Dec/7thNow, we turn our sight toward the system of real numbers. Real number combines both rational numbers and irrational numbers. The rules of real number equally apply to rational numbers and irrational numbers.Why is there an incommensurability? Why cannot people cut things so small to measure all things?The second question, in fact, is the answer of the first one. The incommensurability is caused by the inability to cut the line. We cannot cut lines to parts. All straight lines are commonly measurable by points but a point has no part. This paradox drives us back to Euclid's definition of points and lines. There are infinite space between points and lines but still, calculus does not touch the limit. It builds a connection between finite and infinity. But it is not a countable method for us to touch the end.If people is unable to cut infinite small parts to define a number, it suggests that we cannot define numbers through divisions. Everything is commensurable with itself.
I thought the above proof is an inverse version of the previous statement. The previous one that a cut a splits the straight line into two classes, and here, if we have two classes, there is only one a.Others:imaginary number: square root of 1complex number = imaginary number + any real numberIV: If the system R of all real numbers breaks up into two classes R1, R2, such that every number a1 of the class A1 is less than every number a2 of the class A2 then there exists one and only one number a by which this separation is produced
回应 20161208 08:32 
小魚 (Be brave)
2016/Nov/29th This part, as the subtitle suggested, is talking about the "creation" of irrational numbers, and I believe, the definition of numbers has been refreshed. I am going to discuss this part from the beginning. Dedekind, in fact, starts to prepare his argument from the second chapter, while he is cutting a straight line and talking about the equality of rational numbers. H...20161201 09:31
2016/Nov/29thThis part, as the subtitle suggested, is talking about the "creation" of irrational numbers, and I believe, the definition of numbers has been refreshed. I am going to discuss this part from the beginning.Dedekind, in fact, starts to prepare his argument from the second chapter, while he is cutting a straight line and talking about the equality of rational numbers. Here, to prove rational numbers are discontinuous, he makes cuts and asserts that the cuts could only be nonrational numbers. While a cut divides the whole range into two classes, the cut only could be either the greatest of the first class or the least of the second class. It cannot be both because there are always infinite many rational numbers on the line and the cut can only be counted once and attribute to one class.
To set up a cut and two classes, Dedekind use symbol D(for division?) to represent the square of the cut. Notice that D is NOT the cut, it is the square of the cut. D could be 5,6,7 or 8, but neither 4 nor 10. If D is 5, then the cut, in this case, its value is equal to square root of 5, but it is not the square root of 5 for now since we have not created irrational numbers yet. It is just a cut, but not a number.In the progress of proving the existence of "infinitely many cuts not produced by rational numbers", it requires some algebra skills. For example, when to prove u' is a positive integer and certainly less than u. We should try to evaluate both sides of inequality to get u' > 0 on one side, and u' < u for the other side. I am here skipping all those algebraical content.While the proof verifies Dedekind's hypothesis that there is a cut not produced by rational numbers, it implies that there is no greatest Rational number in the first class, nor least Rational number is in the second class. If the cut is the greatest number of the first class, then the greatest number of the first class is irrational. The least number of the second class could be rational or irrational but we cannot know exactly since there are infinitely many numbers close to each other.The most amazing passage of this chapter is as followings:Let D be a positive integer but not the square of an integer.
Are the numbers the cuts? Or are the numbers uniquely corresponding to each cut? What is the number?Certainly, Dedekind refreshes the definition of numbers. Numbers are numbers, not a ratio for now. It is one number, or a single cut, rather than made by two numbers or a number and a unit. I understand the question whether geometry or number comes first better. To respond this question, people has to state what is the number at the beginning. I do think Dedekind makes a difference here by refining the definition of numbers. Reminding that Dedekind says at the beginning of his paper, that he is working on finding a solid foundation of calculus. Here it shows.There is a phrase which Dedekind uses very frequently, and I was confused at the first time. While going over his word again and passing his new definition, I realized the phrase "not essentially different" indicates the property of numbers. If we would agree that Dedekind thinks every cut is a number or corresponding to a number, there could not be any essential difference between numbers simply because they are both numbers.After the proof, Dedekind also presents a way to define the relationship of two numbers by putting numbers into different classes and comparing to the cut. He also verifies the continuity of numbers.Whenever, then, we have to do with a cut (A1,A2) produced by no rational number, we create a new, an irrational number a, which we regard as completely defined by this cut (A1,A2); we shall say that the number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.
回应 20161201 09:31

小魚 (Be brave)
2016/Nov/17th Open paragraphs About hundreds of years after Leibniz and Newton, Dedekind thinks the geometric foundation is not solid enough to "infinitesimal analysis"(the term he used to indicate differential calculus). In this paper, he aims to provide a "purely arithmetic and perfectly rigorous" scientific foundation of differential calculus. In another word, Dedekind...20161118 03:27 1人喜欢
2016/Nov/17thOpen paragraphsAbout hundreds of years after Leibniz and Newton, Dedekind thinks the geometric foundation is not solid enough to "infinitesimal analysis"(the term he used to indicate differential calculus). In this paper, he aims to provide a "purely arithmetic and perfectly rigorous" scientific foundation of differential calculus. In another word, Dedekind is working on number's property to build his arithmetic foundation. The essential principle he tries to establish here is the continuity of number.About Dedekind's purpose, we need to question firstly why he thinks the geometric foundation of calculus is weak and why he calls the arithmetic foundation is the scientific one. A huge dispute might arise about whether algebra(number) or geometry is the primary beings and what is the relation between algebra and geometry.There could be at least three possibilities for their relation.1, Algebra are truly different from geometry. Thus, we cannot apply geometric inventions directly on numbers. We need a theory of number to develop its “legitimacy” of calculus. 2, Algebra and geometry are the same beings, but they show different ways of behaving and expressing themselves. Thus we need solid foundations for both them to reach the same goal. 3, Algebra and numbers are the same beings, however, the number is the primary existence, and geometry is a second derivative, numberlike thought. If so, geometry should be understood by numbers and thus the geometric foundation of calculus is not solid.From my previous schools' learning experience, algebra was treated as universal math, but really, it is a dispute that whether geometry or algebra appear first. Are they both primary existence or it has to be the case that one appears first and the other derivative from it? I was on the side that geometry appears first in the past. Because I understood numbers as the ratio of geometric units. But I am not sure now. I still incline to believe that geometry is more primary, however, could they both be primary source? It is hard for me to point out a number that existing alone without referring to any unit. I can not see how the number exist by itself naturally. Meanwhile, I think the units, which could be anything, like plants, trees, stones and etc., as an abstract conception, have been extracted from physical beings. (I try to list all natural beings first rather than artificial works like chairs or cups.) A unit could also be considered as not a part of geometric objects. Can we thus neglect geometric consideration and just leave numbers and physical beings here? But there are not any exactly similar natural object exist. By counting one tree or two trees, we still need to distinguish the tree and others, and we also need to make the similarities between two different trees. It seems an abstract understanding jumps in at this moment, and it has to go through a geometric process. Even though the physical beings seems to be more primary than geometric objects, by counting trees, there is an undeclared geometric reference.I also have another conjecture about numbers. For any vocabulary, its meaning changes with time. Could number mean differently in ancient and modern times? Maybe number comes from geometry but then with more property that human added to it, it becomes independent beings. The number that we are talking now is different from the one that Dedekind have learned in his past. Again, the order of our discoveries might not be the same as how the beings exist by nature. (I learned this from Conics.) This can not prove numbers has to be later produced than geometry.If geometry and numbers are both artificial intelligent presentations of nature, and they are not nature, what is the meaning for us to clarify their primary? I may leave this question remain open here and think about it later.Let's assume that we only know it is important for Dedekind to establish an arithmetic foundation for calculus. He thinks the first step would be proven that the numbers are continuous. In his own words, the theorem would be
Numbers are discontinuous before Dedekind's time. Dedekind also thinks the numbers are greater than infinity from reading his friend' paper. This motivates him to build his theory on numbers.every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value.
回应 20161118 03:27 
小魚 (Be brave)
Section I Properties of Rational Numbers 2016/Nov/17th At the beginning of this section, there is an important claim. Dedekind says: I regard the whole of arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the ...20161119 06:30
Section I Properties of Rational Numbers 2016/Nov/17thAt the beginning of this section, there is an important claim. Dedekind says:
Why is Dedekind able to admit the necessity of whole arithmetic from a single act? The tutor said the natural consequence is different from necessity. He suggests that the division, subtraction, and multiplication naturally follow the counting, which is the addition. It does not mean it necessary to explain in this way.Our remaining question would be why Dedekind thinks addition and multiplication are the fundamental ones, and subtraction and division are different from it? He claims:I regard the whole of arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an alreadyformed individual to the consecutive new one to be formed.
This might guide us go back the question "is it acceptable to say adding length make an area ".Others:I forgot why and how the number zero joined the conversation. Maybe when we talk about whether we are able to divide a finite quantity by 0. Any finite quantity divided by infinite small quantities, we get infinity. To express this meaning in numbers, the only way to do is use 0 both represents infinite small quantities and infinity. Actually, this is ridiculous. Thus, here is another suggestion that we are claiming something that we don't know by 0 and saying that we know it. One day, if someone could make numbers to exactly represent dx, ddx such infinite small quantities, maybe we will have a better understanding.2016/Nov/18thI will start with answering the remaining question: why division and subtraction are distinct from addition and multiplication. To discuss this question, I think I should firstly claim the difference between counting and addition, and also explain Dedekind's setting of counting.For counting, as above quoted, Dedekind says a "successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding." A classmates questioned about creation. Are we creating sth. when we count "one tree, two trees, three trees"? Yes and no, yes because we are creating names for numbers to indicate the next one in the preceding process, no, because we do not create trees. The third tree is not created by first two trees. The number three or term third is created by following one and two. There is no limit for counting, it could be extended to infinity. Thus, counting is an endless process. Counting is a natural process no matter number exist or not. To see many trees, we cannot deny that there is a one following it, another one following the last one....Without naming numbers, the successive creation is still there. When a child learn to counting, he points out to a series of trees, he says "1,2,3,4,6". We think he is wrong. His mistake is just about the name of number, instead of the understanding of successive creation. The name of numbers are made for easy communications. A classmate asked how could we suppose everybody is capable to understand such a logic process. I was shocked by her question. For example, she thinks counting "1,2,3" are different from counting "100,567" "100,568" "100,569". People may be unable to apply the same logical process of small numbers on big numbers...I may ask her, if so, how could you distinct women and man, and point out the gender of each individual, since we are all different. By asking her question, she is denying the accessibility for all human to have reasoning faculty. Addition is a simplified action of counting. As Dedekind claimed in his paper, addition is a single act. It is dealing with numbers with gaps, and taking multiplestep counting by one step. However, subtraction and division is not. I will explain with examples:(Multiplication)There are 3 groups and each groups wants 5 apples, how many apples do we need in total?The primal operation would be "1,2,3,4,5" "1,2,3,4,5" "1,2,3,4,5" 5+5+5=15. We indicate each sets first and join them together.(Division)There are 15 apples, and 3 groups, how many apples does each group could have, if equally divided?The primal operation would be 1,1,1; 2,2,2; 3,3,3; 4,4,4; 5,5,5. rather than "1,2,3,4,5" "1,2,3,4,5","1,2,3,4,5"If the above is not clear to you. Thinking there are 16 apples. What about the one left? The reminder of the division would be 1. But if you want to complete a division, you have to divided one by 3 and get 1/3. Such fractional numbers are created by human mind. This is how I understand Dedekind's thought, but I have some objections. He thinks the things can only exist naturally by a unit. What if I cut an apple into third and then want to give each group 3 apples and a 1/3 apple. What would the difference be? For subtraction, if I only have 2 apples, how could somebody take 5 apples from me? 2  5 = nothing (0). Even if I have three apples, 3  5 = nothing (0). 0 would not indicate a number but as a noun means nothing, and the superficial meaning of indicating nothing is the same but the conceal meanings which implies the negative number are different. Negative numbers are created by human imagination too. Here is my objection: This argument is denying the natural relation among people of owing sb sth. Negative numbers are possible to be considered as neutral existence too.To summarize, Dedekind may think "less  greater" and "less / greater" needs human imagination and counting does not make fraction and negative numbers exist. I may only agree with the second part. (Maybe Dedekind also only thinks about the second part and I have misunderstood it.) Let me leave the argument here.Then, Dedekind claims his system will be denote by R. We are using this symbol now to indicate the system of rational numbers. Any numbers that could be presented by a fraction is a rational number. R system has the property of "completeness and selfcontainedness".That classmates asked again why 3.333.....is a rational number and why 3.333... is the same one as 10/3. Firstly, 3.33... and 10/3 are two different expressions of the same number. Then, the way of writing 3.3333 implies another expression 3+1/3. There is still no difference. But I think the numerical expression 3.3333 is pointing out a potentiality, and 10/3 is more like an acutualized number, by using Aristotle's language. In fact, for Aristotle's problem of potentiality and actuality, Galileo decides then make the potential to be acutualized. He used fractional numbers to acutalize. This is an Italian solution to an ancient greek problem.Then he starts his journey of showing his famous 'Dedekind's Principle" in Mathematical Analysis. Section II Comparison of the Rational Numbers with the Points of a Straight LineThe section II looks like that Dedekind is repeating his proving process of the last part of section I with just changing some names of term for geometric indication purpose. He claims this by the first sentence of the section:Addition is the combination of any arbitrary repetitions of the abovementioned simplest act into a single act; from it in a similar way arises multiplication.
Here is an interesting verb "recall". Maybe he is not applying the theory of numbers on geometry. He is writing down something that we have already known. I may have to think why he need this part. What is the purpose here to apply number's theory in a geometric image. Because he will repeat and modify similar proofs again, I want to hold this question until the followings shows up.The abovementioned properties of rational numbers recall the corresponding relations of position of the points of a straight line
回应 20161119 06:30 
小魚 (Be brave)
2016/Nov/22nd I was thinking one question: why is Ď€ incommensurable? While knowing about the Ď€, we have to admit what is one first. Without a unit, any number cannot be expressed. It seems to me there is no number cannot commensurable with unit one. The incommensurability is not a reason for me to recognize the irrational numbers. I found a reason to convince me that irrational numbers ar...20161124 00:35
2016/Nov/22ndI was thinking one question: why is π incommensurable? While knowing about the π, we have to admit what is one first. Without a unit, any number cannot be expressed. It seems to me there is no number cannot commensurable with unit one. The incommensurability is not a reason for me to recognize the irrational numbers. I found a reason to convince me that irrational numbers are different, with the respect to the incommensurability. The irrational numbers, in my way of thinking, can only be measured by unit, there is no another rational number as a mediator to measure it. While reading Dedekind's word, I had another question: if the numbers are the fundamental beings, why do we still need a geometric object as a reference to complete the number theory? Why do we need the creation of new numbers to make the continuity? I will feel much better if Dedekind use his language like "discover new number" rather than make creations. I think there are essencial differences between the act of creation and discovering. The only way that I found might explain his thought is like this: the scientists discovered irrational numbers for years but they never clarified it and they did not treat them as numbers. Thus Dedekind was making numbers that corresponding to these properties. In fact, from another perspective, my question also could not deny the possibility for numbers to be a fundamental being. If geometry really is the layer that covering the beings, there would be connections between the phenomena and the beings. Geometry could be the phenomena drive us towards to the being numbers.
(The footnotes saying that we cannot really know ratios without knowing numbers, which I need to think about.)The essence of the continuity;If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes absolutely necessary that the instrument R constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same completeness, or as we may say at once, the same continuity, as the straight line.
Dedekind will apply this belief later to define the continuity of rational numbers by rational numbers.A classmate asked how we are able to know there are some gaps among rational numbers or how we are able to know the existence of irrational numbers. We know there are gaps and there are irrational numbers because we know there is a group of the number that cannot be express by fractions. People have to find it and symbolize it as irrational numbers.If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.
回应 20161124 00:35 
小魚 (Be brave)
2016/Nov/29th This part, as the subtitle suggested, is talking about the "creation" of irrational numbers, and I believe, the definition of numbers has been refreshed. I am going to discuss this part from the beginning. Dedekind, in fact, starts to prepare his argument from the second chapter, while he is cutting a straight line and talking about the equality of rational numbers. H...20161201 09:31
2016/Nov/29thThis part, as the subtitle suggested, is talking about the "creation" of irrational numbers, and I believe, the definition of numbers has been refreshed. I am going to discuss this part from the beginning.Dedekind, in fact, starts to prepare his argument from the second chapter, while he is cutting a straight line and talking about the equality of rational numbers. Here, to prove rational numbers are discontinuous, he makes cuts and asserts that the cuts could only be nonrational numbers. While a cut divides the whole range into two classes, the cut only could be either the greatest of the first class or the least of the second class. It cannot be both because there are always infinite many rational numbers on the line and the cut can only be counted once and attribute to one class.
To set up a cut and two classes, Dedekind use symbol D(for division?) to represent the square of the cut. Notice that D is NOT the cut, it is the square of the cut. D could be 5,6,7 or 8, but neither 4 nor 10. If D is 5, then the cut, in this case, its value is equal to square root of 5, but it is not the square root of 5 for now since we have not created irrational numbers yet. It is just a cut, but not a number.In the progress of proving the existence of "infinitely many cuts not produced by rational numbers", it requires some algebra skills. For example, when to prove u' is a positive integer and certainly less than u. We should try to evaluate both sides of inequality to get u' > 0 on one side, and u' < u for the other side. I am here skipping all those algebraical content.While the proof verifies Dedekind's hypothesis that there is a cut not produced by rational numbers, it implies that there is no greatest Rational number in the first class, nor least Rational number is in the second class. If the cut is the greatest number of the first class, then the greatest number of the first class is irrational. The least number of the second class could be rational or irrational but we cannot know exactly since there are infinitely many numbers close to each other.The most amazing passage of this chapter is as followings:Let D be a positive integer but not the square of an integer.
Are the numbers the cuts? Or are the numbers uniquely corresponding to each cut? What is the number?Certainly, Dedekind refreshes the definition of numbers. Numbers are numbers, not a ratio for now. It is one number, or a single cut, rather than made by two numbers or a number and a unit. I understand the question whether geometry or number comes first better. To respond this question, people has to state what is the number at the beginning. I do think Dedekind makes a difference here by refining the definition of numbers. Reminding that Dedekind says at the beginning of his paper, that he is working on finding a solid foundation of calculus. Here it shows.There is a phrase which Dedekind uses very frequently, and I was confused at the first time. While going over his word again and passing his new definition, I realized the phrase "not essentially different" indicates the property of numbers. If we would agree that Dedekind thinks every cut is a number or corresponding to a number, there could not be any essential difference between numbers simply because they are both numbers.After the proof, Dedekind also presents a way to define the relationship of two numbers by putting numbers into different classes and comparing to the cut. He also verifies the continuity of numbers.Whenever, then, we have to do with a cut (A1,A2) produced by no rational number, we create a new, an irrational number a, which we regard as completely defined by this cut (A1,A2); we shall say that the number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.
回应 20161201 09:31

小魚 (Be brave)
2016/Dec/8th Before I discuss the new definition of addition set by Dedekind, I recall the definition of irrational numbers: they are defined by a negative way and they are numbers that are not rational. For a detail of the passage: c2 = (a 1/2p) + (B1/2p) As long as the p is wholly distributed into two parts, the equation holds. But p cannot be partly left because then it would be unabl...20161213 00:43
2016/Dec/8thBefore I discuss the new definition of addition set by Dedekind, I recall the definition of irrational numbers: they are defined by a negative way and they are numbers that are not rational.For a detail of the passage: c2 = (a 1/2p) + (B1/2p) As long as the p is wholly distributed into two parts, the equation holds. But p cannot be partly left because then it would be unable to prove the contradiction of definition.The new definition of arithmetic operations enlarges the application of the arithmetic operations from the realm of rational numbers to the realm of all real numbers. The arithmetic operations corresponding to a method of finding the exact cuts on the line. Following by the new definitions of arithmetic operations, Dedekind starts to discuss the meaning of an interval. Interval is defined by bounding limits and a good example would be 2ac < ac + bc < 2bc if a < b. 2ac and 2bc would be two extremities in the case to define the interval. I think the discussion of the interval is a preparation of the following topic: infinitesimal analysis.The infinitesimal analysis is an another name of Calculus. Dedekind is reapproaching the calculus with his new found arithmetic foundations. The system of symbol lim(xoo)F(x) = L follows Dedekind's theory.Does Dedekind successfully build a new arithmetic foundation of calculus? I may say yes. No one has proved or even thought about the question whether the points are continuous in a line. Dedekind proves that numbers are continuous by themselves. Calculus talks about the infinity which is continuous. Algebra now is more solid than geometry as the ground of calculus.Meanwhile, Dedekind proves the selfcompleteness and individuality of the numbers. In the past paper that involves complex calculations, we questioned the meaning of the equations, especially for the ones cannot refer to physics or the geometrical world. But now we know that the algebra equations stand for themselves and there is no need to keep finding the correlations of geometry.Dedekind's paper really resolves me from a longtime confusion about number and geometry. Now I may believe that numbers are ideal and intelligible beings and geometry is the most accessible way for us to approach it. Geometry is more perceivable and they are one of many representations of numbers. There are more secret hiding by numbers.Others: Is geometry purely about the space?Descartes thinks the intuition is an ability to see all the things at once.回应 20161213 00:43 
小魚 (Be brave)
2016/Dec/7th Now, we turn our sight toward the system of real numbers. Real number combines both rational numbers and irrational numbers. The rules of real number equally apply to rational numbers and irrational numbers. Why is there an incommensurability? Why cannot people cut things so small to measure all things? The second question, in fact, is the answer of the first one. The incomme...20161208 08:32
2016/Dec/7thNow, we turn our sight toward the system of real numbers. Real number combines both rational numbers and irrational numbers. The rules of real number equally apply to rational numbers and irrational numbers.Why is there an incommensurability? Why cannot people cut things so small to measure all things?The second question, in fact, is the answer of the first one. The incommensurability is caused by the inability to cut the line. We cannot cut lines to parts. All straight lines are commonly measurable by points but a point has no part. This paradox drives us back to Euclid's definition of points and lines. There are infinite space between points and lines but still, calculus does not touch the limit. It builds a connection between finite and infinity. But it is not a countable method for us to touch the end.If people is unable to cut infinite small parts to define a number, it suggests that we cannot define numbers through divisions. Everything is commensurable with itself.
I thought the above proof is an inverse version of the previous statement. The previous one that a cut a splits the straight line into two classes, and here, if we have two classes, there is only one a.Others:imaginary number: square root of 1complex number = imaginary number + any real numberIV: If the system R of all real numbers breaks up into two classes R1, R2, such that every number a1 of the class A1 is less than every number a2 of the class A2 then there exists one and only one number a by which this separation is produced
回应 20161208 08:32 
小魚 (Be brave)
2016/Nov/29th This part, as the subtitle suggested, is talking about the "creation" of irrational numbers, and I believe, the definition of numbers has been refreshed. I am going to discuss this part from the beginning. Dedekind, in fact, starts to prepare his argument from the second chapter, while he is cutting a straight line and talking about the equality of rational numbers. H...20161201 09:31
2016/Nov/29thThis part, as the subtitle suggested, is talking about the "creation" of irrational numbers, and I believe, the definition of numbers has been refreshed. I am going to discuss this part from the beginning.Dedekind, in fact, starts to prepare his argument from the second chapter, while he is cutting a straight line and talking about the equality of rational numbers. Here, to prove rational numbers are discontinuous, he makes cuts and asserts that the cuts could only be nonrational numbers. While a cut divides the whole range into two classes, the cut only could be either the greatest of the first class or the least of the second class. It cannot be both because there are always infinite many rational numbers on the line and the cut can only be counted once and attribute to one class.
To set up a cut and two classes, Dedekind use symbol D(for division?) to represent the square of the cut. Notice that D is NOT the cut, it is the square of the cut. D could be 5,6,7 or 8, but neither 4 nor 10. If D is 5, then the cut, in this case, its value is equal to square root of 5, but it is not the square root of 5 for now since we have not created irrational numbers yet. It is just a cut, but not a number.In the progress of proving the existence of "infinitely many cuts not produced by rational numbers", it requires some algebra skills. For example, when to prove u' is a positive integer and certainly less than u. We should try to evaluate both sides of inequality to get u' > 0 on one side, and u' < u for the other side. I am here skipping all those algebraical content.While the proof verifies Dedekind's hypothesis that there is a cut not produced by rational numbers, it implies that there is no greatest Rational number in the first class, nor least Rational number is in the second class. If the cut is the greatest number of the first class, then the greatest number of the first class is irrational. The least number of the second class could be rational or irrational but we cannot know exactly since there are infinitely many numbers close to each other.The most amazing passage of this chapter is as followings:Let D be a positive integer but not the square of an integer.
Are the numbers the cuts? Or are the numbers uniquely corresponding to each cut? What is the number?Certainly, Dedekind refreshes the definition of numbers. Numbers are numbers, not a ratio for now. It is one number, or a single cut, rather than made by two numbers or a number and a unit. I understand the question whether geometry or number comes first better. To respond this question, people has to state what is the number at the beginning. I do think Dedekind makes a difference here by refining the definition of numbers. Reminding that Dedekind says at the beginning of his paper, that he is working on finding a solid foundation of calculus. Here it shows.There is a phrase which Dedekind uses very frequently, and I was confused at the first time. While going over his word again and passing his new definition, I realized the phrase "not essentially different" indicates the property of numbers. If we would agree that Dedekind thinks every cut is a number or corresponding to a number, there could not be any essential difference between numbers simply because they are both numbers.After the proof, Dedekind also presents a way to define the relationship of two numbers by putting numbers into different classes and comparing to the cut. He also verifies the continuity of numbers.Whenever, then, we have to do with a cut (A1,A2) produced by no rational number, we create a new, an irrational number a, which we regard as completely defined by this cut (A1,A2); we shall say that the number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.
回应 20161201 09:31 
小魚 (Be brave)
2016/Nov/22nd I was thinking one question: why is Ď€ incommensurable? While knowing about the Ď€, we have to admit what is one first. Without a unit, any number cannot be expressed. It seems to me there is no number cannot commensurable with unit one. The incommensurability is not a reason for me to recognize the irrational numbers. I found a reason to convince me that irrational numbers ar...20161124 00:35
2016/Nov/22ndI was thinking one question: why is π incommensurable? While knowing about the π, we have to admit what is one first. Without a unit, any number cannot be expressed. It seems to me there is no number cannot commensurable with unit one. The incommensurability is not a reason for me to recognize the irrational numbers. I found a reason to convince me that irrational numbers are different, with the respect to the incommensurability. The irrational numbers, in my way of thinking, can only be measured by unit, there is no another rational number as a mediator to measure it. While reading Dedekind's word, I had another question: if the numbers are the fundamental beings, why do we still need a geometric object as a reference to complete the number theory? Why do we need the creation of new numbers to make the continuity? I will feel much better if Dedekind use his language like "discover new number" rather than make creations. I think there are essencial differences between the act of creation and discovering. The only way that I found might explain his thought is like this: the scientists discovered irrational numbers for years but they never clarified it and they did not treat them as numbers. Thus Dedekind was making numbers that corresponding to these properties. In fact, from another perspective, my question also could not deny the possibility for numbers to be a fundamental being. If geometry really is the layer that covering the beings, there would be connections between the phenomena and the beings. Geometry could be the phenomena drive us towards to the being numbers.
(The footnotes saying that we cannot really know ratios without knowing numbers, which I need to think about.)The essence of the continuity;If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes absolutely necessary that the instrument R constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same completeness, or as we may say at once, the same continuity, as the straight line.
Dedekind will apply this belief later to define the continuity of rational numbers by rational numbers.A classmate asked how we are able to know there are some gaps among rational numbers or how we are able to know the existence of irrational numbers. We know there are gaps and there are irrational numbers because we know there is a group of the number that cannot be express by fractions. People have to find it and symbolize it as irrational numbers.If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.
回应 20161124 00:35
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Which one is the primary, geometry of algebra? Are numbers truly different from geometry? What is the number?
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戴德金对无理数的纯算术方法推导。排列有理数用到了数轴。所以数轴不算一个geometrical intuition哦？然后从数轴这个line推断有理数是不连续的，它们的gap里面还有其他数即无理数。这个move不算stretching an analogy哦？
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只读了第一篇和第二篇的前言
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Which one is the primary, geometry of algebra? Are numbers truly different from geometry? What is the number?
1 有用 小魚 20161212
Which one is the primary, geometry of algebra? Are numbers truly different from geometry? What is the number?
0 有用 哈啦哈啦米 20110517
只读了第一篇和第二篇的前言
0 有用 ae 20110503
前一半
0 有用 Caesura 20110518
戴德金对无理数的纯算术方法推导。排列有理数用到了数轴。所以数轴不算一个geometrical intuition哦？然后从数轴这个line推断有理数是不连续的，它们的gap里面还有其他数即无理数。这个move不算stretching an analogy哦？