Richard Dedekind Vincent Shetter


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Richard Dedekind 



 

 

 



 

 

 



 

 

 



 

 

Vincent Shetter 



Math 475: The History of Mathematics 

September 22,2016  

 

 

 



 

 

 



 

 


 

 



 

Julius Wilhelm Richard Dedekind was a German Mathematician born in Brunswick 

(Braunschweig) who contributed to several areas of mathematics.  These areas include analysis, 

modern set theory, number theory, and  abstract algebra.

1

  His most famous contribution, 



however, was the construction of the real numbers from the rational numbers by method of  

“Dedekind Cuts.”

2

 

 



Born in 1831, Dedekind did not seriously pursue mathematics until 1848 while studying 

at Caroline College.  He would study calculus, algebra, and analytic geometry until 1850 when 

he began study at the University of Göttingen.  It was there that Dedekind would study under 

Carl Friedrich Gauss.

3

  Gauss supervised Dedekind’s dissertation which was completed in 1852.



4

  

After completing his second dissertation, Dedekind would remain at the University of Göttingen 



for another four years. Here Dedekind would be strongly influenced by Dirichlet and Riemann.

5

  



Dedekind then moved to Polytechnic in Switzerland, where he would work as a professor.  

Although Dedekind was a great mathematician, he turned down offers for professorships at more 

prestigious universities.

6

  



                                                           

1

 Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of     



Philosophy  (Summer 2016 Edition), Edward N. Zalta (ed.), URL = 

2

 Rogers,Robert and Bowman Eugene, “How We Got From There to Here: A Story of Real Analysis” Open Suny     



Textbooks 2014, http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ 

3

 The Editors of Encyclopædia Britannica, “Richard Dedekind German Mathematician” Encyclopædia Britannica 



Online, https://www.britannica.com/biography/Richard-Dedekind 

4

 Ibid. 



5

 Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of     

Philosophy  (Summer 2016 Edition), Edward N. Zalta (ed.), URL = 

.

 

6



  Gouvêa, Fernando Q., “Was Cantor Surprised?”, The American Mathematical Monthly, Vol. 118, No. 3 (March 

2011) The Mathematical Association of America (2011). URL: 



http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.198 

 

 

 



 

 

The mathematics that lead to Dedekind Cuts date as far back as the Ancient Greeks.  



They knew that the real numbers would be necessary for calculations as soon as they saw that the 

hypotenuse length of a triangle with legs of length one is 

√2  .   In the 1600s, Newton and 

Leibniz were credited with the invention of calculus, though it had no rigorous foundation.  The 

1700s would bring new developments in mathematics.  Major contributors included Taylor, 

Lagrange, Cauchey and Fourier.  During this time continuity was assumed. If the real world were 

made up of discrete points “…then the real world is more like the rational number line (Q)… 

than the real number line (R).”

7

  Assumptions of continuity would continue as most believed that 



a rigorous definition was unnecessary.

8

 “However, the complexities associated with Fourier 



series and the types of functions that they represented caused mathematicians in the early 1800’s 

to rethink their notions of continuity.”

9

  In 1858, while a lecturer of differential calculus at 



Polytechnic, Dedekind realized that there were flaws in the justification of concepts in calculus.  

He believed that the geometric interpretation of continuity and limits, however useful, was 

unsubstantiated.

10

He would discuss his ideas of finding an arithmetic basis of calculus with his 



colleagues, but he would not publish anything on the topic until later.

11

  It was not until 1859 that 



Karl Weierstrass formally defined continuity. Although defining the real numbers seemingly has 

no application, According to Rodgers and Bowman, the real numbers were defined due to the 

                                                           

7

 Rogers, Robert and Bowman Eugene, “How We Got From There to Here: A Story of Real Analysis” Open Suny     



Textbooks 2014, http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ 

8

 Ibid. 



9

 Ibid. 


10

 Jourdain, Philip E. B. "RICHARD DEDEKIND. (1833-1916.)." The Monist 26, no. 3 (1916): 415-27. 

http://www.jstor.org/stable/27900599

 



11

 Ibid. 


 

 



 

exploratory nature of mathematics research.

12

  Dedekind and others of his time would work in 



modern analysis and several mathematicians would go on to construct the real numbers from the 

rational numbers. 

 

While Cantor defined the real numbers from the rational numbers using Cauchey 



sequences, Dedekind developed “Dedekind Cuts.”

13

 “Dedekind Cuts” was first published in 



“Stetigkeit und Irrationale Zahlen” (Continuity and Irrational Numbers) in 1872.  Rodgers and 

Bowman define “Dedekind Cuts” in How We Got From There to Here: A Story of Real Analysis  

 (Figure 1):  

 

Figure 1: Definition of Dedekind Cut



14

 

 



This definition is fairly straightforward. An example to illustrate Dedekind cuts is to define the 

square root of two in terms of two sets of rational numbers.  {

??????|??????

2

< 2} and    {??????|??????

2

> 2}  .  



This would contribute to the reduction of analysis to arithmetic, rather than geometry;

15

 thereby  



relating the discrete to the continuous.   

                                                           

12

 Ibid 


13

 Ibid. 


14

 Ibid.  


15

 Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of     

Philosophy  (Summer 2016 Edition), Edward N. Zalta (ed.), URL = 



 

 



 

 

Cantor and Dedekind would both construct the real numbers from the rational numbers 



in, roughly, the same time period.  Serendipitously, they would meet while they were vacationing 

in Switzerland shortly after completing their respective works.  They would exchange papers on 

the irrational numbers and went on to write one another, sharing their results on set theory.

16

 As 



their  friendship developed, so did their work in set theory.  This friendship would later end, 

when Dedekind declined Cantor’s offer to work with him in Halle, but it would later resume 

after Cantor had his first mental break down.

17

     



 

Dedekind remained at Polytechnic for the majority, of his career.  He would go on to 

contribute to other branches of mathematics as well.  Dedekind was focused on foundation and 

proof.


18

  He worked on rigorous proofs for concepts that seem obvious to most.  He had success 

in shifting mathematical foundations from geometric ideas to arithmetic proofs.  Dedekind died 

in 1916 in Brunswick, the same town in which he was born.   

 

 

 



 

 

 



                                                           

16

 Gouvêa, Fernando Q., “Was Cantor Surprised?”, The American Mathematical Monthly, Vol. 118, No. 3 (March 



2011) The Mathematical Association of America (2011). URL: 

http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.198 

 

17



 Grattan-Guinness, I. "Missing Materials concerning the Life and Work of Georg Cantor." Isis 62, no. 4 (1971): 

516-17. 


http://www.jstor.org/stable/229823

 



18

 Jourdain, Philip E. B. "RICHARD DEDEKIND. (1833-1916.)." The Monist 26, no. 3 (1916): 415-27. 

http://www.jstor.org/stable/27900599

 



 

 



 

Bibliography 

 

Gouvêa, Fernando Q., “Was Cantor Surprised?”, The American Mathematical Monthly, Vol.  



118, No. 3     (March 2011) The Mathematical Association of America (2011). URL:   

http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.198 

 

Grattan-Guinness, I. "Missing Materials concerning the Life and Work of Georg Cantor." Isis 62, 

no. 4 (1971): 516-17. 

http://www.jstor.org/stable/229823

 

Jourdain, Philip E. B. "RICHARD DEDEKIND. (1833-1916.)." The Monist 26, no. 3 (1916): 



415-27. 

http://www.jstor.org/stable/27900599

 

Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford 



Encyclopedia of     Philosophy  (Summer 2016 Edition), Edward N. Zalta (ed.), URL = 

.

 

Rogers, Robert and Bowman Eugene, “How We Got From There to Here: A Story of Real 



Analysis” Open Suny     Textbooks 2014, http://textbooks.opensuny.org/how-we-got-

from-there-to-here-a-story-of-real-analysis/ 

The Editors of Encyclopædia Britannica, “Richard Dedekind German Mathematician” 

Encyclopædia Britannica Online, https://www.britannica.com/biography/Richard-

Dedekind 

 

 




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