1033:
110:
Conversely, if we discover an equivalent form in
Arithmetical Algebra or any other subordinate science, when the symbols are general in form though specific in their nature, the same must be an equivalent form, when the symbols are general in their nature as well as in their
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497:
236:
557:
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173:
One of the main uses of the principle of permanence is to show that a functional equation that holds for the real numbers also holds for the complex numbers.
47:, the principle of permanence was considered an important tool in mathematical arguments. In modern mathematics, arguments have instead been supplanted by
1074:
1098:
942:
832:
104:
Whatever form is
Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote.
1093:
249:. By the principle of permanence for functions of two variables, this suggests that it holds for all complex numbers as well.
602:
76:
1067:
458:
67:, which state that all statements of some language that are true for some structure are true for another structure.
816:
712:
934:
1060:
1040:
179:
17:
888:
447:'s extensions of the natural numbers beyond infinity, neither satisfies both properties simultaneously.
162:
154:
144:
694:
Hence both of these, the early rigorous infinite number systems, violate the principle of permanence.
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29:
509:
503:
64:
993:
859:
802:
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452:
48:
899:. Vol. 1. L'Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi
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133:
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326:
259:
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157:" to describe (and criticize) a method of argument used by 18th century mathematicians like
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562:
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121:
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33:
815:
Wolfram, Stephen. "Chapter 12, Section 9, Footnote: Generalization in mathematics".
711:
Wolfram, Stephen. "Chapter 12, Section 9, Footnote: Generalization in mathematics".
444:
909:
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436:
239:
125:
874:
769:
752:
52:
455:, addition is left-cancellative, but no longer commutative. For example,
60:
799:
On
Symbolical Algebra and its Applications to the Geometry of Position
37:
32:
like addition and multiplication should behave consistently in every
864:
850:
Toader, Iulian D. (2021), "Permanence as a principle of practice",
742:
728:
Toader, Iulian D. (2021), "Permanence as a principle of practice",
1032:
506:, addition is commutative, but no longer left-cancellative, since
59:. Additionally, the principle has been formalized into a class of
43:
Before the advent of modern mathematics and its emphasis on the
929:
Georg Cantor: his mathematics and philosophy of the infinite
51:
built upon axioms, and the principle is instead used as a
252:
For a counter example, consider the following properties
87:
132. Let us again recur to this principle or law of the
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657:{\displaystyle \aleph _{0}+1=\aleph _{0}=\aleph _{0}+2}
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1017:in ordinal and cardinal arithmetic, respectively.
774:History of Science and Mathematics Stack Exchange
165:that was similar to the Principle of Permanence.
897:Cours d'Analyse de l'Ecole royale polytechnique
764:
762:
492:{\displaystyle 3+\omega =\omega \neq \omega +3}
91:, and consider it when stated in the form of a
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8:
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534:
970:The smallest infinite number is denoted by
323:left-cancellative property of addition: if
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789:(J. & J. J. Deighton, 1830). —
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26:law of the permanence of equivalent forms
38:extensions to established number systems
703:
793:(2nd ed., Scripta Mathematica): Vol.1
7:
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1027:
961:, UTM Series, Springer-Verlag, 2001c
833:"Hankel, Hermann | Encyclopedia.com"
231:{\displaystyle e^{s+t}-e^{s}e^{t}=0}
116:The principle was later revised by
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639:
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607:
14:
443:numbers. However, when following
1031:
139:Around the same time period as
75:The principle was described by
552:{\displaystyle x+y=max\{x,y\}}
89:permanence of equivalent forms
1:
801:(1845). Quote from 1830 ed.,
423:Both properties hold for all
36:, especially when developing
1099:History of mathematics stubs
1047:. You can help Knowledge by
176:As an example, the equation
1010:{\displaystyle \aleph _{0}}
256:commutativity of addition:
1115:
1026:
925:Dauben, Joseph W. (1979),
599:is infinite. For example,
770:"Principle of Permanence"
935:Harvard University Press
875:10.1016/j.hm.2020.08.001
753:10.1016/j.hm.2020.08.001
83:(emphasis in original):
1039:This article about the
983:{\displaystyle \omega }
683:{\displaystyle 1\neq 2}
354:{\displaystyle x+y=x+z}
287:{\displaystyle x+y=y+x}
153:, which used the term "
22:principle of permanence
1094:History of mathematics
1041:history of mathematics
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18:history of mathematics
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818:A New Kind of Science
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787:A Treatise on Algebra
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414:
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155:generality of algebra
145:Augustin-Louis Cauchy
141:A Treatise of Algebra
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81:A Treatise of Algebra
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893:"Analyse Algébrique"
852:Historia Mathematica
837:www.encyclopedia.com
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57:algebraic structures
55:for discovering new
30:algebraic operations
28:, was the idea that
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380:{\displaystyle y=z}
313:{\displaystyle x,y}
65:transfer principles
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453:ordinal arithmetic
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592:{\displaystyle y}
572:{\displaystyle x}
134:Alfred Pringsheim
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45:axiomatic method
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821:. p. 1168.
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120:and adopted by
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136:, and others.
122:Giuseppe Peano
118:Hermann Hankel
77:George Peacock
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238:hold for all
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34:number system
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957:Gamelin, T.
953:
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920:
910:Free version
901:. Retrieved
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445:Georg Cantor
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246:
242:
240:real numbers
175:
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169:Applications
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140:
138:
115:
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108:
103:
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99:proportion.
96:
92:
88:
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79:in his book
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698:References
387:, for all
147:published
126:Ernst Mach
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675:≠
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475:ω
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433:rational
294:for all
163:Lagrange
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441:complex
429:integer
425:natural
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71:History
63:called
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