On Information

Concepts in Information Management – blog by Ronald Fuller

Logic, from Philosophy to Mathematics

This post is a collection of quotations from academic writings that illustrate how logic has become seen as more closely related to mathematics than to philosophy. The mathematization of logic did not invalidate or disprove traditional logic, but it did change the character and utility of logic and how it is perceived and taught.


From the Preface to the Handbook of The History of Logic, Volume 3


Virtually every working logician is aware that, after a centuries-long run, the logic that originated in antiquity came to be displaced by a new approach with a dominantly mathematical character. 

...the relationship between mathematics and symbolic logic has been an "uneasy" one, as is the present-day association of mathematics with computing.  Some of this unease has a philosophical texture.


...But for a number of thinkers who took an algebraic approach to logic...(mathematics emerged) as the senior partner.  This was the precursor of the modern view that...logic is indeed a branch of pure mathematics.


From Aristotle’s Prior Analytics and Boole’s Laws of Thought, by John Corcoran


Aristotle achieved logical results that were recognized and fully accepted by subsequent logicians including George Boole. The suggestion that Boole rejected Aristotle’s logical theory as incorrect is without merit or ground...


The publication of Laws of Thought in 1854 launched mathematical logic.


Boole (1854, p. 241) thought that Aristotle’s logic was ‘not a science but a collection of scientific truths, too incomplete to form a system of themselves, and not sufficiently fundamental to serve as the foundation upon which a perfect system may rest. Boole was one of the many readers of Prior Analytics who failed to discern the intricate and fully developed logical system that Aristotle had devised.


It has been said that Galileo’s greatest achievement was to persuade the world’s scientists that physical reality is mathematical, or at least that science should be pursued mathematically.  In his words, ‘The Book of Nature is written in mathematical characters’.  In a strikingly similar spirit, Boole (1854, p. 12) stated 'it is certain that [logic’s] ultimate forms and processes are mathematical'.  Perhaps Boole’s greatest achievement was to persuade the world’s logicians that logical reality is mathematical, or at least that logic should be pursued mathematically.



From Is Modern Logic Non-Aristotelian? by Jean-Yves Beziau


The work of two main figures of the development of Modern logic, Boole and Frege, can be considered as a continuation of the work of Aristotle. The work of Boole can be seen as a mathematization of syllogistic and Frege at the end of the Begriffsschrift presents the square of opposition, to show the harmony of his theory with the Aristotelian tradition.


Boole and Frege didn’t pretend to change the reality of reasoning, they were not proposing a non-Aristotelian Logic. On the other hand people like Henry Bradford Smith or Vasiliev who were using the expression “non-Aristotelian logic”...have had nearly no influence on the development of Modern logic.


Aristotelian logic is a general theory of reasoning supposed to encompass all kinds of reasoning, including mathematical reasoning, but as it is known Aristotle had no special interest for mathematical reasoning and did not pay attention to it. That may explain why Aristotelian logic had absolutely no effect on the development of mathematics.




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