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Wednesday, February 21, 2018

What is mathematics?


Photo source: Wikimedia Commons, Wapcaplet

Photo license: CC B-SA 3.0


What is mathematics? Here is my definition:


Mathematics is the study of numbers, lines and shapes


How does mathematics relate to logic? Logic is the process of forming conclusions using objective reasons or premises. I believe that logic and mathematics overlap when it comes to mathematical equations. For example, imagine that you are trying to figure out the length of the long side on a right triangle and you already know the lengths of the two short sides (a = 4, b = 3). To answer this question, you could use the Pythagorean theorem: a^2 + b^2 = c^2. Now we can solve the equation using logic:


  1. a^2 + b^2 = c^2
  2. a = 4
  3. b = 3
  4. Therefore, c = 25


I believe that solving this equation is simultaneously mathematics and logic. In this example, there are objective principles that form a conclusion and simultaneously there are numbers, lines and a shape.


I also believe that the part of mathematics that is not logic, has to do with numbers, shapes and lines in themselves. For example, think about the mathematical concept of a square. A square in itself is not a process of forming a conclusion. Therefore, shapes in themselves are not logic. The same can also be said for lines and numbers. For example, the number 8 is not a process of forming a conclusion.


To show how logic can stand on its own separate from mathematics, observe this logical argument:


  1. All humans are mortal
  2. Socrates is a human
  3. Therefore, Socrates is mortal


This logical argument is not mathematics because there are no numbers, lines or shapes being referred to. I believe this is a clear example of non-mathematical logic.


The rest of this post is a collection of definitions of mathematics from philosophers. The purpose of showing philosopher definitions is to show the range of potential definitions and how there is no consensus.


Definitions of mathematics from philosophers


Aristotle (384-322 BC):

1. "the science of quantity" (Quoted in A History of Mathematics by Florian Cajori)


Auguste Comte (1798-1857):

2. "the science of indirect measurement" (Quoted in A History of Mathematics by Florian Cajori)


Benjamin Pierce (1809-1880):

3. "the science that draws necessary conclusions" (Quoted in Foundations of Fundamental Concepts of Mathematics by Howard Eves)


Bertrand Russell (1872-1970):

4. "all mathematics is symbolic logic" (The Principles of Mathematics, 1903)


Walter Warwick Sawyer (1911-2008):

5. "the classification and study of all possible patterns" (Prelude to Mathematics, 1955)


Intuitionism (according to Ernst Snapper 1913-2011):

6. "mental activity which consists in carrying out, one after the other, those mental constructions which are inductive and effective" (The Three Crises in Mathematics: Logicism, Intuitionism and Formalism, 1979)


Formalism (according to Ernst Snapper 1913-2011):

7. "the manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules" (The Three Crises in Mathematics: Logicism, Intuitionism and Formalism, 1979)



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