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Wednesday, February 21, 2018

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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:

- a^2 + b^2 = c^2
- a = 4
- b = 3
- 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:

- All humans are mortal
- Socrates is a human
- 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.

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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