Monday, January 7, 2013

Equal and Equivalent are not Equivalent

One of the things that I love most about upper level math is the precision of language. Words have very specific meanings and if you don't say something precisely right, then often it is wrong. This is not referring to the idea that "People like math because there is one right answer." This is literally referring to the words we use to write out mathematical statements.

Although this is always true in math, I did not realize the depth of it until my Linear Algebra class, where by the end of the semester we ended up getting points knocked off for incorrect language. In Linear Algebra you learn matrix mathematics as a way to solve systems of equations. In Linear Algebra you can make a statement like "the columns of matrix A are linearly independent" but you can't make a statement like "matrix A is linearly independent". Linear independence does not have a definition with regards to matrices.

Much of precision in mathematical writing has its roots in logic. For instance "A is true if B is true" is very different than "A is true if and only if B" is true. In this circumstance, the second statement is biconditional, meaning that you could swap A and B and the statement is still true. That is not the case for the first statement.

Today I learned that equal and equivalent are not words that you can use interchangeably in mathematics.  In general, numbers are equal and mathematical statements are equivalent.

To better understand this statement, let's look at what a mathematical statement is. 

A mathematical statement can be defined as any mathematical sentence that has a truth value (meaning, any mathematical sentence that is true or false).


a. 5 is a prime number.

This is a mathematical statement because it has a truth value. This sentence can be either true or false. In this situation it is true.

b. 12 and 13 are solutions to the equation x + 1 = 0

Again, this is a mathematical statement because it has a truth value. In this situation, it is false. A statement does not have to be true for it to be a mathematical statement.

c. 20

The number 20 is not a mathematical statement because it has no truth value. 20 is neither true or false, it just is.

The Oxford-English dictionary defines equal as "being the same in quantity, size, degree or value" (as an adjective) or "to be the same in quantity or amount" (as a verb).  Meaning, equal refers to things that have a measurable size, like numbers. Mathematical statements do not have a measurable size, they merely have a truth value.

Equivalent is defined as "equal in value, amount, function or meaning." In the Oxford-English dictionary we do see some overlap in these definitions. First of all the definition of equivalent has the word equal in it, which I think for the purposes of mathematics, should probably be switched to "being the same in". The definition also reuses the term of value. Nonetheless, we can see the general, albeit subtle, difference between these words. Mathematical statements are statements that have function and meaning (true or false), not a measurable quantity, therefore it is correct to use equivalence rather than equal.

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