# The Case for Formal Reasoning

## Which is longer? • They are the same length
• Our senses trick us into a comprehension of the world which differs from reality (if there is a reality)

## Perception … changes

Some will perceive the dancer spinning clockwise, others counterclockwise.

Your perception might even change over time: you might perceive it rotating in the other direction, after a while (closing eyes might help). ## What number is this?

1,001

What number does it represent?

• One and one thousandth (100 + 10-3)
• One thousand and one (103 + 1)
• Culture and conventions get in the way.

## Language

• This is exactly why we use a language with a grammar and words precisely defined in vocabularies.
• Not enough, I am afraid:
• Two Sisters Reunited after 18 Years in Checkout Counter
• Kids Make Nutritious Snacks

## Mathematics

• Well, this is exactly why we need mathematics:

Unambiguous notation, well-defined concepts, clear meaning, no doubts.

• Promise not met, I am afraid:

Consider the set of all sets that are not members of themselves. Does this set contain itself?

• This sentence can be described in mathematics (informal set theory), still yielding a contradiction.

## Mathematics

• Well, this is why we have theorems and proofs
• Very good, then, what is the value of the following sum?

\begin{equation} \sum_0^\infty (-1)^n \end{equation}
• 0
• 1/2
• 1

## Proof that it is 0

Expanding the sum and using association, we get:

\begin{eqnarray*} (1 - 1) + (1 - 1) + (1 - 1) + ... \end{eqnarray*}

Hence:

\begin{eqnarray*} 0 + 0 + 0 ... \end{eqnarray*}

## Proof that it is 1

Expanding the sum and using association after the first term, we get:

\begin{eqnarray*} 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) ... \end{eqnarray*}

Hence

\begin{eqnarray*} 1 + 0 + 0 + 0 ... \end{eqnarray*}

## Proof that it is 1/2

Let us call:

\begin{eqnarray*} S = 1 − 1 + 1 − 1 + ... \end{eqnarray*}

Hence:

\begin{eqnarray*} 1 − S & = & 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S \\ 1 − S & = & S \\ 1 & = & 2S \end{eqnarray*}