- They are the same length
- Our senses trick us into a comprehension of the world which differs from reality (if there is a reality)

- These are exo-planets
- These are, actually, the bottoms of old frying pans (https://www.boredpanda.com/things-that-look-like-other-things-optical-illusion/) (see also: Amazing ‘space telescope image’ was actually a slice of chorizo)
- The fallacy, thus, works also the other way around: our mental assumptions influence our perception of the world (we “see” planets).

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).

**1,001**

What number does it represent?

- One and one thousandth (10
^{0}+ 10^{-3}) - One thousand and one (10
^{3}+ 1) - Culture and conventions get in the way.

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

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.

- 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}- Answers:
- 0
- 1/2
- 1

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*}

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*}

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*}