The Case for Formal Reasoning
Fausto Giunchiglia, Thomas Trevisan, Adolfo Villafiorita
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)
What are these?
- 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).
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:
- Include your Children When Baking Cookies
- 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}- Answers:
- 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*}