Editing 1208: Footnote Labyrinths

{{comic
| number    = 1208
| date      = May 6, 2013
| title     = Footnote Labyrinths
| image     = footnote_labyrinths.png
| titletext = Every time you read this mouseover, toggle between interpreting nested footnotes as footnotes on footnotes and interpreting them as exponents (minus one, modulo 6, plus 1).
}}

==Explanation==
This is a logic puzzle where the reader has to follow a confusing network of footnotes to determine whether the word "no" is to be ignored or not.

In the following solutions, "right-associative" means that the footnotes are evaluated from right to left or top to bottom, and left-associative from left to right or bottom to top (e.g. (2<sup>6</sup>)<sup><sup>3</sup></sup> is left-associative, and 2<sup>(6<sup>3</sup>)</sup> is right-associative).

The term "ibid." is short for "ibidem", or "at the same place", meaning the reference was noted on the same page just before.

;Interpreting nested footnotes as footnotes on footnotes, left-associative
:no<sup>1<sup>2</sup></sup> = (no<sup>1</sup>)<sup><sup>2</sup></sup> = "ignore this" (it is meaningless to increment a phrase by 2), so the correct statement is "we found evidence for it in our data".

;Interpreting nested footnotes as footnotes on footnotes, right-associative
:"no<sup>1<sup>2</sup></sup>" = "no<sup>1 + 2</sup>" = "no<sup>3</sup>". We turn to the definition of <sup>3</sup>, which is "not true<sup>3<sup>2</sup></sup>" = "not true<sup>3 + 2</sup>" = "not true<sup>5</sup>".
:Now <sup>5</sup> is "true<sup>2<sup>6<sup>3</sup></sup></sup>". The 6 says that the 2 footnote is really 1<sup>2<sup>2</sup></sup> = 1<sup>(4. ibid.)</sup> = 1<sup>3</sup>, but the 3 tells us that the 6 is "not true<sup>5</sup>", getting us into an infinite loop, meaning there is no solution.

The title text suggests interpreting footnotes as exponents (minus one, modulo 6, plus 1). Because applying the operations "minus one, modulo 6, plus 1" to an integer always results in an integer between one and six (inclusive), no sequence of integer exponents will ever end up referencing a footnote that does not exist. In mathematics, nested exponents are exclusively right-associative. "no<sup>1<sup>2</sup></sup>" = "no<sup>1</sup>", so we ignore the "no" and the correct statement is "we found evidence for the data." Meanwhile, <sup>3</sup> becomes "not true<sup>3</sup>", an {{w|infinite recursion}}, and since 2<sup>6<sup>3</sup></sup> mod 6 = 4, we just get "ibid" and the 5 refers back to the 3. Footnote 6 is equivalent to 1<sup>4</sup> = 1 = "ignore this".

The comic [[1184: Circumference Formula]] also plays on the typographical similarity between footnotes and exponents, as well as adding even more ridiculous rules.

==Transcript==
:[Excerpt from what appears to be an academic paper with footnotes.]
:experiments to observe this and we found no<sup>1<sup>2</sup></sup> evidence for it in our data.

:<sup>1</sup>Ignore this
:<sup>2</sup>Increment by 2 before following
:<sup>3</sup>Not true<sup>3<sup>2</sup></sup>
:<sup>4</sup>Ibid.
:<sup>5</sup>True<sup>2<sup>6<sup>3</sup></sup></sup>
:<sup>6</sup>Actually a 1<sup>2<sup>2</sup></sup>

{{comic discussion}}
[[Category:My Hobby]]
[[Category:Footnotes]]
@@ Line 19: / Line 19: @@
 ;Interpreting nested footnotes as footnotes on footnotes, right-associative
 :"no<sup>1<sup>2</sup></sup>" = "no<sup>1 + 2</sup>" = "no<sup>3</sup>". We turn to the definition of <sup>3</sup>, which is "not true<sup>3<sup>2</sup></sup>" = "not true<sup>3 + 2</sup>" = "not true<sup>5</sup>".
-:Now <sup>5</sup> is "true<sup>2<sup>6<sup>3</sup></sup></sup>". Going from top down accordingly, the 3 tells us that the 6 is "not true<sup>5</sup>", getting us into an infinite loop, and if we are restricted to only following the footnotes step-by-step, then we will never reach a solution.
+:Now <sup>5</sup> is "true<sup>2<sup>6<sup>3</sup></sup></sup>". The 6 says that the 2 footnote is really 1<sup>2<sup>2</sup></sup> = 1<sup>(4. ibid.)</sup> = 1<sup>3</sup>, but the 3 tells us that the 6 is "not true<sup>5</sup>", getting us into an infinite loop, meaning there is no solution.
-:Suppose we ignore infinity momentarily, we can say there must be only two possibilities (though we have to be wary of paradoxes) for the meaning of <sup>6<sup>3</sup></sup>: either <sup>6</sup> is true, and thus the 2 is actually a 1 (ibid.), or <sup>6</sup> is false or ignored, and thus the 2 remains a 2. In the latter case, the resolution we see is "true<sup>2<sup>6<sup>3</sup></sup></sup>" = "true<sup>2</sup>". It is meaningless to increment a phrase by 2, so if we also require actions to have meaning, then footnote 2 must change to a 1.
-:Consequently, 6 must be true. We can finally resolve <sup>5</sup> = "true<sup>2<sup>6<sup>3</sup></sup></sup>" = "true<sup>1</sup>" is ignored. Since we now know that footnote 5 is ignored, we then go back to the definition of <sup>3</sup>, which we found was "not true<sup>5</sup>" = "not true".
-:Before returning to "no<sup>3</sup>", we see that this conclusion contradicts our previous assumption that 6 is true. In the definition of <sup>5</sup>, we found that it is ignored since we imposed "true<sup>2<sup>6<sup>3</sup></sup></sup>" = "true<sup>1</sup>", but we also found that the definition of <sup>3</sup> is "not true"; therefore, <sup>5</sup> = "true<sup>2</sup>" since it is not true that 2 is actually 1. Now we have reached our contradiction. The upshot is there really is no solution if we restrict footnotes to have only one meaning/definition (in other words, a footnote cannot be both true and not true).
 The title text suggests interpreting footnotes as exponents (minus one, modulo 6, plus 1). Because applying the operations "minus one, modulo 6, plus 1" to an integer always results in an integer between one and six (inclusive), no sequence of integer exponents will ever end up referencing a footnote that does not exist. In mathematics, nested exponents are exclusively right-associative. "no<sup>1<sup>2</sup></sup>" = "no<sup>1</sup>", so we ignore the "no" and the correct statement is "we found evidence for the data." Meanwhile, <sup>3</sup> becomes "not true<sup>3</sup>", an {{w|infinite recursion}}, and since 2<sup>6<sup>3</sup></sup> mod 6 = 4, we just get "ibid" and the 5 refers back to the 3. Footnote 6 is equivalent to 1<sup>4</sup> = 1 = "ignore this".