explain xkcd - User contributions [en]

2: Petit Trees (sketch)

2022-05-03T22:04:52Z

TheOnlyMrCat: Undo revision 232703 by X. K. C. D. (talk)

<noinclude>:''"2", this comic's number, redirects here. For the comic named "2", see [[2614: 2]].''</noinclude>
{{comic
| number = 2
| date = September 30, 2005
| title = Petit Trees (sketch)
| image = tree_cropped_(1).jpg
| titletext = "Petit" being a reference to Le Petit Prince, which I only thought about halfway through the sketch
}}

==Explanation==
This comic does not present a particular point; it is just a picture drawn by [[Randall]].

''{{w|The Little Prince}}'' (in French ''Le Petit Prince)'' is a novella written by {{w|Antoine de Saint-Exupéry}} in 1943, about the titular Little Prince, who lives on an asteroid and visits other inhabited asteroids and eventually the Earth. The book is filled with drawings of the asteroid, the prince, and the travels they make. It is noted how, on occasion, {{w|Adansonia|baobab trees}} can begin to grow on these asteroids, and should they not be immediately uprooted, the growth of their roots would tear the asteroid apart. In this drawing, the roots are encircling the sphere, rather than piercing it, as Le Petit Prince describes.

''The Little Prince'' has later been referenced both in [[618: Asteroid]] and in [[1350: Lorenz]] at [http://imgs.xkcd.com/comics/a1-2014/VgSdMz8OAHQ8w5Ee432f5Q.png the end] of the space trip branch. It was also referenced to in the What If article [https://what-if.xkcd.com/26/ Leap Seconds].

==Transcript==
:[Two trees are growing on opposite sides of a sphere.]

==Trivia==
*This was the 4th comic originally posted to [[LiveJournal]].
**The previous was [[3: Island (sketch)]].
**The next was [[1: Barrel - Part 1]].
*Original title: "Le Petit"
*Original [[Randall]] quote: "Another fairly old drawing that I scanned."
*This was one of the [[:Category:First day on LiveJournal|thirteen first comics]] posted to LiveJournal within 12 minutes on Friday September 30, 2005.
*This comic was posted on [[xkcd]] when the web site opened on Sunday the 1st of January 2006.
**It was posted along [[:Category:First day on xkcd|with all 41 comics]] posted before that on LiveJournal as well as a few others.
**The latter explaining why the numbers of these 41 LiveJournal comics ranges from 1-44.
*One of the original drawings drawn on [[:Category:Checkered paper|checkered paper]].

{{comic discussion}}
[[Category:Comics posted on livejournal| 04]]
[[Category:First day on LiveJournal| 04]]
[[Category:First day on xkcd]]
[[Category:Checkered paper]]
[[Category:Sketches]]

2610: Assigning Numbers

2022-04-26T09:38:29Z

TheOnlyMrCat:

{{comic
| number = 2610
| date = April 22, 2022
| title = Assigning Numbers
| image = assigning_numbers.png
| titletext = Gödel should do an article on which branches of math have the lowest average theorem number.
}}

==Explanation==
{{incomplete|Created by YÖDA'S COMPLETENESS THEOREM - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

'''This explanation is by mathematical necessity either incomplete or incorrect.'''

[[Cueball]] is falling into a common trap, because a little knowledge is a dangerous thing. Faced with some sort of information, of an unknown kind but seemingly not intrinsically mathematical in nature, he has decided that one possible way to proceed is to somehow translate everything into values which can be combined and compared numerically.

This is a very common thing to do, in fields as diverse as {{w|computational linguistics}} or {{w|sports analytics}}, and can be a powerful tool for understanding and learning new things about a subject as {{w|Data science}} tries to extract knowledge and insights from potentially noisy and disordered facts. But it is also used to implement bad science by using incorrect or misguided ideas about how to represent the source material. While it's possible to casually assign numeric values to random pieces of data, these numbers are generally not meaningful enough to compute with and draw any useful inferences from. It is generally possible to perform statistical analysis only on actual measurements, not on what may effectively be arbitrarily-assigned values.

Machine learning algorithms, which are commonly used by data scientists, typically require all their inputs to be numerical. However, most datasets contains categorical features (e.g. the description of a piece of furniture: chair, table, ...). Data scientists therefore use encoding techniques to convert these categorical features to a numerical form so they can be used as inputs to a machine learning model. For instance, label encoding consists of arbitrarily assigning an integer to a category (chair=0, table=1, ...) which may appear meaningless to most observers. In various cases, they may be right.

So, as well as being the mechanism that underlies one of the most profound theorems of 20th century mathematics, it can be mis-used for all kinds of bad or misguided science. From Cueball's attitude, it is far from clear that his attempt will reliably translate his project into a numerical system, nor that his attempt to "do math on it!" will be any more competent.

One of the major characters who looked at the concept is Kurt Gödel. He introduced the idea of {{w|Gödel numbering}} with his landmark {{w|incompleteness theorems}}. In it a unique natural number is assigned to each axiom, statement, and proof, which might otherwise be difficult to accurately process in any other kind of approach. Instead, it is now possible to create metamathematical statements in the language of mathematics.

This allowed Gödel to make the statement "This statement cannot be proven based on the axioms provided" in a mathematically rigorous way. A simple proof by contradiction shows that the statement cannot be false, and therefore (in most logical systems) must be true. The proof goes as follows: 1. Assume that "This statement cannot be proven from the axioms" (Call this statement G) is false (Call this assumption A). 2. Therefore G can be proven from the axioms (Because the negation of the negation is an affirmation. Based only on A) 3. The axioms exist (Call this assumption B). 4. Therefore, G is true (via {{w|Modus ponens}} applied to 2 and 3, based on A and B). 5. Therefore, G and also not G (via And Introduction applied to 1 and 4, based on A and B). 6. This is a contradiction, and therefore A or B must be wrong. We are not willing to sacrifice assumption B, so we must conclude that A is false, given B ({{w|Reductio ad absurdum}} applied to 1,3, and 5). 7. Therefore, G.

Notice that the truth of Gödel's statement does not depend on any particular set of axioms, and adding axioms (such as "Gödel's particular statement is true") only opens up new iterations of the statement which cannot be proven based on the expanded set of axioms (A statement such as "All statements of a similar nature to Gödel's particular statement" is not precise enough to serve as an axiom.). As such, with a little more legwork, it can be proven that any logical system robust enough to accommodate arithmetic must necessarily contain facts that are true within the system but cannot be proven or disproven within the system. The importance of this result cannot be understated, as it upended the entire philosophy of mathematics. {{w|David Hilbert}}'s famous proclamation "We must know, we will know" is simply incorrect.

The title text suggests that Gödel should perform such an analysis on different branches of mathematics, by calculating the average of all the fields' theorems' Gödel numbers. This is nonsensical for a number of reasons:
:1) Gödel is long dead, and dead people can't write articles;{{Dubious}}<sup> - see [[599: Apocalypse]]</sup>
:2) Gödel numbers grow very large very quickly, and depend heavily on the specific values assigned to each logical operator. Therefore the results could be manipulated simply by changing the numbering order of each operator;
:3) It may be very hard to gather all theorems in a field, or even a representative sample;
:4) Different fields of science, like biology or human behaviour, may not be able to write their theorems in the mathematical language of Gödel's incompleteness theorem
If anyone were to attempt this form of analysis, it would be an example of the bad data science described in the caption.

==Transcript==
:[Cueball holds a hand up to his chin while he ponders the contents of what may be a whiteboard. There are five general lines of unreadable scribbling on the board, and between the two bottom lines, there is a square frame to the right with another scribble to the left. Cueball's thoughts are shown above him in a large thought bubble.]
:Cueball's thinking: If I assign numbers to each of these things, then it becomes '''''data''''', and I can do '''''math''''' on it!

:[Caption beneath the panel:]
:The same basic idea underlies Gödel's Incompleteness Theorem and all bad data science.

{{comic discussion}}
[[Category:Math]]
[[Category:Comics featuring Cueball]]
[[Category:Logic]]

2556: Turing Complete

2021-12-17T22:40:08Z

TheOnlyMrCat: Add link to 2453

{{comic
| number = 2556
| date = December 17, 2021
| title = Turing Complete
| image = turing_complete.png
| titletext = Thanks to the ForcedEntry exploit, your company's entire tech stack can now be hosted out of a PDF you texted to someone.
}}

==Explanation==
{{incomplete|Created by a BOTNET - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

A Turing Machine is a computer (of sorts) that has an infinite tape of ones and zeros and can change values and move up and down this tape. This very simple machine can do every computational task that what we think of as a "computer" can do. Something that is Turing Complete is able to emulate a Turing Machine (though generally with a finite tape), and this means it is also able to do basically every computational task. While many pieces of hardware and software are supposed to be Turing Complete (even Excel, as previously pointed out in [[2453: Excel Lambda]]), this comic implies that this was not what it was designed for. This presumably means Ponytail has found an exploit allowing for arbitrary code execution. This could be harmless and fun, like running Mario on a dishwasher, or a more serious security vulnerability that a nation-state could use to attack you.
==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
[Ponytail and Cueball are standing next to each other]
...Now, it turns out this is actually Turing-Complete...
[caption below the panel:]
This phrase either means someone spent six months getting their dishwasher to play Mario or you are under attack by a nation-state.
{{comic discussion}}