Metamath and Shattered (Star Trek: Voyager): Difference between pages

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Revision as of 16:37, 8 June 2007

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{{ST episode|

== A proof checker very easy to handle ==

name = Shattered|

[[Image:Metamath-aleph-naught.gif|right]]

image = [[Image:ST-VOY_7_11.jpg|270px]]<br />Chakotay is out of phase|

The '''Metamath''' program, devised by Norman Megill, is an automated proof verifier for mathematical theorems.<ref>{{cite web | author=Megill | title=What is Metamath? | work=Metamath Home Page | url=http://us.metamath.org/#faq | accessdate= | accessyear=}} </ref>

series = VOY|

It means that it cannot simply [[Automated theorem proving|prove a theorem automatically]]—it must have a proof entered by a user that it can verify for correctness.<ref>{{cite web | author=Megill | title=Will Metamath help me learn abstract mathematics? | work=Metamath Home Page | url=http://us.metamath.org/#learn | accessdate= | accessyear=}} </ref> Also, unlike proof verifiers like [[Mizar system|Mizar]] that present shorter, more traditional proofs, Metamath presents proofs in a simple, step-by-step format. <ref>{{cite web | author=Megill | title=Are there other sites that formalize math from its foundations? | work=Metamath Home Page | url=http://us.metamath.org/#other | accessdate= | accessyear=}} </ref>

ep_num = 157|

prod_num = 257|

date = [[January 17]], [[2001]]|

writer = [[Mike Sussman (TV series writer/producer)|Mike Sussman]]<br>[[Michael Taylor (screenwriter)|Michael Taylor]]|

director = [[Terry Windell]]|

guest = [[Manu Intiraymi]] as Icheb<br>[[Martha Hackett]] as Seska<br> [[Scarlett Pomers]] as Naomi Wildman<br> [[Martin Rayner]] as Dr. Chaotica|

year = 2377|

stardate = ?|

prev = [[Flesh and Blood (Voyager episode)|Flesh and Blood, Part II]]|

next = [[Lineage (Voyager episode)|Lineage]]

}}

'''Shattered''' is an episode of the TV series ''[[Star Trek: Voyager]]'', the eleventh episode of the seventh season

==Plot summary (a.k.a. spoiler)==

Chakotay encounters [[Icheb]] and [[Naomi Wildman]] in Cargo Bay 2. Naomi is enjoying the puzzle they are playing with, as Icheb misunderstands it as a teaching tool. Chakotay tells Icheb of his cider stash and how he wishes Neelix will not find it. Icheb suggests storing it with the Borg spare parts as Neelix does not care to examine that section.

Given a set of axioms and definitions, Metamath can verify proofs of

mathematical theorems using only one rule, substitution, with optional

provisos (specified in the axioms) for what variables must remain

distinct after a substitution is made. This "substitution" is just the

simple replacement of a variable with an expression and not the proper

substitution described in works on predicate calculus.<ref>{{cite web | author=Megill | title=How Proofs Work | work=Metamath Proof Explorer Home Page | url=http://us.metamath.org/mpegif/mmset.html#proofs | accessdate= | accessyear=}}</ref>

Because of the simplicity of the Metamath language, proof verifiers other than the

Metamath program have been written with as little as 300 lines of Python

code.<ref>{{cite web | author=Megill | title=What other programs have been written for the Metamath language? | work=Metamath Home Page | url=http://us.metamath.org/index.html#language2 | accessdate= | accessyear=}} </ref>

Later, an exploding nebula affects the warp core and Chakotay is hit with a blast of energy.

==A 6,000 theorems database==

Metamath has a rich database of 6,000 theorems including propositional calculus, predicate calculus and set theory. A rich range of concepts are developped : union and intersection, indexed union and intersection, power set, class builder. Cardinals, ordinals and natural numbers are constructed. Recursion, equinumerosity, induction, operations and other concepts of arithmetic can be used. The real and complex numbers are constructed from Dedekind cuts, and the concepts of sequences, limits and so on are developped for them. The exponential and trigonometric functions are implemented. A set of more than 1,000 theorems can allow one to explore the [[Hilbert space|"Hilbert" space]].

A table of contents at the top of the Metamath Proof Explorer gives a comprehensive list of all the brought up topics<ref name="topics">{{cite web | author=Megill | title=Table of contents | work=Metamath Proof Explorer | url=http://us2.metamath.org:8888/mpegif/mmtheorems.html }}</ref>.

He awakens in the Doctor's sickbay and is astonished to discover the Doctor believes it is now several years earlier. Chakotay explores the ship and realizes that it is fractured into many differing time periods. The Doctor's treatment of him, realigning his body with a chroniton serum, allows Chakotay to pass through the time barriers.

[[Image:Metamath-theorem-avril1-indexed.png|thumb|right|Theorem '''avril1''', as shown in the online Metamath Proof Explorer.]]

One noteworthy theorem of Metamath is "Poisson d'Avril's theorem", which spells out "[[April Fool's Day|APRIL FOOL]]" using symbols. Megill claims that the theorem "has left even seasoned professional mathematicians scratching their heads"<ref name="avril">{{cite web | author=Megill | title=Theorem avril1 | work=Metamath Proof Explorer | url=http://us.metamath.org/mpegif/avril1.html | accessdate= | accessyear=}}</ref>, but it is actually quite simple to grasp and prove, since no two sets can be related by the empty set.

During his exploration, he finds Engineering is in control of Seska and Kazon forces. He is beaten and detained by them but manages to escape as the upper level of Engineering is in a different time period all together. The bridge of the ship is a few days before the Caretaker incident, so Chakotay is regarded as a spy. He is detained and taken to the brig. He escapes when the security men who arrest him do not follow him through time as they do not have the serum.

== An electronic treatise of mathematics ==

Whereas the seminal idea that leads Megill to design Metamath was the desire to control very precisely the correctness of some proofs, we can also think that the spirit of Whitehead and Russel animates the growing up of Metamath and its database. the aim of [[Principia Mathematica]] was to give a treatise of mathematics expressed according to Frege's views. And it seems that (as far as possible) a goal of Metamath is this one as well.

He eventually gains more of the serum. He again finds Kathryn Janeway on the bridge. Janeway, a guard and Chakotay talk in her ready room. Despite his knowledge of her past, he is not implicitly believed. He manages to take Kathryn hostage and injects her with the serum. They pass through a time barrier, leaving Janeway's guard Andrews unable to see her. This convinces Janeway to help Chakotay.

== Pedagogy ==

The method of proof used by metamath is far different from what is used in a school context. In school, what is required is the very synthetic method of proof developped by mathematicians since Euclid's time. In Metamath, the method of proof is the symbolic, analytical method of proof invented by [[Aristotle]], [[Leibniz]], [[Peano]] and [[Frege]]. Obviously one can't expect to solve school exercices with Metamath. However, Metamath can be used as a huge example of what a symbolic system means and he can also be useful because its detailed, unambiguous definitions can bring relief when one is in doubt about what a text book, with its informal and sometimes highly ambiguous definitions, means.

Their efforts reveal they must inject the serum into the bio-nueral gel-packs on many areas of the ship. If successful, Chakotay will find himself in his right time and be able to stop the time-fracturing.

Megill gives a list of text books that can be used with metamath and many theorems in the database reference them. So people who want to learn mathematics for themselves can use Metamath in connection with these books ( I know that wanting to learn mathematics for one own's pleasure can seem perverse or even insane but beyond that "people who like this sort of thing will find Metamath the sort of thing they like.")

In the process, he refers to several episodes from the past seven years. Janeway begins to second-guess decisions she hasn't even made yet, especially after learning her human doctor will soon perish. She knows she started with 153 crewmen and struggles with wanting to know how many more she will lose. After a close encounter with the macroviruses, she becomes convinced that entering the Delta Quadrant at all is wrong, as it is a 'gigantic deathtrap'. Chakotay convinces her not to alter the timeline to avoid stranding the ship in the Delta Quadrant.

== Activity ==

Thanks to Norman Megill, the activity is very intense, and new theorems are added to the database everyday. Currently (august 2006), [[topology]] and [[Hilbert space|"Hilbert" space]] are developped. A table of the most recent proofs is maintained <ref name="recent">{{ cite web | title=Recent proofs | author=Megill | work=Metamath Proof Explorer | url=http://us2.metamath.org:8888/mpegif/mmrecent.html }}</ref>.

During the course of this they visit many areas and time periods. Cargo Bay 2 is from when the Borg have formed an alliance with Voyager. They meet [[Seven of Nine]] who is still part of the collective. Astometrics, a section the younger Janeway did not know existed, is staffed by a much older Icheb and Naomi Wildman, both wearing Starfleet uniforms. Both are shocked as, according to their points of view, both Chakotay and Janeway had died long ago. Icheb reveals he had never exposed the ___location of the cider.

==Differences from traditional methods==

Metamath's formal, precise proofs differ from traditional mathematical proofs and can be harder to read.

Both Astrometics and Cargo Bay 2 have provided vital information to the problems, but the next few sections provide many dangers. The Transporter room is filled with angry Maquis, led by Torres, who do not trust the two. The holodeck, which is malfunctioning, is running a dangerous [[Captain Proton]] program, which Janeway must talk herself out of. Again, Chakotay must visit Engineering. He enters it alone. Seska decides to realign Voyager with her time period and use the knowledge of the future to guarantee she will keep control.

As Megill mentions on his website, Metamath's axioms and theorems are actually [[scheme]]s; their variables are generic variables, and any appropriate substitution can be made for them.<ref>{{cite web | author=Megill | title=A Note on the Axioms | work=Metamath Proof Explorer Home Page | url=http://us.metamath.org/mpegif/mmset.html#axiomnote | accessdate= | accessyear=}}</ref>. Some proof checkers don't use schemes but we can't say that schemes are traditional methods however since when one does mathematics with a paper and a pencil one uses schemes without thinking about it. Using schemes is the normal way to do mathematics for a human being.

It does not work. Janeway had organized a resistance movement from all corners of time and ship. Borg-Seven, early Harry and Tom, the Maquis and future Icheb and Naomi, all injected with Chakotay's serum, help defeat Seska's forces and allow the plan to go forward.

Metamath's representations of concepts tend to differ from their traditional representations, sometimes due to technical limitations:

In a few seconds the ship is back to the origanal time and Chakotay manages to stop the distarer. When Janeway asks him why he turned the deflector dish into a giant lightning rod he say he can't tell her because of the [[temporal prime directive]]. Later, Chakotay has dinner with the Janeway, who lets on that she knows the ___location of Chakotay's cider stash but can't tell him how she knows because of the temporal prime directive.

* The inverse of a relation, or "converse" as Megill calls it, is denoted by a u-ish symbol above and to the left of the relation, as opposed to the -1 [[superscript]] used traditionally.<ref>{{cite web | author=Megill | title=Definition df-cnv | work=Metamath Proof Explorer | url=http://us.metamath.org/mpegif/df-cnv.html | accessdate= | accessyear=}}</ref>

* The traditional proper substitution of predicate calculus is not built into Metamath but is a derived notion. Where traditionally φ(y) represents the proper substitution of y for x in φ(x), in Metamath these are denoted [y/x]φ and φ respectively.<ref>{{cite web | author=Megill | title=Definition df-sb | work=Metamath Proof Explorer | url=http://us.metamath.org/mpegif/df-sb.html | accessdate= | accessyear=}}</ref>

* Along the same lines, function values are denoted using a left apostrophe instead of the parentheses around the argument in the literature.<ref>{{cite web | author=Megill | title=Definition df-fv | work=Metamath Proof Explorer | url=http://us.metamath.org/mpegif/df-fv.html | accessdate= | accessyear=}}</ref> The nonstandard, unambiguous notation allows the Metamath program to more easily read proofs.

==Notes==

As a final season episode, "Shattered" was intended as an affectionate look back at the series, referencing various earlier plot lines and bringing back long-gone alien threats like the [[Kazon]] and characters like [[Seska]]. The following past episodes are specifically referenced: "[[Caretaker (Voyager episode)|Caretaker]]", "[[Basics (Voyager episode)|Basics]]", "[[Macrocosm (Voyager episode)|Macrocosm]]", "[[Scorpion (Voyager episode)|Scorpion]]", etc. There are also moments humorously referencing recurring plot elements; in one scene, Chakotay finds the entire crew asleep, and notes that this has happened on a number of occasions.

Whereas the concept of bondage is usually dealt with at a metalogical level in the other proof checkers (i. e. done by the checker in the most cryptic way), all the theorems concerning this property are proved in Metamath. To do that Megill uses an idea by Quine consisting in identifying the "x is bound in φ" proviso by a "( φ -> A. x φ )" statement.

Some Metamath concepts appear rarely in the literature, e.g. the recursive definition generator rec(A,B) (based on [[Willard Van Orman Quine|Quine]]'s book ''Set Theory and its Logic'', published in [[1969]])<ref>{{cite web | author=Megill | title=Definition df-rdg | work=Metamath Proof Explorer | url=http://us.metamath.org/mpegif/df-rdg.html | accessdate= | accessyear=}}</ref> used to assist in defining [[ordinal number]] addition and similar operations. This generator is necessary to avoid adding a recursive proviso to Metamath.

== Other works connected to metamath ==

Using the design ideas implemented in Metamath, Raph Levien has realized what must be regarded as the smallest proof checker in the world. Called Ghilbert<ref>{{cite web | author=Levien | title=Presentation of Ghilbert | work=Asteroid Meta Wiki | url=http://planetx.cc.vt.edu/AsteroidMeta/Ghilbert}}</ref>, this software is only 500 lines of python long. Now Levien would like to realize a system where several people could collaborate together and his work is emphasizing modularity and connection between small theories. Using as well Metamath design principles, Juha Arpiainen has realized his own proof checker called Bourbaki (in common-lisp) and Marnix Klooster has coded a proof checker in Haskell (called Hmm).

The philosophy of Mel O'cat being absolutely existentialist, he has found the bottom-up process of entering proofs in Metamath absolutely inhuman and he designed a system (called mmj2) where proofs can be entered in both directions. There is also a project to add a graphical user interface to Metamath called mmide.

By far the biggest database written for Metamath is set.mm, but there is also a formalization of Peano's arithmetic called peano.mm and and a formalization of natural deduction called nat.mm. You can even find the formal system MIU presented in Gödel, Escher, Bach.

==References==

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<div class="references-small"><references/></div>

==External links==

{{memoryalpha|Shattered|"Shattered"}}

*[http://us.metamath.org/ Metamath Home Page], from which the program can be obtained.

*[http://www.startrek.com/startrek/view/series/VOY/episode/114111.html Episode summary from Startrek.com]

*[http://www.cs.ru.nl/~freek/comparison/comparison.pdf Freek Wiedijk's book ''The Seventeen Provers of the World'']{{PDFlink}}, which compares traditional proofs to seventeen proof languages.

{{Time travel stories}}

[[Category:FreeStar mathematicsTrek: softwareVoyager episodes]]

[[Category:Theorem provers]]