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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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<P>
Metamath is a simple and flexible computer-processable language that supports
rigorously verifying, archiving, and presenting mathematical proofs.
See the <A HREF="#faq">FAQ</A> for more information.
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- Constructs mathematics from scratch, starting from
ZFC set theory axioms. Over 23,000 proofs.
<FONT SIZE=-1>
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HREF="mpeuni/mmtheorems.html">Theorem list</A>
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<A HREF="ileuni/mmil.html"><B>Intuitionistic Logic Explorer</B></A> -
Derives mathematics from a constructive point of view, starting from
axioms of intuitionistic logic.
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<A HREF="nfeuni/mmnf.html"><B>New Foundations Explorer</B></A> -
Constructs mathematics from scratch, starting from
Quine's NF set theory axioms.
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<A HREF="holuni/mmhol.html"><B>Higher-Order Logic Explorer</B></A> -
Starts with HOL (also called simple type theory) and derives equivalents to
ZFC axioms, connecting the two approaches.
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user-contributed proof verifiers, <A HREF="mm_100.html">Metamath 100
list</A>, open problems, other downloads, and miscellany. Filip
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problems, and also his <A HREF="other.html#xpuzzle">XPuzzle</A>
Android app.
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<A NAME="faq"></A>
<CENTER><B><FONT COLOR="#006633" SIZE="+1">Mini FAQ</FONT></B>
</CENTER>
<FONT COLOR="#006633"><B>Q:</B> What is Metamath?</FONT><BR> <B>A:</B>
Metamath is a simple and flexible computer-processable language that supports
rigorously verifying, archiving, and presenting mathematical proofs.
Metamath has a small specification that
enables you to state a collection of axioms (assumptions), theorems, and proofs
(aka a "database").
We have databases for several axiomatic systems.
Metamath provides precision,
certainty, and the elimination of human error.
Through Metamath you can see mathematics developed in complete
detail from first principles, with absolute rigor, in a way
unlike any other system. Hopefully it will
amuse you, amaze you, and possibly enlighten you in its own special way.
<P><A NAME="distinctives"></A><FONT COLOR="#006633"><B>Q:</B> What is
distinctive about Metamath?</FONT><BR>
Metamath is probably unlike anything you have encountered.
Here are some distinctive traits of Metamath:
<OL>
<LI>The Metamath language is extremely simple, with an almost total absence of hard-wired syntax. We believe that it provides about the simplest possible framework that allows essentially all of mathematics to be expressed with absolute rigor.
<LI>Metamath is not tied to any particular set of axioms, instead, axioms are defined in a database.
<LI>Metamath proofs include every step, <i>no exceptions</i>, where each step
is <I>only</i> an application of an axiom or a previously-proved statement.
This is different from
almost all other computer-verifiable proof systems, which allow statements
(like "simp", "auto", or "blast") that don't show the proof steps but
instead ask a computer to try to rediscover the proof steps.
Metamath's unique approach speeds verification, improves archiving, and enables
<I>anyone</I> to follow every proof step.
<LI>Many tools support Metamath, instead of requiring a "canonical" tool.
<LI>Metamath verifiers have been independently
implemented by different people in different programming languages,
reducing the risk of accepting an invalid proof.
<LI>Metamath's fundamental operation (substitution) is easy to understand,
even by those who aren't professional mathematicians.
<LI>Proofs stay proven. In many systems changes to a system's
syntax or tactics cause older proofs to stop being verifiable.
Metamath's approach, which cleanly
separates proof verification from discovery,
provides better support for long-term archives.
<LI>Metamath is one of the top systems in the
<A HREF="https://www.cs.ru.nl/~freek/100/"
>Formalizing 100 Theorems challenge</A>.
The Metamath Proof Explorer (MPE) database alone
has over 23,000 proven theorems.
</OL>
<P>
Of course, other systems may have advantages over Metamath
that are more compelling to you.
Some users of Metamath also use or develop other systems;
we're always delighted to cooperate with other systems' users when we can.
<P><A NAME="discussion"></A><FONT COLOR="#006633"><B>Q:</B> How can I ask
questions or discuss Metamath-related topics?</FONT><BR>
<B>A:</B> The <A HREF="http://groups.google.com/group/metamath">Metamath
Google Group mailing list</A> is a great place to discuss
Metamath. If you have questions, that is a good place
to ask them.
<!-- If you want very old material, here's one place to get it, but
few will care:
The <A
HREF="https://web.archive.org/web/20131219031208/http://wiki.planetmath.org/cgi-bin/wiki.pl/metamath"
>AsteroidMeta</A> [retrieved 22-Sep-2020] wiki was used for many older
Metamath discussions, but is no longer available. Archived discussions
such as this one can be found on archive.org.) -->
<P><A NAME="start"></A><FONT COLOR="#006633"><B>Q:</B> Where do I
start?</FONT><BR>
<B>A:</B> Read Sections <A HREF="mpeuni/mmset.html#overview">1</A>, <A
HREF="mpeuni/mmset.html#proofs">2</A>, and <A
HREF="mpeuni/mmset.html#axioms">3</A> of the <A
HREF="mpeuni/mmset.html">Metamath Proof Explorer</A>. Then look at a
few proofs in Section <A HREF="mpeuni/mmset.html#theorems">4</A> to make
sure you understand how they work.<BR>
Knowledge of mathematics
can be helpful, although it isn't strictly necessary to be able to
mechanically follow the proofs on this site. If you want to start
acquiring a higher-level understanding, a
nice independent introduction to logic is Hirst and
Hirst's <A
HREF="http://www.appstate.edu/~hirstjl/primer/hirst.pdf"><I>A Primer
for Logic and Proof</I></A> <!-- [retrieved 27-Sep-2017] (PDF, 0.5MB)-->; <A
HREF="mpeuni/mmset.html#traditional">compare</A> its axioms to ours.
Wikipedia has an overview of <A
HREF="http://en.wikipedia.org/wiki/Set_theory">set theory</A>.
<!-- [retrieved 4-Aug-2016]-->
The video series
<A HREF="https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX">"Introduction to Higher Mathematics" by Bill Shillito</A>
<!-- [retrieved 27-Sep-2017] --> may also be helpful.
<BR>
<!--
The <I>Stanford Encyclopedia of Philosophy</I> has an informative
overview of <A HREF="http://plato.stanford.edu/entries/set-theory/">set
theory</A> [external]. You can also check out these <A
HREF="mpeuni/mmset.html#read">reading suggestions</A> that closely
follow our material.
-->
<P><A NAME="tool-types"></A><FONT COLOR="#006633"><B>Q:</B> What kinds
of tools support Metamath?</FONT><BR>
Many tools support Metamath. There are three categories of Metamath tools,
though some tools belong to more than one category:
<OL>
<LI><I><A HREF="#proof-assistants">Proof assistants</A></I> help you
interactively create proofs.
<LI><I><A HREF="other.html#verifiers">Verifiers</A></I>
verify that a database is correct - in particular,
that the proofs in a database are correct.
There are over a dozen verifiers available.
Metamath verifiers do not make logical inferences; they just verify
that the proof as stated is correct.
<LI><I><A HREF="other.html#mmtools">Support and other tools</A></I>
perform other Metamath-related tasks.
Some of these tools are <i>also</i> proof assistants and/or verifiers.
For example, the original
<a href="#metamath-exe">metamath-exe</a> tool is a proof assistant,
a verifier, and <i>also</i> provides various support functions such as
generating HTML pages.
<a href="#metamath-knife">Metamath-knife</a>
is a fast verifier in Rust that also has other capabilities.
For more information, see the
<A HREF="other.html#mmtools">list of other tools for Metamath</A>.
</OL>
<P><A NAME="proof-assistants"></A><FONT COLOR="#006633"><B>Q:</B> What
proof assistants support Metamath?</FONT><BR>
To create new Metamath proofs, you'll want to use a proof assistant.
The main Metamath proof assistants are:
<OL>
<li><b><i>mmj2</i></b> - <A HREF="#mmj2">mmj2</A> is currently
the most commonly-used tool. It's written in Java.
David A. Wheeler produced an introductory video, <A
HREF="https://www.youtube.com/watch?v=Rst2hZpWUbU">"Introduction to
Metamath & mmj2"</A>. <!-- [retrieved 4-Aug-2016]-->
<LI><b><i>metamath-lamp</i></b> -
<a href="https://expln.github.io/lamp/latest/index.html"
>metamath-lamp</A>
(Lite Assistant for Metamath Proofs)
is a new Metamath proof assistant by Igor Ieskov
written in JavaScript.
You don't need to install anything to use it, just use your web browser
(including your smartphone web browser) to view the
<a href="https://expln.github.io/lamp/latest/index.html"
>Metamath-lamp application page</A>.
You can learn to use metamath-lamp from the extensive
<A HREF="https://lamp-guide.metamath.org/">Metamath-lamp Guide</A>
by David A. Wheeler.
Software developers may want to see the
<A HREF="https://github.com/expln/metamath-lamp"
>Metamath-lamp source code repository</A>.
<li><b><i>metamath-exe</i></b> - the
<A HREF="mmprog">metamath-exe</A> program is an
ASCII-based ANSI C program with a command-line interface.
This is the original program for processing Metamath; today it's
it's often called "metamath-exe" to distinguish it from other Metamath tools.
<li><b><i>yamma</i></b> -
Yamma is a Metamath proof assistant for Visual Studio Code (VS Code).
It's implemented as a VS Code extension for Metamath.
See the <A HREF="https://github.com/glacode/yamma"
>Yamma source repository</A> for more information.
</OL>
<p>
In addition,
<A NAME="mmpp"></A><i>mmpp</i> is an experimental
proof editing environment for the
Metamath language by Giovanni Mascellani.
See the
<A HREF="https://github.com/giomasce/mmpp">mmpp GitHub project page</A>
for how to build it under Linux, MacOS, and Windows. See also its <A
HREF="https://groups.google.com/d/msg/metamath/UsqyBvemGoY/c82rZAh2BQAJ">Google
group announcement</A>.
<!-- Solitaire requires Java applets, which are generally unsupported:
You can experiment with simple
proofs in the <A HREF="mmsolitaire/mms.html">Metamath Solitaire</A>
applet.
-->
<P><A NAME="learn"></A><FONT COLOR="#006633"><B>Q:</B> Will Metamath
help me learn abstract mathematics?</FONT><BR>
<B>A:</B>
Yes, but probably not by itself.
Metamath uses a single, simple <A
HREF="mpeuni/mmset.html#proofs">substitution rule</A> that allows you to
follow any proof <I>mechanically</I>. You can actually jump in anywhere
and be convinced that the symbol string you see in a proof step is a
consequence of the symbol strings in the earlier steps that it
references, even if you don't understand what the symbols mean. But
this is different from understanding the <I>meaning</I> of the
math that results. Metamath by itself probably will not give you an
intuitive feel for abstract math, in the same way it can be hard to
grasp a large computer program just by reading its source code, even
though you may understand each individual instruction.
However, Metamath combined with other materials can be powerful for
learning.
The <A
HREF="mpeuni/mmbiblio.html">Bibliographic Cross-Reference</A> lets you
compare informal proofs in math textbooks and see all the implicit
missing details "left to the reader."
<P><A NAME="audience"></A><FONT COLOR="#006633"><B>Q:</B> Who is the
intended audience for Metamath?</FONT><BR>
<B>A:</B> Metamath is not for everyone, of course. A person with no
interest in math may find it boring or, optimistically, might find a
spark of inspiration. Professional mathematicians may view it as a
curiosity more than a tool - they need to do things at a high level to
work efficiently. On the other hand, Metamath can appeal to those who
enjoy picking things apart to see how they work. Others may like the
absolute rigor that Metamath offers. Someone new to logic and set
theory, who is still developing the mathematical maturity needed to
follow informal textbook proofs, may find some reassurance in Metamath's
step-by-step breakdown. And anyone who appreciates the austere elegance
of formal mathematics for its own sake might enjoy just casually
browsing through the proofs for their aesthetic appeal.
<!--
<BR> My highest hope is that someone
completely baffled by the idea of a mathematical proof will be able to
say, "Now I understand the underlying symbol manipulation rule. All the
rest is detail." and thus have all of mathematics opened up to them in
principle, limited only by their patience rather than their inherent
mathematical ability.
-->
<P><A NAME="pink"></A><P><FONT COLOR="#006633"><B>Q:</B> I already have
an abstract mathematics background. How can I grasp the key
ideas in a Metamath proof more quickly?</FONT><BR>
<B>A:</B> On the web page with the proof, look at the little <!-- pink
--> colored numbers in the Ref column. The steps with the largest
numbers are usually the ones you want to look at first. The steps with
smaller numbers are typically logic "glue" to tie them together. The
colors follow roughly the rainbow colors as the statement number
increases, so that the largest numbers tend to stand out from the
others.
<!--This feature, combined with the gray indentation levels showing
the tree structure, should help you figure out a higher-level outline of
the proof more efficiently.
<BR>
-->
With a little practice, this feature,
together with the gray indentation levels showing
the tree structure,
should help you
figure out the "important" steps so that you could
write down an informal version of the proof if
you wanted to.
<!--
When studying a proof written by someone else,
occasionally I find it helpful to print out parts of the
proof and highlight the key steps.
I start from the outermost indentation levels and
focus on the steps that aren't obvious, drilling down to underlying
subtheorems if necessary. Eventually, the big picture will become
apparent, so that I could write down an informal version of the proof if
I wanted to.
-->
By the way, it's best not to use the colored numbers
to refer to theorems in an archived discussion, since they change
when new theorems are inserted at an earlier point in the database.
Instead, just use their names.
<P><A NAME="language"></A><FONT COLOR="#006633"><B>Q:</B> What does the
Metamath language look like?</FONT><BR>
<B>A:</B> The precise technical specification of the language is given
in Section 4.1 (p. 112) of the <A HREF="#book"><I>Metamath</I> book</A>
and is about 4 pages long. A simple example is given in Section 2.2.2 (p. 40).
Compare this <A HREF="screen1.html#kedit">source screenshot</A> with
the <A HREF="mpeuni/2eu5.html">generated web page</A>.
As an indication of the language's simplicity,
Raph Levien independently wrote the remarkably small <A
HREF="#mmverify">mmverify</A> proof verifier in Python. He writes,
"I find the whole thing a bit magical. Those 300 lines of code, plus a
couple dozen axioms, effectively give you the building blocks for all of
mathematics."
Bob Solovay wrote a nicely commented
presentation of Peano arithmetic in the Metamath language, <A
HREF="metamath/peano.mm">peano.mm</A>, that is worth reading as a
stand-alone file. <BR>
<BR>
However, <I>you don't have to know or even look at the language</I>
if you just want
to follow the proofs (you can just read pages on this website)
or create new proofs (you would typically use a
<A HREF="#proof-assistants">proof assistant</A>).
<BR>
<!--
An ongoing forum with questions and
answers about the Metamath language can be found at Asteroid Meta's <A
HREF="http://wiki.planetmath.org/cgi-bin/wiki.pl/metamathMathQuestions">
metamathMathQuestions</A> page.
-->
<P><A NAME="otherpgms"></A><FONT COLOR="#006633"><B>Q:</B> What other
programs have been written for the Metamath language?</FONT><BR>
<B>A:</B> Over a dozen proof verifiers for the Metamath language have
been written and are listed at
<A HREF="other.html#verifiers">Known Metamath proof verifiers</A>.
Also, several proof languages have been based on Metamath, and
the software and other documentation for these can be found under
<A HREF="other.html#mmrelated">Metamath-related programs</A>.
<!--
The following programs can verify the proofs in a Metamath
database file (see also
<A HREF="other.html#verifiers">archived versions</A>):
(1) the original <A HREF="#mmprog">Metamath program</A>
(written in C by Norman Megill);
(2) <A
HREF="https://web.archive.org/web/20131219001737/http://wiki.planetmath.org/cgi-bin/wiki.pl/mmj2">mmj2</A> [external]
(written in Java by Mel O'Cat and enhanced by Mario Carneiro,
with a GUI proof assistant that
optionally interfaces with the Metamath program's CLI proof assistant
via the <A HREF="#eimm">eimm</A> program);
(3) <A HREF="https://github.com/getzdan/Metamath.jl">Metamath.jl</A>
[retrieved 12-Jun-2016] (written in <A
HREF="https://en.wikipedia.org/wiki/Julia_(programming_language)">Julia</A>
by Dan Getz);
(4) <A HREF="downloads/mmamm.m">mmamm.m</A> (written in 74 lines,
2885 bytes of Mathematica by Mario Carneiro);
(5) <A HREF="https://github.com/sorear/smm">smm</A> [external]
(written in JavaScript by Stefan O'Rear);
(6) <A HREF="http://mm.ivank.net/">MM Tool</A>
(written in JavaScript by Ivan Kuckir);
(7) <A HREF="https://github.com/Drahflow/Igor">Igor</A> [external]
(written in a custom language by Drahflow; in progress);
(8) <A HREF="#mmverify">mmverify</A>
(written in 350 lines of Python by Raph Levien);
(9) <A
HREF="http://home.solcon.nl/mklooster/repos/hmm/">Hmm</A> [external]
(written in 400 lines of Haskell by Marnix Klooster);
(10) <A
HREF="http://www.fiit.stuba.sk/~kiselkov/Metamath/verify.lua">verify</A>
[external] (written in 380 lines of Lua by Martin Kiselkov; needs 40 min
to verify set.mm, but provides an interesting example for learning Lua);
(11) <A
HREF="http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs">Verifer</A>
[external] (written in 550 lines of C# by Chris Capel);
(12) <A HREF="downloads/checkmm.cpp">checkmm.cpp</A>
(written in C++ by Eric Schmidt);
(13) <A HREF="https://github.com/sorear/smetamath-rs">smetamath-rs</A>
[external] (written in Rust by Stefan O'Rear).
The following
program provides a graphical user interface for displaying the output of
the Metamath program commands:
(14) <A
HREF="http://wiki.planetmath.org/cgi-bin/wiki.pl/mmide">mmide</A>
[external] (written in Python by William Hale).
<BR>
In addition, the
following programs verify proofs in their own languages derived from
Metamath:
(15) <A
HREF="http://ghilbert-app.appspot.com">gh_verify</A>
[external] (for the Ghilbert language; written in Python by Raph
Levien);
(16) <A
HREF="http://home.alamedanet.net/~dan.krejsa/shullivan/shullivan.html">Shullivan</A>
[external] (also for the Ghilbert language; written in C by Dan Krejsa;
loads and verifies the translated set.mm in 500 ms);
(17) <A
HREF="http://wiki.planetmath.org/cgi-bin/wiki.pl/Bourbaki_proof_checker">Bourbaki</A>
[external] (for a custom Lisp-like language; written in Common Lisp by
Juha Arpiainen);
(18) <A
HREF="http://wiki.planetmath.org/cgi-bin/wiki.pl/JHilbert">JHilbert</A>
[external] (written in Java by
Alexander Klauer), a proof verifier for collaborative theorem proving "in the
spirit of Ghilbert" and the engine behind <A
HREF="http://www.wikiproofs.org">Wikiproofs</A> [external]; and
(19) <A
HREF="http://russellmath.org/">Russell</A>
[external] (written in C++ by
D. Yu Vlasov), built upon Metamath
as a high level superstructure with an
automatic proving facility, described in a
<A HREF="http://zbmath.org/?q=an:06055320">paper</A>
[external] (in Russian) and reviewed
<A HREF="http://slawekk.wordpress.com/2012/08/19/the-russell-proof-language">here</A>.
-->
<P><A NAME="confidence"></A><FONT COLOR="#006633"><B>Q:</B> How confident
can I be in the proofs?</FONT><BR>
<B>A:</B> You can be extremely confident that the proofs follow from
their axioms.
All reasoning is done directly in the proof itself
rather than by algorithms embedded in the verification program.
Computer verification programs never get tired and rigorously check every step.
There is the risk that a verifier has a programming bug, but this
is countered by the Metamath language's small size
(this simplicity reduces the likelihood of such bugs) and
by using multiple independently-implemented verifiers
(since it is unlikely that all verifiers will have the same kind of bug).
For example, the
<A HREF="mpeuni/mmset.html">Metamath Proof Explorer</A>
is routinely checked by 5 independently-created verifiers in 5
different programming languages:
<A HREF="#mmprog">metamath</A> (a C verifier by Norm Megill),
<A HREF="#mmj2">mmj2</A> (a Java verifier by Mel O'Cat and Mario Carneiro),
<A HREF="other.html#smetamath-rs">smetamath-rs</A>
(a high-speed Rust verifier by Stefan O'Rear),
<A HREF="other.html#checkmm">checkmm</A> (a C++ verifier by Eric Schmidt).
<A HREF="other.html#mmyverify.py">mmverify.py</A>
a Python verifier by Raph Levien).
In addition, the databases are public and can easily be inspected;
the hypertext links in generated proofs make it especially easy to move
from one theorem to the next.
Metamath enables an extremely rigorous form of peer review.
<P><A NAME="name"></A><FONT COLOR="#006633"><B>Q:</B> Why is it called
"Metamath"?</FONT><BR>
<B>A:</B> It means "metavariable math." See <A
HREF="mpeuni/mmset.html#mmname">A Note on the Axioms</A>. <!-- See also
the Comment in Section C.2.2 (p. 140) of the <A
HREF="#book"><I>Metamath</I> book</A>. -->
Metamath shouldn't be confused
with metamathematics (occasionally abbreviated metamath, metamaths,
or meta math), which is a specialized
branch of mathematics that studies
mathematics itself, leading to results such
as Gödel's incompleteness theorem. An expert in the latter is
called a metamathematician, so to avoid confusion
one might use "metamathician" for someone knowledgeable about Metamath.
<!--
<P><A NAME="if"></A><FONT COLOR="#006633"><B>Q:</B> The symbol
"<IMG SRC='mpeuni/_if.gif' WIDTH=11 HEIGHT=19 ALT='if' ALIGN=TOP>" shows up
in some set theory proofs such as <A HREF="mpeuni/redivclz.html">this
one</A>. What does it mean?</FONT><BR>
<B>A:</B> See the <A HREF="mpeuni/mmdeduction.html#quick">Deduction
Theorem</A>.
-->
<P><A NAME="other"></A><FONT COLOR="#006633"><B>Q:</B> Are there other
sites that formalize math from its foundations?</FONT><BR>
<B>A:</B> Another project that aims to rigorously formalize and verify
math is <A HREF="http://mizar.org/">Mizar</A>. <!-- [retrieved 4-Aug-2016]-->
It is intended to appeal to professional mathematicians and requires a
certain mathematical maturity to be able to follow its proofs. It tries
to mimic mathematical proofs they way they are normally published,
whereas Metamath shows you every detail.<BR>
Some other well-known interactive
theorem provers are
<A HREF="https://leanprover.github.io/">Lean</A>,
<A HREF="http://www.cl.cam.ac.uk/~jrh13/hol-light/">HOL Light</A>,
<!-- [retrieved 4-Aug-2016] --> <A
HREF="http://www.cl.cam.ac.uk/Research/HVG/Isabelle/index.html">Isabelle</A>,
<!-- [retrieved 4-Aug-2016] -->
and <A HREF="http://coq.inria.fr/">Coq</A>. <!-- [retrieved 4-Aug-2016] -->
Freek Wiedijk wrote an interesting collection of <A
HREF="http://www.cs.kun.nl/~freek/notes/index.html">notes</A>
<!--[retrieved 4-Aug-2016] -->
comparing several mathematical proof languages. His book, <A
HREF="http://www.cs.ru.nl/~freek/comparison/comparison.pdf">The
Seventeen Provers of the World</A>,
<!-- [retrieved 4-Aug-2016] (PDF, 0.6MB), -->
compares the
proofs that the square root of 2 is irrational in 17 proof languages,
including Metamath (theorem <A HREF="mpeuni/sqrt2irr.html">sqrt2irr</A>).
The
<a href="mm_100.html">Metamath 100</a> page shows metamath's progress
in
<a href="https://www.cs.ru.nl/~freek/100/">Formalizing 100 Theorems</a>
(a challenge set of theorems for math formalization systems).
For comparisons, see
<a href="#distinctives">what is distinctive about Metamath</a>.
<!-- We covered this above:
Unlike most other systems, Metamath
attempts to use the minimum possible framework needed to express
mathematics and its proofs. Other systems do not consider that aspect
necessarily important, and their underlying computer programs can be
large and complex.
Metamath's proofs are often quite long compared to those of
other systems, but they are completely transparent with nothing hidden
from the user. All reasoning is done directly in the proof itself
rather than by algorithms embedded in the verification program.
Metamath is unique in this sense, offering an alternative approach for
those attracted to its philosophy of simplicity and transparency.
-->
<P><A NAME="contribute"></A><FONT COLOR="#006633"><B>Q:</B> How
can I contribute to Metamath?</FONT><BR>
<B>A:</B>
We'd be delighted if you decide to contribute!
The Metamath community has several
inter-related projects, so you first need to determine which
specific project you want to contribute to.
Here are some common cases:
<OL>
<LI>If you're contributing to "set.mm" (the set of proofs which starts
from ZFC set theory axioms and shown in the "Metamath Proof Explorer"),
the recommended approach is to use its GitHub repository at <A
HREF="https://github.com/metamath/set.mm">https://github.com/metamath/set.mm</A>
(at least as a starting point). For detailed instructions on using
GitHub for this project, read
<A HREF="https://github.com/metamath/set.mm/wiki/Getting-started-with-contributing"
>Getting started with contributing</A> and
<A
HREF="https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md"
>CONTRIBUTING.md</A>.
In practice you'll probably want to chat on the
<A HREF="https://groups.google.com/forum/#!forum/metamath">Metamath mailing
list</A>.
<LI>If you want to modify the mmj2 program (the editor/GUI proof
assistant written in Java by Mel O'Cat and enhanced by Mario Carneiro),
go to its
<a href="https://github.com/digama0/mmj2">mmj2 GitHub repository</a>.
<LI>If you want to modify the metamath-exe program (the original tool
implementation written in C), go to the
<A href="https://github.com/metamath/metamath-exe">metamath-exe
GitHub repository</A>.
<LI>If you want to modify a webpage, you'll need to determine its source.
Changes to top-level web pages go to the
<a href="https://github.com/metamath/metamath-website-seed"
>metamath-website-seed GitHub repository</a>.
Changes to a specific database go to that database's repository, often the <A
HREF="https://github.com/metamath/set.mm">set.mm GitHub repository</A>.
The Metamath website is generated by running scripts that combine
the current metamath-website-seed and database data. In rare cases you
may want to propose changes to those scripts, in which case you can propose
them to the
<A href="https://github.com/metamath/metamath-website-scripts"
>metamath-website-scripts GitHub repository</a>.
</OL>
<P>
When in doubt, ask or post your proposal to
the <A HREF="https://groups.google.com/forum/#!forum/metamath">metamath
mailing list</A>.
We would be delighted to help you.
If you must contact someone privately, please email
<A HREF="mailto:di.gama@NOSPAMgmail.com">Mario Carneiro</A> or
<A HREF="mailto:dwheeler@NOSPAMdwheeler.com">David A. Wheeler</A>
(after removing the NOSPAM markers from the email addresses).
<P><A NAME="contribute"></A><FONT COLOR="#006633"><B>Q:</B> Who
was Norman ("Norm") Megill?</FONT><BR>
<B>A:</B>
Norman "Norm" Dwight Megill, Ph.D. (1950-2021)
was the original creator of Metamath.
He created the Metamath language, developed the first tool to
support it (now called metamath-exe), and first developed axioms
and proofs in this language.
For over 30 years he cultivated an international community of
people with the shared dream of digitizing and verifying mathematics.
His ideas and design have been influential in formal mathematics.
He had an interest in the properties of Quantum logic and Hilbert spaces,
and used Metamath to formalize his investigations.
Norman received his undergraduate degree in Electrical Engineering
and Computer Science from MIT in 1972 and his Ph.D. from the
University of Zagreb, Croatia in 2010.
He died suddenly of natural causes on December 9, 2021, at the age of 71.
He is
<A HREF="https://groups.google.com/g/metamath/c/mN7iFAWttjA"
>sorely missed by all who knew him</A>
(<A HREF="https://www.legacy.com/us/obituaries/bostonglobe/name/norman-megill-obituary?id=31842140">obituary</A>).
The Metamath system he created lives on and is a fitting
memorial to his extensive contributions.
<HR NOSHADE SIZE=1><A NAME="book"></A><CENTER><B><FONT
COLOR="#006633" SIZE="+1">Metamath book</FONT></B>
</CENTER>
<P>
The book
<B><I>Metamath: A Computer Language for Mathematical Proofs</I></B> (248 pp.),
written by Norman Megill
with extensive revisions by David A. Wheeler,
provides an in-depth understanding of the Metamath language,
the metamath-exe program, and an introduction to the set.mm database.
It is also called the <i>Metamath book</i>.
The first part of the book includes an easy-to-read informal discussion of
abstract mathematics and computers, with references to other proof
verifiers and automated theorem provers.
<P>
The Metamath book is available in many forms:
<UL>
<LI><a
HREF="downloads/metamath.pdf">Metamath book in PDF format</A> (free).
<LI><a
HREF="downloads/metamath-german.pdf">German translation of the Metamath book in PDF format</A> (free).
<LI><a href="downloads/metamath-narrow.pdf">Metamath book narrow width in PDF format</a> (free).
This narrow width form is better when reading on small displays
such as smartphones,
<LI><A HREF="http://www.lulu.com/shop/norman-megill-and-david-a-wheeler/metamath-a-computer-language-for-mathematical-proofs/hardcover/product-24129769.html"
>Hardcover version of the Metamath book
(ISBN 978-0-3597-02237)</A>.
If you want a physical copy, it's easy to get one!
This book was released in 2019 and is labeled second edition.
It's available in many other places too, for example, you can get
<a href="https://www.amazon.com/Metamath-Computer-Language-Mathematical-Proofs/dp/0359702236"
>the Metamath book on Amazon</a>.
However, we recommend
<A HREF="http://www.lulu.com/shop/norman-megill-and-david-a-wheeler/metamath-a-computer-language-for-mathematical-proofs/hardcover/product-24129769.html"
>getting the Metamath book from Lulu</a> if it doesn't matter to you
(it tends to be cheaper and faster to get).
<LI><A HREF="latex/metamath.tex">metamath.tex</A>
is the LaTeX source file for the book.
The comment at the beginning explains the steps for how to compile it.
</UL>
<p>
You can also view the <A
HREF="https://github.com/metamath/metamath-book/blob/master/errata.md"
>Metamath book errata</a>.
The book source is maintained on GitHub at
<A HREF="https://github.com/metamath/metamath-book">https://github.com/metamath/metamath-book</A>,
which also provides an archive of older editions.
<!-- In December 2018, the
predicate calculus axioms were renumbered and no longer
match those in Sections 3.3.2 and 3.3.3 of the book. A cross reference
between the new and old numbers can be found in
<A HREF="mpeuni/mmset.html#oldaxioms">Appendix 8</A> of the
Metamath Proof Explorer Home Page.
-->
<p>
The following BibTeX citation is suggested for the printed version.<p>
<TT><FONT SIZE=-1>
@Book{metamath,<BR>
author = {Norman D. Megill},<BR>
author = {David A. Wheeler},<BR>
title = {Metamath: A Computer Language for Mathematical Proofs},<BR>
year = {2019},<BR>
publisher = {Lulu Press},<BR>
address = {Morrisville, North Carolina},<BR>
note = {{\tt http://us.metamath.org/downloads/metamath.pdf}},<BR>
}<BR>
</FONT></TT>
<!--
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Added</FONT> <FONT
SIZE=-1><I>4-Jul-2011</I></FONT> John D. Baker has kindly provided
(<A
HREF="downloads/metamath_kindle.zip">metamath_kindle.zip</A>)
"a modified version of [the] metamath.tex book source that is formatted
for the Kindle. If you compile the document the resulting PDF can be
loaded into into a Kindle and easily read." The compiled PDF file
is also included.
-->
<HR NOSHADE SIZE=1><A NAME="downloads"></A><CENTER><B><FONT
COLOR="#006633" SIZE="+1">Downloads</FONT></B>
</CENTER>
<UL>
<LI><A NAME="mmprog"></A>
<A HREF="downloads/metamath.tar.bz2">metamath.tar.bz2</A> (14 MB) or
<A HREF="downloads/metamath.tar.gz">metamath.tar.gz</A> (17 MB) or
<A HREF="downloads/metamath.zip">metamath.zip</A> (17 MB)
<UL>
<LI><FONT COLOR="#006633"><I>Description:</I></FONT> The
<B>metamath-exe (aka "metamath program")</B> is an
ASCII-based ANSI C program with a command-line interface.
This is the original program for processing Metamath, but since there
are many other tools it's typically called "metamath-exe" to distinguish
it from other tools.
<!--
<B>(5-Apr-2021) The compiled program for Windows, metamath.exe, has been temporarily
removed due to a false positive in an antivirus program. Contact
Norman Megill directly if you need a copy.</B>
-->
</LI>
<LI><FONT COLOR="#006633"><I>Instructions:</I></FONT> <B>1.</B>
Extract all files, which will be contained
in a directory called "metamath". <B>2.</B> For Windows, double-click on
"metamath.exe" and type "read set.mm". For Linux/MacOSX/Unix,
compile with the command "gcc *.c -o metamath" inside the
"metamath" directory, then type "./metamath set.mm" to run.
<B>3.</B> For
all systems, once in the program, use the "help" command to guide you.
Consult the <I>Metamath</I> book (above) for an in-depth understanding.
<!-- (By the way, the web pages on this site were generated with
"show statement * /html".) -->
<LI><FONT COLOR="#006633"><I>To uninstall:</I></FONT> Just delete the
"metamath" directory. Nothing else on your system was touched by the
installation.
</LI>
<!--
<LI><FONT COLOR="#006633"><I>Notes:</I></FONT><UL><LI><FONT
SIZE=-1><B>Quicker installation for Windows users</B> who just want
the main (set.mm) database: 1. Download the
Metamath program
<A HREF="metamath/metamath.exe">metamath.exe</A> (0.5MB)
2. Download the <A HREF="metamath/set.mm">set.mm</A> database (25MB) into the
same folder.
3. Double-click on
"metamath.exe" and type "read set.mm".