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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
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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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list</A>, open problems, other downloads, and miscellany. Filip
Cernatescu's <A HREF="other.html#milp">Milpgame</A> and practice
problems, and also his <A HREF="other.html#xpuzzle">XPuzzle</A>
Android app.
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<A HREF="mpeuni/mmhil.html"><B>Hilbert Space Explorer</B></A> -
<!-- The Hilbert Space Explorer --> Extends ZFC set theory into Hilbert space,
which is the foundation for quantum mechanics. Includes over
1,000 complete formal proofs.
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<A HREF="qleuni/mmql.html"><B>Quantum Logic Explorer</B></A> -
Starts from the orthomodular lattice properties proved in the
Hilbert Space Explorer and takes you into
quantum logic with around 1,000 proofs.
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<A HREF="mmsolitaire/mms.html"><B>Metamath Solitaire</B></A> - A Java
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that demonstrates simple proofs.
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Built-in axiom systems
include ZFC; modal, intuitionistic, and quantum logics; and Tarski's
plane geometry.
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<A HREF="symbols/symbols.html"><B>GIF and PNG Images for Math Symbols</B></A> -
A copyright-free collection of over 1,000 bit-mapped images for
math symbols.
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</A>
</TD></TR></TABLE></TD></TR></TABLE>
</TD>
</TR></TABLE></TD></TR></TABLE>
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<TABLE WIDTH="100%" BORDER=0 CELLPADDING=1 CELLSPACING=0 BGCOLOR="#ACE1AF">
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<TD WIDTH="100%"> <!-- FAEEFF -->
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<TABLE WIDTH="100%" BORDER=0 CELLPADDING=3 CELLSPACING=0 BGCOLOR="#FFFFFF">
<TR>
<TD WIDTH="100%">
<A HREF="mpeuni/mmmusic.html"><B>Metamath Music Page</B></A> - Strictly
for fun. You can listen
<!--
<A HREF="mpeuni/peano2.mid"><IMG
SRC="mpeuni/_note.gif" ALT="MIDI file" BORDER=0 WIDTH=10 HEIGHT=15><FONT
SIZE=-2>0:16</FONT></A>
-->
<!--
<A HREF="mpeuni/sqrth-fshbi.mid"><IMG
SRC="mpeuni/_note.gif" ALT="MIDI file" BORDER=0 WIDTH=10 HEIGHT=15><FONT
SIZE=-2>0:41</FONT></A>
-->
to what mathematical proofs "sound" like!
</TD>
<TD VALIGN=TOP>
<TABLE BORDER=0 CELLPADDING=1 CELLSPACING=0 BGCOLOR="#ACE1AF">
<TR>
<TD>
<TABLE BORDER=0 CELLPADDING=3 CELLSPACING=0 BGCOLOR="#FFFFFF">
<TR>
<TD>
<A HREF="mpeuni/mmmusic.html">
<IMG SRC="_index6mus.gif" HEIGHT=40 WIDTH=60 ALIGN=RIGHT
ALT="Metamath Music Page" TITLE="Metamath Music Page" BORDER=0>
</A>
</TD></TR></TABLE></TD></TR></TABLE>
</TD>
</TR></TABLE></TD></TR></TABLE>
<TABLE><TR><TD HEIGHT=10></TD></TR></TABLE>
<!-- <HR NOSHADE SIZE=1> -->
<!--
<FONT SIZE=-1><I>21-May-2007</I> Some
advanced and difficult <A HREF="award2003.html"> miscellaneous open
problems</A> related to Metamath and other topics on this site.</FONT>
-->
<!--
<P><FORM ACTION="http://us2.metamath.org:8888/thanks.html"
METHOD="GET"><FONT SIZE=-1>Where did you hear about this site (URL is
helpful)? Thanks!</FONT> <INPUT TYPE="TEXT" NAME="where" SIZE=35
VALUE=" "> <INPUT TYPE="SUBMIT" VALUE="Submit"> </FORM> -->
<HR NOSHADE SIZE=1>
<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 tiny language that can express theorems in abstract
mathematics, accompanied by proofs that can be verified by a computer
program. This site has a collection of web pages generated from those
proofs and lets you see mathematics developed in complete
detail from first principles, with absolute rigor. Hopefully it will
amuse you, amaze you, and possibly enlighten you in its own special way.
<!--
<BR>
Metamath also has a
<A HREF="http://en.wikipedia.org/wiki/Metamath">Wikipedia entry</A>.
-->
<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</A> [retrieved 22-Sep-2020] mailing list is being used for
discussion about Metamath. If you have questions, that is a good place
to ask them. (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>
You can experiment with simple
proofs in the <A HREF="mmsolitaire/mms.html">Metamath Solitaire</A>
applet.
To actually create real metamath proofs, you'll want to
<a href="#downloads">download</a> a tool.
A common tool is <A HREF="#mmj2">mmj2</A>.
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].
<!--
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="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.
In order to follow a proof in an advanced math textbook, you may need to
know prerequisites that could take years to learn. Some people find
this frustrating. In contrast, 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 quite different from understanding the <I>meaning</I> of the
math that results. Metamath alone 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, 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.
-->
<BR>
(By the way, it's best not to use the colored numbers
to reference theorems in an archived discussion, since they change
when new theorems are inserted at an earlier point in the database.)
<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>. But <I>you
don't have to know or even look at the language</I> if you just want
to follow the proofs on these web pages.<BR>
<A NAME="langverify"></A> The <A
HREF="#mmprog">metamath program</A> and
<A HREF="#mmj2">mmj2</A> are the main tools for working with
the Metamath language. 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>
<!--
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 4 independent verifiers:
<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), and
<A HREF="other.html#checkmm">checkmm</A> (a C++ verifier by Eric Schmidt).
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 little detail.<BR>
Some other well-known interactive
theorem provers are <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].
There are a few languages based on or derived from Metamath, e.g.,
Raph Levien has developed a related language called <A
HREF="http://ghilbert-app.appspot.com">Ghilbert</A> [retrieved 4-Aug-2016]
that strives to improve upon Metamath by guaranteeing the soundness of
definitions and providing features useful for collaborative work.
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="http://www.cs.ru.nl/~freek/100/">Formalizing 100 Theorems</a>
(a challenge set of theorems for math formalization systems).
<BR>
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 in order to perform mathematical reasoning at a higher
level. 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.
<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 to get your contributions!
The Metamath community has a large set of 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>.
As an alternative to submitting GitHub pull requests (if you don't want to
go through that learning curve in the beginning), you can
email patch files (differences) to
<A HREF="http://us.metamath.org/email.html">Norm Megill</A> or
<A HREF="mailto:[email protected]">Mario Carneiro</A> or even
post to the
<A HREF="https://groups.google.com/forum/#!forum/metamath">Metamath mailing
list</A>.
<LI>If you want to patch the mmj2 program (the editor/GUI proof
assistant written in Java by Mel O'Cat and enhanced by Mario Carneiro),
email <a href="mailto:[email protected]">Mario Carneiro</a> and/or get
yourself added to <a
href="https://github.com/digama0/mmj2">https://github.com/digama0/mmj2</a>.
<LI>If you want to patch the metamath.exe program (the original tool
implementation written in C), send your patch as a "unified diff" ("diff
-u") via email to <A HREF="email.html">Norm
Megill</A>.
<LI>If you want to modify a web page,
send email to <A HREF="http://us.metamath.org/email.html">Norm Megill</A>.
</OL>
When in doubt, ask or post your proposal to
the <A HREF="https://groups.google.com/forum/#!forum/metamath">metamath
mailing list</A>,
and/or privately email
<A HREF="email.html">Norm Megill</A> and
<A HREF="mailto:[email protected]">Mario Carneiro</A>.
<HR NOSHADE SIZE=1><A NAME="downloads"></A><CENTER><B><FONT
COLOR="#006633" SIZE="+1">Downloads</FONT></B>
</CENTER>
<UL>
<LI><A NAME="book"></A><A
HREF="downloads/metamath.pdf">metamath.pdf</A>
(1.3 MB)
<UL>
<LI><FONT COLOR="#006633"><I>Description:</I></FONT> 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
and program.
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.
</LI>
<LI>
A <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 <I>Metamath</I> book
(ISBN 978-0-3597-02237)</A> is also available if you prefer
a printed copy.
This was released in 2019 and is labeled second edition.
</LI>
<LI>A large print and narrow width version of the book,
suitable for reading on small devices such as smartphones, is <A
HREF="downloads/metamath-narrow.pdf">metamath-narrow.pdf</A>. This
version updates the Kindle version provided by John D. Baker in 2011.
</LI>
<LI>You can also view the <A
HREF="https://github.com/metamath/metamath-book/blob/master/errata.md"
>Metamath book errata</a>.
</LI>
<LI>
The
LaTeX source file for the book is <A
HREF="latex/metamath.tex">metamath.tex</A>; the comment at
the beginning explains how to compile it.
The source is maintained on GitHub at
<A HREF="https://github.com/metamath/metamath-book">https://github.com/metamath/metamath-book</A>
[retrieved 6-Feb-2019], which also provides an archive of older editions.
</LI>
<!-- 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.
-->
<LI>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>