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Metamath is a simple and flexible computer-processable language that supports
rigorously verifying, archiving, and presenting mathematical proofs.
See the FAQ for more information.
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Higher-Order Logic Explorer -
Starts with HOL (also called simple type theory) and derives equivalents to
ZFC axioms, connecting the two approaches.
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Mini FAQ
Q: What is Metamath?
A:
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.
Q: What is
distinctive about Metamath?
Metamath is probably unlike anything you have encountered.
Here are some distinctive traits of Metamath:
- 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.
- Metamath is not tied to any particular set of axioms, instead, axioms are defined in a database.
- Metamath proofs include every step, no exceptions, where each step
is only 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
anyone to follow every proof step.
- Many tools support Metamath, instead of requiring a "canonical" tool.
- Metamath verifiers have been independently
implemented by different people in different programming languages,
reducing the risk of accepting an invalid proof.
- Metamath's fundamental operation (substitution) is easy to understand,
even by those who aren't professional mathematicians.
- 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.
- Metamath is one of the top systems in the
Formalizing 100 Theorems challenge.
The Metamath Proof Explorer (MPE) database alone
has over 23,000 proven theorems.
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.
Q: How can I ask
questions or discuss Metamath-related topics?
A: The Metamath
Google Group mailing list is a great place to discuss
Metamath. If you have questions, that is a good place
to ask them.
Q: Where do I
start?
A: Read Sections 1, 2, and 3 of the Metamath Proof Explorer. Then look at a
few proofs in Section 4 to make
sure you understand how they work.
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 Primer
for Logic and Proof ; compare its axioms to ours.
Wikipedia has an overview of set theory.
The video series
"Introduction to Higher Mathematics" by Bill Shillito
may also be helpful.
Q: What kinds
of tools support Metamath?
Many tools support Metamath. There are three categories of Metamath tools,
though some tools belong to more than one category:
- Proof assistants help you
interactively create proofs.
- Verifiers
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.
- Support and other tools
perform other Metamath-related tasks.
Some of these tools are also proof assistants and/or verifiers.
For example, the original
metamath-exe tool is a proof assistant,
a verifier, and also provides various support functions such as
generating HTML pages.
Metamath-knife
is a fast verifier in Rust that also has other capabilities.
For more information, see the
list of other tools for Metamath.
Q: What
proof assistants support Metamath?
To create new Metamath proofs, you'll want to use a proof assistant.
The main Metamath proof assistants are:
- mmj2 - mmj2 is currently
the most commonly-used tool. It's written in Java.
David A. Wheeler produced an introductory video, "Introduction to
Metamath & mmj2".
- metamath-lamp -
metamath-lamp
(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
Metamath-lamp application page.
You can learn to use metamath-lamp from the extensive
Metamath-lamp Guide
by David A. Wheeler.
Software developers may want to see the
Metamath-lamp source code repository.
- metamath-exe - the
metamath-exe 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.
- yamma -
Yamma is a Metamath proof assistant for Visual Studio Code (VS Code).
It's implemented as a VS Code extension for Metamath.
See the Yamma source repository for more information.
In addition,
mmpp is an experimental
proof editing environment for the
Metamath language by Giovanni Mascellani.
See the
mmpp GitHub project page
for how to build it under Linux, MacOS, and Windows. See also its Google
group announcement.
Q: Will Metamath
help me learn abstract mathematics?
A:
Yes, but probably not by itself.
Metamath uses a single, simple substitution rule that allows you to
follow any proof mechanically. 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 meaning 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 Bibliographic Cross-Reference lets you
compare informal proofs in math textbooks and see all the implicit
missing details "left to the reader."
Q: Who is the
intended audience for Metamath?
A: 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.
Q: I already have
an abstract mathematics background. How can I grasp the key
ideas in a Metamath proof more quickly?
A: On the web page with the proof, look at the little 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.
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.
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.
Q: What does the
Metamath language look like?
A: The precise technical specification of the language is given
in Section 4.1 (p. 112) of the Metamath book
and is about 4 pages long. A simple example is given in Section 2.2.2 (p. 40).
Compare this source screenshot with
the generated web page.
As an indication of the language's simplicity,
Raph Levien independently wrote the remarkably small mmverify 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, peano.mm, that is worth reading as a
stand-alone file.
However, you don't have to know or even look at the language
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
proof assistant).
Q: What other
programs have been written for the Metamath language?
A: Over a dozen proof verifiers for the Metamath language have
been written and are listed at
Known Metamath proof verifiers.
Also, several proof languages have been based on Metamath, and
the software and other documentation for these can be found under
Metamath-related programs.
Q: How confident
can I be in the proofs?
A: 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
Metamath Proof Explorer
is routinely checked by 5 independently-created verifiers in 5
different programming languages:
metamath (a C verifier by Norm Megill),
mmj2 (a Java verifier by Mel O'Cat and Mario Carneiro),
smetamath-rs
(a high-speed Rust verifier by Stefan O'Rear),
checkmm (a C++ verifier by Eric Schmidt).
mmverify.py
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.
Q: Why is it called
"Metamath"?
A: It means "metavariable math." See A Note on the Axioms.
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.
Q: Are there other
sites that formalize math from its foundations?
A: Another project that aims to rigorously formalize and verify
math is Mizar.
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.
Some other well-known interactive
theorem provers are
Lean,
HOL Light,
Isabelle,
and Coq.
Freek Wiedijk wrote an interesting collection of notes
comparing several mathematical proof languages. His book, The
Seventeen Provers of the World,
compares the
proofs that the square root of 2 is irrational in 17 proof languages,
including Metamath (theorem sqrt2irr).
The
Metamath 100 page shows metamath's progress
in
Formalizing 100 Theorems
(a challenge set of theorems for math formalization systems).
For comparisons, see
what is distinctive about Metamath.
Q: How
can I contribute to Metamath?
A:
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:
- 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 https://github.com/metamath/set.mm
(at least as a starting point). For detailed instructions on using
GitHub for this project, read
Getting started with contributing and
CONTRIBUTING.md.
In practice you'll probably want to chat on the
Metamath mailing
list.
- 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
mmj2 GitHub repository.
- If you want to modify the metamath-exe program (the original tool
implementation written in C), go to the
metamath-exe
GitHub repository.
- If you want to modify a webpage, you'll need to determine its source.
Changes to top-level web pages go to the
metamath-website-seed GitHub repository.
Changes to a specific database go to that database's repository, often the set.mm GitHub repository.
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
metamath-website-scripts GitHub repository.
When in doubt, ask or post your proposal to
the metamath
mailing list.
We would be delighted to help you.
If you must contact someone privately, please email
Mario Carneiro or
David A. Wheeler
(after removing the NOSPAM markers from the email addresses).
Q: Who
was Norman ("Norm") Megill?
A:
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
sorely missed by all who knew him
(obituary).
The Metamath system he created lives on and is a fitting
memorial to his extensive contributions.
Metamath book
The book
Metamath: A Computer Language for Mathematical Proofs (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 Metamath book.
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.
The Metamath book is available in many forms:
You can also view the Metamath book errata.
The book source is maintained on GitHub at
https://github.com/metamath/metamath-book,
which also provides an archive of older editions.
The following BibTeX citation is suggested for the printed version.
@Book{metamath,
author = {Norman D. Megill},
author = {David A. Wheeler},
title = {Metamath: A Computer Language for Mathematical Proofs},
year = {2019},
publisher = {Lulu Press},
address = {Morrisville, North Carolina},
note = {{\tt http://us.metamath.org/downloads/metamath.pdf}},
}
Downloads
-
metamath.tar.bz2 (14 MB) or
metamath.tar.gz (17 MB) or
metamath.zip (17 MB)
- Description: The
metamath-exe (aka "metamath program") 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.
- Instructions: 1.
Extract all files, which will be contained
in a directory called "metamath". 2. 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.
3. For
all systems, once in the program, use the "help" command to guide you.
Consult the Metamath book (above) for an in-depth understanding.
- To uninstall: Just delete the
"metamath" directory. Nothing else on your system was touched by the
installation.
-
mmj2.zip
(7.2 MB) (latest version, 2.4.1 26-Jan-2016, maintained by
Mario Carneiro)
mmj2-orig.zip
(Mel O'Cat's last official version, 11-Oct-2011)
https://github.com/digama0/mmj2
(development repository)
- Description: Mel O'Cat and
Mario Carneiro's
mmj2 GUI Proof Assistant for
the Metamath language. Includes thorough file validation and
proof verification, syntactic parsing of Metamath formulas
and many other features.
- Instructions: Download
mmj2jar.zip, unzip and read the
enclosed documentation.
David A. Wheeler produced two introductory
videos "Introduction to
Metamath & mmj2" [retrieved 1-Aug-2016] and
"Creating Functions in Metamath" [retrieved 1-Aug-2016].
Some documentation is also available at the (now archived) Asteroid
Meta wiki
mmj2 [retrieved 24-May-2016].
- Quick startup for Windows:
1. Download mmj2.zip and unzip it (wherever)
2. Copy the mmj2\mmj2jar directory to C:
3. Edit C:\mmj2jar\RunParms.txt (with e.g. Notepad).
3a.
The first line will read "LoadFile,set.mm"; change it if necessary
to point to your set.mm file.
3b.
Add the following 2 lines immediately above the last line that reads
"RunProofAsstGUI" (to improve automation in the proof assistant):
ProofAsstDeriveAutocomplete, yes
ProofAsstUseAutotransformations, yes,no,yes
3c.
Add the following 2 lines to the
end of the file (to ensure set.mm definitions are sound):
RunMacro,definitionCheck,ax-*,df-bi,df-clab,df-cleq,df-clel
*done
4. Edit C:\mmj2jar\mmj2.bat. Change "-Xmx256M" to "-Xmx512M"
(to increase heap space for current set.mm size).
Change "C:\metamath" to a directory that exists (to store .mmp worksheets).
5. Start -> All Programs -> Accessories -> Command Prompt
6. Type: java then ENTER. If the response is "'java' is not recognized...",
you need to install the Java runtime system from
java.com
[retrieved 11-May-2016], then exit and reenter the Command Prompt.
7. Type:
CD C:\mmj2jar
mmj2.bat
- Notes:
- The eimm export-import program
links the mmj2 and Metamath proof assistants
without exiting from either program, giving you the features of both
during proof development.
- The mmj2 directory
listing also has the source code, older releases, and MD5 checksums.
-
mpeuni.tar.bz2 (70 MB) or
mpeuni.tar.gz (140 MB) or
mpeuni.zip (180 MB)
- Description: The complete set of
Metamath Proof Explorer web pages.
Includes the Hilbert Space Explorer and the Metamath Music Page. (Does
not include the GIF version of the pages.)
- Instructions: Extract all files
(around 35,000) into a directory
called "mpeuni". The home page is the file "mmset.html". You will need
about 3.5 GB of free space.
-
qleuni.tar.bz2 (1 MB) or
qleuni.tar.gz (2 MB) or
qleuni.zip (4 MB)
- Description: The complete
set of Quantum Logic Explorer web pages.
- Instructions: Extract all
files (around 1,000) into a directory
called "qleuni". The home page is the file "mmql.html".
-
mmsolitaire.tar.bz2 (0.2 MB) or
mmsolitaire.tar.gz (0.2 MB) or
mmsolitaire.zip (0.3 MB)
- Description: The Metamath
Solitaire web page, compiled Java applet,
and applet source code.
- Instructions: Extract all files
into a directory
called "mmsolitaire". Use the page "mms.html" to run the applet.
-
symbols.tar.bz2 (0.2 MB) or
symbols.tar.gz (0.3 MB) or
symbols.zip (0.8 MB)
- Description: A collection of
over 1,000 mathematical symbols
in the form of transparent GIFs that you can use on your own web pages.
- Instructions: Extract all
files into a directory
called "symbols". The home page is the file "symbols.html".
- mmverify.py
(version of 27-Jan-2013)
(previous version)
- Description: Raph Levien's
independently-written Python
proof verifier for the Metamath language.
- Instructions: See the
comments at the top of the program listing.
- eimm.zip (0.1 MB)
- Description: An experimental
proof export-import program (version 0.08 23-Mar-2021) that
translates incomplete proofs in progress between the Metamath program's CLI Proof Assistant
and Mel O'Cat's mmj2
GUI Proof Assistant, without exiting from either proof
assistant, giving you the features of both assistants during proof
development.
- Instructions: Extract all files
into a directory called "eimm". See the readme.txt file for detailed
instructions. A pre-compiled Windows binary is provided; gcc is required
to compile for Linux/MacOSX/Unix.
Metamath program's Proof Assistant (MM-PA> prompt)
| ^
| |
submit eimmexp.cmd /s submit eimmimp.cmd /s
| |
v |
[*.mmp proof worksheet file]
| ^
| |
File/Open File/Save
| |
v |
mmj2 GUI Proof Assistant
- Status: There are no known bugs.
The development of this prototype is believed to be complete. The only
change in the future might be to incorporate the import-export
algorithms natively as Metamath program commands, for convenience.
Suggestions for other possible features are, of course, welcome.
-
finiteaxiom.pdf (0.2 MB)
- Description: Preprint of the
article "A Finitely
Axiomatized Formalization of Predicate Calculus with Equality," which
provides the theoretical basis for Metamath and is referenced on the
Metamath Proof Explorer pages. [This PDF file was generated from the LaTeX
source file finiteaxiom.tex (0.1 MB).]
The correspondence
between the axioms in this paper and the ones in the
set.mm database is described in
Appendix 8 of the
Metamath Proof Explorer Home Page.
See technical note 1 for
some additional notes.
-
weakd.pdf (0.2 MB)
- Description: The article
"Weaker D-Complete Logics," which
is referenced in the Metamath Solitaire applet.
-
Quantum logic papers
- Description: Several papers
on quantum logic,
orthomodular lattices, and Hilbert space
can be downloaded from here.
-
quantum-logic.tar.bz2 (0.05 MB) or
quantum-logic.tar.gz (0.1 MB) or
quantum-logic.zip (0.1 MB)
- Description: Several
programs (lattice.c, latticeg.c,
beran.c, bercomb.c)
referenced in the papers "Algorithms for Greechie Diagrams" and
"Orthomodular Lattices and a Quantum Algebra."
- Instructions: Extract all
files into a directory called "quantum-logic". See the README.TXT
file therein for instructions on compiling and using the programs.
You will need a C compiler such as gcc.
- Note:
- The above programs are frozen at the versions used
for the papers and will reproduce the papers' results exactly.
Each .c file is a stand-alone program. After compiling (under
Linux/Cygwin/MacOSX/Unix) with "gcc program.c -o program",
type "./program --help" for instructions.
-
metamathsite.zip (171 MB)
- Description:
A mirror of the entire Metamath web site
including all the
downloads listed above (that aren't external links).
This can be useful if you have a slow connection
or want to browse the site off-line. A script builds
the site from source files and requires a Linux/MacOSX/Unix operating
system (or the free Cygwin
[retrieved 4-Aug-2016]
for Windows).
About 9 GB of disk space will be needed.
- Instructions:
Extract all files into a directory called "metamathsite". Go to that
directory then type "./install.sh". This
may take several hours to run. The home page (this page) will be
"index.html".
- In Cygwin, to go to a directory, type
"cd c:/tmp/metamathsite" if your directory (folder) is
C:\tmp\metamathsite.
- On MacOSX, select
the Terminal application from
Applications/Utilities to get to the command line.
- To uninstall: Just delete the "metamathsite" directory.
Nothing else on your system was touched by the installation.
- Notes:
- See the README.TXT file that
accompanies the download for more detailed
instructions.
- Another way to install your local copy is with rsync (on
Linux/MacOSX/Unix or Cygwin). The download will be compressed to about
2GB and automatically expanded to about 3.5GB. Create and go to the
metamathsite directory, then type (including the last period):
rsync -vrltS -z --delete --delete-after
rsync://rsync.metamath.org/metamath .
Rerunning this same command
periodically will also keep your copy updated, downloading only the
files that changed. Note that you need twice the disk space
during rsync, i.e. 7GB. - A third way to install your local copy is with wget (see the Download and Extraction Help below). The full
uncompressed 3.5GB site will be downloaded, so it will take a long time,
depending on your connection speed. Create and go to the metamathsite
directory, then type:
wget -nH --mirror
"http://us.metamath.org/index.html"
- If you would like to set up a mirror site for public access, read
the instructions in mirror.txt.
Note: Some of the links in the section below are obsolete. Let me
know if you have current links. --NM 16-Feb-2013
|
|---|
| Directories |
 Wikipedia |
 Drexel University's Math Forum Internet
Mathematics Library (another
mention) |
 Government of Australia
Education Portal |
 Encyclopædia Britannica "approved
iGuide site" (Oct. 11, 2006) (free
set theory
full text article) |
|
|---|
| Awards |
 The Golden House Sparrow Award:
Site of the Day (Jul. 20, 2000) (check out their eclectic current
page) |
 Scout
Report for Science and Engineering Selection (Jul. 19, 2000) |
 Knot a Braid
of Links "Cool
math site of the week" (Jul. 7-13, 1998) |
 Rated by JARS (Apr. 26, 1998) |
This page was
last updated on 7-Aug-2021.
|
Mirror Site
Selection
