As a current description, by default people work on their own
often-overlapping
topics, with the help of the community answering any questions. I'm
unsure so
I'll leave the question of contributors open.
The project proposed here seems highly interesting. I think the main
part
of the work is simply defining all the functions, since many
concepts haven't
been defined yet.
One potential difficulty is that Metamath currently does not have
much tooling.
So computer-generated proofs are ironically tedious and manual to
prove by
default (not sure if https://us.metamath.org/mpeuni/631prm.html was done
manually; https://us.metamath.org/mpeuni/ax-bgbltosilva.html is also
illustrative).
Computer-generated proofs are rare in math, luckily. But if the plan
is to prove a
very large amount of sequences then I imagine it would be more
efficient to make
the tooling for it first. Or (dun dun dun?) ask the Lean community.
Teamwork
makes the dream work, in this case the sequences are
math-community-wide so
I'd imagine the larger amount of people doing Lean proofs would make
it easier.
I don't see any other strategy than starting with some sequence, and
start seeing
what needs to be done (a lot). As a start, here's A1, informally the
(number of groups of order N), the definition is
(number of groups of order N, mod equivalence)
a1 = ( n e. NN |-> ( # ` ( { g e. Grp | ( od ` g ) = n } /. ~=g ) ) )
On Thursday, November 21, 2024 at 10:54:30 AM UTC-6 Peter Dolland wrote:
Hello Metamath Community,
I'm still new here. So let me explain my intention, what I want
here. My
original motivation comes from combinatorics. I am interested in
the
relationship between the specification of an algebraic structure
(such
as directed or undirected graph, magma resp. groupoid,
semigroup, group,
topological space, partition, etc.) and the sequence that
indicates how
many non-isomorphic instances of the structure there are over a
set with
n elements, where n=0,1,2,3,.... . These sequences are published on
oeis.org <http://oeis.org>: A273, A88, A1329, A27851, A1, A1930,
A41,... . For some of
them, a generating algorithm (in some programming language) is also
given. However, in none of these sequences is a formal
specification of
the underlying algebraic structure given, although this is often
much
simpler than the generating algorithm.
My impression so far is that the desired specifications can be
created
with MetaMath without any effort, as long as they are not already
contained in set.mm <http://set.mm>. A real challenge would
probably be to verify the
specified algorithms against the specifications. It seems to me
that a
basis of theorems would have to be created first. In particular,
Polya's
counting theorem would need to be formalised. But perhaps one of
you
knows what already exists on this topic and can be used.
Irrespective of this, it also seems worthwhile to me to create a
bridge
between MetaMath and OEIS in such a way that MetaMath
specifications are
added to corresponding sequences. The aim here could be to
improve the
findability of sequences. For example, the additional criterion
that the
elements of the carrier set are to be regarded as different
(labelled)
could be achieved by adding an independent complete order
relation. For
example, compare the sequences A88 with A53763, A273 with A2416,
A41
with A110 or A1930 with A798. So you can see the first sequence
is for
each pair the unlabelled version. And the second ones yield the
labelled
version.
I see a need for action that goes far beyond my areas of different
expertises. The question now arises as to whether there are
people in
your community who are willing and able to support me in my
search of a
bridge as mentioned above. I have to admit that I have not yet
delved
very deeply into MetaMath and set.mm <http://set.mm>. If I find
someone here, I really
will have chance in order to acquire the necessary knowledge and
skills.
I am hopefully very convinced that MetaMath makes a decisive
contribution to the further development of mathematics and computer
science by formalising important statements.
With best wishes,
Peter Dolland
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