
Where is the fashionable mathematics? - karlicoss
https://xenaproject.wordpress.com/2020/02/09/where-is-the-fashionable-mathematics/
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dwheeler
Article says:

> 1\. There is a large community of mathematicians out there who simply cannot
> join these communities because the learning curve is too high and much of
> the documentation is written for computer scientists.

> 2\. Even if a mathematician battles their way into one of these communities,
> there is a risk that they will not find the kind of mathematics which they
> are being taught, or teaching, in their own department, and there is a very
> big risk that they will not find much fashionable mathematics.

> My explicit question to all the people in these formal proof verification
> communities is: what are you doing about this?

I think many of the formal proof verification communities are trying to
address these. I will focus on Metamath, especially its set.mm database that
focuses on classical logic + ZFC (
[http://us.metamath.org/mpeuni/mmset.html](http://us.metamath.org/mpeuni/mmset.html)
), but much of the following applies to all of them.

I think the main problem is that too many mathematicians expect computer
systems to have all the capabilities of a well-trained graduate mathematician.
Yet the problem is hard. Computers are much better at some things (they don't
get bored or sleepy), and humans are much better at other things (seeing the
big picture & having insights into how to combine "unrelated" ideas). Much
would be better if the formalization and traditional mathematics communities
had more "meetings of the minds" & communication in general.

Focusing on these questions:

1\. The high learning curve is true for all systems, that's true. To be fair,
mathematics has a high learning curve, you don't learn how to do it in a week.
The problem is that although tools are very good at _verifying_ formalized
proofs, they are not great at coming up with the proofs themselves. I do agree
that the documentation could be improved so "non computer scientists" could do
things more easily. I think much of what's needed is for traditional
mathematicians to engage with the formalization community, to make it clearer
what's missing. There also needs to be work (and funding) on improved tooling,
which implies a need for more funding (I'll discuss that below).

2\. "They will not find much fashionable mathematics." There's a chicken-and-
egg problem here. It takes a while to get proofs up to the "basics" of
mathematics as expected by people work on "fashionable" mathematics. Here I
think the solution is to have many people working to build up those basics.
Take a look at my visualization of Metamath set.mm; note that it took many
years for it to build up, and only when many people joined did it start
seriously growing:
[https://www.youtube.com/watch?v=XC1g8FmFcUU](https://www.youtube.com/watch?v=XC1g8FmFcUU)

The _real_ bottom line is that there really needs to be more funding on tools
& formalized systems. Metamath isn't funded at all to my knowledge. Lean and
Coq have some funding, but nothing like the funding of many other fields. We
should be impressed about how far they've come in spite of that.

A broader problem is that mathematicians typically accept a proof if a paper
_seems_ to be okay to another mathematician. When the math and proofs were
simpler, that was probably fine, and since computers weren't very capable
that's all we could do anyway. But today's math (and proofs) are far more
complex, and it's becoming absurd to leave that as our standard of proof.
Computers are available now; we should be using them.

~~~
kragen
If I understand the context correctly, Buzzard wrote this post as a step
toward solving the problems you state (mathematicians are unconvinced of the
usefulness of proof assistants, many people need to join in to build things up
in a proof assistant, documentation needs work, input from mathematicians is
needed).

------
kragen
Kevin Buzzard is doing really interesting work here, and I think that it's
quite plausible that having fashionable mathematicians formalizing fashionable
math in whatever proof assistant they choose will represent a significant step
forward both for mathematical research and for the practice of programming,
which is unavoidably formal in precisely the way math historically isn't. By
formalizing it to that level, we will substantially increase our intellectual
powers and make vast new fields tractable for formal reasoning, which also
makes them tractable for programming.

I'm not sure whether the beauty-contest winner will be LEAN (as Buzzard is
promoting in this blog post), Metamath, or what. It may turn out that, say,
Isabelle-HoTT or HoTT/Agda or something to pull ahead of LEAN, for example —
certainly HoTT is a lot more fashionable among mathematicians than the CoC,
and that might turn out to be either for a good reason (that manifests in
technical advances in HoTT resulting in easier proofs) or a sufficiently
strong social push to overcome the added friction.

It's sure going to be interesting, though.

~~~
prox
One of these questions I never dared to ask; What does formal mean in the area
of mathematics? Is it like finalizing a formula / proof?

~~~
kragen
I'm not the ideal person to ask, never having published a novel mathematical
result and not having particularly deep knowledge of math, but I understand
"formal" in this context to mean "mechanical". A formal manipulation of
symbols is, as I understand it, one that you can carry out without attaching
any _meaning_ to the symbols you are manipulating.

This is not the same thing as "rigorous", because it's entirely possible for a
formal manipulation to be unsound (with respect to some theory). For example,
suppose you want to prove that a left multiplicative identity is also a right
identity; that is, ∀ _x_ : 1· _x_ = _x_ ⇒ ∀ _x_ : _x_ ·1 = _x_. You can apply
commutativity and rearrange the symbols to get a proof — that's a purely
formal technique, and it's correct and rigorous for, say, complex-number
multiplication. But if you're thinking of some other multiplication
operations, like that of square matrices of some size, or of quaternions,
that's not a rigorous proof, because those multiplication operations aren't
commutative. (But in those cases it turns out to be a correct theorem anyway,
because they _are_ associative.)

Compiling a program is another operation that is formal but rarely rigorous.

The hope of using proof assistants in math is that by _formalizing_ our
reasoning, we can _also_ make it more rigorous, with the aid of computers and
a great deal of cleverness. Most mathematicians are not yet convinced.

~~~
lonelappde
That's not quite right.

Rigor means not hand-waving about things that a skeptic might doubt, not
skipping corner cases, etc.

Formal means doing things like a computer -- strictly symbolic analysis that
doesn't claim to do the impossible task accurately map back to our informal
ideas of what things mean.

Formality is across a chasm -- it's the most accurate kind of math we can do,
but it can't be trusted to say that a formal proof about say complex numbers
actually applies to what you are thinking about when you day "complex
numbers".

As is said, "It is impossible to pass from the informal by purely formal
means."

Here's a better explanation: [https://www.quora.com/What-is-difference-
between-rigorous-an...](https://www.quora.com/What-is-difference-between-
rigorous-and-formal-mathematics)

~~~
kragen
This is in accordance with my understanding, but since you seem to think I was
saying something different, I think your expression of it is clearer than mine
was.

------
zozbot234
Not very surprising. When it comes to formal verification, you get the biggest
bang for the buck (by far) via focusing on what the nLab wiki calls
'synthetic' mathematics, viz. fairly self-contained subfields where the 'rules
of the game' may be somewhat complex in their own terms, but can be stated
without relying on a massive amount of prereqs. 'Fashionable' math tends to be
just the opposite: easy, logically-simple entailments, but building on very
complex prereqs.

It's obvious why formalizing the latter is comparatively hard: you need to
work on the prerequisites first, since your formalization won't be usable
without them! Also, since the formalized-math field is still quite fragmented,
large projects (such as formalizing a big chunk of some basic curriculum) are
discouraged to an even greater extent - quite simply, it can't be assumed that
others will be building upon that work.

------
kenkubota
My comment on Kevin Buzzard's intervention:
[https://groups.google.com/d/msg/metamath/Fgn0qZEzCko/bvVem1B...](https://groups.google.com/d/msg/metamath/Fgn0qZEzCko/bvVem1BZCQAJ)

Link list of the discussion threads: [https://owlofminerva.net/kubota/update-
to-the-foundations-of...](https://owlofminerva.net/kubota/update-to-the-
foundations-of-mathematics/)

------
amvalo
The answer is simple: these systems aren't mature enough to formalize the
modern fashionable math. They need better ergonomics and perhaps better
underlying theory before we attempt that.

~~~
kragen
What would the better ergonomics look like? Do you have an idea what might be
wrong with the theory?

~~~
amvalo
The biggest pain point with the theory was its handling of equality, which
HoTT fixes.

Ergonomically.. well it's hard to describe TBH, the easiest way to see is to
just download one of these systems and try using them. You try to prove a
theorem and everything just ends up taking way longer than you'd expect.
Mostly because you can't gloss over small details the way mathematicians will
do informally. Every small turn of phrase like "for large enough N" or
"without loss of generality" can become dozens of extra lines of code.

~~~
kragen
I didn't mean to doubt your claim — my limited experience is that proof
assistants are totally inscrutable, although I've been inspired by some of the
Lean and Agda stuff I've been seeing lately. I just wanted to ask for your
perspective, since it's probably more informed than mine!

It seems to me that if you want to _formalize_ the small details, you will
necessarily have to do something different with some of those small details,
won't you? Maybe a tactic search can find a formal and rigorous proof without
you having to _write_ those dozens of extra lines by hand, but simply glossing
over them seems like it would defeat the goal of formalization.

------
dang
Related from last year:
[https://news.ycombinator.com/item?id=21200721](https://news.ycombinator.com/item?id=21200721)

[https://news.ycombinator.com/item?id=20909404](https://news.ycombinator.com/item?id=20909404)

------
kalyantm
A little off topic, I'm genuinely interested in diving into Math again. Never
appreciated it in college (Comp Sci Engineering, had the first year with some
engineering maths) but now i really want to get into it again. (Calculas,
trignometry and statistics) Can anyone point me to resources/path on how/where
to begin?

~~~
drchewbacca
If you can program you might like Metamath, it feels quite a lot like writing
code.

Here is the main site.
[http://us.metamath.org/index.html](http://us.metamath.org/index.html)

Here is the book which can help with understanding.
[http://us.metamath.org/downloads/metamath.pdf](http://us.metamath.org/downloads/metamath.pdf)

Here are some tutorials for MMJ2 which is the main proof assistant to use,
[https://www.youtube.com/playlist?list=PL1jSu6GGefBm7RBP0Id2S...](https://www.youtube.com/playlist?list=PL1jSu6GGefBm7RBP0Id2Sa9uyVuyhioAC)

it can be found here,
[http://us.metamath.org/#mmj2](http://us.metamath.org/#mmj2)

Here are some beginners proof exercises which are a good place to start out

[http://us.metamath.org/mpegif/mmtheorems289.html#mm28844b](http://us.metamath.org/mpegif/mmtheorems289.html#mm28844b)

I will warn you though it is a bit like the wild west, it is not easy to
accomplish anything and it is exciting to be on the frontier.

The community is really cool, you can chat with them here.

[https://groups.google.com/forum/#!forum/metamath](https://groups.google.com/forum/#!forum/metamath)

------
booleandilemma
An article about fashionable mathematics and no mention of category theory?

~~~
neonate
In his famous talk from a few months ago he dismisses it as not-real-
mathematics. I found it hard to tell whether he was being unironically
provocative, trollish, or just cheeky.

~~~
kevinbuzzard
I was being intentionally provocative. On the other hand I feel like there are
plenty of people in my (mathematics) department who would say that "normal"
fields like geometry, topology, algebra, number theory and analysis are where
the action is happening, and category theory is just a tool which we use to
get "normal" maths done. On the other hand now Scholze is beginning to use
infinity categories more in his work, this might change -- but it might not.
Maybe in 10 years time there will be a book "infinity categories for the
working mathematician" which we all read the first ten pages of and this is
all that most of us need. Note that category theorists like Hyland and
Johnstone have retired from Cambridge now and have not been replaced -- in the
UK now you are more likely to find a category theorist working in a computer
science department than a mathematics department. Whether or not it is "real
mathematics", it is certainly a fact that in the UK at least it is an
extremely small community, whereas our departments are full of number
theorists, geometers, topologists, analysts and algebraists all of whom need
to know essentially no category theory beyond the basic language of adjoint
and representable functors.

------
Koshkin
Looks like Geometric Algebra is something that has been talked about quite a
lot lately. (It is "vector algebra done right.")

~~~
madhadron
Geometric algebra is something that hasn't found a "killer app," if you will.
The largest users of vector calculus are physicists and engineers, and those
communities have 1) enormous existing literature using Gibbs-Heaviside
vectors, and 2) enormous institutional investment in teaching them across
departments. We haven't found a justification for switching that would
outweigh the amount of inertia involved.

It's hardly a new thing. David Hestenes has been writing books about its
advantages in physics since the 1960's.

------
nathias
HoTT so hot right now...

~~~
auggierose
But its inventor is really cold.

------
yters
I feel like mathematicians should make the same effort for non mathematicians.
Why do all these weird terms even matter to anyone else besides a self
selected group of mathematicians? If they don't, why should anyone care about
such things, just as the author asked why mathematicians should care about
formal proof systems? Academia in general is so used to not having to justify
their interests to anyone else that many seem to live in their own isolated
little worlds. Gone are the days when the 'uni' in university meant a unified
realm of knowledge. We should rename 'university' to be 'diversity.'

However, the origin story of academia with Greek philosophy sought to not
merely subsist in rapidly fracturing groups of special interest, but to also
seek the unifying underlying ideas. Similarly with the scholastics in the
medieval era, which actually birthed our university system.

I believe academia has lost its way, which may spell its end. Which is
unfortunate for our civilization, as it is so fundamentally tied to the quest
for wisdom and knowledge.

~~~
miscPerson
Mathematics is modern ontology.

We’re not great at predicting which parts of ontology are eventually useful in
other fields — mostly physics and other hard science, but more recently
computer science, economics + finance, and even things like sociology and
linguistics.

So we let the people who self-select to be ontologists guide what the field
researches — and this has generally been fairly effective. Certainly more
effective than if we’d only looked at things which had immediate, obvious use.
Complex numbers, widely used in science and engineering, were once regarded as
suspect abstract nonsense. That’s why they’re called “imaginary numbers”: it
was a pejorative name that stuck.

We have cryptography, computers, modern physics, and modern finance to show
for our efforts, among other things.

It simply takes time (like, decades to centuries) for new ontological ideas to
propagate to other fields. We’re hoping that formalizing into HoTT and other
computer friendly systems will allow us to align with software development,
and speed the process up.

That seems to be going well, and at an accelerating pace.

The hope is that HoTT and category theory give us a framework to do exactly
what you propose — more easily specify and interlink knowledge.

Expect results around 2050.

It took around 40-60 years for category theory to have a big impact — but now
it is in fields as far away from mathematics as linguistics. Hopefully HoTT
will get there a little faster, but it’s still going to take decades to go
from niche research to widely used in mathematics to widely used across
disciplines.

So, to summarize:

1\. Because we’re bad at predicting the future and abstract math has often
turned out to be useful later.

2\. Mathematics is trying to do exactly what you propose with knowledge, via
exactly the programs this blog is talking about.

~~~
yters
I disagree that all knowledge, or even the most important items of
knoknowledge, are reducible to mathematics.

