
Some Fundamental Theorems in Mathematics - postit
https://arxiv.org/abs/1807.08416
======
throwawaymath
The second theorem (the generalized Pythagorean theorem, which uses inner
product spaces) feels incomplete to me. The exposition is fine, but I don't
think the explanation should have concluded with the regular

    
    
        a^2 + b^2 = c^2
    

equality. You wouldn't use that in the context of an inner product space. The
generalized Pythagorean theorem actually looks like

    
    
        ||x + y||^2 = ||x||^2 + ||y||^2 + 2<x, y>.
    

I think it would be better if there's less editorializing of the equalities
and more relating of their abstract forms to their "simple", familiar forms.
Otherwise I don't really understand the exercise - if the first page has a
theorem involving inner product spaces, this is clearly targeted at people
with at least a full course of linear algebra under their belt. Unless the
audience is already aware of how that second equality implies the first
equality, this explanation doesn't capture the heart of it.

As another commenter said, this is not a self-contained exposition (and I
don't think it realistically could be). But if it's not going to be self-
contained, I think it could be improved by more completely showing how the
elegant abstractions imply the things we're already familiar with in a neat
way.

~~~
andrepd
The Pythagorean theorem is about orthogonal vectors, so it would just be

    
    
      |x + y|^2 = |x|^2 + |y|^2
    

Plus the line just above that tells you: let

    
    
      a = |x|, b = |y|, c = |x-y|
    

I don't think they are obscuring anything. I think they are showing how the a
b and c in the familiar a^2+b^2=c^2 can be generalised to |x|, |y|, |x-y| for
any orthogonal vectors x and y in an inner product space.

------
skh
Number 20 in logic is incorrect. It’s surprising how often this is misstated.
There are lots of complete axiom systems. Indeed, take as your axioms all true
statements in a given model. Voila, you now have a complete axiom system. It’s
not just a useful one but it is complete. What isn’t complete is a recursively
enumerable axiom system that contains arithmetic of the standard model of
integers.

Overall it is a nice list and well done.

EDIT: Deleted a sentence that was pointed out to me that was wrong in the
context of the assumptions made in the paper.

~~~
romwell
>Number 20 in logic is incorrect.

>What isn’t complete is a recursively enumerable axiom system that contains
arithmetic of the standard model of integers.

That's exactly what #20 says. The "axiom system" there, by definition,
includes Peano axioms (2nd sentence of #20).

~~~
skh
I didn’t read that part but the statement is still wrong. Take as your model
the standard model of integers. Collects all true statements of this model.
Make this your axiom system. It is now complete and contains the first order
Peano Axioms.

~~~
gnulinux
Then that system is not finitely axiomatizable (there are infinitely many
truths in that model) so the original claim is still true. It's true for all
finitely axiomatized theories.

~~~
andrepd
Not finite, but recursively enumerable, no? Since you can just start from PA
and derive conclusions forever, so this is an effective procedure in that any
true statement will eventually be thus produced.

~~~
krusch
No, even if you start with the axioms of PA and enumerate all theorems
provable, you'll miss some true sentences, one of them being the Goedel
sentence of PA.

------
thearn4
I came into the link ready to nitpick the choices (for some reason which I
really should examine), but it's actually a nice birds-eye-view of the field
that is rare to see. I feel like mathematicians, perhaps more than most, tend
to stay siloed within our specialized subtopics.

~~~
gowld
I belive it's because math is such a "clean" field that it doesn't take many
people to 'clear' a topic and advance a subfield or branch into sub-subfields.
So you don't have a large number of people all rooting around in the same area
speaking the same language.

With complex sciences like biology, or worse, wet lab experimental science, it
takes so much drudgery just to get data to work with, that you can have 100x
people working in the same area, trying to cover the data/processing
requirements to support theorizing and cataloging discovered entities.

------
tmyklebu
This list is fun, but it's a little bit sloppy:

\- The central limit theorem relies on the mean and variance of your random
variable existing; there are random variables for which mean and variance
don't exist.

\- The definition of "measure-preserving" in the ergodic theorem statement is
missing the measure.

\- dim (ran A)^perp = dim ker A^T can be strengthened by dropping the dim's.
If v in ker A^T, then A^T v = 0. Pick any Ax in ran A; then <v, Ax> = <A^T v,
x> = <0, x> = 0, so v is in (ran A)^perp.

\- Others pointed out that 20 looks busted.

\- The definition of Haar measure needs to fix the measure of some nonzero-
measure set, or "unique" needs to be replaced with "unique up to scaling"
otherwise the theorem isn't true.

\- "Sounders Mac Lane"

\- 25: x_0 came out of nowhere; it's unclear from the text which "invertible"
is meant; the basin of convergence for Newton's method for finding a local
continuation can be very small indeed.

\- 36: Why is the adjoint of A named T^*?

\- 46: You need some assumptions about f.

~~~
FabHK
I think you're being a bit harsh.

The central limit theorem does hold for iid rv (independent identically
distributed random variables) with finite mean and variance. Now, those
assumptions can be relaxed (the rv need not be independent, but they can't be
too dependent, and they need not have finite variance, but they can't be too
"far out"), and some of the pertinent proofs are only a few decades old; but
you can hardly expect a survey with 135 proofs to cover all the subtleties.

Some of the other points may be more egregious howlers, but, again, come on -
this is not the definite reference for any one of those theorems.

~~~
tmyklebu
There's more than enough neat stuff I don't know in there. Unfortunately it's
tough to trust a source with so many errors in the stuff I do know.

I wasn't aware of any CLT for iid random variables with infinite variance. Do
you have references?

~~~
FabHK
Check Valentin Petrov _Limit Theorems of Probability Theory_.

Here [1] is a CLT for RV with infinite variance, Prop 3.1.12, but notice the
(larger) scaling coefficient (1/sqrt(n log n)).

Also see the second answer on SO here [2].

[1]
[https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes....](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)

[2] [https://stats.stackexchange.com/questions/169611/the-role-
of...](https://stats.stackexchange.com/questions/169611/the-role-of-variance-
in-central-limit-theorem)

EDIT to add: Having said that, the Lindeberg-Feller and the Lyuapunov
formulation of the CLT do require finite variance, so maybe I was too quick in
stating that that assumption can be relaxed.

------
Topolomancer
I love Oliver Knill! Not only does he have very broad mathematical interests,
he also has this crazy YouTube Channel
([https://www.youtube.com/user/knill101/videos](https://www.youtube.com/user/knill101/videos))
where he sometimes uses a synthetic voice (!) to give talks---supposedly
because he cannot talk as fast as he wants to.

He really is quite an interesting charactern!

------
triska
Thank you for sharing! One thing I noticed is the statement in 20. Logic:

 _An axiom system is neither complete nor provably consistent._

I think this would benefit from some rephrasing, because such a system could
for example be both complete and provably consistent (in the sense given in
the paper) if it were, in fact, inconsistent.

Also, IUT is not "Inner-Universal Teichmuller", but rather _inter_ -universal
Teichmüller theory.

~~~
btilly
That one also bothered me, because it is very much wrong.

The correct result is

 _Any consistent axiom system that describes the natural numbers cannot be
complete or provably consistent._

You pointed out that an inconsistent axiom system is neither. And at the
opposite extreme, Gödel proved that the axioms for the real numbers are
provably consistent. (However those axioms do not allow for induction, and
therefore do not allow one to describe the natural numbers!)

~~~
skh
This is incorrect as well. Take the standard model of the integers. Now take
all true statements in this model. Let this be your axiom system. It is
complete. It is not recursively enumerable and therefore not very useful. You
can also have second order systems that are categorical. For instance the
second order Peano axioms. I believe they are complete.

~~~
btilly
Point.

But honestly, nobody ever convinced me that second order logic wasn't made up
BS. :-)

Or that reasoning of a form that can't be verified, even in theory, actually
is sensible. (Why yes, I do have Constructivist tendencies, why do you ask?)

~~~
skh
Before I took Mathematical Logic in graduate school I believed all kinds of
nonsense about Godel’s theorems. Even after the class I still had false ideas
about what it really meant. It took a while for the the recursively enumerable
part to sink in as to why it is important.

I’ve heard what you say about second order axiom systems before but I don’t
know enough about the subject. I naively think, “Why not just use the Second
Order Peano Axioms?”. But people far more knowledgeable than me don’t like
them so I defer to their judgement.

~~~
btilly
I don't qualify as far more knowledgeable than you, in fact I'm probably less
knowledgeable but with a different point of view.

However with that disclaimer, the fundamental challenge is that when we start
reasoning about reasoning, things that "should obviously work" run into
trouble. (Most commonly due to variations on the liar's paradox.)

First order reasoning within a first order logic system that we think is good
is "obviously correct" reasoning.

Second order reasoning introduces as much reasoning about reasoning as we
think we can get away with, hopefully without causing problems. The result is
that second order reasoning lets us talk about what kinds of unverifiable
statements we can discuss the absolute truth of. But I'm uncomfortable with
talking about the absolute truth of any unverifiable statements, which makes
it feel like discussing the subtle shades of BS we fool ourselves into
believing.

------
graycat
Gorgeous.

IIRC S. Eilenberg once said "Elegance in mathematics is directly proportional
to what you can see in it and inversely proportional to the effort required to
see it.". This Knill book is _elegant_.

Topic by topic for a wide range of pure and well polished applied math, Knill
is able and does go right to the most important results and for each gives a
short outline of the needed definitions and of a proof, usually with good
references to full details, and then states the theorem precisely. Net, for
nearly any topic in pure/applied math, go there first.

For many of the topics, I studied them carefully from some of the best
sources, but just on a short scan this Knill book often has a better
treatment, e.g., has a super nice, simple, practical statement of a
fundamental theorem in Fourier series, a super nice statement of the Lebesgue
decomposition from the Radon-Nikodym theorem, some nice stuff from Zorn's
lemma, and much more, e.g., often some historical notes.

I have to like the connection with Zorn's lemma: At one point I went to a
lecture at Indiana University and to the tea before, and an older professor
introduced himself as Max Zorn. Since the previous summer I'd taken a course
in axiomatic set theory on an NSF summer program at Vanderbilt, I was nearly
floored! All I could do was blurt out, "What did Paul Cohen prove?". The next
day Zorn gave me his copy of Cohen's paper. I still have it!

At one point I saved FedEx by pleasing two guys from crucial investor General
Dynamics by doing a revenue projection with the differential equation

y'(t) = k y(t) (b - y(t))

with y(0), k, and b given. So, the solution is a _lazy S_ curve that starts at
t = 0 at y(0) and as t increases rises, has an inflection point, and rises
from below asymptotically to b. So, it is a case of growth to a market size of
b. With y(t) the current revenue and (b - y(t)) the market revenue yet to be
obtained, has the growth rate y'(t) proportional to the current revenue y(t)
(number of current customers talking) and the remaining customers (hearing the
talking) (b - y(t)). So, it's a simple model of _viral_ growth.

Well, Knill has this differential equation with b = 2 for a case of biological
growth!

------
romwell
A supremely amazing collection! Succinctly written, and a joy to read. The
curation is fantastic. Some would disagree with the choices, but the task was
herculean. (e.g. geometric group theory is not really represented there - but
one can't expect to have all of mathematics in 60 pages!)

I ended up going through the list, categorizing these theorems into know/don't
know/should know categories. Time well spent.

\-----------

Mandatory "Hey, there's a typo there" note:

In #77: HOMFLYPT polynomial's list of authors is missing Y: David N. Yetter, a
professor in Kansas State University. Hope this omission will be fixed!

I learned knot theory and did research with Yetter in 2007 (which resulted in
a paper on Vassiliev Invariants, a subject also mentioned in #77); that was a
pivotal point in my life.

Besides mathematics, Yetter's other major interests, to my knowledge, are
Orthodox Christianity and anime.

------
uptownfunk
As a math major, this makes me giddy. It's like a family reunion, all my old
friends in one place.

------
khawkins
The Control Theory (129) section does a real disservice to the field by
insinuating it is little more than the study of Kalman filters. While an
important model, the core results of control theory are without a doubt
stability criterions.

The Lyapunov stability criterion is even quite elegant, despite being
powerful: A system is asymptotically stable if there exists a function V where
for x/=0, V(x)>0 and [Sum]_i (dV/dx_i) (dx_i/dt) < 0.

~~~
jjoonathan
Every single subject in the overview is summarized with similar brutality, yet
it still reads like an encyclopedia. There's a lot of math out there.

------
nilkn
It's amazing to see so many things included and yet so many missing. It really
speaks to the breadth and depth of the subject. For instance, for elliptic
curves, only a single result is mentioned (their points can be endowed with an
abelian group structure), and it's really one of the most basic ideas known
about these objects.

~~~
jjoonathan
Right. If I pick one of these subjects where I know, say, 100x more than what
is present, and know of the existence of 1000x more than is present, I can't
help but multiply the number of subjects in this overview by that that same
1000x, and then my mind explodes.

------
js8
Other lists in the similar spirit:

[http://www.cs.ru.nl/~freek/100/](http://www.cs.ru.nl/~freek/100/)

[https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK)

------
maxiepoo
The Topos theory example is completely wrong though? > For example, if E is
the topos of sets, then the slice category is the category of pointed sets:
the objects are then sets together with a function selecting a point as a
“base point”.

There's no object X where E/X is pointed sets. If you take the co-slice of a
one point set 1\E that would be the category of pointed sets.

The right way to think about E/X is that its objects are "families of sets
indexed by X", because a function f : Y -> X defines for each x in X a set
f^{-1}(x). If you know about dependent types, this is basically a type
dependent on X.

------
prostitute
Fundamental Theorem of Group Homomorphisms:

If f is a homomorphism from a group G_1 onto the group G_2, then G_1/ker(f) is
isomorphic to G_2

...and that's the tea.

------
waynecochran
The Church-Turing Theorem is not really a mathematical theorem; It is more
like F = Ma in Physics. It's an empirical result.

~~~
kccqzy
The statement that "f is generally recursive iff it is Turing computable" is a
theorem that can be proven, because we can rigorously define what it means to
be Turing computable (computable by a Turing machine which is precisely
defined) and generally recursive (which is the smallest set of functions
closed under projection, composition, primitive recursion and minimalization).
The theorem was proven by Turing and Kleene soon after the concept of Turing
computable was introduced.

But if you remove the word "Turing" it is no longer a theorem. It becomes a
thesis because we don't have a definition of what it means to be computable.
It is in fact an innovation of Turing to identify computable with Turing
computable, but that's by no means universal.

~~~
ccalf
I think it's clearer to understand the Church-Turing thesis as two theses, not
one. Turing's thesis was that (intuitive) computability equals (formalizable)
Turing computability.

So you could completely ignore the recursive functions part (and so, Church's
Thesis) and there would still be the same sort of lay confusion about the
computability thesis.

------
weerd
From #113, Artificial Intelligence

 _The lebowski theorem of machine superintelligence: No AI will bother after
hacking its own reward function._

 _once the AI has figured out the philosophy of the “Dude” in the Cohen
brothers movie Lebowski, also repeated mischiefs does not bother it and it
“goes bowling”._

I'm relieved to learn that this is a theorem. :D

~~~
HONEST_ANNIE
The Lebowski theorem: "No superintelligent AI is going to bother with a task
that is harder than hacking its reward function" is surprisingly deep.

You can see the theory in action in human history. We have drives programmed
by evolution, but we try to to shortcut them. Drugs are deliberate hack for
example. TV, entertainment and games can provide rewards faster than effort
towards real life goals.

We fully understand that there is difference between the drive's purpose and
what we do, but but we don't actually value the purpose of our reward
function, only the reward. Catholic church tries to insist that people should
limit sexual pleasure to the purpose it was created (by god or evolution) but
people don't listen.

------
Retra
>There is no apparent “fundamental theorem” of AI, (except maybe Marvin
Minsky’s ”The most efficient way to solve a problem is to already know how to
solve it.”

Personally, I consider Richard Feynman's method (1. Write down the problem. 2.
Think real hard. 3. Write down the solution.) to be just as fundamental.

The reason being that steps 1 and 3 are actually _very_ important. Step 1 asks
that you model the problem you're solving, otherwise you might solve some
other problem instead. Step 3 asks that you model the domain of solutions.
This helps ensure that it is possible for solutions to even exist.

One of the big problems with unresolved and open-ended philosophical
quandaries is that people never specify what they expect a solution to look
like. Like if you don't expect a solution to look like a string of plain
words, and instead expect it to look like some mystical revelation, then you
can't honestly expect to get an intelligent resolution communicated to you.

I'd also say it is integral to understanding the difference between simply
solving a problem and intentionally solving a problem.

------
luminati
Naive question: How would one go about learning all about this (or even a
subset of this) for the mathematically inclined working engineer? Where does
one even start? Just take each theorem from each section and try to follow
through the proof? What are the prerequisites required?

~~~
jacobolus
The theorems in this list are in many cases central topics in 1+ semester
courses, and aren’t really worth worrying about out of context. Overall you
are looking at more than 5 years of full-time (40 hours/week) study, maybe
more than 10 years. Unless you have extensive background reading and writing
proofs, self-study is not likely to be very effective. You could conceivably
make your way through with textbooks and the help of an expert tutor/mentor.

For most people the easiest method would be to enroll in an undergraduate
mathematics degree at a decent university.

If you wanted to make your way through more of the topics in this list than an
undergraduate degree covered, you could follow-up by enrolling in a PhD
program.

------
1024core
I think it's a great list but the typos bothered me. To me, they seem to
indicate sloppiness; and if there is sloppiness, could the mathematical
results be sloppy too?

e.g., on page 2 itself:

"Anticipated by _Babylonians Mathematicians_ in examples..."

"Let f be a function of one _variables_..."

------
dclowd9901
Does anyone have a cheat sheet of sorts for the symbols and syntax used in
these?

~~~
jordigh
It doesn't work like that. Some of the same symbols are used in two different
places with vastly different meanings, even within this paper.

To give an imperfect programming language analogy, imagine that every symbol
is implicitly defined and has lexical scope. The grammar is as far from being
context-free as possible. What may sometimes look like individual symbols to
you are in fact a collection of inseparable symbols with its own semantic
parse.

~~~
pantalaimon
That's not really helpful. I struggle with the same problem, in fact the
notation intimidates me to a point where I decided to forego getting deeper
into theoretical computer science and haunts me every time I try to read a
paper.

A resource that would explain formal mathematical notation would be great,
much like a dictionary.

You can't google those symbols properly either.

~~~
jordigh
Did you know what the words meant? Can you read this and except for the
symbols understand everything else?

I don't think the symbols are the problem. They're a very superficial
obstacle, at least in this paper.

This isn't meant as a put-down. It's just that I really do think the actual
material requires a lot more preparation that consists in a lot more than just
learning the definition of the symbols. I could rewrite the whole paper using
only text and no symbols. The symbols are just shorthand for English words. I
don't think if I were to replace all of the symbols with the English words
that they stand for, you would be much closer to understanding.

~~~
dclowd9901
"The symbols are just shorthand for English words."

OK, so then why would it be so difficult to produce a veritable cheat sheet to
aide in reading a proof?

~~~
jordigh
Because the symbols' translation to English words changes on context, culture,
and even author. It's like trying to translate the word "rain" only by hearing
it without having the context if it really meant "reign" or "rein". With some
mathematical training, you don't really need the symbols to know what's being
talked about, because you already have enough context to know what should be
talked about and what fits into the context, regardless of how it's written
down.

Anyway are symbols really the main obstacle for you? When you read "every
second countable regular Hausdorff space is metrizable" (theorem 59 in this
paper), you have no problem understanding what this means?

~~~
dclowd9901
Fair point, but you know very well that, for true illustration, you'd have to
expand out the meaning of the words "countable", "regular", "Hausdorff",
"space" and "metrizable". And I bet you could. I get the sense you're being
obstinate for a point.

~~~
jordigh
And don't forget the word "second". I'd have to explain that one too.

And then I'd have to explain the words I used to explain these words.

It all just takes a while. I might be able to do it, but we'd both need to
spend some time understanding all of this together, going back and forth,
considering examples, and building the foundations.

------
angel_j
Some Fundamental Theorems in Mathematics are bigger than others. Some
Fundamental Theorems in Mathematics' Mothers are bigger than other Fundamental
Theorems in Mathematics' Mothers.

------
krambs
Can you really trust any mathematician's judgment who doesn't use serial
commas? ;)

------
amelius
This is not a self-contained exposition, and a lot of the interesting
implications of the theorems seem to be omitted (why are these theorems
fundamental; and fundamental to what?) It begs the question: why not instead
post a list of links to Wikipedia articles? That would allow the reader to
click through to definitions that they are not familiar with, as well as
explore what these theorems are (practically or theoretically) useful for.

