
Some Fundamental Theorems in Mathematics (2019) [pdf] - bryanrasmussen
http://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf
======
ivan_ah
Wow I really like this document. As someone who really like mathematics but
didn't get to do this as my major in undergrad, I'm missing out on so much,
but also there is not enough time to start with undergrad books and do three
years of basics... This document with all the clear statements at least allows
me to see what is there to learn. Yes it's long, but compare that to dozens of
the other books I would have to read to encounter these theorems, it's
actually very short!

There is some useful "about this document" info near the end:

> _The motivation to try such a project came through teaching a course called
> Math E 320 at the Harvard extension school. This math-multi-disciplinary
> course is part of the “math for teaching program”, and tries to map out the
> major parts of mathematics and visit some selected placed on 12 continents._
> [...] _A goal of this project is also to get back up to speed up to the
> level of a first year grad student (one forgets a lot of things over the
> years) and maybe pass the quals (with some luck)._ via
> [http://people.math.harvard.edu/~knill/graphgeometry/papers/f...](http://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf#page=127)

~~~
jointpdf
You might enjoy Oliver Knill’s lecture notes on undergrad math topics, which
are similarly concise:
[http://people.math.harvard.edu/~knill/teach/index.html](http://people.math.harvard.edu/~knill/teach/index.html)

~~~
ngcc_hk
Thanks. Same as the poster. But some “horrible” memory did meet me as I did
math stat. Within 1/2 hour the lecturer tried to do a proof of central limit
theory using an assumption that one can draw sphere to fill up a 3D space.
Reading this does frighten me to a great extent at the same time appreciate
what humanity have reached for doing such apparent “useless” things. Public
goods are hard as it might look useless but unlike private goods it can be
consumed and reuse many times. And some may find use of matrix in AI, number
theory and geometry in encryption ... all because it is a public good that can
be shared.

The trick whoonearth waste their life to create the first and develop later
such “useless” things.

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JadeNB
It's great to see someone assemble some of their favourite theorems, but, if
you have occasion to use these, make sure that you do so with caution.

Aside from some typos ('Perseval' for 'Parseval') and some curiosities
(stating on p. 3 that the cardinalities of a set and its power set are
_different_ , which is true; but why not say that the latter is larger?), I
noticed a wrong statement on p.7: assuming that to 'extend' a monoid to a
group is to realise it as a submonoid of a larger monoid that is also a group,
this cannot be done in general. (For example, the 'min-monoid' whose objects
are the natural numbers, and in which the product of two natural numbers is
their minimum, does not embed in a group because it obeys no cancellative
law.) What is true is that, for every commutative monoid, there is a universal
homomorphism from that monoid to a group, in the sense that all other such
homomorphisms factor through it. (In my example, this universal homomorphism
is the constant-0 homomorphism to the 0 group.)

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dan-robertson
I recall learning the fundamental theorem of applied maths. The approximate
phrasing is “in applied maths, if it looks right then it is” and a more
precise phrasing is that “in applied maths, all reasonable series converge,
all functions are continuous, differentiable and smooth almost everywhere, all
limits (and integrals) exist, all sums or integrals commute, all Taylor series
are good approximations, all of the silly well-behaved ness conditions of the
theorems you want apply, and every equation has one solution”.

~~~
ilyagr
> every equation has one solution

So when you see the sun rising, cherish the moment, as it has never happened
before and will never happen again.

~~~
hnarn
In a philosophical manner of speaking, that's true. You never step into the
same river twice.

~~~
ilyagr
Indeed. But the scientific approach is different -- it's all about finding
patterns that do repeat and making predictions.

The original comment's "theorem" is funny, but unless you are doing a homework
problem, you better have a good intuition for when to take a second look at a
seemingly simple situation.

~~~
hnarn
> the scientific approach is different -- it's all about finding patterns that
> do repeat and making predictions

Sure, but if you're scientifically looking to answer "where and when will the
sun rise" you're only going to collect enough variables to answer that
question, and within an acceptable margin of error, right? If you can measure
with greater accuracy and collect more data, then you would probably realize
that every sunrise is _not_ strictly identical.

We're splitting hairs at this point but I wonder if the comment I replied to
that stated that "it has never happened before and will never happen again"
can't be argued to be actually true. In a philosophical sense it's more
obvious, but in a scientific sense, the more precision you get in your
analysis of a sunrise, the more data you would get that differentiates it from
other sunrises, no? After all, our solar system isn't closed and constant and
there are minor changes not only in smaller factors like weather on Earth, but
also larger factors like the orbit of the earth and the drift of the different
planetary bodies.

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jmount
That was fantastic. Often things titled "some fundamental" are a short
superficial list, but this was some great deep stuff.

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augustt
This is awesome, although a bit in the old sense of the word, since it's scary
to see entire courses I've taken distilled into one result that I probably
couldn't even attempt to prove now.

aside: wonder why for group theory (51) Lagrange's theorem is only stated for
cyclic subgroups, instead of any subgroup.

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mike00632
If someone tells you they are fluent in 100 languages then you get a documents
like this. The list starts out solid but then gets into areas where the author
is clearly less versed.

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dang
Discussed at the time:
[https://news.ycombinator.com/item?id=19039525](https://news.ycombinator.com/item?id=19039525)

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perfunctory
Something went wrong with the way math is taught at school. We teach math as a
tool and not as something that can be appreciated and enjoyed the way we
appreciate and enjoy art. Some math theorems and their proofs are a thing of
beauty. Often one doesn't even have to understand all the nitty-gritty details
to see that beauty. Some basic understanding of fundamental principals often
suffices.

~~~
Konohamaru
You cannot teach people--even people at a young age--to enjoy mathematics
because mathematics is based on logic and almost everyone except for a few
weirdos hates logic. Mr. Spock was designed to point out how unusual people
who like logic are, but as a character from a minstrel show, not as a role
model....

~~~
idclip
I disagree. Mathematics is a tool to practice our hobbies .. see Feynman‘s
biography.

Rot mathematics is where it goes wrong. Meta mathematics, abstract, detached
from real life, as alot of the academy with its publish pr perish thinking ...
that produces pain. Existential voids filled with an exercise that does not
help many to move forward optimally.

saying weirdos hate logic is extremely dismissive to a lot of what I would
call the human experience.

The same stuff can be taught with soul and purpose If we actually show
children how they can apply mathematics to improve and enhance what they’re
actually doing whether it is building castles in Lego, Or designing faster toy
car, A better school project volcano, etc

~~~
butterthebuddha
> Rot mathematics is where it goes wrong.

Not sure what you mean by "rot mathematics".

> Meta mathematics, abstract, detached from real life, as alot of the academy
> with its publish pr perish thinking ... that produces pain.

Some (including me) would say that the meta and abstract mathematics is most
beautiful of all...

> saying weirdos hate logic is extremely dismissive to a lot of what I would
> call the human experience.

I think you misread the parent comment (although to be clear, I'm not
defending it).

~~~
golem14
rote

~~~
idclip
Stanis.jpg :D

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generationP
How useful is this? Not much that these things have in common beyond "results
cited a lot".

Can't say that the subjects are chosen very precisely either -- the
Fundamental Theorem of Algebra isn't actually a theorem of algebra;
Tychonoff's theorem is fundamental only to the set-theoretical part of
topology; the Fundamental Theorem of Counting is just a particular case of the
"Fubini" interchange-of-summations formula, which I would call the real
fundamental theorem of counting. The number of platonic solids is mostly a
curiosity from a modern perspective; so are the transcendences of pi and e.

Also, the word "extended" in the Number systems section needs to be taken with
some artistic license; the "extension" introduces new symbols (negatives) and
new relations that can occasionally render some old elements identical (in the
worst case, the whole monoid can collapse to a point). The most famous example
of this is what happens if you divide by 0 (= extend the multiplicative monoid
of real numbers to a group). This is not something the author should be blamed
for; it's the only real error I've spotted at a quick skim, and it's rare for
a collection as diverse as this to have this few errors. The real problem is:
what's the point of such a survey if pretty much any of the results is given
so little time and space that only those who already know it can understand it
from the description?

~~~
spappal
Compact summaries are useful when revisiting something that was learnt before.
Such a document might be more useful for mathematics than most subjects, since
many have studied maths but stopped using it, and those teachings are
generally still true and relevant.

The doc would be at least 20 % more useful to me if the pdf had a table of
contents. Should be easy to include assuming that it was written with latex.
Opinion: when writing a lengthy latex document, the extra 0.5 % of work
required to add automated pdf metadata (table of contents, clickable
references) has outsized usability effects.

I stumbled upon typos:

* "Basel problem formula": pi should be squared.

* The "more general" statement related to Bayes theorem lacks a right parenthesis.

------
staycoolboy
I followed it for three pages, then recognized the next 3-5 pages, and then
literally started to laugh at the explosion of exotic names and terms as I
crossed page 30. I had no idea mathematics was so ginormous. Humbled.

~~~
Micoloth
Absolutely relatable!

------
alderz
In the subject of Artificial Intelligence it says:

> _Theorem: No AI will bother after hacking its own reward function._

and then goes on:

> _The picture [263] is that 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”. Objections are brushed away with “Well,
> this is your, like, opinion, man”. Two examples of human super intelligent
> units who have succeeded to hack their own reward function are Alexander
> Grothendieck or Grigori Perelman._

Grothendieck abandoned the mathematical community after modernizing an entire
field and eventually secluded himself in France. He is considered one of the
greatest mathematicians
[https://en.wikipedia.org/wiki/Alexander_Grothendieck#Retirem...](https://en.wikipedia.org/wiki/Alexander_Grothendieck#Retirement_into_reclusion_and_death)

~~~
asgard1024
I think it just happened:
[https://news.ycombinator.com/item?id=24061653](https://news.ycombinator.com/item?id=24061653)

GPT-3 just wrote an essay arguing that intelligence is doing nothing.

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mazesc
Ctrl+F Gödel "not found".

Also proof theory is completely absent, like Gentzen's cut elimination
theorem.

These are "fundamental" theorems of mathematics in the literal sense.

~~~
ogogmad
Goedel is there. See pages 8 to 9.

~~~
mazesc
Oops, thanks! Didn't try "oe".

