
Lost in Math? - ernesto95
https://cacm.acm.org/magazines/2019/3/234913-lost-in-math/fulltext
======
throwawaymath
_> About 10 years ago, in the wake of the 2008 financial crisis, the Nobel
Laureate economist Paul Krugman made the same point with respect to economics
and mathematics in an influential article titled "How Did Economists Get It So
Wrong?" His main answer was: mistaking mathematical beauty for truth. "As I
see it," wrote Krugman, "the economics profession went astray because
economists, as a group, mistook beauty, clad in impressive-looking
mathematics, for truth."_

I appreciate the point the article is trying to make, but I think this example
is shoehorned in. You can misuse math without it being because you're "seduced
by the beauty" of it.

I do agree with the author's example in physics. I have seen a lot of
beautiful math in physics; look at lie algebras, monstrous moonshine and
representation theory. Quite a bit of modern physics PhD dissertations are
actually just math dissertations, and the same holds for a significant amount
of new research in the field.

On the other hand I haven't seen that in finance. Highly exotic (read:
"beautiful") mathematics is extremely rarely used in financial engineering.
Pricing derivatives is decidedly mundane work compared to the brain-meltingly
abstract mathematics deployed in high energy particle physics research. That's
not to say it isn't difficult - it is difficult! But difficulty is better
described by the word "complex" rather than "beautiful", and then _of course_
financial engineering is complex. Then we should be talking about how getting
mired in complexity can be bad for accountability and transparency.

This is a different thesis than the one presented by the author. Being led
astray because you've built extremely brittle financial products using layers
of complicated math is _not_ the same as being more preoccupied with the
elegance of a grand unifying theory than its agreement with reality.

But hey, maybe I'm just being pedantic. You can misuse mathematics in a lot of
ways.

~~~
roenxi
It is possible to be too critical about a magazine article, but it can't be
stated enough that the evidence that economists are abusing maths is weak.

Most of the evidence points to the economists abusing _assumptions_ , which is
hardly a mathematics problem. Most assumptions can lead to elegant math. The
biggest problem in modern economics as practiced is the tacit assumption that
because practically all people would like to be able to consume more the
system should favour consumers over savers. Which is a logical non-sequitur,
so that can't be pinned on mathematics.

They may as well call the modern approach to interest rates the "Global War on
Savers". Anyone attempting to save without moving into stocks & other assets
will be wiped out long term.

The risk from using maths is irrelevant compared to the damage done by
assuming a bad value structure - and there are so many forces influencing the
value structure (particularly political ones) that I don't see how
mathematical beauty could be a problem for economics as a discipline.

~~~
neokantian
> Most of the evidence points to the economists abusing assumptions, which is
> hardly a mathematics problem.

Agreed. Economics is about the real world. Therefore, it has to be empirical.
That means that axiomatically deriving conclusions from assumptions is not
legitimate in economics. Still, in the context of empirical knowledge, we have
only two usable methods: the scientific or the historical one. Economics
cannot be validated by testing experimentally. Hence, economics cannot
possibly rest on a sound method. Therefore, economics is fundamentally not a
legitimate academic discipline.

~~~
Nasrudith
There is a difference between non-repeatable in same state and non-scientific.
Applying absolute standards of rigor is ironically also unscientific.

We know that hyperinflation is a way to screw over an economy utterly. It can
and will fail and in the best case be the equivalent of dissolving the
currency and going bankrupt.

The most benign form of it that may not techically count would involve massive
growth as well and the devaluation wouldn't be a pathology but a reflection
that yes, a well honed spear, flint knives, a badket, and a few carved bone
pieces of jewelry may have been respectable wealth for nomadic hunter-gathers
but aren't really worth anything compared to even the contents of a jalopy in
the great depression.

Just a steel knife or pot would be grand artifacts because they are better in
performance than anything else they could find.

That their old currency isn't worth anything is reflective of the fact that
past production has been rendered obsolete and the old goods are worth little.

~~~
neokantian
> There is a difference between non-repeatable in same state and non-
> scientific.

No, there isn't.

> Applying absolute standards of rigor is ironically also unscientific.

The rules governing science are not determined by science itself. Science
experimentally tests propositions about facts. Rules about science are
propositions about other propositions. Hence, science has absolutely nothing
to say about its own rules. Therefore, propositions about the scientific
method are necessarily unscientific.

------
BucketSort
Related: Dijkstra's comments on mathematics and CS -
[http://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/EW...](http://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/EWD1243a.html).

I've personally been getting a lot of satisfaction from learning Haskell and
seeing how the functional programming community is taking ideas from abstract
mathematics, like category theory, and is turning them into practical ways of
thinking about programming.

~~~
tmountain
Me too. Listened to a podcast recently where they were talking about things
like, "once you know what a monoid is, you start seeing them everywhere". I've
tried to express these benefits to coworkers recently. Leveraging ideas from
math allows you to take advantage of many decades of research and provides a
structural foundation that's substantially more robust than things you might
find in the gang of four book or other "design pattern" resources. I think
there's still a lot of opportunity to bring these ideas to the masses in the
way that Evan Czaplicki , the author of Elm, is trying to do. The challenge in
doing so seems to be finding the right level of abstraction to expose people
to with the goal of maximizing benefit while simultaneously minimizing the
prerequisite knowledge you're requiring folks to take on to participate.

~~~
BucketSort
I think it depends on what you are doing. If you are just doing API plumbing,
the idea of introducing more rigor into the process may seem somewhat absurd.
If you are doing real programming, however, Haskell allows you to elegantly
structure and think about a problem. Certainly in parsing applications Haskell
is a no brainier, Pandoc is written in Haskell for example. People are saying
Rust is going to be the chosen one that brings it all together.

For anyone interested in Haskell, I recommend starting with
[http://learnyouahaskell.com/](http://learnyouahaskell.com/) (very friendly,
intuitive and light) then doing these exercises:
[https://github.com/data61/fp-course](https://github.com/data61/fp-course).

~~~
mywrathacademia
Isn't the book you linked a bit dated?

~~~
BucketSort
The main concepts are presented well, but people have complained about it
being dated. What in particular do you find outdated?

~~~
endgame
I saw someone confused on haskell-cafe yesterday because it doesn't cover
Applicative so their Monad instance was invalid.

~~~
BucketSort
Oof that is bad. I admit that I only used it to learn the basic concepts that
just did exercises and read the prelude documentation ( which is fantastic ).
Thanks for letting me know.

------
dkarl
_But complexity theory aims at describing the performance of A over the space
of all problem instances and it does so by abstracting away from individual
problem instances._

I appreciate the effort to extend the story into CS, but I wonder if you have
to be familiar with the particular work he's alluding to. The charge (as
leveled against theoretical physics) is not that some people do pure
mathematical work for the sake of beauty. The charge is that people who are
_supposed_ to be applying mathematics to reality are instead prioritizing
mathematics and neglecting reality. To extend the analogy to CS, he must be
talking about researchers supposedly trying to model real systems but instead
just chasing beautiful math, but he isn't specific. Is it obvious to people in
the know who or what he's talking about?

For practical programmers, I think the problem is the reverse of being "lost
in math." Practical programmers use extremely general theoretical results
because they _don 't want to do_ math, not because the math is more beautiful.
If they applied the information they know about their particular problem, they
could get more useful mathematical results, but since they want to stay as far
away from (doing) theory as possible, they use whatever facts they remember
from class, which are ironically the most purely theoretical ideas because
those are the simplest and easiest to remember.

~~~
UncleMeat
SAT is NP-Complete. In principle, this means that SAT solvers don't scale. If
we stopped here then we would never have developed symbolic execution. It
turns out that SAT and SMT solvers _do_ scale for lots of real world inputs.
Cook's proof is amazingly elegant and powerful but fails to inform real
development.

~~~
tikhonj
SAT solvers only scale _sometimes_. Complexity theory is a starting point for
understanding why this "sometimes" is inevitable, but it never even _implies_
that NP-complete problems are never tractable. Complexity theory gives us an
understanding of how powerful and expressive SAT is which very much _does_
inform real development.

Arguing that "SAT is NP-complete and therefore useless" is not misusing
complexity theory, it's _misunderstanding_ complexity theory—not
misunderstanding some deep result or non-trivial consequence of complexity
theory, but misunderstanding the fundamentals that are covered in the first
lecture of the first class on the topic.

~~~
bonoboTP
It's common. My lecturer in the first lecture gave a motivation why we learn
about this saying:imagine your boss proposes you compute <whatever>, then you
- informed by this lecture - will recognize it belongs to an infeasible
complexity class and can tell your boss it won't be possible.

We did learn the definitions but "worst case" was never highlighted as such,
it was naturally assumed. The closest we got was the discussion that constant
factors may matter for small problems and O analysis masks those differences.

~~~
dllthomas
> tell your boss it won't be possible

I mean, if your boss is demanding a solution that's both accurate and fast on
every input, you _can_ tell your boss that showing P=NP is probably out of
scope of your project. But that's the start of a discussion about how to step
back from that ideal, not the end of a discussion about the potential product.

------
dnautics
I haven't been in the developer industry for too long, but excepting the
haskell community, I would say that the way CS tends to treat math is as
guardrails, as in, "you can't do that because of the halting theorem". "you
might be butting up against computational complexity if you try doing it this
way". "reconstruction of this data shard is impossible because you don't have
enough points to determine the equation".

In the FP communities, I _do_ sometimes see people overoptimize for TCO. Your
datastructure is never going to be more than 10-100 deep. Don't worry about
it. Just write the most legible recursive algorithm, not the most performant.

~~~
tathougies
> Your datastructure is never going to be more than 10-100 deep. Don't worry
> about it. Just write the most legible recursive algorithm, not the most
> performant.

What? Try computing the 1000th fibonacci number.

~~~
chopin
I agree as it is important to know your boundaries. But I also like the GP
prefer legible over performant code whenever it is obvious that I am not going
to need more performance. In almost all cases I encountered the necessity to
optimize code the reason was I/O bound. I don't recall any instance of
algorithmic performance being a problem.

~~~
tathougies
Tco has little to do with timing performance and all to do with keeping code
legible in functional languages.

~~~
dnautics
In FP that implements actor model, for example, you will be in a world of hurt
if you don't TCO your actor - has nothing at all to do with legibility.

------
paulpauper
_But the seductive power of mathematical beauty has come under criticism
lately. In Lost in Math, a book published earlier this year, the theoretical
physicist Sabine Hossenfelder asserts that mathematical elegance led physics
astray. Specifically, she argues that several branches of physics, including
string theory and quantum gravity, have come to view mathematical beauty as a
truth criterion, in the absence of experimental data to confirm or refute
these theor_

Her criticism has gotten more attention than justified by its merits. No one
has argued that beauty holds precedent over truth

~~~
sls
In a recent blogpost [1], Hossenfelder responds to a review of her book. All
but the first two paragraphs are basically a response to this idea. A brief
excerpt:

> In most cases, however, physicists are not aware they use arguments from
> beauty to begin with (hence the book’s title). I have such discussions on a
> daily basis.

> Physicists wrap appeals to beauty into statements like “this just can’t be
> the last word,” “intuition tells me,” or “this screams for an explanation”.
> They have forgotten that naturalness is an argument from beauty and can’t
> recall, or never looked at, the motivation for axions or gauge coupling
> unification. They will express their obsessions with numerical coincidences
> by saying “it’s curious” or “it is suggestive,” often followed by “Don’t you
> agree?”.

[…]

> What physicists are naive about is not appeals to beauty; what they are
> naive about is their own rationality. They cannot fathom the possibility
> that their scientific judgement is influenced by cognitive biases and social
> trends in scientific communities. They believe it does not matter for their
> interests how their research is presented in the media.

Have you read the book or any of her posts about the ideas in the book? There
are in fact a lot of people who do claim that research programs and research
dollars should be prioritized because of ideas like naturalness or beauty,
even when decades of increasingly expensive and time-consuming work has led to
no support for the natural or beautiful hypothesis.

[1] [http://backreaction.blogspot.com/2019/02/a-philosopher-of-
sc...](http://backreaction.blogspot.com/2019/02/a-philosopher-of-science-
reviews-lost.html)

~~~
calf
I find Hossenfelder problematic because if you look at the rhetoric the blog
post itself is relying on intuition and in essence a beauty argument in order
to make a sociological point. I think Hossenfelder has a point about the state
of professional physics, but the way this is communicated is unclear and lacks
this rhetorical introspection as well. It's why her posts draw so much
attention; the strong assertions, the arguing, and how the unrigorous writing
itself obscures this.

------
mikorym
> the theoretical physicist Sabine Hossenfelder asserts that mathematical
> elegance led physics astray

I feel the other way around: Applied mathematicians and physicists led the
pure mathematicians astray. But I say this for a different reason. I feel that
physics has become convoluted with a plethora of theories where this same
beauty is interpreted wildly differently by different people. In other words,
people all have different ways of thinking, different notions of beauty, and
ultimately, this manifests into different (competing) notions in physics.
These may even be equivalent notions and they may embody the desired
perspective on beauty, but instead of consolidation there is extended
differentiation.

My point is that physics has become ugly exactly because of physicists'
ignorance towards mathematical elegance in favour of personal beauty. I don't
think physics can become consolidated without exactly a stark appreciation for
elegance.

IMO this is why category theory took so long to start appearing in physics:
The physicists are caught up in their own idea of _beauty_ rather than the
mathematical tradition of finding the minimal sufficient proofs and theories
(which I call elegance).

------
drilldrive
Coming from a mathematics background, I personally do not care so much for the
'beauty' of mathematics, and am moreso interested in clarity of properly
abstracting and insight to the resulting formal theory. I feel that physicists
care more about such intuitive ideas than anybody else.

Regardless, you only can become lost in math if you have bad premises.
Mathematics is a relative subject, abstracting the arbitrary of reality to
axioms. And if the axioms do not hold, the theory is bunk. It will always be
the case that more granularity is required in real-life situations.
Mathematics is precise and sound; it's not gospel.

~~~
mikorym
> And if the axioms do not hold, the theory is bunk.

You assume the axioms hold. That is what an axiom is: an assumption.

> Mathematics is precise and sound; it's not gospel.

We do think that mathematics is precise. We don't really know if it is sound
(unless I am missing something). For example, what _is_ a set? It is not a
"collection of things". Rather, it is an object in some mathematical setting.

The topic of whether mathematics is "correct" is something else. We can all go
out and build a DIY logical machine from scratch and see for ourselves that
the mathematics we use give the results that we expect. Applied mathematics in
this sense concerns itself with mathematics that suitably describe real
situations.

~~~
drilldrive
>We don't really know if [mathematics] is sound. My apologies, I was hoping to
be concise. I meant to say that mathematics is sound relative to the
assumptions, which is exactly correct. But if the assumptions do not hold in
reality (and they never quite do), then the theory as a whole is slightly off.
I mean to say that mathematics is never 'correct' with reference to reality,
but of course is always 'correct' with reference to the axioms.

------
salty_biscuits
If you count all the work in AI/ML then the criticism has been overwhelmingly
in the other direction, i.e. to much "just trying stuff to see what happens"
and not enough "really understanding what is going on". Always seemed like a
weak criticism to me honestly. You can advance theory, or you can advance
through experimental insight. Neither is the right or wrong path, just
whichever seems like the best way to make progress given the state of current
knowledge.

~~~
kevinventullo
Yeah, I feel ML would have been a more apt analogy than complexity theory. For
some problems it's an ugly, "brute force" approach that works really really
well.

------
andrewmatte
I have long felt like "beauty" in mathematics is just oversimplification.

Ironic that intelligent mathematics types get caught up in what could be
analogous to socially hurtful stereotypes.

I am going to follow the author.

~~~
lalalandland
Is't a large part of the history of physics about doing away with wrong
assumptions based on beauty? Circular orbits of planets etc.

~~~
masteranza
Except that you've got the point completely upside down! At first people
didn't believe in heliocentrism because it used circular orbits, and instead
used to believe that the universe is better described from the geocentric
point of view which was one ginormous mess from the mathematical point of
view. Then people realized that the beautiful addition of conical curves
solves the problem completely. I say 'beautiful' because clearly you don't
have a sense of mathematical beauty. Mathematical beauty is not only the
simplicity of the solutions, but also it's robustness and fluid way in which
it fits into the rest of our knowledge. The real simplification that came from
it is far deeper than any 'circular orbits' picture you imagine. Naturalness
and search for beauty can never be proved wrong. Those principles can perhaps
only be applied prematurely, before enough data is collected.

------
karozagorus
It's terrible that nobody got punished for it, I hope Bitcoin will solve all
of our problems soon.

