
Mathematics as thought - magoghm
https://aeon.co/essays/the-secret-intellectual-history-of-mathematics
======
hodgesrm
I was a bit disappointed that this article did not offer more practical
examples of the ties between the development of mathematics and the history of
ideas in general.

For instance, the vision of mathematicians like Hilbert and Russell of a
cohesive mathematics defined from first principles seems a very Victorian
notion that the universe is knowable if you simply search hard enough. It's
the same enlightment frame of mind that resulted in intellectual creations
like the Oxford English Dictionary.

Similarly the results of Gödel, Church, Turing, and others showing the limits
of mathematical consistency seem of a piece with the confounding discoveries
of Quantum physics, which replaced the static models of classical physics.
They seem to correspond with a darker vision of the limitations of human
intellect that emerge in abstract art and poetry like Eliot's 'The Wasteland'
around the same time.

If you push this too far you end up sounding like an idiot (perhaps this is
already there) but there are clearly discernible connections across science,
art, literature, and politics both now and deep into the past. It might be a
personal projection but Brahms concertos always strike me as the music of a
people who believed in an orderly, Newtonian universe and upheld the static
political order of the Ancien Regime.

At the end of the day most intellectuals read the same books and attend the
same salons, however defined. There is continuous cross-fertilization between
different realms of thought. The resulting connections are there if you just
look for them.

~~~
pegasus
Related book: [https://www.amazon.com/Mathematics-Roots-Postmodern-
Thought-...](https://www.amazon.com/Mathematics-Roots-Postmodern-Thought-
Vladimir/dp/0195139674)

------
Meai
"How mathematics works rests on no absolute timeless standard, despite what
many assume today, given its precision and efficacy"

Exactly this is key to teach people to like math, stop trying to train me in
it. It's like math has become so saddled with politics or ego that people want
to train everybody in the same system of values and they are willing to lie to
get you to use to the same symbols, the same way of thinking, etc. Disclaimer,
I'm really bad at math. But I hate how math and physics are taught, it should
always be taught from a spirit of exploration as if you are discovering new
islands or continents and you get to name the things you discover. Try getting
a math professor to admit that he doesn't actually have an intuitive
understanding for any of the stuff he teaches. I believe the vast majority
probably don't have any such understanding but they are understandably scared
to admit it. It's a cycle of lies. Who are you to say that I have to use that
particular symbol or term to describe this math? People have a large stake in
interopability between mathematicians so they teach us these symbols as if
they are facts of nature when in reality they are just arbitrary drawings that
people invented to name stuff that they found in nature. Fields of math stack
on top of each other because they happen to work together but in reality we
dont know why. They just do and it's interesting that they do so we keep doing
it that way because it works and hasnt broken yet and it's useful. There were
cultures that did math with a completely different set of symbols, not just
symbols but even way of thinking. Multiplication, addition, entire base ideas
were thought of differently. Yet our current way of doing math is taught to us
as dogmatic, accept it or you are a troublemaker. No wonder people dont like
math, nobody wants to be a slave to somebody else's ideas and value system.
Every math class should start with something like "This is the symbol for
addition: '+'. This was invented by some person, you could use something else,
you could try to find some other way to add numbers but the way we teach you
to think has proven to be fairly fast and convenient so we teach it." That in
my opinion is how you make people interested in math because you treat them as
equals instead of subjects to be trained in your favorite cultural way of
doing math. Math should be seen like a weird natural phenomenon that we
observe, from the getgo. From childhood on. Explain it like that and I believe
kids (and adults) will love to discover more about it.

~~~
jackpirate
_Try getting a math professor to admit that he doesn 't actually have an
intuitive understanding for any of the stuff he teaches. I believe the vast
majority probably don't have any such understanding but they are
understandably scared to admit it._

Mathematicians are famous for embracing the fact that they have no intuition.
For example, here's a famous quote from Geoff Hinton:

 _To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space
and say 'fourteen' to yourself very loudly. Everyone does it._

And here's a famous quote from G.H.Hardy:

 _In mathematics, you never understand things; you just get used to them._

The idea behind both of these quips is that there is no intuitive way to
visualize these weird mathematical objects. The main reason people struggle
with math (in my experience) is that they assume there must be an intuition.
There's not. Math is weird. And all mathematicians feel that way about math.

~~~
romwell
>Mathematicians are famous for embracing the fact that they have no intuition.

That's just plainly false. Mathematics is _all_ about intuition.

That's how we all _know_ that Riemann hypothesis is true, regardless of
whether there's a proof of it. Or that Fermat's Last Theorem holds.

That's how we still _do_ math, even though we never really had a solid system
of axioms for it -- and Godel showed that, in some sense, it's not even
possible.

That's how even famous theorems were proven after many attempts, with people
accepting proofs with errors in them.

That's how Calculus was invented before the notions of the limit, derivative,
and integral -- the most fundamental ones! -- were solidly written down.
Newton and Leibniz built up the math upon heresy, and everyone took it,
because it _felt right_.

What you are writing about is _surprise_ that mathematics still gives to
practicing mathematicians.

And your first quote is a trick on how to _extend_ one's intuition, not
_abandon_ it!

~~~
nimonian
We know that FLT is true because Andrew Wiles has proven it. We don't know
that RH is true. We have strong reasons to believe it is true since we have
discovered directly a large number of roots on Re(z) = 1/2 and none anywhere
else. I do believe intuition exists in the process of doing mathematics, but I
don't think either of these are good examples.

~~~
trukterious
How about in the process of reading a proof? Each single step is validated by
the intuition of the reader.

------
bmc7505
Nice parallel with Iverson's 1979 Turing Award Lecture, "Notation as a Tool
for Thought":
[http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pd...](http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pdf)

------
blt
Opening up with the claim that computer software is "based on the ideas of
Claude Shannon" is strange. Does anyone seriously believe that Shannon's
contributions to the foundations of computing stand so far above others'?

~~~
TheOtherHobbes
Turing - On Computable Numbers, with an Application to the
Entscheidungsproblem - 1937

Shannon - A Mathematical Theory of Communication - 1948

Shannon calculated the entropy of arbitrary symbol systems over noisy
communication channels, and made some first steps towards practical data
compression algorithms.

Turning formalised the concept of computability using an abstraction of a
mechanised logic system.

I'd say Turing wins easily - we still say logical systems are Turing Complete
rather than Shannon Compatible.

~~~
mbrock
Don’t forget Shannon’s 1937 master’s thesis, _A Symbolic Analysis of Relay and
Switching Circuits_.

~~~
yesenadam
Yeah, he actually had the idea that you could use the 19th C Boolean logic and
switching circuits to do mathematical operations, making modern computers
possible. It's disappointing that he's not better known for this, to say the
least. e.g. that _anyone_ on here doesn't know his huge importance.

------
astroalex
> For instance, say you’re in a race at school. You do surprisingly well and
> beat most of your classmates. All things being equal, the next time around,
> you’re actually not likely to do as well, relative to the other runners.

If the events are independent (as I believe the author is assuming at this
point), then your performance in the first race has no effect on your
performance in the next race, right? Maybe it's just a poor choice of wording,
but it comes dangerously close to reinforcing a common misconception about
probability (Gambler's fallacy).

~~~
btilly
No, the author is not assuming that. And in fact the opposite is true.

The assumption is that your performance on any given race is some combination
of luck and ability. If you perform extremely well, it should be assumed that
both were in your favor that time. The next time you'll still have whatever
ability is in play, but you are unlikely to have luck.

The result is that your performance is likely to be good, though not stellar.
You racing one day or the next are not independent events - your ability
creates a correlation. But your performance one day is also not a linear
prediction of your expected performance the next.

This is called regression to the mean. It comes into play in everything from
stock picking to poker players.

~~~
antidesitter
You are committing the gambler’s fallacy. There’s no reason to expect your
“luck” (whatever you mean by that) to decrease rather than increase the next
time you race.

This is _different_ from saying that you are likely to win again. You’re just
as likely to win as you were the first time around, which may be high or low
depending on ability.

~~~
btilly
No, I am not committing the gambler's fallacy. Though I can see why you might
think that. There are enough things that sound similar.

I'm instead talking about a more subtle detail. Which is that selecting a
group based on performance results in selecting people in part for having been
lucky. They were lucky to have done that well that time, were lucky to be in
your group, and their future performance probably won't be as good.

The result is called regression to the mean. See
[https://academic.oup.com/ije/article/34/1/215/638499](https://academic.oup.com/ije/article/34/1/215/638499)
for an example of statisticians talking about it. See
[http://onlinestatbook.com/stat_sim/reg_to_mean/index.html](http://onlinestatbook.com/stat_sim/reg_to_mean/index.html)
for a more introductory tutorial. And see
[http://wmbriggs.com/post/63/](http://wmbriggs.com/post/63/) for an example of
an article discussing this counterintuitive phenomena in the context of
sports. (Namely the "Sports Illustrated curse" \- the future performance of
athletes whose performance was good enough to get them on the cover of Sports
Illustrated drops after their article appears.)

~~~
antidesitter
I think I see what you mean now.

If you roll 20 dice and select the highest, the next time you throw that very
same die it’ll probably have a lower value, simply as a consequence of
outcome-dependent selection + a fixed probability. It’s not that the
probability has changed, just that you didn’t choose the first outcome
according to the true generating distribution.

Comparing this highest value with the highest value across _all_ coins when
you toss them all again a second time is a different matter. It’s no more
likely to be lower than higher.

It's best to avoid using the term “luck” altogether in discussions of
probability, since it’s easy to misinterpret.

~~~
btilly
That's exactly the right idea.

------
allemagne
I thought this was a really beautiful article. All I have to add is that it
seemed to avoid discussing philosophy as a discipline, even when it seems
impossible to ignore. For example:

>Perhaps more than any other subject, mathematics is about the study of ideas.

