

There’s more to mathematics than rigour and proofs - jonnybgood
https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs

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analog31
_The transition from the first stage to the second is well known to be rather
traumatic, with the dreaded “proof-type questions” being the bane of many a
maths undergraduate._

I was fortunate that my K-12 math curriculum included proofs and derivations.
We learned about sets in first grade. This was a suburban public school using
textbooks from a mainstream publisher, in the 70s. Likewise college physics.
As a result, there was no transition, and my college math grades went up when
the courses were primarily about proofs.

It seems like a couple of things have happened since then, good and bad. What
constitutes "math" has been reformulated to give kids a better chance of
learning it. In my generation, we did proofs, but it was OK if 90% of the
students failed to learn math at all. Math has multiple roles in society, as
do science and computer programming, and preparing kids for professional
careers in those subjects is not the only reason to teach them.

On the other hand, "math" has also been reformulated to manipulate
standardized test scores.

~~~
clebio
I may have inadvertantly down-voted when I meant to up-vote, because mobile
phone interfaces are still terrible. HN doesn't let me change the vote. So I
want to apologize and say that your comment is spot-on. There's a lot of
specifics I'd reply to... if I was at my computer.

~~~
fsk
It sounds like they need a UI change for mobile.

It should be

upvote arrow, username, downvote arrow

instead of the way it is now

Then there would be no accidental downvotes.

~~~
clebio
Yeah, or like on the Stack Exchange sites, where you can undo the actions.

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MichaelBurge
I think something similar happens with programmers:

* I vaguely remember ambling around when I first started writing C programs, using a debugger whenever it broke, and generally just slapping for loops and variables together until it worked. I had some intuition for how things worked, but it often took hours of hard debugging. I perceived code purely operationally - "This line sets this variable to 5".

* At some point, you start structuring things into modules, subroutines, etc. You might learn 'best practices' and try intently to practice each one. You'll unit test everything, or otherwise hold strong opinions on how things should be done. Debugging is approached in a more scientific manner. You start to feel like you can get a handle on just about any bug, since you have tools and methods for reasoning about them. There is a lot of focus on verifying code, but not so much on validating. I perceived sections of code as implementing some intuitive concept - "This class represents a bullet, and its methods are reasonable things for working with bullets"

* Now, I rarely spend any time worrying about the best practice of the day just for the sake of it. I generally focus on getting the current task done, but I intuitively know when to apply a certain practice. For example, I might write a test or two after writing algorithmic code. I rarely spend any time at all debugging my own code, since it tends to just work. Since verification is so easy, most of my focus is on validation: "Am I even building the right thing? Will this way of laying things out have acceptable performance characteristics? Does this way of laying out the types mirror how I think about the problem? Is everything I'm writing necessary to solve the problem?" I perceive most code as a formless soup of syntax trees with an operational semantics, and reason about the segments of the trees themselves.

I imagine that Terence's breakdown is relevant to many professions besides
just mathematics.

~~~
mannykannot
For many professional programmers, the rigorous stage of their education never
got to the techniques for proving the correctness of programs, or even a
realization that such techniques exist. We see this in the discussion of
design issues, where informal arguments based on rules-of-thumb,
reasonableness, best practices and aphorisms (e.g. "code smells") predominate.
There is a tendency for dogmatism to arise in this environment ("thou shalt
never write the same expression more than once - always factor it into a
function.") (dogmatic thinking is probably confined to a small minority, but
because they are vocal, they tend to dominate the discussion.)

I am not one of those who think a transition to formal methods is feasible,
let alone that it would solve software's problems. I am also aware that many
programmers are effective in putting their general critical thinking and
reasoning skills to work in writing correct code (this is one of the reasons
why some people without any formal CS education can become excellent
programmers.) Nevertheless, and speaking only from my own experience, I think
some exposure to more rigorous approaches would help programmers tackle tricky
problems, such as those that arise in matters of concurrency or security.

------
mhartl
My favorite example of mathematics that blends intuition and rigor comes from
the works of Archimedes of Syracuse: the _mechanical method_ used to intuit
results such as the area of a parabola, followed by the _method of exhaustion_
used to prove the result rigorously.

In the mechanical method, Archimedes imagined, for example, a section of a
parabola balancing with a triangle, using the law of the lever (which he also
discovered) to derive the area necessary to achieve said balance. He then used
inscribed and circumscribed polygons to prove upper and lower bounds on the
area thus derived, with (in modern parlance) the two bounds converging in the
limit _n_ → ∞, thereby establishing the result. The rigorous method of
exhaustion (due originally to Archimedes' predecessor Eudoxus) is effectively
equivalent to integral calculus (~2000 years ahead of its time), but guessing
the right answer would in many cases have been difficult or impossible without
the non-rigorous mechanical method.

Incidentally, the mechanical method itself might have been lost to history had
it not been for the discovery of the _Archimedes Palimpsest_ in a medieval
prayer book [1], which contains the only known copy of the work describing it.
Often called simply _The Method_ , it takes the form of a letter from
Archimedes to Eratosthenes, the chief librarian of the Library of Alexandria.
When I had occasion to see some original pages of the palimpsest last year (at
The Huntington Library in San Marino, adjacent to Pasadena, California), I was
struck by the collegial tone of the letter, whose genuine human warmth was
instantly recognizable even across two millennia.

[1]:
[http://en.wikipedia.org/wiki/Archimedes_Palimpsest](http://en.wikipedia.org/wiki/Archimedes_Palimpsest)

------
Jun8
The three levels Tao lists (roughly paraphrasable as novice - master - true
master) can be applied to many other endeavors, e.g. art, martial arts,
programming, as other point out in other comments. What I'd like to stress is
the damage that's generally done in going from Level 1 to 2 in math teaching.

As a specific example take Analysis, generally taught using Rudin. I took this
course in my graduate EE studies and _detested_ it. I later thought about this
and came to the conclusion that the main reason was the manner of exposition
in Rudin where the classical approach is used: each chapter contains an
endless sequence of lemmas, minor theorems, etc., one after the other with no
discernible purpose and at the end of the chapter you get to prove a big
result by using all that machinery. This approach, which dates back to at
least Gauss (commonly attributed quote, which I couldn't find the source: "no
self-respecting architect leaves the scaffolding in place after completing the
building") not only is backward to the real course of events, it sucks the
motivation by being so. At least it did so for me and for some other otherwise
intelligent friends.

~~~
anatoly
It's interesting that you say that. I'm trying to square this with my own
experiences - I absolutely loved Rudin and to this day it's one of my favorite
textbooks; but at the same time I recognize what you're saying about hiding
the scaffolding and I share your distaste towards that.

I guess what I love in Rudin is that he gives the Level 2 details in such a
lucid logical manner, with nothing missing and yet relatively tersely, all the
details interlocking together. I got a sense of real beauty from reading him
as an undergraduate that I did not get from other textbooks. I think that
enjoyment ranks much higher from me than the disappointment from not getting
the motivation early, which I do try to give myself when I teach something.

I wonder if these two can be separated: if a textbook could be Rudin-style in
logical unity, terseness, and beauty and yet not "hide the scaffolding". To
some degree, I thought Stephen Abbott's "Understanding Analysis" was a step in
that direction, though it was still too wordy and meandering compared to
Rudin, for me.

------
jeeyoungk
I can't believe that none of the people have mentioned Lakatos's "Proofs and
Refutations" \- such a good work, which shows the organical refactoring of a
vague mathematical intuition into a formal system.

------
wwosik
There's a lot in this article that reminds me of Bruce Lee's:

"Before I learned the art, a punch was just a punch, and a kick, just a kick.
After I learned the art, a punch was no longer a punch, a kick, no longer a
kick. Now that I understand the art, a punch is just a punch and a kick is
just a kick."

------
j2kun
I'm curious about HN readers so I made a poll: where you do see yourself in
Tao's hierarchy?

[http://goo.gl/forms/KuXqj0wgZu](http://goo.gl/forms/KuXqj0wgZu)

Edit: for those who are interested in the results, here's a one-click link
[https://docs.google.com/forms/d/1_RkOUYEoC4m9mYQZKexp2uSuj63...](https://docs.google.com/forms/d/1_RkOUYEoC4m9mYQZKexp2uSuj63wnq24QEGND6xBZ2M/viewanalytics?usp=form_confirm)

~~~
analog31
Post rigorous, heading towards rigor mortis. ;-) This doesn't mean I'm a
professional caliber mathematician -- I'm not, and a lot of my math education
was in studying physics. It just represents which stages I've passed through,
if those stages are valid

It would be interesting to know if there are corresponding hierarchies in
subjects such as science, computer programming, or even music.

~~~
cossatot
My field is earth science which is, at its best, highly empirical. The stages
are a little different: 1) Learning to observe and learning the basics of the
theory; 2) Learning the theory quite a bit more and getting more practiced
about observing with theory in mind; and 3) Hammering on, and molding or
breaking, the theory with more observations or other theory.

Unlike math, it's just really, really difficult to ascertain that the theory
is correct and it's never correct at all levels. I have always called geology
'the science of exceptions' because many of our laws are actually tendencies,
and given 4.6 billion years, any bug or loophole in the theory will be
exploited--and this is when you really learn stuff.

In practice, I actually think the three steps there are kind of fractally
embedded throughout in the stages of one's career, one's individual research
project, or the overarching evolution of the science.

It's pretty fun.

------
im3w1l
I think I remember hearing the same breakdown of martial arts before.

------
iamcurious
Intuition sounds nice until you realize it means "there are things happening
here which I cannot explain nor verify nor teach.".

~~~
mannykannot
Once you know and respect the difference between intuition and an explanation,
it can be very helpful - but you can never trust in it working.

------
justifier
this sort of exposition leaves me feeling ill

why base your theoretical structure on satire? ..unless it also is meant to be
taken as satire?

why the need to blanket your own experience onto others?

math stands alone, and the curiosity therein also stands alone

i'm arguing myself whether it be systemic or ironic that an academic
mathematician would use a limited set to try to categorise mathematical
curiosity

why do academics build so many walls and then wonder why they feel so alone?

all you need to be a mathametician is curiosity

------
hyh1048576
Mods, Please mark this as 2009.

