There’s more to mathematics than rigour and proofs
terrytao.wordpress.com
terrytao.wordpress.com
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.
It should be
upvote arrow, username, downvote arrow
instead of the way it is now
Then there would be no accidental downvotes.
My secondary school went for SMP maths (I suspect because the deputy principal was part of the project). I'm very glad they did. It helped later.
A longitudinal survey of the destinations of those who did A level Maths in the 1970s tracking who did the SMP syllabus and who did a traditional syllabus might be interesting...
It included some interesting things, but left out a lot of traditional pure and applied maths (especially mechanics).
The group theory stuff made things a lot easier for me later.
I imagine this is all fairly random.
* 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.
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.
But I agree that this breakdown will likely apply to many other professions. The reason being that mathematics is in essence the art of working with rules (in an abstract sense). Since almost any professions requires you to work with some rules (some very concrete, others more subjective) learning that part will resemble learning mathematics.
Unfortunately you're very unlikely to encounter this part of mathematics if you don't go beyond high school level. I think some people want to introduce it through programming, but I'm not sure if that'll be any more effective.
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.
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.
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.
"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."
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...
I'm pretty comfortable with probability and optimization, which is my day-to-day. I will probably never be better at topology than I was when finishing my undergrad degree. And in my one brush with differential geometry, I struggled to reach the rigor.
And even then it's more complicated, because human intuition is notoriously bad in probability.
It's very exciting, I feel like I'm finally able to play by the rules of math, and I'm looking forward to the places this newfound knowledge can take me.
It would be interesting to know if there are corresponding hierarchies in subjects such as science, computer programming, or even music.
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.
I think that you could probably apply the labels to just about any profession, or even hobby (I'm thinking of rock climbing as one close to my heart), and it might even be interesting, but it probably wouldn't have quite the same resonance: I don't think that there are many other fields where being rigorous is seen as an end in itself, and people already tend to think of post-rigorous proficiency in, say, music as a desireable state to reach.
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