🔖 What is good mathematics? | Terry Tao

What is good mathematics? by Terence Tao(arxiv.org)
Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.

This looks like a cool little paper.

Update 3/17/17

Some thoughts after reading

And indeed it was. The opening has lovely long (though possibly incomplete) list of aspects of good mathematics toward which mathematicians should strive. The second section contains an interesting example which looks at the history of a theorem and it’s effect on several different areas. To me most of the value is in thinking about the first several pages. I highly recommend this to all young budding mathematicians.

In particular, as a society, we need to be careful of early students in elementary and high school as well as college as the pedagogy of mathematics at these lower levels tends to weed out potential mathematicians of many of these stripes. Students often get discouraged from pursuing mathematics because it’s “too hard” often because they don’t have the right resources or support. These students, may in fact be those who add to the well-roundedness of the subject which help to push it forward.

I believe that this diverse and multifaceted nature of “good mathematics” is very healthy for mathematics as a whole, as it it allows us to pursue many different approaches to the subject, and exploit many different types of mathematical talent, towards our common goal of greater mathematical progress and understanding. While each one of the above attributes is generally accepted to be a desirable trait to have in mathematics, it can become detrimental to a field to pursue only one or two of them at the expense of all the others.

As I look at his list of scenarios, it also reminds me of how areas within the humanities can become quickly stymied. The trouble in some of those areas of study is that they’re not as rigorously underpinned, as systematic, or as brutally clear as mathematics can be, so the fact that they’ve become stuck may not be noticed until a dreadfully much later date. These facts also make it much easier and clearer in some of these fields to notice the true stars.

As a reminder for later, I’ll include these scenarios about research fields:

  • A field which becomes increasingly ornate and baroque, in which individual
    results are generalised and refined for their own sake, but the subject as a
    whole drifts aimlessly without any definite direction or sense of progress;
  • A field which becomes filled with many astounding conjectures, but with no
    hope of rigorous progress on any of them;
  • A field which now consists primarily of using ad hoc methods to solve a collection
    of unrelated problems, which have no unifying theme, connections, or purpose;
  • A field which has become overly dry and theoretical, continually recasting and
    unifying previous results in increasingly technical formal frameworks, but not
    generating any exciting new breakthroughs as a consequence; or
  • A field which reveres classical results, and continually presents shorter, simpler,
    and more elegant proofs of these results, but which does not generate any truly
    original and new results beyond the classical literature.
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Why some infinities are bigger than others | Aeon Essays

Why some infinities are bigger than others by A. W. Moore(Aeon Essays)
Georg Cantor showed that some infinities are bigger than others. Did he assault mathematical wisdom or corroborate it?

Continue reading “Why some infinities are bigger than others | Aeon Essays”

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Emily Riehl’s new category theory book has some good company

Emily Riehl's new category theory book has some good company.
Emily Riehl's new category theory book has some good company. It's a beautiful book by the way
Emily Riehl’s new category theory book has some good company. It’s a beautiful book by the way.

Instagram filter used: Clarendon

Photo taken at: UCLA Bookstore

I just saw Emily Riehl‘s new book Category Theory in Context on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one wouldn’t purchase the physical version.

I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.

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Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

An informal orientation for those taking math classes from Dr. Michael Miller through UCLA Extension.

Congratulations on your new math class, and welcome to the “family”!

Beginners Welcome!

Invariably the handful of new students every year eventually figure the logistics of campus out, but it’s easier and more fun to know some of the options available before you’re comfortable halfway through the class. To help get you over the initial hump, I’ll share a few of the common questions and tips to help get you oriented. Others are welcome to add comments and suggestions below. If you have any questions, feel free to ask anyone in the class, we’re all happy to help.

First things first, for those who’ve never visited UCLA before, here’s a map of campus to help you orient yourself. Using the Waze app on your smartphone can also be incredibly helpful in getting to campus more quickly through the tail end of rush hour traffic.

Whether you’re a professional mathematician, engineer, physicist, physician, or even a hobbyist interested in mathematics you’ll be sure to get something interesting out of Dr. Miller’s math courses, not to mention the camaraderie of 20-30 other “regulars” with widely varying backgrounds (from actors to surgeons and evolutionary theorists to engineers) who’ve been taking almost everything Mike has offered over the years (and yes, he’s THAT good — we’re sure you’ll be addicted too.) Whether you’ve been away from serious math for decades or use it every day or even if you’ve never gone past Calculus or Linear Algebra, this is bound to be the most entertaining thing you can do with your Tuesday nights in the Autumn and Winter. If you’re not sure what you’re getting into (or are scared a bit by the course description), I highly encourage to come and join us for at least the first class before you pass up on the opportunity. I’ll mention that the greater majority of new students to Mike’s classes join the ever-growing group of regulars who take almost everything he teaches subsequently.

Don’t be intimidated if you feel like everyone in the class knows each other fairly well — most of us do. Dr. Miller and mathematics can be addictive so many of us have been taking classes from him for 5-20+ years, and over time we’ve come to know each other.

Tone of Class

If you’ve never been to one of Dr. Miller’s classes before, they’re fairly informal and he’s very open to questions from those who don’t understand any of the concepts or follow his reasoning. He’s a retired mathematician from RAND and long-time math professor at UCLA. Students run the gamut from the very serious who read multiple textbooks and do every homework problem to hobbyists who enjoy listening to the lectures and don’t take the class for a grade of any sort (and nearly every stripe in between). He’ll often recommend a textbook that he intends to follow, but it’s never been a “requirement” and more often that not, the bookstore doesn’t list or carry his textbook until the week before class. (Class insiders will usually find out about the book months before class and post it to the Google Group – see below).

His class notes are more than sufficient for making it through the class and doing the assigned (optional) homework. He typically hands out homework in handout form, so the textbook is rarely, if ever, required to make it through the class. Many students will often be seen reading various other texts relating to the topic at hand as they desire. Usually he’ll spend an 45-60 minutes at the opening of each class after the first to go over homework problems or questions that anyone has.

For those taking the class for a grade or pass/fail, his usual policy is to assign a take home problem set around week 9 or 10 to be handed in at the penultimate class. [As a caveat, make sure you check his current policy on grading as things may change, but the preceding has been the usual policy for a decade or more.]

Parking Options

Lot 9 – Located at the northern terminus of Westwood Boulevard, one can purchase a parking pass for about $12 a day at the kiosk in the middle of the street just before Westwood Blvd. ends. The kiosk is also conveniently located right next to the parking structure. If there’s a basketball game or some other major event, Lot 8 is just across the street as well, though it’s just a tad further away from the Math Sciences Building. Since more of the class uses this as their parking structure of choice, there is always a fairly large group walking back there after class for the more security conscious.

Lot 2 – Located off of Hilgard Avenue, this is another common option for easy parking as well. While fairly close to class, not as many use it as it’s on the quieter/darker side of campus and can be a bit more of a security issue for the reticent.

Tip: For those opting for on-campus parking, one can usually purchase a quarter-long parking pass for a small discount at the beginning of the term.

Westwood Village and Neighborhood – Those looking for less expensive options street parking is available in the surrounding community, but use care to check signs and parking meters as you assuredly will get a ticket. Most meters in the surrounding neighborhoods end at either 6pm or 8pm making parking virtually free (assuming you’re willing to circle the neighborhood to find one of the few open spots.)

There are a huge variety of lots available in the Village for a range of prices, but the two most common, inexpensive, and closer options seem to be:

  • Broxton Avenue Public Parking at 1036 Broxton Avenue just across from the Fox Village and Bruin Theaters – $3 for entering after 6pm / $9 max for the day
  • Geffen Playhouse Parking at 10928 Le Conte Ave. between Broxton and Westwood – price varies based on the time of day and potential events (screenings/plays in Westwood Village) but is usually $5 in the afternoon and throughout the evening

Dining Options

More often than not a group of between 4 and 15 students will get together every evening before class for a quick bit to eat and to catch up and chat. This has always been an informal group and anyone from class is more than welcome to join. Typically we’ll all meet in the main dining hall of Ackerman Union (Terrace Foodcourt, Ackerman Level 1) between 6 and 6:30 (some with longer commutes will arrive as early as 3-4pm, but this can vary) and dine until about 6:55pm at which time we walk over to class.

The food options on Ackerman Level 1 include Panda Express, Rubio’s Tacos, Sbarro, Wolfgang Puck, and Greenhouse along with some snack options including Wetzel’s Pretzels and a candy store. One level down on Ackerman A-level is a Taco Bell, Carl’s Jr., Jamba Juice, Kikka, Buzz, and Curbside, though one could get takeout and meet the rest of the “gang” upstairs.

There are also a number of other on-campus options as well though many are a reasonable hike from the class location. The second-closest to class is the Court of Sciences Student Center with a Subway, Yoshinoya, Bombshelter Bistro, and Fusion.

Naturally, for those walking up from Westwood Village, there are additional fast food options like In-N-Out, Chick-fil-A, Subway, and many others.

Killing Time

For those who’ve already eaten or aren’t hungry, you’ll often find one or more of us browsing the math and science sections of the campus bookstore on the ground level of Ackerman Union to kill time before class. Otherwise there are usually a handful of us who arrive a half an hour early and camp out in the classroom itself (though this can often be dauntingly quiet as most use the chance to catch up on reading here.) If you arrive really early, there are a number of libraries and study places on campus. Boelter Hall has a nice math/science library on the 8th Floor.

Mid-class Break Options

Usually about halfway through class we’ll take a 10-12 minute coffee break. For those with a caffeine habit or snacking urges, there are a few options:

Kerckhoff Hall Coffee Shop is just a building or two over and is open late as snack stop and study location. They offer coffee and various beverages as well as snacks, bagels, pastries, and ice cream. Usually 5-10 people will wander over as a group to pick up something quick.

The Math Sciences Breezeway, just outside of class, has a variety of soda, coffee, and vending machines with a range of beverages and snacks. Just a short walk around the corner will reveal another bank of vending machines if your vice isn’t covered. The majority of class will congregate in the breezeway to chat informally during the break.

The Court of Sciences Student Center, a four minute walk South, with the restaurant options noted above if you need something quick and more substantial, though few students use this option at the break.

Bathrooms – The closest bathrooms to class are typically on the 5th floor of the Math Sciences Building. The women’s is just inside the breezeway doors and slightly to the left. The men’s rooms are a bit further and are either upstairs on the 6th floor (above the women’s), or a hike down the hall to the left and into Boelter hall. I’m sure the adventurous may find others, but take care not to get lost.

Informal Class Resources

Google Group

Over the years, as an informal resource, members of the class have created and joined a private Google Group (essentially an email list-serv) to share thoughts, ideas, events, and ask questions of each other. There are over 50 people in the group, most of whom are past Miller students, though there are a few other various mathematicians, physicists, engineers, and even professors. You can request to join the private group to see the resources available there. We only ask that you keep things professional and civil and remember that replying to all reaches a fairly large group of friends. Browsing through past messages will give you an idea of the types of posts you can expect. The interface allows you to set your receipt preferences to one email per message posted, daily digest, weekly digest, or no email (you’re responsible for checking the web yourself), so be sure you have the setting you require as some messages are more timely than others. There are usually only 1-2 posts per week, so don’t expect to be inundated.

Study Groups

Depending on students’ moods, time requirements, and interests, we’ve arranged informal study groups for class through the Google Group above. Additionally, since Dr. Miller only teaches during the Fall and Winter quarters, some of us also take the opportunity to set up informal courses during the Spring/Summer depending on interests. In the past, we’ve informally studied Lie Groups, Quantum Mechanics, Algebraic Geometry, and Category Theory in smaller groups on the side.

Dropbox

As a class resource, some of us share a document repository via Dropbox. If you’d like access, please make a post to the Google Group.

Class Notes

Many people within the class use Livescribe.com digital pens to capture not only the written notes but the audio discussion that occurred in class as well (the technology also links the two together to make it easier to jump around within a particular lecture). If it helps to have a copy of these notes, please let one of the users know you’d like them – we’re usually pretty happy to share. If you miss a class (sick, traveling, etc.) please let one of us know as the notes are so unique that it will be almost like you didn’t miss anything at all.

You can typically receive a link to the downloadable version of the notes in Livescribe’s Pencast .pdf format. This is a special .pdf file but it’s a bit larger in size because it has an embedded audio file in it that is playable with the more recent version of Adobe Reader X (or above) installed. (This means to get the most out of the file you have to download the file and open it in Reader X to get the audio portion. You can view the written portion in most clients, you’ll just be missing out on all the real fun and value of the full file.) With the notes, you should be able to toggle the settings in the file to read and listen to the notes almost as if you were attending the class live.

Viewing and Playing a Pencast PDF

Pencast PDF is a new format of notes and audio that can play in Adobe Reader X or above.

You can open a Pencast PDF as you would other PDF files in Adobe Reader X. The main difference is that a Pencast PDF can contain ink that has associated audio—called “active ink”. Click active ink to play its audio. This is just like playing a Pencast from Livescribe Online or in Livescribe Desktop. When you first view a notebook page, active ink appears in green type. When you click active ink, it turns gray and the audio starts playing. As audio playback continues, the gray ink turns green in synchronization with the audio. Non-active ink (ink without audio) is black and does not change appearance.

Audio Control Bar

Pencast PDFs have an audio control bar for playing, pausing, and stopping audio playback. The control bar also has jump controls, bookmarks (stars), and an audio timeline control.

Active Ink View Button

There is also an active ink view button. Click this button to toggle the “unwritten” color of active ink from gray to invisible. In the default (gray) setting, the gray words turn green as the audio plays. In the invisible setting, green words seem to write themselves on blank paper as the audio plays.

History

For those interested in past years’ topics, here’s the list I’ve been able to put together thus far:

Fall 2006: Complex Analysis
Winter 2007: Field Theory
Fall 2007: Algebraic Topology
Winter 2008: Integer Partitions
Fall 2008: Calculus on Manifolds
Winter 2009: Calculus on Manifolds: The Sequel
Fall 2009: Group Theory
Winter 2010: Galois Theory
Fall 2010: Differential Geometry
Winter 2011: Differential Geometry II
Fall 2011: p-Adic Analysis
Winter 2012: Group Representations
Fall 2012: Set Theory
Winter 2013: Functional Analysis
Fall 2013: Number Theory (Skipped)
Winter 2014: Measure Theory
Fall 2014: Introduction to Lie Groups and Lie Algebras Part I
Winter 2015: Introduction to Lie Groups and Lie Algebras Part II
Fall 2015: Algebraic Number Theory
Winter 2016: Algebraic Number Theory: The Sequel
Fall 2016: Introduction to Complex Analysis, Part I
Winter 2017: Introduction to Complex Analysis, Part II

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I’m a sucker for references to math and pastry

Math is about understanding processes and not just eating end results.

W

hat can I say? I’m a sucker for references to math and pastry.

 

“One of the wonderful features of math is that, like with pastry, it can use quite simple ingredients to make very complicated situations. This can also make it rather offputting, like making puff pastry. Actually, I don’t think puff pastry is that difficult if you follow the instructions carefully. But even if you don’t want to try doing it yourself, perhaps you can still enjoy the fact that such simple ingredients can turn into delicious puff pastry. Math is about understanding processes and not just eating end results.”

Eugenia Cheng, mathematician, amateur chef
in How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics (Basic Books, 2015)
 
How to Bake Pi
How to Bake Pi
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Why Aren’t Math Textbooks More Straightforward?

Math textbooks often seem difficult, obtuse, and often incomprehensible. Here are some hints and tips for making the situation better for all students.

Some General Advice for Math Students of All Ages

I recently saw the question “Why aren’t math textbooks more straightforward?” on Quora.

In fact, I would argue that most math textbooks are very straightforward!

The real issue most students are experiencing is one of relativity and experience.  Mathematics is an increasingly sophisticated, cumulative, and more complicated topic the longer you study it. Fortunately, over time, it also becomes easier, more interesting, and intriguingly more beautiful.

As an example of what we’re looking at and what most students are up against, let’s take the topic of algebra. Typically in the United States one might take introductory algebra in eighth grade before taking algebra II in ninth or tenth grade. (For our immediate purposes, here I’m discounting the potential existence of a common pre-algebra course that some middle schools, high schools, and even colleges offer.)  Later on in college, one will exercise one’s algebra muscles in calculus and may eventually get to a course called abstract algebra as an upper-level undergraduate (in their junior or senior years).  Most standard undergraduate abstract algebra textbooks will cover ALL of the material that was in your basic algebra I and algebra II texts in about four pages and simply assume you just know the rest! Naturally, if you started out with the abstract algebra textbook in eighth grade, you’d very likely be COMPLETELY lost. This is because the abstract algebra textbook is assuming that you’ve got some significant prior background in mathematics (what is often referred to in the introduction to far more than one mathematics textbook as “mathematical sophistication”, though this phrase also implicitly assumes knowledge of what a proof is, what it entails, how it works, and how to actually write one).

Following the undergraduate abstract algebra textbook there’s even an additional graduate level course (or four) on abstract algebra (or advanced subtopics like group theory, ring theory, field theory, and Galois theory) that goes into even more depth and subtlety than the undergraduate course; the book for this presumes you’ve mastered the undergraduate text and goes on faster and further.

A Weightlifting Analogy

To analogize things to something more common, suppose you wanted to become an Olympic level weightlifter.  You’re not going to go into the gym on day one and snatch and then clean & jerk 473kg! You’re going to start out with a tiny fraction of that weight and practice repeatedly for years slowly building up your ability to lift bigger and bigger weights. More likely than not, you’ll also very likely do some cross-training by running, swimming, and even lifting other weights to strengthen your legs, shoulder, stomach, and back.  All of this work may eventually lead you to to win the gold medal in the Olympics, but sooner or later someone will come along and break your world record.

Mathematics is certainly no different: one starts out small and with lots of work and practice over time, one slowly but surely ascends the rigors of problems put before them to become better mathematicians. Often one takes other courses like physics, biology, and even engineering courses that provide “cross-training.”  Usually when one is having issues in a math class it’s because they’re either somehow missing something that should have come before or because they didn’t practice enough in their prior classwork to really understand all the concepts and their subtleties.  As an example, the new material in common calculus textbooks is actually very minimal – the first step in most problems is the only actual calculus and the following 10 steps are just practicing one’s algebra skills. It’s usually in carrying out the algebra that one makes more mistakes than in the actual calculus.

Often at the lower levels of grade-school mathematics, some students can manage to just read a few examples and just seem to “get” the answers without really doing a real “work out.” Eventually they’ll come to a point at which they hit a wall or begin having trouble, and usually it comes as the result of not actually practicing their craft.  One couldn’t become an Olympic weightlifter by reading books about weightlifting, they need to actually get in the gym and workout/practice. (Of course, one of the places this analogy breaks down is that weightlifting training is very linear and doesn’t allow one to “skip around” the way one could potentially in a mathematics curriculum.)

I’m reminded of a quote by mathematician Pierre Anton Grillet: “…algebra is like French pastry: wonderful, but cannot be learned without putting one’s hands to the dough.” It is one of the most beautiful expressions of the recurring sentiment written by almost every author into the preface of nearly every mathematics text at or above the level of calculus. They all exhort their students to actually put pencil to paper and work through the logic of their arguments and the exercises to learn the material and gain some valuable experience.  I’m sure that most mathematics professors will assure you that in the end, only a tiny fraction of their students actually do so. Some of the issue is that these exhortations only come in textbooks traditionally read at the advanced undergraduate level, when they should begin in the second grade.

Patissiere puts their hands to a pie crust in a rustic kitchen
“Algebra is like French Pastry: wonderful, but cannot be learned without putting one’s hands to the dough.” -Pierre Anton Grillet

“It’s Easy to See”

A common phrase in almost every advanced math textbook on the planet is the justification, “It’s easy to see.” The phrase, and those like it, should be a watchword for students to immediately be on their guard!  The phrase is commonly used in proofs, discussions, conversations, and lectures in which an author or teacher may skip one or more steps which she feels should be obvious to her audience, but which, in fact, are far more commonly not obvious.

It’s become so cliche that some authors actually mention specifically in their prefaces that they vow not to use the phrase, but if they do so, they usually let slip some other euphemism that is its equivalent.

The problem with the phrase is that everyone, by force of their own circumstances and history, will view it completely differently.  A step that is easy for someone with a Ph.D. who specialized in field theory to “see” may be totally incomprehensible for a beginning student of algebra I in the same way that steps that were easy for Girgory Perelman to see in his proof of the Poincaré conjecture were likewise completely incomprehensible for teams of multiple tenured research professors of mathematics to see. (cross reference: The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea (Walker & Co., 2007))

How to Actively Read a Math Text

The Problem

So how are students to proceed? It will certainly help to see a broader road map of what lies ahead and what the expected changes in terrain will look like. It will also help greatly if students have a better idea how to approach mathematics for themselves and even by themselves in many cases.

In my opinion, the most common disconnect occurs somewhere between high school mathematics and early college mathematics (usually a calculus sequence, linear algebra) and then again between linear algebra/differential equations (areas which usually have discussion followed by examples and then crank-out problems) and higher abstract mathematical areas like analysis, abstract algebra, topology (areas in which the definition-theorem-proof cycle of writing is more common and seemingly more incomprehensible to many).

The first big issue in early college mathematics is the increased speed at which college courses move. Students used to a slower high school pace where the teachers are usually teaching to the middle or lower end of the class get caught unaware as their college professors teach to the higher ability students and aren’t as afraid to leave the lower end of the spectrum behind. Just like high school athletes are expected to step up their game when they make the transition to college and similarly college athletes who go pro, mathematics students should realize they’re expected to step up their game at the appropriate times.

Often math students (and really any student of any subject) relies on the teacher assigning readings or problems from their book rather than excersizing their curiosity to more avidly and broadly explore the material on their own.  If they can take the guidance of their teacher as well as that of the individual authors of books, they may make it much further on their own.  High school teachers often skip sections of textbooks for time, but students should realize that there is profitable and interesting material that they’re skipping.  Why not go and read it on their own?

Earlier I mentioned that an average undergraduate abstract algebra textbook might cover the totality of a high school algebra textbook in about three pages.  What does this mean for upper level mathematics students?  It almost always means that the density of material in these books is far greater than that of their earlier textbooks.  How is this density arrived at?  Authors of advanced textbooks leave out far more than they’re able to put in, otherwise their 300 page textbooks, if written at the same basic level as those that came before would be much more ponderous 1000+ page textbooks.  What are they leaving out?  Often they’re leaving out lots of what might be useful discussion, but more often, they’re leaving out lots of worked out examples. For example, a high school text will present a definition or concept and then give three or more illustrations or examples of problems relating to the concept.  The exercises will then give dozens of additional drill problems to beat the concept to death.  This type of presentation usually continues up to the level of calculus where one often sees massive tomes in the 800+ page length. Math texts after this point generally don’t go much over 300 pages as a rule, and it’s primarily because they’re leaving the examples out of the proverbial equation.

The Solution

How does one combat this issue?  Students need to more actively think back to the math they’ve taken previously and come up with their own simple examples of problems, and work though them on their own.  Just because the book doesn’t give lots of examples doesn’t mean that they don’t exist.

In fact, many textbooks are actually presenting examples, they’re just hiding them with very subtle textual hints. Often in the presentation of a concept, the author will leave out one or more steps in a proof or example and hint to the student that they should work through the steps themselves. (Phrases like: “we leave it to the reader to verify” or “see example 2.”) Sometimes this hint comes in the form of that dreaded phrase, “It’s easy to see.” When presented with these hints, it is incumbent (or some students may prefer the word encumbering) on the student to think through the missing steps or provide the missing material themselves.

While reading mathematics, students should not only be reading the words and following the steps, but they should actively be working their way through all of the steps (missing or not) in each of the examples or proofs provided. They must read their math books with pencil and paper in hand instead of the usual format of reading their math book and then picking up paper and pencil to work out problems afterwards.  Most advanced math texts suggest half a dozen or more problems to work out within the text itself before presenting a dozen or more additional problems usually in a formal section entitled “Exercises”. Students have to train themselves to be thinking about and working out the “hidden” problems within the actual textual discussion sections.

Additionally, students need to consider themselves “researchers” or think of their work as discovery or play.  Can they come up with their own questions or exercises that relate the concepts they’ve read about to things they’ve done in the past? Often asking the open ended question, “What happens if I…” can be very useful.  One has to imagine that this is the type of “play” that early mathematicians like Euclid, Gauss, and Euler did, and I have to say, this is also the reason that they discovered so many interesting properties within mathematics.  (I always like to think that they were the beneficiaries of “picking the lowest hanging fruit” within mathematics – though certainly they discovered some things that took some time to puzzle out; we take some of our knowledge for granted as sitting on the shoulders of giants does allow us to see much further than we could before.)

As a result of this newly discovered rule, students will readily find that while they could read a dozen pages of their high school textbooks in just a few minutes, it may take them between a half an hour to two hours to properly read even a single page of an advanced math text. Without putting in this extra time and effort they’re going to quickly find themselves within the tall grass (or, more appropriately weeds).

Another trick of advanced textbooks is that, because they don’t have enough time or space within the primary text itself, authors often “hide” important concepts, definitions, and theorems within the “exercises” sections of their books.  Just because a concept doesn’t appear in the primary text doesn’t mean it isn’t generally important.  As a result, students should always go out of their way to at least read through all of the exercises in the text even if they don’t spend the time to work through them all.

One of the difficult things about advanced abstract mathematics is that it is most often very cumulative and even intertwined, so when one doesn’t understand the initial or early portions of a textbook, it doesn’t bode well for the later sections which require one to have mastered the previous work.  This is even worse when some courses build upon the work of earlier courses, so for example, doing well in calculus III requires that one completely mastered calculus I. At some of the highest levels like courses in Lie groups and Lie algebras requires that one mastered the material in multiple other prior courses like analysis, linear algebra, topology, and abstract algebra. Authors of textbooks like these will often state at the outset what material they expect students to have mastered to do well, and even then, they’ll often spend some time giving overviews of relevant material and even notation of these areas in appendices of their books.

As a result of this, we can take it as a general rule: “Don’t ever skip anything in a math textbook that you don’t understand.” Keep working on the concepts and examples until they become second nature to you.

Finally, more students should think of mathematics as a new language.  I’ve referenced the following Galileo quote before, but it bears repeating (emphasis is mine):

Philosophy is written in this grand book, the universe which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.

Galileo Galilee (1564–1642) in Il saggiatore (The assayer)

Though mathematical notation has changed drastically (for the better, in my opinion) since Galileo’s time, it certainly has its own jargon, definitions, and special notations.  Students should be sure to spend some time familiarizing themselves with current modern notation, and especially the notation in the book that they choose.  Often math textbooks will have a list of symbols and their meanings somewhere in the end-papers or the appendices. Authors usually go out of their way to introduce notation somewhere in either the introduction, preface, appendices, or often even in an introductory review chapter in which they assume most of their students are very familiar with, but they write it anyway to acclimate students to the particular notation they use in their text.  This notation can often seem excessive or even obtuse, but generally it’s very consistent across disciplines within mathematics, but it’s incredibly useful and necessary in making often complex concepts simple to think about and communicate to others. For those who are lost, or who want help delving into areas of math seemingly above their heads, I highly recommend the text Mathematical Notation: A Guide for Engineers and Scientists by my friend Edward R. Scheinerman as a useful guide.

A high school student may pick up a textbook on Lie Groups and be astounded at the incomprehensibility of the subject, but most of the disconnect is in knowing and understanding the actual language in which the text is written.  A neophyte student of Latin would no sooner pick up a copy of Cicero and expect to be able to revel in the beauty and joy of the words or their meaning without first spending some time studying the vocabulary, grammar, and syntax of the language.  Fortunately, like Latin, once one has learned a good bit of math, the notations and definitions are all very similar, so once you can read one text, you’ll be able to appreciate a broad variety of others.

Actively Reading a Mathematics Text Review:

  • Work through the steps of everything within the text
  • Come up with your own examples
  • Work through the exercises
  • Read through all the exercises, especially the ones that you don’t do
  • Don’t ever skip anything you don’t fully understand
  • Math is a language: spend some time learning (memorizing) notation

Exceptions

Naturally there are exceptions to the rule. Not all mathematics textbooks are great, good, or even passable.  There is certainly a spectrum of textbooks out there, and there are even more options at the simpler (more elementary) end, in part because of there is more demand.  For the most part, however, most textbooks are at least functional.  Still one can occasionally come across a very bad apple of a textbook.

Because of the economics of textbook publishing, it is often very difficult for a textbook to even get published if it doesn’t at least meet a minimum threshold of quality. The track record of a publisher can be a good indicator of reasonable texts. Authors of well-vetted texts will often thank professors who have taught their books at other universities or even provide a list of universities and colleges that have adopted their texts. Naturally, just because 50 colleges have adopted a particular text doesn’t necessarily mean that that it is necessarily of high quality.

One of the major issues to watch out for is using the textbook written by one’s own professor.  While this may not be an issue if your professor is someone like Serge Lang, Gilbert Strang, James Munkres, Michael Spivak, or the late Walter Rudin, if your particular professor isn’t supremely well known in his or her field, is an adjunct or associate faculty member, or is a professor at a community college, then: caveat emptor.

Since mathematics is a subject about clear thinking, analysis, and application of knowledge, I recommend that students who feel they’re being sold a bill of goods in their required/recommended textbook(s), take the time to look at alternate textbooks and choose one that is right for themselves.  For those interested in more on this particular sub-topic I’ve written about it before: On Choosing Your own Textbooks.

Parting Advice

Often, even with the best intentions, some authors can get ahead of themselves or the area at hand is so advanced that it is difficult to find a way into it. As an example, we might consider Lie groups and algebras, which is a fascinating area to delve into. Unfortunately it can take several years of advanced work to get to a sufficient level to even make a small dent into any of the textbooks in the area, though some research will uncover a handful of four textbooks that will get one quite a way into the subject with a reasonable background in just analysis and linear algebra.

When one feels like they’ve hit a wall, but still want to struggle to succeed, I’m reminded of the advice of revered mathematical communicator Paul Halmos, whose book Measure Theory needed so much additional background material, that instead of beginning with the traditional Chapter 1, he felt it necessary to include a Chapter 0 (he actually called his chapters “sections” in the book) and even then it had enough issueshewas cornered into writing the statement:

…[the reader] should not be discouraged, if on first reading of section 0, he finds that he does not have the prerequisites for reading the prerequisites.

Paul Halmos in Measure Theory(1950)

This is essentially the mathematician’s equivalent of the colloquialism “Fake it ’til you make it.”

When all else fails, use this adage, and don’t become discouraged. You’ll get there eventually!

 

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