
What is algebra? - ColinWright
http://profkeithdevlin.org/2011/11/20/what-is-algebra/
======
asolove
As good a time as any to go re-read "A mathematician's lament" [0] which
begins:

A musician wakes from a terrible nightmare. In his dream he finds himself in a
society where music education has been made mandatory. “We are helping our
students become more competitive in an increasingly sound-filled world.”
Educators, school systems, and the state are put in charge of this vital
project. Studies are commissioned, committees are formed, and decisions are
made— all without the advice or participation of a single working musician or
composer.

[0]: [http://worrydream.com/refs/Lockhart-
MathematiciansLament.pdf](http://worrydream.com/refs/Lockhart-
MathematiciansLament.pdf)

~~~
purplelobster
Very interesting read. It's not only a problem with mathematics though, it's
nearly every subject in school, whether it's science, languages or math. It
seems to me that the problem is that to teach well, you need good, autonomous
teachers, and that just doesn't scale. Like the author said, you can't teach
teaching.

I'm of the unpopular opinion that digital learning is the only possible
solution for this. I'm talking about a digital tutor with infinite patience,
with the best qualities of the best teachers. I'm not talking about a robot
that passes the Turing test, just one that passes the test for the narrow
field of teaching a specific subject.

People say you need real teachers, you need the human touch, but if the
average teacher sucks (and they do), then I'd rather be taught by software.
And I think it's just one of those things people say because it sounds true:
"you need the human touch", just like people said they prefer real books over
e-books, face to face conversations over texting, navigating by feel over
using the GPS etc. These sentiments almost always turn out to be wrong.

~~~
Volpe
> Like the author said, you can't teach teaching.

Yes you can. I mean sure there is an explanatory gap with things like empathy
and identifying scaffolding opportunities. But that just means you can't teach
ALL of teaching with a book, experience and guidance is also required.

> I'm talking about a digital tutor with infinite patience...

One that can empathise with a students particular background/learning style
and understand what might work to convey a novel concept better?

> People say you need real teachers, you need the human touch, but if the
> average teacher sucks (and they do), then I'd rather be taught by software.

It's not about the human touch. It's about empathy, and expert diagnosis of
learning conditions, which the worst teacher does a better job of than the
best computer/software. Just because most of our teachers suck doesn't mean
software is the better solution. Better teachers are the solution, perhaps
that is better education for teachers, or perhaps it's more communication from
the realm of pedegogy down to the teaching curriculum that teachers are taught
from.

Software can assist teachers, and can even replace certain aspects of
teaching. But you will leave a lot of students behind if you try to replace
teachers completely.

~~~
wtallis
You can't teach _teaching_ in isolation. A math teacher has to know math and
teaching, but that's not really how they're trained - an undergraduate Math
Education curriculum diverges from the Math curriculum shortly after calculus,
which is far too soon.

~~~
Volpe
An undergraduate math education is more than enough to teach the current high
school math curriculum.

From my point of view though, if my statement above is true, that is very sad.
As I don't think it should be. The math curriculum is horribly deficient. But
software doesn't seem to be the answer to fixing the curriculum.

~~~
wtallis
> _An undergraduate math education is more than enough to teach the current
> high school math curriculum._

Is that an undergraduate Math degree, or a Math Education degree? If the
former, you don't get training in pedagogy. If the latter, you don't get
enough proof-based classes to properly understand what math is or to fully
understand the subjects you'll teach.

------
electrograv
_> ... numbers in general, not particular numbers. And the human brain is not
naturally suited to think at that level of abstraction._

This is so wrong (at least as a stereotype).

All throughout my early education I _HATED_ arithmetic, and found almost
everything about it mind-numbingly boring and repulsively repetitive. At that
point in my life, I hated math. The moment I encountered algebra though, it
was "love at first sight", and ever since I've absolutely been fascinated and
engaged with every type of high-level math I encounter (the more abstract, the
better). And not just "fascinated" in the "I like it" sense -- math, CS, etc.
is more easy/natural to me than most humanities subjects, by far.

So although I can only speak for myself, I quite disagree with any claim that
the brain isn't naturally suited to abstract thinking. While I know not all
people think the way I do, certainly quite a few do.

~~~
ctdonath
Watching my kids play DragonBox, methinks a major problem with teaching
algebra is the insistence on forcing steps from arithmetic to algebra, bogging
down in numeric & non-numeric symbols which students have little or no
cognitive relationship with at that age. Starting with pure algebraic
concepts, devoid of explicit numeric meaning, may be much easier to absorb
_then_ transition into meaning-laden symbols.

~~~
pbhjpbhj
Is that learning mathematics though or is it just learning symbolic
manipulations by rote.

Yes some rote learning is necessary - and I'd warrant very useful in maths.
However, generally in order to build on what you're learning you need to
understand why you should perform certain actions.

There seems little point in learning to simply mechanically do the actions
necessary to solve an equation. It is the meaning that is the reason for doing
the learning and I worry that the last transition will be missed and make the
entire prologue void of worth.

~~~
ctdonath
_generally in order to build on what you 're learning you need to understand
why you should perform certain actions._

Before knowing _why_ you should perform certain actions, it helps to know that
you _can_ perform certain actions. Methinks getting these basic concepts
("combine something with its inverse and it disappears", "thing over same
thing is 1", "1 'times' something is that something", ...) into a kid's head
_very_ early is a good thing - may not yet understand why, but it's a mental
tool that can be applied. Don't underestimate the value of having tools even
if you don't know why/how they work; give a kid a hammer and he'll figure out
it's for pounding nails.

~~~
pbhjpbhj
> _give a kid a hammer and he 'll figure out it's for pounding nails._ //

How old is the kid? I think you'll just end up with everything broken, unless
you present it with the nails then the combination of the two is unlikely to
happen naturally I feel until the child is quite old (if then).

~~~
ctdonath
[http://steelwhitetable.org/media/images/cavlin-banging-
nails...](http://steelwhitetable.org/media/images/cavlin-banging-nails-in-
table.gif)

I don't think the basics of algebra will have such results, but both are
indeed tools that a kid will figure out a good use for _if_ such tools are on
hand and have been played with enough.

I could credit a lot of my creative skill to spending inordinate time as a kid
just fiddling with a broad range of tools in the basement, regardless of
whether I understood them at the time.

------
ctdonath
Compare the article's premise to the HN-presented app _DragonBox_
[https://www.hnsearch.com/search#request/all&q=dragonbox](https://www.hnsearch.com/search#request/all&q=dragonbox)
which abstracts the concepts of algebra into a tile-based game which young
children (3yo even!) can learn and enjoy.

~~~
kemayo
I will second this. My kid was able to play it at 3, and do decently, despite
the concepts involved being (approximately) identical to solving equations.

------
pflats
My favorite way to define "algebra" to my students is to go back to its
etymology, from the Persian textbook "Al-kitāb al-mukhtaṣar fī ḥisāb _al-ğabr_
wa’l-muqābala", or "The Compendious Book on Calculation by Completion and
Balancing".

Specifically (borrowing from Wikipedia):

"The al-ğabr (in Arabic script 'الجبر') ("forcing " or "restoring") operation
is moving a deficient quantity from one side of the equation to the other
side. In an al-Khwarizmi's example (in modern notation), "x^2 = 40x − 4x^2" is
transformed by al-ğabr into "5x^2 = 40x"."[1]

I'll also tell them that Algebra is solving equations through the use of
inverses. I admit that both definitions are reductive but mostly to the point.

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

------
matho
I am not sure of the pedagogical benefit of what is being described here - a
separation of arithmetical and algebraic thinking at the level of school
mathematics.

By the distinction that Prof Devlin tries to make, primary school subtraction
is typically taught/learned in an 'algebraic' way (logical reasoning to invert
addition). This makes it difficult to understand what he is trying to say.

I most strongly doubt the claim that students who are strong in arithmetic
find it harder to learn algebra. It's obvious that students with good
arithmetic skills are more quickly able to find value and purpose in algebra.

~~~
bitwize
It may be taught that way, but do kids learn it that way? Kids learn a bunch
of "facts", and are tested on their capability to repeat those facts in a
timely manner. Then they are taught rote-level procedures for combining those
facts in order to do arithmetic on multi-digit numbers. The fact that
subtraction is the opposite of addition gets demoted to "interesting bit of
trivia".

For most school kids, algebra is the first place where the art of mathematics
comes into play: where you are given a problem, and are not told specifically
how to get from here to a solution. You have to figure that out yourself with
the aid of mathematical tools.

For most people, who aren't used to the art of logical thinking, that is
crushing.

~~~
saraid216
Slightly tangential, but every time I work with someone on basic kinematics
and they say, "Oh, well, if I just flip the sign from positive to negative, I
get the right answer! I'm done!" a little piece of me dies.

It's like... you're not even trying to understand the concepts, are you?

~~~
matho
I think this is hard because it's not immediately obvious that direction is
such an important concept in kinematics. It's probably the most important
thing to get across when teaching the subject, but you have to battle against
prior knowledge:

Math students (that is, everyone) earlier learned that this sign-flipping
strategy is ok and perhaps an encouraged shortcut when e.g. subtracting.

(see also that Khan Academy video - "first do the multiplication, then think
about the sign")

------
cromwellian
I think the claims that X is not necessary for everyday living ignores one of
the purposes of education, which is to expose you to a diverse array of
subjects and teach you _ways of thinking_ about things.

Maybe you took algebra and calculus, but you never actually use them as you
did in math class in your everyday life. But you learned problem solving in
instances where you don't have all the data, you learned things like limits,
infinite sums, and the relationship between lines, equations, areas and sums.

Although you don't use the explicit rule based symbols, perhaps your brain got
wired from the overall experience to think somewhat differently when
encountering certain situations.

Perhaps you'll understand your mortgage a little bit better because of your
exposure to geometric series. Perhaps when looking at your finances, the
concepts of slopes and tangents will re-emerge.

The same goes for learning history, or philosophy, or english literature. It's
not that you have to "use" your known of the Civil War or of Shakespeare in
everyday activities, but as a functional citizen, having been exposed to those
things, perhaps when you are asked to evaluate what's happening on the nightly
news, you will have a deeper perspective to draw on?

Personally, I use math all the time. I was a math major and I love it, so of
course, it's my standard tool. But I don't just use it in coding, I use it
thinking about art and lots of other things. I tend to "see math" all
throughout the universe and human experience.

------
StandardFuture
I did a quick google search for: High School Algebra Curriculum and got this:
[http://www.mathsisfun.com/links/curriculum-high-school-
algeb...](http://www.mathsisfun.com/links/curriculum-high-school-algebra.html)

I can say that while this covers a few algebraic topics I strongly believe
that the class title should be changed to: Practical Mathematics.

If High Schools _truly_ wanted to introduce Algebra to students in a "pure"
form then Math teachers would need to know/understand Abstract Algebra. And
then be able to teach introductory ideas to Abstract Algebra which, in all
honesty, _anyone_ can come to understand.

Btw I encourage everyone to learn at least some basic Abstract Algebra to see
the beauty behind what pure algebra is. :)

EDIT: It seems the curriculum is generalized to "set" the students up for
future multiple areas of Mathematics at once. Ranging from analysis &
differential equations to more advanced algebraic topics. But it certainly
introduces students to basic topics (sets, polynomials, functions) albeit in a
rather restricted (and not necessarily well taught) sense.

EDIT2: Not only should the title of the class be changed, but possibly the
class curriculum should focus even more on "Practical Mathematics" as well??

~~~
pflats
The best place to look at what is being taught as "Algebra" at the high school
level is to go to the (new) source: the Common Core State Standards
Initiative[1]. 45 states (and DC) are using the standards. The best places to
start are Math Appendix A[2], which explains the Initiative's recommendations
for a High School curriculum, and the Introduction to the Algebra Strand[3],
which describes (in brief) the goals for actual algebra instruction.

>If High Schools truly wanted to introduce Algebra to students in a "pure"
form then Math teachers would need to know/understand Abstract Algebra. And
then be able to teach introductory ideas to Abstract Algebra which, in all
honesty, anyone can come to understand.

They don't. Not on the general level. You're worrying overmuch about
semantics. The name "Algebra" isn't ever going away; no high school guidance
department wants to explain to every single college that their program teaches
the same thing as a normal "Algebra 1" class but just calling it "Practical
Mathematics". (Incidentally, "Practical Mathematics" would imply much more
basic mathematics, your traditional "Home Economics" class with taxes,
investments, credit cards, balancing checkbooks, etc.)

Beyond that, abstraction is much, much, much, harder for the average student
than you realize. Students have trouble seeing the relationship between the
Distance Formula and the Pythagorean Theorem. Some have trouble even
manipulating basic formulas, such as solving the Ideal Gas Law for a given
variable. They will insist on plugging in the numbers into PV=nRT every time,
and then solving the equation over and over again. Dividing by 5 is tangible
and easily visualized to these students, dividing by R is not.

Math teachers take the job because we love math and want to share that with
our students, but practical concerns come first. Symbolic manipulation is much
more widely used in the average high school student's future education than
pure math. I, and many teachers, include facets of pure mathematics in our
courses. When my Honors Geometry students begin working with infinity, I have
them read Strogatz's excellent piece on Hilbert[4], which is one of the most
popular assignments each year. (And yes, there is next to no Geometry in
there, but you have to keep minds sharp somehow.) I also throw in some basic
Real Analysis when discussing the concept of rigor in proofs. When my Algebra
2 classes have to trudge through a brief review of Algebra 1, we spend the
time talking about why closure matters, and what number systems are and are
not closed over what operations. My PreCalc classes do a decent amount of
Number Theory.

Finally, most schools offer Discrete Mathematics (i.e., an introduction to
Pure Math) as a senior-level math elective. However, it's competing for the
brightest minds with AP Calculus and AP Statistics. If you'd like to see more
students study Pure Mathematics, the best way to do that would probably be to
gather a group of like-minded educators and petition the College Board to
create an AP course covering said material

[1][http://www.corestandards.org](http://www.corestandards.org)

[2][http://www.corestandards.org/assets/CCSSI_Mathematics_Append...](http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf)

[3][http://www.corestandards.org/Math/Content/HSA/introduction](http://www.corestandards.org/Math/Content/HSA/introduction)

[4][http://opinionator.blogs.nytimes.com/2010/05/09/the-
hilbert-...](http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-
hotel/?_r=0)

------
bitwize
The art of transforming mathematical statements in a way that preserves their
truth value. That's one guess at it.

~~~
gizmo686
But algebra does not preserve truth values. It guarantees that if you start
with a true statement you will end with a true statement. If you begin with a
false statement, it is still valid to end with a true statement. Simple
examble:

1=2 | false

0 * 1=0 * 2 |Multiply both sides by 0

0=0 | true

~~~
bitwize
Multiplying both sides by zero yields an equation which is vacuously true, and
is not useful for solving the problems algebraists set out to solve.

~~~
gizmo686
True, but it is still valid algebra. There are other examples where one might
turn a false statement into a true statement without doing something as
pointless as multipling by zero. A common example is squaring an equation.
Consider a system of equation in which it can be shown that x=3. Within this
system, the statement x=-3 is false, however the statement x^2=9 is true.

Again, this specific example is contrived, but this does come up annoyingly
often.

------
theaeolist
Algebra is not just about numbers. It can be about diagrams or knots or
whatever: sets where the elements have operations that fit certain axioms.

~~~
cpa
Well, he only talks about school algebra. I'm not familiar with the US high
school curriculum but I doubt it goes much beyond polynomials… FTA (3rd
paragraph): > (I should stress that in this article I’m focusing on school
arithmetic and school algebra. Professional mathematicians use both terms to
mean something far more general.)

~~~
Someone
That's almost clear from the question. If this was about higher math, the
question would likely have been "What is _an_ algebra?", as "What is algebra?"
has the simple but almost content-free answer "The study of algebras."
([http://mathworld.wolfram.com/Algebra.html](http://mathworld.wolfram.com/Algebra.html))

------
jfarmer
Hmm, I think this is a fine essay for people who have already internalized the
notion of algebra _per se_ , but it's not great for a student who has to yet
to do so and is asking the titular question "What is algebra?"

Here's my attempt for, say, a relatively smart high school student. Consider
this a rough draft; I'm writing it from start to finish without editing. I'd
love, love, love feedback, though.

Today even schoolchildren understand arithmetic. We have these things called
"numbers." There are different types of numbers like natural numbers,
integers, rational numbers, and irrational numbers. We have rules for
manipulating these numbers like addition, subtraction, multiplication, and
division.

This wasn't always the case, however. I don't mean that humans couldn't always
add, but I do mean that it took humans thousands of years and multiple false
starts to come up with a sensible way to represent numbers and these
operations. Think about trying to do division with Roman numerals, for
example. It'd be a nightmare!

There was a time when numbers like "4/5", "-2", "0", and "√5" made people
freak out because we didn't have the symbols to represent them and it wasn't
obvious what they corresponded to "in real life," if anything. Imagine
yourself living in a world like the ancient Greeks, for example, where you
represented numbers by talking about lines of a given length. It was very hard
for you to talk about numbers _per se_ without drawing a shape that somehow
encoded that number. Now, pop quiz: how do you represent something like "0" or
"-2" in this world? If you were a Roman using Roman numerals, how would you
represent "4/5"?

This is just a story to highlight that although we take our numbering system
and arithmetic for granted, this was not the case for most of human history,
even most of recorded human history. The way we do arithmetic was an
_invention_ that both helped us do arithmetic and helped us understand WTF a
number even is.

By the way, if you freak out at the idea of imaginary numbers or the idea of a
number i which satisfies the property i^2 = -1, this is _no different_ than
the kind of freaking out the ancient Greeks did when they first encountered √2
or other cultures tried to make sense of negative numbers.

Now, let's think about what we really did by inventing arithmetic as we
understand it today. You can't point to the number "5" anywhere in the world,
right? Even the symbol 5 isn't five _per se_ , any more than "five", "fünf",
"|||||", "V", or "五" are five _per se_. But with one symbol "5" we can
represent this abstract thing.

Then something like "5 + 4" might represent the length of a line segment made
from concatenating a line segment of length 5 and a line segment of length 4,
the age of a 5-year-old in 4 years, the volume of water made from pouring five
buckets of water into a pool and then four buckets of water, the number of
apples shared between a 5-apple basket and a 4-apple basket, and so on. So,
these abstract things we call "numbers" and "arithmetical operations" can
represent many more concrete things.

Let's call this process "abstracting." Algebra is what you get when you treat
numbers as the concrete thing and apply this same process. With arithmetic, we
want to talk about numbers divorced from a particular concrete realization.
That is, we want to talk about the number 5 without having to talk about a
basket of five apples. With algebra, we want to talk about numbers _per se_
divorced from a particular concrete number.

Remember, when we invented arithmetic we had to invent a bunch of symbols to
represent the abstract thing. We do that in algebra, too. Often we use single-
letter symbols like x and y, but we could use anything like ☃, ☂, or zorpzop.
These symbols "stand in" some number in the same way that the symbol 5 "stands
in" for all the things 5 could possibly represent in the world. We can talk
about 5 without talking about the things it might represent.

So, we say things like "let x be a number" or "let x be a positive number" or
"let x be a rational number." What can we say about x in each of these
situations?

For example, if x, y, and z are all standing in for some number, we can say
the following:

    
    
        x + 0 = 0 + x = x          regardless of what number x is
        x + y = y + x              regardless of what numbers x and y are
        x + (y + z) = (x + y) + z  regardless of what numbers x, y, and z are
        x*1 = 1*x = x              regardless of what number x is
        x*(y + x) = x*y + x*z      regardless of what numbers x, y, and z are
    

These are true because of what we mean when we say "number" and what we mean
when we say "addition." It's not as if these are true for some numbers and not
all, nor is it as if we know these are true because we've "checked all the
numbers." That's impossible because there are an infinitude of numbers.

So, now we might ask things like, "Are there any numbers x such that x^2 + 1 =
0? How about x^2 + x - 1 = 0?" These questions might be hard to answer, but we
have now at least invented a language where we can ask them, whereas before
"abstracting" arithmetic into algebra we had no easy and succinct way of
asking them.

This is no different than not being able to easily ask, "Can we construct an
equilateral triangle with side lengths of π?" before abstracting from more
concrete things into numbers. Without a symbol for π we have to say things
like "the constant that is the ratio formed between the circumference and
diameter of a circle." This is how mathematics was done for thousands of
years. It was tough going, as you can imagine, and we missed many things that
would seem "obvious" to people using our notation.

You can continue this process further, by the way, and abstract further from
algebra. This is what mathematicians call _abstract algebra_
([http://en.wikipedia.org/wiki/Abstract_algebra](http://en.wikipedia.org/wiki/Abstract_algebra)).
In this context there are multiple algebras and the "algebra of numbers"
becomes the concrete thing in this new system. Linear algebra is a different
algebra, for example, with a different sets of "numbers" and a different set
of "arithmetical operations" that don't always correspond 1-to-1 with the
numbers and operations we find in arithmetic.

Often, when presented with a new physical system of objects that interact in a
certain way, we can try to abstract these objects and operations into symbols
and derive rules about these abstract symbols and operations that correspond
to the workings of the physical system.

We might call this symbolic system "an algebra." For example, Claude Shannon
invented an algebra for relay and switching circuits that allows us to
understand how they operated and how to combine them without actually building
physical circuits. See
[http://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf](http://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf)

~~~
themodelplumber
Division with Roman numerals being a nightmare was not obvious to me. That was
my first friction point here :-)

> _There was a time when numbers like "4/5", "-2", "0", and "√5" made people
> freak out because we didn't have the symbols to represent them_

Who freaked out, and how could they freak out if they could not behold these
numbers in the first place (i.e. no symbolic representations)?

> _without drawing a shape that somehow encoded that number_

Recommend you use "represented that number" because encoding has a strong,
separate type of meaning to me.

> _Now, pop quiz: how do you represent something like "0" or "-2" in this
> world? If you were a Roman using Roman numerals, how would you represent
> "4/5"?_

For "0" I would have made an empty box, out of strings if necessary. For the
-2 I would have placed two of the strings in a different location. For the
Roman numeral conundrum I would have placed IV and then a line and then a V
below that. Now, I just solved your conundrums. Or didn't I? To really speak
to beginners it's important to delve into this stuff. So far I have not
discovered why our number system is so great, and I don't feel like I'm
freaking out about anything in particular. :-)

Thanks for your writeup, though. I hope you can turn it into a book for people
like me.

\--Designer guy who is not super great at math

~~~
andolanra
> Who freaked out, and how could they freak out if they could not behold these
> numbers in the first place (i.e. no symbolic representations)?

The story goes that Hippasos of Metapontum discovered the existence of
irrational numbers like √2 while at sea, and his fellow Pythagoreans threw him
overboard, because it proved that there were aspects of the world that could
not be represented with rational numbers. That said, the notion of a square
root had been established for at least a thousand years by that point (by the
Egyptians and Sumerians), so the freaking out was less about their
representation and more about their properties.

> For "0" I would have made an empty box...

Your representations are parasitic on the fact that you already have a deeply
ingrained representation for those concepts and are comfortable manipulating
those concepts. It's like suggesting that you'd reinvent the wheel if you
lived in a civilization with no wheels merely because it seems obvious to you
looking at it today. Really it's not at all obvious, as no New World
civilization ever developed the wheel, and most Old World civilizations
borrowed it rather than inventing it independently.

Think of it this way: if you understood fractions, but nobody else did, how
would you represent them symbolically such that everyone else would? I'm of
the opinion that our typical notation for calculus is spectacular (especially
compared to Newton's original notation—the d/dx notation used today comes via
Leibniz) but if you don't understand calculus, how does that help you? You
understand negative numbers, but to a person with no concept of debt, a person
for whom numbers represent 'how many sheep you have', what does that mean, and
how do you symbolize it so that they do understand?

------
VLM
From the article, which runs off the rails after this "In today’s world, most
of us really do need to master algebraic thinking."

In practice its a class based (several meanings of class...) sorting/filtering
path for students... this group will successfully take orders from above to
handle unusual logical problems with some stability, discipline, and a self
directed very short term plan (aka future supervisor material) and this group
of kids can't/won't and will end up working underneath the first group as
proles.

Unless you think most people will become supervisory workers or above, its
just not necessary.

~~~
Steuard
The explicit example in the article was "Creating formulas in a spreadsheet"
(paraphrasing a bit). Is that really a task that will never be required of
non-supervisory workers?

I'm really not sure where you're going with this, or what it is that you see
as "run[ning] off the rails".

------
brudgers
Algebra is an abstraction over arithmetic, which is an abstraction layer over
counting. Or at least that is the abstraction I am using today after reading
the article.

As a nitpick, I am not convinced that arithmetic is prior to geometry - e.g.
Stonehenge and celestial navigation require geometric abstractions but not
arithmetic. "Fabricate bricks until I tell you I have enough" is a more
reliable logistical model for the construction of Rome's aqueducts than
"provide _x_ bricks for each of the _y_ miles."

~~~
dcga
This is funny because the construction of Rome's aqueducts was anything but
"... until I have enough".

~~~
Zircom
This is funny, because the text you quoted wasn't referring to the
construction of the aqueducts, only the fabrication of the bricks used in the
construction.

------
GreyZephyr
Tangentially related to the article, but for many years I struggled with
understanding what algebra was an more importantly why it was of interest. On
one hand you had groups and rings that sort of seemed related, but they
weren't algebra's. Then there were special techniques like linear algebra and
I could see that they were all sort of similar but why they were of interest
and all considered to be part of the same subject escaped me. Sure they were
interesting in a way and group theory was sort of cool but what was the point?
Analysis on the other hand made sense, it was the study of how continuous
things changed. As a result I though of algebra as a sort of collection of
things that you got when you discretized continuous objects. Lie algebra's and
groups came as a bit of a shock, but they were basically still thought of
algebra as discrete.

Many of my friends raved about its beauty, but I still had no intuition as to
why it was interesting. I went on a bit of a reading binge, and eventually
ended up bumping into some papers of Shafarevich[0] which in turn lead me to
his delightful book on algebra [1]. In it he defines algebra as the
construction and study of systems of measurement. For example counting is the
simplest such system and gives the natural numbers. Attempts to describe and
measure the diagonal of the unit square gave irrational numbers. As we
explored the world we needed to construct new systems. A more recent example
is provided by quantum mechanics, where numbers are insufficient to describe
our observations, however Hilbert space provides a natural setting. Attempts
to describe what a simultaneous measurement of two quantities is find a
natural description as commuting operators. He provides many other examples in
his very readable book. For whatever reason this definition of algebra
resonated with me and if this was a Zen koan I would say I was enlightened.

This is really just a very long winded way of saying that I think
Shafarevich's book in which he defines algebra as the study of measurement is
lovely. You should read it if such things interest you.

[0][https://en.wikipedia.org/wiki/Igor_Shafarevich](https://en.wikipedia.org/wiki/Igor_Shafarevich)
[1]
[https://encrypted.google.com/books/about/Basic_Notions_of_Al...](https://encrypted.google.com/books/about/Basic_Notions_of_Algebra.html?id=CHadhWO4DWYC)

------
saraid216
I really like the "algebra is like cheating for math" epiphany. It gets to the
core of both higher mathematics and the problems in our educational approach.

