Ask HN: How to self-study mathematics from the undergrad through graduate level?
Part I<p>I have some opinions on the question in
the OP:  I&#x27;m heavily self-taught from
independent study in math; that study
helped me with, and at times was part of
my good career in, applied math and
computing before my Ph.D. in pure&#x2F;applied
math and helped a lot for my Ph.D.<p>For a <i>curriculum</i>:<p>(1) Do the standard high school math,
Algebra I, Plane Geometry (based on
proofs), Algebra II, Trigonometry, and
Solid Geometry (based on proofs).  In each
of these, it is sufficient (A) to work
most of the more challenging exercises and
(B) try to think a little about what is
going on.  For (C), it would be good
occasionally, say, 1-4 times a year, to
chat about your progress with a good
mathematician, Ph.D., likely a college
prof.  To find a Ph.D. who will give you
an hour or so per visit, maybe do some
<i>networking</i>. If you don&#x27;t like the first
choice, e.g., if they are not encouraging
and helpful, then try a different Ph.D.
mathematician.<p>(2) Do Analytic Geometry and Calculus.
Use a good college textbook.  Since highly
polished texts have been widely available
for decades, there&#x27;s no need to pay big
bucks for new copies of the latest.  So
get a good used text.  Better yet, get 1-4
good texts, use your favorite one as your
primary source and the others for more
content, alternate descriptions, evidence
of what is more&#x2F;less important, etc.<p>Sets:  Now in an important sense,
essentially all of math is heavily about
<i>sets</i>.  For first calculus, you don&#x27;t
need to know much about sets; the
following should be nearly sufficient:<p>By 1900 or so, a lot of math was known,
but there was a nagging, somewhat
philosophical, question of what math
really <i>is</i>.  The answer of <i>pure</i> math
was to construct a <i>foundation</i> in, say,
the deep basement, that would answer the
question but not much change the rest.
The answer was to start with <i>sets</i> and
then define everything else in terms of
sets.  In particular do some somewhat
tricky and obscure work to define some
sets that look, work, walk, talk, etc.
like the numbers on the line, the <i>real</i>
numbers.  We don&#x27;t think of the real
numbers this way, but the effort addressed
the philosophical question -- or, if I can
get you to believe in sets, then I&#x27;ve got
you for the rest of math.  Much of the
early work on sets was from G. Cantor.
Yes, soon there were some issues from
Bertrand Russell and then Kurt Gödel and,
later, Paul Cohen.<p>For a long time in math, you can regard a
<i>set</i> as a conceptual (imaginary)
collection of some kind, e.g., all the
peas in a little jar in the back of the
refrigerator, all the people with an
iPhone, all the real numbers on the line,
all the water molecules in the earth&#x27;s
oceans, all the parabolas, all the trials
you might do in a lab experiment, and much
more -- really, from the philosophical
approach to what math <i>is</i> for all of
math.  Or intuitively <i>sets</i> are the
containers for what we are thinking about.<p>Functions:  Much of math is about
<i>functions</i>.  E.g.,<p><pre><code>     f(x) = 2x^2 + 3x - 4
</code></pre>
So, in <i>pure</i> math, the function is f. The
x is the <i>argument</i> of the function, i.e.,
a <i>variable</i> which is essentially always
in math from,<p><pre><code>     &quot;Think of something; call it x.&quot;
</code></pre>
where at least in calculus usually the
thing are thinking about is a number.  Or,
to &quot;solve for x&quot;, think<p><pre><code>     &quot;I suspect there is a number such
     that these conditions hold; I want to
     find if there is such a number and if
     so what its actual numerical value
     is.&quot;
</code></pre>
Now you understand what math means when it
talks about variable x and function f.
Then, given variable (e.g., number) x and
function f, f(x) is the value of f at x.<p>E.g., for this set stuff, a function gets
defined in terms of sets, e.g., the set of
all <i>ordered pairs</i><p><pre><code>     (x, f(x))
</code></pre>
for x in some set called the <i>domain</i> of
the function.  In calculus, the domain is
usually the set of real numbers, that is,
the points on the number line (yes, that&#x27;s
not the tricky set theory definition of
the real numbers).<p>So, with the ordered pairs<p><pre><code>     (x, f(x))
</code></pre>
we have a very precise definition of a
function in terms of sets (ordered pairs
can also be defined in terms of sets).<p>Of course, in actual work, especially in
applied math, nearly no one actually
thinks of a function as a set of ordered
pairs.<p>Completeness:  There is an old joke,
partly appropriate, that<p><pre><code>     &quot;Calculus is the elementary
     consequences of the completeness
     property of the real number system.&quot;.
</code></pre>
Intuitively <i>completeness</i> is, if you are
converging to something by more and more
accurate approximations, then there really
is something there for you to converge to.
This property does NOT hold for the
rational numbers -- the <i>rational numbers</i>
are all the numbers of the form (that can
be written as) p&#x2F;q for p and q integers
(whole numbers, positive, negative, and zero) with q not zero.  Why?
E.g., can use the rational numbers to
approximate as closely as we please the
square root of 2, but the square root of 2
can&#x27;t be a rational number.  Why not?
Because if the square root of 2 were
rational, then we would also have for
integers p and q, q not zero<p><pre><code>     (p&#x2F;q)(p&#x2F;q) = 2
</code></pre>
or<p><pre><code>     (p)(p) = 2(q)(q)
</code></pre>
So the left side has some even number
(possibly 0) factors of 2 while the right
side has an odd number of factors of 2,
and this violates the fundamental theorem
of arithmetic that each integer can be
factored in only one way as a product of
prime numbers.<p>Well, a big deal about the real numbers is
that they are <i>complete</i>; i.e.,
intuitively, if something converges, then
there is something to converge to.
Completeness is a big deal because it
generalizes, especially to Hilbert space
that you may come to.<p>So, in calculus often we make
approximations that can become as accurate
as we please, and we want the completeness
property to know that what these
approximations approach, <i>limits</i>, can
exist.<p>E.g., with the real numbers, if we make
more and more accurate approximations to
the square root of 2, then we know that
our approximations will <i>converge</i> to the
actual square root of 2 which DOES exist
as a real number.<p>Sets get to be quite important in math in
the last years of college math and beyond.
But for calculus, nearly all of that
subject, especially what learn in a first
calculus course, was quite solid well
before G. Cantor.  So, in practice, when
studying first calculus, we don&#x27;t see much
about sets.  So, for first calculus, sets
remain mostly a topic in the deep basement
that addresses an old philosophical issue.
So, for first calculus, sets are no big
issue.<p>Analytic Geometry:  Here you study the
<i>conic sections</i>.  They are a bit amazing
and important in physics, mechanical
engineering, and more and something
definite to work with while learning
calculus.  So, imagine a cone, say, an ice
cream cone.  Take two of them that are the
same and put the points together so that
they are both on the same <i>axis</i> through
the points (I&#x27;ll let you get a precise
definition of <i>axis</i>).  Now imagine a
sharp sword, say, as in the old John
Belushi Samurai Tailor skit and use the
sword to slice the cones and look at the
cut edges.  Right, we assume that the
sword moves in a plane.  Depending on
where you slice, you will get (A) just a
point (where the two cones touch
together), (B) two lines crossed as in an
X (the cross in the X is at the point
where the two cones come together), (C) a
circle (we are awash in circles), (D) a
parabola (to a fairly good approximation,
a grand slam baseball follows a parabola),
(E) an ellipse (to a quite good
approximation, each planet goes around the
sun in an ellipse), (E) the two parts of a
hyperbola (an electron shot at a negative
charge will follow one of the halves of a
hyperbola as it avoids the negative
charge).<p>First Calculus in a Nutshell:  Then for
first calculus, there are two big topics,
differentiation and integration.
Differentiation is finding rate of change,
intuitively the slope (as in high school
algebra) of a tangent line to the graph of
a curve (defined in terms of a function
with domain some or all of the real
numbers).  E.g., for time t, let d(t) be
the function that gives us our distance
from home at time t. Then the derivative
of function d at time t is the velocity,
say, function v, at time t. So, in a car,
the odometer gives d and the speedometer
gives v. That is, at time t we have
traveled distance d(t) and are moving a
velocity v(t).  Here to be simple, I do
not make a careful distinction between
what physics calls speed versus velocity.<p>What differentiation does, integration
reverses, undoes.  So, we can use
integration on function v to recover
function d. We have now introduced the
fundamental theorem of calculus.<p>Right, v is the function, and v(t) is its
value at time t, but in first calculus it
is common to drop the distinction between
v and v(t) and just say &quot;the function
v(t)&quot; and mostly avoid mentioning v by
itself.