
What is math? - fogus
http://scienceblogs.com/goodmath/2009/12/what_is_math.php?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+scienceblogs%2FCyKN+%28Good+Math%2C+Bad+Math%29
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lmkg
(I have a math degree)

I think the OP still commits the common error of focusing too much on what
math studies rather than how it studies it, to a mild extent. Math, to me, is
a style of thinking, and the corpus of math knowledge is this style of
thinking applied to particularly well-suited domains.

That style of thinking is characterized by a few things. One, obviously, is
rigor: every step is justified by formal rules, there are no appeals to
intuition or leaps of logic. Most people's impression of mathematics stops
there, that it's a list of rules like you're taught in grade school (because
that style of memorization is the only way your teacher got through math in
college). There's a lot more to it though.

For example, there's a heavy focus on interpreting a single fact in many ways.
There's a lot of bridging knowledge between different sub-disciplines of math,
and they work a lot by interpreting a mathematical structure in new ways.
There's a lot of treating abstractions as new concrete topics of study.
There's a lot of exploration.

Abstractions are well-suited to math because you can make them clean and
simple enough to be amenable to complete rigor. This doesn't mean that math-
style thinking is limited to worlds populated only by platonic ideals. And I
think that's another core disconnect between the potential of math, and its
realization in the education system, is that the arbitrary rules of symbol-
manipulation are described as applying to illusionary ideals, not real things.
There needs to be a bigger focus on motivating the ideals as modeling
something useful, because they do, and that connection to the concrete helps a
lot at getting a grasp on the topics.

I don't do very much "math" in my day job (other than working with numbers ;P)
but that style of thinking is still a great benefit. The real world doesn't
often allow math-level rigor, but you can still show exactly where the rigor
is missing in your line of reasoning, which lets you identify assumptions, and
then challenge them. The focus on abstraction and fundamental principles of
systems is also useful, especially because it helps to apply experience from
one domain onto another.

~~~
timwiseman
_One, obviously, is rigor: every step is justified by formal rules, there are
no appeals to intuition or leaps of logic._

I am not certain that is entirely true. In areas that are long settled, this
has often been made true, but that is often long after the fact, and often
done through a process of "monster barring".

That is someone proposes a definition upon which theorems are built, and
someone else later will present an object which meets the technical definition
but which clearly did not meet the intent of the definition. So the technical
definition is revised to exclude that. And the process repeats, narrowing the
definition to remove that problem but often in the process making it more
difficult to come to an intuitive understanding of that formal definition. One
historical example of this was in the definition of polygon.

Areas of active research do tend to appeal to intuition, at least to a degree,
until the formalism can be completed. Historically that formalism has often
come from other hands than the original researchers and often either in the
process of "monster barring" or as the final touch after several iterations of
it.

In practice, even in areas where full rigor is absolutely possible, it can be
terious beyond any use if there is no debate or challenge. Mathematical papers
often do not have complete formalism and even text books on fully settled
areas often do not try to justify every single step if there is no reason for
it. Perhaps a better definition is that it must be possible in principle to
justify every step by formal rules.

~~~
lmkg
You are correct that the definitions are usually motivated by a particular
problem or attempt at modeling something. However, if a mathematician arrives
at a "bad" result, the reaction is not always to revise the definition.

Sometimes we think it's really awesome that the definition is general enough
to capture additional phenomena in the same framework. Sometimes these
degenerate ideas lead to interesting areas of research in their own right,
like how adding some strange Godel-related axiom to number theory leads to
Supernatural Numbers. Sometimes the theorems that are developed are not
applicable to their original subject of study, but can be applied to another
by re-interpreting their meaning, like non-Euclidean geometry. Sometimes the
unintuitive result is simply accepted, like Godel's theorem. Sometimes our
intuition is simply wrong, so unintuitive edge cases like Dirac spikes end up
being fundamental tools in physics and signal analysis.

Intuition is often used to motivate the definitions or the axioms, but they
are clearly labeled as definitions and axioms. What I was trying to say is
that the chain from axioms to conclusion must be complete, with every step
explained and approved. It's this method of explicit exposing your thought
process, to yourself as well as to others, that I've found really useful
outside of just the realm of mathematics.

~~~
timwiseman
_However, if a mathematician arrives at a "bad" result, the reaction is not
always to revise the definition._

Certainly not, I meant that as an example, not as a full description of all
cases.

 _What I was trying to say is that the chain from axioms to conclusion must be
complete, with every step explained and approved._

But this is what I am trying to say does not always happen, or at least does
not always happen right away. Another commenter, mreid cited Thurston's essay
"On proof and progress in mathematics" which discusses this in far more detail
and far more authoratitively than I possibly could. Hardy also touches on it
in his "A Mathematician's Apology".

Mathematics which is long settled, the type of thing generally studied in high
school and for most of undergraduate mathematics education, is often (but not
always) presented in this format, with axioms and definitions cleanly laid out
and tiny steps form one to the other to arrive at the theorem.

Mathematics for working mathematicians does indeed involve this, but it also
often involves intuitive leaps, thoerems where the first proof is only vaguely
sketched out, and reasoning from diagrams. They normally try to clean this up
by the time it is published, but for a variety of reasons even there it is
often not perfectly rigorous and hardly ever in tiny steps with every step
justified by axiom or previous proof.

There are many things held in common knowledge that are never cited. This is
of course fine if the proofs are available, but steps away from that school
house rigour immediately. Even beyond that, it is not uncommon for some ideas
to only be sketched, and occassionally errors are found in the omitted steps
in those sketches years after they have been formally peer reviewed and
published.

These sketches are used in proof of absolute formalism often just for the sake
of time and space in the journal, but occassionally because no one has ever
actually gone through and filled in every detail at all, trusting to the more
intuitive logic of the sketch.

The assumption in most cases is that it could be made perfectly rigorous if
the need arised, but that is often only actually done if there is some
question or debate.

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btilly
The best description of math that I have ever encountered of what math is is
the book _The Mathematical Experience_ by Davis and Hersch.

Don't be scared by the fact that it is a book. It is very accessible since it
is divided into little articles on different areas of math. A lay person can
read it and gain a sense of what this math stuff is and how varied it is. A
PhD can read the same material and will appreciate it on a much deeper level.

Let me give one data point. I know more mathematicians who became interested
in math because of reading _The Mathematical Experience_ than any other book.
Seriously.

~~~
ubernostrum
Hm. I've not read it, but I have read a rather scathing review of it by Martin
Gardner (whose opinions I usually respect). disagreeing with it on fundamental
philosophical grounds.

~~~
RiderOfGiraffes
That's interesting. Gardner describes himself as a journalist interested in
math and science, but specifically not a mathematician. It's interesting that
he disagrees with two mathematicians on the question of what math is.

I find MG fantastic, but sometimes his articles and books lack depth. He is a
wonderful writer and populariser, but I wonder whose side I would take if this
were a real debate.

But it isn't. There's room for both opinions. I've never met Gardner and am
now unlikely to do so, but I know a few people who know him fairly well. I'll
ask their opinion when I get the chance.

Do you have a reference for that review?

~~~
slackenerny
_Do you have a reference for that review?_

[http://www.cs.nyu.edu/pipermail/fom/1997-November/000128.htm...](http://www.cs.nyu.edu/pipermail/fom/1997-November/000128.html)

~~~
RiderOfGiraffes
Cool - thanks.

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tokenadult
Professor John Stillwell writes, in the preface to his book Numbers and
Geometry (New York: Springer-Verlag, 1998):

"What should every aspiring mathematician know? The answer for most of the
20th century has been: calculus. . . . Mathematics today is . . . much more
than calculus; and the calculus now taught is, sadly, much less than it used
to be. Little by little, calculus has been deprived of the algebra, geometry,
and logic it needs to sustain it, until many institutions have had to put it
on high-tech life-support systems. A subject struggling to survive is hardly a
good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but
also algebra, geometry, and the whole idea that mathematics is a rigorous,
cumulative discipline in which each mathematician stands on the shoulders of
giants.

"The best way to teach real mathematics, I believe, is to start deeper down,
with the elementary ideas of number and space. Everyone concedes that these
are fundamental, but they have been scandalously neglected, perhaps in the
naive belief that anyone learning calculus has outgrown them. In fact,
arithmetic, algebra, and geometry can never be outgrown, and the most
rewarding path to higher mathematics sustains their development alongside the
'advanced' branches such as calculus. Also, by maintaining ties between these
disciplines, it is possible to present a more unified view of mathematics, yet
at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of
"elementary" topics in the rest of his book, which is a delight to read and
full of thought-provoking problems.

<http://www.amazon.com/gp/product/0387982892/>

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michael_nielsen
Math seems to me a bit like love, you have to experience it to understand it.
Definitions and discussions like the OP might help a little, but don't get at
the essence. No-one who reads such a definition will suddenly come away
knowing what mathematics is.

I had a few epiphanies in my life that made me understand what mathematics is.
One of them was understanding why it is that there are infinitely many primes
- I think I learnt more about what mathematics is in the moment that proof
clicked than I have from reading hundreds of discussions of "what is
mathematics". The experience was, simply, miraculous.

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vitaminj
Also check out the reddit thread that inspired this post -
[http://www.reddit.com/r/AskReddit/comments/abiax/can_someone...](http://www.reddit.com/r/AskReddit/comments/abiax/can_someone_explain_mathematics_to_me/)

Some very eloquent responses in there. I was moved.

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ez77
Does not quite answer the question, but a true ode to mathematics:

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a
beauty cold and austere, like that of sculpture, without appeal to any part of
our weaker nature, without the gorgeous trappings of painting or music, yet
sublimely pure, and capable of a stern perfection such as only the greatest
art can show. The true spirit of delight, the exaltation, the sense of being
more than man, which is the touchstone of the highest excellence, is to be
found in mathematics as surely as in poetry."

Betrand Russell: "Mysticism and Logic and Other Essays".
[<http://www.gutenberg.org/etext/25447>]

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jmatt
I agree. The way most people understand what math is and the way we teach math
is horrible. The math curriculum is a disservice to everyone that's been
exposed to it. I think mathematicians and scientists alike are realizing this
and beginning the long process of fixing it in schools and eventually the
culture. Some of the most creative people in the world today are not musicians
or artists but rather mathematicians. (and as a corollary computer scientists,
engineers, etc.)

I recommend "A Mathematician’s Lament" by Paul Lockhart [1]. He further
investigates what math is, how we can better teach it and how it's portrayed
in our culture.

[1] <http://www.maa.org/devlin/LockhartsLament.pdf>

EDIT: I also posted this in the linked article. But wanted to provide this
great paper to the HN crowd too.

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maurycy
For me, math is all about consequences. I'm extremely poor mathematician and
my background is much more about computer science, yet I'm fascinated with
this simple fact.

You define few simple structures, and, in consequence, get an enormous amount
of structures' properties, more abstract structures and facts about newly
created structures.

Thing like an algebraic structure can be understood by a clever seven years
old. However, it simply takes stating structure such as an algebraic structure
combined with a space, to enable existence of many other structures, more
abstract or more specific, with their own properties, and similarities.

This is a very powerful thing. For me, math is all about this 'generative' way
of thinking.

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hegemonicon
My favorite definition of math, courtesty of [http://www.win-
vector.com/blog/2008/08/what-is-mathematics-r...](http://www.win-
vector.com/blog/2008/08/what-is-mathematics-really/#more-21)

"Mathematics is the minimal environment to preserve ideas."

------
bokonist
My own definition: "Math is a language for describing phenomena that can be
precisely defined."

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presidentender
Math is the study of verifiable, undebatable truth. If a system is fully
understood, it is susceptible to mathematical study.

~~~
jpeterson
_Math is the study of verifiable, undebatable truth._

A certain Mr. Gödel would disagree with you there.

~~~
ubernostrum
No, I don't think he would. He'd point out that there are things which cannot
be placed in the realm of verifiable and undebatable, but that doesn't mean
there are no things which can be placed in that realm.

~~~
jpeterson
Gödel's intent was to show that there are things that are "true" but not
"provable". So to say that math is the study of things that are verifiably and
undebatably true is at odds with that.

~~~
Jimmy
Can math study everything that is true? No. As far as the things it studies
goes, are they true? Yes.

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DanielBMarkham
Since everybody has a definition for math, it seems, I'll give it a whirl.

Math is the act of creating a self-consistent system of symbology and symbolic
abstractions that, at some place, exactly mirrors reality (even if that place
is only the counting numbers). The belief is that since nature is also self-
consistent and ontologically hierarchical that by working inside a parallel
symbolic system we can discover aspects of nature we never knew existed. Note
that the systems have gotten so complex that in many fields math is entirely
theoretical and never "closes the loop" back to observed reality. In fact,
some would argue that applied math is more in the realm of physics, whereas
"true" math is entirely abstract.

Is that anywhere close?

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wkdown
baby don't hurt me ...

don't hurt me ...

no more ...

