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I think a good way to connect to the mathematical side
of your brain is to find a hobby or something that you
are passionate about and are willing to tinker with.
Some people will like playing mathematically-themed puzzles,
like Sudoku puzzles, very popular.
As a kid, I liked playing with computer games and logic
puzzles.
If you like fixing up old machinery or something,
the challenge of making some broken piece of machinery work,
that kind of problem solving but that
shows up in that it actually is very
parallel to mathematical thinking.
If you are a cook and you're following a recipe
and the recipe is four people but one day you
have to cook for six people and you
have to change all the amounts of ingredients,
or maybe the recipe requires you to cook something
for an hour and a half, but you only have an hour.
That's a mathematical problem.
How can you use resources most efficiently?
I think we bump up against mathematics all the time.
Now often you can kind of wing it.
An experienced chef may have some rules of thumb.
But having some mathematical training
can help you avoid a disaster I think.
At least give you some sort of first approximation
as to how to adapt to an unexpected situation.
I think we could all play a little bit more
with trying little tasks first before moving on
to big high stakes tasks.
I think mathematics is such an extreme example
where it doesn't matter how many times you
fail to solve the math problem that there's
no real penalty other than waste of time.
But it's not even really a waste as long
as you learn something from it.
The more you expose yourself to doing tasks in a fun,
challenging way, enjoying just the challenge.
You know, I enjoy the challenge of assembling furniture
from very unreadable instructions.
I mean, I don't want to do it for a living or anything,
but almost anything in real life can be
turned into a little problem.
And finding situations in life where the stakes are low,
it's OK if you screw up, you just start over.
And just getting into a mindset where the goal is not
necessarily to solve the problem quickly or efficiently,
but to enjoy yourself and to draw lessons from it.
One of the basic techniques in mathematics
is, if you have a complicated problem,
you isolate a simpler version of the problem.
Solve that first, and then once you
know how to solve various simpler sub
cases of the problem, try to put them back together again
and solve the full problem.
Just to give you one example from my own personal
experience, I once had to put up some curtains on one
of my windows in my house.
And you had to stand on a chair and there's this heavy rod
and there's a bunch of curtain rings
and there's a heavy curtain.
But maybe because of my mathematical training, the way
I approached this was I first tried to assemble
the curtain on the ground.
And then I practiced how to assemble each piece
on the chair.
So I would take just the rod and see
how to put the rod on the hooks and how
to put the curtain on the rod.
And only after I had practiced each individual piece
of the problem would I try to put it all together.
That took longer than if I had tried to assemble
the curtain directly.
But I'm pretty sure if I try it directly,
I would have messed up one or two of the stages.
And I think this process was actually better in the end.
So is mathematics a creative subject?
Definitely.
When you see mathematics in a school context,
it's often presented in a somewhat dry manner
that there's certain recipes that you have to follow
in order to solve a problem.
And if you deviate, then you get your marks deducted
or something.
But when you're a research mathematician
you are solving problems that standard techniques don't quite
apply.
Because it's so abstract and not necessarily
tethered to reality, it allows you
to be very creative and very flexible.
In the real world, you may have a problem where you have say,
some finite number of resources.
You only have X amount of dollars to thwart a problem.
You may only have so much time.
But in mathematics, you can change the parameters.
You may say, OK, what have I had a billion dollars?
Could I solve this problem?
Or what if I had an infinite amount of manpower?
It gives you a lot of flexibility
to change the problem into one that maybe you can solve first
and then you can then from there, go back
to solve the actual problem.
And that's a freedom that you just
don't have in the real world.
You can't, you can't just say, oh,
before I solve this problem, can I first have a billion dollars
to experiment.
Yeah, the abstraction that mathematics
has afforded a lot of creative freedom.
Many people, they had a problem to solve
and they basically did mathematics
even without knowing that they did mathematics.
There's one story I quite like.
So there was this mathematician named Johannes Kepler
in the 17th century.
And he once had to purchase wine for I
think his daughter's wedding.
And the way that the wine merchants
would price these barrels was that there was a little bung
hole in the middle of the barrel and there will be this stick.
And they would poke the stick through the hole
to the opposite corner of the barrel and measure the length.
And based on that length, they would say, OK, this is how many
ducats or whatever worth of wine.
When Kepler saw this, he was amazed.
I mean, normally, you would think
that if you want to measure how much wine there is in a barrel,
you'd have to measure how many gallons fill it up
or something.
And this was a very quick way.
And for these wine merchants, it worked apparently.
And they couldn't explain why it worked.
They just empirically worked out what lengths of this stick
corresponded to how much, what quantity of wine.
So Kepler got very intrigued and he approached the bottle
mathematically.
He considered wine barrels of various widths and heights
and computed the lengths of the sticks and their volumes.
And he did indeed see that there was actually
this very nice relationship.
The barrels that were made in Austria,
they were all roughly the same shape.
And it turned out that if you made the barrel 5%
narrower or 5% wider, the relationship between volume
and the length of stick was roughly the same.
The dependence of volume on the shape of the barrel
was actually at near local maximum.
Local maximum refers to a situation
in which any small change in the unknown variables,
in this case, the shape of the barrel,
would lead to a reduction in the quantity being studied, which
in this case, is the volume of the barrel.
And so small changes in the shape
did not actually make big changes in the volume.
And in fact, Kepler developed some
of the early tools of calculus in order to solve this problem.
And it was actually quite important for the development
of mathematics.
So there's a lot of mathematical problems
that we just encounter in our everyday lives
and we often just have to solve them
with or without formal mathematical training.
But once you have the training, you
can explain why these funny tricks
that people have in their various professions actually
work.
So what are some components of practical mathematical
thinking?
Well, they include experimentation, a willingness
to fail, proactive deconstruction, and remembering
sometimes to zoom out, apply abstraction, choose
a plan of action, then act accordingly.
So keep an eye on the big picture.