For all those who feel baffled by maths. Don’t be.
The teaching
of math is a disaster: we have such an interesting and beautiful subject —the
jewel in the crown of human thought— being taught through wearisome repetition
and recipes, and it just ends up intimidating most people.
What is mathematics?
There is an
ancient debate that continues to this day: that of whether mathematics is
invented or discovered. The arguments are interesting and in the second case,
many think that "the universe is made of mathematics", and that we
are simply discovering it. It is an enticing, beautiful idea, but we are not
going to go into it here. For our purposes we are going to explain it from the
other point of view:
Mathematics
is a language, just like Spanish or French: we can confidently declare
something like "I understand Spanish, English and mathematics."
There are two main
differences between math and natural languages: the first is that the former is very
strict and precise, and the latter can be vague, with many exceptions to their
rules. The second is that math is not a language for two-way conversation but for one-way expression, in a way akin to art. Still, math and natural languages are both means of communicating ideas.
Math and
grammar
Let's take
math and English. Both are languages that allow us to communicate
ideas, and they both have a grammar: that is, a set of
rules. Those are the similarities. Now let's see the differences between
them.
Math is a
language that can express ideas in a unique, simple way, and crucially: without
confusion or ambiguity. English, just like any natural language, can express
ideas clearly, but it can also do so in approximate, contradictory, and even in
intentionally vague ways.
We construct
sentences according to grammatical rules. For example: "Anna and Julia ate
a fish" indicates a simple fact, but if we say "Anna ate a fish and
Julia", you are either making a grammatical mistake or talking about a
particularly gruesome incident.
In math we construct
equations just like we make sentences in natural language. The rules (the
grammar) are simple and clear. For example:
1 + 2 = 3;
3 - 2 = 1;
3 = 3
They are all
simple and “grammatically correct” mathematical sentences. And as with natural
language, we cannot do certain things, like writing “3 2 1 = -” because then it
is no longer understandable.
Mathematics and translation
Mathematics and translation
So now that
we know that math is a language: what do we do with it? Well, what we do
most of the time is basically translating work:
When we see
a complex problem in the real world, we measure it and translate it into
mathematical language, which is very reliable and easy to manipulate. Once
translated to this language, we solve it there and we then use that answer to translate
it back to the real situation.
Let's see an
example: one of those classic riddles of recreational math. This is the
problem stated in natural language:
Jean and Grace
are sisters. Jean is older than Grace. If we take Jean’s age and subtract Grace’s
age, we get 6. If we take Grace’s age, multiply it by 2, and add it to Jean’s
age, we get 18. How old are Jean and Grace?
So we have a few
numbers that we know, and a couple that we don't. Those that we don't
know, we represent by using a convention of the language of mathematics: the
letters x and y. If we say that x
is Jean's age, and y is Grace's age, then we can do the following:
We take the
fact that “If we take Jean’s age and subtract Grace’s age, we get 6” and write
it like this:
x - y = 6
then we take
the fact that “If we take Grace’s age, multiply it by 2, and add it to Jean’s
age, we get 18” and write it thus:
2 x y + x = 18
Looks a bit
confusing, what with the use of the multiplication sign and our x,
which we are looking for. So we use another of math’s grammar rules, which
always want to make things simpler, and instead of “2 x y”, we simply write 2y.
So we have
these two equations:
x - y = 6
2y + x = 18
And we just
do one more thing: since in mathematics we like to put things as symmetrically
as possible for clarity, we swap the places of the terms of the second
equation, so that they are end up like this:
x - y = 6
x + 2y = 18
Aaand we’re
done! Our English-Math translation is complete!
Our
translation represents the problem in a simple and complete way, without any
unnecessary elements: for example, the fact that Jean and Grace are sisters.
And if we have our problem expressed in this language, we can use more of our
"math grammar" —such as addition, subtraction, multiplication or
grouping together— to solve it. In this simple case, we can use two different
techniques to solve: Elimination or Substitution of variables. But the point I
want to make here is not on solving equations, but on the fact that many extraordinarily
complex problems from the real world, can be expressed with them.
Vocabulary
If we
understand math as translation, we have understood not only the concept of
almost all classical mathematics, but we can get rid of almost all our fears:
learning math is learning a language.
Of course,
there are simple and complex problems. Learning to translate a problem like the
one above is like learning to say “Bonjour” in French or “¿Cómo estás?” in Spanish:
a basic level. In order to translate more complex problems, it is necessary to
develop our vocabulary and our understanding of grammar.
Math and
poetry
The polar
opposite of mathematics is poetry, because while we use mathematics to express
clear and unambiguous relationships, poetry does exactly the opposite: it
expresses subtle, metaphoric relations in many different levels.
For example,
if we say “your eyes are like the wings of a raven” or “your eyes are like the
night” it is clear that we are using a simile to imply the color black. But if
we say “your eyes are like a whisper”: it is something poetically valid, but it
cannot be expressed in any other way. It is something that is left to the
reader's imagination.
Mathematics
is not crated to express those flights of imagination and emotion, but in its own
way, it can do something similar. In addition to “translating the world”, as we
said before, you can also imagine with mathematics: when you have mastered the
vocabulary and the grammar, you can “create” new things and find new
relationships between them, in the same way that imagination can create a
unicorn or a dragon. A lot of modern mathematics does this: exploring infinite
numbers, creating geometrical systems that you can mention but not actually
visualize, or devising “monster groups” of fabulous symmetrical objects that
exist in a 100 thousand dimensions.
Thus we have
that math is not as cold or remote or as intimidating as it seems: we just
have to see it in the right way.
With math
we can not only translate the world into a more practical and manageable
language, but we can also use them to exercise our imagination and create strange
beasts and chimeras that others can paint upon, twist around and develop
further. And like the best poetry, sometimes mathematics can show us real world
relationships so surprising —such as the fact that algebra and geometry are two
sides of the same coin— that we cannot help but be moved by amazement and
excitement.
This
excitement comes from knowing that an invention of our intellect in fact can
open the door to the understanding of the universe.
Math is our
jewel in the crown, don't let it scare you away.
