In my last video, I took a look at a document which showed some of the math that Terence
Tao could do when he was seven years old.
But there was one page I didn't show you.
And that's this one, a list of some of the books that Terence had read as a child.
These books are from a longer list of 22, that Terence had read, by the time he was
eight.
It says that he liked to read entire books, and that he was keen for advice on what to
read next.
It also says that he had a remarkable memory for things that he read.
Terence is now a professor of math at UCLA.
And he's 46.
So some of these books are fairly old.
But I managed to find some of them at the library.
I've got Flatland by Edwin Abbott.
I've got The Elements of Logic by S Barker.
And I've got Calculus Pure and Applied by Sherlock.
And I thought, just out of curiosity, I'd take a look at what these books are like.
Before I get stuck into them, I will say these books contain hard stuff.
The textbooks are university level.
And the idea that an eight year old would read them doesn't even really make sense to
me.
But it's true that most of us weren't given these books as eight year olds, so we will
never really know if we could have read them.
Of course, young Terence was lucky that his family was able to support his talents.
I believe that genius is not just a function of your mind, but also your environment.
But that shouldn't take away from how impressive Terence is.
To have the concentration and discipline to finish these books at any age is awesome.
Even with the incentive of filming this video, I have probably spent more time on TikTok
than I have reading these books over the past week.
And for that I do look up to the enthusiasm of an eight year old.
Flatland is the only one of the books that I had read already.
And unlike the others, which are textbooks, this one is fiction.
It's a story about shapes, which I love.
I think it's clever, insightful, and it made me think about dimensions in a way I never
had before.
But honestly, I didn't love everything about it.
There are some parts that I found hard to read because of the social commentary about
gender and class that just doesn't land well today.
The story is set on a two dimensional plane.
And the narrator, a self described gentleman, is a square.
Shapes with more sides are higher in class.
Hexagons are noble, and something with more sides than that will approach the priestly
order.
Lower class workman such as Square's Butler are triangles.
Woman are a single straight line.
There's his wife right there.
Square's house is a Pentagon but this top down view is not how he sees it.
To him all things look like lines of varying size.
But he can feel the vertices and determine the angle.
Square visits Lineland, which is only one dimensional, and wishes that he could communicate
to the inhabitants that there is a second dimension out there.
But when Square is visited in Flatland by a sphere from a third dimension, Square finds
it hard to accept because he can't imagine a third dimension.
Square wonders if this is true, then what would come next? and asks Sphere if there
is a fourth dimension.
Sphere says "no, that's ridiculous".
But really, they're having the same problem as Square, they simply cannot imagine a dimension
above their own.
Overall, it's a worthwhile mind bender.
And it shows that even a book about math can't be totally separated from the culture in which
it was written.
Moving on to Calculus.
This is what seems like a fairly standard calculus textbook.
The first line of it is 'any situation involving a rate of change is described mathematically
in the language of calculus.'
It starts with the idea of a gradient then works through differentiation, areas under
a curve, and integration.
And we've got circular functions, motion terms and parametric equations, a visit to infinite
series and exponentials and logarithms.
There's also differential equations, hyperbolic functions and curves and surfaces.
Some of the stuff I met at the end of high school or at university and much of it I've
forgotten.
I've slept on my integration so long I can barely do the basics.
So this book would do me well as a refresher, it seems clear enough although from what I
can see, it doesn't always explain too much of the why, behind some of the definitions.
It gives revision questions after every few chapters with answers, which I like.
Someone who checked this out in the past, maybe this year, or maybe 39 years ago, when
it was published, has made a lot of annotations in pencil.
I find these comforting.
Amongst all the seriousness of the math is a human reminder and a source of company.
It's a reminder as I tried to decipher each example, that I'm not the only one who's been
here.
They even left a little To Do List: 'Saturday avo, read notes.'
This book would have just been published when Terence read it, which is maybe why he picked
it.
It is my opinion that when it comes to introductory calculus textbooks, they all pretty much cover
the same sort of stuff.
So if your goal is to learn calculus, then the best book for you is probably just the
one that you have access to.
Or it might be worth checking out a few sample pages from a few options to see which of the
explanations you prefer.
On young Terence's reading list, there is also Matrices and Vectors, Numbers, Inequalities
and Linear programming, Readings in mathematics, and I found this picture of him reading Continued
Fractions by Olds at age nine.
Also on the list is the International Mathematics Olympiad, a book of past problems.
And this is interesting because Terence would go on to become the youngest competitor in
this Olympiad.
At age 10.
He won a bronze, silver and then a gold medal at age 13, making him the youngest winner
of each of these categories.
And his two brothers, Nigel and Trevor have also represented Australia at this competition.
This next one is The Elements of Logic, and it's got a pretty nice cover.
It goes through an introduction to logic as a means to make deductive and inductive arguments,
to understand truth functions, translating words into symbols, fallacies, and applications
to philosophy.
It says that logic can be intellectually rewarding as knowledge for its own sake.
But it is also practical in helping us to become more effective in recognising and avoiding
mistakes and reasoning.
And a person who can recognise and avoid these logical mistakes in reasoning, will be able
to think more clearly and correctly and more soundly and surely, regardless of what subjects
they're thinking about.
If you are wondering what logic has to do with philosophy, then consider some of the
questions of philosophy.
Like what is the difference between good and evil?
Is it possible to have knowledge?
And can we ever prove that a thing is beautiful?
Well, logic also deals with very general questions, like what good reasoning is, and the difference
between correct and incorrect steps in thinking.
And in both philosophy and logic, no experiments are necessary.
Before we can consider what is good and bad reasoning, we need to consider what reasoning
itself is.
And it says here that reasoning is starting with certain statements that you assume to
be true, then in thinking, moving on to a consequence that you think follows from them.
So what we're doing is making an inference, constructing an argument and trying to give
a proof.
What we start with are called premises.
And what we reach at the end is the conclusion.
And the book goes on to explore all the different forms of reasoning.
We start to see some symbols here with truth tables.
This line means NOT.
And the symbol v means OR, so for example, in the top line, if p is true, and q is true,
then p AND q has to be true, but NOT (p AND q) would have to be false.
NOT p is false, NOT q is false, NOT p AND NOT q is false, and NOT p OR NOT q is also
false.
And from there, we work up to the elementary form of a valid argument.
Each line here represents a step of logic that we can use to construct an argument.
This first bit is read, if p then q.
So if p is true, therefore q is true.
Kind of like when we're doing algebra, we can replace something with an equivalent form.
We can do that here with logical expressions.
These are a list of statements that are equivalent to each other.
So we can put it all together in an example like this one here, we start with these first
three premises, we then make a series of logical arguments, and end with forming a conclusion.
This book was written in 1965 and there is one application of logic that isn't mentioned,
but is now central to modern computing.
And that is logic gates.
This part of the video is sponsored by Brilliant and I'm going to work through their quiz on
logic gates to see how they work.
We see the symbols for AND, OR, and NOT, and we can see what these basic gates would look
like.
AND outputs a one if both inputs are one, OR outputs a one if at least one input is
one, NOT can take a one to a zero, or a zero to a one.
To combine these into a circuit means we can chain a logical expression.
The circuit here represents A AND NOT B, or NOT A AND B.
We're asked what the output of the circuit will be, and let's go through it.
This first gate will output a one since both inputs are one, that one would be combined
with a zero as input into the second AND, and that would output a zero, since an AND
requires that both inputs are one for the output to be one.
Circuits, like logical expressions, can be equivalent.
For example, these two circuits are equivalent as two consecutive NOT gates have no effect.
And this is especially important for circuits as we generally want to build ones that are
as small as possible.
Combining logic gates, we can perform all sorts of operations, including things like
the addition of numbers.
If you would like to continue this course yourself, you can head to brilliant.org/Tibees.
And that link will be down in the description below.
Thank you to my Patreon supporters for making these videos possible.
And a special shout out to today's Patron Cat of the Day, Feldman.
Thanks for watching.