Transcript
DR JOHN WATT
This week, we meet the unsung hero of the science world – mathematics. It’s the drive behind the engineering that makes the world tick.
VOICE-OVER
From construction sites to traffic lights to air traffic control. But this week, we’re looking at how maths governs how nature works as well and can even be used to predict how we as human beings behave. First, I’m off to Victoria University to meet mathematician Dillon Mayhew. He’s going to try to convince this chemist that maths is the most special of sciences.
DR JOHN WATT
Dillon, what is the difference between mathematics and other sciences?
DR DILLON MAYHEW
Mathematics doesn’t rely upon experimentation, so we don’t get our knowledge from running experiments and making observations. We get our knowledge from making logical deductions. So I can illustrate the difference between experimentation and deduction with an example. Let’s do some sums. What’s 1 + 2?
DR JOHN WATT
Three.
DR DILLON MAYHEW
Yes, that’s very good. And what’s 1 + 2 + 3?
DR JOHN WATT
Six.
DR DILLON MAYHEW
Excellent! And 1 + 2 + 3 + 4?
DR JOHN WATT
That’s 10.
DR DILLON MAYHEW
You’re really very good at this.
DR JOHN WATT
Yes, I am.
DR DILLON MAYHEW
But what if I asked you to add up 1 + 2 + 3 all the way to 100. What would you do then?
DR JOHN WATT
I’d look to you.
DR DILLON MAYHEW
Yeah, and you would hope that there’s a formula.
DR JOHN WATT
Yeah.
DR DILLON MAYHEW
And there is. Let’s see if we can figure out what it is. So 3 is equal to 2 times 3 divided by 2.
DR JOHN WATT
Right.
DR DILLON MAYHEW
Six is equal to 3 times 4 divided by 2, and 10 is equal to 4 times 5 divided by 2.
DR JOHN WATT
Right.
DR DILLON MAYHEW
And notice that the first number here is always the last number in the sum. So that lets us predict that the general formula, if I take the numbers 1 + 2 + 3 all the way up to some unspecified number, which I’ll just call n, the answer is going to be that last number, n, times the next number, n + 1, divided by 2. Do you want to try it out?
DR JOHN WATT
All right, let’s do it. So what shall we go to?
DR DILLON MAYHEW
Let’s say n is equal to 7.
DR JOHN WATT
So 1 + 2 is 3, + 3 is 6, + 4 is 10, + 5 is 15, + 6 is 21, + 7 is equal to 28. So using that 7 times 8 divided by 2 is equal to 56, divided by 2 is equal to 28. So you’ve got 28.
DR DILLON MAYHEW
It’s like magic, isn’t it?
DR JOHN WATT
Voila! So how do you know this is always going to work?
DR DILLON MAYHEW
If mathematics were an experimental science, all we could do is test the first 100 cases or maybe test the first 1 billion cases. But would that be enough?
DR JOHN WATT
Well, it tells you that it’s always true for those numbers.
DR DILLON MAYHEW
Right. But how would we know that it works for 1 billion and 1?
DR JOHN WATT
I’d look to you.
DR DILLON MAYHEW
Yeah, exactly. Experimentation cannot tell us that this always works. Instead, we have to use deductible logic. So here’s how we can show that it always works. I’m going to take these numbers, 1 + 2 + 3 all the way up to n, and I’m going to write the same series of numbers underneath, but in reverse order. So I’ll write n + (n - 1) + (n - 2) all the way down to 1.
Now notice that this top row adds to exactly the same quantity as the bottom row. They’re the same numbers, just in a different order. So now I’m going to add the top and the bottom row together, and I’m going to do it column by column. What’s the sum of these two numbers?
DR JOHN WATT
n + 1.
DR DILLON MAYHEW
n + 1.
VOICE-OVER
In fact, every single column adds to n + 1.
DR DILLON MAYHEW
OK, so what’s this total? I’ve got how many collections of n + 1?
DR JOHN WATT
Well the amount of numbers, so whatever n is.
DR DILLON MAYHEW
Exactly, whatever n is. So I’ve got n, lots of n + 1, so that’s n times n + 1. But the sum of the top row is equal to half the sum of the bottom row, so I take that quantity and divide by 2, which is that one. It’s exactly the formula we predicted. So now we know that it’s always going to work.
DR JOHN WATT
Dillon, this is great for a bunch of numbers, but what does this tell me about the real world?
DR DILLON MAYHEW
A mathematician can do mathematics for exactly the same reasons that an artist can do art – to make something beautiful to add to our heritage. So if you think this is beautiful, well, that’s enough. But of course, mathematics is also useful, and it’s useful because we can use it to model the natural world. For reasons that I don’t really understand, it seems as though the natural world runs on mathematics. So, if we want to model the process in the natural world, then mathematics is the best language to do it.
DR JOHN WATT
So what kind of examples can you give me?
DR DILLON MAYHEW
Well, suppose that I want to drop an object from a height and I want to figure out how long it’s going to take to hit the ground. How can I do that?
DR JOHN WATT
You’d have to know a few things first.
DR DILLON MAYHEW
You need to know the formula, and it’s a mathematical formula because the natural world seems to run on mathematics. So here’s the formula. So the time it takes for the object to fall is equal to the square root of 2 times the height you drop it from divided by 9.81.
DR JOHN WATT
Which is nice and simple. Shall we test it?
DR DILLON MAYHEW
I don’t see why not. Let’s go.
DR JOHN WATT
It’s cold.
DR DILLON MAYHEW
Well it’s cold and windy.
DR JOHN WATT
But in the name of science, I guess …
DR DILLON MAYHEW
In the name of science. So what we’re going to do in honour of Isaac Newton is drop this apple from this balcony and I’m going to use that formula to predict how long it takes to hit the ground.
DR JOHN WATT
Right. What do we need to know?
DR DILLON MAYHEW
We need to know a couple of things. We need to know how high the drop is. To figure that out, I need you down there with a tape measure.
DR JOHN WATT
All right, done!
VOICE-OVER
The mathematical truths that Newton discovered in the 17th century are just as true today and will remain true forever – a certainty that is seldom achieved in any other field.
DR JOHN WATT
All right, 9.6 metres.
VOICE-OVER
Now we have the height of the drop, Dillon can solve his equation.
DR DILLON MAYHEW
Take the square root … 1.4 seconds. So you’re ready?
DR JOHN WATT
Yeah, go for it!
DR DILLON MAYHEW
This is the moment of truth.
DR JOHN WATT
OK.
DR DILLON MAYHEW
On 3.
DR JOHN WATT
OK.
DR DILLON MAYHEW
1, 2, 3.
DR JOHN WATT
I got 1.36 seconds.
DR DILLON MAYHEW
Well I predicted 1.4.
DR JOHN WATT
So that’s close as!
DR DILLON MAYHEW
That’s pretty good.
DR JOHN WATT
That was a little bit of physics to get us started, but up next, we meet a brave mathematician taking on the terrifyingly complex world of biology in an attempt to answer some of evolution’s biggest questions.
Go to Part 2
