There's a fundamental difference between the heuristics and knowledge derived through mathematics versus that which comes from the scientific method. As for Engineering, that's all about best practices.
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One of my favorite jokes, it's an old joke, involves the different perspectives of mathematicians, scientists, and engineers.
And it goes like this.
Say you present a mathematician, a scientist, and an engineer, with a little red ball.
And you ask them, what is the volume of this ball?
Well, the mathematician pulls out his ruler, measures the diameter, and then with a bit of math, 4/3 pi r cubed, he gives you the volume.
The physicist, meanwhile, takes the ball, dunks it in a beaker of water, and measures how much water has been displaced by the ball.
Then you go up to the engineer, and upon studying it, he pulls out his big red book of little red balls and looks up the serial number, then reads the volume out to you.
I want to explore this concept in this video.
I want to talk about the differences between mathematical truth, scientific truth, and engineering best practice.
Because I think it's an important distinction to keep in mind when we're talking about what is true, what is objective reality.
It's important to keep in mind what domain we're operating within.
And so I'm going to illustrate this with how truth would be applied in each one of these fields.
Starting with mathematics.
Now, one of the early questions in math was: are there an unlimited number of prime numbers?
In the natural number system, do we run out of prime numbers eventually?
Do we just get to numbers that are so big they're divisible by everything?
Or are there an infinite number of these prime numbers?
And Euclid came up with a very easy to understand, very intuitive proof that there's infinite.
There is no limit to prime numbers.
And the proof is this simple.
You take the factorial of a prime number.
And the factorial, for the record, factorial is when you take a number and multiply it by all the numbers that are smaller than it.
So if you're doing factorial 5, it's 5 times 4 times 3 times 2 times 1.
So you take a prime number, you factorial it, and then you add 1.
And if that number isn't a prime number, it's divisible by a prime number larger than the one you started off with.
So let's run through this.
So we're going to take the number 2, and we're going to factorial it.
2 times 1 is 2.
Add 1, you have 3, another prime number.
Factorial 3.
3 times 2 times 1 is 6.
Add 1 to it, you have 7, another prime number.
Now it's factorial 7.
7 times 6 times 5 times 4 times 3 times 2 times 1, and you get 5,040.
Add 1 to that, 5,041.
Now, let's think about this for a second.
5041 is not divisible by 7, or 6, or 5, 4, 3, 2.
It's not divisible by any of those.
Either it is only divisible by itself and 1, or there's a higher prime number.
Because what happens if we go past 7?
Let's say the number 8, well, that's divisible by 2.
The number 9, divisible by 3.
The number 10 divisible by 5.
Well, 11, there you have another prime number.
12, 13, 14.
So either 5041 is divisible by 11 or 13, or it's, well, it can't be divisible by 14 or 10, because then it would be divisible by a 7 or 5.
So there must be a larger prime number than 7 if 5041 is not itself a prime number.
And sure enough, 5041 is the square of 71, the 20th prime number.
Keep running through it.
Factorial 71, and the number you get plus 1 is not divisible by any of the numbers below 71.
So it's either only divisible by itself, or it's divisible by itself in another prime number.
Do you see this?
This mathematical proof?
You now know, without a doubt, you know that there's an infinite number of primes out there.
And this can't change.
You know this beyond all doubt.
Once you understand this simple little proof, you know it.
And it's never going to be adjusted, it's never going to be modified.
You know there's an infinite number of primes out there.
So when we talk about mathematical proof, that's what we're talking about.
Okay, when something is proven in mathematics, it is rock solid.
It is an absolute certainty.
You have no doubt in your mind when you have a mathematical proof.
This is why mathematics is considered the hardest of the disciplines.
Now, let's take science.
Let's say you're trying to figure out gravity.
You're trying to understand what gravity is doing.
Well, you make some observations.
When you drop an object, in the first second, it falls 4.9 meters.
And after the second second, it's fallen 19.6 meters.
You do a little bit of math, a little bit of calculus, and you figure out that objects accelerate towards the Earth at 9.81 meters per second squared.
Excellent.
And everywhere you go, you get the exact same observation, 9.81 meters per second squared.
So you come up with a theory of gravitation.
And your theory is that things are pulled down towards the Earth, and that the acceleration, the law, the relationship, is 9.81 meters per second squared.
And things are going absolutely great until one day you drop an object and it falls at 1.6 meters per second squared because you're on the moon.
Well, now we've got an issue.
Clearly, gravitation is not as simple as things falling towards the Earth.
There's a little bit more going on.
So you expand the law.
You introduce the gravitational constant.
So now the acceleration due to gravity is g times mass 1 times mass 2 over the radius squared.
And if you calculate that, when you're at sea level on Earth, you get 9.81 meters per second squared.
You have the exact same observation.
Observations haven't changed, but the law has changed.
Your understanding of the universe has changed.
Now you have a more complex theory, a more subtle theory, and a more complex law to describe the relationship.
It still gives you the same results that you had with your simple law, but it's far more accurate for a variety of circumstances.
And things are going just fine and dandy with your new gravitational constant when you notice that light bends around gravity wells.
So now, now, nope, it's not simple mass attracting mass in a Euclidean geometry space.
No, now you have to bend space-time to explain how something like a photon, which has no mass, is going to curve around mass.
Again, you're just getting the same results.
You're getting the same observations, but each new observation adds a layer of subtlety.
So with math, the proof is the proof is the proof, and it's done.
With science, you never really prove anything with science.
In fact, all you can do is disprove things.
You know, you start off with the simple law.
Things fall to the Earth at 9.81 meters per second, and you do 100 experiments, and you fail to disprove that for 100 experiments.
So everything's good, you're happy.
And then you do the 101st experiment on the moon, and you disprove it.
You get a result that does not match what you thought you were going to get.
So now you have to readjust everything.
So that's science.
Science is always learning more.
And it's not contradicting the old observations.
It's not like science just changing its mind like it's fickle or anything like that.
But it's not complete.
Mathematical proofs are complete.
They are done.
You understand them inside and out intuitively.
In fact, once you understand a mathematical proof, you can't not understand it.
You can't imagine the universe being any other way.
Whereas with science, there's always more mystery that we are discovering with it.
And then you get to engineering.
Now, engineering is not so much concerned with the truth.
It's concerned with getting stuff done.
So let's say you were building a building as an engineer, and for whatever reason, you were concerned about things falling off the side of the building.
This was important for your blueprints.
Well, what are you going to assume is the acceleration of these things falling off the building?
I'll tell you what, you're not going to say that things fall at 9.81 meters per second squared.
No, no, that's way too many assumptions there.
The engineer is going to design this building so that things that fall off of it will fall anywhere between 9 and 11 meters per second squared.
Now, why would the engineer put this variance in?
After we know it's 9.81 meters per second.
Well, you know, that's the physics joke.
The physicist designed the perfect milk barn, but it only works for spherical cows in a vacuum.
The engineer is dealing with the messiness of the real world, and so he puts margin of error.
He doesn't really care why things work the way they do.
He just cares that they work.
And so the reason he'd put this margin of error in there is, you know, take the old riddle.
What weighs more?
A pound of bricks or a pound of feathers?
For you American viewers, feathers, that's the English word for bird leaves.
So what weighs more?
A pound of bricks or a pound of bird leaves?
Well, they both weigh a pound, but which one's going to fall faster?
Obviously, the bricks are going to hit the ground way before the bird leaves do.
So, this is why he has that margin of error.
9.81 meters per second squared is under ideal conditions.
You know, maybe the thing falling off the side of the building has a lot of drag.
Or, alternatively, maybe the shape of this object has some sort of sail or fin on it that the crosswinds going past the building actually serve to accelerate its descent.
So, the engineer builds things with margins of error.
And if you look into, and this is why the engineer has his big red book of little red balls.
And so, the mathematician measuring the ball will give you the definite answer.
The physicist is going to tell you how much water was deployed.
He's going to describe the consequences of it.
All right?
Mathematician is giving you the definite answer.
Scientist is describing the consequences, the relationships.
You know, it displayed this volume of, displaced this volume of water, so it must have this volume.
The engineer, when he looks up in his big red book of little red balls, is going to say the volume of this little red ball is anywhere between this value and that value.
Because again, margins of error.
And the big red book of little red balls has been so thoroughly tested that he's not going to run a mathematical experiment.
He's not going to run a scientific experiment to figure out the ball.
He is going to trust the values that have been reached after a great deal of effort.
So that's the difference between these three fields.
Mathematics is ontological.
Okay, once you understand math, you can't not understand math.
Science is based upon observations and some guesswork, some intuition, trying to figure out why we observed what we observed and trying to predict how it's going to operate in the future.
And it never proves anything.
All it does is it eventually disproves it and forces us to think even harder.
And when it comes to engineering, you're talking about practical applications with reasonable margins of error.
The engineer doesn't assume that he's dealing with spherical cows in a vacuum.
He assumes the world is messy and unpredictable, and that even if steel has a melting point of this, well, maybe it's a really hot day and that melting point changed.
Who knows?
So he puts margins of error into everything.
So yeah, folks, truth, well, it's not always truth.
Not all forms of truth are the same.
So keep that in mind, and don't mistake a scientific theory for an absolute fact.
And don't mistake engineering practices for ontology.
