Mathematical Truth VS Scientific Truth
Episode Stats
Summary
In this episode, we explore the difference between mathematical and scientific truth, and how they are different from each other. In fact, there is no such thing as an infinite number of prime numbers, and there can be no limit to prime numbers.
Transcript
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One of my favorite jokes, it's an old joke, involves the different perspectives
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Say you present a mathematician, a scientist, and an engineer with a little red ball,
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and you ask them, what is the volume of this ball?
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Well, the mathematician pulls out his ruler, measures the diameter, and then with a bit of math,
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four-thirds pi r cubed, he gives you the volume.
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The physicist, meanwhile, takes the ball, dunks it in a beaker of water,
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and measures how much water has been displaced by the ball.
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Then you go up to the engineer, and upon studying it, he pulls out his big red book of little red balls,
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and looks up the serial number, then reads the volume out to you.
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I want to talk about the differences between mathematical truth, scientific truth, and engineering best practice.
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Because I think it's an important distinction to keep in mind when we're talking about what is true,
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It's important to keep in mind what domain we're operating within.
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And so I'm going to illustrate this with how truth would be applied in each one of these fields,
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Now, one of the early questions in math was, are there an unlimited number of prime numbers?
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In the natural number system, do we run out of prime numbers eventually?
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Do we just get to numbers that are so big they're divisible by everything?
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Or are there an infinite number of these prime numbers?
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And Euclid came up with a very easy to understand, very intuitive proof that there's infinite.
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factorial is when you take a number and multiply it by all the numbers that are smaller than it.
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So if you're doing factorial 5, it's 5 times 4 times 3 times 2 times 1.
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So you take a prime number, you factorial it, and then you add 1.
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And if that number isn't a prime number, it's divisible by a prime number larger than the one you started off with.
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So we're going to take the number 2, and we're going to factorial it.
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7 times 6 times 5 times 4 times 3 times 2 times 1.
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5,041 is not divisible by 7, or 6, or 5, 4, 3, 2.
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Either it is only divisible by itself and 1, or there's a higher prime number.
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Let's say the number 8, well, that's divisible by 2.
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Or it's, well, it can't be divisible by 14, or 10, because then it would be divisible by 7, or 5.
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So there must be a larger prime number than 7, if 50, 41 is not itself a prime number.
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And sure enough, 50, 41 is the square of 71, the 20th prime number.
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Factorial 71, and the number you get plus 1 is not divisible by any of the numbers below 71.
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So it's either only divisible by itself, or it's divisible by itself in another prime number.
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You now know, without a doubt, you know that there's an infinite number of primes out there.
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Once you understand this simple little proof, you know it.
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And it's never going to be adjusted, it's never going to be modified.
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You know there's an infinite number of primes out there.
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So when we talk about mathematical proof, that's what we're talking about.
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Okay, when something is proven in mathematics, it is rock solid.
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You have no doubt in your mind when you have a mathematical proof.
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This is why mathematics is considered the hardest of the disciplines.
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You're trying to understand what gravity is doing.
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When you drop an object, in the first second, it falls 4.9 meters.
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And after the second second, it's fallen 19.6 meters.
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You do a little bit of math, a little bit of calculus.
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And you figure out that objects accelerate towards the Earth at 9.81 meters per second squared.
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And everywhere you go, you get the exact same observation.
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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.
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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.
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Clearly, gravitation is not as simple as things falling towards the Earth.
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So now, the acceleration due to gravity is g times mass 1 times mass 2 over the radius squared.
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And if you calculate that, when you're at sea level on Earth, you get 9.81 meters per second squared.
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Observations haven't changed, but the law has changed.
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Your understanding of the universe has changed.
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Now you have a more complex theory, a more subtle theory, and a more complex law to describe the relationship.
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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.
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And things are going just fine and dandy with your new gravitational constant when you notice that light bends around gravity wells.
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So now, now, nope, it's not simple mass attracting mass in a Euclidean geometry space.
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No, now you have to bend spacetime to explain how something like a photon, which has no mass, is going to curve around mass.
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Again, you're getting the same results, you're getting the same observations.
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But each new observation adds a layer of subtlety.
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So with math, the proof is the proof is the proof, and it's done.
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With science, you never really prove anything with science.
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Things fall to the Earth at 9.81 meters per second.
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And you do 100 experiments, and you fail to disprove that for 100 experiments.
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And then you do the 101st experiment on the moon, and you disprove it.
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You get a result that does not match what you thought you were going to get.
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So that's science. Science is always learning more.
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And it's not contradicting the old observations, okay?
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It's not like science just changing its mind like it's fickle or anything like that.
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You understand them inside and out intuitively.
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In fact, once you understand a mathematical proof, you can't not understand it.
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You can't imagine the universe being any other way.
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Whereas with science, there's always more mystery that we are discovering with it.
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Now, engineering is not so much concerned with the truth.
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So let's say you were building a building as an engineer.
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And for whatever reason, you were concerned about things falling off the side of the building.
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Well, what are you going to assume is the acceleration of these things falling off the building?
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You're not going to say that things fall at 9.81 meters per second squared.
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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.
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Now, why would the engineer put this variance in after we know it's 9.81 meters per second?
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The physicist designed the perfect milk barn, but it only works for spherical cows in a vacuum.
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The engineer is dealing with the messiness of the real world.
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He doesn't really care why things work the way they do.
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And so the reason he put this margin of error in there is, you know, take the old riddle.
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What weighs more, a pound of bricks or a pound of feathers?
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For you American viewers, feathers, that's the English word for bird leaves.
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So what weighs more, a pound of bricks or a pound of bird leaves?
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Obviously, the bricks are going to hit the ground way before the bird leaves too.
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9.81 meters per second squared is under ideal conditions.
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You know, maybe the thing falling off the side of the building has a lot of drag.
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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.
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So the engineer builds things with margins of error.
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And if you look into, and this is why the engineer has his big red book of little red balls.
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And so the mathematician measuring the ball will give you the definite answer.
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The physicist is going to tell you how much water was deployed.
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Mathematician is giving you the definite answer.
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The scientist is describing the consequences, the relationships.
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You know, it displayed this volume of water, so it must have this volume.
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The engineer, when he looks it up in his big red book of little red balls,
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is going to say the volume of this little red ball is anywhere between this value and that value.
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And the big red book of little red balls has been so thoroughly tested that he's not going to run a mathematical experiment.
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He's not going to run a scientific experiment to figure out the ball.
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He is going to trust the values that have been reached after a great deal of effort.
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So that's the difference between these three fields.
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Once you understand math, you can't not understand math.
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Science is based upon observations and some guesswork, some intuition,
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trying to figure out why we observed what we observed and trying to predict how it's going to operate in the future.
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All it does is it eventually disproves it and forces us to think even harder.
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And when it comes to engineering, you're talking about practical applications with reasonable margins of error.
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The engineer doesn't assume that he's dealing with spherical cows in a vacuum.
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He assumes the world is messy and unpredictable and that even if steel has a melting point of this,
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well, maybe it's a really hot day and that melting point changed.
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So yeah, folks, truth, well, it's not always truth.
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So keep that in mind and don't mistake a scientific theory for an absolute fact.
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And don't mistake engineering practices for ontology.