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<v Speaker 1>Think about the software you rely on the most today, right,

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<v Speaker 1>Like we look at these sleek interfaces on our phones,

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<v Speaker 1>the seamless notifications, the smooth scrolling, and we view them

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<v Speaker 1>as finished, polished machines.

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<v Speaker 2>Yeah, totally.

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<v Speaker 1>But back in nineteen seventy six, one of the absolute

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<v Speaker 1>pioneers of computer science argued that presenting software this way

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<v Speaker 1>is well, not just wrong, he said, it makes it

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<v Speaker 1>completely unfit for human appreciation.

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<v Speaker 2>Which is I mean, it is a staggering critique to

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<v Speaker 2>level at an entire industry. Oh absolutely, And that tension

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<v Speaker 2>really sits right at the core of our deep dive

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<v Speaker 2>today into Edgar W. Dykstra's seminal nineteen seventy six book

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<v Speaker 2>It's called a Discipline of Programming.

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<v Speaker 1>Yeah, in Dykster he was looking at this rapidly expanding

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<v Speaker 1>field back then, right and realizing that by focusing solely

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<v Speaker 1>on the final executable code, we were basically hiding the

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<v Speaker 1>actual art exactly.

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<v Speaker 2>We were burying the messy, beautiful mathematical design process that

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<v Speaker 2>you know the software in the first place.

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<v Speaker 1>Which is like it's like looking at the final score

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<v Speaker 1>of a chess game without ever being allowed to see

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<v Speaker 1>the brilliant strategic moves that were actually played on the board.

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<v Speaker 2>Right, you know who won, but you're entirely blind to

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<v Speaker 2>the intellect that got them there exactly.

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<v Speaker 1>And to fix this, Dykstra goes to some pretty wild

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<v Speaker 1>extremes in this book. I mean, he actually refused to

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<v Speaker 1>use any of the standard quote unquote higher level programming

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<v Speaker 1>languages of his era.

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<v Speaker 2>Yeah, he hated them. He argued that they're excessive features,

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<v Speaker 2>like all those bills and whistles we love. They actually

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<v Speaker 2>belong to the solution set rather than the problem set.

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<v Speaker 1>They're just distractions, total distractions.

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<v Speaker 2>So he just designs his own minimalist mini language specifically

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<v Speaker 2>for this text. He forces you to strip away the

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<v Speaker 2>hardware and focus purely on the logic.

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<v Speaker 1>Okay, let's unpack this because the way he visualizes a

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<v Speaker 1>program without using actual code the cardboard computer.

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<v Speaker 2>It's just the cardboard machine. Yeah. He introduces this concept

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<v Speaker 2>called executional abstraction. It's basically how we map different computations

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<v Speaker 2>onto each other so our human brains can actually grasp them.

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<v Speaker 2>And to explain this without any digital hardware, he uses

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<v Speaker 2>the classic math problem of finding the greatest common divisor

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<v Speaker 2>the GCD of two numbers.

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<v Speaker 1>Right, like say one eleven and two fifty nine exactly.

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<v Speaker 2>But he solves it using literal pieces of cardboard. He

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<v Speaker 2>asked you to imagine two completely different cardboard machines.

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<v Speaker 1>Okay, so the first machine, it's deceptively simple. It's just

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<v Speaker 1>two pieces of cardboard on top of each other. The

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<v Speaker 1>top piece says GCD one eleven two hundred and fifty

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<v Speaker 1>nine equals and when you lift that.

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<v Speaker 2>Flat, the bottom piece just says thirty seven.

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<v Speaker 1>Right, just the answer.

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<v Speaker 2>It's the ultimate black box. I mean, it gives you

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<v Speaker 2>the answer instantly, zero friction, which sounds great, right, it

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<v Speaker 2>sounds perfect, Yeah, but it possesses a catastrophic flaw for

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<v Speaker 2>a mathematician. It requires absolute blind faith in the manufacturer.

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<v Speaker 1>Yeah, you just have to trust that whoever printed that

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<v Speaker 1>cardboard didn't like make a TYPEO exactly.

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<v Speaker 2>There is zero visibility into the logic, and worse, it

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<v Speaker 2>only solves that one hyper specific iteration of the problem.

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<v Speaker 1>You're completely blind to the logic. You just have to

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<v Speaker 1>trust the system. And trust is terrible mathematics.

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<v Speaker 2>It really is, which is why Dykstra immediately replaces that

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<v Speaker 2>black box with the second machine. And this one is

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<v Speaker 2>very different, very this one is a massive piece of

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<v Speaker 2>cardboard featuring a colossal grid of two hundred and fifty

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<v Speaker 2>thousand points. You think of an X and Y axis,

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<v Speaker 2>both stretching from zero all the way to five hundred

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<v Speaker 2>and cutting right down the middle of this grid is

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<v Speaker 2>a diagonal answer line where the X coordinate perfectly equals

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<v Speaker 2>the Y coordinate.

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<v Speaker 1>And to operate this massive grid, you play what he

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<v Speaker 1>calls the pebble game, which I love.

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<v Speaker 2>It's so good.

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<v Speaker 1>You take a pebble and it represents your two starting numbers,

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<v Speaker 1>so X is two fifty nine, Y is one eleven,

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<v Speaker 1>and you place the pebble on that specific coordinate on

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<v Speaker 1>the grid right. Then you follow one strict geometric rule

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<v Speaker 1>as long as the pebble is not resting on that

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<v Speaker 1>diagonal answer.

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<v Speaker 2>Line, which it wouldn't be since two fifty nine does

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<v Speaker 2>an equal one to eleven exactly.

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<v Speaker 1>So you look at your axe is you form the

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<v Speaker 1>smallest equilateral right triangle you can with them, and you

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<v Speaker 1>move the pebble to the sharp angle of that triangle.

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<v Speaker 2>And the geometry here it's just a beautiful physical manifestation

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<v Speaker 2>of eucliding division.

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<v Speaker 1>It's actually kind of wild to picture.

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<v Speaker 2>It is because when you drop a perpendicular line to

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<v Speaker 2>the axes and form that triangle, moving to the sharp

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<v Speaker 2>angle is literally just subtracting the smaller number from the

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<v Speaker 2>larger number.

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<v Speaker 1>Ah.

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<v Speaker 2>So, if you are at two fifty nine to one eleven,

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<v Speaker 2>the smaller side is one eleven. Moving the pebble physically

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<v Speaker 2>shifts your ex coordinate down by one eleven, landing you

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<v Speaker 2>at one forty eight.

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<v Speaker 1>You are physically watching subtraction happen in space.

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<v Speaker 2>Yes, and you repeat that geometric slide over and over.

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<v Speaker 2>You keep moving the pebble down and over, down and

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<v Speaker 2>over until it eventually hits that diagonal line where X

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<v Speaker 2>and Y are I.

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<v Speaker 1>Tentacle, and wherever it lands on that line, that coordinate

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<v Speaker 1>is your greatest common divisor exactly. Now, I have to

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<v Speaker 1>be honest, My initial reaction to this was pure pushback.

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<v Speaker 1>Oh really, yeah, I mean, wait, so, so instead of

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<v Speaker 1>just giving me a calculator, you've made me play a

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<v Speaker 1>tedious board game across a quarter of a million potential

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<v Speaker 1>starting squares just to get the same results. Right.

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<v Speaker 2>It feels like way more work.

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<v Speaker 1>But then the genius of the trade offf hits you.

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<v Speaker 2>Because the tradeoff is the entire point. By establishing that

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<v Speaker 2>one single compact rule the mathematical invariant, Right, you haven't

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<v Speaker 2>just solved the problem for two fifty nine and one eleven.

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<v Speaker 2>You have instantly proven the outcome for all two hundred

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<v Speaker 2>and fifty thousand possible starting positions on that board.

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<v Speaker 1>You trade the immediate, unverified convenience of the first machine

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<v Speaker 1>for the undeniable, mathematically proven truth of the second machine.

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<v Speaker 2>Exactly, a single elegant argument governs every possible game played

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<v Speaker 2>on that cardboard.

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<v Speaker 1>And that is the true essence of an algorithm. It's

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<v Speaker 1>not about rushing to an output. It's about establishing absolute

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<v Speaker 1>confidence in the rules that dictate the output.

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<v Speaker 2>Right. But if the immense power of an algorithm comes

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<v Speaker 2>from that absolute mathematical certainty, you face a new problem.

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<v Speaker 1>How do you actually communicate those rules?

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<v Speaker 2>Exactly? You cannot build a system of flawless, rigid logic

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<v Speaker 2>using a medium that is fundamentally designed for ambiguity. And

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<v Speaker 2>this lead Dikstra to deliver a massive warning about what

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<v Speaker 2>we currently call programming languages.

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<v Speaker 1>Yeah, he argues that calling them languages is actually a

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<v Speaker 1>really dangerous mixed blessing. It's a huge trap because we

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<v Speaker 1>use natural languages you know, English, Manner and whatever specifically

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<v Speaker 1>because they are vague. The imprecision of human language is

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<v Speaker 1>a feature, not a bug.

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<v Speaker 2>Right. It allows for poetry and context and nuance reading

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<v Speaker 2>between the lines.

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<v Speaker 1>But formal notation has to be the exact opposite.

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<v Speaker 2>It has to be perfectly rigid. If you write the

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<v Speaker 2>rules of the pebble game using informal words like Euclid

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<v Speaker 2>originally did thousands of years.

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<v Speaker 1>Ago, which was just in Greek, right, right, it.

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<v Speaker 2>Remains completely unfit for formal mathematical treatment by an automaton.

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<v Speaker 2>If we want to create automatic computers, we need unambiguous

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<v Speaker 2>formal notation.

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<v Speaker 1>Calling them programming languages tricks us into reading them like conversations.

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<v Speaker 1>And you see this constantly today with large language models.

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<v Speaker 2>All the time people try to.

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<v Speaker 1>Prompt an AI to write highly specific, flawless code using

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<v Speaker 1>just conversational English, and they get incredibly frustrated when the

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<v Speaker 1>AI you know, hallucinates or produces unpredictable garbage.

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<v Speaker 2>Because the natural language is too vague to hold the

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<v Speaker 2>strict and variant logic. We are literally using the wrong

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<v Speaker 2>tool for the job.

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<v Speaker 1>And Dykstra saw this trap forming decades ago.

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<v Speaker 2>He really did and he pairs this linguistic warning with

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<v Speaker 2>a sharp critique of early computer history, because in the

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<v Speaker 2>early days of computing, the hardware was the bottleneck.

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<v Speaker 1>It was astronomically expensive and massive, massive.

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<v Speaker 2>Highly limited, just incredibly clunky. Consequently, efficiency became the absolute

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<v Speaker 2>supreme quality criterion for programming.

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<v Speaker 1>So whatever made the vacuum tubes or the you know,

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<v Speaker 1>the magnetic drums spin the fastest was ground the quote

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<v Speaker 1>unquote best.

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<v Speaker 2>Code, which is exactly the problem. Because physical hardware is

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<v Speaker 2>inherently messy. It's full of bizarre physical quirks. So programmers

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<v Speaker 2>started writing code that catered to those specific physical anomalies.

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<v Speaker 1>They were optimizing for the machine, not for the math.

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<v Speaker 2>Right, the code became a truthful reflection of the hardware's weirdness,

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<v Speaker 2>completely sacrificing the logical beauty of the algorithm.

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<v Speaker 1>We were hacking the limitations of the cardboard instead of

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<v Speaker 1>focusing on the pebble game itself.

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<v Speaker 2>That's a great way to put it, and Dykstra argues

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<v Speaker 2>vehemently that we have to redress this balance. We have

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<v Speaker 2>to put the pure algorithm back in the center of

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<v Speaker 2>the universe.

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<v Speaker 1>I love how he phrased it in his view, the

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<v Speaker 1>fact that a piece of metal and silicon can actually

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<v Speaker 1>execute the algorithm is just a lucky accidental circumstance.

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<v Speaker 2>It's such a radical reframing. The execution is secondary, the

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<v Speaker 2>mathematics are primary.

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<v Speaker 1>So what does this all mean? If we accept that

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<v Speaker 1>the physical machine is just a lucky accident, we have

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<v Speaker 1>to ask how a program actually organizes its reality in.

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<v Speaker 2>The abstract, right, how does it manage what we call

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<v Speaker 2>a state space?

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<v Speaker 1>And the way he describes building this state space is

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<v Speaker 1>fascinating because it's historically humans named numbers haphazardly. Right, We

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<v Speaker 1>just invented new words as we needed them.

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<v Speaker 2>Like a child counting, just making up sounds.

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<v Speaker 1>Exactly, which is impossible to scale. It wasn't until we

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<v Speaker 1>adopted systematic nomenclature rules that let you drive the next thing,

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<v Speaker 1>like the decimal system, that mathematics really exploded.

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<v Speaker 2>Because systematic nomenclature is about defining boundaries rather than trying

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<v Speaker 2>to memorize infinite lists. And in computing we build state

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<v Speaker 2>spaces the exact same way.

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<v Speaker 1>Okay, walk us through that.

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<v Speaker 2>So Deekster ask us to imagine the mechanical register. Picture

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<v Speaker 2>a machine with eight wheels, and each wheel holds ten

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<v Speaker 2>digits zero through nine.

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<v Speaker 1>Like a combination lock.

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<v Speaker 2>Exactly like a lock, but simply combining those constraints you

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<v Speaker 2>create a Cartesian product. You instantly define a rigid state

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<v Speaker 2>space containing exactly one hundred million distinct possible states.

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<v Speaker 1>Wow, and once you have that perfectly defined mathematical space,

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<v Speaker 1>you don't have to wander through it blindly. The source

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<v Speaker 1>provides this incredibly elegant example of using equations what he

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<v Speaker 1>calls predicates to define specific conditions within that massive space

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<v Speaker 1>without ever listing them all out.

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<v Speaker 2>Right, because listing one hundred million states would be impossible.

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<v Speaker 1>Yeah, so you don't have to list out all three

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<v Speaker 1>hundred and sixty five days of the year to tell

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<v Speaker 1>a system what Christmas is.

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<v Speaker 2>Well, you just define the state, right, You just.

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<v Speaker 1>Define it as month equals December and day equals tots

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<v Speaker 1>five or day equals twenty.

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<v Speaker 2>Six, or defining payday is simply day equals twenty three.

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<v Speaker 2>It is a compact, flawless way to slice through a

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<v Speaker 2>massive Cartesian product. It's so clean it is. And this

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<v Speaker 2>exact mechanism leads to Diykstra's first major surprise while writing

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<v Speaker 2>this book, The immense power of predicate transformers.

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<v Speaker 1>Okay, now this sounds incredibly theoretical, but the application is

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<v Speaker 1>deeply practical, isn't it.

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<v Speaker 2>Oh completely, Because in the early days, programmers would get

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<v Speaker 2>lost trying to track every single intermediate state a machine

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<v Speaker 2>passed through during execution.

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<v Speaker 1>Like tracking a pinball bouncing around exactly.

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<v Speaker 2>It was just total mental overload. Credicut transformers allow a

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<v Speaker 2>programmer to skip that MetalMan. They let you define a

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<v Speaker 2>direct relation between your starting state and your final state. Okay,

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<v Speaker 2>They introduce the formal concept of ROUP, which stands for

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<v Speaker 2>the weakest precondition. The notation literally looks like rupip and

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<v Speaker 2>then in parentheses S comma R, where.

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<v Speaker 1>S is the system or the program you are running,

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<v Speaker 1>and are is the result like the post condition you're

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<v Speaker 1>desperately trying to achieve.

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<v Speaker 2>Right, So calculating the weakest precondition is essentially asking what

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<v Speaker 2>is the absolute broadest, most lenient safety net I need

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<v Speaker 2>to string up before I hit GO to mathematically guarantee

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<v Speaker 2>that this system will land exactly on my target.

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<v Speaker 1>So you are fundamentally working backwards, entirely backwards. Instead of

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<v Speaker 1>starting a program and just hoping it gets to the

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<v Speaker 1>right answer, you start with the undeniably true answer and

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<v Speaker 1>calculate the exact conditions necessary to make it inevitable.

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<v Speaker 2>Yes. And this mathematical rigorousness brings us to property one

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<v Speaker 2>of his predicate transformers, and I just have to say

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<v Speaker 2>it possesses quite possibly the most poetic name in the

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<v Speaker 2>entire history of computer science.

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<v Speaker 1>Oh, it really does. The law of the excluded miracle.

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<v Speaker 2>Law the excluded miracle.

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<v Speaker 1>Which, beyond just sounding beautiful, it perfectly grounds this abstract

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<v Speaker 1>math in an unbending reality because it states that if

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<v Speaker 1>your final goal, your desired post condition are is a

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<v Speaker 1>logical impossibility, which is represented by the state of false right,

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<v Speaker 1>then there is no starting condition in the universe that

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<v Speaker 1>can guarantee you will get there.

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<v Speaker 2>In formulaic terms, wop of s false equals false.

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<v Speaker 1>You cannot ask the machine for magic.

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<v Speaker 2>No, if your desired destination contradicts logical reality, no amount

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<v Speaker 2>of clever programming, no perfect starting conditions, nothing will bend

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<v Speaker 2>the universe to reach it. The system will fail. The

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<v Speaker 2>miracle is excluded.

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<v Speaker 1>Which is deeply comforting in a way. You know it

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<v Speaker 1>provides a bedrock of certainty. But all of this flawless

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<v Speaker 1>invariant logic relies on one massive assumption. What's that It

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<v Speaker 1>assumes that if you set the starting conditions perfectly, the

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<v Speaker 1>mechanism itself will carry out the steps exactly as planned.

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<v Speaker 1>But anyone who has ever used a piece of technology

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<v Speaker 1>knows that things don't always go according to plan.

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<v Speaker 2>The hardware betrays the math.

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<v Speaker 1>Exactly the hardware betrays the math.

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<v Speaker 2>And this realization leads directly to Dykstra's second and perhaps

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<v Speaker 2>most profound surprise in the book, Because up to this

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<v Speaker 2>point in computer science history, deterministic machines were the unquestioned holy.

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<v Speaker 1>Grail, deterministic meaning perfectly predictable.

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<v Speaker 2>Right, The entire discipline was built on the assumption that

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<v Speaker 2>identical initial states must one hundred percent of the time

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<v Speaker 2>yield identical outcomes, utter unbreakable predictability.

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<v Speaker 1>But the physical world doesn't really like perfect predictability, does

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<v Speaker 1>it not at all?

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<v Speaker 2>And Dykester shares this genuinely harrowing story from nineteen fifty eight.

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<v Speaker 1>Yeah, this part blew my mind.

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<v Speaker 2>He was working with a new machine that featured an

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<v Speaker 2>io interrupt, basically an external hardware signal that could asynchronously

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<v Speaker 2>pause the current process. Right, And because this interrupt was

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<v Speaker 2>tied to physical timing rather than logical steps. It caused

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<v Speaker 2>irreproducible nondeterministic behavior.

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<v Speaker 1>Meaning the machine would execute differently even when provided the

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<v Speaker 1>exact same initial state exactly.

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<v Speaker 2>A hardware glitch or you know, a microsecond timing delay

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<v Speaker 2>would evoke an erratic error that you couldn't reliably reproduce

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<v Speaker 2>to study. It was a ghost in the machine, a

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<v Speaker 2>total ghost, and Dyscher was paralyzed by this. He literally

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<v Speaker 2>referred to the machine as an intractable beast.

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<v Speaker 1>Intractable beast.

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<v Speaker 2>He was so terrified by the prospect of having to

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<v Speaker 2>build reliable, flawless software on top of a fundamentally unpredictable

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<v Speaker 2>foundation that he delayed a major software design decision for

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<v Speaker 2>three entire months.

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<v Speaker 1>Wait, three whole months, three months over one unpredictable interrupt.

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<v Speaker 1>That is just It really illustrates how foundational perfect determinism

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<v Speaker 1>was to their entire worldview.

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<v Speaker 2>Back then, it was everything to them.

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<v Speaker 1>Unpredictability wasn't just a bug, It was the colle apse

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<v Speaker 1>of mathematical certainty itself.

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<v Speaker 2>Right. But eventually he has this massive psychological and mathematical breakthrough.

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<v Speaker 2>He realizes that his terror wasn't because the machine was

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<v Speaker 2>inherently broken. His terror stemmed entirely from a lack of methodology.

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<v Speaker 1>He didn't have the mental models to process the unpredictability.

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<v Speaker 2>Exactly, and when he finally develops those models, he flips

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<v Speaker 2>the entire paradigm of computer science on its head. He

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<v Speaker 2>comes to regard nondeterminacy as the normal default situation of

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<v Speaker 2>complex systems.

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<v Speaker 1>He decides that perfect determinism, the thing they had worshiped

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<v Speaker 1>for decades, is actually just a special, highly restricted, and

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<v Speaker 1>in his own words, not even very interesting case of nondeterminacy.

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<v Speaker 2>It completely shatters the illusion. It's like accepting that friction

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<v Speaker 2>is the reality of the physical world and a frictionless

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<v Speaker 2>vacuum is just a highly specialized laboratory condition.

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<v Speaker 1>That's a perfect analogy.

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<v Speaker 2>And once Diyster accepts this, he begins building constructs that

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<v Speaker 2>actively embrace a level of nondeterminacy rather than fighting it.

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<v Speaker 2>He stops overspecifying how the machine.

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<v Speaker 1>Should behave Here's where it gets really interesting. The source

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<v Speaker 1>gives a brilliant example of this with his concept of

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<v Speaker 1>concurrent assignment. Yes, the concurrent assignment, because in traditional highly

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<v Speaker 1>deterministic programming, if you want to swap the values of

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<v Speaker 1>variable x and variable Y.

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<v Speaker 2>You can't just swap them, No, you can't.

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<v Speaker 1>You have to create a temporary third variable. You move

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<v Speaker 1>x to tempt, then you move Y to X, then

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<v Speaker 1>you move temp to hy. It is a rigid, hyper

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<v Speaker 1>sequential step by step march.

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<v Speaker 2>It forces a strict chronological sequence where none is logically necessary.

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<v Speaker 2>You're basically over constraining the machine.

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<v Speaker 1>Right, and Deekstra's concurrent assignment elegantly sidesteps this whole dance

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<v Speaker 1>by simply writing X comma Y becomes y comma x right.

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<v Speaker 2>Just swap them simultaneously.

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<v Speaker 1>You define the post condition, the values are swapped, and

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<v Speaker 1>you leave the exact mechanical sequencing slightly ambiguous. You allow

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<v Speaker 1>for nondeterminacy in the how safely bounded by the mathematical

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<v Speaker 1>invariant of the what.

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<v Speaker 2>You stop micromanaging the intractable beast and you start guiding

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<v Speaker 2>it with flawless logic.

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<v Speaker 1>Which perfectly synthesizes the massive journey Diekstra takes us on

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<v Speaker 1>in this book. I mean he wanted to pull us

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<v Speaker 1>away from being glorified mechanics just endlessly catering to the

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<v Speaker 1>physical anomalies and sequential limitations of whatever metal box happen

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<v Speaker 1>to be sitting on our desks.

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<v Speaker 2>We start with that massive cardboard grid, realizing that true

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<v Speaker 2>algorithmic beauty isn't about getting a fast answer, It's about

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<v Speaker 2>establishing a mathematical invariant that proves the truth across a

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<v Speaker 2>quarter of a million possibilities right.

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<v Speaker 1>And we learn to abandon the dangerous vagueness of natural languages,

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<v Speaker 1>recognizing that human ambiguity has absolutely no place in the

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<v Speaker 1>rigid architecture of an automaton.

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<v Speaker 2>Exactly, we arm ourselves with predica transformers. We learned to

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<v Speaker 2>work backwards from our desired reality to find that weakest precondition,

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<v Speaker 2>always always respecting the law of the excluded miracle.

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<v Speaker 1>We cannot ask for magic.

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<v Speaker 2>We cannot ask for magic.

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<v Speaker 1>And finally, we watch one of the greatest minds in

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<v Speaker 1>the field suffer a three month panic attack over an

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<v Speaker 1>unpredictable interrupt, only to emerge with the absolute revelation that

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<v Speaker 1>rigid determinism is a myth.

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<v Speaker 2>Nondeterminacy is the true nature of complex systems, and true

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<v Speaker 2>programming elegance is learning to mathematically guide that unpredictability rather

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<v Speaker 2>than fearing it.

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<v Speaker 1>He elevates software from a mechanical chore to a goal

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<v Speaker 1>directed activity of profound mathematical beauty. He asked us to

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<v Speaker 1>look past machine and marvel at the logic itself.

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<v Speaker 2>It's a beautiful way to look at the world, which leaves.

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<v Speaker 1>You, the listener, with a really fascinating lens through which

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<v Speaker 1>to view the technology surrounding you today. Because we live

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<v Speaker 1>in an era that is increasingly dominated by massive AI

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<v Speaker 1>models and neural networks that are fundamentally, by their very architecture,

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<v Speaker 1>non deterministic.

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<v Speaker 2>They absolutely are.

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<v Speaker 1>They hallucinate, They surprise us. They produce irreproducible outputs. If

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<v Speaker 1>a pioneer like Diikstra realized way back in nineteen seventy

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<v Speaker 1>six that unpredictable, non deterministic systems are the normal state

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<v Speaker 1>of reality rather than a glitch to be feared.

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<v Speaker 2>How should that change your expectations exactly.

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<v Speaker 1>When an algorithm surfaces a bizarre result, or an AI

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<v Speaker 1>gives you an answer you didn't anticipate. Perhaps the system

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<v Speaker 1>isn't broken. Perhaps we are simply still stubbornly demanding the

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<v Speaker 1>flawless cardboard determinism of the past from the wonderfully and

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<v Speaker 1>tractable beasts running our future.