WEBVTT 00:00:00.000 --> 00:00:02.520 Welcome to the Deep Dive. I'm really glad you're 00:00:02.520 --> 00:00:04.160 joining us today because we are looking at something 00:00:04.160 --> 00:00:06.599 that is going to fundamentally change the way 00:00:06.599 --> 00:00:08.259 you look at a simple string of numbers. Yeah, 00:00:08.279 --> 00:00:10.500 it really shifts your entire perspective. It 00:00:10.500 --> 00:00:13.599 does. Today we're exploring a single remarkably 00:00:13.599 --> 00:00:16.539 dense source that you shared with us. It's a 00:00:16.539 --> 00:00:19.480 comprehensive overview of a concept called Polish 00:00:19.480 --> 00:00:22.199 notation. Which is just a brilliant piece of 00:00:22.199 --> 00:00:24.640 history. Right. And our mission for this deep 00:00:24.640 --> 00:00:27.859 dive is to explore how simply, you know, rearranging 00:00:27.859 --> 00:00:29.920 the order of mathematical symbols completely 00:00:29.920 --> 00:00:33.039 eliminates the need for parentheses. We're going 00:00:33.039 --> 00:00:35.539 to look at how it revolutionized early formal 00:00:35.539 --> 00:00:39.259 logic and how it ultimately became the secret 00:00:39.259 --> 00:00:41.659 language of the computers we rely on every single 00:00:41.659 --> 00:00:43.820 day. To really understand the origins of this, 00:00:43.920 --> 00:00:46.020 you have to imagine the intellectual atmosphere 00:00:46.020 --> 00:00:50.479 of a, well, a classic 1920s European library. 00:00:50.780 --> 00:00:53.700 Oh, I love that visual. Right. Picture towering 00:00:53.700 --> 00:00:56.820 oak bookshelves, academics debating in the halls, 00:00:56.920 --> 00:00:59.340 and tables covered in these beautifully complex, 00:00:59.560 --> 00:01:03.000 dusty logic manuscripts. That was the exact environment 00:01:03.000 --> 00:01:06.239 where this concept was born. Today we are going 00:01:06.239 --> 00:01:08.840 to synthesize the history, the mechanics, and 00:01:08.840 --> 00:01:11.500 the modern applications of this notation, showing 00:01:11.500 --> 00:01:14.159 not just what it is, but why it actually matters 00:01:14.159 --> 00:01:16.590 to you. I want to set the stage for you, the 00:01:16.590 --> 00:01:18.510 listener, for a second. Think about the last 00:01:18.510 --> 00:01:20.590 time you tried to write out a really complex 00:01:20.590 --> 00:01:24.709 math problem or a dense logic puzzle. Or even 00:01:24.709 --> 00:01:28.049 just a long formula in Excel. Yes, exactly. Imagine 00:01:28.049 --> 00:01:30.510 you're typing a convoluted equation into a spreadsheet. 00:01:30.870 --> 00:01:32.730 You probably find yourself completely bogged 00:01:32.730 --> 00:01:34.829 down by brackets, parentheses, and the strict 00:01:34.829 --> 00:01:37.090 rules of the order of operations. It gets messy 00:01:37.090 --> 00:01:39.260 fast. It really does. You know the drill making. 00:01:39.379 --> 00:01:41.400 Sure, you have exactly three closed brackets 00:01:41.400 --> 00:01:43.680 at the end of your formula just to perfectly 00:01:43.680 --> 00:01:45.640 match the three open brackets at the beginning. 00:01:45.799 --> 00:01:48.359 And if you miss one? Right. A single typo breaks 00:01:48.359 --> 00:01:50.680 the entire formula. The software just yells at 00:01:50.680 --> 00:01:53.219 you. But what if there was a cleaner, completely 00:01:53.219 --> 00:01:55.079 linear way? What if you could just write the 00:01:55.079 --> 00:01:57.219 instructions straight across left to right and 00:01:57.219 --> 00:02:00.920 the math just worked? That is the exact problem 00:02:00.920 --> 00:02:03.659 Polish notation solves. It strips away all of 00:02:03.659 --> 00:02:06.599 that visual clutter and relies purely on the 00:02:06.599 --> 00:02:08.860 sequence of the information. Okay, let's unpack 00:02:08.860 --> 00:02:12.000 this. We all grew up on standard infix notation. 00:02:12.319 --> 00:02:15.180 That means the operator, the plus sign, the minus 00:02:15.180 --> 00:02:17.319 sign, the multiplication symbol, it's trapped 00:02:17.319 --> 00:02:19.639 right in the middle of the numbers it's operating 00:02:19.639 --> 00:02:22.759 on, like 1 plus 2. Right, the operator is fixed 00:02:22.759 --> 00:02:25.719 in the center. Exactly. But Polish notation... 00:02:25.949 --> 00:02:28.469 which is also called prefix notation, rips that 00:02:28.469 --> 00:02:31.330 operator out and puts it at the very front. So 00:02:31.330 --> 00:02:33.770 it becomes plus 1, 2. And while putting the plus 00:02:33.770 --> 00:02:35.710 sign before the numbers might look a little strange 00:02:35.710 --> 00:02:38.509 for simple addition, its true power reveals itself 00:02:38.509 --> 00:02:40.509 when the equations become layered and highly 00:02:40.509 --> 00:02:43.610 complex. Right. Let's look at a harder problem. 00:02:43.750 --> 00:02:46.129 Let's challenge you, the listener, to visualize 00:02:46.129 --> 00:02:49.349 this expression, open parenthesis, 5 minus 6, 00:02:49.590 --> 00:02:53.349 close parenthesis, multiplied by 7. So standard 00:02:53.349 --> 00:02:56.819 infix notation. Right. Because we use standard 00:02:56.819 --> 00:02:59.580 infix notation, we absolutely need those parentheses. 00:02:59.979 --> 00:03:03.199 If we drop them, the order of operations forces 00:03:03.199 --> 00:03:06.240 us to multiply the 6 and the 7 first, completely 00:03:06.240 --> 00:03:08.840 changing the math. What's fascinating here is 00:03:08.840 --> 00:03:12.080 how Polish notation elegantly bypasses that entire 00:03:12.080 --> 00:03:15.750 problem. In this prefix system, the operator 00:03:15.750 --> 00:03:19.310 always precedes its operands. So if I have that 00:03:19.310 --> 00:03:20.969 same equation, I'd start with the multiplication 00:03:20.969 --> 00:03:23.610 because that's the outermost operation. I write 00:03:23.610 --> 00:03:26.530 the multiplication sign first. But where does 00:03:26.530 --> 00:03:29.310 the subtraction go if I just write multiply 5 00:03:29.310 --> 00:03:32.629 minus 6, 7? That just looks like a jumbled mess 00:03:32.629 --> 00:03:34.870 of characters. How does the structure actually 00:03:34.870 --> 00:03:37.080 hold together without the brackets? That's the 00:03:37.080 --> 00:03:39.819 trick. You work strictly by operations. You have 00:03:39.819 --> 00:03:42.020 your multiplication sign. It needs two operands. 00:03:42.120 --> 00:03:44.120 The first operand isn't a single number. It's 00:03:44.120 --> 00:03:46.919 the result of 5 minus 6. Oh, I see. So immediately 00:03:46.919 --> 00:03:48.620 after the multiplication sign, you write the 00:03:48.620 --> 00:03:50.620 subtraction sign, followed by the 5 and the 6. 00:03:50.819 --> 00:03:53.180 Then finally, you put the 7 at the very end. 00:03:53.219 --> 00:03:55.580 The sequence becomes multiply minus 5, 6, 7. 00:03:55.960 --> 00:03:58.659 And because every operator has a strict rule 00:03:58.659 --> 00:04:01.240 about how many numbers it needs, the brackets 00:04:01.240 --> 00:04:04.849 just vanish. Precisely. That property is called 00:04:04.849 --> 00:04:07.710 its arity. A subtraction sign has an arity of 00:04:07.710 --> 00:04:10.409 two. It always needs exactly two numbers following 00:04:10.409 --> 00:04:13.210 it. A multiplication sign also has an arity of 00:04:13.210 --> 00:04:16.750 two. The structure regulates itself. No arbitrary 00:04:16.750 --> 00:04:19.930 order operations to memorize and no hunting for 00:04:19.930 --> 00:04:22.290 missing parentheses. But wait, what about order? 00:04:22.730 --> 00:04:24.269 If I'm multiplying, it doesn't really matter 00:04:24.269 --> 00:04:27.649 if I say 3 times 4 or 4 times 3, but division 00:04:27.649 --> 00:04:30.250 and subtraction are noncommutative. The order 00:04:30.250 --> 00:04:32.629 matters immensely. Does shifting everything to 00:04:32.629 --> 00:04:35.269 a linear line create confusion there? Not at 00:04:35.269 --> 00:04:37.730 all. The source provides clear examples to ensure 00:04:37.730 --> 00:04:40.709 there is zero ambiguity. When dealing with noncommutative 00:04:40.709 --> 00:04:42.769 operations, the sequence of the operands directly 00:04:42.769 --> 00:04:44.990 follows how the operator takes its arguments, 00:04:45.189 --> 00:04:47.410 moving strictly left to right. So the order is 00:04:47.410 --> 00:04:49.819 logged in. Exactly. If you write the division 00:04:49.819 --> 00:04:52.379 symbol followed by 10 and then 5, it strictly 00:04:52.379 --> 00:04:55.259 means 10 divided by 5. If you write the minus 00:04:55.259 --> 00:04:58.199 sign followed by 7 and then 6, it strictly means 00:04:58.199 --> 00:05:01.399 you subtract 6 from 7. The action word dictates 00:05:01.399 --> 00:05:03.420 exactly what happens to the sequence that follows. 00:05:03.639 --> 00:05:05.779 Imagine writing out instructions in your daily 00:05:05.779 --> 00:05:08.339 life where the action always dictates exactly 00:05:08.339 --> 00:05:11.740 what follows, eliminating any ambiguity. You 00:05:11.740 --> 00:05:14.810 never have to... backtrack or second guess which 00:05:14.810 --> 00:05:16.709 part of the instruction takes precedence. It's 00:05:16.709 --> 00:05:20.029 just a pure, linear flow of logic. It is incredibly 00:05:20.029 --> 00:05:22.610 elegant. And to understand where this pursuit 00:05:22.610 --> 00:05:24.709 of pure logic came from, we have to travel back 00:05:24.709 --> 00:05:28.110 to the vibrant intellectual era of 1920s Poland. 00:05:28.430 --> 00:05:31.389 The history here is fascinating. The Polish in 00:05:31.389 --> 00:05:33.589 Polish notation refers to the nationality of 00:05:33.589 --> 00:05:36.430 its inventor, a Leducian named Jan Łukasiewicz. 00:05:36.610 --> 00:05:38.889 He invented this parenthesis -free notation in 00:05:38.889 --> 00:05:41.829 1924. Right. He actually mentioned it in a lithograph 00:05:41.829 --> 00:05:45.009 report in Polish in 1929 and then formally detailed 00:05:45.009 --> 00:05:48.610 it in a 1931 paper. It's gone by a few names 00:05:48.610 --> 00:05:50.449 over the years. You'll hear it called Normal 00:05:50.449 --> 00:05:53.009 Polish Notation or NPN. Sometimes it's called 00:05:53.009 --> 00:05:55.569 Warsaw Notation or Eastern Notation. We should 00:05:55.569 --> 00:05:58.250 provide a bit of historical context, too. While 00:05:58.250 --> 00:06:01.089 Ukaziewicz created the first linearly written 00:06:01.089 --> 00:06:03.730 parenthesis -free notation, he wasn't the only 00:06:03.730 --> 00:06:06.989 one chasing this kind of logical purity. Moses 00:06:06.989 --> 00:06:09.310 Schoenfinkel, whose work was edited by Heinrich 00:06:09.310 --> 00:06:12.430 Behmann in 1924, had the idea of eliminating 00:06:12.430 --> 00:06:14.850 parentheses in logic formulas. Oh, true story. 00:06:15.029 --> 00:06:18.529 Yeah, and even further back, in 1879, Gottlob 00:06:18.529 --> 00:06:21.689 Friege proposed a visual parenthesis -free notation 00:06:21.689 --> 00:06:24.490 called Bigriff's Rift. But Friege's system was 00:06:24.490 --> 00:06:26.829 visual, right? It relied on two -dimensional 00:06:26.829 --> 00:06:29.689 lines and trees, which made it an absolute nightmare 00:06:29.689 --> 00:06:32.310 to actually typeset and print in a book. Oh, 00:06:32.350 --> 00:06:34.490 completely impractical for mass printing. Yeah. 00:06:35.600 --> 00:06:37.360 version struck a chord because it was linear. 00:06:37.480 --> 00:06:39.639 It solved the typesetting problem for logic. 00:06:39.920 --> 00:06:42.899 Exactly. It was so impactful that Alonzo Church, 00:06:43.139 --> 00:06:46.459 an absolute giant in mathematical logic, specifically 00:06:46.459 --> 00:06:49.300 praised Ukasiewicz's parenthesis -free notation 00:06:49.300 --> 00:06:53.019 in his classic 1944 book. Church called it worthy 00:06:53.019 --> 00:06:55.420 of remark, contrasting its elegance directly 00:06:55.420 --> 00:06:57.579 against the heavyweights of the time, Alfred 00:06:57.579 --> 00:06:59.779 North Whitehead and Bertrand Russell, and their 00:06:59.779 --> 00:07:02.980 notoriously dense masterwork, Principia Mathematica. 00:07:03.240 --> 00:07:06.060 That is a huge endorsement. Principia Mathematica 00:07:06.060 --> 00:07:09.220 is famous for taking hundreds of pages of incredibly 00:07:09.220 --> 00:07:12.399 dense, bracket -heavy notation just to prove 00:07:12.399 --> 00:07:14.860 that 1 plus 1 equals 2. It's infamous for it. 00:07:14.980 --> 00:07:18.300 Yeah. To have Church point to Ukasiewicz's linear 00:07:18.300 --> 00:07:20.279 string and say, look at how much cleaner this 00:07:20.279 --> 00:07:23.899 is, speaks volumes. It does. And we should note 00:07:23.899 --> 00:07:26.379 that Ukasiewicz didn't just use this notation 00:07:26.379 --> 00:07:29.759 for standard arithmetic. His primary focus was 00:07:29.759 --> 00:07:32.699 using it for complex formal logic. If you were 00:07:32.699 --> 00:07:34.879 to look at his original chalkboards, you wouldn't 00:07:34.879 --> 00:07:37.459 see math symbols. You'd see capital letters based 00:07:37.459 --> 00:07:39.420 on Polish words. Let's break those down because 00:07:39.420 --> 00:07:40.920 the language connection is really interesting. 00:07:41.060 --> 00:07:44.220 For the concept of negation, like saying not 00:07:44.220 --> 00:07:47.240 true, he used the letter N, which stands for 00:07:47.240 --> 00:07:49.699 the Polish word negacja. Makes perfect sense. 00:07:49.939 --> 00:07:51.779 Right. For the material conditional, which is 00:07:51.779 --> 00:07:55.060 an if -then statement, he used C for implicacja. 00:07:55.470 --> 00:07:59.569 For disjunction, or or, he used a for alternativa. 00:07:59.689 --> 00:08:02.810 And for conjunction, meaning and, he used k for 00:08:02.810 --> 00:08:05.930 konimcha. So you could write out massive, sprawling 00:08:05.930 --> 00:08:08.370 logical arguments just using strings of letters 00:08:08.370 --> 00:08:11.129 and variables with zero punctuation. But that 00:08:11.129 --> 00:08:13.649 inevitably leads to some academic friction, doesn't 00:08:13.649 --> 00:08:16.430 it? When you start claiming letters for specific 00:08:16.430 --> 00:08:19.209 logical functions, what happens when someone 00:08:19.209 --> 00:08:21.589 else wants to expand the system? This is where 00:08:21.589 --> 00:08:24.730 we run into a wonderfully quirky historical contradiction. 00:08:25.329 --> 00:08:27.790 Another logician by the name of Bocheski came 00:08:27.790 --> 00:08:31.629 along and expanded Ukasevich's system. He wanted 00:08:31.629 --> 00:08:34.629 to include all 16 binary connectives for classical 00:08:34.629 --> 00:08:38.389 propositional logic. But in doing so, his notation 00:08:38.389 --> 00:08:41.470 clashed horribly with Ukasevich's work in a different 00:08:41.470 --> 00:08:43.970 area called modal logic. Because they were both 00:08:43.970 --> 00:08:46.090 trying to use the same letters for entirely different 00:08:46.090 --> 00:08:48.240 things. You can see if it was using the letter 00:08:48.240 --> 00:08:51.039 M to mean possibility from the Polish word, we 00:08:51.039 --> 00:08:52.919 leave Osh and the letter L to mean necessity 00:08:52.919 --> 00:08:57.159 from Koneczny precisely. But Bocheski, building 00:08:57.159 --> 00:08:59.980 his own massive extension of the very same parenthesis 00:08:59.980 --> 00:09:02.860 free notation, decided to use M and L for completely 00:09:02.860 --> 00:09:04.659 different things, specifically non implication 00:09:04.659 --> 00:09:07.379 and converse non implication in classical logic. 00:09:07.600 --> 00:09:10.620 You had these two brilliant minds working in 00:09:10.620 --> 00:09:13.620 the exact same ecosystem, using the exact same 00:09:13.620 --> 00:09:16.610 letters to mean entirely different logical. concepts. 00:09:16.690 --> 00:09:19.970 It highlights just how specialized and dense 00:09:19.970 --> 00:09:22.590 this field of study was becoming. Here's where 00:09:22.590 --> 00:09:24.710 it gets really interesting, though, because despite 00:09:24.710 --> 00:09:27.730 this being such a beautifully pure system, such 00:09:27.730 --> 00:09:29.970 an elegant way to write out logic without any 00:09:29.970 --> 00:09:33.529 brackets, human beings actually ended up hating 00:09:33.529 --> 00:09:36.879 Polish notation. It is the ultimate irony. The 00:09:36.879 --> 00:09:39.379 system was designed by a human mind to make human 00:09:39.379 --> 00:09:42.299 logic more streamlined, but the human brain just 00:09:42.299 --> 00:09:44.559 doesn't process information that way very well. 00:09:44.720 --> 00:09:46.860 It's fascinating because researchers later found 00:09:46.860 --> 00:09:49.379 that humans actually strongly prefer visual formatting. 00:09:49.820 --> 00:09:53.039 A 2011 paper by Martinez -Nava, highlighted in 00:09:53.039 --> 00:09:55.080 the materials, points out that Polish notation 00:09:55.080 --> 00:09:58.539 fell into disuse, precisely because of how hostile 00:09:58.539 --> 00:10:01.399 it is for human reading. Our eyes naturally want 00:10:01.399 --> 00:10:03.379 to look at a spatial relationship. We want that 00:10:03.379 --> 00:10:05.720 visual balance. Yes, we want to see a number, 00:10:05.820 --> 00:10:07.399 an operator in the middle, and another number. 00:10:07.519 --> 00:10:09.860 That's how we balance concepts in our head. Reading 00:10:09.860 --> 00:10:12.220 a string of prefix commands requires you to hold 00:10:12.220 --> 00:10:14.519 a tremendous amount of abstract information in 00:10:14.519 --> 00:10:17.000 your short -term memory. You have to stack all 00:10:17.000 --> 00:10:19.059 the operations in your mind before you ever get 00:10:19.059 --> 00:10:22.250 to the numbers they apply to. The research specifically 00:10:22.250 --> 00:10:24.509 highlights how incredibly difficult this makes 00:10:24.509 --> 00:10:27.289 reading for individuals with dyslexia. Right. 00:10:27.409 --> 00:10:29.889 It removes all visual anchors. But you know what 00:10:29.889 --> 00:10:32.870 doesn't need visual anchors? A microchip. If 00:10:32.870 --> 00:10:35.450 we connect this to the bigger picture, this failure 00:10:35.450 --> 00:10:37.870 in the realm of human cognition paved the way 00:10:37.870 --> 00:10:40.509 for its greatest triumph. While humans struggled 00:10:40.509 --> 00:10:43.269 immensely to read a sequential, parenthesis -free 00:10:43.269 --> 00:10:46.330 stream of logic, computers absolutely loved it. 00:10:46.549 --> 00:10:49.580 Humans need spatial cues. The computer processor 00:10:49.580 --> 00:10:51.759 just wants a line of instructions it can chew 00:10:51.759 --> 00:10:54.480 through one by one. The material details what's 00:10:54.480 --> 00:10:57.019 called the evaluation algorithm. And this is 00:10:57.019 --> 00:11:00.009 where Polish notation becomes a superpower. Let 00:11:00.009 --> 00:11:02.049 me explain how this algorithm works, as it is 00:11:02.049 --> 00:11:05.230 foundational to modern computing. Prefix expressions 00:11:05.230 --> 00:11:07.750 can be evaluated by a computer moving left to 00:11:07.750 --> 00:11:10.230 right or right to left using something called 00:11:10.230 --> 00:11:13.490 a stack or a push -down store. Imagine a spring 00:11:13.490 --> 00:11:16.190 -loaded stack of cafeteria trays. Okay, I like 00:11:16.190 --> 00:11:18.590 that. You push a new tray down onto the top, 00:11:18.710 --> 00:11:21.090 and when you need one, you pop the top one off. 00:11:21.230 --> 00:11:23.289 Okay, I can picture that, but how does the computer 00:11:23.289 --> 00:11:25.649 actually use that tray dispenser to do math? 00:11:25.929 --> 00:11:29.100 The computer... reads the input string token 00:11:29.100 --> 00:11:32.980 by token, a token being either an operator like 00:11:32.980 --> 00:11:36.019 a plus sign or an operand like a number. Let's 00:11:36.019 --> 00:11:39.700 say it is evaluating from left to right. It pushes 00:11:39.700 --> 00:11:42.399 these tokens onto the stack one by one. It keeps 00:11:42.399 --> 00:11:44.379 pushing them down until it realizes that the 00:11:44.379 --> 00:11:46.840 top entries of the stack perfectly match the 00:11:46.840 --> 00:11:49.080 requirements, the arity of the operator sitting 00:11:49.080 --> 00:11:52.259 just below them. So if the computer pushes down 00:11:52.259 --> 00:11:54.600 a multiplication sign, it knows it needs two 00:11:54.600 --> 00:11:57.299 numbers. It pushes down a five. It pushes down 00:11:57.299 --> 00:11:59.940 a six. Suddenly it says, wait, I have an operator 00:11:59.940 --> 00:12:02.779 and the exact number of operands it needs. Exactly. 00:12:02.960 --> 00:12:05.779 Boom. It executes the multiplication right then 00:12:05.779 --> 00:12:07.840 and there, takes the 5, the 6, and the multiplication 00:12:07.840 --> 00:12:10.200 sign off the stack, and replaces them with the 00:12:10.200 --> 00:12:12.200 result 30. Then it just keeps reading the rest 00:12:12.200 --> 00:12:14.259 of the string. But wait, that makes sense for 00:12:14.259 --> 00:12:16.419 simple math, but what happens when the computer 00:12:16.419 --> 00:12:19.460 hits a massive nested equation? Doesn't the stack 00:12:19.460 --> 00:12:21.580 get overloaded, holding all those operators waiting 00:12:21.580 --> 00:12:24.139 for numbers? That's the brilliance of it. It 00:12:24.139 --> 00:12:26.299 doesn't overload because the moment any local 00:12:26.299 --> 00:12:29.200 condition is met, it collapses the data into 00:12:29.200 --> 00:12:31.639 a single result. The computer doesn't need to 00:12:31.639 --> 00:12:33.740 scan back and forth across a massive equation 00:12:33.740 --> 00:12:36.059 looking for brackets to figure out what to do 00:12:36.059 --> 00:12:38.679 first. It just acts immediately. Yes. It doesn't 00:12:38.679 --> 00:12:40.919 need the capability of arbitrary stack inspection. 00:12:41.080 --> 00:12:44.679 It just reads sequentially, pushes data onto 00:12:44.679 --> 00:12:47.379 the stack, and executes the moment the already 00:12:47.379 --> 00:12:49.659 conditions are met. It is incredibly efficient. 00:12:50.169 --> 00:12:52.210 This is why for programming language interpreters, 00:12:52.250 --> 00:12:55.049 Polish notation is effortlessly parsed into abstract 00:12:55.049 --> 00:12:58.289 syntax trees. Yes. An abstract syntax tree, or 00:12:58.289 --> 00:13:01.809 AST, is how a compiler understands code. It builds 00:13:01.809 --> 00:13:04.070 a tree where the operator is the root and the 00:13:04.070 --> 00:13:06.769 numbers are the branches. With normal human math, 00:13:06.929 --> 00:13:08.870 building that tree is computationally expensive. 00:13:09.330 --> 00:13:12.370 You have to program complex rules for precedence 00:13:12.370 --> 00:13:14.090 and parentheses. Which takes time and memory. 00:13:14.289 --> 00:13:16.649 Right. But Polish notation essentially is the 00:13:16.649 --> 00:13:20.120 tree just written on a single line. perfect one 00:13:20.120 --> 00:13:22.419 -to -one mathematical representation of the code 00:13:22.419 --> 00:13:25.200 structure. It strips away all the ambiguity of 00:13:25.200 --> 00:13:28.120 human language, leaving only the pure, executable 00:13:28.120 --> 00:13:30.419 logic. Which brings us to where we are today. 00:13:31.039 --> 00:13:33.879 I want to ask you, the listener, have you ever 00:13:33.879 --> 00:13:36.440 tinkered with coding? Have you ever written a 00:13:36.440 --> 00:13:39.919 script to automate a task or used a highly advanced 00:13:39.919 --> 00:13:42.639 engineering calculator? Because if you have, 00:13:42.759 --> 00:13:45.399 you have interacted with the ghost of Jan Ukasiewicz. 00:13:45.960 --> 00:13:48.580 Polish notation didn't die out. It just went 00:13:48.580 --> 00:13:50.580 behind the scenes. The modern implementations 00:13:50.580 --> 00:13:53.500 are vast, and they show just how integral this 00:13:53.500 --> 00:13:56.120 logic remains. Prefix notation is still heavily 00:13:56.120 --> 00:13:58.840 used in Lisp programming with its S expressions. 00:13:59.440 --> 00:14:02.379 Now, it's worth noting that Lisp does use parentheses, 00:14:02.659 --> 00:14:05.100 which seems counterintuitive to what we've been 00:14:05.100 --> 00:14:06.820 discussing. A bit of a paradox, yeah. But that's 00:14:06.820 --> 00:14:09.340 a unique case because Lisp functions are treated 00:14:09.340 --> 00:14:12.559 as first -class data. and are variatic, meaning 00:14:12.559 --> 00:14:14.500 they can take a variable number of arguments. 00:14:14.759 --> 00:14:17.620 The parentheses just help group them. But the 00:14:17.620 --> 00:14:20.360 core structural logic is still entirely prefix. 00:14:20.679 --> 00:14:22.620 And we see it in data querying as well. If you 00:14:22.620 --> 00:14:24.820 work in IT, you are likely familiar with LDD 00:14:24.820 --> 00:14:27.980 filter syntax. Yes. LDAP is a great example. 00:14:28.179 --> 00:14:30.639 If you need to search a massive company database 00:14:30.639 --> 00:14:33.440 for a specific user profile, you have to group 00:14:33.440 --> 00:14:36.179 complex logic together. Find someone in marketing, 00:14:36.320 --> 00:14:40.830 ND, who is either a manager or a director. LDAP 00:14:40.830 --> 00:14:43.529 uses prefix notation to cleanly encapsulate that 00:14:43.529 --> 00:14:46.269 logic without needing arbitrary precedence rules 00:14:46.269 --> 00:14:49.669 for AND versus OR. It keeps it clean. You also 00:14:49.669 --> 00:14:51.450 see it in the TCL programming language through 00:14:51.450 --> 00:14:53.850 its MathOp library, the AMBI programming language, 00:14:54.029 --> 00:14:56.690 and even CoffeeScript, which allows prefix notation 00:14:56.690 --> 00:14:58.950 for function calls. But we also have to talk 00:14:58.950 --> 00:15:00.909 about its mirror image. While prefix notation 00:15:00.909 --> 00:15:03.350 puts the operator first, there is a variation 00:15:03.350 --> 00:15:06.470 called reverse Polish notation, or RPN. This 00:15:06.470 --> 00:15:09.210 is post -fix notation. In RPN, the operators 00:15:09.210 --> 00:15:11.220 follow... the operands. So instead of plus one 00:15:11.220 --> 00:15:13.659 two, you write one two plus. And RPN is everywhere 00:15:13.659 --> 00:15:15.539 in the hardware and stack -oriented software 00:15:15.539 --> 00:15:18.179 world. It is heavily used in stack -oriented 00:15:18.179 --> 00:15:20.259 languages like PostScript, which is the foundational 00:15:20.259 --> 00:15:22.759 language that powers modern printing, and the 00:15:22.759 --> 00:15:25.100 fourth programming language. But its most famous 00:15:25.100 --> 00:15:28.299 application is arguably in Hewlett -Packard calculators. 00:15:28.519 --> 00:15:31.259 If you have ever seen an engineer or a finance 00:15:31.259 --> 00:15:34.720 professional using a classic HP calculator, you 00:15:34.720 --> 00:15:37.460 were watching reverse Polish notation in action. 00:15:38.000 --> 00:15:40.340 They loved those machines because when you were 00:15:40.340 --> 00:15:43.220 calculating load -bearing stress or complex compound 00:15:43.220 --> 00:15:46.320 interest, you were dealing with massive formulas. 00:15:46.659 --> 00:15:48.980 You'd get lost on a normal calculator. Exactly. 00:15:49.039 --> 00:15:52.379 With a standard calculator, you need a notepad 00:15:52.379 --> 00:15:55.330 to write down intermediate results. Or you need 00:15:55.330 --> 00:15:57.789 to perfectly track eight levels of parentheses. 00:15:58.389 --> 00:16:00.769 With an RPN calculator, you just keep pushing 00:16:00.769 --> 00:16:02.610 numbers onto the stack and applying operations. 00:16:03.129 --> 00:16:06.009 It felt like you were manually driving the computer's 00:16:06.009 --> 00:16:08.649 processor. It operates at a very low level. Post 00:16:08.649 --> 00:16:10.549 -fix operators are even used by stack machines 00:16:10.549 --> 00:16:13.110 like the Burroughs Large Systems. And right toward 00:16:13.110 --> 00:16:15.669 the end of our analysis, there is an incredibly 00:16:15.669 --> 00:16:18.669 dense mathematical rule that proves why all of 00:16:18.669 --> 00:16:20.830 this works so flawlessly. It's regarding the 00:16:20.830 --> 00:16:23.509 equation of returns. Yes. It states that the 00:16:23.509 --> 00:16:25.919 number... of return values of an expression equals 00:16:25.919 --> 00:16:27.840 the difference between the number of operands 00:16:27.840 --> 00:16:30.620 and the total arity of the operators minus the 00:16:30.620 --> 00:16:32.860 total number of return values of the operators. 00:16:33.159 --> 00:16:35.960 So what does this all mean? That is a very dense 00:16:35.960 --> 00:16:38.919 sentence to process audibly. Let's actually test 00:16:38.919 --> 00:16:41.940 that to give you, the listener, a concrete idea 00:16:41.940 --> 00:16:43.580 of what's happening under the hood. Let's track 00:16:43.580 --> 00:16:46.559 it in real time using our earlier example. Multiply 00:16:46.559 --> 00:16:50.299 minus 5, 6, 7. First we count the operands, the 00:16:50.299 --> 00:16:53.690 numbers. We have 5. Six and seven. That's three 00:16:53.690 --> 00:16:56.129 operands. Okay, three operands. And then we look 00:16:56.129 --> 00:16:58.429 at the total arity of our operators. We have 00:16:58.429 --> 00:17:00.570 a multiplication sign and a subtraction sign. 00:17:00.809 --> 00:17:03.309 They each require two inputs, so the total arity 00:17:03.309 --> 00:17:05.890 is four. Exactly. And those two operators will 00:17:05.890 --> 00:17:08.769 eventually spit out two outputs or two return 00:17:08.769 --> 00:17:12.170 values total. What that complex formula essentially 00:17:12.170 --> 00:17:14.710 proves is that when you map all those input requirements 00:17:14.710 --> 00:17:16.690 and output values together against the starting 00:17:16.690 --> 00:17:19.849 numbers, the entire sequence perfectly collapses 00:17:19.849 --> 00:17:22.690 to exactly one final answer. It mathematically 00:17:22.690 --> 00:17:25.650 balances out. There are no dangling numbers left 00:17:25.650 --> 00:17:27.750 on the stack, and there are no missing variables 00:17:27.750 --> 00:17:31.150 causing a crash. It is a perfect, unbreakable 00:17:31.150 --> 00:17:34.279 mathematical loop. apparently verifies itself 00:17:34.279 --> 00:17:36.759 without needing any outside punctuation to hold 00:17:36.759 --> 00:17:39.119 it together. It is the ultimate expression of 00:17:39.119 --> 00:17:41.720 logical self -sufficiency. And that brings us 00:17:41.720 --> 00:17:44.019 to the end of this journey. To summarize our 00:17:44.019 --> 00:17:46.059 deep drive today, we explored the brilliant, 00:17:46.119 --> 00:17:50.000 elegant concept of Polish notation. Born from 00:17:50.000 --> 00:17:53.819 the mind of Jan Ukasiewicz in 1920s Poland as 00:17:53.819 --> 00:17:56.960 a way to simplify and purify formal logic, it 00:17:56.960 --> 00:17:59.339 proved to be a bit too alien for the spatial 00:17:59.339 --> 00:18:01.799 processing brains of human beings. It just didn't 00:18:01.799 --> 00:18:04.119 catch on for everyday writing. Right. It was 00:18:04.119 --> 00:18:06.299 largely abandoned as a written language because 00:18:06.299 --> 00:18:08.380 it was just too hostile to human reading habits. 00:18:08.839 --> 00:18:11.859 But its pure, parenthesis -free, linear structure 00:18:11.859 --> 00:18:14.839 turned out to be the exact perfect match for 00:18:14.839 --> 00:18:17.359 the way computer processors operate, making it 00:18:17.359 --> 00:18:20.119 the foundational high - efficient logic language 00:18:20.119 --> 00:18:23.299 that quietly powers modern computer parsing, 00:18:23.319 --> 00:18:26.279 algorithmic stacks, and engineering calculators 00:18:26.279 --> 00:18:28.619 to this day. It is a remarkable testament to 00:18:28.619 --> 00:18:30.480 the idea that just because a system doesn't work 00:18:30.480 --> 00:18:32.480 well for human eyes doesn't mean it lacks intrinsic 00:18:32.480 --> 00:18:35.259 perfection. I want to thank you, the listener, 00:18:35.380 --> 00:18:37.539 for coming along with us on this deep dive. And 00:18:37.539 --> 00:18:39.539 as we conclude, I want to leave you with a final 00:18:39.539 --> 00:18:41.880 thought to mull over. This raises an important 00:18:41.880 --> 00:18:44.819 question. Today, we've seen how standard mathematical 00:18:44.819 --> 00:18:47.519 notation, with all its messy parentheses and 00:18:47.519 --> 00:18:50.019 precedence rules, is actually less efficient 00:18:50.019 --> 00:18:52.980 and less logical than a parenthesis -free linear 00:18:52.980 --> 00:18:56.039 string. We only use infix notation because it 00:18:56.039 --> 00:18:58.119 is what we are used to. It caters to our visual 00:18:58.119 --> 00:19:01.099 habits. So if we are still clinging to a less 00:19:01.099 --> 00:19:03.759 efficient math notation out of sheer habit, what 00:19:03.759 --> 00:19:05.960 other foundational systems in our daily lives 00:19:05.960 --> 00:19:08.490 or workplace... or our communication are we stubbornly 00:19:08.490 --> 00:19:11.289 holding onto out of comfort long after a vastly 00:19:11.289 --> 00:19:13.109 more logical alternative has been discovered.