WEBVTT 00:00:00.000 --> 00:00:02.779 Welcome to the Deep Dive. Today we are exploring 00:00:02.779 --> 00:00:06.660 a truly fascinating Wikipedia article on a mathematical 00:00:06.660 --> 00:00:09.800 concept called Polish notation. Yes, and it is 00:00:09.800 --> 00:00:12.759 a fantastic topic. It really is. We are going 00:00:12.759 --> 00:00:15.880 on a journey today to discover how moving a single, 00:00:16.000 --> 00:00:19.039 simple math symbol from the middle of an equation 00:00:19.039 --> 00:00:22.460 to the absolute front completely revolutionized 00:00:22.460 --> 00:00:26.239 logic. Eventually built the invisible architecture 00:00:26.239 --> 00:00:28.920 for the entire field of computer science. Which 00:00:28.920 --> 00:00:30.739 is wild when you think about it, just shifting 00:00:30.739 --> 00:00:33.840 one symbol. Right. I want you to think back for 00:00:33.840 --> 00:00:35.659 a moment to your days in middle school algebra 00:00:35.659 --> 00:00:38.820 class. Have you ever looked at a long, complicated 00:00:38.820 --> 00:00:41.280 math equation on a chalkboard and just gotten 00:00:41.280 --> 00:00:44.060 completely lost in a sea of parentheses? Absolutely. 00:00:44.240 --> 00:00:46.460 Everyone has. You probably had brackets inside 00:00:46.460 --> 00:00:48.439 of brackets. You're trying to remember PEMDAS 00:00:48.439 --> 00:00:50.659 and the order of operations. The whole thing 00:00:50.659 --> 00:00:52.659 just felt incredibly cluttered and overwhelming. 00:00:53.420 --> 00:00:55.200 Well, what if there was an entirely different 00:00:55.200 --> 00:00:57.799 way to write math that deletes those confusing 00:00:57.799 --> 00:01:00.340 parentheses completely? It sounds almost like 00:01:00.340 --> 00:01:02.520 a magic trick when you first encounter it to 00:01:02.520 --> 00:01:05.180 just, you know, erase the brackets and still 00:01:05.180 --> 00:01:07.599 have the math perfectly hold together. But it 00:01:07.599 --> 00:01:10.840 is a very real, highly structured and incredibly 00:01:10.840 --> 00:01:14.069 elegant logical system. Consider this deep dive 00:01:14.069 --> 00:01:16.329 your ultimate shortcut to being well -informed. 00:01:16.629 --> 00:01:18.969 By the time we wrap up this conversation, you 00:01:18.969 --> 00:01:21.409 are going to understand the secret language that 00:01:21.409 --> 00:01:24.230 powers the calculators you use and the computer 00:01:24.230 --> 00:01:26.750 programs you interact with every single day. 00:01:27.069 --> 00:01:29.549 You will honestly never look at a simple addition 00:01:29.549 --> 00:01:32.799 problem the same way again. And beyond just math, 00:01:33.000 --> 00:01:35.079 it really forces you to change your perspective 00:01:35.079 --> 00:01:38.019 on how we communicate instructions. It highlights 00:01:38.019 --> 00:01:40.459 the massive gap between how the human brain wants 00:01:40.459 --> 00:01:42.920 to read a sentence and how a machine needs to 00:01:42.920 --> 00:01:45.500 process data. Okay, let's unpack this. The core 00:01:45.500 --> 00:01:47.900 concept here starts with how we do everyday math. 00:01:48.099 --> 00:01:50.439 The system you and I grew up learning in school 00:01:50.439 --> 00:01:53.739 is called infix notation. That word infix just 00:01:53.739 --> 00:01:55.900 means the operator, the plus or minus sign is 00:01:55.900 --> 00:01:57.799 fixed right in the middle of the numbers. So 00:01:57.799 --> 00:01:59.920 if you want to add one and two, you write the 00:01:59.920 --> 00:02:01.680 number one, then the plus sign, then the number 00:02:01.680 --> 00:02:04.480 two, one plus two. Simple enough. But then our 00:02:04.480 --> 00:02:07.480 sources introduce us to tollish notation. You 00:02:07.480 --> 00:02:09.479 might also see it referred to as normal Polish 00:02:09.479 --> 00:02:12.860 notation, Warsaw notation. Polish prefix notation, 00:02:13.159 --> 00:02:16.639 or Eastern notation. In this system, the operator 00:02:16.639 --> 00:02:19.500 actually precedes the operands. It goes right 00:02:19.500 --> 00:02:21.560 at the very front. Right at the front. So instead 00:02:21.560 --> 00:02:23.520 of writing 1 plus 2, you would write plus 1, 00:02:23.560 --> 00:02:26.580 2. What's fascinating here is the why behind 00:02:26.580 --> 00:02:29.560 this shift. Why bother moving the plus sign to 00:02:29.560 --> 00:02:31.680 the front at all? Because it feels completely 00:02:31.680 --> 00:02:34.080 counterintuitive to how we speak. Yeah, it really 00:02:34.080 --> 00:02:36.479 does. But the genius of the system is that as 00:02:36.479 --> 00:02:38.879 long as an operator has a fixed number of operands, 00:02:38.919 --> 00:02:41.240 meaning a plus sign always needs exactly two 00:02:41.240 --> 00:02:44.060 numbers to add together, Polish notation requires 00:02:44.060 --> 00:02:47.580 absolutely zero parentheses. None at all. Wow. 00:02:47.680 --> 00:02:50.139 You could write out the most complex, massive, 00:02:50.259 --> 00:02:52.879 multi -layered mathematical formula in the world, 00:02:52.939 --> 00:02:55.539 and you will never need a single bracket to tell 00:02:55.539 --> 00:02:58.009 you what part to solve first. That's just wild 00:02:58.009 --> 00:02:59.370 to think about. I want to make sure we're really 00:02:59.370 --> 00:03:01.870 picturing this correctly. Let's walk through 00:03:01.870 --> 00:03:03.969 a specific visualization from our source material 00:03:03.969 --> 00:03:07.129 so you, listening right now, can really see how 00:03:07.129 --> 00:03:10.330 this plays out. Let's take a standard math equation. 00:03:10.650 --> 00:03:14.349 Open parenthesis, 5 minus 6, close parenthesis, 00:03:14.389 --> 00:03:17.110 multiplied by 7. Okay, got it. In the standard 00:03:17.110 --> 00:03:20.819 infix... math, we all know, you need those parentheses 00:03:20.819 --> 00:03:22.919 around the five minus six. So, you know, to do 00:03:22.919 --> 00:03:24.879 that subtraction first, right? For multiplying 00:03:24.879 --> 00:03:26.840 the whole thing by seven. But let's translate 00:03:26.840 --> 00:03:29.159 that into Polish notation. We start with the 00:03:29.159 --> 00:03:31.800 innermost part, the five minus six. Since the 00:03:31.800 --> 00:03:34.039 operator moves to the front, that simply becomes 00:03:34.039 --> 00:03:36.300 minus five, six. And that neat little package, 00:03:36.439 --> 00:03:39.659 the minus five, six becomes the new single unit 00:03:39.659 --> 00:03:42.560 that we want to multiply by seven. Exactly. So 00:03:42.560 --> 00:03:44.819 to finish translating the equation, you take 00:03:44.819 --> 00:03:46.840 your multiplication sign and you stick that at 00:03:46.840 --> 00:03:49.340 the very front. the entire string. So the finished 00:03:49.340 --> 00:03:53.099 equation is just times minus 5, 6, 7. Or if you're 00:03:53.099 --> 00:03:55.280 looking at it on a page of symbols, a multiplication 00:03:55.280 --> 00:03:58.639 sign, a subtraction sign, a 5, a 6, and a 7. 00:03:58.719 --> 00:04:01.900 You read it, and the order of operations is perfectly 00:04:01.900 --> 00:04:05.599 clear without a single bracket. And we can contrast 00:04:05.599 --> 00:04:08.039 that by moving the parentheses in our original 00:04:08.039 --> 00:04:11.000 math problem. What if the equation was 5 minus, 00:04:11.199 --> 00:04:14.659 open parenthesis, 6 times 7, close parenthesis. 00:04:15.050 --> 00:04:17.269 In Polish notation, you just shift the operators 00:04:17.269 --> 00:04:19.949 at the front. It simply becomes minus 5 times 00:04:19.949 --> 00:04:23.529 6, 7. Just by changing the sequence of the operators 00:04:23.529 --> 00:04:26.269 and the numbers, the innermostness of the math 00:04:26.269 --> 00:04:28.670 is perfectly conveyed to the reader. It's brilliant. 00:04:28.970 --> 00:04:30.790 There is a crucial nuance here that we have to 00:04:30.790 --> 00:04:32.490 point out, though, especially when we're dealing 00:04:32.490 --> 00:04:36.050 with non -commutative operations. Non -commutative, 00:04:36.149 --> 00:04:38.329 meaning? Operations where the order of the numbers 00:04:38.329 --> 00:04:40.410 completely changes the answer, like division 00:04:40.410 --> 00:04:43.829 or subtraction. 5 minus 6 is very different from 00:04:43.829 --> 00:04:46.129 6 minus 6. minus 5. So you have to coordinate 00:04:46.129 --> 00:04:48.269 the sequential arrangement perfectly. Right. 00:04:48.329 --> 00:04:50.670 The order really matters. Exactly. In Polish 00:04:50.670 --> 00:04:53.110 notation, reading left to right, if you see the 00:04:53.110 --> 00:04:55.990 division symbol followed by a 10 and a 5, so 00:04:55.990 --> 00:04:59.490 a division sign, then 10. Then 5, that strictly 00:04:59.490 --> 00:05:02.529 means 10 divided by 5. The number to the left 00:05:02.529 --> 00:05:04.509 is the dividend. The number to the right is the 00:05:04.509 --> 00:05:07.930 divisor. Got it. Similarly, a minus sign followed 00:05:07.930 --> 00:05:11.149 by 7, then 6 strictly means 7 minus 6. You are 00:05:11.149 --> 00:05:13.930 subtracting the 6 from the 7. The structure leaves 00:05:13.930 --> 00:05:18.009 zero room for ambiguity. It's so incredibly rigid, 00:05:18.089 --> 00:05:20.649 but in the best possible way. Now, let's talk 00:05:20.649 --> 00:05:22.129 about where this came from, because the history 00:05:22.129 --> 00:05:24.689 outlined in the article is fantastic. The Polish 00:05:24.689 --> 00:05:27.189 in Polish notation refers to the nationality 00:05:27.189 --> 00:05:29.910 of a brilliant logician named Jan Jukasiewicz. 00:05:30.129 --> 00:05:32.230 A very influential figure. He invented the system 00:05:32.230 --> 00:05:36.029 in 1924. But here's the historical detail I absolutely 00:05:36.029 --> 00:05:39.129 love. He wrote a paper in 1931 where he casually 00:05:39.129 --> 00:05:41.610 stated that he actually first used this revolutionary 00:05:41.610 --> 00:05:45.110 parenthesis free notation tucked away in a mere 00:05:45.110 --> 00:05:48.129 footnote. A footnote. Yes, specifically a footnote 00:05:48.129 --> 00:05:50.670 on page 610 of a lithographed report he published 00:05:50.670 --> 00:05:54.069 in 1929. Just imagine inventing a whole new way 00:05:54.069 --> 00:05:56.550 to write math, a system that would go on to fundamentally 00:05:56.550 --> 00:05:59.589 shape modern computer science. And you debut 00:05:59.589 --> 00:06:02.310 it in a footnote on page 610. It really speaks 00:06:02.310 --> 00:06:05.370 to how these monumental shifts in academic thought 00:06:05.370 --> 00:06:07.370 sometimes enter the world with a whisper rather 00:06:07.370 --> 00:06:09.709 than a shout. But to provide some broader historical 00:06:09.709 --> 00:06:12.889 context, Yukasiewicz wasn't operating in a total 00:06:12.889 --> 00:06:14.949 vacuum here. Right, there were others working 00:06:14.949 --> 00:06:17.790 on it. This was a time of immense exploration 00:06:17.790 --> 00:06:21.370 and debate in mathematical logic. A German mathematician 00:06:21.370 --> 00:06:24.329 named Heinrich Biemann actually had a somewhat 00:06:24.329 --> 00:06:26.970 similar idea for eliminating parentheses and 00:06:26.970 --> 00:06:29.610 logic formulas right around the same time in 00:06:29.610 --> 00:06:32.329 1924. Interesting. And if we look even further 00:06:32.329 --> 00:06:35.629 back into the sources, the famous logician Gottlob 00:06:35.629 --> 00:06:39.790 Friedge proposed his own heavily structural parenthesis 00:06:39.790 --> 00:06:42.310 -free notation called the Begriffschrift way 00:06:42.310 --> 00:06:45.980 back in 1879. The desire to get rid of the clunky 00:06:45.980 --> 00:06:48.160 brackets had been brewing in the academic community 00:06:48.160 --> 00:06:52.300 for quite a while. Precisely. However, Lukasiewicz's 00:06:52.300 --> 00:06:54.500 system was a massive breakthrough because it 00:06:54.500 --> 00:06:57.500 was the first linearly written, incredibly compact 00:06:57.500 --> 00:07:00.980 version. Friedge's older system was two -dimensional. 00:07:01.259 --> 00:07:02.959 What do you mean by two -dimensional? It looked 00:07:02.959 --> 00:07:05.540 almost like a massive sprawling family tree with 00:07:05.540 --> 00:07:07.800 branches going up and down, and it took up a 00:07:07.800 --> 00:07:09.939 huge amount of physical space on the page. You 00:07:09.939 --> 00:07:11.779 couldn't just easily type it out. Oh, I see. 00:07:12.060 --> 00:07:13.980 Ukasevich gave the world something you could 00:07:13.980 --> 00:07:16.240 just type out sequentially on a single line of 00:07:16.240 --> 00:07:19.970 paper. In fact, his notation was so highly regarded 00:07:19.970 --> 00:07:23.350 that the legendary logician Alonzo Church specifically 00:07:23.350 --> 00:07:25.889 praised it in his classic book on mathematical 00:07:25.889 --> 00:07:29.170 logic. Church called it worthy of remark and 00:07:29.170 --> 00:07:31.889 contrasted it favorably with the incredibly dense, 00:07:31.970 --> 00:07:34.670 complicated notation used by Alfred North Whitehead 00:07:34.670 --> 00:07:37.370 and Bertrand Russell in their massive foundational 00:07:37.370 --> 00:07:40.230 work, the Principia Mathematica. Here's where 00:07:40.230 --> 00:07:43.399 it gets really interesting. Lukasiewicz didn't 00:07:43.399 --> 00:07:45.699 actually invent this just to do basic arithmetic 00:07:45.699 --> 00:07:48.639 like addition and subtraction. He was a logician 00:07:48.639 --> 00:07:51.720 first and foremost. He used this system to map 00:07:51.720 --> 00:07:54.259 out complex philosophical and logical arguments. 00:07:54.560 --> 00:07:57.199 And because he was working in Poland, the letters 00:07:57.199 --> 00:07:59.480 he used in his notation actually stood for specific 00:07:59.480 --> 00:08:02.220 Polish words. It creates a wonderful collision 00:08:02.220 --> 00:08:04.839 of linguistics and pure mathematics. Let's walk 00:08:04.839 --> 00:08:07.060 through some of the vocabulary he created. Instead 00:08:07.060 --> 00:08:09.500 of using traditional math symbols, he used capital 00:08:09.500 --> 00:08:12.259 letters as his operators. So he used the letter 00:08:12.259 --> 00:08:15.480 N for negation, because the Polish word is negacja. 00:08:15.620 --> 00:08:18.240 Makes perfect sense. He used C for the material 00:08:18.240 --> 00:08:21.439 conditional, which is a fancy way of saying if 00:08:21.439 --> 00:08:23.620 this, then that, because the Polish word for 00:08:23.620 --> 00:08:27.339 implication is implicacja. He used K for conjunction. 00:08:27.930 --> 00:08:30.889 meaning the word and, from the Polish word koniokcia. 00:08:30.970 --> 00:08:33.590 Right. He even ventured into modal logic, which 00:08:33.590 --> 00:08:36.250 deals with philosophical concepts like possibility 00:08:36.250 --> 00:08:39.389 and necessity. So he used M for possibility, 00:08:39.649 --> 00:08:42.509 from the Polish word muliwosz, and he used L 00:08:42.509 --> 00:08:45.190 for necessity, from the Polish word konieczność. 00:08:45.610 --> 00:08:47.649 And whenever you have different brilliant academics 00:08:47.649 --> 00:08:50.409 building on a single system, you inevitably get 00:08:50.409 --> 00:08:52.889 some quirky historical contradictions. Oh, of 00:08:52.889 --> 00:08:55.330 course. Another prominent logician named Bocheski 00:08:55.330 --> 00:08:57.789 came along later and loved this system so much 00:08:57.789 --> 00:09:00.210 that he expanded it. He introduced a version 00:09:00.210 --> 00:09:03.850 of Polish notation that named all 16 binary connectives. 00:09:03.909 --> 00:09:05.950 To break that jargon down a bit for you listening, 00:09:06.129 --> 00:09:08.529 a binary connective is just a way to logically 00:09:08.529 --> 00:09:11.769 link two ideas together. Things like Andy, REO, 00:09:11.889 --> 00:09:15.820 or IFNN. Exactly. Bocheski mapped out all 16 00:09:15.820 --> 00:09:18.840 possible ways to do that. But here's the funny 00:09:18.840 --> 00:09:21.779 part. Bocheski and Yukasiewicz ended up using 00:09:21.779 --> 00:09:24.580 the exact same letters, specifically L and M, 00:09:24.740 --> 00:09:27.759 to mean completely incompatible things in their 00:09:27.759 --> 00:09:30.960 respective systems. Oh, wow. I can only imagine 00:09:30.960 --> 00:09:33.480 the absolute chaos that must have caused at mathematics 00:09:33.480 --> 00:09:36.019 conferences in the 1930s. It sounds like the 00:09:36.019 --> 00:09:38.940 academic equivalent of saving over someone else's 00:09:38.940 --> 00:09:41.139 shared spreadsheet and ruining all their formulas. 00:09:41.419 --> 00:09:44.299 It probably caused quite a few headaches. Lukasiewicz 00:09:44.299 --> 00:09:46.899 was using L and M for modal logic necessity and 00:09:46.899 --> 00:09:49.600 possibility. But Bocheski was using L and M in 00:09:49.600 --> 00:09:51.820 classical logic to represent non -implication 00:09:51.820 --> 00:09:54.840 and converse non -implication. It's a great example 00:09:54.840 --> 00:09:57.259 of how even perfectly logical systems can get 00:09:57.259 --> 00:09:59.340 a little messy when different human beings start. 00:09:59.470 --> 00:10:01.330 adapting them for their own specific overlapping 00:10:01.330 --> 00:10:04.169 needs. So what does this all mean? We have this 00:10:04.169 --> 00:10:07.669 elegant, linear, parenthesis -free system born 00:10:07.669 --> 00:10:10.350 in the 1920s in Poland. It makes a big splash 00:10:10.350 --> 00:10:12.330 in the world of formal logic. But our sources 00:10:12.330 --> 00:10:14.470 note that today, Polish notation is actually 00:10:14.470 --> 00:10:16.750 no longer used very much in formal logic itself. 00:10:17.029 --> 00:10:19.669 No, not really. The logicians largely went back 00:10:19.669 --> 00:10:21.909 to using infix notation and all those parentheses. 00:10:23.019 --> 00:10:25.279 But the notation didn't die. It found its true 00:10:25.279 --> 00:10:28.340 home, its ultimate destiny in the realm of computer 00:10:28.340 --> 00:10:30.360 science. If we connect this to the bigger picture 00:10:30.360 --> 00:10:32.659 of the 20th century, the invention of Polish 00:10:32.659 --> 00:10:35.759 notation perfectly predated the dawn of the computing 00:10:35.759 --> 00:10:38.980 age. And it solved a massive foundational problem 00:10:38.980 --> 00:10:41.860 for early computer engineers. Think about how 00:10:41.860 --> 00:10:45.539 a machine actually reads information. Explaining 00:10:45.539 --> 00:10:48.259 the human order of operations to a computer using 00:10:48.259 --> 00:10:50.840 parentheses is incredibly difficult. Right, because 00:10:50.840 --> 00:10:53.980 of the nesting. Exactly. If you feed a computer 00:10:53.980 --> 00:10:56.740 an equation with brackets inside of brackets, 00:10:57.100 --> 00:10:59.340 you have to program the computer to scan all 00:10:59.340 --> 00:11:01.639 the way ahead, find the deepest set of brackets, 00:11:01.940 --> 00:11:04.360 hold all the other ignored information in its 00:11:04.360 --> 00:11:07.080 temporary memory, do the deepest math, and then 00:11:07.080 --> 00:11:09.779 work its way backward. It takes a ton of memory 00:11:09.779 --> 00:11:11.860 and processing power, which early room -sized 00:11:11.860 --> 00:11:14.600 computers simply did not have. I was reading 00:11:14.600 --> 00:11:16.919 about this in the source material, and they mentioned 00:11:16.919 --> 00:11:21.240 that Polish notation... solves this by allowing 00:11:21.240 --> 00:11:23.220 computers to use something called an evaluation 00:11:23.220 --> 00:11:26.580 algorithm or a stack algorithm. But I want to 00:11:26.580 --> 00:11:28.120 make sure I am picturing this correctly because 00:11:28.120 --> 00:11:30.159 the technical descriptions get pretty dense. 00:11:30.659 --> 00:11:33.139 If I am the computer reading a string of Polish 00:11:33.139 --> 00:11:36.159 notation from right to left, what exactly am 00:11:36.159 --> 00:11:38.179 I doing with these numbers? How does the stack 00:11:38.179 --> 00:11:40.600 actually work? it is beautifully simple when 00:11:40.600 --> 00:11:43.519 you break it down imagine the computer is a worker 00:11:43.519 --> 00:11:46.620 in a cafeteria and the stack is literally one 00:11:46.620 --> 00:11:49.460 of those spring -loaded plate dispensers okay 00:11:49.460 --> 00:11:51.940 picture that the computer reads the equation 00:11:51.940 --> 00:11:54.659 from right to left every time it sees a number 00:11:54.659 --> 00:11:56.919 it writes that number on a plate and pushes it 00:11:56.919 --> 00:11:59.419 down onto the stack so if the equation is a plus 00:11:59.419 --> 00:12:02.960 sign then a two then a three reading right to 00:12:02.960 --> 00:12:06.399 left It sees the three, puts it on a plate, and 00:12:06.399 --> 00:12:09.200 pushes it down. Then it sees the two, puts it 00:12:09.200 --> 00:12:11.559 on a plate, and pushes it down on top of the 00:12:11.559 --> 00:12:13.539 three. Okay, so I have a two sitting on top of 00:12:13.539 --> 00:12:16.259 the three. Correct. The magic happens the moment 00:12:16.259 --> 00:12:18.120 the computer reads an operator, like a plus sign. 00:12:18.419 --> 00:12:21.340 It knows a plus sign needs two numbers. It doesn't 00:12:21.340 --> 00:12:23.799 look around. It doesn't scan ahead. It simply 00:12:23.799 --> 00:12:27.450 pops the top two plates off the stack. the two 00:12:27.450 --> 00:12:29.190 and the three adds them together to get five 00:12:29.190 --> 00:12:31.809 writes the number five on a brand new plate and 00:12:31.809 --> 00:12:34.350 pushes that single new plate back onto the stack 00:12:34.350 --> 00:12:36.190 and then it just moves on to the next symbol 00:12:36.190 --> 00:12:39.049 in the string that completely eliminates the 00:12:39.049 --> 00:12:42.309 need for parentheses it just keeps stacking plates 00:12:42.309 --> 00:12:45.570 and every time an operator shows up it consolidates 00:12:45.570 --> 00:12:47.629 the top plates into a single new number there's 00:12:47.629 --> 00:12:50.559 no scanning ahead And no backtracking. And crucially, 00:12:50.639 --> 00:12:52.600 there is no need for what computer scientists 00:12:52.600 --> 00:12:55.700 call arbitrary stack inspection. Which means? 00:12:56.019 --> 00:12:58.039 That is just a fancy way of saying the computer 00:12:58.039 --> 00:12:59.840 never has to dig around in the middle of the 00:12:59.840 --> 00:13:02.039 pile of plates. It only ever looks at the plate 00:13:02.039 --> 00:13:04.419 sitting right on the very top. The sheer efficiency 00:13:04.419 --> 00:13:07.480 of that is beautiful. A simple push down stores 00:13:07.480 --> 00:13:10.220 all you need to implement this parsing. It saves 00:13:10.220 --> 00:13:13.259 a massive amount of computational memory. Because 00:13:13.259 --> 00:13:16.370 it is so lightweight and efficient. This concept 00:13:16.370 --> 00:13:19.450 actually lives on in the real world today. It 00:13:19.450 --> 00:13:21.309 isn't just a museum piece of early computing 00:13:21.309 --> 00:13:24.370 history. For instance, there is a very famous, 00:13:24.450 --> 00:13:26.690 very influential programming language called 00:13:26.690 --> 00:13:29.629 Lisp. It uses something called S -expressions, 00:13:29.649 --> 00:13:33.049 which rely heavily on prefix notation. Yes, Lisp 00:13:33.049 --> 00:13:35.789 is a classic example. Now, it is kind of ironic 00:13:35.789 --> 00:13:37.889 because if you look at Lisp code, it actually 00:13:37.889 --> 00:13:40.950 does use a ton of parentheses. Our sources explain 00:13:40.950 --> 00:13:43.690 that this is because in Lisp, the functions act 00:13:43.690 --> 00:13:46.519 as data. And they are variadic. To translate 00:13:46.519 --> 00:13:48.720 that jargon for you listening, variadic just 00:13:48.720 --> 00:13:51.100 means a function can take any number of inputs. 00:13:51.259 --> 00:13:53.580 It isn't fixed. Right. Unlike standard addition. 00:13:53.940 --> 00:13:56.720 Exactly. Simple addition takes exactly two numbers. 00:13:56.899 --> 00:13:59.200 But a variadic addition function is like throwing 00:13:59.200 --> 00:14:01.100 ingredients into a blender. You can throw in 00:14:01.100 --> 00:14:03.919 two numbers or five numbers or 50 numbers and 00:14:03.919 --> 00:14:06.360 it will add them all up. Because the operator 00:14:06.360 --> 00:14:08.940 doesn't have a fixed number of operands, Lisp 00:14:08.940 --> 00:14:10.899 has to use parentheses to show the computer where 00:14:10.899 --> 00:14:13.039 the list of ingredients finally ends. It has 00:14:13.039 --> 00:14:15.340 to have some way to know it's done. Right. But 00:14:15.340 --> 00:14:17.620 the core Polish structure, putting the operator 00:14:17.620 --> 00:14:20.659 at the very front, remains the absolute foundation 00:14:20.659 --> 00:14:23.100 of the language. Lisp is certainly one of the 00:14:23.100 --> 00:14:26.159 most famous examples, but the influence of Jukasiewicz's 00:14:26.159 --> 00:14:28.879 work goes far beyond that. I was amazed at the 00:14:28.879 --> 00:14:31.919 list of applications. The TCL programming language. 00:14:32.539 --> 00:14:34.820 uses Polish notation for its math operations. 00:14:35.440 --> 00:14:38.179 There is a language called Ambi that uses it 00:14:38.179 --> 00:14:41.179 for arithmetic and program construction. If you 00:14:41.179 --> 00:14:43.340 work in corporate IT, you might have wrestled 00:14:43.340 --> 00:14:46.179 with LDAP filter syntax that uses Polish prefix 00:14:46.179 --> 00:14:49.139 notation to search directories. Oh, definitely. 00:14:49.360 --> 00:14:51.919 Even more modern web -focused tools like CoffeeScript 00:14:51.919 --> 00:14:54.500 allow functions to be called using this prefix 00:14:54.500 --> 00:14:57.320 notation we also absolutely have to mention the 00:14:57.320 --> 00:14:59.539 very famous sibling of this system which the 00:14:59.539 --> 00:15:01.940 article covers in detail if you can move the 00:15:01.940 --> 00:15:04.200 math operator to the absolute front of the equation 00:15:04.200 --> 00:15:06.480 you can logically also move it to the absolute 00:15:06.480 --> 00:15:09.720 end that variation is called reverse polish notation 00:15:09.720 --> 00:15:12.940 or rpn it is also known as post fixed notation 00:15:12.940 --> 00:15:15.899 post fix meaning after exactly so instead of 00:15:15.899 --> 00:15:17.539 writing plus one two you would write one two 00:15:17.539 --> 00:15:20.179 plus and i imagine the cafeteria plate stack 00:15:20.179 --> 00:15:22.259 algorithm works exactly the same way you just 00:15:22.259 --> 00:15:24.059 process this string from left to right instead 00:15:24.059 --> 00:15:25.779 of right to left. You hit the nail on the head. 00:15:25.940 --> 00:15:28.639 The mechanics are identical, just mirrored, and 00:15:28.639 --> 00:15:31.259 reverse Polish notation powers a massive amount 00:15:31.259 --> 00:15:34.059 of technology that we rely on. It is the basis 00:15:34.059 --> 00:15:36.860 for stack -oriented programming languages like 00:15:36.860 --> 00:15:39.240 PostScript, which is the hidden language used 00:15:39.240 --> 00:15:41.960 in desktop publishing to tell your printer exactly 00:15:41.960 --> 00:15:45.440 how to draw fonts and graphics on a page. I had 00:15:45.440 --> 00:15:47.740 no idea. It is the foundation of the programming 00:15:47.740 --> 00:15:50.519 language Forth. It was heavily used in the architecture 00:15:50.519 --> 00:15:53.799 of Burroughs' large systems mainframes back in 00:15:53.799 --> 00:15:56.379 the day. The article also points out an application 00:15:56.379 --> 00:15:58.720 that everyday consumers might actually recognize, 00:15:59.000 --> 00:16:01.279 especially if they work in finance or engineering. 00:16:01.850 --> 00:16:05.549 Reverse Polish notation is the chosen default 00:16:05.549 --> 00:16:08.629 notation for high -end calculators, most notably 00:16:08.629 --> 00:16:11.009 the scientific and financial calculators manufactured 00:16:11.009 --> 00:16:13.899 by Hewlett Packard. Those HP calculators have 00:16:13.899 --> 00:16:16.279 an incredibly loyal cult following for a very 00:16:16.279 --> 00:16:19.539 specific reason. Engineers and finance professionals 00:16:19.539 --> 00:16:21.919 swear by them because once your brain gets used 00:16:21.919 --> 00:16:24.480 to thinking in reverse Polish notation, you can 00:16:24.480 --> 00:16:27.600 punch in massive, complex calculations significantly 00:16:27.600 --> 00:16:30.419 faster than you ever could on a standard calculator. 00:16:30.679 --> 00:16:32.860 Because of the keystrokes. Exactly. You never 00:16:32.860 --> 00:16:35.019 have to waste keystrokes manually typing out 00:16:35.019 --> 00:16:37.440 open and close parentheses. And you never have 00:16:37.440 --> 00:16:38.860 to worry about the calculator misunderstanding 00:16:38.860 --> 00:16:41.379 the order of operations. You just type the numbers. 00:16:41.710 --> 00:16:44.830 type the operator, and the internal stack handles 00:16:44.830 --> 00:16:47.929 the rest instantly. It is amazing to wrap up 00:16:47.929 --> 00:16:49.950 this journey and look at the vast trajectory 00:16:49.950 --> 00:16:53.710 of this one single idea. We started with Jan 00:16:53.710 --> 00:16:56.809 Jukasiewicz, a Polish logician in the 1920s, 00:16:56.809 --> 00:16:59.190 who was simply trying to clean up his logic equations 00:16:59.190 --> 00:17:01.809 to make them a little easier to type out on a 00:17:01.809 --> 00:17:04.309 single line of paper. Just a pragmatic solution. 00:17:04.710 --> 00:17:07.990 And he drops this parenthesis -free idea in a 00:17:07.990 --> 00:17:12.319 footnote on page 610. a 1929 report. And by doing 00:17:12.319 --> 00:17:14.799 that, he accidentally created the invisible, 00:17:15.059 --> 00:17:17.759 highly efficient mathematical architecture that 00:17:17.759 --> 00:17:19.779 allowed early computers with terrible memory 00:17:19.779 --> 00:17:22.549 to process complex math. An architecture that 00:17:22.549 --> 00:17:24.670 still powers incredibly sophisticated programming 00:17:24.670 --> 00:17:27.230 languages, printers, and calculators today. It's 00:17:27.230 --> 00:17:29.529 incredible. So the next time you use a computer 00:17:29.529 --> 00:17:31.789 to crunch some numbers or you hit print on a 00:17:31.789 --> 00:17:34.930 PDF document, remember that a version of Ukasiewicz's 00:17:34.930 --> 00:17:38.170 elegant parenthesis -free stack is quietly stacking 00:17:38.170 --> 00:17:40.450 plates behind the scenes to make it all happen. 00:17:40.569 --> 00:17:43.089 It serves as a brilliant testament to the lasting 00:17:43.089 --> 00:17:46.009 power of pure logic. And looking at all of this, 00:17:46.089 --> 00:17:47.750 this raises an important question, something 00:17:47.750 --> 00:17:49.829 for you to think about long after this deep dive 00:17:49.829 --> 00:17:51.940 is over. Let's hear it. We've seen today that 00:17:51.940 --> 00:17:55.039 prefix and postfix notations are incredibly elegant. 00:17:55.279 --> 00:17:57.880 They are completely unambiguous. They are hyper 00:17:57.880 --> 00:18:00.519 -efficient for computers to process. They completely 00:18:00.519 --> 00:18:03.440 eliminate the need for arbitrary rules like PEMDAS 00:18:03.440 --> 00:18:06.299 or messy confusing brackets. But humans inherently 00:18:06.299 --> 00:18:08.859 think in a subject -verb -object pattern. We 00:18:08.859 --> 00:18:11.819 want to say 1 plus 2. So what does our stubborn 00:18:11.819 --> 00:18:13.960 attachment to writing in -fix notation fixing 00:18:13.960 --> 00:18:15.920 that plus sign in the middle say about how the 00:18:15.920 --> 00:18:17.799 human brain instinctively categorizes the world 00:18:17.799 --> 00:18:20.380 versus the ruthless, mechanical, efficient... 00:18:20.329 --> 00:18:21.950 of a perfectly logical machine.