How Polish notation eliminated parentheses

0:00You're listening to a podcast right now, driving, working out, walking the dog. If you're into podcasts, chances are you have something to say too. With RSS.com, starting your own is free and easy. Upload an episode, and we distribute it to Apple podcasts, Spotify, Amazon Music, and hundreds more. Track your listeners, see where they're from, and start earning from ads like this. Even with just 10 listeners a month. If you've been thinking about starting a podcast, this is your sign. Start free at RSS.com. Welcome to The Deep Dive. Today we are exploring a truly fascinating Wikipedia article on a mathematical concept called Polish notation. Yes, and it is a fantastic topic. It really is. We are going on a journey today to discover how moving a single, simple math symbol from the middle of an equation to the absolute front completely revolutionized logic and eventually built the invisible architecture for the entire field of computer science.

1:03Which is wild when you think about it, just shifting one symbol. Right. I want you to think back for a moment to your days in middle school algebra class. Have you ever looked at a long complicated math equation on a chalkboard and just gotten completely lost in a sea of parentheses? Absolutely. Everyone has. Brackets inside of brackets, you're trying to remember PEMDOS and the order of operations and whole thing just felt incredibly cluttered and overwhelming. Well, what if there was an entirely different way to write math that deletes those confusing parentheses completely? It sounds almost like a magic trick when you first encounter it to just, you know, erase the brackets and still have the math perfectly hold together. But it is a very real, highly structured and incredibly elegant logical system. After this deep dive, your ultimate shortcut to being well informed. By the time we wrap up this conversation, you are going to understand the secret language that powers the calculators you use and the computer programs you interact with every single day. You will honestly never look at a simple addition problem the same way again.

2:06And beyond just math, it really forces you to change your perspective on how we communicate instructions. It highlights the massive gap between how the human brain wants to read a sentence and how a machine needs to process data. OK, let's unpack this. The core concept here starts with how we do everyday math. The system you and I grow a learning in school is called in fixed notation. That word in fixed just means the operator, the plus or minus sign is fixed right in the middle of the numbers. So if you want to add one and two, you read the number one, then the plus sign, then the number two, one plus two, simple enough. But then our sources introduces to totalish notation. You might also see it referred to as normal Polish notation, Warsaw notation, Polish prefix notation, or Eastern notation. In this system, the operator actually precedes the operands. It goes right at the very front. Right at the front. So instead of writing one plus two, you would write plus one two. What's fascinating here is the why behind this shift. Why bother moving the plus sign to the front at all? Because it feels completely counterintuitive to how we speak.

3:08Yeah, it really does. But the genius of the system is that as long as an operator has a fixed number of operands, meaning a plus sign always needs exactly two numbers to add together. Polish notation requires absolutely zero parentheses, none at all. You could write out the most complex, massive, multi-layered mathematical formula in the world, and you will never need a single bracket to tell you what part to solve first. That's just wild to think about. I want to make sure we're really picturing this correctly. Let's walk through a specific visualization from our source material. So you, listening right now, can really see how this plays out. Let's take a standard math equation. Open parenthesis, five minus six, close parenthesis, multiplied by seven. Okay. Got it. In the standard-infix math, we all know, you need those parentheses around the five minus six. So you know, to do that subtraction first, right, for multiplying the whole thing by seven. But let's translate that into Polish notation. We start with the innermost part, the five minus six. Since the operator moves to the front, that simply becomes minus five six.

4:09And that neat little package, the minus five six, becomes the new single unit that we want to multiply by seven. Exactly. So to finish translating the equation, you take your multiplication sign and you stick that at the very front of the entire string. So the finished equation is just times minus five six seven. Or if you're looking at it on a page of symbols, a multiplication sign, a subtraction sign, a five of six and a seven. You read it. And the order of operations is perfectly clear without a single bracket. And we can contrast that by moving the parentheses in our original math problem. What if the equation was five minus open parenthesis, six times seven, close parenthesis? In Polish notation, you just shift the operators at the front. It simply becomes minus five times six seven. Just by changing the sequence of the operators and the numbers, the innermostness of the math is perfectly conveyed to the reader. It's brilliant. There is a crucial nuance here that we have to point out, though, especially when we're dealing with non-commutative operations.

5:09Non-commutative, meaning operations where the order of the numbers completely changes the answer, like division or subtraction, five minus six is very different from six minus five. So you have to coordinate the sequential arrangement perfectly. Right. The order really matters. Exactly. In Polish notation, reading left to right, if you see the division symbol followed by a ten and a five, so a division sign, then ten, then five, that strictly means ten divided by five, the number to the left is the dividend, the number to the right is the divisor. Got it. Similarly, a minus sign, followed by seven, then six, strictly means seven minus six. You are subtracting the six from the seven. The structure leaves zero room for ambiguity. It's so incredibly rigid, but in the best possible way. Now let's talk about where this came from, because the history outlined in the article is fantastic. The Polish in Polish notation refers to the nationality of a brilliant logician named John Eukacheiewicz. A very influential figure. He'd vented the system in 1924. But here is the historical detail I absolutely love.

6:11He wrote a paper in 1931 where he casually stated that he actually first used this revolutionary parenthesis free notation tucked away in a mere footnote. A footnote. Yes. Specifically, a footnote on page 610 of a lithographed report he published in 1929. Just imagine inventing a whole new way to write math, a system that would go on to fundamentally shape modern computer science, and you debut it in a footnote on page 610. It really speaks to how these monumental shifts in academic thought sometimes enter the world of the whisper rather than a shout. But to provide some broader historical context, Eukacheiewicz wasn't operating in a total vacuum here. All right. There were others working on this. This was a time of immense exploration and debate in mathematical logic. A German mathematician named Heinrich Beeman actually had a somewhat similar idea for eliminating parenthesis and logic formulas right around the same time in 1924. Interesting. And if we look even further back into the sources, the famous logician Gottlob Friedge proposed

7:11his own heavily structural parenthesis free notation called the Big Rift Shrift way back in 1879. So the desire to get rid of the clunky brackets had been brewing in the academic community for quite a while. Precisely. However, Eukacheiewicz's system was a massive breakthrough because it was the first linearly written, incredibly compact version. Friedge's older system was too dimensional. What do you mean by too dimensional? It looked almost like a massive sprawling family tree with branches going up and down, and it took up a huge amount of physical space on the page. You couldn't just easily type it out. Oh, I see. Eukacheiewicz gave the world something you could just type out sequentially on a single line of paper. In fact, his notation was so highly regarded that the legendary logician Alonzo Church specifically praised it in his classic book on mathematical logic. Church called it worthy of remark and contrasted it favorably with the incredibly dense, complicated notation used by Alfred North Whitehead and Bertrand Russell in their massive foundational

8:12work, The Principia Mathematica. Here's where it gets really interesting. Eukacheiewicz didn't actually invent this just to do basic arithmetic like addition and subtraction. He was a logician first and foremost. He used this system to map out complex philosophical and logical arguments. And because he was working in Poland, the letters he used in his notation actually stood for specific Polish words. It creates a wonderful collision of linguistics and pure mathematics. Let's walk through some of the vocabulary he created. Instead of using traditional math symbols, he used capital letters as his operators. So he used the letter N for negation, because the Polish word is negacha. He used C for the material conditional, which is a fancy way of saying if this than that, because the Polish word for implication is implication. He used K for conjunction, meaning the word and from the Polish word koniokcha. He even ventured into modal logic, which deals with philosophical concepts like possibility and necessity.

9:12So he used M for possibility and the Polish word Mniegołczyś. And he used L for necessity from the Polish word koniecznus. And whenever you have different brilliant academics building on a single system, you inevitably get some quirky historical contradictions. Of course. Another prominent logician named Boczeski came along later and loved the system so much that he expanded it. He introduced a version of Polish notation that named all 16 binary connectives. To break that jargon down a bit for you listening, a binary connective is just a way to logically link two ideas together. Things like Andy Rio or IF DanN. Exactly. Boczeski mapped out all 16 possible ways to do that. But here's the funny part. Boczeski and Yukasiewicz ended up using the exact same letters, specifically L and M, to mean completely incompatible things in their respective systems. Oh, wow. I can only imagine the absolute chaos that must have caused at mathematics conferences in the 1930s. It sounds like the academic equivalent of saving over someone else's shared spreadsheet

10:14and ruining all their formulas. It probably caused quite a few headaches. Yukasiewicz was using L and M for modal logic necessity and possibility. But Boczeski was using L and M in classical logic to represent non-implication and converse non-implication. It's a great example of how even perfectly logical systems can get a little messy when different human beings start adapting them for their own specific overlapping needs. So what does this all mean? We have this elegant linear parenthesis-free system born in the 1920s in Poland. It makes a big splash in the world of formal logic. But as sources note that today, Polish notation is actually no longer used very much in formal logic itself. No, not really. The logicians largely went back to using infix notation and all those parenthesis. But the notation didn't die. It found its true home, its ultimate destiny in the realm of computer science. If we connect this to the bigger picture of the 20th century, the invention of Polish notation perfectly predated the dawn of the computing age. And it solved a massive foundational problem for early computer engineers.

11:15Think about how a machine actually reads information. Explaining the human order of operations to a computer using parenthesis is incredibly difficult. Right. Because of the nesting. Exactly. If you feed a computer and equation with brackets inside of brackets, you have to program the computer to scan all the way ahead. Find the deepest set of brackets, hold all the other ignored information in its temporary memory, do the deepest math, and then work its way backward. It takes a ton of memory and processing power, which early room size computers simply did not have. I was reading about this in the source material. And they mentioned that Polish notation solves this by allowing computers to use something called an evaluation algorithm or a stack algorithm. But I want to make sure I am picturing this correctly because the technical descriptions get pretty dense. If I am the computer reading a string of Polish notation from right to left, what exactly am I doing with these numbers? How does the stack actually work? It is beautifully simple when you break it down.

12:16Imagine the computer is a worker in a cafeteria. And the stack is literally one of those spring-loaded plate dispensers. Okay, picture that. The computer reads the equation from right to left. Every time it sees a number, it writes that number on a plate and pushes it down onto the stack. So if the equation is a plus sign, then a two, then a three, reading right to left, it sees the three, puts it on a plate and pushes it down. Then it sees the two, puts it on a plate, and pushes it down on top of the three. Okay, so I have a two sitting on top of the three. Correct. The magic happens the moment the computer reads an operator, like a plus sign. It knows a plus sign needs two numbers. It doesn't look around. It doesn't scan ahead. It simply pops the top two plates off the stack. The two and the three adds them together to get five, writes the number five on a brand new plate and pushes that single new plate back onto the stack. And then it just moves on to the next symbol in the string. That completely eliminates the need for parentheses. It just keeps stacking plates and every time an operator shows up, it consolidates the

13:20top plates into a single new number. There's no scanning ahead and no backtracking. And crucially, there is no need for what computer scientists call arbitrary stack inspection. Which means? It's just a fancy way of saying the computer never has to dig around in the middle of the pile of plates. It only ever looks at the plate sitting right on the very top. The sheer efficiency of that is beautiful. A simple push down store is all you need to implement this parsing. It saves a massive amount of computational memory. Because it is so lightweight and efficient, this concept actually lives on in the real world today. It isn't just a museum piece of early computing history. For instance, there is a very famous, very influential programming language called Lisp. It uses something called S expressions, which rely heavily on prefix notation. Yes. Lisp is a classic example. Now, it is kind of ironic because if you look at Lisp code, it actually does use a ton of parentheses. Our sources explain that this is because in Lisp, the functions act as data and they're very attic.

14:20To translate that jargon for you listening, very attic just means a function can take any number of inputs. It isn't fixed. Right. Unlike standard addition. Exactly. Simple addition takes exactly two numbers, but a very attic addition function is like throwing ingredients into a plender you can throw in two numbers, or five numbers, or 15 numbers, and it will add them all up. Because the operator doesn't have a fixed number of operands, Lisp has used parentheses to show the computer where the list of ingredients finally ends. It has to have some way to know it's done. Right. But the core Polish structure, putting the operator at the very front, remains the absolute foundation of the language. Lisp is certainly one of the most famous examples, but the influence of your Kasević's work goes far beyond that. I was amazed at the list of applications. The TCL programming language uses Polish notation for its math operations. There is a language called ambi that uses it for arithmetic and program construction. If you work in corporate IT, you might have wrestled with LDP filter syntax that uses Polish prefix notation to search directories.

15:22Oh, definitely. Even more modern web-focused tools like coffee script allow functions to be called using this prefix notation. We also absolutely have to mention the very famous sibling of this system, which the article covers in detail. If you can move the math operator to the absolute front of the equation, you can logically also move it to the absolute end. That variation is called reverse Polish notation, or RPN. It is also known as post-fixed notation. Post-fixed meaning after. Exactly. So instead of writing plus one, two, you would write one, two, plus. And I imagine the cafeteria plate stack algorithm works exactly the same way you just process the string from left to right instead of right to left. You hit the male on the head. The mechanics are identical, just mirrored. And reverse Polish notation powers a massive amount of technology that we rely on. It is the basis for stack-oriented programming languages, like post-cript, which is the hidden language used in desktop publishing, to tell your printer exactly how to draw fonts and graphics on a page. I had no idea. It is the foundation of the programming language for.

16:23It was heavily used in the architecture of Burrow's large systems mainframes back in the day. The article also points out an application that everyday consumers might actually recognize, especially if they work in finance or engineering. Reverse Polish notation is the chosen default notation for high-end calculators, most notably the scientific and financial calculators manufactured by Hewlett-Packard. These HP calculators have an incredibly loyal cult following for a very specific reason. Engineers and finance professionals swear by them, because once your brain gets used to thinking in reverse Polish notation, you can punch in massive, complex calculations significantly faster than you ever could on a standard calculator. Because of the keystrokes. Exactly. You never have to waste keystrokes manually typing out open and closed parentheses. And you never have to worry about the calculator misunderstanding the order of operations. You just type the numbers, type the operator, and the internal stack handles the rest instantly. It is amazing to wrap up this journey and look at the vast trajectory of this one single

17:25idea. We started with Jan Yukosiewicz, a Polish legician in the 1920s, who was simply trying to clean up his logic equations to make them a little easier to type out on a single line of paper. Just a pragmatic solution. And he drops this parenthesis-free idea in a footnote on page 610 of a 1929 report. And by doing that, he accidentally created the invisible, highly efficient mathematical architecture that allowed early computers with terrible memory to process complex math. An architecture that still powers incredibly sophisticated programming languages, printers, and calculators today. It's incredible. So the next time you use a computer to crunch some numbers or you hit print on a PDF document, remember that a version of Yukosiewicz's elegant parenthesis-free stack is quietly stacking plates behind the scenes to make it all happen. It serves as a brilliant testament to the lasting power of pure logic. And looking at all of this, this raises an important question, something for you to think about long after this deep dive is over. Let's hear it. We've seen today that prefix and post-fixed notations are incredibly elegant.

18:29They are completely unambiguous. They are hyper-efficient for computers to process. They completely eliminate the need for arbitrary rules like pimp does or messy, confusing brackets. But humans inherently think in a subject-verb object pattern. We want to say one plus two. So what does our stubborn attachment to writing infix notation fixing that plus sign in the middle say about how the human brain instinctively categorizes the world versus the ruthless mechanical efficiency of a perfectly logical machine? You're listening to a podcast right now, driving, working out, walking the dog. If you're in a podcast, chances are you have something to say too. With RSS.com, starting your own podcast is free and easy. Upload an episode and we distribute it to Apple Podcasts, Spotify, Amazon Music, and more. Track your listeners, see where they're from, and start earning from ads just like this. If you've been thinking about starting a podcast, this is your sign. Order your new podcast for free today at rss.com.