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. Eventually built the invisible architecture
for the entire field of computer science. Which
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. You probably had brackets inside
of brackets. You're trying to remember PEMDAS
and the order of operations. The 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. Consider 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. And 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. Okay, let's unpack this. The core
concept here starts with how we do everyday math.
The system you and I grew up learning in school
is called infix notation. That word infix 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 write the
number one, then the plus sign, then the number
two, one plus two. Simple enough. But then our
sources introduce us to tollish 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 1 plus 2, you would write plus 1,
2. 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. Yeah, 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. Wow.
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, 5 minus 6, close parenthesis,
multiplied by 7. 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. And 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. the entire string. So the finished
equation is just times minus 5, 6, 7. Or if you're
looking at it on a page of symbols, a multiplication
sign, a subtraction sign, a 5, a 6, and a 7.
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 5 minus,
open parenthesis, 6 times 7, close parenthesis.
In Polish notation, you just shift the operators
at the front. It simply becomes minus 5 times
6, 7. 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. Non -commutative,
meaning? Operations where the order of the numbers
completely changes the answer, like division
or subtraction. 5 minus 6 is very different from
6 minus 6. minus 5. 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 10 and a 5, so
a division sign, then 10. Then 5, that strictly
means 10 divided by 5. The number to the left
is the dividend. The number to the right is the
divisor. Got it. Similarly, a minus sign followed
by 7, then 6 strictly means 7 minus 6. You are
subtracting the 6 from the 7. 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 Jan Jukasiewicz.
A very influential figure. He invented the system
in 1924. But here's the historical detail I absolutely
love. He 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 with a whisper rather
than a shout. But to provide some broader historical
context, Yukasiewicz wasn't operating in a total
vacuum here. Right, there were others working
on it. This was a time of immense exploration
and debate in mathematical logic. A German mathematician
named Heinrich Biemann actually had a somewhat
similar idea for eliminating parentheses 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 his own heavily structural parenthesis
-free notation called the Begriffschrift way
back in 1879. The desire to get rid of the clunky
brackets had been brewing in the academic community
for quite a while. Precisely. However, Lukasiewicz's
system was a massive breakthrough because it
was the first linearly written, incredibly compact
version. Friedge's older system was two -dimensional.
What do you mean by two -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.
Ukasevich 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
work, the Principia Mathematica. Here's where
it gets really interesting. Lukasiewicz 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 negacja.
Makes perfect sense. He used C for the material
conditional, which is a fancy way of saying if
this, then that, because the Polish word for
implication is implicacja. He used K for conjunction.
meaning the word and, from the Polish word koniokcia.
Right. He even ventured into modal logic, which
deals with philosophical concepts like possibility
and necessity. So he used M for possibility,
from the Polish word muliwosz, and he used L
for necessity, from the Polish word konieczność.
And whenever you have different brilliant academics
building on a single system, you inevitably get
some quirky historical contradictions. Oh, of
course. Another prominent logician named Bocheski
came along later and loved this 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, REO,
or IFNN. Exactly. Bocheski mapped out all 16
possible ways to do that. But here's the funny
part. Bocheski 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 and ruining all their formulas.
It probably caused quite a few headaches. Lukasiewicz
was using L and M for modal logic necessity and
possibility. But Bocheski 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 our 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 parentheses.
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. Think about how
a machine actually reads information. Explaining
the human order of operations to a computer using
parentheses is incredibly difficult. Right, because
of the nesting. Exactly. If you feed a computer
an 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 -sized
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 imagine 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 top 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?
That is 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 stores
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 are variadic. To translate
that jargon for you listening, variadic 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 variadic addition function is like throwing
ingredients into a blender. You can throw in
two numbers or five numbers or 50 numbers and
it will add them all up. Because the operator
doesn't have a fixed number of operands, Lisp
has to use 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 Jukasiewicz'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 LDAP filter syntax that uses Polish prefix
notation to search directories. Oh, definitely.
Even more modern web -focused tools like CoffeeScript
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 fix 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 this string from left to right instead
of right to left. You hit the nail 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
PostScript, 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 Forth. It was heavily used in the architecture
of Burroughs' 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. Those 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 close 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 idea. We started with Jan
Jukasiewicz, a Polish logician 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. 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 Ukasiewicz'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 postfix notations are incredibly elegant.
They are completely unambiguous. They are hyper
-efficient for computers to process. They completely
eliminate the need for arbitrary rules like PEMDAS
or messy confusing brackets. But humans inherently
think in a subject -verb -object pattern. We
want to say 1 plus 2. So what does our stubborn
attachment to writing in -fix notation fixing
that plus sign in the middle say about how the
human brain instinctively categorizes the world
versus the ruthless, mechanical, efficient...
of a perfectly logical machine.