Welcome to the Deep Dive. I'm really glad you're
joining us today because we are looking at something
that is going to fundamentally change the way
you look at a simple string of numbers. Yeah,
it really shifts your entire perspective. It
does. Today we're exploring a single remarkably
dense source that you shared with us. It's a
comprehensive overview of a concept called Polish
notation. Which is just a brilliant piece of
history. Right. And our mission for this deep
dive is to explore how simply, you know, rearranging
the order of mathematical symbols completely
eliminates the need for parentheses. We're going
to look at how it revolutionized early formal
logic and how it ultimately became the secret
language of the computers we rely on every single
day. To really understand the origins of this,
you have to imagine the intellectual atmosphere
of a, well, a classic 1920s European library.
Oh, I love that visual. Right. Picture towering
oak bookshelves, academics debating in the halls,
and tables covered in these beautifully complex,
dusty logic manuscripts. That was the exact environment
where this concept was born. Today we are going
to synthesize the history, the mechanics, and
the modern applications of this notation, showing
not just what it is, but why it actually matters
to you. I want to set the stage for you, the
listener, for a second. Think about the last
time you tried to write out a really complex
math problem or a dense logic puzzle. Or even
just a long formula in Excel. Yes, exactly. Imagine
you're typing a convoluted equation into a spreadsheet.
You probably find yourself completely bogged
down by brackets, parentheses, and the strict
rules of the order of operations. It gets messy
fast. It really does. You know the drill making.
Sure, you have exactly three closed brackets
at the end of your formula just to perfectly
match the three open brackets at the beginning.
And if you miss one? Right. A single typo breaks
the entire formula. The software just yells at
you. But what if there was a cleaner, completely
linear way? What if you could just write the
instructions straight across left to right and
the math just worked? That is the exact problem
Polish notation solves. It strips away all of
that visual clutter and relies purely on the
sequence of the information. Okay, let's unpack
this. We all grew up on standard infix notation.
That means the operator, the plus sign, the minus
sign, the multiplication symbol, it's trapped
right in the middle of the numbers it's operating
on, like 1 plus 2. Right, the operator is fixed
in the center. Exactly. But Polish notation...
which is also called prefix notation, rips that
operator out and puts it at the very front. So
it becomes plus 1, 2. And while putting the plus
sign before the numbers might look a little strange
for simple addition, its true power reveals itself
when the equations become layered and highly
complex. Right. Let's look at a harder problem.
Let's challenge you, the listener, to visualize
this expression, open parenthesis, 5 minus 6,
close parenthesis, multiplied by 7. So standard
infix notation. Right. Because we use standard
infix notation, we absolutely need those parentheses.
If we drop them, the order of operations forces
us to multiply the 6 and the 7 first, completely
changing the math. What's fascinating here is
how Polish notation elegantly bypasses that entire
problem. In this prefix system, the operator
always precedes its operands. So if I have that
same equation, I'd start with the multiplication
because that's the outermost operation. I write
the multiplication sign first. But where does
the subtraction go if I just write multiply 5
minus 6, 7? That just looks like a jumbled mess
of characters. How does the structure actually
hold together without the brackets? That's the
trick. You work strictly by operations. You have
your multiplication sign. It needs two operands.
The first operand isn't a single number. It's
the result of 5 minus 6. Oh, I see. So immediately
after the multiplication sign, you write the
subtraction sign, followed by the 5 and the 6.
Then finally, you put the 7 at the very end.
The sequence becomes multiply minus 5, 6, 7.
And because every operator has a strict rule
about how many numbers it needs, the brackets
just vanish. Precisely. That property is called
its arity. A subtraction sign has an arity of
two. It always needs exactly two numbers following
it. A multiplication sign also has an arity of
two. The structure regulates itself. No arbitrary
order operations to memorize and no hunting for
missing parentheses. But wait, what about order?
If I'm multiplying, it doesn't really matter
if I say 3 times 4 or 4 times 3, but division
and subtraction are noncommutative. The order
matters immensely. Does shifting everything to
a linear line create confusion there? Not at
all. The source provides clear examples to ensure
there is zero ambiguity. When dealing with noncommutative
operations, the sequence of the operands directly
follows how the operator takes its arguments,
moving strictly left to right. So the order is
logged in. Exactly. If you write the division
symbol followed by 10 and then 5, it strictly
means 10 divided by 5. If you write the minus
sign followed by 7 and then 6, it strictly means
you subtract 6 from 7. The action word dictates
exactly what happens to the sequence that follows.
Imagine writing out instructions in your daily
life where the action always dictates exactly
what follows, eliminating any ambiguity. You
never have to... backtrack or second guess which
part of the instruction takes precedence. It's
just a pure, linear flow of logic. It is incredibly
elegant. And to understand where this pursuit
of pure logic came from, we have to travel back
to the vibrant intellectual era of 1920s Poland.
The history here is fascinating. The Polish in
Polish notation refers to the nationality of
its inventor, a Leducian named Jan Łukasiewicz.
He invented this parenthesis -free notation in
1924. Right. He actually mentioned it in a lithograph
report in Polish in 1929 and then formally detailed
it in a 1931 paper. It's gone by a few names
over the years. You'll hear it called Normal
Polish Notation or NPN. Sometimes it's called
Warsaw Notation or Eastern Notation. We should
provide a bit of historical context, too. While
Ukaziewicz created the first linearly written
parenthesis -free notation, he wasn't the only
one chasing this kind of logical purity. Moses
Schoenfinkel, whose work was edited by Heinrich
Behmann in 1924, had the idea of eliminating
parentheses in logic formulas. Oh, true story.
Yeah, and even further back, in 1879, Gottlob
Friege proposed a visual parenthesis -free notation
called Bigriff's Rift. But Friege's system was
visual, right? It relied on two -dimensional
lines and trees, which made it an absolute nightmare
to actually typeset and print in a book. Oh,
completely impractical for mass printing. Yeah.
version struck a chord because it was linear.
It solved the typesetting problem for logic.
Exactly. It was so impactful that Alonzo Church,
an absolute giant in mathematical logic, specifically
praised Ukasiewicz's parenthesis -free notation
in his classic 1944 book. Church called it worthy
of remark, contrasting its elegance directly
against the heavyweights of the time, Alfred
North Whitehead and Bertrand Russell, and their
notoriously dense masterwork, Principia Mathematica.
That is a huge endorsement. Principia Mathematica
is famous for taking hundreds of pages of incredibly
dense, bracket -heavy notation just to prove
that 1 plus 1 equals 2. It's infamous for it.
Yeah. To have Church point to Ukasiewicz's linear
string and say, look at how much cleaner this
is, speaks volumes. It does. And we should note
that Ukasiewicz didn't just use this notation
for standard arithmetic. His primary focus was
using it for complex formal logic. If you were
to look at his original chalkboards, you wouldn't
see math symbols. You'd see capital letters based
on Polish words. Let's break those down because
the language connection is really interesting.
For the concept of negation, like saying not
true, he used the letter N, which stands for
the Polish word negacja. Makes perfect sense.
Right. For the material conditional, which is
an if -then statement, he used C for implicacja.
For disjunction, or or, he used a for alternativa.
And for conjunction, meaning and, he used k for
konimcha. So you could write out massive, sprawling
logical arguments just using strings of letters
and variables with zero punctuation. But that
inevitably leads to some academic friction, doesn't
it? When you start claiming letters for specific
logical functions, what happens when someone
else wants to expand the system? This is where
we run into a wonderfully quirky historical contradiction.
Another logician by the name of Bocheski came
along and expanded Ukasevich's system. He wanted
to include all 16 binary connectives for classical
propositional logic. But in doing so, his notation
clashed horribly with Ukasevich's work in a different
area called modal logic. Because they were both
trying to use the same letters for entirely different
things. You can see if it was using the letter
M to mean possibility from the Polish word, we
leave Osh and the letter L to mean necessity
from Koneczny precisely. But Bocheski, building
his own massive extension of the very same parenthesis
free notation, decided to use M and L for completely
different things, specifically non implication
and converse non implication in classical logic.
You had these two brilliant minds working in
the exact same ecosystem, using the exact same
letters to mean entirely different logical. concepts.
It highlights just how specialized and dense
this field of study was becoming. Here's where
it gets really interesting, though, because despite
this being such a beautifully pure system, such
an elegant way to write out logic without any
brackets, human beings actually ended up hating
Polish notation. It is the ultimate irony. The
system was designed by a human mind to make human
logic more streamlined, but the human brain just
doesn't process information that way very well.
It's fascinating because researchers later found
that humans actually strongly prefer visual formatting.
A 2011 paper by Martinez -Nava, highlighted in
the materials, points out that Polish notation
fell into disuse, precisely because of how hostile
it is for human reading. Our eyes naturally want
to look at a spatial relationship. We want that
visual balance. Yes, we want to see a number,
an operator in the middle, and another number.
That's how we balance concepts in our head. Reading
a string of prefix commands requires you to hold
a tremendous amount of abstract information in
your short -term memory. You have to stack all
the operations in your mind before you ever get
to the numbers they apply to. The research specifically
highlights how incredibly difficult this makes
reading for individuals with dyslexia. Right.
It removes all visual anchors. But you know what
doesn't need visual anchors? A microchip. If
we connect this to the bigger picture, this failure
in the realm of human cognition paved the way
for its greatest triumph. While humans struggled
immensely to read a sequential, parenthesis -free
stream of logic, computers absolutely loved it.
Humans need spatial cues. The computer processor
just wants a line of instructions it can chew
through one by one. The material details what's
called the evaluation algorithm. And this is
where Polish notation becomes a superpower. Let
me explain how this algorithm works, as it is
foundational to modern computing. Prefix expressions
can be evaluated by a computer moving left to
right or right to left using something called
a stack or a push -down store. Imagine a spring
-loaded stack of cafeteria trays. Okay, I like
that. You push a new tray down onto the top,
and when you need one, you pop the top one off.
Okay, I can picture that, but how does the computer
actually use that tray dispenser to do math?
The computer... reads the input string token
by token, a token being either an operator like
a plus sign or an operand like a number. Let's
say it is evaluating from left to right. It pushes
these tokens onto the stack one by one. It keeps
pushing them down until it realizes that the
top entries of the stack perfectly match the
requirements, the arity of the operator sitting
just below them. So if the computer pushes down
a multiplication sign, it knows it needs two
numbers. It pushes down a five. It pushes down
a six. Suddenly it says, wait, I have an operator
and the exact number of operands it needs. Exactly.
Boom. It executes the multiplication right then
and there, takes the 5, the 6, and the multiplication
sign off the stack, and replaces them with the
result 30. Then it just keeps reading the rest
of the string. But wait, that makes sense for
simple math, but what happens when the computer
hits a massive nested equation? Doesn't the stack
get overloaded, holding all those operators waiting
for numbers? That's the brilliance of it. It
doesn't overload because the moment any local
condition is met, it collapses the data into
a single result. The computer doesn't need to
scan back and forth across a massive equation
looking for brackets to figure out what to do
first. It just acts immediately. Yes. It doesn't
need the capability of arbitrary stack inspection.
It just reads sequentially, pushes data onto
the stack, and executes the moment the already
conditions are met. It is incredibly efficient.
This is why for programming language interpreters,
Polish notation is effortlessly parsed into abstract
syntax trees. Yes. An abstract syntax tree, or
AST, is how a compiler understands code. It builds
a tree where the operator is the root and the
numbers are the branches. With normal human math,
building that tree is computationally expensive.
You have to program complex rules for precedence
and parentheses. Which takes time and memory.
Right. But Polish notation essentially is the
tree just written on a single line. perfect one
-to -one mathematical representation of the code
structure. It strips away all the ambiguity of
human language, leaving only the pure, executable
logic. Which brings us to where we are today.
I want to ask you, the listener, have you ever
tinkered with coding? Have you ever written a
script to automate a task or used a highly advanced
engineering calculator? Because if you have,
you have interacted with the ghost of Jan Ukasiewicz.
Polish notation didn't die out. It just went
behind the scenes. The modern implementations
are vast, and they show just how integral this
logic remains. Prefix notation is still heavily
used in Lisp programming with its S expressions.
Now, it's worth noting that Lisp does use parentheses,
which seems counterintuitive to what we've been
discussing. A bit of a paradox, yeah. But that's
a unique case because Lisp functions are treated
as first -class data. and are variatic, meaning
they can take a variable number of arguments.
The parentheses just help group them. But the
core structural logic is still entirely prefix.
And we see it in data querying as well. If you
work in IT, you are likely familiar with LDD
filter syntax. Yes. LDAP is a great example.
If you need to search a massive company database
for a specific user profile, you have to group
complex logic together. Find someone in marketing,
ND, who is either a manager or a director. LDAP
uses prefix notation to cleanly encapsulate that
logic without needing arbitrary precedence rules
for AND versus OR. It keeps it clean. You also
see it in the TCL programming language through
its MathOp library, the AMBI programming language,
and even CoffeeScript, which allows prefix notation
for function calls. But we also have to talk
about its mirror image. While prefix notation
puts the operator first, there is a variation
called reverse Polish notation, or RPN. This
is post -fix notation. In RPN, the operators
follow... the operands. So instead of plus one
two, you write one two plus. And RPN is everywhere
in the hardware and stack -oriented software
world. It is heavily used in stack -oriented
languages like PostScript, which is the foundational
language that powers modern printing, and the
fourth programming language. But its most famous
application is arguably in Hewlett -Packard calculators.
If you have ever seen an engineer or a finance
professional using a classic HP calculator, you
were watching reverse Polish notation in action.
They loved those machines because when you were
calculating load -bearing stress or complex compound
interest, you were dealing with massive formulas.
You'd get lost on a normal calculator. Exactly.
With a standard calculator, you need a notepad
to write down intermediate results. Or you need
to perfectly track eight levels of parentheses.
With an RPN calculator, you just keep pushing
numbers onto the stack and applying operations.
It felt like you were manually driving the computer's
processor. It operates at a very low level. Post
-fix operators are even used by stack machines
like the Burroughs Large Systems. And right toward
the end of our analysis, there is an incredibly
dense mathematical rule that proves why all of
this works so flawlessly. It's regarding the
equation of returns. Yes. It states that the
number... of return values of an expression equals
the difference between the number of operands
and the total arity of the operators minus the
total number of return values of the operators.
So what does this all mean? That is a very dense
sentence to process audibly. Let's actually test
that to give you, the listener, a concrete idea
of what's happening under the hood. Let's track
it in real time using our earlier example. Multiply
minus 5, 6, 7. First we count the operands, the
numbers. We have 5. Six and seven. That's three
operands. Okay, three operands. And then we look
at the total arity of our operators. We have
a multiplication sign and a subtraction sign.
They each require two inputs, so the total arity
is four. Exactly. And those two operators will
eventually spit out two outputs or two return
values total. What that complex formula essentially
proves is that when you map all those input requirements
and output values together against the starting
numbers, the entire sequence perfectly collapses
to exactly one final answer. It mathematically
balances out. There are no dangling numbers left
on the stack, and there are no missing variables
causing a crash. It is a perfect, unbreakable
mathematical loop. apparently verifies itself
without needing any outside punctuation to hold
it together. It is the ultimate expression of
logical self -sufficiency. And that brings us
to the end of this journey. To summarize our
deep drive today, we explored the brilliant,
elegant concept of Polish notation. Born from
the mind of Jan Ukasiewicz in 1920s Poland as
a way to simplify and purify formal logic, it
proved to be a bit too alien for the spatial
processing brains of human beings. It just didn't
catch on for everyday writing. Right. It was
largely abandoned as a written language because
it was just too hostile to human reading habits.
But its pure, parenthesis -free, linear structure
turned out to be the exact perfect match for
the way computer processors operate, making it
the foundational high - efficient logic language
that quietly powers modern computer parsing,
algorithmic stacks, and engineering calculators
to this day. It is a remarkable testament to
the idea that just because a system doesn't work
well for human eyes doesn't mean it lacks intrinsic
perfection. I want to thank you, the listener,
for coming along with us on this deep dive. And
as we conclude, I want to leave you with a final
thought to mull over. This raises an important
question. Today, we've seen how standard mathematical
notation, with all its messy parentheses and
precedence rules, is actually less efficient
and less logical than a parenthesis -free linear
string. We only use infix notation because it
is what we are used to. It caters to our visual
habits. So if we are still clinging to a less
efficient math notation out of sheer habit, what
other foundational systems in our daily lives
or workplace... or our communication are we stubbornly
holding onto out of comfort long after a vastly
more logical alternative has been discovered.