Physics taught me I was counting wrong 1.pdf

Physics taught me I was counting wrong
This is about counting. Yes the same counting you learned probably as your first ever math
lesson. Except this is a bit adult stuff where your intuition will be pushed to the limit. It is a math
article although a good portion of it toes the line between math and physics. However it is
written as a standalone math article. So besides the obvious demographic, I very much
intended this for physicists too. If you are a mathematician reading this and have a physicist
friend, please make sure your friend reads this. As for the rest, anyone who enjoys critical
thinking, you are more than welcome to read. The prerequisites I ask for in order to appreciate
the content of the article are as follows: high school physics knowledge; you do not need to
know any math of quantum mechanics, but I expect you to know the events that shaped this
branch in story format. These are the events in particular you should know: the UV catastrophe,
the double slit experiment, Rutherford's gold foil experiment, and the Stern-Gerlach experiment
(just familiarize yourself with the bizarreness). On the math front, high school math knowledge
(esp. rational, irrational numbers, the real number line, concept of an interval, the concept of a
function), only the concept of limit (nothing else about calculus), basic operations on sets,
counting elements of a set and the axiom of choice.
I wrote it as if I'm having a conversation with you where you get to follow my thought process.
None of the figures, sections or the references in this article is numbered. The reason is part of
the article. The references in the text are only marked with random letters [* ]. They are listed at
the end of the article. Single quote has been used to highlight a term and double quote as air
quotation.
Without further ado, let us begin.
Cantor's insight
The argument goes like this: we pick all the distinct numbers in random from the interval (0,1)
simultaneously (the axiom of choice [g]) and assume that this sequence is the complete list of
all the numbers in that interval. Since the sequence is distinct we can always make a one-to-
one correspondence with the set of integers. If you want a visual―think of a gigantic rake with
many, many tines, as many as the number of integers, and you stab the interval (0,1) with it. All
the numbers (points really) stabbed will be simultaneous as well as one-to-one with all the
integers. An example of the list is given below:

Note the underlined digits: first digit (count starts after the decimal) of the first number, second
of the second and so on. In general, consider the i-th digit of i-th number. Written as above,
they are lined across the diagonal. And now comes Cantor's ingenious insight [s]. Let's define a
function f(n) as,
f(n)={
who takes those diagonal (underlined) digits as its inputs. I'll call this special function, diagonal
function or diag. fn in short throughout this article. We will create a new number with the digits
outputted by the function by putting them one after the other after the decimal. In this particular
example, the number is 0.01100⋯ (since in the example, the diagonal numbers: odd, even,
even, odd, odd). This new number can't be anywhere on the list. Because if it did its k-th
position (assuming it's the k-th number) will either be 0 or 1, which the function will flip (since 0
is even and 1 is odd), making the resulting number differ at k-th position. This new number is
also in between 0 and 1. Why stop here? We can create as many new numbers as we want by
changing the function, none of which will be found in the list. This contradicts our assumption
that the list consists of all the numbers in (0,1). Since the list is a random collection of numbers
and arbitrary, it will be true for any and all lists (that there are always numbers left over) making
it impossible to establish a one-to-one correspondence with the integers. Hence uncountable.
As a fun exercise you can try with digits in other position to see how only the diagonal position
guarantees a number that's not on the list, or you can take almost a century's worth of
mathematicians' word for it.
The algorithm
It was either May or June of 2018 (I'm old, I forget). Just finished my bachelor's. Unemployed.
Given up on life. I was reading a math book (either on analysis or on fractals), cause I enjoy
reading math. This proof popped up and I was as awed by the brilliance of it as I was the first
time when I learned it. How does someone come up with it? I thought nothing more of it and I
moved on. A few days passed. One evening, I was staring at the blank wall contemplating the
point of my existence, an incongruous thought intruded. It was about the proof. What if those
1↔ 0.1
–
50164786⋯
2↔ 0.34
–
1865142⋯
3↔ 0.586
–
213754⋯
4↔ 0.0421
–
02654⋯
5↔ 0.98203
–
9586⋯
⋮
0,if n isodd
1,if n iseven

left-over numbers were appended to finish the list. They are distinct as well to establish one-to-
one correspondence with the integers. But the integers are all being occupied by the original
list; how do I fix that? I needed a rearrangement. After a couple of minutes I came up with it.
This was my line of thinking: I took a finite list of numbers and applied the diag. fn to create a
new number which I added to the list at the end. Then I increased the number of numbers in the
list and did the same thing. I noticed how the position of the new number increased with the
longer list—the generated number got pushed back. I based my algorithm on this.
I'm committed to the rake bit, so I'll explain using that analogy. Each prong of the rake, which
represents all the integers, picks up distinct points from (0,1). I have a diag. fn, the number
created by which points toward a different distinct point in the interval. I need to make room for
this surplus since all the spots are filled. Let's remove all the numbers skewered on the prongs
and leave them on the ground. Rake is empty now, put the new number on its second teeth.
Pick any of the old numbers (if you have kept them in order, you can start with the first number;
it doesn't matter) and add it on the empty first tine. At the same time, the created number that's
on the second tine, shifts by one, moving to the third, leaving second one empty. Pick up
another number from the ground, place it on the empty second one, while the pariah shifts to
the fourth slot. And on and on. This new list will be one-to-one with the set of integers. Because
by design, the old list contains all the distinct numbers, and the number produced by the diag.
fn is also distinct. The position of the unlisted number in the new list gets shifted by one when
adding the numbers from the previous list which are placed on distinct prongs maintaining their
one-to-one correspondence with the integers. Adding 1 to an integer always produces another
distinct integer, meaning the newfound number maintains a distinct position too. No two
numbers will ever share the same tine (integer) even in the presence of this new addition.
Observe the bizarreness: the maximum number of distinct numbers you can select at one go
has to be the number of all integers, surely one more number than that has to share a room
with one of the existing selections. But if you follow my method, you'll see there's a distinct
empty room to house this apparent homeless number. Unless of course you can prove that we
will run out of integers if we keep adding 1, or that there exists an integer which equals its next-
door neighbor.
I'll use one more number. From the old first list, using another diag. fn, a second new number is
generated. Same as before: empty the tines, put the first new number on the second tine and
the second number on the third. Put the numbers from the old list back starting on the first tine,
shifting the new two simultaneously by one. This second new list will be one-to-one with the
integers as usual. Keep doing it to all the excess numbers to establish a one-to-one
correspondence with the set Z. Let me write this algorithm mathematically.

Explanation: start with a list as long as the list of all integers (you can't do better than that).
Have all possible diag. fn's point toward all the surpluses using that list. Do that with all possible
arrangements (permutations) of that list, since using the same function will produce a different
number from the same list with a different arrangement. This will exhaust all the extras that can
be reached using that list. Put them on the f2 list using the recurring j index. Then fill up the f1
list of original numbers using i index in recursion. When concatenated into the list f1+2 , all the j
index of extras of the f2 list will be offset by the f1 list, according to the algorithm. f1+2 , now, is
the revised list with all the old numbers with the extras. Continue until all the numbers are
exhausted. The list remains one-to-one with the integers at any given point.
This algorithm has flexibility. Do it like this: start with a collection of numbers and do all the
usual shenanigan to create a different collection. Find all the different collections. Then pick up
the maximum number of numbers at one go (which is the total number of integers), and put it in
any list, it doesn't matter (also the numbers don't have to be from any particular collection, just
pick whatever). Pick up again from the rest of the numbers—you got a different list. Now you
can either treat the previous list as the f1 list, and the current as f2 , and follow the indexing
rule, or you can treat the current list as the f1 , and the previous f2 , and proceed from there.
Bottom line is that you don't have to do any of this following in any particular order. As long as
you have the numbers, you can do whatever.
Let's make it a bit more general. Understand that the diag. fn isn't helping "invent" new
numbers. Those numbers already exist and always existed. The function merely helps us see
that there are numbers beyond the list. Knowing that, I can enlist in this way: take a list of
numbers, call it A1 . Then from the set (0,1)∖A1 , take another collection, say A2 , making sure
A1∩A2=∅. Similarly, from (0,1)∖(A1∪A2) , make A3 . In general,
(0,1)∖(A
1∪A
2∪A
3∪⋯), where, A
i∩A
j=∅, ∀ i≠j
You continue doing this until it exhausts all the numbers from the interval. Then following the
algorithm, make the one-to-one correspondence with the integers. Diag. fn's aren't needed.
f1≡f1, i=f1, i−1+1;f1, 0=0,i=1,2,3,⋯,1stlistenumeration
f
1+2≡f
1+f
2=f
1, i+f
2, j; the2ndrevisedlist
f
2≡f
2, j=f
2, j−1+1;f
2, 0=1,j=1,2,3,⋯,
⋮
f1+2+⋯+n≡fn−1+fn=fn−1, m+fn, k;then-threvisedlist
fn≡fn, k=fn, k−1+1;fn, 0=1,k=1,2,3,⋯,
⋮

Here's how things get interesting. Given any list, no matter how exhaustive it is, the diag. fn will
say there are numbers unlisted. And my algorithm is specifically designed to address that issue.
But at the end of the day, it still produces a list, and according to the diag. fn, there will be
numbers left over. At the same time my algorithm will say, no, those numbers are in a different
sub-list. It seems like I've reached a paradox. An unstoppable force met an unmovable object. I
came up with all this within a few minutes, it's not that complicated. If only I had spent a little
more than 10 minutes to furnish my argument better I would've had an answer. But I didn't. Why
would I? I had absolutely no intention of going against the brightest mathematical minds for
over a century all of whom conceded on the proof that the set of real numbers in (0,1) is
uncountable. Why would they listen to a 22-year-old nobody? I'm sure it worked for a lot of
famous people. I'm not them. I'm disputing an established and accepted proof. Scary would be
an understatement.
Having convinced myself thusly, I left the topic. That same evening, however, an hour or so
later, I decided to do/think something practical, so of course I started thinking about time travel
(jokes aside, I do write fiction, and I was looking for a story idea; don't ask, all my stories are
unfinished). From that point on, to find some answers in physics, a cavalcade of weird physics-
related events forced my hands to return to this math problem. Never in a million years would I
have guessed that something as abstract as counting numbers will have anything to do with
physics. But here I am.
Before I return to the topic, I need to mention that I didn't like the algorithm, because it has
moving parts. Or in other words, it's a dynamic algorithm, and I'd prefer a static one.
An unhinged function
Let's jump 5 years into the future. To 2023. Yes, I also hate time jump in a story, but this article
isn't a bildungsroman. That is part of the bigger story involving physics. My plan was to write
only that half page algorithm and release it to the world. They could tell me if my line of thinking
was correct or if that paradox had any solution. Having written it down, I felt it was hollow,
barely anything in there and I could almost see people telling me that I only proved the
uncountability of the reals in a slightly involved way. So I decided I'll think about it actively for
10−15 minutes, if I could find something, great, else I'll show my hollow work and take the
humiliation.
Right off the bat, I noticed a few trivial things. Let's say the first list is A
1 , and the set of
numbers pointed toward by the diag. fn's from A
1 is the list A
2 . If these two are concatenated
using my method into the larger list A
1+2 , then wrt (with respect to) A
1+2 I can say the diag. fn's
are pointing toward the numbers within the same list. It looks like cheating, but it really isn't,
because this is how I got my first key insight. If I use the diag. fn on the A
1+2 , my algorithm will
say all the excesses are listed on the sub-list A
3 , both of which are part of the list A
1+2+3 . So

on and so fourth. The algorithm acts something like a nesting doll, where A1 being the
innermost doll/list. From the pov of the whole list then, it looks like the diag. fn's are always
pointing from one sub-list toward other sub-lists. No matter how much I try, I can't apply the
diag. fn to the whole list, because according to my algorithm it's always a sub-list of a bigger
list, whatever "bigger" means in this context—none of the sub-lists individually nor their
aggregation exceed the total number of integers. Like I said before, these numbers have always
existed and if they can be enumerated, they already are. They are not waiting for someone to
make a list. The thing is that if someone does try, some of them always slip through the gaps
like sand. That's what mathematicians concluded as being uncountable. However, my method
changes this conclusion: it says that the whole list might never be accessible, but that doesn't
necessarily imply that the set is uncountable. So I wanted to investigate the function's fishy
behavior when the whole interval is considered. That's when I discovered the first fatal flaw.
Flaws:
My initial conclusion was that being able to form numbers using the diag. fn is not necessarily
an indication of an incomplete list, nor do the function necessarily point toward the excess
numbers. I wanted to test it.
Let's consider the whole interval (0,1). It doesn't matter if all the existing numbers in it are
countable or not. Each is of the format 0.d
1d
2d
3⋯ , and exists next to one another
(geometrically the interval is a line segment). There's no ambiguity in considering the
diagonal digits from this set of numbers who come after one another (for visualization,
instead of thinking the interval a horizontal line segment, think it's vertical, or if you are
comfortable with horizontal, think the decimal representations are sprawled one side of the
line, vertically up/down). I will use the diag. fn on them. It'll output a number which will be
between 0 and 1 (by design of the function), i.e. ∈(0,1). But that's impossible. If it's in (0,1),
then one of its digit has been the input of the function which, by virtue of the function, will
change. No number ∈(0,1), can be equal to the number formed from the outputs of the
function and the function insists on its being in (0,1), which is impossible since each
number in the interval took part in it. Where is this number? You might say that I made a
mistake taking a completed list. But I want to remind you the argument used to prove the
uncountable nature of the reals in the interval: it started with an assumed completed list and
then the number created by the outputs of this function using that list was what used to
prove the incompleteness of the list. No restriction was ever imposed on this function such
as it can't be used on a completed list. If that were the case, you couldn't prove that the
reals in (0,1) are uncountable.
I did something simple. Let's define the interval (0,1)∖{π/10}. I'll consider the whole
interval as before, i.e. the diagonal digits of the decimal representation of all the numbers.
Then it's valid to expect that all the diag. fn's would point toward π/10 with certainty. Except

All of the scenarios above held my conclusion true. So my 10 minutes of investigation turned
out to be fruitful and I decided to figure out what it all meant. I wrote them in a draft back in 2023
, which of course I didn't show to anyone. In it, the segue from here to the next section I
remembered being clunky. So looking for a smoother transition, in this 2025 draft, I found even
more anomalies. I am mentioning it because this next part of the writing is anachronistic,
contrasting my attempt to capture my chronological thoughts.
I have read this proof a thousand times; let me present it here in verbatim. It's from Calculus Vol
2 by Tom M. Apostol [m]:
The set of real x satisfying 0<x<1 is uncountable.
Proof. We assume the set is countable and arrive at a contradiction. If the set is countable we
may display its elements as follows: {x
1,x
2,x
3,⋯}. Now we shall construct a real number y
satisfying 0<y<1 which is not in this list. For this purpose we write each element x
n as a
decimal: x
n=0.a
n, 1 a
n, 2 a
n, 3⋯, where each a
n, i is one of the integers in the set {0,1,2,⋯,9}
. Let y be the real number which has the decimal expansion y=0.y
1 y
2 y
3⋯, where
y
n={
Then no element of the set {x
1,x
2,x
3,⋯} can be equal to y, because y differs from x
1 in the
the example diag. fn from earlier points to a number consisting of 0's and 1's. This number
is nonsensical, because it can't be in the current interval.
Let's give it a breathing room, and consider the interval (0,0.5). The entire other half of the
original interval is left as the diag. fn's playground. Using the example diag. fn on all the
numbers here produces the same result, the number isn't in (0.5,1).
Next, I considered 5 subintervals, viz. (0,0.2), (0.3,0.4), (0.5,0.6), (0.8,0.9), (0.9,0.95). Note
they don't add up to (0,1). I will consider all the numbers from (0.3,0.4)∪(0.8,0.9). Let me
use a different diag. fn just for fun. Say, it outputs 3 when the digit is divisible by 4, or else it
gives out 8. In this case the number would be a string of 3's and 8's, i.e.
∈(0.3,0.4)∪(0.8,0.9). It doesn't matter how finely you partition the interval, or how many
subintervals you create, or whether they add up to the whole interval, used on all the
numbers from any of these subintervals or the union of any of them, there exists some diag.
fn's that will always create some paradoxical numbers.
I thought maybe the rationals in this interval are causing some mischief. Rationals are
countable, so if I don't consider them, they shouldn't affect any outcome. This time the
interval is (0,1)∖Q. If I were to use either of the example diag. fn's on all the irrationals,
then the resulting number would be either a string of 0's and 1's, or 3's and 8's, non
repeating and non terminating—an irrational number which once again cannot exist. You
can try with all the above cases using only the irrationals, you'll get the same nonsensical
behavior.
1,if a
n, n≠1,
2,if an, n=1 .

first decimal place, differs from x2 in the second decimal place, and in general, y differs from xk
in the k-th decimal place. (A situation like xn=0.249999⋯ and y=0.250000⋯ cannot occur
here because of the way the yn are chosen.) Since this y satisfies 0<y<1, we have a
contradiction, and hence the set of real numbers in the open interval (0,1) is uncountable.
Never did it catch my eyes until this time it did: the author's choice of the numbers
xn=0.249999⋯ and y=0.250000⋯ as an example to show, I believe, that no rationals come
up in the whole procedure. What I didn't understand was why the author bothered to show what
xn might not be. Only one digit of any xn's are used in the diag. fn, so why does it matter what x
n's representation looks like? Even more baffling was the y's example expression: why did he
choose 0's when yn can only be 1 or 2 (I pretended the 5 wasn't there)? If he needed to show a
rational number, he should've picked some string of repeated 1's and/or 2's.
That was it.
There is no restriction imposed on the order of the sequence {x1, x2, x3,⋯}. If the sequence
was such that all the diagonal digits, an, n≠1, then y would be 0.111⋯ . Similarly it can be
0.222⋯ by having all the diagonals equal to 1. These are valid lists, and the diag. fn is also
valid. But it's locating rational numbers. Rationals are proven to be countable. I checked with
both the example diag. fn's from earlier, and in a similar vein it can be shown that they produce
either 0.000⋯, or 0.111⋯, or 0.333⋯, or 0.888⋯ .
Perhaps it's true for any diag. fn. I tried to prove it.
The proof turned out to be very simple. Since it is a function, it will not have two distinct
outputs for a single input (definition of a function). This removes ambiguity. Fix one of its
outputs and arrange the list accordingly such that all the diagonal digits produce said
output. Since both the domain and the range of any diag. fn are {0,1,⋯,9}, it is always
possible to arrange a list this way. The number formed will have the desired digit in
repetition. In words, given a diag. fn, there exists an arrangement from which the function
produces a rational number. Weaker version.
Since a diag. fn will have at least 2 conditions, it will produce 2 different rationals for each of
those conditions. Then, there exists at least two arrangements for a given diag. fn that
generate rationals. A bit stronger.
I realized I could make a much stronger version. Take any natural number n. Have the first
n numbers in the list be in a way that their diagonal digits when used as the inputs of a diag.
fn do not create any pattern in the outputs. The rest of the numbers are arranged fixing a
particular output of the function. The number produced thus will have the repetition of the
said output digit starting from position n+1. All the permutations of the first n numbers
(without touching the rest) will produce n! different arrangements from which the diag. fn
produces rational numbers. There exists at least n! arrangements. Now I can choose any
arbitrary large number as n and arrange the list this way so that the diag. fn produces a
rational. Which implies, given a diag. fn, the possible number of lists, all of which are

In light of this discovery, I need to modify the case where I considered only the irrationals. It will
be: when all the irrationals only are being used in the diag. fn (their diagonal digits), there exists
at least one arrangement of the list which will produce a nonsensical irrational number that can't
exist.
While thinking about the modified statement above, something drew my attention about the
original proof (reals are uncountable). The proof doesn't involve separating the rationals from
the irrationals. All types of numbers are considered regardless. And I found a devastating
implication of that.
guaranteed to produce rational numbers through the function, can be made as big as
possible.
You might say that these aren't an example of a random list. The thing is that the proven
existence of at least one list, let alone as many as you want, is enough to weaken the
randomness of it all. Given a random list, then, the diag. fn is not guaranteed to produce an
irrational number.
If the extra numbers alluded by the diag. fn's are rationals, that is not a problem, since
rationals are countable. My initial conclusion about the function continues to hold true.
In current mathematical literature, the rationals are proven to be countable and the reals to
be uncountable. I will only focus on the interval (0,1). Since the rationals are countable, a
list can be created without controversy. I'll use the diag. fn on this list. As I showed earlier,
given a diag. fn there are at least some arrangements of the list from which the diag. fn will
produce a rational number. But wait, where is this rational number? By virtue of a diag. fn,
the number produced by using it, will never be on the list used to produce it. That would
imply you cannot account for this new rational number. You can create as many rational
numbers as you want.
I only realized while writing it that this is just a special case of the first flaw I detected when I
considered all the numbers in (0,1). Using the proof that there exists at least a few
arrangements of any list for each diag. fn that will produce rational numbers, I can generate
rational numbers that cannot be in the interval. Implying that there are rationals always
remaining unaccounted for, making the set of rationals in (0,1) uncountable as well.
I will mention one more special case for future reference. Consider the list of all the
numbers possible to form using only 0's and 1's in (0,1). The smallest number would be
0.00⋯ (that's just 0) and the biggest 0.111⋯ (or in fraction 1/9). Every number in between
made from only 0 and 1. The diag. fn changes a 0 to a 1 and vice versa. If I use the diag. fn
on the diagonal digits of this list, it will produce a string of 0's and/or 1's. As usual, this
number is nonsensical since by construction, I have considered all the possible
permutations of 0's and 1's in the list. Mathematicians can already guess why I chose this
special case and what I'm alluding to.

The behavior of this unhinged function is bizarre to say the least. I was having an existential
crisis on behalf of this function, and doing a lot of introspection about the nature of numbers and
functions and all that. This led me to spot the mistake even more bizarre than the function's
behavior. (I have to mention here that, that bigger physics discussion I had been having, helped
me where to look)
What exactly is this function even doing? Nothing other than telling us which digits to use when
creating the decimal representation. The list is merely dictating the writing order of those digits.
But that's just a typical decimal expression. For instance, take the representation
0.01001100011100001111⋯, I can create a list and a diag. fn which will result in that expression.
This can be said about any and all numbers. The function isn't creating anything exotic. It's the
same hodgepodge of digits for all. And among those, if there exists a collection of digits which
gives an incomprehensible decimal representation, then that would also invalidate all the other
collection of digits as a decimal representation. There is nothing for me to show favoritism for.
The fault must lie in decimal expressions.
Now, proving that is a tall order.
Hello physics people, I want you to pay extra attention from here on out.
Race to 0
There are two pressing issues at the moment: whether there's a way to know if the list is
complete and the problem with decimal expressions. These two birds can be killed with the
stone, 'the expression for the point right next to 0'.
In mathematics, a line segment is considered to be consisted of a collection of points. The
interval is represented with a straight line segment, and a complete list means that it has all the
points/numbers representing them. It is easy to locate the points that represent rational
numbers. E.g. the middle point of the line segment represents 0.5. However, the perplexing
mystery is that nobody knows the decimal representation for the point right next to this point (on
either side). It's not like those points don't exist because the line segment is continuous. It
happens because these numbers represent a distance from 0, and nobody knows how to define
what the distance should be for two points right next each other. Intuition says the distance
should be zero since they are touching each other, but the sum of a number and 0 is just that
number, pointing toward itself instead of the next distinct point. If the points are distinct, their
numerical representations should be distinct too. This is true for all the known points we can
locate, viz. the rational representatives. We can't locate any irrational numbers since their full
representation is unknown; so it is obvious that we wouldn't know their next point
representations either. My idea was to focus on a single known point and gain insight about the
representation of its next point. Understanding one case will explain all the other. I chose 0
because it'll be less messy.

If I were to hazard a guess about the representation of the point right next to 0, I would say it'd
be a non-terminating sequence of digits after a decimal. Mathematicians use the concept of
limit to get close to a point. I don't need the definition in its full form, just this simplified version:
the point 1 is at a distance 1 from 0, 0.1 at 1/10-th of the previous, 0.001 at 1/1000-th of 1, 10
−100
at 1/10
100
-th of 1 and so on (in math jargon, I'm approaching 0 from the right). The sequence
(0.1, 0.001, 0.0001,⋯,10
−10
,⋯,10
−100
,⋯) can have arbitrarily small number of your
choosing. As you can see all those points get increasingly closer to 0; you can move as close
as you desire without any hitch since the line segment is continuous. But whichever point you
choose, however small, there are an obscene number of points in between you and 0. So none
of these points of your choosing is the next point to 0. How do I resolve this?
While tackling the physics problems, I got used to thinking in terms of reference frames: a thing
being farther from something than the distance between two other things, or something being
hotter wrt something else etc. Getting used to the relative sense that something can't be more
without implicating its reference being less—it's a two way street. So when I was thinking about
limit in this case, I did the relative thing out of habit, and a beautiful picture presented itself to
me.
The distance between 0 and 1 is 1. Since I'm writing in base 10, in order to get close to 0, my
next step would be to divide the distance into 10 equal parts and taking the first position. My
current position in decimal representation: 0.00⋯,0.100⋯(me),1.00⋯ . Not close enough, so
10 equal division of the distance between my current position and 0, and taking the first position
again: 0.00⋯,0.0100⋯(me),0.100⋯,1.00⋯ . I want to get closer: I have to repeat the
second step. All the subsequent processes will be the same if I want to continue getting closer. I
noticed getting closer to 0 implies going farther away from 1. In my first step, I moved 9 times
away from 1, in the second 99, third 999, and so on. This is where I got a clear sense of what
was actually going on. I had the whole mental picture, so I was shifting my perspective left and
right (in this case literally). Something funny was going on on 0's side. From my current position,
I am always dividing the in-between distance into 10 equal parts. Which makes my distance
from 0 to be 10 divisions away. In each iteration, I will have to do this. So no matter the number
of iteration, the closest point to 0 remains 10 times farther away from it. No matter what, this
number remains fixed, as if I'm running on a treadmill toward 0, running without going
anywhere. This is the quintessential relative sense of size. You can keep a distance "fixed" and
wrt some other distances, it's possible to make that "fixed" distance seem getting smaller. In this
scenario, from this pov, the distance between between 0 and 1 keeps increasing 10 fold with
each iteration, giving a sense that the closest reachable point to 0 is getting closer to it.
Perhaps (most definitely) you would say that this is not what's happening. The distance
between 0 and 1 is kept fixed and dividing a fixed distance into more and more in each iteration
makes the subdivisions, in any one step, smaller than the previous step. I'm aware of that.
Please bear with me as I haven't concluded anything yet.

In this case, the closest reachable point was seen to be 10 times away from 0, due to the
number of subdivisions made. However, 10 is just a dummy number, it can be anything. In those
cases, the distance of the closest reachable point to 0 (from 0 obviously) will be that particular
number of times far. 10 seems a bit high, so let's try with some small numbers. The least
number of subdivisions you have to do is 2. There is no other choice. You can't do fractions
because fraction is what I'm trying to define here. Subdividing 2 will give a base 2
representation, but that doesn't matter in the slightest. What matters is that you either choose to
stay at 1, or you have to move to the middle to get "closer" to 0. You have to. You can choose to
stay at this new position, or you have to consider going to the middle point between you and 0.
There's no other choice except just jumping to 0. It started with 1 being at a distance 1 from 0
(only one part), but the moment you divide the distance, wrt the new subdivisions the distance
reads 2. And on and on with each step. You want to scream at me, don't you? You want to tell
me that the process of dividing doesn't make it 2, rather 1/2. I understand. For now, I only ask
you to focus on the distance measurement through subdivision count. Try to get this odd
sensation of Sisyphus pushing the rock without getting anywhere "close" to the summit. It is
futile to move.
I'll use base 10 for cleaner representation. The distance between 0 and 1 isn't 1 if measured
using the subdivision count. The numerical distance representation would be 10000⋯ (note
how it is the same as the decimal representation, 1.000⋯, without the decimal). It is dependent
on the subdivision, and since there is no restriction on how many subdivision you can create, it
turns into this absurdity.
Looks neat, doesn't it? That's what I should expect. Except it's nonsense. What if I told you that
the "closest" reachable point to 0 is 1 (on the right)? But you know 'show, don't tell' and what
not. Let's see what's going on.
Since there is no restriction on the total number of subdivisions between any two points, by that
count the points 1.0, 0.1, 0.01, 0.001 etc lose their distinction. Traditionally, their relationships are
as follows:
So the subdivision count should follow suit as it's measuring the same thing. Except what
exactly is the 10, or 100, or 1000 times 1000⋯ , where the 0's following 1 never end?
Conversely, I would expect that there would be 1 less number of 0's in the point 0.1 than there
00000⋯  → zerothposition
01000⋯  → a“close”enoughpointtozero
⋮
10000⋯  → 1’sposition
0.1×10=1
0.01×100=1
0.001×1000=1, andsoon

are in 1; 2 less in 0.01 etc. What does that even mean when the 0's never end? They all have
the same numerical representation according to subdivision count.
1000⋯=01000⋯=001000⋯=⋯=00⋯100⋯
I only put the zeros on the left for a lame attempt to "distinguish", except 0's on the left of whole
numbers doesn't do anything, just how 0's on the right (after all the non-zero digits), after a
decimal doesn't do anything. The only way these can be made distinct if it were possible to
know the number of 0's following 1. Do you think a randomly placed dot fixes all this?
I wish the damage were contained to this.
Can I write the following?
200⋯>100⋯
The only way to make sure the above relation holds if I knew the number of 0's trailing 1 and 2.
Since the subdivision count removes any and all distinction among the points, 0.2, 0.002, 20, or
2000, as well as, 0.001, or 10, or 1000, it also removes the fundamental property of real
numbers: order. I cannot write anything like that. The only relation I can write is: 100⋯≠200⋯
. And since order has been destroyed, there's no point talking about getting closer to anything.
This will be true for all the numbers.
I started this section with the assumption that there might be some flaw in decimal
representation. Then the whole discussion was about showing all the problems risen by them. It
doesn't say anything about the cause. So what is the mistake? Using subdivision count? But
that's literally how all the numbers are represented on a line. How do you represent the number
10 on a number line? You would have to count 10 divisions from 0 to reach the point 10. That's
also how a fraction is defined. That's how we measure any length. It's intuitive. That's not the
issue. I didn't have to go such a roundabout way via diag. fn to show the anomaly in decimal
representation. Anybody could see the problem. But what specifically is causing the flaw?
If you think I'm in the wrong here and that these sums of subdivisions "converge" instead of
diverging like I have shown here, then please proceed to the next section.
If not, then before you read further, I kindly ask you to try to figure out some semblance of what
might have caused all of this. Here are some key points you might want to consider:
What is everything?
Think what a number (just the natural ones) really means, and what connection it has with
the concept of counting.
Length measurement is interval counting; how does this connect with the previous point?
What exactly is measurement? Aka, what does it mean to measure something?

I developed this section in parallel to the previous. The foregoing section focused on numbers
with geometry as a companion; this section presents the converse. This section is a bit hectic
as I have juggled with several closely related concepts. It also reflects how I developed it even
though this is a cleaner version of a much much messier process.
I wanted the point right next to 0, so I needed to get an idea geometrically. It started simple: I
imagined a disk (a filled out/solid circle) to represent a point when taken to its limit. I put a
second disk next to it (on the right from my pov), touching to make them continuous. The
distance measurement would be just counting all the touching points (I'll use and mean points
and disks interchangeably) between any two points. According to this definition, I can say the
distance between two adjacent points is 0, and that at least 3 points are needed to create a
distance of 1. Next I wanted to see how I can fill space by these disks. Common sense dictated
me to define triangular and rectangular areas. Easier said than done. I ran into a problem:
where to place the third disk. It could be either as 'triangle-like' or 'line-like' (according to the
figure).
The triangle-like position made a new problem: all three disks touching one another means
each at a distance 0 from the other two but at the same time they are occupying a space. My
definition of distance makes it difficult to define area: how can an area have its perimeter be 0?
And if I try to define the area being 3, then each point gets an area measurement, whatever that
means (because in the limit it has to be 0, a point doesn't occupy area). On top of that disks
don't fill up space, there's always inter-disk 'space' (means gap here, unfortunately the word
space is used to express different meanings). Granted they should go away in the limit, but if
the behavior is alien at the limit, I don't know how to reconcile the normal behavior with the
limiting behavior. This was not going well.
I experimented with other shapes. For one thing, polygons (solid/filled out of course) do fill
space. Doing this, made me notice something. Take regular hexagons for example. There can
be only 6 of those fitted around a central hexagon. Similarly, 3 equilateral triangles fit around an
equilateral triangle, and 4 squares around a square. The number of these shapes that can be
fitted around a similar shape are size independent. Making them smaller wouldn't allow for
more of the same shapes to be fitted around a central similar shape. This is odd because in the
limit all of these shapes will be points, and there's no restriction on how many points that can
surround a central point (all points touching the center). But what limit? If these shapes were to
be "smaller", what are they getting smaller with respect to? Then I noticed another thing I was
doing subconsciously: I was filling space. Here, I was trying to fill the 2-D space, which enabled
me to create the 'triangle-like' situation. Okay, so why not place the third disk on top (or bottom)

of the two disks? In this case the third one is being placed along the third dimension, and will
overlap the other two (a different complication). Why stop there? I can put the third disk along
the fourth dimension. Well I can't physically do it, nor can I imagine but the observation here is
that the disk can be placed along any direction based on the dimension I want to consider. This
was one of those moments when you feel something is wrong but you can't verbalize it.
My initial two conclusions from all this:
Hold on. I need to make things clearer. Since counting gets involved, I need to define how to
count. I'll use the fundamental intuitive understand of counting. An object is defined/identified
first, then we see whether this object is being repeated distinctly; counting gives the measure of
that distinct repetition. An apple is an object that can be defined distinctly, can be repeated
distinctly, so apple can be counted. When the object called apple is being counted, you will not
get the result of how many oranges are present. So the identity of an object is of utmost
importance to counting. When I say there are 5 apples, it means that there exists 5 distinct
copies of the object called apple, but not potatoes. Water (please consider room temperature)
cannot be counted even though it can be defined because its repetition is not distinct. However,
if I say water molecule, then it becomes countable since water molecule can be defined
distinctly and its repetition is also distinct. The identified object which has a distinct existence
and can be repeated distinctly, I'll call it a unit object/unit. If something is countable, that means
it's being counted wrt a unit object. These are the concepts that cannot be defined and have to
be understood from intuition: being distinct, repeating, understanding the concept of counting,
object. I have considered the concept of numbers under the concept of repetition, but I'll
elaborate. These symbols called numbers (natural numbers) are standard to represent the
distinct repetitions of a unit object: 1,2,3,4,⋯ . From now on by numbers I'll explicitly mean
natural numbers only unless otherwise mentioned. These numbers are related via the
undefined concept of 'having more copies, or having less copies of a unit'. In familiar terms,
these numbers are related by addition and subtraction (but I'll mostly talk about addition). Our
understanding of the distinction among the states—more, less and equal—is also undefined. If
from the 5 apples, you took 2, then the states—total number of apples, the number you took
and the remaining—are all distinct. The number you took and the remaining are both less than
the total number of apples at the beginning. The same can be put differently—the total number
of apples are more than both the number you took and the remaining. This is the relative sense
of size, one will always imply the other. If you give one apple to your friend, then you both have
the equal number of apple. You have the freedom to define a unit object. For example, water is
uncountable, but you can count water as one of many distinct liquids. In this case you changed
the unit object to liquid, wrt which water can be defined distinctly, and each liquid can be
repeated distinctly. Interesting thing to note here is that unlike apple, where all the repetitions of
the information of 'next place' depends on the dimension,
a point, whatever this object is, cannot be approximated with other objects (as in cannot be
thought of an object taken to its limit).

the apples are the still apples, the liquids themselves are different, as in water is not the same
as milk. But since the unit is defined as liquid, their particular names don't matter anymore, as
they are both liquids. To parallel with the apple example, it would be as if you decided to give
each apple a different name.
This is my working principle of counting. As usual I'll modify it if I find any error as I go.
I want to return to the topic of making them smaller. In the limit all of these solids turn into
points: that's the idea, right? You cannot simultaneously shrink all the objects. Because you'll
create a reference frame problem. With respect to what are they shrinking—it cannot be defined
without a reference. So you have to shrink all the other shapes leaving one unchanged. To use
the disk example (in 2-D), a shrinking can be defined keeping a disk fixed. Let's fix the central
disk and shrink all the others that surround it (each touching the center and its two neighbors).
Here, it is possible to shrink the surrounding disks as much as you desire. Meaning the number
of disks can also be increased as many as possible (please remember that the central disk
cannot be counted as it is not a replica of the other disks). In the limit, they do mimic the
behavior of points; however, the surrounding disks turn into the points of the circumference of
the central disk. This can never be changed since the central disk cannot shrink. Which means
this also fails to approximate point behavior of how many points fit around a central point (in 2-
D). The same argument can be applied in other dimensions as well. You can think of this in a
slightly different way: if the relative size of all the disks are kept same wrt to one other, then
even with an external reference frame, the number of disks surrounding a central one will never
change at any stage of shrinking. There has to be a relative difference in size among the disks
themselves.
This led me to the second conclusion, the behavior of 2-D solids do not match with the behavior
of a point. You can see that it applies to 3-D solids as well: only a certain number of solids can
be fitted around a central one. Note here, since there is a count measure present, as in the
number of solids, I don't need to mention that they all are similar (copies) as it is being implied
by the count (you can't count apples to oranges). What about irregular solids? Can they
approximate points? The copies of an irregular solid unit would also surround the unit only a
certain number. Same issue. Sometimes the regular units (e.g. disk or sphere) or the irregular
units might leave gaps when fitted. If those gaps are filled with a different type of solids, then I
have different types of objects and not a particular unit being repeated: counting becomes
tricky. On top of that, in the limit they should all become points. Points are all the same,
meaning a point is a unit. So how do different types of units become the same (apples and
oranges turning into bananas)? Where does the distinction go?
What exactly is a point? Or for that matters, what is a straight line, or other solids? Most
importantly, why is the assumption that everything is consisted of points? If that's true why I
failed to reach point from those different objects?
One more thing before I move on to my next plan of action. According to the working principle of
counting, it might not be far-fetched to say that the real numbers are countable. Since the points

in a line segment (take any line segment and write 0 and 1 on both its ends) represent the reals,
and a point can be considered as a unit, which can be repeated distinctly, it must be countable.
But that's not enough to conclude anything. I was still on the hypothesis of reals being
uncountable. I had had a few hiccups, which led to this investigation. The reason for me to bring
this up is that the working principle I am using is the underlying intuition behind counting. And
following that, it paints a clear picture of what it means to be uncountable. E.g. water or air
being not countable the same way apples are. But the reals resemble the apples more than
they do water or air. If the reals turn out to be uncountable, how would they connect with our
intuitive understanding of uncountable objects?
Since dimensions turned out to be inevitable in understanding point behavior, I shifted my
attention toward it; started with what a dimension is. Intuition says a dimension is an object's
degrees of freedom. A 1-D object has 2 degrees of freedom, a 2-D one has 4, 3-D one 6 and so
on (I was thinking in terms of standard x, y, z coordinates). I was imagining moving various
objects to understand their degrees of freedom in different dimensions. It got me nowhere. So I
changed my perspective ever so slightly. Instead of moving the particular object, I focused on
stopping the object: given an object, what is the least number of objects that need to be placed
touching the target object to trap it, i.e. it won't be able to move without disturbing any of the
other objects? (please keep in mind whenever I am using a count measure, that implies all the
objects under consideration are copies of a particular unit object) The target object is also not
allowed to deform since all the objects are copies of one another, and deforming one will
change all the others. The picture looks like this: a 1-D object has the freedom to move in one
direction, so a replica of the object is placed on that side to prevent any movement, and another
replica is placed on the opposite side of the target to stop its movement in the opposite
direction, thereby locking the target object's relative position wrt the two replicas. Same for
other two dimensions (the ones I can visualize). A 2-D object has two free directions, block
those with 2 replicas of the target and the opposite of those two directions the same way. 3-D
will be the same. You get the idea. I fidgeted here for a while failing to see anything of
substance. But then I remembered that I was trying to solve the mystery of sizes. In 1-D case it
will be length. I wanted to create a length and this is how I did it.
Take two 1-D objects (I'm not yet calling them straight line) that are touching each other at one
end. Here, it doesn't matter if one is a copy of the other, meaning they are of unknown lengths.
The counting 'two' is happening wrt just 'object', nothing specific, as long as they are objects.
From this arrangement, you either keep them right next to each other, implying there's nothing
in between them, or you move one and create a gap. I can treat this gap as a new 1-D object
and replace the objects on its both end with this new one's replicas. What I realized was that I
couldn't create anything "smaller" than this. Because I started with the objects touching,
meaning no gap, no object in between. There's no object to start the comparison with (the
starting objects are dummy objects for visual reference only, remember I am trying to create a
length). So I have to move them in order to create the gap to create a length at all. So "smaller"

than this in the starting step is impossible. However, once there's a starting length, it is possible
to compare.
Look at the figure 'length-formation'. I have three 1-D objects next to one another without any
gap, created by copying the unit I just created above. I add a fourth copy DE , and remove the
object CD (D is just a symbolic representation of where they are touching, I'm not talking about
points here, I'll not talk about points for now). There's a gap and I can move DE toward BC to
create a smaller length than the pieces themselves. The question now is: how do I establish a
relationship between these two lengths? I didn't have a clear verbalization of my argument at
this point, but I knew which direction I was going.
I progressed to 2-D to create an area. I could've just started with an area surrounded by 4 of its
copies, all touching the central one, fixing its relative position wrt them. Using the argument as
before that I need to have at least an area to start comparison with. However I was tinkering
with it mechanically by moving areas, creating gaps, just like the 1-D case. I imagined 4
rectangular areas meeting at corners, all touching, making a '+' (these are dummy areas for
visualization, they aren't copies of anything, count 4 wrt just areas, nothing specific). I moved
one on the bottom right laterally, resulting in a downward slit. This is where I noticed that
moving an area this way always involves both the directions, i.e. the gap is also 2-D.
Remember, the gap is merely a device to create a solid, and it's creating a 2-D solid. This was
all I needed.
I need some working definitions of solids in different dimensions.
A 1-D object is one which has length, and whose freedom can be impaired by two of its copies
on its both end.
A 2-D object is one which has area, and whose movement can be blocked by 4 of its copies
placed on its left, right, above and below.
A 3-D object is one which has volume, and can be trapped by 6 of its copies placed along the
standard x, y, z and their opposite directions.
Solids in other dimensions follow suit, and the volume will be called n-dimensional volume.
Undefined concepts are our understanding of length, area and volume, things touching/putting
next to one another, being in between and our innate sense of direction.
Returning to the 2-D case. Our ability to put those 4 other objects to cease a 2-D object's motion
comes from the fact that the target 2-D object gives the information of where those 4 copies
would be placed. Each of those other ones as well gives the complementary positional
information wrt the target. So a 2-D object is the representation of the directions that make the
two dimension, two dimension. In figure '2D arrangement', if I were to talk about the position of
the solid 'r' in itself, it would be meaningless.

But the existence of the object 'm' gives meaning to r's position: it's on the right of m.
Conversely, r has the positional information of left, which complements m's position wrt to r.
Same can be said about each of these objects. Bottom line is that each 2-D object has the full
description of the directions which can be used to place them accordingly wrt one another. But
is the figure a proper representation of 2 dimension?
A 2-D object is a complete description of the dimension, having all the directions using which
you can place another 2-D object next to it (it doesn't have to be right next to). Tell me then, you
can see the figure, can't you? It's in front of you on the screen, is it not? But each of those
object has a complete directional information of 2-D, none of which is the direction along which
you are looking. None of those 2-D objects define the direction 'away from the screen' (in this
situation where the 2-D is along the screen). More generally, they cannot define any other
direction other than the ones that are necessary to define that 2-D. So the often used examples
of the top of a table or a book, or a page as a representative of 2-D is incorrect. Those objects
are 3-D and the directional information of 'top' is one of its 3-D directions. If those objects were
to represent 2-D along their surface, they wouldn't define the direction "top/bottom of the
surface".
It is quite a commonplace thing to hear that being in higher dimension will allow someone to be
able to look inside of an enclosure in lower dimension. If you are confined in a room, then a
being existing in higher dimension than us would be able to look through the walls without
breaking them. But it's a theoretical impossibility. Say a direction is defined along which a 6-D
being can tell where you are. That will undeniably tell you the direction of where this being is as
well. Still not convinced? Try defining right without implicating left. I'm not even talking about
light's behavior or anything. I'm talking about the fundamental way to define direction and the
reciprocity of direction is unequivocal. Which means you would also know exactly where this 6-
D being is wrt you and be able to turn in that direction. And if my assumptions are correct, none
of us know any other direction that doesn't involve the standard 3-D directions. 3-D objects do
not define the directional information of any other dimension. In fact, it is impossible for us to
see 1-D or 2-D either. Their definition prevents it.
Let's see what other repercussions it has. Continuing with 2-D. A common representative would
be a surface, and in mathematics, there are two normal directions considered coming out of the
surface. That would be incorrect. Which means, it will not be theoretically possible to stack
more surfaces along one of those normal directions to create a 3-D volume. In math jargon: the
boundaries of 3-D objects are not 2-D. Different phrasing: it is not theoretically possible to
enclose any 3-D space using 2-D objects. Generalizing: it is not possible to enclose space in a

particular dimension using objects from any other dimension. It implicates the converse to be
true as well. This is how I can show it. Think of a 3-D volume and cut it randomly. Each piece's
position can be defined wrt any other pieces completely in 3-D by the pieces themselves.
Meaning, there's no other dimension hiding in the object. In words, no amount of cutting or
trying to reduce the size of an object in a certain dimension will produce an object in any other
dimension. It matches my earlier observation of how the solids weren't properly approximating a
point. But I will not talk about points yet.
Let's see what happens when I apply this new insight in 1-D. First and foremost, drawing a
straight line on anything will not make it a true representative of 1-D, and it's not due to the
usual reasoning that any drawing would have a little breadth to it along with the length. It's
because if the 1-D object is spanning left and right, then it cannot represent any other direction,
making it impossible to put on a paper. But what about a straight line going into (or coming out,
you know, reciprocity) the paper? In this case the the direction is defined since the paper is at
one end of the straight line. What I'm asking here is, mathematically put, what happens when
two objects in different dimensions intersect. The answer is very simple, yet I was confused and
couldn't figure it out at that moment. I focused on the things I did understand. I added figures for
some semblance of visualization. None of these represent 1-D. (all the straight lines are
assumed to be stretching left-right)
The conception of 2-D as extending a straight line to its perpendicular direction is incorrect,
as a 1-D object doesn't define that direction.
A straight line cannot be morphed into a curved line. So saying 1-D objects to be 'straight'
line is redundant. It doesn't have any other variant. However, it doesn't prove that the
curved lines cannot exist on their own in 2-D. This simply says that no line can be bent to
create a curve and conversely a curve cannot be bent to a line.
A line doesn't divide the 2-D plane into 2 parts. Same reasoning. As the halves of the plane
will exist in opposite direction of each other, that direction isn't defined by the line.
Lines cannot be stacked along the perpendicular direction in theory to create a 2-D solid.
But that's not the only way to create solid in theory. Rotation is another way: a line can be
rotated to create a circle. You can see that's not possible. Once again, it doesn't say
anything whether a circle can exist on its own in 2-D.

2-D (assuming a 2-D surface along the screen you are reading this):
3-D (where we live and breathe):
Then there is pizza slice created by rotating a line to a certain degree. That is also
impossible.
I was looking at the picture for a little while when the full realization hit me. One of the lines
is existing in a position which cannot be defined by the other line. Implying that two lines
cannot intersect at an angle. I have found many an odd thing thinking about the physics
problems which left me flabbergasted since 2018. But this, when I found this in late 2023, it
left me the most shocked of all. It still has me shocked while I'm writing this in 2025. The fact
that two lines can intersect is the most basic thing in geometry and probably one of the
oldest known geometrical fact. Even a baby learning figures understands this. I don't think
anybody ever questioned the validity of the concept of an angle. Unfortunately it turns out to
be a theoretical impossibility. It is the simplest most boring things nobody cares about that
leave you the most bewildered.
It's clear from all the above that parallel lines are also a theoretical impossibility.
Impossibility of line intersections at angle implies the impossibility of all polygons (convex or
otherwise).
Let's take a quick look at how things are in other two dimensions.
A 2-D surface cannot be bent out (or in) of its plane of existence (doesn't say anything about
the existence of spherical shells in 3-D). Implying that 2-D surfaces do not have 3-D
orientation.
2-D solids cannot have straight edges as perimeter.
Parallel planes cannot exist (one formed by placing two planes along any other higher
dimension).
A plane cannot divide a volume (regardless of its dimension) into two parts.
A surface cannot be revolved out of its plane of existence. Thereby two planes cannot
intersect this way. Implying no angles can be formed this way either.
Previous point makes polyhedra a theoretical impossibility.
A 3-D volume cannot be bent out (or in) of its volume of existence (doesn't say anything
about the existence of higher dimensional spherical shells). Implying that 3-D volumes do
not have orientation in any other higher dimension.
3-D solids cannot have 2-D objects as boundary.

Things I don't know yet: how the 2-D planes or 3-D volumes intersect? As in what does their
intersections look like? Or if it is possible at all. What about inter-dimensional intersection? Not
only these, but also how do the planes or volumes look like? Which shapes are allowed and
which aren't? For example, if I were to think of a cube, why can't I? Polyhedrons cannot be
formed because the 2-D planes cannot be moved out of their plane of existence. But is it
possible for the cube to exist on its own in 3-D? After all it doesn't need permission from us to
exist. And no, still no part of the cube will contain any 2-D surfaces. The tops where you can put
your imaginary hands on is due to the fact it's 3-D and the directional definition is valid. What
isn't valid is to separate it from the solid to represent the 2-D.
For all of this, I need to return to the other half of the picture, what I was talking earlier. Let's
make things smaller.
Magnitude of moving pieces
Before I start, I want to reiterate that in the figure, 'A', 'B', 'C', 'D', 'E' do not represent points.
Generally in math, 'B', 'C', 'D' are considered to be shared points between line segments. But
that's not the case here. Here, a line segment has been copied a few times and they are placed
next to one another so that one's end touches another's. Since copying creates a distinct
segment, no part of it is being shared with any other. They are existing separately. The sole
purpose of the chosen labeling is to make things less cluttered. With this out of the way, let us
start.
I started with a line segment, since without it there's no comparison. Let it be AB. AB has been
copied three times and placed according to the figure 'length movement'. I removed the piece
CD making room for ED to move toward BC. As in the figure I have moved the piece ED, to
newly ED
′
, ED=ED
′
, and since there is a reference now, it will be possible for me to say the
gap can be smaller than the reference length of AB. Let the representation of this new length
be CD
′
. In order to establish a relation between the reference and the new length, I have to do
the usual. The number of divisions of AB will depend on the base number system, but I will use
the bare minimum here as in dividing it into 2 equal parts. This was where I faced the problem.
If I'm saying that the length of 2 lines are equal, then there is an implicit implication of a different
Parallel volumes cannot exist (one formed by placing two volumes along any other higher
dimension).
A 3-D volume cannot divide any higher dimensional volume into two parts.
A volume cannot be revolved out of its volume of existence. Thereby two volumes cannot
intersect this way.
Previous point makes higher dimensional polyhedra a theoretical impossibility.

reference. They are equal to something. In other words, a particular length has been copied to
make sure of this equality. So equal division is equivalent to starting with a length and then
copying it and place them next to each other. But that's exactly what I have been doing from the
start. Division should make something "smaller" wrt to their parents and adding copies should
make it "bigger" than the copies. What eluded me is the reconciliation between these two, 'top-
down' and 'bottom-up', pov's. I felt like I was this close yet I couldn't put it into words. And then I
was thinking about a tomato and everything became clear.
A tomato when repeated creates a count measure. The count gives the number of repetition of
the tomato. However, I noticed how the tomato itself remained undefined and how the repetition
is being defined wrt a unit but also how the repetition is defining the unit. Then I was thinking
about iron. In itself, iron isn't something you can count. But make an iron ball and repeat the
ball, and notice how magically it is gaining a measure: the number of balls. It can be any shape
as long as it is distinct and has distinct repetition. No counting can be defined on water, but
make ice cubes from it and suddenly it has a count measure you can define. Think about clay.
Undefined. Make some clay dolls and you have a measure. The key point being 'as long as it
can be distinctly repeated, it creates a measure'. I used this observation. A length, or a distinct
line segment, itself is undefined, but when you repeat it you have a count measure. Not length
measure yet. For that you have to put the copies end to end. If there are 5 pieces in total, the
number/length 5 is meaningful only wrt to the reference line. The line that is being repeated that
becomes the unit. But only wrt to the repetition. Otherwise no measure can be assigned to it.
So just because I know algebra, I can't start with 'let's assume the length of this line to be x'
without implying that with respect to a unit it has been repeated x times and the copies have
been placed end to end. Both the unit and the number of repetition of it are defining each other
and very much like direction, 'left cannot be defined without implicating right', you cannot define
one without implicating the other.
Let's get back to the main course.
(I'll be talking about length measure, which will imply that the lengths are placed after one
another)
When I have a unit and repeated 5 times and that same unit repeated again 3 times, the length
spanned by the former will be longer than the latter. This is possible because of our intuitive
understanding the concept of 'more' and that there's more of one particular thing. If I started
with two unknown lengths and repeated one 5 times and the other 3 times creating length
measurements of 5 and 3 respectively, then can I say 5>3? Of course not. Because here 1≠1
. There's no way of knowing how much the length of each unit is. The only thing I can say is that
there are more number of copies of the first unit than the second. That's count measure
relation, not length measure, and the count measure is happening wrt to just general length as
unit object. Like if you have 2 apples and 4 pencils, you can say you have 2 objects and that
you have more of one object than the other. Because you are free to choose wrt what the count
happens, aka the unit object. In this case it was changed to generic object. When I was

"dividing" AB into 2 equal parts, I was basically picking some length, copying it twice, placing
them next to each other and calling the whole thing AB. The new length that I just chose to do
this with, calling it "smaller" than AB completely hid the part where I first made AB bigger than
that piece by making more of the piece. If I want to "divide" the new length, I'll have to do the
same thing again. None of these stages are making the newer pieces "smaller", rather, what's
happening is that I'm constantly changing units and then I'm just repeating more of that unit.
When you say you can make a length smaller by "dividing" it, you are misleading me with a
false start, because you hid the real start when you built the length up from some other length.
Here's the big revelation: it is impossible to be smaller than unity. Or in other words, unity
cannot be fractured/fractioned. You need to have a unit to create any length at all. This unit isn't
representing 1 the way you typically think. There is no meaning of 1; unity is undefined. But if
you repeat it, a beautiful co-dependent relationship emerge, where the unit and the repetition
define each other. Let's look at the top-down pov and where the problem lies. If you continue
"dividing" and never ever stopping then it is equivalent to saying that you do not have any
length to begin with. So when looking from the bottom-up pov, it means that there is no object
(because you have to stop, I cannot stress this enough) which is being copied making the
meaning of number 2 meaningless (in general whatever number of equal "divisions" you were
doing). You have to have something that you are counting. You do not have a choice in this
matter. This was the discrepancy I felt earlier but couldn't verbalize. The top-down and the
bottom-up pov's have to be complementary, not contradictory. This answers the irrational
number mystery.
What about the rational number decimal problem? This was the most difficult part to "see",
because seeing/my casual intuition, developed by centuries of mathematics was biasing me
from seeing or listening to what math was actually saying.
You understand 1,2,3,4,⋯ . You also understand that that sequence has no upper limit. And all
these numbers are additive. You can also define 1 to be 1/2 or 1/3 or 1/4 or 1/500, or
1/(totalnumberofintegers) etc—in a multiplicative way (I explained the reason as unit and its
repetition can define each other, in this case the repetitions are used to define 1 as a ratio). But
is 1 related to these numbers in a multiplicative way? In this way adding 1/4 and 1/5 gives 2.
That's exactly what happens when you divide 1 into 10 equal parts; they are related in a
multiplicative way, even though you are counting those parts in an additive way. You are
counting 10 pieces yet saying 1/10, because a unit and its repetition define each other and you
chose to define the unit wrt the repetition. It's not a problem as long as you remember how
counting works, or it's very easy to defy the meaning of counting. Adding another of the new
piece will result in 11 wrt the pieces, but writing as 11/10 wrt 1 will defy the meaning of counting.
Because the object defined as 1 cannot give the counting result of a different object. And if I'm
adding 1 to 1, all the pieces it's consisted of will have to repeat too because the counting is
happening wrt 1 and not wrt the pieces it's made of. You cannot add one unit with another
different unit. Wrt a unit all the other units lose distinction.

This concept might be a bit hard to grasp. Let me summarize it here properly so the coming
explanations become easier to follow. Any measurement happens wrt something undefined.
When that is repeated, a count measure is created. That measure is only definable wrt that
undefined thing; from that the repetition (the number) gets its meaning. Now say you have a
length and that's repeated a certain number of times, you have a length measure (placing them
one after the other). You want to equate it to another length measure with a different number of
repetition defined wrt another unit. Since you never have any grasp of how much the first length
measure actually is, you can equate it to any other measurements because the later
measurements also have the same problem. None giving you any idea of how much they
actually mean. They only have meaning wrt their respective units. This makes the meaning of
numbers completely meaningless since you can say any number is equal to any other number.
It's not a fault of mathematics, rather it's our negligence in understanding what measurement
really is and how it works. Measurement isn't something absolute—it is only meaningful wrt its
unit. That undefinable nature is the most delicate part of it. Handling it callously will result in
paradoxes. I used the length measurement for easy explanation but it works for all
measurements. This is the reason if you start with a unit length, you cannot divide it. Because
looking at it from the bottom-up pov, it will make the very concept of numbers and
measurements meaningless. Don't worry, the rest of the article is more elaboration of this.
I'll explain the above abstract concept using some analogies. Let's use the clay doll. When the
doll is repeated you get only the result of how many dolls you have, nothing else. Let's say this
doll is made out of a certain number of pieces. Say you have 5 dolls and some pieces in front of
you and if I ask how many dolls do you have in front of you what would you say? Would you say
5, or would your answer also involve the pieces? It won't because I asked how many dolls are
in front of you. Let me take the dolls away, only the pieces are kept in front you, and I ask again
how many dolls you have. Wouldn't your answer be 0? Because the doll is being counted. The
doll can be made out of whatever number of pieces for all I care, when the doll is counted, the
pieces lose distinction with respect to the doll. Because counting the dolls gives the result of
the distinct number of dolls with respect to a doll itself, not wrt anything else at all. Now if I ask
about the pieces, the doll loses its distinction wrt the pieces, because counting of pieces can
only happen wrt the pieces. A bit more practical example. Carbon atom has 6 protons, and
hydrogen has 1. If in a box there are 6 hydrogen atoms, would you say there is a carbon atom
in the box? If another box contains 2 carbon atoms, would you say there are 12 hydrogen atoms
in it? No. Carbon can be consisted of 6 protons, but when counting carbon atoms, you don't get
the answer of how many hydrogen atoms there are. Neither when you're counting hydrogen
atoms, do you say n/6 (n number of hydrogen atoms) carbon atoms present. Hydrogen is not
defined as 1/6th carbon, or 1/2 helium, or 1/26th iron. If the box contains 5 carbon and 2
hydrogen atoms, I cannot say there are 5
1
3
carbon atoms or
7
26
iron atoms in total. I can't count
carbon and hydrogen together. Another example to emphasize a different aspect of counting.
This article has a certain number of words when submitted. You can't add or subtract words
without changing the identity of the article. Words are distinct and can be repeated distinctly (as

long as they are words, I haven't said a particular word is being repeated). This makes it
possible to define a count measure on them. Not only that, I can change the number of
repetitions of words (both increase and decrease) indefinitely. This is another feature of
counting measure. However, if words of this article is under consideration, I cannot do that. I
cannot change the number of words in this article (finished) without changing the identity of the
article. Just like how you can change the number of hydrogen, or carbon atoms you have as
many times as you want, but you cannot change the number of protons in either of them without
changing their identity. While the words are distinct in this article, the word count of this article
does not create a well defined unit wrt which the counting can be defined. You can count words
wrt words, you can count articles wrt articles, but words wrt an article is not defined. Two
different units are always independent even though one makes up the other. Whenever a unit is
considered, everything else loses their distinction. And unity is the theoretical lower limit of
counting.
Now I will show how counting also helps establish directions to some extent. I'll start with an
example. Apples are countable. Meaning each addition of an apple changes the total number of
apples. Let's say there are 3 apples. I'll consider this as a state 1. Adding an apple will change
the number to a 4, also changing the previous state. I'll call this state 2. So there is a property,
viz. the number of apples, that can be changed. The apple itself doesn't have a value but put it
after 3, it helps create the meaning of 4. Its presence alone wrt its peers defines the change. So
the apple is the inseparable part of that change. The change will always incorporate the object
causing the change. I can represent this change with an arrow depicting as a direction.
State1→State2
The number is the representative of the change here. Count measure is the bare minimum of all
different types of measurements. I can use this bare minimum measure to represent a direction.
I can now easily represent this measurement in 1-D, where each value (each single apple) can
represent a line, and the value implies change and the change gives meaning to direction, even
though counting apples has no meaning in spatial sense. Now think about points. I will consider
them as distinct mathematical objects first and nothing else. I can copy them distinctly. Meaning
I have the bare minimum property (distinction for counting) to establish a measure. Having
followed the same way as the apple counting, I can represent these point objects as 1-D
objects, where each object representing the value of 1. Going from one object to another
changes a state reflected by their value. Okay, if a point has the minimum property of being
distinct, it can be represented by a value, thereby a change, thereby direction which we can
associate with length from our intuitive description of length. You already know where I'm going
with this. A point is defined to have no measurement in any direction. A point cannot have
length. (the article 'a' is misleading, but I'm grammatically obligated, sorry about that) If a point
doesn't even have the bare minimum for a measurement, I cannot show that the object has any
distinct self. Or in other words, there cannot be established a direction (which is represented by
the value) along which "two" points can be shown to be distinct. I cannot define a count
1

measure on the mathematical object called point. Point(s) with the properties given to it by
mathematics, cannot represent physical space the way we understand it.
For communication purposes only, I'm telling you to assume an object called line, X, spreading
left-right on the screen (in no way it represents a line, just a reminder). A distinct copy of it, M,
would increase its separation by the value of X (colloquially length, I didn't call it a unit since it
is already being implied by the copies) from another copy, P, right next to M, by crossing X.
Such a convoluted (mathematically abstract) way of describing a length. Imagine Another copy,
L, existing out of the screen. If it crosses X to get to the other side of the screen and gain a
separation value of X, from a different replica, C, then I would be able to define a direction (via
the change) by the value of X, along in and out of the screen. But according to the definition of
length, it is not allowed have any other value in any other manner. So a line can also not define
any other direction other than that along its length. In other words, lines cannot be distinguished
along any other direction except their length.
I needed to do it formally since I know it is a common practice in math to attribute properties to
objects that might not belong to them.
This helped me define dimension in a more abstract way. My working definitions:
An object is 1-dimensional if its existence defines 1 simultaneous change among its copies.
An object is 2-dimensional if its existence defines 2 simultaneous distinct changes among its
copies.
An object is 3-dimensional if its existence defines 3 simultaneous distinct changes among its
copies.
Generally, an object is n-dimensional if its existence defines n simultaneous distinct changes
among its copies.
(In these definitions, the identity/definition of the object is of utmost importance) In the
definitions, you still need to know which way to arrange the objects to represent the distinct
changes. So while the abstraction doesn't need to "know", we do need to know to work with
them. Which is why the intuition still remains.
I also noticed how the term scalar quantity makes no sense. Since having a value is enough to
define a change which can be used to define a direction. Direction isn't some disembodied
apparition that can be latched on to a value to create a different quantity. Separating them only
creates redundancy. There's no direction without value and vice versa.
I was having a lot of difficulty imagining space abstractly without being interfered by the thought
of physical space. When I was trying to answer the question if a line can displace any length
(besides its own length) wrt another line, I was drawing it on paper which was providing me an
unnecessary reference frame. My judgement was being severely distorted by it and
subconsciously I kept thinking about 'size' in terms of the paper instead of abstractly. In order to
overcome that I went back to counting. So I'll be counting apples again. Each apple represents
a value of 1 representing the change in the number-of-apple property. These apples can be

placed anywhere in the physical space, doesn't matter, but with an apple, the number-of-apple
property goes up by only one. This was really helpful since the backdrop of physical space is
meaningless here. I can think of the number of apples represent an abstract 'apple-space'
where addition of one apple increases the total number of apples. When I say there are 6
apples, that means first I have to build up to that number by placing them one by one. The fact
that I can reach 'a place' in apple-space, it implies that the place has been built up first, to make
sure of its existence. Slightly different words, the place has been defined. I applied this same
concept to length measurement. Once I have a well-defined 1-D mathematical object (just hold
on one moment from imagining it as a length), it can be distinctly replicated to create the bare
minimum measure. Just like the apples, although they can be placed anywhere in physical
space (to fight off our imagination biases), in that particular object space (the apple space),
each object (apple) increases (when adding) the value by 1 from their counting numbers
property. From this, I only need to add the property of length to make it a length measurement.
These mathematical objects have a property called length, the total value count of which will go
up by the property value of an object with each addition. Note how I wrote 'property value of an
object' instead of saying 'one'. I really want to emphasize the object's 'undefined nature' and
how just by a simple act of repetition you can associate values with them, thereby creating a
measure. See now I don't need to care about the where, since they have to follow the counting
rule along with the intuition of length. The space is built up this way. Whenever I'm trying to put
that mathematical object (the line under consideration) somewhere in that space, the space will
define its position with ease. Because you can't put an object somewhere that isn't defined.
Let's do it in layman's terms: the total length count value will go up by one with each addition
translated in our intuitive sense of length is that the lengths have to be placed one after the
other (end to end). No gap is possible during the build up process since the gap will represent a
different type of object (principle of counting coupled with our intuitive understanding of length
takes care of this). Having the space, we can place the lengths accordingly. We can create
gaps since we know where we are putting the objects. You can easily see that all the distances
(in case you want to create gaps between two lines) will be some integer multiples of the length.
And interpretation of this isn't that length is discrete/quantized as the space created is clearly
continuous (no gap possible during the build up), rather, it is the nature of measurement and
that the existing perception of continuity is theoretically incorrect.
Okay, now let's get used to it. I have three copies of a line, j k l, placed in a way that they create
a length measure of 3 (i.e. end to end placement, I'm not drawing them, their positions are in
accordance with the order of the letters). What it means that if another copy, x, is placed on the
left of j without gap, and similarly another, y, on the right of l, then the distance between x and y
is 3 wrt the defined unit. A reference frame is always needed to make sense of a measurement.
Let me show you what I mean by this. What is the distance between two adjacent lengths? If
you said 0, I understand why you said that. I did the same. I'll use the setup, x j k l y, and try to
find the distance between k l. When I say k l represents a distance of 2, what remains
understood is that it is a representation of the distance between j and y, the reference. This is

how it can be thought: from x, j has to shift 2 units to be next to y, meaning taking the place of l.
Two things: k l's distance representation isn't self-referential. The distance has meaning only
wrt to something. k being right next to l means how much k needs to shift from j, to be in place
of l, or, how much l needs to shift from y to be in k's position. Notice how either k or l can't
reference their own positions (that is obvious but still wanted to point out). Since they can't
reference their own positions, a distance between k and l cannot be defined (in simple words
there is no between, neither k nor l are in between anything, and in themselves they don't mean
anything). Zero doesn't mean undefined. This comes from the property of length. It implies an
enclosure (the reference). It is a general property of the space we are familiar with. Not only
that it is the general meaning of measurement as well, representing a change. So it's a
narrative issue where we aren't so pedantic about reference frames.
Having understood this, I need to slightly modify the build up process of the 1-D space. I just
need to add a reference. I have to start by defining the length. 3 copies are needed for this and
the length between the two copies on the extreme defines our starting length. The total value
count of the separation length between the extremities will go up by the length value of an
object with each addition.
A weird situation arises from the fact how unit objects cannot be fractured. What it means is that
you cannot think in terms of parts of these objects either. In the context of the lines, when we
say lines placed end to end, we visualize that one end of a line is touching another end of an
adjacent line. But I shouldn't be able to point out the end of a line since it implies I'm able to
separate a part of the line from the rest. This is also backed up by the fact that an object cannot
reference itself (if it could, no reference frame is needed to define motion), implying that the unit
cannot reference its own parts. So even though I can assume the unit to be spanned left-right,
there's no "part" of it is on the left from any other "part" (you provide the reference of left-right in
your imagination). It can only reference another distinct object's position and vice versa (since
direction always reciprocates). Although it has the information of left and right which enables it
to reference others but not itself.
This is what helped me understand abstract picture better. Even though I created the 1-D space
from scratch using a length, I could not, for the love of me, justify why a different length (this is a
distinction of different kind; each oranges is distinct from other oranges, but an orange is also
distinct from an apple; I'm referring to the latter kind of distinction) cannot just be placed next to
the starting length. After all, that's what intuition says, right? But it can't be done because a
length (the unit) doesn't have an end to put next to. Okay no. Let me start over. A length is an
undefined object. Any undefined object which is possible to repeat distinctly creates the bare
minimum measure of counting. On the other hand, a length refers to a space between two
reference objects. I can assume the length to be an object and need to repeat it (distinct
repetitions of course) in order to create the bare meaning measure of counting, which will allow
me to associate values with these objects. In order to increase the separation between the
references (in simpler terms increase the length), I need to insert another of those length

objects. But these length objects themselves have no value, implying putting two distinct kind of
lengths wouldn't create any meaning to the word 'value'/ 'measurement'. The length won't
increase as no order can be established without 'value'. My last resort to establish any measure
would be a count measure. Now whichever object insertion would increase the count value by 1
, that object would be my pick. And that's possible if and only if I pick a replica of the starting
separation length between the references. I didn't say anything about placing them next to or
anything. Since a length object can't reference its own ends, the count measure is the only way
I can make sense of the information of 'next where'. If the length is replicated, it also makes me
possible to assign a value to the undefined object, length, as well as the positional value it
references. So only with the help from the copies I can define a meaning of where next is. In
other words, two distinct kinds of objects do not hold the information of next in general. 3 apples
always come after 2 apples, but never after 5 potatoes. Even though these counts don't have
spatial meaning, spatial information was possible to extract from the counting. I missed this
detail while I was building up the space. It was there, if you reread you'll see where I say 'the
total value count of the separation length between the extremities will go up by the length value
of an object with each addition' (since the object doesn't get any length value until it's repeated,
distinct type of objects fail to constitute any length measure). Still too difficult to imagine? Okay,
think of it in this way: the starting length being undefined means you don't know how far you
need to go to reach its cozy end, and since the object can't reference itself either, you have
nothing. So no, you cannot put a different kind of unit "next" to it. Notice how this also forces the
reference objects to be the copies of the length too (I didn't start with that). And this was the
final nail in the coffin of imagining 1-D objects with lines. Because those lines were the most
confusing parts biasing me with false reference. Also, our intuition wasn't incorrect. Our intuition
has been shaped from drawing lines on paper or sand, or screen, or imagining sticks as lines.
Everything we interacted with are made out of same unit objects, viz. electron, proton, neutrons.
Of course they make it possible to add lines of "different lengths". But when I talk about two
distinct lengths as mathematical objects, our intuition wrongfully applies the "length addition"
part from real life forgetting the nuance that the building blocks of real life are all the same. Our
intuition was just shallow. All of these follow in all the other dimensions as well.
So what does happen when there are different 'types' of mathematically distinct units?
Something very interesting. Each object has its own distinct space and they can coexist without
any problem. They are not against some common backdrop of space where they are fighting for
their places. No. One is oblivious to the others' existence, not affecting in any way. This is
extremely common in the real world: it's not like part of an electron has mass and a different
part reserved for charge. Everywhere in this universe gravitation and electromagnetism coexist
without vying for some "physical space". It is possible for different types of units to exist in
combination. From the previous discussion it is obvious that if there are different types of units,
then they are non-additive. Please take this literally, by non-additive, I am not implying
multiplicative or divisive; the only thing I can say for certain is that they are present in a non-
additive way. The amount of the constituent units doesn't matter. Their mere existence is

enough. No one unit can be expressed in terms of the other unit since their distinction is
independent. You can say this is like a combined unit—combined in the sense that they are
always present together and the object's behavior is affected by all of them under any
circumstances. One close real life analog: if the unit of electric force is a combination of two
charge units (at least two are needed, or else you do not have reference) and their distance
unit, then all three would be combined in a non-additive way. I cannot derive the actual
expression from any of this but I can tell which way they should not be combined. Electric
forces, charges, distances individually will be additive as usual (same units), but not the way
they combine. So you can tell the superposition of electric forces. In the second chapter of
Introduction to Electrodynamics by David J. Griffiths [u], the author writes (as a footnote):
The principle of superposition may seem obvious to you, but it did not have to be so simple: if
the electromagnetic force were proportional to the square of the total source charge, for
instance, the principle of superposition would not hold, since (q1+q2)
2
≠q
2
1
+q
2
2
(there would
be cross terms to consider). Superposition isn't a logical necessity, but an experimental fact.
Actually, superposition is a logical necessity and the expression of the electric force will not
include the sum of anything else besides itself (when adding multiple forces of the same type).
It comes from our fundamental understanding of counting. All measurements are just counting
with a specific condition attached representing the property being measured. We do not
understand how else to measure.
Electric and magnetic forces add because they are the same thing. What about electromagnetic
and gravity? Do they add? You might say, of course, they are both forces. That's true that they
add. But not because of that. The problem lies in the narrative that you think you can measure
them separately. Except if I were to follow physics' model of these forces, they are omnipresent.
And everything that has ever been experimented with, interacts with both of these forces.
Meaning, all the measurements have always been influenced by both of them. Implication is
that you have always only worked with them combined, rather than them individually. Which is
why you can add. But you do not know how they actually combine. Whether it is possible to
separate the two is an entirely different story which I'll not be talking about here. (Funny how
physics is all about combining the two while I'm asking the exact opposite question of how to
separate them. Also you cannot use the reasoning 'gravity is a very weak force', since the
"weakness" of it doesn't matter at all; mere existence of the constituents is enough)
Big caveat: mathematics has glaring flaws when it comes to representing the real world. So
please do not use math to explain away physical phenomena. I'll elaborate this later on.
I want to give another practical example of combined unit to make another point. Let's mix
water and milk. Their individual amounts needed to make the mixture is irrelevant since the
constituents lose their individuality wrt the combined unit. Say I mixed m volume of water with n
volume of milk. Then the amount of water and milk wrt the mixture are
m
m+n
and
n
m+n
respectively. Note how the individual units are being mixed in a non-additive way (ratio in this
case). Adding water and milk, two apparent different units, is possible because they are both

made out of the same base materials, viz. e
−
,p
+
,n. Which is why the total mixture of the
volume increases too. But this method will not apply to truly distinct types of units. Based on
this, I need to modify these axioms (these are common axioms in both math and physics):
There's another situation I want to talk about. I'll start with an example. I'll assume everything is
made out of e
−
,p
+
,n. Even though all the elemental atoms are made out of those, each atom
acts as if they are units. The counting rules apply to them. Not only that each compound
molecule made out of elements also acts like units. If you put hydrogen and oxygen together,
then their combined gas volume does increase additively (they are made out of the same things
and measurement is wrt 'gas volume', so it is defined), but that doesn't say anything about
water though. Water is separated. However in order to create more water you do need more
hydrogen and oxygen, so water is dependent (as expected since these are not mathematically
distinct objects) on them. At the same time you can just treat water on its own without having to
worry about its constituents like a true unit. These types of units, which aren't really units but
made up of some same type of true units, I'll call them 'pseudo units'. Mathematically I don't
know how to create them: it has to be mathematical equivalent of chemical reaction. What I can
tell though is that the units don't just add up to form these pseudos—the relationship is once
again non-additive. You can't have a 6p
+
,6e
−
,6n in a box and call it a carbon. Only under some
special circumstances they can be shown to be additive. E.g. the gas volume or water milk
mixture volume etc. The special circumstance is that you still need to change the unit of
measurement as gas volume or liquid volume which is made possible by the same true units
they are made of. The rest is the same. You always need to add the mix in the same way. If you
change that you have created a new pseudo unit. If you mix the new mixture with the old, you
have an entirely new pseudo unit etc. The bottom line is that extreme caution is needed when
selecting a unit. General rule of thumb would be that units don't mix as we desire. Pseudo units
Two objects cannot occupy the same space. First I need to modify what space means in
this context. Proper way of saying it would be an object cannot be in another's place without
replacing the second object. Now the modified version: a unit object and its copies are all
distinct in the space defined by them. They are all in the same space, but, in this space, a
copy cannot be in another copy's place without replacing the second copy. Distinct types of
units define their own spaces, neither of which are dependent on any other. More than one
type of unit can be combined in the sense that they are always present together keeping
their individual spaces distinct. In other words, the combination of different types of units
does not create a common space upon which the individual types of units become
dependent.
The whole is always bigger than its parts. Modified version: comparison is possible if and
only if, the objects being compared are built up from the same unit. The comparison is done
via checking the number of repetitions of the unit. All objects are built up from something,
called a unit object and a unit object cannot be fractured. It is the theoretical lower limit. Two
distinct types of units cannot be compared.

also define their own space. Their relation with the space defined by the units are very
interesting (since you already know these aren't additive) and I'll be talking about it after I finish
discussing the unit spaces.
Understanding this I'll talk about one common math practice. Imagine a solid sphere. If
someone asks you to imagine a different sphere that is twice the volume of the first sphere,
what would you imagine? Two of the first sphere or a singular bigger sphere? I am assuming
the latter. You can't do that with unit objects. If you want to represent a volume twice of some
unit, it needs to have exactly two copies of the unit volume. This is very important. If you
replace with one volume then you have created an entirely different unit with no relation to the
prior. The definition of a unit is of utmost importance. You can never turn a 2 into a 1. These
numbers are meaningful only within their own context and their description is complete.
Mathematical relations are permanent. In real, if somewhere there is a 2e charge, it doesn't
mean there is one big electron with that amount of charge. There are two electrons there. If an
object alone has that charge, it will not be identified as an electron. Identity matters the most.
So if you were to create a pseudo unit from some true units, the pseudo unit will always contain
the true units in that exact number and the pseudo unit cannot replace it with one big different
thing. Even in pseudo unit the true units remain distinct.
I noticed another thing. Traditionally, different lengths can be constructed using geometric
relations. E.g. right angle triangle sides relation. Then I observed how an irrational number
always associates with the measurements of different types of 2-D polygons—measurements
defined by lengths. I know now that those shapes are incorrect in theory. I realized how
irrational numbers were math's way of telling us that there were some conceptual errors (not
that anyone saw any problems with irrational numbers to understand that there were any
conceptual errors). But yes, if there is some conceptual errors they will show up everywhere. I
concluded that no 1-D measures can be used in measurements of objects in different
dimensions. It will lead to paradoxes. Generally the relation
(volume)
n
∝(length)
n
, n=n-thdimension
will not hold.
This tiny sentence 'unit objects cannot be fractured/fractioned' is so dense with information that
it feels like I still haven't fully unpacked it (even after writing this article).
If a unit cannot be fractured that means, for units in all the dimensions higher than 1-D cannot
have perimeter/boundary. Because the unit cannot define its own inside. Generally, the
narrative is that the boundary encloses the object "inside". But an enclosure can only be
defined wrt some references. The object can't be its own reference. Its existence is distinct from
the references. Weirdest implication of this: when two units are in front of each other, they are in
front of each other in their entirety, or each unit "sees" all of the other unit as neither of them
has any "inside". They are completely exposed to each other even though they are solids. Units

cannot have holes either because that would require a fraction of the unit to be discarded. In
plain language, it implies that a unit cannot be any closed loop (2-D) or shell type enclosure (3-
D or higher dimensional shells) (you can show it conversely: assume a hollow shell to be a unit;
since units cannot be fractured, along with its inability to reference itself, you'll fail to define
where the hollowness exists). Since the units do not reference themselves, nothing can be said
about their shapes. If electron is considered to be a unit, there's no way to know what its shape
is. It also answers my early wondering if a cube can exist on its own right. For the same above
reasons it is impossible to define rotation on unit objects, i.e. unit objects cannot rotate on
themselves. An electron can't rotate on itself like the earth does. Unit objects cannot be
reflected either. Unit objects cannot intersect with their replicas. It cannot be defined.
Speaking of intersections, let's talk about it.
Two 1-D objects cannot intersect at an angle since no distinction can be established along
the direction needed for the intersection rendering the significance of 'two' meaningless. Do
two 1-D objects meet at a point (when placed along their lengths obviously)? I already
proved point(s) do not exist anywhere. I need to modify. Can two 1-D objects share anything
when they are placed along their length? Since replications of a unit are distinct, no part of
an object share any part of its replica. They all exist distinctly or else the word distinct would
lose all its meaning. So no two 1-D objects share anything when they are next to each
other.
What about 1-D object with a 2-D one? The answer is obvious. I'll still explain. All of 2-D
space is defined by a starting area. If a 1-D unit were to exist in that space, that would imply
that a direction is possible to define on it that is impossible to define (a value other than its
length gets defined). In fact, no object in any certain dimension can exist in any other
dimension. Or in other words, no one dimension is embedded in other dimension. It
should've been obvious from all the foregoing discussions but I discovered it in the context
of intersection. If you have difficulty visualizing, just notice how in the space we walk
around, any place here can always be defined using 3 directional sense from anywhere
else. There's no place here you can point to where it will have any loss of directional
information. The very act of "pointing" will foil any and all such attempts. This is what I said
earlier: 'no amount of cutting or trying to reduce the size of an object in a certain dimension
will produce an object in any other dimension'.
I can also show that you cannot even imagine any lower dimensions either. Here's how.
Can you imagine a line for me? How did you imagine it? Something in front of you stretched
left-right or up-down etc? Can't be. There is no orientation of a line. Perhaps you imagined
looking at it through one of its end? So you imagined something like a point. Which means
you have left emptiness surrounding it. You can't define surrounding of a 1-D object. You are
left with nothing. Similarly you can see for yourself that imagining 2-D is also impossible. In
fact you cannot even imagine a true 3-D unit as a volume since it has no shape, no inside,

I need to briefly talk about the space build up in 2-D even though it's almost similar to 1-D.
There are some nuances introduced due to the presence of more than one simultaneous
distinct change. From 2-D onward, all the other dimensions will have the same nuances. The
usual things are usual: only one type of unit define a space for its copies, no two distinct type of
units can be placed "next" to one another, distinct type of units are independent. Let's start as
usual by defining the area. It will be the area defined by its copies on 4 sides, left, right, above,
below. In the figure '2D space build up', 'm' being defined by 'l', 'r', 'a', 'b' (always copies of m).
These obviously aren't "disks" since unit objects do not have any shapes, and the figure is
incapable of representing 2-D. This is purely an attempt for me to communicate with you in a
simple way.
From that, the total area count of the enclosure can be increased by 2 through 2 distinct
changes using m by putting n1 and n2. In the abstraction, I don't need to specify the directions,
since the directions will be defined by those 2 simultaneous distinct changes that m's existence
creates. Notice how I talked about the movement in terms of area measure. Traditionally this
would be described as some "linear" shift and the measure would be a length measure. But
lengths have no meaning in 2-D. By the way, if I put n1 in a position where the area between l
and r increases by 2, that wouldn't be considered 2 distinct types of changes. I'll explain in
layman's terms: since the change represents the direction, n2's existence represents increase
on the right side wrt m, and wrt n2, m represents the left side increase. So putting n1 on the left
of m wouldn't be considered a distinct change since n2 already does the same job. Now let's
focus on n2. It is a 2-D object, so it can be used to define 2 distinct changes. Here's the problem
though. n2 cannot reference itself. So organically I can only put another of the copies
somewhere (e.g. on the right of n2) to define an increase in area between m and r. But not
anywhere else since n2 itself cannot define it. If you know where to put it (i.e. above n2), that's
due to the fact that you are another reference frame who can see the whole system and the
screen is providing a backdrop (which is extremely misguiding). You are not one of the pieces.
This isn't a big issue for math however. Since these are areas, you can always associate each
with a reference. Yes n2 might only be "bound" by m and r wrt us, but since this is an area,
no boundaries. You can only create shapes using them since we understand 3-D
positioning. Our field of imagination is purely 3-D.
Since units do not intersect, if two objects (made up of units) intersect, they will share a unit
or some integer multiples of the unit. An analogy would be how the electrons are shared in a
chemical bond as if the atoms are intersecting. The electrons don't get cut.

whether we want to acknowledge all the references or not, it always is being defined by all the
references necessary. The space is defined, these are copies, so they automatically place
themselves accordingly. I can always imagine a reference like a2, wrt which I can place v1 and
v2 organically in the context of mathematics. I'm sure you can guess I brought it up because it
does something freaky. Before I get to that, I want to lay a little groundwork.
For starters, there are no "gaps" when the whole space is considered unlike the figure
suggests. It is continuous. In fact I can claim the human brain is incapable of conceiving the
idea of a discontinuity/an isolation (I can prove this, but not here of course). Let's talk about
positions. Units can see each other fully having no inside or boundary. If there are only two
units, they will always be positioned in front of each other. There will be no other position. Do
not try to think against the backdrop of anything. It might not be easy to imagine, but you'll get
there. This would imply l, r, a, b are all in front of m (the first diagram). But, l is at the back of m
wrt r, or r is at the back of m wrt l. So I need two reference frames to make sense of the
opposite direction since m cannot reference any direction by itself. Similarly, a is at back of m
wrt b or vice versa. The ability to switch between the two pov's, as in 'a is at back of m wrt b or
vice versa', is solely due to the fact that we are providing another reference wrt which the whole
system is "in front of". Let's be each of the piece and see from their perspectives. m "sees" all
four at the same time. So it will be impossible for m to rotate (because the four always are in
front of m and m has no parts). I mentioned this earlier about unit objects. Each respective
piece also "sees" m. But r can't "see" l, nor can a, b.
Now the weird part. This time you are not allowed to add pieces to create references. I will only
talk about the pieces present in the figure. Do you think r can "see" a? Remember you cannot
join a "line". There is no "line of sight". If I were to create something akin to light in this scenario,
then the direction will be defined by the change. Direction will not be on anybody's whim. You
saw that in the elaborate preceding discussions. Summarization will be that change always
happens in front of the unit. That is the only definable direction. So the direction of the
equivalent of light will be defined by that. Following this, I can say r can't "see" a or b. Same
with all the respective pieces. m can't "see" a1 or v1 (the third and fourth diagram respectively).
I can't define "diagonal" since I have lost the concept of angles—change isn't defined in that
"direction". However, if I remove all the other pieces and only consider m and a1 or, m and v1,
they will be "visible" to each other since they are the only two there, so they will be in front each
other. Do you see how unit objects' "visibility" literally depends on the number of references, i.e.
the number of units present? I'm tempted to say that if I started with m and a1 then the space
defined by them would look different. But I can't say that because a1 being at the position a1 is
only due to the current set-up. a1 doesn't define an absolute position and the reason I think I
can put it there is because of the background that's giving me reference. You aren't entirely free
to put anything just anywhere and then connecting the dots as coordinate geometry promised. If
something is somewhere then it first needs to go there. There are other weird things. If I ask
you to switch r and l in the first diagram, you can do it without a problem. What you didn't take

into account is that your ability to do so comes from the fact that you and the screen provides
two more references that cannot exist. In the first diagram, where there are exactly 5 units, l
and r cannot switch wrt m. I need more references for that. Same for a and b, wrt m. In the
fourth diagram, a switch can be defined between m and v2 wrt n2. This is how: in the current
position, n2 is at the back of v2 wrt r; at the switched position, n2 will be at the back of v2 wrt l.
By the way, all these eccentricities are also present in all the higher dimensions.
When there are only 2 units, no other references are present, it is impossible to define a
rotation. This is true for all the dimensions. Oh this one's the worst and the most important in
terms of implication. Since there are only 2, the one and only conceivable direction is toward
each other. As the units cannot reference themselves, even the-away-from-each-other direction
isn't defined. The only definition is that they are in front of each other. As before, I'll assume
proton and electron to be units. My question would be: how do these two physically create the
hydrogen atom? The current model suggests that the electron orbits around the proton. In this
description, no other reference frame is utilized. The description is complete. But that's
theoretically impossible. You might suggest that the electron is a wave. That's immaterial. You
still have to define a surrounding wrt the proton and the proton itself cannot reference anything
making it impossible for any one object's motion to surround the proton. Maybe quantum is too
complicated. Let me give you something classical. The sun and the earth are by no means unit
objects. They are made out of them and for now, let's also assume those constituent units (think
they are atoms) are aggregated volume wise (in simple words, volume units are piled on to
create the earth and the sun, so units are additive here). I will also consider them to be rigid, i.e.
they are not breaking apart—the relative positions of the units wrt one another are fixed. Now
remove everything else from the universe. These two are the only two objects left. Would the
earth still rotate the sun? If so how can you prove it? Once again, there are no other objects
present, and you cannot break the earth or the sun to create other objects, since by
assumption, they are rigid. Solely these two objects present. That also implies that you cannot
observe the system as a third object. You either have to be the sun or the earth. Can you still
define an orbital motion?
I will not answer those questions (there are some other nuances in the situations in the question
above that I'm deliberately leaving behind since this is still a math article). But I will discuss
some things. Humans figured out the earth's orbital motion around the sun by observing other
celestial bodies. They used those as reference frames. If the earth moves around the sun, that
means that its motion is constantly being defined by all those references. Implying the earth's
orbital path is by no means an absolute thing. This is a common narrative in physics that the
orbital motion of the earth isn't much affected by the objects beyond the solar system due to
physics' favorite line of argument 'gravity gets weaker with distance, hence negligible'. Except
magnitude plays second fiddle to the concept of direction (reference frames). Also that narrative
assumes magnitude to be an isolated phenomenon from direction. Everything needs to define
earth's trajectory immaterial of how far they are. If even one of the references stray, the current

path will also need to change accordingly. What I'm getting at is that the distribution of the
objects is hugely influential in determining the motion of any of those objects which in turn helps
us measure forces associated with them. We first need to build up to that distribution. Physics
would say that a particular distribution is the result of a force (assume that there's only one
force involved) among objects. At the same time how the force acts depends on the distribution.
So which one comes first? (not going to answer this one either)
I'll go back to rotation. I'll focus on 2-D since all the other dimensions will follow. Along with the
fact that units cannot be rotated, there's another thing about axis of rotation. Clearly I cannot
use lines anymore to represent them. In addition, I cannot define an "axis" coming out of a
plane of existence either (i.e. if the "plane" is along the screen, then no "axis" can be along the
in-out direction of the screen). Rotation can be defined wrt objects. Here's an outline of what it
might look like:
In the figure 'rotation', x rotates around m wrt 1, 2, 3, 4 (these are at relative rest with m). In the
first step, m and 2 are both in front of x (different phrasing: m is at the back of x wrt 2). x is
defining the change (by being in front of m) from m toward 2 (since 2 is in front of m in the
absence of x). If I want x to move wrt m then I can only move it to a place where m defines
change which becomes distinct wrt the references (without them, x can only be in front of m
and cannot be moved). So in the second step, x moves in front of 1, the other front of m (just a
reminder, m isn't referencing, it can't, 1 is). And the process goes like that. (As usual, the space
is defined, so whatever distribution it starts with, was first built up to that accordingly, and as I
said it's an outline and not a true representative). Notice how m can always observe x. But it
has a price though: x is never moving wrt just m. The movement is always happening wrt the
external references with m as an anchor; m is the one plays a pivotal role in where x should go
next. However, the reference units cannot fully track the trajectory of x. You can see how
depending on the observer the same situation can be wildly different.
It still has a problem. Yes, m defines and decides where x should go next but how does m
disseminate that information to x? But this isn't within the purview of mathematics. I just wanted
to bring it up for the physicists.

Let's add a new reference 5 as shown in the figure 'rotation with new reference'.
You'd think that the reference 5 would help trace out the rotation better more "continuously". But
it won't. You can explain it in two ways. One way, the anchor of the rotation is m, so the
changes have to include m. Even though the position shown in the figure is valid (5 is in front of
x, so it defines the direction x can move), x isn't in front of m anymore—the change is not
happening wrt m. The second way, in the new position, x cannot move in the direction shown in
the figure since there is no external reference and x isn't self-referential.
The space is defined but that doesn't allow an object to move freely. It needs guidance.
References are those guides.
Non-sequitur: In traditional math, the order of rotational symmetries of a square is 4, and that of
a cube is 24. While the axis of rotation of the square is defined at the center of it existing out of
the plane of the square in 3-D, the axis of rotation defined on the cube does not exist in 4-D.
The rotation of the square happens in 2-D wrt an axis existing in a different dimension
altogether from its own plane of existence but the 3-D cube's rotation and its axis of rotation
both exist in the same dimension. Why this inconsistency? Has anyone ever tried to define a 2-
D rotation of a square wrt a 2-D axis of rotation consistent with the 3-D case? If you try to rotate
a square wrt a 2-D axis, the axis will pass through two of its side and you have to rotate the
square out of its plane. Something has to be in 3-D. Why doesn't it happen in 3-D? Why doesn't
the cube fall out of 3-D when rotated, like in 2-D? I don't know if anybody ever tried to apply the
logic consistently, but if they did, they should have at least noticed some problems regarding
the way mathematics handles dimension.
Let's see how a rigid body rotates. The constituents of a rigid body, by definition, is fixed wrt one
another. Here, the units are aggregated to create the rigid body, i.e. units are additive. The
figure 'rigid body rotation' shows the rigid body and 1,2,3,4 as the references.

The setup has been built up as usual. I'll try to define the rotation of the rigid body in the current
position in first diagram. Notice something first: in the absence of the references, the rigid body
itself cannot rotate (because m1 and m2, as well as m3 and m4, cannot interchange between
themselves(pairwise); at the same time m1 and m4, as well as m2 and m3, cannot define each
another's position (pairwise)). In presence of the references, however, it still doesn't rotate the
way we think it should. m cannot rotate under any circumstances, implying that the outer layer
has to rotate in tandem for the whole object to rotate at all. The third diagram shows the rotated
body. You might think that it's violating the premise of a rigid body since part of the body is
rotating wrt to another part of it. Here's the fun part: even though that's what's happening wrt the
external references, at no stage of the rotation any of the units are changing their relative
position wrt one another (notice the use of different references). Because the body's
constituents will always be at fixed relative position wrt one another in the absence of external
references. That's the first thing I drew your attention toward. And it doesn't violate the definition
of rigidity if the position is changing wrt something else. About the mechanics of the rotation.
The corner pieces do not "see" the external references since their changes are defined along
the sides of the rigid body, e.g. the upper right corner piece is in front of both m4 and m2. It has
to be maintained since the body is rigid. The middle diagram shows that the reference piece 2
comes in front of that piece. But that can't be since the corner piece can only "see" pieces that
are along m4 and m2, nothing in between(in other words, the pieces have to be along m2 and
m4 to be in front of the corner piece). So the middle stage is not defined. If the rigid body is
rotated wrt the arrangement of external references depicted in the figure, it goes from stage 1 to
stage 3. You'd think it "jumped" or showing "discontinuous" behavior but you're oblivious to the
fact that you are an observer that cannot observe 2-D—a 2-D rotation doesn't happen wrt 3-D,
thereby your intuition is misplaced. Also notice, how the object rotates and doesn't rotate at the
same time. It is entirely observer (the references) dependent. You are not one of its observer.
And the observers can't be just anywhere.
Doing this drove me insane. If you feel dizzy reading this, I want you to know you are not alone.
Key takeaway from this: the definition of space is not sufficient to define movements of
objects, the references are indispensable, and with the increase of the number of objects,

more reference frames come into play and they change the way objects move. Something I
alluded to earlier about how the distribution matters. It is one of the many confusing matters
about quantum mechanics: how an "ordinary" object, when reduced to a certain size, behaves
differently. Better way to put it would be: with each aggregation of the units, the object created
in each of these step is in some way different from its previous step. It is never the same even
though it seems like "just addition". The 'macro' object's classical behavior comes from the fact
that it is much more difficult to change references of macro objects than that of units. Because
the very make-up of macro objects (that they are an aggregation of units (simplified)) provides
an enormous supply of references. I will not say more than this here without turning this article
into a full blown physics discussion. Our intuition is misplaced, not that it is incorrect. But wait,
please do not use mathematics to justify physical phenomena. I can say it is an outline of an
argument; not the complete argument. I will talk about one more bizarre observation before
explaining what math's shortcomings are.
Let me ask you this: in the figure, what does the outer layers of the rigid body "see" in the
absence of the external references? You might've said something along the lines, on one side
they "see" their fellow units, i.e. the inside and on the other, they "see" emptiness. I thought so
too but immediately realized that's my pov wrt the screen, not theirs. They don't have those two
references. I went back to the way two units (of the same types obviously) "see" each other.
They "see" each other in their entirety. Translating in our language to have any semblance of
understanding would be that it's as if they are "surrounded" by each other. Even though there
are only two, the "field of vision" of each is "filled" with the other. This is only for our
understanding, which is why so many air quotes. A unit cannot reference its own surrounding. I
can apply this to aggregated objects. In the absence of external references, the units in the
outer layer still only "see" the object (aka their neighbors) as if they are "surrounded" by the
object even though they are the "outer layer". Seems counterintuitive, no? That's because you
are not an observer of 2-D. So let me use something where we as observers are properly
defined.
The units cannot reference any direction themselves. However, what a unit's existence does
say in the space defined by it, is where another unit is not allowed. It's like 'hey this seat's
taken'. It references its existence via a negation. The objects made out of aggregating the same
units continue to preserve this reference, which gets you to the axiom how 'an object cannot
take the place of a second without replacing' (I used the modified axiom). So when there are a
bunch of such aggregated objects scattered around, in the space they created, each of these
objects can be thought of as the boundary of that space. Notice how it subverts the way we
think of boundaries as nice clean "lines" and/or "surfaces". Let's apply this to our universe.
You'd immediately notice how everything in the universe acts as its boundary. To make matters
simpler, I'll just consider the boundaries of each of these aggregated objects are the boundaries
of the universe, such as the surface of the stars, or planets or asteroids, or us when we are
outside under the sky and not inside any enclosure (a room), or the trees, or animals etc; you

get the point. Since boundaries of each aggregated objects can only "see" the boundaries of
other objects (think how each unit can only "see" the other units), it is impossible for an object
(unit or otherwise) to not be surrounded by all the other objects in the universe at all times.
Even if I take away the simpler version, this will hold. The simpler version is for easy
visualization. It means no matter where you go, however far, you'll always be surrounded by the
objects in the universe. Everything always will be "inside" the universe. Note how I used an air
quote. Because the concept of inside or outside is an example of misplaced intuition. All you
can say for certain is that 'nobody can take my place; everything else is elsewhere where I'm
not'. Please do not think that being surrounded by means the object "distances" vanish.
Everything is surrounding you from "afar"; in technical terms, 'surrounding you from where you
are not'. Also, this is why you'll always create paradoxes if you try to define the "shape of the
universe" like the way you envision the shape of everyday object.
With all that, let's see how much mathematics is capable of representing the physical world. I
noticed it before but a formal argument formulated when I understood how two distinct types of
units cannot define each other. Physics' assumption is that all phenomena can be represented
wrt a general physical space. It was obvious to me how that is impossible given that gravity and
electromagnetism are two separate things. The concept of general physical space from math is
used in this way: mass is a property that resides inside a volume. So there is another unit at
play here, i.e. the volume.
In math, the volume units can be placed next to one another, permanent and stable. They do
not have any set behavior from the outset; they act however we want. Any volumes can be
placed next to another without needing a "glue". Physics' premise is not so. The behavior of all
the objects in the universe is assumed to be dictated by some sort of force. If two objects are
placed "next" to one another, they need a "glue" (force) to keep their positions fixed. So how
does a mathematical volume represent a physical object? By considering force to be another
mathematical object that is applied on those volume objects? Except this creates a direct
contradiction. According to math, volumes do not need any force to move around. Physics says
they do. Which one should I follow? Because even when I'm using force objects to move
volumes, it does not take away the fact that the volumes can be moved without the force. If I
hold on to mathematics, it invalidates physics from the outset and if I follow physics it
invalidates mathematics. There's no reconciliation between existence and not existence. It's as
if trying to prove 1=0. Also I don't understand physics' pov. Mathematics is imaginary; I get it.
But physics is all about physical proof. Can you prove the existence of a mathematical volume?
I can prove that you can't with one sentence: a volume, by definition, doesn't interact with any
force for anyone to experimentally interact with a volume. Then how can any physical object be
defined using mathematical volumes? In other words, if volume and force both exist on their
own separately, where does a relationship between them come to existence? So trying to
represent gravity and electromagnetic force as a function of general space will always lead to
paradoxical situations. It makes the existence problem (how everything comes to exist) worse.

Let me show you how following mathematics distorts our interpretation of reality through some
famous experiments.
Electrons, and alpha particle passing through gold foil means atoms are mostly "empty".
Except that's not the case at all. An electron moves in a space defined by them, alpha
particles move in a space defined by them and the gold atoms define a space for
themselves. An alpha particle (He ion) and a gold atom are two pseudo units, meaning they
are not volumetrically additive (i.e. 2 neutrons and 2 protons in a box wouldn't make it a
helium nucleus). So they themselves act almost like a unit on their own. And an electron is
already a unit. Since they are all separate, there's no reason for their respective spaces to
coincide. Although still having the same building blocks, each space is capable of
influencing the other. Simple terms, they will be able to collide, or absorb, or deflect. When
the electrons and the alpha particles pass through the gold foil, they are simply following
their trajectory in their respective spaces (according to the references of course, everything
is a reference, not just what we consider to be as). They are not "seeing" the same obstacle
as we are. But since in math, you are capable of placing volumes next to one another
without a "gap", of course your interpretation will be the atoms have "gap". Except atoms do
not follow volume addition rule. An atom is formed with e
−
, p
+
, n; where each is positioned
will be determined by their behavior of how they come together. I only know that it's not
volume addition. Then the atom will define the information of the 'next' atom according to
the rule their behavior defines. I mean how is it not obvious? Why would a real object with
rules follow a hypothetical object with no rules? It cannot depend on a general physical
space (as in it won't be a function of physical space as math conceives). I cannot say an
atom is "mostly empty". It's not filling a hypothetical volume (which by definition cannot
interact with an atom, thereby impossible to fill with anything). Looking at this way, you'll
also realize that the concept of stability is superfluous. It is generally said about an atomic
or a molecular or a lattice structure. Think simply in this way: say something is pressing on
a lattice structure. Obviously the atoms in the structure will move accordingly under the new
circumstances in front of new references. When that is removed the lattice go back to its
"stable" structure. The easiest interpretation is that the behavior of units or pseudo units
depends on the circumstances. Or when a neutron is "added" to a nucleus and it becomes
"unstable". The neutron isn't just being "added", rather according to the rule, under the
circumstance of having an extra neutron warrants a new behavior and it simply follows that.
There's no absolute structure like volumes are. Even the "stable" structures in real life are
under a specific circumstances. Everything is always wrt something else. (Rutherford's
gold foil experiment [v])
Electrons passing through two holes. The explanation is similar to the previous one.
Electrons do not "see" the holes we are seeing. Their trajectories are defined by their own
space but also influenced by the space defined by the obstacle. Not only that. A defined
space is not sufficient to move around, references are an absolute necessity to define that
motion as I demonstrated in the rotation examples. These two spaces don't coincide since

The reason what I said in the last point is because the explanations presented here does not
involve time. There's no physics without time. The arguments presented here are outlines and
show that something so artificial as mathematics also needs to follow the rules of the physical
worlds. It happens because the intuition used to model math, are from the physical world. So
these are not the same things. Electrons do not need to be waves. No, I cannot say what
the trajectory would look like. How can I? My intuition will be misplaced if I model electron's
behavior based on aggregated object (almost; nothing in real life add entirely based on their
volumes, volumes are fictitious, there are always forces at play) I interact in my day-to-day
life. All I can tell that it will be different. The rest needs to be figured out from the actual
experiment. Intuition misplaced, not counterintuitive. (Electron double slit experiment [y])
Silver atoms trajectories influenced by an electromagnetic field, once again, cannot be
determined by the general space. The atom space under the influence determines what the
trajectory of an atom should be. So they do not need to be deposited on the plate
"continuously" according to the math definition of it. They are exactly where their space is
dictating them to be (and the references). Change the circumstances. Say you are making a
silver sheet. Do you think the sheet will have a big hole in the middle since the silver atoms
deposit "discretely"? Of course not. Silver atoms will behave differently when they are being
arranged in a lattice structure to form a sheet. Both of these are happening in the space
defined by the atoms. But as I have mentioned, space is not enough. The circumstances,
aka, the references will have the final say. Also they will recreate the same trajectories only
under the exact same circumstances. If the circumstances are changed the result will adjust
accordingly without any memory. Because memory will suggest an absolution. The
"memory" seen in macro objects is due to their having way more references, changing all of
which altogether isn't as easy as changing a lone unit's reference. (Stern-Gerlach
experiment [a])
The same reason why it is so difficult for particles to collide in a particle accelerator. It has
nothing to do with their size. They are simply moving in different spaces according to
references. They will collide only where the spaces coincide which is possible since they all
are governed by the same forces.
I have mentioned to some extent how trajectories are observer-dependent. Even when you
have the space defined, the movement is guided by references. When you include more
and more reference frames, object behavior changes. But it is not for forever. There is a
limit. I will not explain more than this here.
You can make a rough outline about the uncertainties of measurement from this discussion,
but I'd suggest you not to do that as this article doesn't have enough ingredients for that.
A good portion of this article is an explanation of "quantization" of measurement. It doesn't
make the universe discrete. Only proves the mathematical misrepresentation of it. But
please do not use the argument presented here as a full explanation. I can show it using
physical argument. That's part of my physics story. (the UV catastrophe [n])

whatever attempt there have been to rid of the physicality from math to make it truly abstract, it
will always fail. I can show how mathematical theory utopia cannot exist on its own.
I'm about a good century late to predict any of the above things. Those have already been
shown many many times experimentally. I guess I am providing some theoretical explanation.
When I was talking about combined unit, I gave a close real life analog of electric forces. I said
it was close because the force unit is described in terms of a common space (the "distance"
unit), which is not correct and it will lead to paradoxical situation.
In physical measurements, the rounding off values work because it was a faulty theoretical
premise that created those irrational numbers. They were never there to begin with. So when
the "rounding off" happened, it happened in our mind wrt our math model. In reality, nothing was
being rounded off. (There's another type of negligence argument used in physics. Friction is
one of those culprits. Why this negligence works for a lot of measurements is a different
mystery)
In real life day-to-day measurement, fractions might not cause much of a problem (for the most
part) since the reality isn't governed by our rules. We can interact with objects. And since we
deal with mostly aggregated (for simplicity I assumed the aggregation) objects, the fractions are
justified. But only up to a certain extent. Fractions are not allowed when we are building up
abstract mathematical measures. I gave a day-to-day example below where I showed fractions
break math.
I will not talk about relativity here. That's in the physics story.
One more thing. Every mathematical object and relation is permanent. Permanent has meaning
only wrt time. But if time is represented in math, then with respect to what would the time object
be permanent?
The definitions I used for dimensions are based on everyday object intuition and borrowed from
Descartes' coordinate system. I cannot say anything about the 1-D or 2-D since as a 3-D being I
cannot even imagine them. What I can say is that natural units will not follow the 3-D definition.
That is pretty obvious at this point. I cannot even verify if my definition is correct as there are no
volumes in real life. The closest I can get to would be lattice structures. Using experiments, it
needs to be figured out which unit "sees" what. Or the nucleus of an atom also comes close but
of course they are held together by a "glue" since these are real objects. The mathematical
definitions also give the wrong impression of making everything "knowledge-based". You need
to start by declaring that you are using a 3-D unit and then you assign "directions" to it, known
from using existing 3-D objects. In reality, how does the unit's behavior come to exist? Do they
"know" they are in 3-D? Without intending to, math imparts sentience on the universe. The n
unit coordinate vectors in Vn are considered to be linearly independent [j]. In plain 3-D terms it
means that no one direction can define the other two. I based my own definition on this too by
saying 3 distinct changes. But is that correct? Math has no answer to this question since math
just assumes that. There's no question of verification since everything starts after that. If you

say it's built from intuition, then I'd say you probably know by now that our intuition isn't always
properly applied. The thing is that I can prove that in reality they are not independent. That
require physics which will also show how objects do not need "knowledge" to navigate. This will
also help explain to some extent how a fixed dimension our universe came to exist in. Anyway,
those fictitious definitions did shed some light on the supposed bizarreness of unit objects.
Speaking of light, it doesn't move in space following a straight line. Also the way reflection (or
refraction, basically any light related behavior) is represented needs to be changed. This article
is not the place for that.
Some more subtleties of counting.
This might sound a bit confusing but I'll try my best to illustrate. I'll speak in the wrong terms
for easy understanding. Gravity and the electromagnetic force coexist in the "physical
space", i.e. they can "take each other's place without replacement". There is no problem for
two distinct units to be taking up the "same space". These two units are of distinct type.
(They are actually not taking up the same space; that comes from the misconception of
assuming that everything is happening against the same backdrop of space.) However, if I
change my counting unit to be the types of distinction, then in that types-of-distinction space
gravity and electromagnetism both belong where they cannot take each other's place
without replacement.
I explained how counting comes to be and how it is the bare minimum measure. But there's
a little subtlety how counting makes something measurable. Say your hard drive got
corrupted and you lost a lot of your important work. You'd feel sad (I'm assuming). But if you
lost a loved one, I'm sure you'd be sadder. These two events are distinct and the sadness is
repeated. Yet the repetition isn't the cause of your being more sad. You can assign a count
measure on the number of times you've been sad. However, since the repetitions do not
create the sense of being 'more' sad, this count measure doesn't make sadness
measurable. So the question is, what makes something measurable? This article deals with
already known measurable objects and only points out given something is measurable how
to establish a measure.
No figures, sections, or references in this article are numbered. These do not constitute a
count with respect to this article. All I need is to mark them to show which one I'm using
when but they should not be numbered as if they are being counted. Of course, you can
mark using the usual numbers, since they are symbols too. But I don't think anybody is
seeing a number and not using it in a counting sense. This is why I chose to label this way.
It seems innocuous but this is how intuition is misplaced that leads to paradoxes. They are
distinct but not countable in this context. If you do count then you are just counting that will
not require involving the article in any way. Units are independent of other units. This is
something I mentioned many times. Page number of a book is also not countable since you
cannot add or remove pages from the book without changing its identity. But books in a
library are countable since adding more books wouldn't change it from being a library. The

I will not answer these questions here because they are all tied to the measurable nature of our
universe. And that is the central part of my physics story where I try to explain how our universe
becomes measurable.
word count of a dictionary is also defined, since adding words to it won't stop it from being a
dictionary. I also said how in real life nothing just "adds up". Take a sentence. It is a human
creation, right? Artificial. And yet you can't just put word after word to create a sentence.
They are added via grammatical rules. Sentences haven't just been "added" to create this
article (or any piece of writing basically). They are added in a way to create a 'meaning',
whatever that underlying mechanism would be. Library books or dictionary words addition
have rules too. Only math adds things without any underlying rule. So the question is how
are these rules come to exist in reality?
Mathematical counting doesn't come with a ceiling. It goes on forever. In reality, however, all
measurements have some sort of upper limit. Once again, let's ignore nature cause it's
mysterious and all, and take a human created thing, like the English alphabets. It stops at
26. Or take the words in any language. There's always a certain number. It doesn't have to
but it does. I showed the theoretical lower limit of measurement. What about an upper limit?
Where does this upper limit come from and what role does it play in measurement? Of
course math doesn't have any answer because math was designed to be estranged from
reality. But it sure does stand in the way to understand nature.
I have only assumed electromagnetic force and gravity in this article. So that should be 2
units. In reality, all matter are created with 3 distinct objects, even though they all follow the
rules of the two forces. Why does these 2 units have 3 avatars? In general, the question
would be what creates this types of distinction? (Each copy of an apple or an orange is
distinct. But I am asking how the distinction of an apple and an orange is created) Does this
have anything to do with measurements?
The day-to-day example I mentioned earlier. Let's say there's a tax law where people with
incomes up to 100 units (including) have to pay a 10% tax, and up to 200 units (including) a
20% tax (unit as in any monetary unit). Who's paying more tax based on their income
bracket? You'd obviously say it's the latter group since within 100, 20 is bigger than 10. Let's
see what the theory says. I have a monetary unit, undefined of course. If it's repeated I have
a count measure. Whoever has the more repetition of the unit than someone else, that
person has more money. That's the order. Now I want to take it and turn it into a 100. But
that's the problem. That means I need to start with a new unit and I'll lose the meaning of
comparison. Because the numbers only have meaning in a certain context and they do not
translate into any other measurement. I have thoroughly explored this in this article.
Remember when I said if I ask you to imagine a sphere and then imagine double its
volume? I told you not to imagine a singular sphere? The same thing here. If the
measurement is happening in a monetary unit, then you cannot turn that into anything else.
You cannot consolidate it into something new. You will lose the ability to compare. All

The supposed non-existent elephant in the room
What does 0 represent? If you said nothingness, I'd agree. Then how do you add nothingness
to something? Because by virtue of something's existence, it nullifies nothingness. Existence is
independent of nothingness, but nothingness isn't independent in our narrative. When I choose
to count pens, all the numbers will represent its repetition. If I choose books, then all the
numbers will represent its repetition. 4 books≠4 pens. But 0 can represent the absence of both
indistinguishably. You may say the absence of books doesn't guarantee the absence of pens.
That's true. I will counter you with the presence of the books also doesn't guarantee the
presence of the pens. Wrt the books the pens are always at 0 and vice versa. That's the thing I
repeated like a thousand times throughout this article that units are independent of one another.
This way of interpretation doesn't make 0 represent nothingness. If it is representing
nothingness then 0 books=0 pens=0 apples=0 monkeys⋯ etc. It loses distinction. Which is
obvious. How can you distinguish one nothingness with another? The illusion of distinction is
happening via the existence where distinction is perfectly defined. (Empty set isn't capturing the
meaning of nothingness either, it still creates an object called set; nothingness is not and
should not be made equivalent to an object). Then what is the meaning of this definition:
n×0=0, or, n⋅0=0 (where n is any natural number)? What exactly am I counting? Because
counting cannot be established on nothingness. Am I counting the symbol '0'? In that case the
value will be n and not 0. But this is harmless, isn't it? Treating 0 as any other numbers? Then
it's natural to ask what happens when divided by 0. This question exists solely because 0 has
been given the status of any other number even though there is no logical justification to do so.
This article showed how you can't even divide with just any number as you please. So the rules
of counting are much stricter where 0 is nothing but a logical time-bomb just waiting to disrupt
everything.
This is almost the end of the article. Many of the conclusions in the previous section I first found
proofs in physics. Especially the big two, measurements having a lower limit (came up with this
back in 2018 in the same evening I found the first algorithm) and dimensions not existing like a
nesting doll. But I didn't know how they would look like in math. They looked very different in
physics context. However, knowing what my goal is to prove here helped me a lot developing
because the unit of any measurement is always undefined. So if you sleep well knowing the
rich pay more tax than you, it's due to a theoretical misunderstanding. It is incomparable. If
you and your friend are reading two different length books and both of you have read 50% of
your respective books, that doesn't mean you two have read the same amount (this
becomes very easy to understand once you change the unit of counting to words).
What about probability? It's better if I talk about it in a physical context.

the explanation in math specific way. Although I'd argue that it blurs the line between math and
physics.
The draft for this article I wrote back in 2023 got lost when my hard drive got corrupted. I had to
rewrite this from scratch in 2025. I remembered having a section in the old draft about the
definition of set and how it sent math down a rabbit hole rife with paradoxes. I omitted it here
because, even though you might think a math article is the perfect place for talking about sets, it
couldn't be more out of place. I understood what the mistake is in defining a set from a
measurement perspective which came from thinking about the physical world. The groundwork
needed for that isn't present here. It wouldn't justify the insanity I had to go through to put all the
pieces together to just write the conclusion in this article. This article is a chapter in that bigger
physics story that I have mentioned so many times. Which I also lost. It wasn't finished of
course but I lost the progress. I even lost all the notes. Now I have nothing and I have to start
over. I'm dreading to do that for obvious reasons. You can understand the slight bit of insanity
from this article. I went against the most fundamentals of mathematics. The physics story is
wilder. Because it involves way more than physics. Who in their right mind goes against the
smartest people in the world? I don't know what to do. Especially after having lost the work, I
lost my motivation to write again. I was interested in writing it as a story and publishing it (I
know nothing about publishing). So consider this article as one of those previews, 'first 10
minutes of a movie', upload (except this isn't the first chapter of the story). I am submitting this
in a science (mostly math and physics) explanatory competition, 'Summer of Math Exposition',
held by a YouTube channel called 3blue1brown [f]. This provided me some motivation to
present my work for the first time in front of people.
If you are still interested in what more I have to say, that'll probably give me some motivation to
tell the whole thing. By no means, it is a complete story. It will be just a few insights I had that I
think answers a few questions. That's all I could do. By the way, the story involves biology (and
some splashes of philosophy) and I'm pretty sure biologists would be interested too. So
basically I'm letting all disciplines of science arrange my funeral. The only thing left is the body
(of writing). If anyone can help me with publishing, then I guess I'll provide the body. If nothing,
at least I can brag about being a published author. LOL.
After everything, I want to conclude this article by presenting some other counting algorithms
that I came up with during all this since I didn't like my first one. Yes after all this I am still going
to count "real" numbers. I'm possessed. I came up with all these algorithms in the bathroom in
one shower session.
The big sort
This is simple. Start with a list. Then the outputs of the diag. fn generates a new decimal
number (I don't care anymore if it's rational or irrational, it can be whatever). Start a new list with
this diagonal number in position 1. From the old list pick a number. If it's smaller than the diag.

number then it increases its index by 1, the smaller number takes diag. number's old index, and
if the picked number is bigger, the diag. number's stays in its position and the bigger number is
added in the next position. All the other indexes are adjusted accordingly to make room. You
can clearly see that the new list including the diag. number will be one-to-one with the integers
since all I'm doing is adding 1. You can do this with as many new crazy numbers as possible.
Continue doing this until the interval is exhausted. It is always possible to make the list one-to-
one with the integers.
This algorithm is still dynamic. I didn't like it either.
The zigzag
I was staring at the bathroom tiling. It had some square patterns where I was visualizing the
number line. I thought the list places the numbers on the positive integer side. The whole
negative integer side is open to admit a plethora of new numbers. But what about new numbers
created when both sides are filled? I was stumped for a second when it occurred to me: just
rotate the whole thing and I will have plenty of space. The zigzag emerged. This is how it works.
Start with a list and keep it like a column. Then the next column includes all the diag. numbers
created from the list. Create more diag. numbers from the second column. And on and on. Join
them according to the figure 'the zigzag'.
The boxes represent the numbers obviously. No matter how many numbers you present to me,
it will always be countable with this method. This one's my favorite algorithm.
The big O

First all the possible lists are created like a beads on a line. Then one end of each of these lines
are added like rays being radiated from a "point" like in the figure 'The big O' (circles are the
numbers in the figure). After that the counting happens as shown in the figure. The counting
spirals out from the center (I haven't shown it in the figure because it looked messy). "In the
limit", the "points will fill up" the 2-D plane. So accidentally I created an algorithm to count
"points" in 2-D plane.
As I said before, I was having an existential crisis on behalf of the diagonal function. I never
stopped thinking about it. While writing this I found some thing a little redeeming about the
function. This is a last minute addition. The number of digits after a decimal (on the right) is one-
to-one with the integers. I didn't see anyone pointing it out in any other way. The first list also
contains the same amount of numbers as it is one-to-one with the integers. Together they
create a square (imagine the digits after the decimal on the right creates the width and the list
below creates the length of the square). When used a diag. fn to create a diag. number from the
list, this number will also have the same number of digits after the decimal. If this number is
added to the list, then the diag. fn cannot be used in the new list since there are no more diag.
digit as it's a rectangle now. The redeeming quality is that the diag. fn was never possible to use
on the whole list. Don't worry it doesn't fix anything. If the diag. fn isn't possible to use at some
point, that means the list is complete and you can use any of my algorithms to show a one-to-
one correspondence with the integers. Which means the diag. fn is now possible to use in this
list. But the function just indicated that the list is complete. A contradiction. It didn't remove the
core flaw. If the fundamental premise is wrong, it will always, always, lead to paradoxes. I still
wanted to point it out since I didn't feel good about bashing the function that badly. Cantor's
ingenious insight should bid goodbye on a high note.

You can easily count all the reals on the real number line now. In fact, you can count all the
"points" in any dimension as well. I will not show as I don't see the point anymore.
There's one last thing about the power set of the set of all integers being uncountable. This
proof also uses a diag. fn approach. You can prove it incorrect (use that special case I
mentioned when I was pointing out the flaws of the diag. fn, where I created a list only using 0
and 1). I can show you how the counting works as well. It's easy. But I'm not going to since I
have a problem with the definition of set and thereby the concept of power set.
The article concludes here. Thank you for going along this crazy journey with me.
Author: Gracie
contact: [email protected]
References
[g] K. Kunen, Set theory, U.K.: College publications, 2011.
[m] T. M. Apostol, Calculus, Vol. II: Multi-Variable Calculus and Linear Algebra with Applications
to Differential Equations and Probability, 2nd ed. New York: Wiley, 1969.
[j] T. M. Apostol, Calculus, Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra,
2nd ed. New York: Wiley, 1967.
[u] D. J. Griffiths, Introduction to Electrodynamics, 4th ed. Boston, MA: Pearson 2013.
[s] G. Cantor, 'Über eine elementare Frage der Manningfaltigkeitslehre', Math. Ann., vol. 38, pp.
547-572, 1891.
[n] M. Planck, 'On the law of distribution of energy in the normal spectrum', Ann. Phys., vol. 4,
no. 553, pp.1-10, 1901.
[v] E. Rutherford, 'The scattering of α and β particles by matter and the structure of the atom',
Philos. Mag., vol. 21, pp. 669-688, 1911.
[y] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 3.
Reading, MA: Addison-Wesley, 1965.
[a] W. Gerlach and O. Stern, 'Der experimentelle Nachweis der Richtungsquantenlung im
Magnetfeld', Z. Phys. vol. 9, no. 1, pp. 349-352, 1922.
[f] 'Summer of Math Exposition 2025', 3Blue1Brown, 2025. [Online]. Available:
https://some.3b1b.co/

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