SNERX.COM/MATH Last Updated 2022/8/1 • Read Time 22min • Discord
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I never learned math in school, so as an adult I have been trying to teach it to
myself. The things I have found on this page are probably wrong, and the things I am
not wrong about were likely discovered long ago, but I'm just doing this for fun.
Infinite Series & Randomness: If you have an infinite series of whole numbers from one
to positive infinity, and you randomly select a number from that series, then the
length (number of digits) of that selected number will also be infinite. This is
because the average number of digits for numbers in a series of integers from one to
infinity is itself infinite. By definition, average means half the numbers in the
series must be longer than their infinite middle, and half must be shorter than their
infinite middle. But half of infinity is still infinity, and half of that is infinite
still, thus all the numbers available to pick must be infinite in length. Of course,
natural numbers cannot be infinite in length, so we have a problem here.
Conversely, there is no actual randomness in infinity. If you are to randomly pick a
number from one to ten, each number has a one-out-of-ten chance of being picked, but
if you are to randomly pick a number from one to infinity, each number has infinitely
low odds of being picked. I posit that since these odds asymptotically approach zero,
the odds of randomly picking any number out of an infinite series is actually zero,
meaning you simply could not pick a number. As further explanation, James Torre
pointed me to the impossibility of a measure for the Naturals for why this random
selection fails.
Goldbach's Conjecture: If not already familiar, read this. All even numbers are
predicated off of 2, and 2 is a prime, so all reduction of numbers predicated by 2 can
be reduced to primes of the predication value, which in this case is 2 (so 2 primes).
The framework then is such that any member of a factor-tree based on some number X
will inherit properties of X and will be reducible to an X-unit-count that share one
of those given properties.
The way this looks for the existing conjecture: Every number factorable by 2, past 2,
can be expressed by 2 units of some property of 2 (namely primehood).
The way this looks for new conjectures:
1 - Every number factorable by 1 (all whole numbers), past 1, can be expressed by 1
unit of some property of 1 (namely empty product or unity/identity).
3 - Every number factorable by 3, past 3, can be expressed by 3 units of some property
of 3 (namely merseene primes or fermat primes); effectively the weak conjecture.
π - Every number factorable by π, past π, can be expressed by π units of some property
of π (namely irrational numbers or transcendental numbers).
2 - Every even integer greater than 2 can be expressed as the DIFFERENCE of 2 primes.
--> Goldbach's Conjecture modified to show that unit relations are arbitrary so long
as the relations are of non-arbitrary units themselves.
If the form of this is valid, then its reduction to symbolic logic will return its
formal proof, but I'll let someone else do that work.
Invented, Not Discovered: Numbers don't real and Platonism is a cope. Mathematicians
and physicists alike can't fathom why an invented description of the world would give
accurate predictions of it, despite almost every other framework we invented also
giving accurate predictive power in their relevant domains (like economic, political,
and psychological theory, etc.), so they stand in awe of mathematics' predictive power
like feeble-minded infants and loath basic questions like, "Why is math true?" or,
"What is it's subject of study?"
Peek cope looks like this, where established scientist Max Tegmark says in more or
less words, "All frameworks of mathematics are true, there's a place where Euclidean
space is real, just not here." At the very beginning of that video he also says we
aren't free to invent a sixth regular 3D shape because it doesn't exist, however you
absolutely are free to invent one. People did this with imaginary numbers, we could
easily do it with imaginary regular shapes and variable the number of sides each flat
surface has, kinda like analytic continuation but applied to basic geometry.
The problem is even worse than I am making it seem. Consider that discoveries map
one-to-one on the world, e.g. if I have discovered a hidden chamber below the Sphinx,
then I have discovered a hidden chamber below the Sphinx - I have distinctly not
discovered a room floating in space several lightyears away from the Sphinx. A
discovery about X returns X, not not X. Discoveries don't return contradictions.
However, when asking two different frameworks of mathematics the same question, they
return different answers. Let's take two frameworks that are supposedly 'objective'
and ostensibly even about the same exact kind of phenomenon, like Leibnizian Calculus
and Newtonian Calculus. If you were to ask these two frameworks of calculus where the
center of mass of the Moon will be in one day's time, they will return two different
points in space to you. This is saying the Moon will be at X and not X at the same
time in the same regard - a contradiction. Since discoveries don't return
contradictions, one or both of these mathetmatical frameworks has to be invented.
Contrast math (also known as second-order-logic) with formal logic (first-order). Here
we start to see what's really going on. Formal logic only has one framework, and
that's the one Aristotle discovered 2400 years ago, remarkably unchanged and more
remarkably unappended since then; Aristotle completed the discipline. Sure, there are
different ways people try to symbolize logic as predicate, existential, modal, and so
on, but the rules of logic are finished, and they're finished in a single package. The
same cannot be said about math because math isn't an actual component of reality.
My real thoughts on the ontological status of math are a lot more charitable than what
I've said here, but it's worth noting that even if one of the frameworks of
mathematics we currently have ends up being discovered rather than invented, it is
still the case that all the others weren't.
Serious Problems for Cantor's Diagonalization: The conclusions drawn here are that Z
and R are not of separate cardinalities. There are three issues I've found with
diagonalization while trying to do the metalogic for it. The first occurs when
swaping the lists; if you put the natural numbers in a list denominated by the real
numbers instead of the other way around (the way Cantor does it), you attain the same
outcome of new infinites, meaning the conclusion would be that the set of natural
numbers is larger (and uncountable) than the set of real numbers (which are then the
countable set), a contradiction since this set contains the natural set. This is of
course the opposite of what Cantor concluded. To demonstrate this, look at the
following images modified from Veritasium's video Math Has a Fatal Flaw:
Above, you see all I have done is swapped the natural index numbers on the left with
the list of real numbers on the right from 0.0 to 1.0. The randomized list of natural
numbers is enumerated just the same in the right list as the real numbers are in the
left list. All we do now is apply the diagonalization technique Cantor uses on the new
list the same as the old list, shown below.
What you see here is that the new natural number we generate from the diagonalization
method similarly 'does not appear in the list', much the same as the new real number
Cantor generates from the diagonalization method. My critique is directly addressed
here and the above issue is considered a non-problem because natural numbers cannot be
infinite in length. However, it's not impossible to have infinitely long numbers since
we do have some numbers that are infinite in length, like the decimals used in the set
of real numbers listed by Cantor, and further, the first section on this page about
infinite series and randomness suggests we might in fact have infinitely long natural
numbers. So I don't consider this issue a closed matter until someone has a counter
that doesn't rely on the finitude of natural numbers.
What this means is that had Cantor started with indexing the Reals instead of the
Naturals, he would have concluded that the infinite set of natural numbers was
qualitatively larger than the set of real numbers. I believe he purposefully avoided
doing it this way for the reason in the last paragraph - that natural numbers are
finite in length - but I have doubts about whether that is a good enough reason.
Even if the above is ultimately a non-problem, it is only one of the three issues I
have with diagonalization. For the second issue, I believe his derivation of different
cardinalities is due to his mixing of potential infinity with actual infinity,
something already known to be improper in metaphysics as far back as their discovery
by Aristotle. Whether you use my reversed list or Cantor's original list, both ways
require the use of potential infinity for the index numbers and actual infinity for
the numbers enumerated to the right of the index. So of course this would appear to be
different kinds of infinity, because you have begged the question and baked a
pre-supposed conclusion into the formulation of its proof. This seems like an obvious
problem to me, but if it is not clear to you the reader as to why this is wrong or how
this works, then we'll look at third issue with diagonalization.
Cantor's diagonalization requires the ordering of his real numbers to be random, as we
shall see that reordering the list from smallest to largest real numbers demonstrates
the diagonalized new number in fact already appears on the list. If Cantor's list was
in order from smaller to larger infinite decimals, we would see the list ordered as
0.111, then 0.112, 0.113, and so on. If we then applied diagonalization we would get
0.2 from the first real, 0.02 from the second, 0.002 from the third, and so on,
resulting in 0.22 as our newly generated real number. But of course, when ordered from
smallest to largest decimal, 0.22 would obviously appear later in the list.
I contend that if you don't believe I applied Cantor's diagonalization properly, then
you have not been careful to note the kind of infinities I used. In my example, I do
not mix potential with actual infinity, and thereby the list of 0.11n's we enumerated
earlier results in an infinite number of reals that lead with 0.11, meaning we
generate an infinite series of 2's. If you think this problem is resolved by baking
the idea of uncountability back into real numbers, then simply swap the naturals with
the reals as I did in the pictures earlier and you'll see the problem I just described
reoccurs.
So even if one or two of these issues can be resolved, the third will still re-assert
itself, making the notion of distinct cardinality between sets of infinites in math an
untennable position. If I am horribly wrong about this, and I hope I am, I would
greatly appreciate someone who knows better taking the time to explain to me why that
is the case.
Separate Cardinalities of Zero: After thinking a lot about Cantor's infinite sets, I
started to notice that infinity and zero share a lot of properties in common. You can
multiply or divide infinity by any number and it is still infinity. You can multiple
or divide zero by any number and it is still zero. Infinity and zero are both
non-numbers since the first is a non-finite quantity and the second is the lack of a
Above, you see all I have done is swapped the natural index numbers on the left with
the list of real numbers on the right from 0.0 to 1.0. The randomized list of natural
numbers is enumerated just the same in the right list as the real numbers are in the
left list. All we do now is apply the diagonalization technique Cantor uses on the new
list the same as the old list, shown below.
What you see here is that the new natural number we generate from the diagonalization
method similarly 'does not appear in the list', much the same as the new real number
Cantor generates from the diagonalization method. My critique is directly addressed
here and the above issue is considered a non-problem because natural numbers cannot be
infinite in length. However, it's not impossible to have infinitely long numbers since
we do have some numbers that are infinite in length, like the decimals used in the set
of real numbers listed by Cantor, and further, the first section on this page about
infinite series and randomness suggests we might in fact have infinitely long natural
numbers. So I don't consider this issue a closed matter until someone has a counter
that doesn't rely on the finitude of natural numbers.
What this means is that had Cantor started with indexing the Reals instead of the
Naturals, he would have concluded that the infinite set of natural numbers was
qualitatively larger than the set of real numbers. I believe he purposefully avoided
doing it this way for the reason in the last paragraph - that natural numbers are
finite in length - but I have doubts about whether that is a good enough reason.
Even if the above is ultimately a non-problem, it is only one of the three issues I
have with diagonalization. For the second issue, I believe his derivation of different
cardinalities is due to his mixing of potential infinity with actual infinity,
something already known to be improper in metaphysics as far back as their discovery
by Aristotle. Whether you use my reversed list or Cantor's original list, both ways
require the use of potential infinity for the index numbers and actual infinity for
the numbers enumerated to the right of the index. So of course this would appear to be
different kinds of infinity, because you have begged the question and baked a
pre-supposed conclusion into the formulation of its proof. This seems like an obvious
problem to me, but if it is not clear to you the reader as to why this is wrong or how
this works, then we'll look at third issue with diagonalization.
Cantor's diagonalization requires the ordering of his real numbers to be random, as we
shall see that reordering the list from smallest to largest real numbers demonstrates
the diagonalized new number in fact already appears on the list. If Cantor's list was
in order from smaller to larger infinite decimals, we would see the list ordered as
0.111, then 0.112, 0.113, and so on. If we then applied diagonalization we would get
0.2 from the first real, 0.02 from the second, 0.002 from the third, and so on,
resulting in 0.22 as our newly generated real number. But of course, when ordered from
smallest to largest decimal, 0.22 would obviously appear later in the list.
I contend that if you don't believe I applied Cantor's diagonalization properly, then
you have not been careful to note the kind of infinities I used. In my example, I do
not mix potential with actual infinity, and thereby the list of 0.11n's we enumerated
earlier results in an infinite number of reals that lead with 0.11, meaning we
generate an infinite series of 2's. If you think this problem is resolved by baking
the idea of uncountability back into real numbers, then simply swap the naturals with
the reals as I did in the pictures earlier and you'll see the problem I just described
reoccurs.
So even if one or two of these issues can be resolved, the third will still re-assert
itself, making the notion of distinct cardinality between sets of infinites in math an
untennable position. If I am horribly wrong about this, and I hope I am, I would
greatly appreciate someone who knows better taking the time to explain to me why that
is the case.
quantity. Both represent limits of calculation since infinity is an upper bounds of
quantity that cannot go any higher and zero is a lower bounds of quantity that cannot
go any lower. Further, dividing any number by zero returns infinity and vice versa
dividing any number by infinity returns zero. There are other relations they share but
I think you get the point.
Given the similarities, I wondered, despite my complaints in the prior section, if
Cantor is right about the difference in kinds of infinites, then maybe there is also a
difference in kinds of zeros. You would think we couldn't use the same diagonalization
method for zeros that Cantor uses for infinites since infinite lists can be enumerated
by all the real numbers and zero is just a single instance of zero, but if
infinitesimals, or infinite-period decimals approaching zero that are considered equal
to zero are used, like 0.001 or 0.0099, then we have an infinitely enumerable number
of zeros we can index and apply diagonalization to just the same as Cantor did.
The objection to this would be that infinite-period decimals approaching zero are not
distinct values and do not qualify as being distinct in identity, however that is only
true if the next section is wrong. So depending on how the next section works out, we
may have a means to prove that there are distinct cardinalities of zeros. But I have
no idea how far off base any of my ideas are without talking to real mathematicians,
so I'm probably wrong about everything anyways.
Multiple-Infinite Decimals: 1/3 = 0.33, and 2/3 = 0.66, but what numbers out of a
whole give us the other repeating decimals? If we wanted 0.66 out of 1 whole instead
of 2, we get the following.
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1 / x = 0.66
1 = 0.66 • x
x = 1 / 0.66
x = 1.5
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But I contend that this number is actually 1.50015. Why and how? Dividing 1 by 0.666,
we get 1.5015, by 0.6666 we get 1.50015, by 0.6666666666 we get 1.50000000015, and so
on. By 0.66 what we get is an infinite series of infinites, namely the infinite bar
between 15's, that is 1.50015001500, repeating. This is the same as saying 1.5 with an
infinite series of zeros following it, and then after infinite zeros there is a 15
followed by another infinite series of zeros, and so on. Another way of saying this is
that as the antecedent (divisor) grows in decimal length, so too do the number of
zeros between the numbers of the decimal of the consequent (quotient). Therefore with
an infinitely-repeating-decimal divisor you get infinitely repeating zeros followed by
a finite series of numbers, the set of which itself then infinitely repeats, in the
quotient.
I've had people argue with me that, "This is not how fractions work," and if we were
using whole-number fractions, they would be right, as one divided by two-third's
becomes three over two and then cleanly resolves as one-and-a-half. But we aren't
concerned with whole-number fractions here; the property I describe shows that the
numbers in decimal format are not 'cleanly' divided. 1 divided by 3 gets you 0.33, but
as now described, 1 divided by 0.33 does not seem to get you 3.
The closest thing to this I have been able to find online are Shanks' numbers, but
those are distinct in scope and application, so I call these other numbers Snax's Bar
Numbers lol. I have written out some of the bar numbers below so you can see their
weird properties.
0.11 is 1/3 of 0.33 so 3 by 3 should mean 9, and in fact we see that 1/9 does equal
0.11; this then works as a grounding for the others as the others are multiples of
this first one.
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1 / x = 0.11
1 = 0.11 • x
x = 1 / 0.11
x = 9.00900900 repeating
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This means 900 is the bar number attained from 0.11. To re-iterate why this happens we
can follow the non-bar series, which results in:
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1 / 0.1 = 10
1 / 0.11 = 9.0909
1 / 0.111 = 9.009009
1 / 0.1111 = 9.00090009
1 / 0.11111 = 9.0000900009
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You keep adding zeros per n decimals of 1 from here, ultimately giving us the bar number
we attained, 900, for the series of 0.11. A list of these follows.
For 0.11 we get 9.009, or 900 as the bar number.
For 0.22 we get 4.50045, or 4500 as the bar number.
For 0.33 we get 3.003, or 300 as the bar number.
For 0.44 we get 2.2500225, or 22500 as the bar number.
For 0.55 we get 1.80018, or 1800 as the bar number.
For 0.66 we get 1.50015, or 1500 as the bar number.
For 0.77 we get 128571428571428571428700, as the bar number (see below).
For 0.88 we get 1.125001125, or 112500 as the bar number.
For 0.99 we get 1.001, or 100 as the bar number.
N.b., 0.11 and 0.99 are inverses of each other but there are no other inverses. Notice
also the strangness of 0.77's bar number and how no other bar creates the same level of
noise so far. 0.77 is more dynamic and there appears at first to be no upper bounds on
the series length or mutations; it does resolve, but I don't know how many digits out
it takes to resolve since I only tried up to 12 and then skipped to 30.
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1 / 0.7 = 1.42857142857
1 / 0.77 = 1.29870129870
1 / 0.777 = 1.28700128700
1 / 0.7777 = 1.28584287000128584287000
1 / 0.77777 = 1.28572714298571557144142870000128572714298571557144142870000
1 / 0.777777 = 1.28571557142985714414285842857271428700000128571557142985714414285842857271428700000
1 / 0.7777777 = 1.28571441428572714285842857155714287000000128571441428572714285842857155714287000000
1 / 0.77777777 = 1.28571429857142870000000128571429857142870000000
1 / 0.777777777 = 1.28571428700000000128571428700000000
1 / 0.7777777777 = 1.28571428584285714287000000000128571428584285714287000000000
1 / 0.77777777777 = 1.28571428572714285714298571428571557142857144142857142870000000000
1 / 0.777777777777 = 1.28571428571557142857142985714285714414285714285842857142857271428571428700000000000
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
This is strange since it resolves to an infinite period in the bar number and further
there are two infinite sequences within the general infinite sequence (demarcated by
the double-bar). I believe this also serves as proof that the numbers past the
infinite repetitions are non-trivial since there is not an infinite sequence of '00'
at the tail of 142857, but rather '14287' instead. The bar number for 0.77 does not
have a finite period length, even though all the others do.
You might say, "Okay whatever, but what practical application could this possibly
have?" And to this I say that fractional divisors in equations for physical systems
that result in very fuzzy statistical outcomes (like in quant) could probably be
cleaned up by acknowledging that the quotients are not so 'clean' and this
'infinite-zero-finite-sequence-repeating' property (name pending) should not be
ignored since infinite values appear often in some systems. From a metalogical
perspective, when this is factored in, it does result in 'clean' outcomes.
As proof for this, look to the convention that the number 0.001 is equal to 0. Since
there are different kinds of infinities in math, not including my objections to the
counter in a previous section on this page, the countable infinity in the number 0.001
will be overcome when divided by a number that is an uncountable infinity. The
infinite part of 0.001 can be skipped over by an uncountable infinity, leaving the 1
at the end as a non-arbitrary part of the divisor. This makes what I've been calling
the 'bar' numbers above worth considering for application in cleaning up infinites.
"But what about 0.12, or 0.69, or 4.20?" Most of the numbers I've looked at don't
result in much of anything interesting, e.g. if we look at 1.11 we get 0.9009 (or
900), totally in line with what we've already seen. However, some numbers truly have
unique properties, like 1.22, which resolves as follows:
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1 / 1.2 = 0.833
1 / 1.22 = 0.819672131147540983606557377049180327868852459016393442622950
1 / 1.222 = 0.818330605564648117839607201309328968903436988543371522094926350245499181669394435351882160392798690671031096563011456628477905073649754500
1 / 1.2222 = 0.818196694485354279168712158402880052364588447062673866797578137784323351333660612011127475045000
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
So what is the bar number for 1.22? Part of why this one is so weird to me is that the
bar number is not constant. The number we get changes depending on if the infinite
series of decimals is even in length or odd in length. If the infinite series of
decimals for 1.22 is even in length, then the bar number resolves as 819669421... and
if the ifninite series of decimals for 1.22 is odd in length, then the bar number
resolves as 818330578.... In both cases, WolframAlpha suggests there is a repeating
period length for the sequence that follows 81 but the complete sequence is infinite
and so WolframAlpha does not say what that period length is.
The series' period that follows 81 grows rapidly as you include more decimals and the
same number that follows 81 resolves granularly to a definite series, yet is assumedly
infinite in length at its absolute resolution. The fact that the number it resolves to
also alternates depending on the even-ness or odd-ness of the infinite decimal series
for 1.22 is strange in itself but what makes this more challenging for me is that
given the metaepistemics of maths (or what I know of this subject in limited fashion)
is that a 'true' calculation of 1 divided by 1.22 could not actually resolve to any
number since not only can the series not be determined in finite time but the series
alternates its determination dependant on how the bar 'feels' as a function of
even-ness or odd-ness. I need someone much smarter than me to explain this.
We have of course only looked at repeating decimals divided out of 1, and could go
through the same infinite list of decimals and divide them out of 2, or 94, or π, and
get new infinite lists of bar numbers, most of which would probably never be touched
or be useful to anyone or anything. But I think it's neat.
Tarski & Gödel: I wish to attack asepects of the incompleteness theorem and show that
since set theory is not a properly formal part of first-order-logic, that its
implications in second-order-logic are also malformed, pulling from the undefinability
theorem to justify this, as well as recursively seating the incompleteness theorem
inside itself to invalidate itself, showing internal inconsistency similar to a
halting problem, but it will be a while before I get to writing this out so this is a
placeholder section until then. I want to also make a physical computational
implementation of the halting problem to help illustrate this issue, and that will be
added to the /engineering page when that happens.
Circles: I have been told that it is valid in math to have a circle with diameter
zero. But, does this mean that a circle with radius zero is half as small as a circle
with diameter zero? It seems to me that a circle with a diameter of zero would also
have a radius of zero, since half of zero is still zero. And if this is all true, then
it follows that this circle must also have a circumference of zero. My understanding
of a 'point' is that points have no diameter, no radius, no circumference, and so on,
so why is this circle not just a point? I cannot find good online resources for this.
Unrelated, if we remade the unit circle at base 100 instead of base 360, the numbers
work out a lot cleaner and base 100 means you can use percent values to ascertain
positions on a grid. This is much easier to visualize mentally for most people and
it's objectively much faster to parse through. This also makes it possible to give
%-%-% formated coordinates for points in 3D space. Again much easier to cognize than
using 360 or π. As an important irony, this turns the unit circle into an actual
single unit since 100% equals 1 whole, instead of "2 units of π," which by it's very
description is not a unit circle but a two-unit circle. By happenstance this is also
better for relativistic frameworks used in mapping galaxies. I assumed someone had
already made this but I couldn't find it online so I did it myself and it gets used
sparsely in some of the games we've dev'd on Snerx. If you want to look at other dumb
shit we've done with percent-based relativistic frameworks you can check some of those
out here.
I Need Help Learning: If any of you know a real mathematician, I need help
understanding things that I can't find papers or videos for online. Things like why
isn't there a constant for primes or coprimes despite so many theorems showing regular
sets wherein primes occur (like 10 mod 1, 10 mod 3, 10 mod 7, and 10 mod 9)? Why are
primes a function of division only, why isn't there an analogue set of prime-like
entities for multiplicatives? Since there isn't an analogue, is this just asymmetric
logical operation in math? I am dumb and don't know where to look to find the answers.