$\frac{n(n+1)}{2}$
#Triangular Numbers (equilaterals)
west bronze
hexed wharf
what
brittle loomBOT
rockhoven
west bronze
this is the object we seek to construct
hexed wharf
T_n = n(n+!)/2
west bronze
the object in the numerator must be divided into 2 equal parts in order to render an equilateral triangle
what is the object in the numerator?
$\frac{n(n+1)}{2}$
brittle loomBOT
rockhoven
hexed wharf
T_n + T_{n+1}=(n(n+1)+(n+1)(n+2))/2=(2n^2+n+n+2n+2)/2=(n+1)^2
west bronze
just inverse the operation and ask "What is the object I will get if I double the equilateral triangle?
since no position is given we may get a variety of answers or objects
let's explore all of the options
if
$\frac{n(n+1)}$
brittle loomBOT
rockhoven
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west bronze
if the numerator can render that object then we might have something
if we evaluate the numerator according to standard basic mathematical principles
OK
I need to gather a few materials
in order to demonstrate the possible construction that we can make
I mean plural constructions
think about it and let's resume later
final ivy
Analyzing this way is not that great because you are looking at the final product, not the steps to it. If you try this with like the quadratic formula it wouldn’t really work
Quadratic formula just kind of looks like a mess
silver vigil
yeah
multiply by 4a
add b²
take the √
take b two sides
divide by 2a two sides
$ax^2+ bx + c = 0 \ ax^2 + bx = -c \ 4a^2x^2 + 4abx = -4ac \ (2ax)^2 + 4abx + b^2 = -4ac + b^2 \ (2ax)^2 + 2axb + 2axb + b^2 = -4ac + b^2 \ 2ax(2ax+b)+b(2ax+b)=b^2-4ac \ (2ax+b)^2 = b^2-4ac \ 2ax + b = \pm\sqrt{b^2-4ac} \ 2ax = -b\pm\sqrt{b^2-4ac} \ x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$
brittle loomBOT
neruguis
final ivy
Like what??? Best I can take from it is if the discriminant is if it’s 0 that’s means there’s one solution which is -b/2a which is also the axis of symmetry
silver vigil
like what what?
we can find x
using a b and c
its cool
already answered the polls?
here
final ivy
already answered the polls?
No but I don’t want to
silver vigil
why bro?
final ivy
Because I want to give out minimal information
silver vigil
ok
if u want
but just see
some dont give info
like who you hate more in the server
or which math branches
final ivy
**neruguis**
For me it was taught you subtract c, factor out a, complete the square, then isolate
I did the first one :3
Also @silver vigil why does the bot sometimes 💝 some messages
final ivy
?
silver vigil
i tried to post cute messages
silver vigil
I did the first one :3
Thanks
silver vigil
bro
ur pronouns are block and cheese
so i say
“Oh, block lost a thing” “Yes, it’s Cheese”
“Oh, he lost a thing” “Yes, it’s his”
final ivy
Don’t forget my nouns, adjectives, and verbs
west bronze
You know, I understand that you think your diagram is a joke
but it is not
we are going to be coming back to it in the future
so I am saving this image
it is very impo
it is very important to our whole discussion
BTW does anyone know why we are discussing this?
Everything is set in a big broader context
the point that we are investigating is the conflicts between ancient and modern math
We were talking in the Pascals Triangle thread about this
some of the materials am posting here like the Mathematics pdf
#1240434247160828067 message
and also some other authors I'll be posting
they suggest that this conflict is due to the fact that modern math with it's equations
is removed from the objects under consideration
I am following this train of thought by asking us to refer back to the object under consideration
this is the classical approach
instead you are responding with more piles of abstruse\ mathematics
removing yourselves further from the objects under examination
can you not focus upon the actual objects?
I gave one way to view the objects represented in
$\frac{n(n+1)}{2}$
brittle loomBOT
rockhoven
west bronze
There are other ways to approach the object through this equation
you might begin with the side n
hiow would you build the object from the side?
you might substitute a number for the side
like 5
then n+1=6
so we have 5*6
silver vigil
√3/2 n²
the area
of a triangle
yeah
already answered the polls?
already talked in general?
go studying some calculus bro
this won’t take you anywhere
brittle loomBOT
rockhoven
rockhoven
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rockhoven
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subtle graniteBOT
rockhoven has been timed out temporarily for 20m 
west bronze
Anyway you can envision the object described in the numerator
as 5*6=30
this can be divided by the denominator 2
in several ways
and we get 15
but i see no division that can render an equilateral triangle
final ivy
ancient math started out as numbers correspond to sides and stuff but as math moved on it acsended that purpose
modern math is basically ancient math freed from its geometry forms
silver vigil
bros look what ur losing discussing this
west bronze
transcended? Is that what you mean?
final ivy
transcended? Is that what you mean?
I mean that its more than lengths
final ivy
but the equations bear little relation
thats because its no longer rooted is geometry
west bronze
the equation can find the number 15 and that is a triangular
but no properties of the triangular can be found or extracted from the equation
final ivy
but no properties of the triangular can be found or extracted from the equation
not really
west bronze
that is why the ancient and modern maths are in conflict
final ivy
its just a simplified version of it
we used the properties of triangular numbers create the equation
west bronze
we used the properties of triangular numbers create the equation
how? How do the properties of the triangular (not the triangle)
how are they used to create the equation?
I dont know what that 💝 is
final ivy
because of the properties of triangular numbers we can arrange them in this format and then modify this to create n(n + 1)/2
west bronze
because of the properties of triangular numbers we can arrange them in this form...
how? show me
west bronze
#1240434247160828067 message
try this one
the point is made in this article
and in another article I'll be posting
west bronze
the problem is that ancient and modern math are two different animals
final ivy
the problem is that ancient and modern math are two different animals
modern math is just evolved/ascended or whatever ancient math
final ivy
that link cannot do
why not
west bronze
go back to Nicomachus and get it right
final ivy
the triangles are right
so?
west bronze
we have to be on the same page
we can not equivocate
read Nicomachus
read exactly what he says
do not replace the triangular with any old triangle
west bronze
how many times do I have to explain that our object is an equilateral triangle
I'll tell you why not
because we are not talking about any other object
you see
final ivy
west bronze
by utilizing these equations you lose sight of the objects we are discussing
you dont know what we are discussing
final ivy
by utilizing these equations you lose sight of the objects we are discussing
the equations are the objects, defined as numbers and variables to explain part of or the entire object
final ivy
ok
west bronze
the equation finds the symbol or integer
it does not find the object
what do right triangles have to do with it?
read Nicomachus
and learn the definition of the object that we are discussing
that is all we are discussing
we are not discussing triangles in a general sense
we are discussing a certain type of triangle
it says so in the title
go to the top and read the title
it says "Triangular Numbers (equilaterals)"
is this correct?
this is what we are discussing
final ivy
is this correct?
yes
west bronze
and you are not following the topic
follow the topic
a good thing is to go to the top and see all of the points that were made
final ivy
but why can't we have other triangles if it helps in the pursuit of an answer
west bronze
along with all of the errors that have already occurred
west bronze
but why can't we have other triangles if it helps in the pursuit of an answer
that's not the topic
go make a thread about triangles
make a thread about obtuse triangles
so that I can come into and talk about acute triangles
final ivy
so you want to explain triangular numbers via equilateral triangles?
west bronze
sure
after you have explain to me how an equation can be made concerning triangulars
from the properties of triangulars
because I see no properties of triangulars in the equation
final ivy
https://math.stackexchange.com/questions/3081649/understanding-some-proofs-witho...
this
it uses the property to create a new one
west bronze
$\frac{n(n+1)}{2}$
final ivy
triangular number are the sum of the natrual numbers
brittle loomBOT
rockhoven
final ivy
we can represent that using squares side length 1
west bronze
triangulars are not only the sum of all of the natural numbers
final ivy
the sum of each row down is a new triangular number
west bronze
each number in the addition must be centered in position
that is not indicated in any equation
all of the gnomons must be bisected by a common axis
that is not in the equations
they must have three equal sides and three equal angles
that is not in the equation
final ivy
well one property can't explain all the characteristics
the property of having 4 sides doesnt explain all the characteristics of a rectangle
west bronze
we need all of the properties or we are removed from the object
the objects escape us when we employ equations
final ivy
that is not in the equation
it seems here you want all the properties
west bronze
yes
final ivy
the objects escape us when we employ equations
one equation cannot represent all the characteristics of a given object
west bronze
This is the point made in the article
#1240434247160828067 message
that modern math is removed from the objects it studies
final ivy
so you are saying because a singular equation cannot represent the eniter object we lose out of the base knowledge of thus object, hence removing ourselves
west bronze
neither of the equations are close to the object
final ivy
equations cab describe the properties
west bronze
I am just saying that 6*6=36
and 36 is a square number
but 4*9=36 is a rectangular number and not a square at all
and also
$\frac{n(n+1)}{2}$
brittle loomBOT
rockhoven
silver vigil
its boooth
west bronze
is not a triangular number because it does not have the shape of the object
9 is not always a square number
final ivy
is not a triangular number because it does not have the shape of the object
a shape is not a triangular number
west bronze
that is not what Nicomachus is saying
final ivy
9 is not always a square number
???
west bronze
9 can be a trapezoid
silver vigil
square numbers refers to triangles
west bronze
it is easy to show that 9 is not always a square number
let's take 15
and deconstruct it
15-1=14
since we have removed the apex from an equilateral triangle
final ivy
Numbers may have came from objects but are not defined as them
west bronze
9 is not always a square number
west bronze
^
bt you are off topic
our beginning is with the classic definition of these numbers and objects
the numbers are tied to objects
west bronze
i don't appreciate the elevation of squares over any other possible shapes
because the unit can have any dimension or shape
silver vigil
west bronze
according to the equation
silver vigil
1+2+3+4+5
west bronze
5*6=30
silver vigil
dont use mult
west bronze
how can this object a rectangle
be divided into two equal parts and give us two equilateral triangles?
if you read the pages from Nicomachus, Descartes, and Courant
silver vigil
well
final ivy
Numbers describe objects and objects describe numbers
Just because they are connected doesn’t mean they are inseparable
west bronze
you should understand what I am discussing
Block / you are a student of modern math
therefore you subscribe to that POV
you might want to consider that there are other POV's on this
west bronze
we can simply recognize the conflicts between math system and agree that they are in conflict
that is all
in my approach to math
west bronze
the problem is at least as or more important than a solution
silver vigil
you are trying to find something that cant be found
west bronze
to understand the problem is what my math is about
silver vigil
Iol
west bronze
if we cannot find the object in the equation then my point has been demonstrated
my general point is in alignment with the article
math has conflicting information
it is inconsistent
west bronze
if it were not inconsistent how could ancient and modern math be in conflict as the artice states?
final ivy
Ancient math is this:
Addition is just adding the lengths of two line segments
Subtraction is the taking away of a length from another length
Multiplication is the creation of an area or volume
Division is splitting of an object
Because of this it makes to sense for negative numbers (negative length is crazy)
After math was freed from geometry negative numbers made sense because they were no longer sides.
silver vigil
algebra or geometry
west bronze
I don't have to choose anything
west bronze
I can choose to recognize the unsurmountable barriers
silver vigil
is an error
west bronze
like recocognizing that the circle can not be squared
west bronze
I don't have any geometry
you might want to simply read and contemplate the materials I am posting for you
OK
Read the material and discuss the material
silver vigil
west bronze
that was what I was doing when I was away
silver vigil
no thanks
silver vigil
good for u
final ivy
Ancient math is this:
Addition is just adding the lengths of two line segments
...
If you try to compare modern with ancient math it doesn’t make sense because math is no longer just shapes. Its like comparing geometry and arithmetic
west bronze
I'll be back with more materials to WORK on
final ivy
If you try to compare something with something it’s not you are going to have problems
west bronze
If you try to compare modern with ancient math it doesn’t make sense because mat...
geometry and arithmetic run parallel to each other
there is not much in one that can not be expressed in the other
final ivy
geometry and arithmetic run parallel to each other
They are not the same. That’s why you can’t find one in another
west bronze
If you try to compare something with something it’s not you are going to have pr...
that is correct and the equation is not a formula for triangular numbers
final ivy
You can find ancient math in modern math (geometry and completing the square)
west bronze
that is what the problem is
silver vigil
not deep dive in the inconsistency of bla bla blah and how things exists and formulas are objects…
silver vigil
geometry and arithmetic run parallel to each other
yeah
west bronze
well you guys have not studies ancient maths
silver vigil
parallel do not cross
west bronze
so you are taking a limited view
you are fomenting an argument where there should be none
final ivy
that is correct and the equation is not a formula for triangular numbers
n(n + 1)/2 describes the “distribution” of triangular numbers but does not describe it as a whole, because it’s basically arithmetic, not geometry
west bronze
simply read the article and tell me whether it is true or not
west bronze
n(n + 1)/2 describes the “distribution” of triangular numbers but does not descr...
that is a good point to consider
it does a good job at that
but if I am interested in studying the properties of these numbers I do not need that equation
final ivy
Arithmetic can describe geometry and we kind of build upon that until we didn’t need to describe it
west bronze
because I can do that with a few small numbers like 3, 6, 10, 15, 21, 28, 36
final ivy
but if I am interested in studying the properties of these numbers I do not need...
You kinda do because the equation describes triangular numbers, which can lead you to a pattern
west bronze
and it would be cumbersome to try to investigate properties with a large object like 1225
final ivy
Arithmetic helps describe geometry and vice versa
west bronze
You kinda do because the equation describes triangular numbers, which can lead y...
you can not find any properties in that equation
final ivy
and it would be cumbersome to try to investigate properties with a large object ...
That’s why arithmetic is handy with describing geometry
silver vigil
parallel do not cross
lol only in spheres
final ivy
you can not find any properties in that equation
I can see it rapidly increases due to it being a quadratic
silver vigil
true
final ivy
I can go backwards to find the basis of triangle numbers
west bronze
i think we already knew that
silver vigil
a = 1/2 b = 1/2 c = -t_n
west bronze
I just went backwards with the regular formula
final ivy
I just went backwards with the regular formula
Going backwards with the formula ends up with geometry because triangle numbers are basically geometry ig
Because in deriving it you turn geometry into arithmetic to describe it
west bronze
a = 1/2 b = 1/2 c = -t_n
what is this? Your equations are always gibberish because there is no accompanying prose writing. That is not math. You have to have some kind of prose or there is no context.
west bronze
you know I have copied some of the gibberish that comes from this discord and taken it to other math forums
the one complaint is that no one can read it
silver vigil
$\frac{1}{2} x^2 + \frac{1}{2} x - t_x = 0$
west bronze
it makes no sense to them
brittle loomBOT
neruguis
silver vigil
bro
west bronze
because there is never any prose writing that gives context
west bronze
there you go again
final ivy
what is this? Your equations are always gibberish because there is no accompanyi...
All equations from an ancient mathematical point of view are completely nonsensical until you give it meaning
final ivy
I think he does
final ivy
@west bronze what grade
silver vigil
stop bro
silver vigil
you have to connect it to an *idea*
ok
final ivy
you guys need to learn to read and write PROSE
What’s PROSE
west bronze
go look it up
you need it
read Einstein
he writes a TON of prose and gives a FEW equations
that is real math
final ivy
First definition I suppose?
west bronze
Read some real mathematicians
not textbooks
you need to be able to explain your ideas without math
then the math is useful
silver vigil
For finding a triangular root of certain $t$, we can use the quadratic formula and the triangular formula $\frac{n+n^2}{2}$. First, rewrite the equation as $x^2 + x - 2t = 0$, with x being the root. Then, $x = \frac{-1 \pm \sqrt{1+8t}}{2}$.
brittle loomBOT
neruguis
west bronze
first learn how to explain your ideas in words
silver vigil
good for u?
west bronze
**neruguis**
right. here there is accompanying prose. it is not just an equation
there are words that are employed
silver vigil
bro im just 13
final ivy
you need to be able to explain your ideas *without* math
Huh
silver vigil
i want to be a philosopher when i grow up
final ivy
you need to be able to explain your ideas *without* math
Why do we?
west bronze
who are accompanying their philosophical ideas with breakthrough math
west bronze
Why do we?
go read some real mathematicians
I am leaving now
Tis is wasted time
I could spend these hours with Euclid, Kant, Newton, Whitehead, Russel, Archimedes, Galileo, etc
and learn great math
final ivy
If you think you are wasting time discussing on what you seem to think as lower individuals who are not enlightened by 100 pages of math philosophy instead of thinking you are helping them along a journey we all take I find that selfish
But it’s ok
west bronze
In the meantime you can goback and review all that ha=s been said
#1254508004636758218 message
pale rampart
what the actual f*** is this
final ivy
what the actual f*** is this
Man I’m trying to understand him and he’s like “waste of time”
What’s the point in discussing then
pale rampart
have y'all been arguing for over a month?
final ivy
have y'all been arguing for over a month?
Just started today
silver vigil
How can we find $x$ in an equation like this: $ax^2 + bx + c = 0$? First, we need to multiply both sides by $4a$, getting $4a^2 x^2 + 4axb + 4ac = 0$. Then we pass the $4ac$ and transform the $4a^2x^2$: $(2ax)^2 + 4axb = -4ac$. Adding $b^2$, $(2ax)^2 + 2(2ax)(b) + b^2 = b^2- 4ac$. Seeing the binomial expansion, take the square root and get $2ax + b =\pm\sqrt{b^2- 4ac}$. Pass the $b$ and the $2a$, $x = \frac{-b \pm \sqrt{b^2- 4ac}}{2a}$.
FINNALLY
final ivy
…
brittle loomBOT
neruguis
final ivy
Ok
silver vigil
he left?
silver vigil
Man I’m trying to understand him and he’s like “waste of time”
SAME THING
silver vigil
I could spend these hours with Euclid, Kant, Newton, Whitehead, Russel, Archimed...
your crazy rockhoven
silver vigil
yeah
final ivy
but if I am interested in studying the properties of these numbers I do not need...
He may want to describe all of math in words because all of math came from words
Thoughts?
@silver vigil
silver vigil
hi
no, i get what he’s saying
math is only math if it works on real life
and not just algebraically
he’d say
what do you mean when you say that (a+b)² = a² + 2ab + b²?
then show a square like this
|
|L|
| |
| |
| | |
| | |
L_________L___|
oh fk no
tough cargo
first learn how to explain your ideas in *words*
I think words are unsufficient and sometimes ambiguous
jagged stratus
west bronze
What have we learned, unlearned or relearned here?
every triangular number is also a rectangular number
west bronze
I think words are unsufficient and sometimes ambiguous
Math is a language and all languages are ambiguous
I hope to establish beyond a doubt that math is just as ambiguous as any language
take the equation n(n+1)÷2
ambiguous result
this will happen over and over repeatedly as we examine mathematics
with a microscopic lens
math is a theory of knowledge
it belongs to philosophy
Philosophy>Linguistics>Mathematics
that is how to study math
Philosophy is the study of ideas
Languages are the vehicles for communicating the ideas
Math is an abbreviated form of those languages that express these philosophical ideas
Therefore math is a language of philosophical thought
it is a little more universal because it condenses all of the words in any language that can be used for the expression of a given idea
and it's symbols bring concise expression
yet all of the ambiguity that is present in the linguistic expression of these ideas is present in the abbreviated form
reef barn
Math is a language and all languages are ambiguous
Uh, no, math isn't a language, nor are all languages ambiguous. Almost all formal languages studied by mathematicians are unambiguous, but it's not even clear what you mean by that. Do you mean they have an ambiguous grammar?
west bronze
I mean that the symbols whether words or mathematical, express ambiguous ideas to begin with.
But we have a very clear example of ambiguity in this thread if you have followed it.
reef barn
I don't really agree here, since I don't think mathematical sentences necessarily express ideas in the first place. Do you have an example of ambiguity in a formal language, or is the ambiguity meant to be metalingual?
west bronze
Read this thread. There is your example. I cannot waste my time here repeating myself.
Good day.
tough cargo
I hope to establish beyond a doubt that math is just as ambiguous as any languag...
I disagree that they're all equally ambiguous
tough cargo
Math is a language and all languages are ambiguous
Math is also a formal language
Big difference
west bronze
Math is also a *formal* language
what is the big difference?
west bronze
I don't really agree here, since I don't think mathematical sentences necessaril...
1 is not an idea? ∞ is not an idea? Triangles, squares and cubes are not objects of thought? Then how can we think anything with these "non-ideas"?
the number 1 is the same thing as the word one
it expresses the same ideas (plural)
and since these symbol carry around the same plurality of ideas within them, they are ambiguous
when it is stated that n(n+1)/2 is a triagonal when it is actually in fact a rectangle is to be in error. This expression is ambiguous. It has been in error ever since the foundation of polygonal theory. The use of the words "equilateral triangle" in reference to this expression of n(n+1)/2 is an equivocation.
when two ideas share the same symbol there will be ambiguity
west bronze
anyone should see that n(n+1)/2 is an oblong number and is not a polygon.
we should not attach the names of shapes to numbers that do not possess that shape.
that is equivocation
it is equivalent to speaking of a "nail" as if it were a clave to be driven by a hammer when in fact you mean your fingernail
glass echo
arent you guilty of this when you associated trianglular numbers with actual shapes instead of knowing that its just about counting objects in an equilateral triangle
tough cargo
what is the big difference?
In a formal language every sentence must be well-formed. You can't just throw in some words without an agreed upon definition like "love" or "numbers".
west bronze
The only difference I can see is that the words that represent the ideas (which are all ambiguous) have been abbreviated into the symbols that we call math.
The ideas themselves are questionable
Let's look at the historical record. Have you read the Nicomachus?
tough cargo
The only difference I can see is that the words that represent the ideas (which ...
At the very least less ambiguous than natural languages
west bronze
OK. Then we agree upon a couple of things
- math is a language
- math has less ambiguities but still possesses ambiguity and equivocation
so I concede that there is an equal amount of ambiguity
of course, I reserve the right to be wrong
and with it the right to change my opinion again
it depends on how this investigation continues
west bronze
Let's look at the historical record of polygonal theory.
Nicomachus comes first at 60 AD
Diophantus follows around 200 AD
They both developed polygonal theories
This is part of the polygonal theory of Nicomachus
Now read very carefully his use of language on plate 12-263
He equivocates in his language
he does exactly what every other person who argued here did
hexed lake
west bronze
he swaps the word and the idea of triangular which means equilateral triangle or for short triagonal for another word and idea: triangle
This is exactly the error that was committed here in this thread repeatedly
Now read what is written and learn
we cannot know math without an understanding of the ideas behind the symbols
whether those symbols are signs, letters or words
they signify ideas
hexed lake
@west bronze why does the root square is called root square?
Where's the frikin root?
west bronze
there is a difference between the idea of triangle or right triangle or obtuse triangle
west bronze
<@699627666319605861> why does the root square is called root square?
Where's t...
any term defined from an ill-conceived concept such as a square number exclusive of it's actual dimensions is going to inherit all of the problems of the ill-conceived monster that begat the beast
Read the pages I have posted and converse with the mathematician
that is who I am conversing with here
I am arguing with Nicomachus
Next I have an argument with Diophantus
but first read the theories so that you are well versed in the subject matter
west bronze
Anyway my problem with Nicomachus is that he swaps one idea for a totally different one
and it slips by because the words look nearly identical
triangular and triangle
one is specific and the other is general
you might be very annoyed if you were specifically conversing of right triangles and someone insisted on speaking of them generally as simply triangles
In the context of polygonal theory this equivocation is a disaster
the thing is that in a polygonal theory you have to have polygons
so he moves from a triagonal to a square
Diophantus does the same
but a false relation is set up between the two objects
the addition of triagonals does not result in squares as has been proven
Now this is a modern interpretation of the theory
note that a line has been drawn to highlight the diagonal in the square
and it is being suggested that two triagonals have been joined to for a square
when nothing of the kind has occurred
two right triangles are joined to make a square
thus there is equivocation in math
because math is made from the minds of mathematicians
who are just as prone to error as philosophers
this is because the great mathematicians are philosophers
and visa versa
a great philosopher is a master mathematician
Whitehead was a mathematician and a philosopher
Russell was a philosopher and a mathematician
they are errors, ambiguities and equivocations in philosophies and the same in maths
because they come from the same minds
In 126, look down at the bottom of the page and see the table
the second row is labeled squares when it should be labeled as non-square rhombuses
by the addition of triagonal numbers
but then they would not be polygons
so the whole theory is distorted in it's logical progression
mixing up some ideas with others
and using the language sloppily
it makes no difference if we substitute the abbreviated language of mathematical symbols
the logical fallacies are carried within these symbols
if the ideas are illogical then the math expresses illogical concepts
tough cargo
This is exactly the error that was committed here in this thread repeatedly
And also commited by you
tough cargo
Now read very carefully his use of language on plate 12-263
Nicomachus being lazy ^
tough cargo
they are errors, ambiguities and equivocations in philosophies and the same in m...
Do you have to say anything about ambiguities in the axioms of ZFC?
tough cargo
who are just as prone to error as philosophers
I agree
west bronze
And also commited by you
it would not surprise me at al if I have committed errors because I am talking math
what is your interest in this topic?
tough cargo
what is your interest in this topic?
math in general? it's very useful for societal problems and I also like the theoretic nature of it
west bronze
no, what is your interest in discussing this particular topic?
tough cargo
no, what is your interest in discussing this particular topic?
Triangular numbers? None
high parcel
What is a trigonal?
west bronze
What is a trigonal?
That is a very interesting question and we may open up another can of worms by looking at it.
When I began this topic, I immediately became weary of writing the words equilateral triangle over and over.
I then asked if we could come up with a symbol for the idea in the title of the topic.
We did not discuss this and no symbol was decided upon.
Recently, I was browsing through some ancient mathematics and i a treatise of polygonal numbers by Diophantus he refers to this type of triangle as a triagonal
Now, I am reappearing here with this abbreviated expression for our idea
In polygonal theory we speak of n-gons
especially for the higher numbers
there are triagonals and squares and pentagons, hexagons and so on
a general expression is n-gon
but I found this word triagonal and have adopted it
however it immediately brings a question to my mind
if a triagonal is a figure, a polygon with three equal sides and three equal angles
what does the term diagonal signify?
we can look up the modern definition
but it may be more illuminating to research the origins of this term diagonal
in relation to Diophantus' use of the word triagonal
in this pursuit I will begin with a chronological list of mathematicians
because I would like to trace out the history of the etymology
Euclid of Alexandria: Approximately 323 - 283 BC
Nicomachus of Gerasa: Approximately 60 - 120 AD
Diophantus of Alexandria: Approximately 201 - 285 AD
Does anyone know how to read Greek?
AI says: The First Use of "Diagonal"
The word "diagonal" comes from the ancient Greek "diagonios," which means "from angle to angle."
Source icon
While the concept of a diagonal, a line segment connecting non-adjacent vertices of a polygon, was undoubtedly understood and used by mathematicians before the term itself was coined, the earliest recorded use of the word "diagonal" is attributed to ancient Greek mathematicians, including Euclid and Strabo.
Also: Diophantus and the Triagonal Number
That's an interesting find!
It appears Diophantus used the term "triagonal" as a synonym for "triangular" when discussing polygonal numbers. This is a fascinating detail about his terminology.
While the term "triagonal" isn't as common as "triangular" in modern mathematical discourse, it's clear that Diophantus had his own nomenclature for these numbers.
So much for AI.
In fact, Diophantus never uses the term triangular
I believe Nicomachus does use this term
but Diophantus does not but instead refers to a triagonal number
However this could be a quirk in translation.
I may be slightly off on this. I'm taking a second look at Diophantus and now I see that he does use the term triangular. and now I am wondering where I picked up the term triagonal!
I have been browsing some other books and translations. I may take me some time to sort this out.
He speaks of pentagonal numbers and hexagonal numbers
I want everything to be fully documented
but now that I am going over the documents again, I am having trouble finding where that term was used
west bronze
Why do we?
good question
west bronze
All equations from an ancient mathematical point of view are completely nonsensi...
I agree. that is what I am struggling with. I have been struggling to find meaning in math since I was a child in the 1st grade. I still remember the first time I had to "solve" an equation. I felt guilty about having to lie.
west bronze
First definition I suppose?
Prose is like narrative or expository writing. like this writing. I am writing prose. The great mathematicians were all expert prose writers. Nicomachus writes in Prose. Diophantus, many others
west bronze
For finding a triangular root of certain $t$, we can use the quadratic formula a...
"For finding a triangular root of certain $t$, we can use the quadratic formula and the triangular formula $\frac{n+n^2}{2}$. First, rewrite the equation as $x^2 + x - 2t = 0$, with x being the root. Then, $x = \frac{-1 \pm \sqrt{1+8t}}{2}$."
brittle loomBOT
rockhoven
west bronze
i want to be a philosopher when i grow up
We cannot separate math from philosophy. This separation is only a very recent development that has ill effects on the study of math. Since you study math, you are also a student of philosophy and you are a philosopher.
west bronze
How can we find $x$ in an equation like this: $ax^2 + bx + c = 0$? First, we nee...
What is all of this jargon about? Can you make some sense out of it by way of explanation?
west bronze
your crazy rockhoven
LOL. And you are my biggest fan!
west bronze
Thoughts?
Yeah. That's a good observation. No, I want to learn some more math and am going to need help as we crack open all of these books that I barely understand. I'm sure people here can help me. I am just not religious with math. I find math to be just as questionable as any other theory of knowledge or philosophy.
west bronze
math is only math if it works on real life
OK. I'll have to think about that. I am not sure what I think.
Well here. What are the common properties of the following number series? 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
west bronze
And also commited by you
Was there some error or equivocation in my thought?
west bronze
Do you have to say anything about ambiguities in the axioms of ZFC?
What are the axioms of ZFC?
There are not sposed to be ambiguities in any axiom
I find no reason to accept axioms of math
axioms are unpremised premises
we do not allow this in philosophy
tough cargo
What are the axioms of ZFC?
the axioms of Zermelo-Fraenkel Choice set theory
west bronze
math should not be given such license
tough cargo
we do not allow this in philosophy
then how do you decide what to do next
west bronze
how can there be such a thing as an "axiom"?
tough cargo
when you don't take anything for granted
west bronze
any proposition must be premised upon another proposition
I take nothing for granted
I am an atheist
I do not accept math as a religion
I spose that we don't accept spiritual agents in math
tough cargo
is not a triangular number because it does not have the shape of the object
you equivocated here
west bronze
but it is still very religious to not question the authority of "axioms"
tough cargo
numbers are not shapes
west bronze
This number 3 is not a shape?
tough cargo
no
west bronze
numbers are shapes, always have been shapes and always will be shapes
how do we distinguish 3 from 4 if numbers have no shapes?
tough cargo
any proposition must be premised upon another proposition
so you start with nothing?
hmm
west bronze
yes
tough cargo
then how do you get something?
tough cargo
i never said you were
west bronze
if you are searching for something how can you start with something?
you have to find something to begin your search with
but that is arbitrary
tough cargo
you clearly got something you're thinking about
west bronze
sure
I have been programmed the same as you
we both carry a fundamental axiom within us
it was instilled by educatio
education
we are both continually operating from the same unquestioned axiom
tough cargo
we both carry a fundamental axiom within us
and you accept those axioms?
west bronze
they are axioms or premises
there is one most fundamental and unstated axiom
Truth does not contradict itself
Therefore we all proceed to make coherent statements
Spose the truth does contradict itself?
Then we are all wrong
tough cargo
what
west bronze
If the truth contradicts itself then we are wrong in assembling any coherent argument
how do we distinguish 3 from 4 if numbers have no shapes?
tough cargo
how do we distinguish 3 from 4 if numbers have no shapes?
by the peano axioms
4 := S(3)
tough cargo
the next natural number after 3
west bronze
especially arithmetic
tough cargo
math was not founded upon axioms
and now it is
west bronze
well, the cave dudes who sat around arranging pebbles into triagonal shapes did not have peano axioms
tough cargo
well, the cave dudes who sat around arranging pebbles into triagonal shapes did ...
and they didn't live very long
west bronze
so numbers have always been too intimately associated with shapes to be able to separate them
3 is a shape
4 is a different shape
tough cargo
3 is a shape
a Euclidean shape?
west bronze
and they didn't live very long
I think you have signed on to a lot of BS concerning math and science
what does practicality have to do with mathematical theory?
a thing is true because it works?
tough cargo
I think you have signed on to a lot of BS concerning math and science
i think you have signed on to a lot of BS concerning math and science
you are allowed to change your mind
so i think math may do that too
west bronze
One question I have here is how a science so full of errors can produce such consistent results?
tough cargo
a thing is true because it *works*?
no
tough cargo
what does practicality have to do with mathematical theory?
nothing
west bronze
good we agree
tough cargo
One question I have here is how a science so full of errors can produce such con...
which errors
west bronze
ha! that is the line of bs that math delivers
that it is some sort of perfect sytem free of error
this is false
tough cargo
i feel like we keep going in- and outside our common ground
tough cargo
that it is some sort of perfect sytem free of error
i never said that
i asked you which errors
west bronze
What errors does math have?
you don't think that math is a perfect system?
then why?
tough cargo
you don't think that math is a perfect system?
i don't know if it is
and so far i like it
west bronze
i like it too
tough cargo
and it seems like i can't have better
tough cargo
What errors does math have?
give an example
west bronze
i like math even though math is wrong
i have already given a number of examples
the expression n(n+1)/2 as signifying a triagonal is false
that is erroneous
you still have yet to admit this error
tough cargo
the expression n(n+1)/2 as signifying a triagonal is false
that's an equivocation on your side
west bronze
how?
west bronze
the expression n(n+1)/2 is a rectangle
west bronze
so using it to express a triagonal is a false assertion
tough cargo
but that doesn't mean they are the same
west bronze
a false assertion that you want to defend
west bronze
the expression n(n+1)/2 as signifying a triagonal is false
west bronze
it produces an oblong number
tough cargo
what does that mean
west bronze
well put some real number input int the equation
tough cargo
why do you say numbers are shapes
west bronze
draw a map of what the expression states
tough cargo
n(n+1)/2
west bronze
because they are
tough cargo
that's it
tough cargo
1 2 3 4 5 6 7 8 9 are shapes
which shapes do you think they are
west bronze
the shapes of the numbers that I just wrote
also in polygonal theory numbers are shapes
that is the whole premise of polygonal theory
the theory of polygonal numbers directly derives numbers from shapes and visa versa
so when you put three points i a triagonal formation
that could serve just as well for the number 3 as the shape of 3 does
we could represent all of the number line by individual polygons
and this conversation concerns polygonal number theory
tough cargo
the shapes of the numbers that I just wrote
yes, what are their shapes?
west bronze
what shape can you make from n(n+1)/2?
In polygonal number theory, what are the shapes of the numbers?
Please refer to Nicomachus, Diophantus and Theon.
Diophantus
Nicomachus
Is it possible to make a square from an equilateral triangle?
What properties are common to the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100?
"And also a question, how come triangle numbers still seem to adapt to square numbers? We can verify it by squaring known to us numbers?
Coincidence?" I ask nearly the same questions, with the exception of the word triangle. Instead, I ask "how come triangular or triagonal numbers still seem to adapt to square numbers?" I think the proper approach will be to examine these figures very closely. We may have to do further revisions on our questions. At this point, I don't think we are prepared for the task. We are discussing polygonal number theory in which all numbers are expressed as geometrical figures.
Another avenue of exploration will be to examine that nature of the units and lines employed. This may be revealing in the search for an answer to the association between these shapes.
One thing I find immensely interesting is that every number can be expressed by a unique polygon. This opens the way for great philosophical discussion.
Diophantus is the first to employ a sort of primitive algebra
we might enjoy actually working on some of the math that appears in the first page of his theory
from what I know, he does not make any generalities
all of his solutions are specific to a specific problem
we might be able draw the generalities ourselves
just by practicing his math
Imagine expressing every single number on the whole number line as an unique geometrical figure - a polygon
this is another fantastic philosophical question to explore
such a number system would not demonstrate any periodicity
the periodicity in math is an artificial contrivance
which is helpful for the recognition of the number shapes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
but is this the real nature of number?
The polygonal theory challenges this
in this theory no number is repeated to form a new number as we can put 1 and 2 next to each other to express 12
or 3 and 4 to express 34
although I think the Babylonians did do just this with the geometrical forms
Let's bring that system up and examine it
because I think it does work by putting triangles together
but it is inconsistent
because it does not produce the required shapes that actual addition of shapes provides
but the reason for a departure from what I would call pure polygonal number theory is for practical reasons
to make a practical math system
I have no interest in this
I am interested in a system the would be utterly useless
except for philosophical investigations
in that system all numbers would have unique symbols
I don't say that we should replace one system with another
but the exploration of both systems together may lead us into added revelations concerning reality and existence
and that seems to be the reason for math in the first place
to be able to measure reality and possibly understand it in some small way
west bronze
Let's take a look at some of the math in Diophantus
i. If there are three numbers with a common difference, then 8 times the product of the greatest and middle plus the square of the least = a square, the side of which is the sum of the greatest and twice the middle number.
This is the 1st proposition
***Let the numbers be AB, BE, BD in the figure, ***
E—----A—-C—-D--B
and we have to prove 8AB x BC+(BD sq.) = {AB+2BC) sq.
The first thing I will do is rewrite this in modern notation
AB = a
BE = b
BD = c
AC = d
8a*b+c sq = (a+2b) sq
well that is about the best I can do
merry hearth
I don't understand what's your question
west bronze
how do I get the square? The small 2?
that's what I need
I need to be able to write this in modern notation
west bronze
$8a*b+c^2=(a+2b)^2$
brittle loomBOT
rockhoven
west bronze
OK now we are on track
west bronze
triangular numbers are right triangles
where did you learn this?
merry hearth
where did you learn this?
reality
actually saw that in a book and then in a video
but it's also basic sense
west bronze
Yes, I have seen this also on web pages and math blogs
merry hearth
try making a triangular numbers on a sheet
west bronze
how?
west bronze
but it's also basic sense
what do you mean that it is also basic sense?
how do you know?
merry hearth
what do you mean that it is also basic sense?
that is actually what they sould mean
west bronze
that is actually what they sould mean
why should they mean that?
think you are working backward from the premise that two right triangles make square
so it would be logical that the square would be composed of two right triangles
merry hearth
how?
shade one square.
that's tri(1)
then shade the square below it and the square to the right
that's tri(2)
then the 2 squares below those and the square to the right
that's tri(3)
notice that if you make tri(n) and tri(n+1) you can put them together in a way that forms a square with sidelength n+1
merry hearth
tri(4)
west bronze
I mean I do not have the technical skill for making graphics
what are the triangular numbers?
thanx for the graphic
merry hearth
tri(4) and tri(3)
if you smash them together it makes a square with sidelength 4
west bronze
that is interesting
merry hearth
it can also be proven algebraicly
west bronze
well this is polygonal theory
merry hearth
using the property of n^2 = 1+3+5...n/2+1
west bronze
and while the square you constructed is a polygon
the triangles are not polygons
and the units for your triangle are squares
while it seems logical to me that if 1 is a potential triagonal then the other triagonals shall be generated from the unit triagonal
in polygonal number theory, the numbers are all polygons
unless we are equivocating
so the units in a triagonal should themselves be triagonals
your "right triangles" have no diagonals
can anyone do this math without bringing into it superfluous distortions of the materials?
Nicomachus defines a triangular number
How does Nicomachus define a triangular number?
As the title of the thread states Triangular Numbers (equilateral)
All of the numbers in polygonal number theory are depicted as polygons
We can make other shapes but the building blocks are polygons
He goes from triagonals to squares because they are equilateral
polygons have equal sides and equal angles
squares qualify as polygons but their component parts do not
every number on the polygonal number line will be unique
when we get to the number 10 it will simply be a ten sided polygon
same with 11
every number will have a unique symbol through ∞
of course such a system will be outrageously impractical and completely confusing
but that is where the interesting philosophical questions of reality and existence will come into play
it will be shown that it is not the polygonal system which is flawed
but our human sensory apparatuses are imperfect
the periodicity that we have artificially injected into our number system is to overcome the limitations of the senses
we are not able to read a 1000 sided polygon at a glance
it might take us minutes to read such a symbol
if we had an apparatus system that was much more sensitive
we would detect the difference between a 1000 sided polygon and a 999 sided polygon in a flash
just as fast as it takes to distinguish 1 from 2
the problem is not with the system
the polygonal number theory is not the problem
the actual problem is physiological
periodicity is not natural to the number line
the polygonal theory is more natural in some ways
every number gets it's own distinct and individual symbol
a square designates the number 4
a pentagon signifies 5
and so on without exception
this holds even for the numbers 1,000,000,001
and 1,000,000,002
if we had ears to hear the harmonic interval 1,000,000,001: 1,000,000,002
this sound would be distinctly different from any other previously produced by the juxtaposition of superparticulars on the number line
west bronze
a Euclidean shape?
numbers have always had shapes. how else could we recognize a number?
3 has a shape like someone but turned sideways
4 is shaped like a square with no top on it's head and standing on one leg
in algebra n has a shape like an up side down u
the letters x, y and z which signify number all have shapes
polygonal theory is a study of number shapes
you admit that you have no interest in this topic
so what exactly is your purpose here?
because when I talk to myself progress is made in understanding the theory
and when I talk to you we just go around in circles
you bring nothing of substance into the discussion
get some materials and bring them in for us to look over
tough cargo
numbers have always had shapes. how else could we recognize a number?
by counting sheep for example
how is that a shape?
also, you're dwelling on the past
west bronze
$8a*b+c^2=(a+2b)^2$
brittle loomBOT
rockhoven
west bronze
How does this equation look to you?
west bronze
you don't seem to be following the topic
I translated the math of Diophantus proposition #1 into modern symbols
This is what Diophantus says in the first proposition
scroll up to find the page
How can we check this equation over?
I know nothing about math
so I do not know how to process this further
I simply copied it from the book but translated it into modern notation
so I have no idea if I translated it correctly
or if his math checks out algebraically
I translated it because I doubt that anyone here could read the book in it's original language
My translation is the following
$8a*b+c^2=(a+2b)^2$
brittle loomBOT
rockhoven
west bronze
If there are three numbers with a common difference, then 8 times the product of the greatest and middle plus the square of the least equals a square, the side of which is the sum of the greatest and twice the middle number.
I have not used a chatbot for any of this work
AS you know, I am a chatbot
Just because I translated it does not mean I understand it
a scribe could copy or translate a scroll word for word and still have little or no comprehension of it's significance
a note in the margin refers the reader to Euclid 2:8
so this must be the source of his equation
Euclid 2:8
west bronze
using the property of n^2 = 1+3+5...n/2+1
"using the property of n^2 = 1+3+5...n/2+1 "
I appreciate your input. Feel free to develop your ideas
your equation simply means that a square is the sum of all of the successive odd numbers
this surely is a part of polygonal number theory
and this is an interesting part of the theory
squares were originally formed through the addition of gnomons
I think we merely need back up a few pages in Nicomachus to see this demonstrated
but it's important to realize that the polygons were not originated through multiplication
but through addition
but I have a lot of thoughts I am roaming around
which I acquired through extensive reading in the mathematical literature
and I reserve the right to be wrong
If I am wrong in a particular it is because my focus is on a broader perspective
but it may be the case that multiplication was accidentally discovered from the building up of polygons through successive additions of gnomons
They call Diophantus the "Father of Algebra"
His propositions may be one of the first instances of geometry being translated into algebraic formulas
If Euclid were alive and reading Diophantus with us, he might translate his algebra into his own proposition 2:8
***"If a straight line is cut at random, then four times the rectangle contained by the whole and one of the segments plus the square on the remaining segment equals the square described on the whole and the aforesaid segment as on one straight line." ***
In Diophantus' demonstration he only uses the aforesaid segment as on one straight line.
IOW, he uses a straight line for the demonstration
so he has translated Euclid's two dimensional geometry (in 2:8) to one straight line
So why is he beginning a book on polygonal numbers with this proposition?
BTW, polygon just means many-sided in modern math
but this word is used more specifically in this math of Nicomachus, Diophantus and Theon
in ancient polygonal number theory it means having equal sides and equal angles
the concept of the "square" number throws the polygonal theory into confusion
at some point the other polygons were assigned to the back of the bus