#372 AP book of science--- High School Logic textbook

[Archimedes Plutonium]

I thought I needed rest and relaxation from writing Logic textbooks. But instead I seemed to have found enough energy to complete this 5 textbook series.

#366 Elementary Logic

#369 History of Logic

#370 Intermediate Logic

#371 Advanced Logic

#372 High School Logic textbook

While still fresh in mind, I better finish the series.

I have learned from this experience, that writing a Logic Textbook is the hardest textbook to write, for the simple reason, the talk of Logic, in addition the book itself has to be logical. 

If I were writing a physics or math textbook, there is ample leisure in jumping around on topics. But in a logic textbook, the book itself has to be written logically and that is why it has taken me so long to complete this task.

AP

[Archimedes Plutonium]

I start this textbook by referring to Mathematics as a model. For it is commonsense that Mathematics is the playing around with numbers. If we replace numbers with "ideas" and play around with ideas, is a short definition of Logic.

Math plays around with numbers. While Logic replaces numbers and plays around with "ideas".

And since math is a subset of Logic, for ideas encompass numbers of math and encompass geometry of math, we can expect that whatever exists in mathematics, must exist in Logic. Sort of like the idea that Chemistry is a subset of Physics. So whatever exists in chemistry, also exists in physics.

Now, since mathematics has 6 basic operators, means, Logic must have 6 basic operators, only we call them "connectors".

For Math we have add, subtract, multiply, divide, derivative, integral. For Logic we have AND, OR, Equal-Not, If-->then, Existential quantifier and Universal quantifier.

So to teach High School students beginners logic, I constantly refer to the mathematics.

AP

[Archimedes Plutonium]

Now I can remember well, why I love Winter as the best season of the year, although in the middle of Winter, I complain and say it is the worst season of the year.

 I love Winter because you can spend the whole day on just book writing; not have to go outside for work and chores and by the time you have work done, too tired to write books.

A never ending saga for me--- when Winter comes I complain, when Spring comes, I complain about too many flies. When Summer comes, you name it, and I complain about it. Autumn is nice-- bugs are disappearing and things are shutting down. But Winter, well, that is the supreme best time to write science books.

But let me get started on High School Logic textbook.

I know as a fact, I would not be in science at all, if not for Logic. Logic is what gave me the Plutonium Atom Totality theory in 1990. Without my logic, I would not be a scientist at all.

But the state of condition of Logic, even today in 2026 is a awful and even I would say degenerate state and condition. Just look at College Logic classrooms. They teach OR as add with their truth tables, when even Humpty Dumpty would have enough common sense to realize that AND is "add". They teach that there are two types of OR, an exclusive and a inclusive. Which is so ignorant--- how ignorant is that???? Well, what if math teachers taught you that multiplication has two different types, or that subtraction has two different types???? These are supposed to be teachers, teaching logic and how logical is it for two types of OR, as if they commit a fallacy already. Have they not heard of "science needs unique operators"???????

I am going to teach this High School Logic textbook in the most simple manner I can think of. By directly talking to the reader-student.

I was once a High School teacher of mathematics, and the thing I hated most--- is teaching over the heads of students. I wanted clarity and understanding of students as they learn. And science is easy, super easy to talk over the heads of students. Feynman's Lectures on Physics is over the heads of undergraduate physics. My own High School textbook on physics PSSC was over the heads of High School students. Most math classes teach with a textbook that is over the heads of undergraduate college students.

This book must not be over the heads of any High School student. So I am going to talk directly to the student-reader.

For Logic is one of the most important of all academic subjects for it helps us live our lives better, if properly used.

Here is a outline of the chapters in this book.

For High School logic, I list and talk about the 6 connectors. Reminiscent of operators in math such as add, subtract, multiply, divide.

0) What is Logic?

1) AND which is add
2) OR which is subtract (remove)
3) Equal-Not which in math is multiply
4) If-->then which in math is divide
5) Existential quantifier which in math is seen as the calculus derivative (do not be scared, I will teach you this calculus supereasy)
6) Universal quantifier which is the math calculus called the integral (do not be scared-- it is easy)

Keeping it simple and easy with examples out of what a High School student experiences.

Then I list and talk about some great principles in Logic.

7) Well-defined terms or concepts

8) Non-contradiction
9) Symmetry
10) Non-Sequitur
11) Ad Hominem
12) Mis-identification
13) Consistency
14) Occam's Razor-- a better name for this is Experiment Completeness

15) Completeness

So, lets get started and let the talk teaching begin.

0) What is Logic

_________________

Mathematics is well developed and a perfect science to model Logic. So we use mathematics to help us understand what logic is. Math plays with numbers and geometry figures. In the same way, Logic is a play with ideas. Instead of numbers, logic plays with ideas. That is about as simple as to say what Logic is. Logic tries to make ideas clear, straight, and correct to help form conclusions from ideas.

Math is numbers. Logic replaces numbers with "ideas".

1) AND in Logic is Add of math

----------------------------------------------------

[Archimedes Plutonium]

On Wednesday, June 10, 2026 at 3:07:47 AM UTC-5 Archimedes Plutonium wrote:

(snipped)

So, lets get started and let the talk teaching begin.

0) What is Logic

_________________

Mathematics is well developed and a perfect science to model Logic. So we use mathematics to help us understand what logic is. Math plays with numbers and geometry figures. In the same way, Logic is a play with ideas. Instead of numbers, logic plays with ideas. That is about as simple as to say what Logic is. Logic tries to make ideas clear, straight, and correct to help form conclusions from ideas.

Math is numbers. Logic replaces numbers with "ideas".

So, what is an "Idea" which is the fundamental unit of Logic. The fundamental unit of mathematics is numbers and geometry figures.

An Idea in Logic can be as simple as a single word or complicated as a sentence or even a paragraph long. But, also, an idea can be a single picture or sequence of pictures or images.

Math is numbers and geometry figures.

Logic is word or words and picture-images.

[Archimedes Plutonium]

1) AND in Logic is Add of math
----------------------------------------------------

So we have an idea, an idea of food and we say a single word.

Spaghetti.

Now we have another new idea of Meatball.

So we say: "For dinner tonight, I want spaghetti and meatballs.

We can replace "and" with the math term "add". In fact, whenever we say "and" in a sentence we can replace it with "add".

Math has numbers and you have to combine numbers by using operators of math such as add, subtract, multiply, divide. Same thing goes for Logic, we have "ideas" and now we want to play around with ideas to form new ideas or to form a conclusion from ideas.

When I was in High School, my favorite dinner was spaghetti and meatballs and a cola drink with ice cubes. That was then in the 1960s but now, I can only eat and drink organic food and I no longer drink soft drinks as too much sugar.

Homework: Write a Logic sentence or paragraph of your favorite dinner using the "And" connector many times. Then replace the And with Add in that same sentence or paragraph.

A more complex logic sentence/paragraph is now shown.

(A) Physics is the science of matter and motion, and force and energy, and chemistry is the science of matter and the chemical bond.

We can break down that complex sentence into this.

Physics is the science of matter.

And

Physics is the science of motion.

And

Physics is the science of force.

And

Physics is the science of energy.

AND

Chemistry is the science of matter.

And 

Chemistry is the science of the chemical bond.

For Homework: Analyze this complex logic sentence/paragraph, and break it down into individual sentences.

(B) Astronomy is the science of all astronomical bodies in Space and whether they are stars like the Sun, and planets and their satellites, and asteroids and comets, and other stars in the Milky Way galaxy and other galaxies. And Geology is the science of just planet Earth and its surface, and mantle, and two cores.

Break that complex paragraph into individual sentence ideas.

[Archimedes Plutonium]

Now in many chapters I need to caution and warn the students where Old Logic failed and has huge mistakes. Mistakes in textbooks and on Internet and Wikipedia.

The big mistake in Old Logic with AND is that the pioneers of logic Boole and Jevons thought AND is subtraction and that OR is addition. A grotesque mistake which college textbooks like Copi's still abide by and that colleges and universities across the world still teach Old Logic that AND is a form of subtraction while OR is addition.

You can see this on the inside cover of Copi, Introduction to Logic, 4th edition, 1972 where Copi wrote this.

9. Addition (Add.)

p

therefore p OR q

The special reason AP is writing 5 textbooks on Logic, is because current logic taught in schools , colleges, universities are a cesspool sewer of error.

[Archimedes Plutonium]

2) OR which is subtract (remove)

---------------------------------------------------

We come to the second Logic connector which happens to be the math operator subtract. In fact, in the history of math, they sadly put the name subtraction to that of remove. They should have used the word Remove and ditched the word "subtract". For remove is more precise in what is going on. If I said 5 remove 3 equals 2 makes far more commonsense than if I say 5 subtract 3. Reason: if you never learned math before, you would know what "remove" means but you would not know what subtract means. And also, if we see 5 dogs in the yard and asked you to Remove 6 dogs, you would say impossible, but because Old Math had subtract that 5 - 6 =-1 and up pops a bad and lousy concept that really does not exist-- the negative numbers. If Math had never used subtract, then it may have been saved and spared of the error of negative numbers which are as false as saying witches exist and fly around on broomsticks. We see how important it is to properly name things. 

AND was Add as a logic connector, and now we have the reverse of AND which is OR, the remove. One is add, or join, while OR is remove.

A) For dinner tonight, I cook and eat a hamburger OR I fix a peanut butter jelly sandwich. 

Analysis:: Cook and eat a hamburger Remove peanut butter jelly sandwich. Remove cook and eat hamburger; fix a peanut butter jelly sandwich.

Homework: student make up your own OR argument and provide an analysis.

B) Smilodon, the saber toothed cat was genuine a cat with oversized canine upper jaw teeth OR it was a normal cat whose upper jaw was found fossilized with walrus tusks and where the museum screwed or glued the walrus tusks onto a normal cat jaw. To prove one way or the other, either a scientist measures the DNA of the saber tooth to be of a walrus, or a cat.

Analysis:: Smilodon, the saber toothed cat when DNA tested of the upper jaw and the saber teeth, if found to be both cat DNA then Remove walrus tusk idea. Smilodon, the saber toothed cat when DNA tested of the upper jaw and the saber teeth, if found that the teeth are walrus DNA, then Remove the idea that Smilodon was a saber tooth tiger.

Homework: student make up a science OR argument and provide an analysis.

[Archimedes Plutonium]

The subject of Logic is very picky about order and sequence. What follows next. Logic is order.

We start the connectors of Logic with AND as add or join and then we followed with OR, subtract -- remove.

Now we go, like in math we go to multiply and then divide.

But there is a problem here, for multiply in logic connectors is Equal-Not, and divide is If-->Then. 

It was easy to see that AND is Add and that OR was subtract --- remove.

Very difficult to see that Equal-Not is multiply, or that If-->Then is divide.

So at this moment in time of teaching Logic, we must start to learn what Calculus is. And I will teach you the easiest way to learn calculus.

For the integral is Fast Add which is multiply; while the derivative (differentiation) is fast subtract which is divide.

So here, I stop and teach function, integral calculus for multiply and derivative calculus for division.

It is impossible to properly teach Logic without teaching rudimentary calculus.

3) Equal-Not which in math is multiply
4) If-->then which in math is divide
5) Existential quantifier which in math is seen as the calculus derivative (do not be scared, I will teach you this calculus supereasy)
6) Universal quantifier which is the math calculus called the integral (do not be scared-- it is easy)

Before I explain function, derivative, integral, let me show you pictures which sums it all up. These pictures are actually a proof of the biggest theorem-proof in calculus, called the Fundamental Theorem of Calculus.

From this:
        B
        /|
      /  |
 m /----|
  /      |
|A      |
|____|
a      b


The trapezoid roof has to be a straight-line segment (the derivative)
so that it can be hinged at m, and swiveled down to form rectangle for
integral. The area of the rectangle is the integral.

To this:

__m__
|         |
|         |
|         |
---------
a        b


So, we have two items in Calculus for this theorem, we have a derivative, the straight line segment A to B with m in the middle. And we have the rectangle area which we call the integral.

We draw in m, the midpoint because that is where we have a hinge, we imagine a hinge there. In fact, some teachers will build this model in wood working class just to use in math class.

So, Calculus has two items-- the derivative which is the rooftop, the straightline. And the other item, the integral which is the rectangle area.

So, what is this theorem all about?

Well, it says that-- if you have a rectangle with a midpoint on its top side.

__m__
|         |
|         |
|         |
--------- 

a       b

That you can cut a right triangle from the midpoint

__m__
|  /      |
|/        |
|         |
--------- 

a       b

Cut that right triangle and swivel it up to make the trapezoid

        B
        /|
      /  |
 m /----|
  /      |
|A      |
|____|
a      b
Or, you can start with that trapezoid and swivel the right triangle downwards to make the rectangle


__m__
|  /      |
|/        |
|         |
---------
a       b

And, basically that is the Calculus at its most simple form. Where the slanted line is the derivative and the rectangle area is the integral. So, there, 15 year olds, you have just learned the fundamental basics of Calculus. Take a rectangle, swivel the right triangle and you have a derivative. Take the trapezoid, swivel the right triangle to form a rectangle area and you have the integral.

Basically, that is all that Calculus is.

Homework:: Take a sheet of paper, a used sheet, for no need to ruin a fresh sheet of paper. And find the midpoint and form the right-triangle and swivel up the right triangle forming the trapezoid.

[Archimedes Plutonium]

I am trying to make the world's most simple explanation of Calculus, for High School students.

I am sure it must be a Geometry picture.

An explanation that even those who hate math, can understand.

Calculus is the science of motion, of change.

Calculus involves division in the derivative-- the slope, as we carve out the right triangle inside the rectangle and lift it up at the midpoint.

Calculus involves multiplication as the integral --- the rectangle (sometimes a square).

[Archimedes Plutonium]

So, I am striving for the most simple of all explanations of Calculus. The Finest- Simple Teaching of Calculus.

And what I need to accomplish that are 3 lines of thought. The first line of thought is what the true numbers of mathematics are. They are not the Reals with their continuum. You cannot have a calculus when the numbers form a continuum. You need empty space from one number to the next number.

The second line of thought to teach the most simple and easy Calculus is to well define what a Function is. Old Math did a good enough job in defining a function, but they should have carried it further to the idea that the Polynomial Function is the only valid function in all of mathematics, and that any of the other so called functions are just horrible silly and stupid aberrations. I say this because anyone that has completed first year college calculus, can testify along with me, that the Polynomial because it obeys the Power rule for derivative and integral--- just add or subtract 1 from exponent, yet every other so called (idiot function) has no easy rule--- especially the awful ugly trig functions. So, well, Stewart's beloved Old Math Calculus book is approaching 1500 pages, and where some girls would have a hard time of even picking up the tome. When if you made POLYNOMIALS the only valid function in all of math, Stewart, I am guessing could have written a Calculus textbook covering all of what he covered in just 200-250 pages.

It amazed me in Freshman Calculus class at University of Cincinnati, 1968-1972, how thoroughly easy Calculus is, if the only function was the Polynomial. Because of its charming easy Power Rules-- add or subtract 1 from exponent. And I am sure, although I as a teenager would not have known this in 1968, I am sure that the professors of math in most every college and university across the world would have encountered and known of the Lagrange Interpolation. This is a fact that any function that is not Polynomial, is easily transformed into being  a polynomial. Can we say the reverse of that is true?????? 

What I mean is can we say that any function can be transformed into a trigonometry function???? Or a logarithmic or exponential function????????

Is it true only for the Polynomial Function--- that hand me any obnoxious, obscene, dirty, defiling, pornographic function such as trigonometry, exponential, logarithmic, hyperbolic, and turn all other functions into that particular type of function, or is it the case that only POLYNOMIAL functions have that supreme feature and characteristic of turning stupid idiotic other functions into being another polynomial.

Third, is the picture diagram given above of the rectangle and a midpoint on top and then carve out a right triangle for which when hinged (up or down) is the derivative while the rectangle is the integral.

It makes sense, that true calculus is taught not in math classes in Colleges and Universities from math professors who know little to no logic, but that the true calculus is taught by a Logician in his 5 book series of Logic textbooks.

So, I need to cover 3 concepts (1) true numbers of math (2) what are functions and what is the Polynomial function (3) the geometry picture diagram of the proof of the Fundamental Theorem of Calculus.

I already have the picture diagram, and now talk about the true numbers and the function concept.

AP, King of Science

[Archimedes Plutonium]

I am mindful that this is High School.

1) True numbers of mathematics. In the year 1900 a famous physicist announced to the world what would later be known as the birth of Quantum Mechanics. this physicist was Max Planck and what he discovered is that physics comes only in discrete quantities. The word "quantum" means discrete. We could today call it Discrete Mechanics.

Now if physics finds that the world of physics is all quantized, then what should have happened after the year 1900, is that all math professors and the college and university math departments should have payed attention to physics and asked the question--- are the numbers of mathematics quantized also???? But no, that never happened because mathematicians from year 1900 onwards were mostly dull and stupid as they further dived deeper into continuums of their Real numbers with rationals and irrationals all mixed together, cobbled together along with negative numbers, and where Paul Cohen dives deeper into quagmire of a "continuum hypothesis".

Not until 2013 does a mathematician, AP, while writing his book "True Calculus" does it become apparent that no Calculus can exist when the numbers of math are a continuum, for the simple reason, the derivative when hinged up at midpoint, must fall on the very next number that the function graph pinpoints. The derivative intercepts the next coordinate point of the function graph itself. If the numbers of math form a continuum, the derivative cannot fall on a ---- next coordinate point---.

So the true numbers of math have to have gaps and holes in between one number and the next number. The true numbers of mathematics are the Decimal Grid Numbers and the smallest of these is the 10 Grid, next comes the 100 Grid, next the 1000 Grid.

Here is a picture of the Decimal 10 Grid.

9.0, 9.1, 9.2, 9.3, 9.4, 9.5 9.6, 9.7, 9.8, 9.9, 10.0
8.0, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9,
7.0, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9,
6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9,
5.0, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9,
4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9,
3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9,
2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9,
1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,
0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 

There are exactly 100 numbers not counting 0 in the Decimal 10 Grid.

If I wrote out the 100 Grid it would start with 0. 0.01, 0.02 and end in 99.98, 99.99, 100.

So, between 0 and 0.1 exists no other number but is empty space. Same for 0.1 to 0.2. 

In order for Calculus to exist at all, you need empty space from one number to the next number. For the derivative in the picture diagram spans with a straight line the next point on the function graph. This is why physics needs calculus derivative for it predicts the next point of the function and is part of the function graph itself. Old Math was too stupid and they believed the derivative is a "tangent line to a point on the function graph". No, the derivative connects the previous point to the next point of the function graph and is part and parcel of the function itself.

Now we talk about the concept of Function.

Old Math got this correct in the idea that the Function is a correspondence of a given x-value to a unique one of a kind y-value.

I like to view a Function as a Motion, and calculus derivative as "in motion".

So in true math, there is first quadrant only for there are only positive numbers, no negative numbers. In Old Math, math professors had a difficult time of gaining fame and fortune. So, whenever a professor wants some fame and fortune, if they dream up some silly outlandish idea--- "hey, negative numbers exist". They get publicity and fame and fortune follows. Not content and happy to make a career in teaching just True Math, no, they want fame and fortune which then pollutes math.

In True Math we have only positive numbers and need for a graph to use only 1 quadrant.

Here is the start of 1st Quadrant where x and y axes are Decimal 10 Grid.

 ^

.4|

.3|

.2|

.1|

0|___________________>

       .1   .2   .3    .4

Now for teaching purposes, I will do the Integers only in 10 Grid and graph the function f(x) = x^2. We write a function as f(x), but I prefer to write a function as x^2 --> Y, or Y--> x^2.

The function x^2--> Y is a motion across the x-axis, starting at 0 and taking in every x-value number. Remember that the definition of function is there is a unique y-value given a x-value. A circle cannot be a function because many x values have 2 y-values. A half circle avoids this problem.

So looking at the function x^2 -> Y and we make a table in 10 Grid integers only.

x^2 -> Y
x       y
0      0
1      1
2      4
3      9

4      16

5      25

6      36

7      49

8      64

9      81

10    100

So we plug into the x^2 all the 10 Grid integer values on the x-axis and start making a table.

Then, we graph our table.

y-axis
^
|
|
|
                             
9                         /| 9
                            |
                           |
                           |
                         / |
                           |
                           |
                       /   |
                           |
                    /     |
                          |         
4            4/ |        |
                 |        |
           /     |        |
                 |        |
1    /  |1      |        |
   /     |       |        |
------------------------------------------------> x-axis
0      1       2       3

So let us summarize what a function is.

A function is a assigning a number on x axis with a unique number on y-axis. We denote a function with the symbol of an arrow ->. This symbol comes from the if-->then logic connector. The equality symbol = comes from equal-not in the logic connectors. We generally write a function with the arrow from the x axis to the y axis value, such as x -> Y, or x^2 +1 -> Y.
A function has a Y value side and the other side is the x value.
We use up every number in the x-axis of a Grid System to make our table. That is important, we use up every number on the x-axis to make our table. For this gives the function that of --- Motion---. Motion as it moves from 0 to every x value point.
Once we made our table we plot the function as a graph.
A function has a restriction, though. A function must have one y value for any given x-value. So for example a half a circle can be a function but not a full circle because most x values have two y values.

[Archimedes Plutonium]

Now, the student can see and sense the motion of a function, it always moves from 0 to the next point and picks up a y value then it moves to the next point of x axis and picks up a new y-value. This is motion of a function. And the student can see that in the function x^2 --> Y that the first coordinate point is (0,0), the next is (1,1), the next is (2,4) and the next is (3, 9)

For homework, the student figures out what the following x value coordinate points are for 4, 5, 6, 7, 8, 9, 10.

Now, we bring in our picture diagrams of derivative and integral, but we must stop here and talk about midpoints of cells.

Take a look at the graph above of x^2-->Y. Now we are going to ignore the interval for x = 0 to x=1, for the simple reason that multiplication of small numbers yields an even smaller number. What is the midpoint of 1 to 2???? That would be 1.5. Now plug into x^2 --> Y that of 1.5 and that gives me 2.25. 

Homework:: Get a sheet of graph paper and mark out the function x^2 --> Y with midpoint of 1 to 2 is 1.5 and draw in the rectangle of 1 to 2 width with height 2.25, and draw in the right triangle that will span from (1.5, 2.25) to reach (2,4). Compute the rectangle for interval 2 to 3 with midpoint 2.5 and draw in the rectangle of 2 to 3 width with height of ___ and draw in the right triangle that is going to reach (3,9).

[Archimedes Plutonium]

Philosophy of Math problems I had while in High School

----------------------------------------------------------------------------------------

Well in New True Math we have no negative numbers and that causes far less problems when the only numbers that exist are positive numbers. When you have the Axiom that you cannot subtract more than what is available, you are released from a super burden of nonsense.

In High School we spent a-lot of time on solutions of the Quadratic equation ax^2 + bx + c = 0, which is a polynomial but that time spent was mostly a brainwash waste of time. Old Math never understood that 0 is a special number and it should never be all alone on the right side of the equation. Instead, a positive Decimal Grid Number should be alone at all times on the rightside of the equation. When this happens, the solution for the Quadratic equation is as easy as organic cherry pie with organic whipped cream on top. Another missing axiom in Old Math as you can never have an equation with 0 all alone on the rightside.

But there was one monster hardship that was extremely difficult for me to cope with while in High School. It is true but very difficult to explain. It occurs when you multiply two fractions of 1 together and you end up with a number that is smaller than either of the multipliers.

For example: 1/2 x 1/2 = 1/4 or in decimals 0.5 x 0.5 =0.25. The problem comes in when presented with multiplication as area.

So I have a square here.

_________

|                |

|                |

|_________|

Now I am told the side is 3, fine, the area inside is 3 x 3 =9. No problems there.

I am now told the side is 1, no problems there for 1 x 1 =1 for interior area. Still no problems come to mind.

However, now I am told the side is 0.5 and the interior area is 0.25. Here I have a problem in the mind. Same square as 3x3 or 1x1 but the side is 0.5 x 0.5 and the area inside is LESS than either of the sides.

In High School, the books and teacher try to justify or reconcile this problem by saying by saying area of 0.5 x 0.5 =0.25 for Area is measured in how many unit-squares are inside a square. There is only 1/4 of a unit square in 0.5 x 0.5.

But still that is unsatisfying to the mind as a answer.

Another explanation goes on the lines of ---- if you multiply a number less than 1 by another number less than 1, your answer must be smaller than either of your multipliers.

I bring this Conundrum up because the Integral of Calculus as the area inside each cell, starts to make Sense once we reach 1 and go beyond 1, while if we focus on the integral in all the cells of 10 Grid before we reach 1 with 0 to .1 or .1 to .2, or .2 to .3 etc. We run into this difficulty that the derivative has a hard time of spanning to where the next coordinate point of function graph lies.

It is appropriate that Logic Class fixes this problem for the math professors of the world could not fix their wrong and muddleheaded Calculus, for they never trained in Logic and have no logical minds to fix math conundrums.

AP, King of Science

[Archimedes Plutonium]

On Saturday, June 13, 2026 at 6:24:12 PM UTC-5 Archimedes Plutonium wrote:

Philosophy of Math problems I had while in High School

----------------------------------------------------------------------------------------

Well in New True Math we have no negative numbers and that causes far less problems when the only numbers that exist are positive numbers. When you have the Axiom that you cannot subtract more than what is available, you are released from a super burden of nonsense.

In High School we spent a-lot of time on solutions of the Quadratic equation ax^2 + bx + c = 0, which is a polynomial but that time spent was mostly a brainwash waste of time. Old Math never understood that 0 is a special number and it should never be all alone on the right side of the equation. Instead, a positive Decimal Grid Number should be alone at all times on the rightside of the equation. When this happens, the solution for the Quadratic equation is as easy as organic cherry pie with organic whipped cream on top. Another missing axiom in Old Math as you can never have an equation with 0 all alone on the rightside.

But there was one monster hardship that was extremely difficult for me to cope with while in High School. It is true but very difficult to explain. It occurs when you multiply two fractions of 1 together and you end up with a number that is smaller than either of the multipliers.

The human mind is shaped to think that when you multiply two numbers together, the answer must always be larger than either of the multipliers. This is the psychology of the problem. But we can reconcile that by thinking, fractions of 1 as multiplier reduces the end result.

But when teaching True Calculus, we have an added difficulty, in the numbers from 0 to 1 on the x-axis. Because the Derivative needs to span a distance to get to the next coordinate point as we carve out a right triangle in the Rectangle that is the integral.

You see, in my example of the function x^2 --> Y, I started with 1 and went with cells from 1 and beyond 1.

If I asked for the integral in cell 0 to 0.1 or from cell 0.1 to 0.2, the problem is--- can the derivative of a right triangle carved out of a rectangle, can it actually reach the next coordinate point.

For that same example from 1 to 2 cell we took the midpoint as 1.5 which is 2.25 and we can carve a right triangle out for the derivative to span from (1,1) to that of (2,4). Same goes for x= 3 the midpoint of 2 to 3 is 2.5 and that would be x^2 = 6.25 and easy to carve out a right triangle in that rectangle to span from (2,4) to reach (3,9).

But, once again, the cells of the function graph in 10 Grid or 100 Grid, from 0 to 1, the problem is, can the integral rectangle be enough to carve out a right-triangle for a derivative to span to reach the next point.

For example: 1/2 x 1/2 = 1/4 or in decimals 0.5 x 0.5 =0.25. The problem comes in when presented with multiplication as area.

So I have a square here.

_________

|                |

|                |

|_________|

Now I am told the side is 3, fine, the area inside is 3 x 3 =9. No problems there.

I am now told the side is 1, no problems there for 1 x 1 =1 for interior area. Still no problems come to mind.

However, now I am told the side is 0.5 and the interior area is 0.25. Here I have a problem in the mind. Same square as 3x3 or 1x1 but the side is 0.5 x 0.5 and the area inside is LESS than either of the sides.

In High School, the books and teacher try to justify or reconcile this problem by saying by saying area of 0.5 x 0.5 =0.25 for Area is measured in how many unit-squares are inside a square. There is only 1/4 of a unit square in 0.5 x 0.5.

But still that is unsatisfying to the mind as a answer.

Another explanation goes on the lines of ---- if you multiply a number less than 1 by another number less than 1, your answer must be smaller than either of your multipliers.

I bring this Conundrum up because the Integral of Calculus as the area inside each cell, starts to make Sense once we reach 1 and go beyond 1, while if we focus on the integral in all the cells of 10 Grid before we reach 1 with 0 to .1 or .1 to .2, or .2 to .3 etc. We run into this difficulty that the derivative has a hard time of spanning to where the next coordinate point of function graph lies.

It is appropriate that Logic Class fixes this problem for the math professors of the world could not fix their wrong and muddleheaded.

You see, the Calculus adds a new dimension to the confounding conundrum of multiplication of fractions of 1.

Whenever we multiply two numbers, both of which are either 1 or larger than 1, we always have an answer that is either 1 or larger.

Can you proof that in a theorem??????????? 

[Archimedes Plutonium]

I have been stuck for 3 days now, over this one issue.

I am not going to do truth tables in High School Logic. The trouble with that omission is explaining how Equal-Not connectors is Multiplication. With Truth Tables I am able to easily show Equal-Not is multiplication.

AND is add (join); OR is subtract (remove); If--> Then is seen as divide. Leaving us with Equal-Not. Not intuitive at all that Equal-Not is multiplication.

Then I add on Derivative which is again division, especially knowing the very definition of derivative is dy/dx. And integral as area under the function graph is seen as multiplication for area of rectangle.

But still, Equal-Not seems far distant in saying it is multiplication.

By elimination, Equal-Not is multiplication since it is the only connector remaining. But that is not satisfying.

Again, if I brought in the Truth Tables, I can affirm Equal-Not is multiplication. But I do not want to do that.

So, I give it a few more days to somehow explain Equal-Not is Multiplication.

Can I link Equal-Not to Integration????? If so, well, I would then have the explanation.

AP, King of Science

[Archimedes Plutonium]

The fourth day is a wonder.

Solved!!!! The answer is somewhat Cool.

Question was. Can we explain that Equal-Not of the 6 connectors of Logic is multiplication of math????

Logic

---------

AND

OR

Equal-Not

If-->Then

Existential quantifier

Universal quantifier

Paired up with mathematics operators

------------------------------------------------------------

add

subtract (remove)

multiply

divide

derivative 

integral

Without using Truth Tables, the pairing of AND with add and of OR with subtract was easy to see.

Then the difficulty set in--- how on Earth is Equal-Not that of multiplication?????

Divide was somewhat easy as we use Calculus derivative for it is defined as dy/dx, the slope of the function graph. It is division. Then we look at the Integral for it is area under the function graph. The area inside small slender rectangles I call cells. In Decimal 10 Grid each successive number 0, .1, .2, .3, . . . ,9.8, 9.9, 10 forms a cell that is at least 0.1 wide and how long it is depends on the y-value.

But the hardship was that of Equal-Not. That is not intuitively multiplication. If I ask anyone in college today--- is it intuitive that multiplication is Equal-Not?????? No. But it is intuitive that AND is add or join while OR is subtract or remove.

Solution: It is not obvious that Equal-Not is multiplication, unless we can say that the Integral is Equal-Not. For the Integral is area under the function graph. Area is multiplication of length times width of rectangle.

When you study True Calculus, you will learn that the derivative of the integral returns us to the original function graph. And the integral of the derivative also returns us to the original function graph. When the only valid function in all of mathematics is the Polynomial function and the most simple of polynomial functions are constant functions such as Y--> 2 or Y--> 10 a flat straight line when graphed. The next most simple function after constant functions is the Identity function Y--> x which is a diagonal upward line that bisects the 1st Quadrant in half.

^

|    /

| /___>

In Calculus with polynomials the only valid functions, for if not a polynomial already, you can turn it into a polynomial. That the derivative of all polynomials and the integral of all polynomials obey what is called the Power Rule.

For the derivative the Power Rule involves a subtraction of 1 from exponent and for integral involves a addition of 1 onto exponent.

The identity function Y--> x has a derivative being 1 while the integral of x is (1/2) x^2.

If I take the integral of 1 it turns out by the Power Rule to be x.

If I take the derivative of (1/2)x^2 it turns out to be x.

Here is the Power Rule for derivative and for integral.

--- quoting my 45th book of science TEACHING TRUE MATHEMATICS: Volume 2 for ages 5 to 18, math textbook series, book 2

by Archimedes Plutonium

Written in 2 May, 2019 ---

The highlight of this year is the Power formula for Calculus. So this means quite a bit of Algebra.

We learn for the Derivative Power formula of a polynomial x^n that the derivative is n(x^n-1).

So for example the function x^2 -> Y its derivative using the power formula is 2 (x^2-1) = 2x

Do you see how we got that?? Probably not, so let us do it in slow-motion.

We have a function x^2 -> Y and asked to do the Calculus derivative upon x^2. We use the power formula which says, we drop that exponent number down to be a coefficient. The exponent is 2 so we drop it down

2 (?)

Now the rule tells us to do a n-1 on the exponent n. Our exponent in x^2 is 2, so what is 2-1 ? and it is 1.

So our answer is 2x.

Now try another function say x^3 -> Y, and so our exponent is 3 and we drop it down

3 (?)

Now the rule says do a n-1 on that exponent and so we do a 3-1 and get 2. So our final answer is

3x^2

Try another, say our function is 3x^2 -> Y. What is our exponent? It is 2 and we must drop it down as being a coefficient.

3x2 (?)

Now what is n-1 ? It is 2-1 = 1 so our final answer is :

3 times 2x which is 6x.

The Integral Power Formula is sort of the opposite, actually the reverse of the derivative formula so for polynomial x^n that the integral is (1/(n+1)) times (x^(n+1)). In the derivative we subtract, in the integral we add. For example the integral of x^2 -> Y is (1/(2+1)) times (x^(2+1)) = 1/3x^3.

Let us try another integral of x^3 -> Y. What is our exponent? It is 3, so our n+1 is 3+1 = 4 and that gives us 1/(n+1) as being 1/4.

1/4(?)

Now what is our new exponent of x^(n+1) and it is x^4 so our final answer is :

1/4x^4

The derivative is subtraction of 1 from exponent, the integral is addition of 1 to exponent.

And that is all there is to Calculus, provided that our functions, all functions are polynomials.

So let us do many exercises.

The important idea to learn is the Power Formula so you an easily do all of Calculus, all of Calculus once we have all functions converted to polynomials.

Power formula for Differentiation x^n ->Y then nx^(n-1) -> Y'

Power formula for Integration x^n -> Y then Integral is (1/(n+1))* x^(n+1) -> Y_int

--- end quoting my 45th book of science---

SOLUTION to Equal-Not being Multiplication.

The solution involves CONSISTENCY. The only way you can have Calculus where the derivative ---of--- is multiply integral returns to the original funciton graph and where the integral ---of--- is multiply derivative returns to the original function graph, is when Multiplication comes from Equal-Not.

AP

[Archimedes Plutonium]

Yes, in my 45th book of science,  45th book of science TEACHING TRUE MATHEMATICS: Volume 2 for ages 5 to 18, math textbook series, book 2  by Archimedes Plutonium  Written in 2 May, 2019 , I teach High School students the basic foundation of Calculus even before they reach college or university. I teach the Polynomial is the only valid function with the True numbers of math as Decimal Grid Numbers and I teach the Power Rules formula for derivative and integral.

When your math is all lousy, fake and flawed, you can not teach Calculus in High School because you have the ignorant stupid Limit concept which is a fakery to cover up the fake math taught in schools.

Old Math saying that derivative is a tangent line to function graph when in truth, the derivative is part and parcel of the function graph itself.


It is easy to understand that Logic has to cover everything done in Math
---------------------------------------------------------------------------------------------

Math has 6 crucial operators-- add, subtract, multiply, divide, derivative, integral.

That means Logic must have at least 6 crucial connectors-- AND, OR, Equal-Not, If-->Then, Existential quantifier, Universal quantifier.

It is super easy to see AND is add and OR is subtract or remove.

But then we were stuck with Equal-Not, for how on Earth is that multiply?????

It is easier to see that If-->Then is divide and that Existential quantifier is also divide for the derivative is defined in calculus as a division dy/dx which is called the "slope" in geometry.

And it is known that the integral in calculus is area of rectangle of length times width, a multiplication.

But how did we reach the conclusion that Equal-Not is multiplication. That was not clear.

But we know that the derivative of a integral returns us to the original function and that the integral of derivative must also return to the original function graph.

For example the Function x^2 --> Y. Its derivative from power-rule is 2x and its integral from power-rule is (1/3)x^3.
If we take the derivative of (1/3)x^3 by power-rule, we return to x^2, and if we take the integral of 2x by power rule is 2(1/2)x^2 we again return to x^2 the original function.

So from this observation we can see that Equal-Not is Multiplication for the derivative times (multiply) integral must equal original function graph and integral times (multiply) derivative must equal original function graph.

Equality-Not is tied into multiplication to make Calculus consistent.

AP, King of Science

[Archimedes Plutonium]

So when I went to University of Cincinnati, 1968 to major in mathematics, in Freshman Calculus was presented to us the "composite function". My textbook back then was Fisher & Ziebur "Calculus and Analytic Geometry" 1965, and on page 40, Fisher & Ziebur introduces the "composite function" but they never give an example in polynomials. By that time in my life in mathematics, I had turned off on trigonometry for trig is only a tool in math to be never used in calculus or math itself, but for such purposes as engineering or surveying when you know a few numbers and angles and need trigonometry to find the other missing numbers.

You see, trigonometry is graffiti math and what I mean by that is the fact that you cannot graph trig functions in mathematics by having angles be the x-axis and numbers be the y-axis. Math must always be consistent. If you have numbers on Y-axis, you must have numbers on x-axis. Look in any Old Math textbook for a graph of sine or cosine and you see these funny squiggly up and down curves called the sinusoid. Most mathematicians never take Logic while in school to learn that you cannot have the x-axis be apples and the y-axis be raccoons. So, when you have a math teacher who has taken logic in college and teaching trigonometry, they should have the good sense to know that when you graph sine or cosine and the axes are both numbers, that the sine is a semicircle function along with cosine and ____not sinusoid___. And this makes absolute common sense reasoning because the sine and cosine are defined from inside a circle of right triangles going around inside the circle. 

Got it!!! ---- If the definition of sine and cosine are from right triangles sweeping out the inside of a circle----- Then, therefore, the graphs of sine and cosine involves semicircles, and never Sinusoid.

You see, when you lack Logic, it is easy to trip and fall all over oneself when doing math or science. For the person cannot think straight, clearly or truthfully or correctly.

So let me switch over to my favorite Old Math textbook on Calculus--- that of Stewart. Mind you, Stewart did a great job in writing calculus, but still--- his is illogical for it uses the wrong numbers of math, and uses thousands of stupid functions when math has only one valid function--- the polynomial. And Stewart never had a geometry proof of Fundamental Theorem of Calculus to warn him that the derivative is actually part and parcel of the function graph itself when Space is discrete and numbers have holes and gaps in between one number and the next number.

Stewart "Calculus" 5th edition, 2003, on page 44 says: Find the composite function f(x) = x^2 and g(x) = x-3, where both functions are polynomials. And Stewart goes on to show that f*g = (x-3)^2 while g*f = x^2 - 3. Stewart goes on to say that in general f*g is Not Equal to g*f.

AP, King of Science

[Archimedes Plutonium]

I wrote:

Stewart "Calculus" 5th edition, 2003, on page 44 says: Find the composite function f(x) = x^2 and g(x) = x-3, where both functions are polynomials. And Stewart goes on to show that f*g = (x-3)^2 while g*f = x^2 - 3. Stewart goes on to say that in general f*g is Not Equal to g*f.

Now, earlier, on page 42-43, Stewart describes how you can take two functions and add, subtract, multiply, divide them.

So, the question is, can we say that the composite function f*g is a form of multiplication or in a category all by itself????

When I was revising the Maxwell Equations of physics, I often said that the multiplication was "generalized multiplication" of using either normal multiplication, scalar, or vector dot product or vector cross product.

Is that what we have here? That we have two forms of multiplication of functions--- the normal multiplication and the composite function? 

Well, the product of two functions is not the same as the composition (composite) function. The two are distinct. But can the two be called forms of multiplication like my three forms of multiplication of scalar, dot product, cross product???

I believe so, that it is the case where Function product is one form of multiply and composite another form of function multiply. I say this because of what is known as the Chain Rule of calculus that involves composite functions. For the derivative of the composite function F*G is the product of the derivatives of F and G.

But all of this above is beyond High School, and Elementary Logic and will be taken up in Intermediate Logic and Advanced Logic textbooks.

AP, King of Science

[Archimedes Plutonium]

Interesting that the Set theory has different forms of multiplication in the Cartesian Product and then there is the Power Set of 2^n for all its subsets.

[Archimedes Plutonium]

Continuing onwards....

3) Equal-Not which in math is multiply.

An important feature of combining Equal with Not as one connector of Logic is the concept of Contradiction which plagues clear thinking.

When a person makes a contradiction, they end up saying or believing that "the sky is blue" and also the "sky is not blue".

Logic abhors the contradiction, and whenever a contradiction arises, we must stop everything we are doing and fix the problem before resuming.

[Archimedes Plutonium]

I backtrack here.

For decades now, I have been struggling with 3D calculus. It is obvious the "classical calculus" is 2 dimensional. And I have tried to make it natural in 3D, with limited success.

Funny, that here in writing High School Logic, that I run into this idea of 3 types of multiplication in physics-- scalar, vector dot product, vector cross product.

Perhaps I have been far too hard on Trigonometry. For trig is involved with dot product and cross product.

Trig is necessary in the Faraday Law Structure and in Optics of Snell's Law Structure and refraction.

So, to be prudent, I need to re-look this problem of the 3D Calculus for we have 3 types of multiplication.

Maybe the 3D Calculus involves 2 dimensions and a third dimension of trigonometry and angles. 

Re-investigate the 3D Calculus.

AP, King of Science

[Archimedes Plutonium]

This goes to show, that sometimes, when trying to teach a simple subject matter--- Logic to High School, that the exercise may bring forth the most profound ideas in science.

For decades now, I have been trying to solve ---- What is 3D Calculus, for classical calculus is all in 2nd dimension.

I feel I am coming closer to a solution, and, would you not know it, it involves saving and using trigonometry.

The Multiplication in Physics can be the normal multiplication, scalar quantities but it can be a Cross product which involves Sine or it could be a Dot product which involves cosine. 

Physics needs trigonometry for the Faraday Law-structure of the thrusting bar magnet through coil, where the magnetic lines of force of magnet count only those perpendicular to the coil versus magnet and thus trigonometry is required. And in Snell's law in Optics requires sine trigonometry for refraction.

If Physics needs a piece of math, we can never throw it out.

This discussion, of course belongs in my Intermediate Logic and Advanced Logic textbooks, not my High School textbook.

So, what I am thinking at this moment if the formula of a Torus has the Sine and Cosine in it and according to Wikipedia, it surely does so.

--- quoting Wikipedia on torus formula---

Wikipedia, the free encyclopedia

Not to be confused with Taurus.

This article is about the mathematical surface. For the volume, see Solid torus. For other uses, see Torus (disambiguation).

A ring torus with a selection of circles on its surfaceAs the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a double-covered sphere.A ring torus with aspect ratio 3, the ratio between the diameters of the larger (blue) circle and the smaller (red) circle. The two radii coordinates are shown as well. The radius denoted by capital, R, is the distance from the geometric center of the outer ring lying outside the volume, to the center of the inner ring. The radius denoted by lower case, r, is the distance from the inner ring's center to the surface of the torus.

In geometry, a torus (pl.: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of tori include ring tori, horn tori, and spindle tori. A ring torus is sometimes colloquially referred to as a doughnut.

If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a toroid, as in a square toroid.

Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. 

A torus is different than a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S¹ × S¹, which is sometimes used as the definition. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S¹ in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space.

In the field of topology, a torus is any topological space that is homeomorphic to a torus.^[1] The surface of a coffee cup and a doughnut are both topological tori with genus one.

An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Klein bottle).

Etymology

Torus is a Latin word denoting something round, a swelling, an elevation, a protuberance.

Geometry

Bottom-halves and
vertical cross-sections

R > r: ring torus or anchor ring

R=r: horn torus

R < r: self-intersecting spindle torus

A torus of revolution in 3-space can be parametrized as:^[2]using angular coordinates , representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius Ris the distance from the center of the tube to the center of the torus and the minor radius r is the radius of the tube.^[3]

The ratio R/r is called the aspect ratio of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2.^{[citation needed]}

An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is

Algebraically eliminating the square root gives a quartic equation,

An apple and a lemon derived from a spindle torus with proportions of a vesica piscis

The three classes of standard tori correspond to the three possible aspect ratios between R and r:

• When R > r, the surface will be the familiar ring torus or anchor ring.
• R = r corresponds to the horn torus, which in effect is a torus with no "hole".
• R < r describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple.
• When R = 0, the torus degenerates to the sphere radius r.
• When r = 0, the torus degenerates to the circle radius R.

When R ≥ r, the interiorof this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem, giving:^[4]

These formulae are the same as for a cylinder of length 2πR and radius r, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.

--- end quoting Wikipedia on torus formula---

Notice in the parametric equation, it has both sine and cosine.

In the Cartesian coordinate equation torus is a Quartic equation.

Now I am going to be sidetracked for several days while working on this, before I come back and resume High School Logic.

AP, King of Science

[Archimedes Plutonium]

So, what I am thinking here, is that a 3 Dimensional Calculus does not exist, and that Calculus tops off at 2D.

A 3D calculus would register Volume under a Function Graph, where the graph is a surface. But the derivative would be a surface also, making it impossible for a x to reach two coordinate points of y, z.

I am not sure yet, but I suspect 3D calculus is impossible.

If so, then this is another proof that 4th dimension is a fakery.

AP