My #372 science book-- #372 AP book of science--- High School Logic // Logic science by Archimedes Plutonium

[Archimedes Plutonium]

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

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Archimedes Plutonium<plutonium....@gmail.com>

Jun 8, 2026, 2:13:32 AM

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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<plutonium....@gmail.com>

Jun 8, 2026, 2:22:31 AM

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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<plutonium....@gmail.com>

Jun 10, 2026, 3:07:47 AM (12 days ago) 

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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<plutonium....@gmail.com>

Jun 10, 2026, 4:13:59 PM (11 days ago) 

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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<plutonium....@gmail.com>

Jun 10, 2026, 4:49:19 PM (11 days ago) 

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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<plutonium....@gmail.com>

Jun 11, 2026, 4:38:15 AM (11 days ago) 

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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<plutonium....@gmail.com>

Jun 11, 2026, 5:03:41 AM (11 days ago) 

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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<plutonium....@gmail.com>

Jun 11, 2026, 5:43:36 PM (10 days ago) 

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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<plutonium....@gmail.com>

Jun 12, 2026, 5:23:56 PM (9 days ago) 

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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<plutonium....@gmail.com>

Jun 13, 2026, 3:48:01 AM (9 days ago) 

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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<plutonium....@gmail.com>

Jun 13, 2026, 4:43:12 AM (9 days ago) 

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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<plutonium....@gmail.com>

Jun 13, 2026, 5:28:50 AM (9 days ago) 

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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<plutonium....@gmail.com>

Jun 13, 2026, 6:24:12 PM (8 days ago) 

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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<plutonium....@gmail.com>

Jun 13, 2026, 6:45:19 PM (8 days ago) 

to Plutonium Atom Universe

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<plutonium....@gmail.com>

Jun 14, 2026, 4:10:23 AM (8 days ago) 

to Plutonium Atom Universe

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<plutonium....@gmail.com>

Jun 14, 2026, 6:53:53 PM (7 days ago) 

to Plutonium Atom Universe

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]

I was writing this book in a different thread and then a insight came to me about 3D Calculus. One in science, never knows when insights strike. And as the insight struck, I devoted my attention to 3D Calculus and ignoring the High School Logic, until today where I separated out the 3D Calculus, and will continue to finish this High School Logic textbook.

Some more posts need to be recovered, such as this one of June 15.

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]

Another post to keep.

Archimedes Plutonium<plutonium....@gmail.com>

Jun 16, 2026, 4:47:20 AM (6 days ago) 

to Plutonium Atom Universe

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 need to finish this textbook on High School Logic as top priority, which thus unleashes my 5 textbook series of Logic. By far, the most difficult books I have ever written so far. It is not only because Old Logic is riddled in error. But because of the fact that when you write a textbook of Logic, seems as though every sentence, every paragraph needs to written Logically itself. When writing math or physics textbooks, you can be somewhat sloppy, but not logic textbooks.

Archimedes Plutonium<plutonium....@gmail.com>

Jun 21, 2026, 9:42:47 PM (5 hours ago) 

to Plutonium Atom Universe

I am never going to finish my 5 textbook series of Logic, if whenever I get sidetracked and spend time on that sidetrack. So, to remedy that problem I am carrying the conversation of 3D Calculus to two other books I wrote in the past on that very same subject of 3D Calculus.

And now, let me continue with High School Logic textbook.

I believe I left off with the chapter on Equal-Not. Let me pick up from there.

AP

[Archimedes Plutonium]

So today is 22June 2026 and on the chapter of Equal-Not. What distracted me before was 3D Calculus. But before the distraction, I was saying that I wanted Not to include Truth Tables into High School logic. Save truth tables for College logic.

But I had a problem. AND is easy to see it is Add and OR is easy to see it is subtract (I like better to say remove). But then the next connector of Logic is Equal-Not and extremely difficult to intuit that is multiplication, extremely difficult to think that is multiply. With the truth tables it is easy to show it is multiplication, but without truth tables, extremely difficult. So what can I do???

I ended up showing that Equal-Not is multiplication by a important concept of Logic, one of the most important concepts in all of Logic and called "Consistency". The Not in Equal-Not is the bearer of two important concepts, the Contradiction and the Consistency issue.

But, to make a long story short, Equal-Not, without the use of truth-tables has to be Multiplication because Calculus has derivative as divide and has integral as area which is multiplication. And that --- thus--- Equal-Not has to be multiplication for the integral is multiply and that the Fundamental Theorem of Calculus, FTC, Equates derivative as inverse to integral and Equates integral as inverse to derivative.

So we have Equal in the calculus, as equality in inverses and we have Not in calculus in that the Calculus must be consistent for the Fundamental Theorem of Calculus to be true.

Can you see how the Calculus forces Equal-Not to be multiply?? The integral is multiply in area is length times width. The derivative is division in dy/dx is defined as the derivative slope. The FTC equates derivative with integral as inverses, meaning Equal-Not is forced to be multiply.

AP

[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.

But also, very important is Consistency. Both contradiction and consistency have shades of NOT permeating them. A contradiction has a 100% Not to it. For example: The land is dry and the land is wet. Both cannot be true.

For example consistency: a notorious inconsistency in society was when women were paid less yet did the same amount of work as a man in that same job.

Now, some people define Consistency as a concept that contains no contradictions. But I believe a definition of consistency requires much much more than to say it absent of contradictions.

AP, King of Science 

[Archimedes Plutonium]

Now Yogi Berra, the baseball star made fun with comical sayings and some of them involve consistency.

Here are two of them ascribed to Yogi Berra. 

" I never said most of the things I said."

And this one.

" Nobody goes there anymore, It's too crowded."

" Baseball is ninety percent mentalm and the other half is physical."

" It's tough to make predictions, especially about the future. "

For homework talk about the inconsistency in each statement. Then fix it so that it is not inconsistent.

AP

[Archimedes Plutonium]

Actually, I need not even talk about consistency to show that Equal-Not is Multiplication.

I can make a direct proof that Equal-Not is Multiplication, for which it is understandable by High School students.

Let me get started on that.

Statement:: the Equal-Not connectors of Logic is the Multiplication operator of mathematics.

Proof:: 

1) We saw that from calculus, derivative is defined as dy/dx division and that integral is defined as multiplication for area of rectangle in each cell.

2) We saw that derivative is inverse to integral and integral is inverse to derivative, landing us back to the original function graph.

3) We know that division is inverse of multiplication and that multiplication is inverse of division. For example 1/2 multiply 2 = 1 is the same as 2 multiply 1/2 = 1.

4) We know that Fundamental Theorem of Calculus is true from the picture diagram proof and so an Equality was established between derivative and integral as inverses. They are Equal, and not-unequal.

5) AND is add, OR is subtract, If-->Then is derivative which is divide, thus, Equal-Not has to be multiply.

Note: Apparently I need to teach If-->Then is divide.

[Archimedes Plutonium]

So here I am giving a proof, a formal proof, but a simple proof to High School students. Many of them probably took geometry where they had experience in making a proof of triangle congruence, remember the side-angle-side SAS or ASA proofs? And I probably criticized this geometry in the past saying too much time spent on congruence proofs. But here I come to praise teaching congruence proof. For then my proof that Equal-Not of Logic is that of multiplication in math.

AP

[Archimedes Plutonium]

Let me do some polishing here of this proof. Keeping in mind that High School students rarely see proofs except maybe in geometry classes of side-angle-side or angle-side-angle triangle congruences (triangle equalities).

Let me polish the proof given above until it is fit to teach in High School.

Statement:: the Equal-Not connector of Logic is the Multiplication operator of mathematics.
Proof::
1) We saw that from calculus, derivative is defined as dy/dx, the slope, a form of division, and that integral is defined as multiplication for area of rectangle in each cell of calculus.

2) We saw that derivative is inverse to integral and integral is inverse to derivative, landing us back to the original function. For example in math 1/2 is the inverse of 2 so that if we multiply 1/2 times 2 we end up with 1 and if we multiply 2 times 1/2 we end up with 1. Thus we can say that (1/2) x 2 is Equal to 2 x (1/2) is Equal to 1. 

3) We know that division is inverse of multiplication and that multiplication is inverse of division. For example 1/3 multiply 3 = 1 is the same as 3 multiply 1/3 = 1.

4) We know that Fundamental Theorem of Calculus is true from the picture diagram proof and so an Equality was established between derivative and integral as inverses. When we apply derivative onto integral we get the same result if we apply integral on derivative, and both return to the original function. The derivative on integral = integral on derivative are Equal as the same original function results, and not-unequal---- but instead Equal.

5) AND is add, OR is subtract, If-->Then is derivative which is divide, thus, Equal-Not has to be multiply in order for derivative on integral equals integral on derivative equals original function.

Here all I am doing is trying to make it as understandable as possible for the High School student.

AP

[Archimedes Plutonium]

Now in this chapter of Equal-Not, we need to discuss in some detail some varieties of Not.

The worst type of Not for logic is when you have a Contradiction.

A contradiction is when you completely oppose a idea.

For example: Rutherford, Geiger, Marsden, Bohr in early 1900s said this idea:: Atoms have a Nucleus composed as a tight ball of protons mixed with neutrons.

In 2017:: AP said this idea--- Atoms have No Nucleus but rather they have a proton torus in the center of the atom with Neutrons outside the proton torus arranged in a configuration of a parallel plate capacitor (a battery).

In this example we can simplify what Rutherford, et al said to this more compact idea.

a) Atoms have a nucleus.

 

And AP's idea can be simplified also.

b) Atoms have no nucleus.

These two ideas are contradictory. 

A contradiction in Logic is when you have one idea that says (a) Atoms have a nucleus. Along with another idea that is directly opposite (b) Atoms have no nucleus.

When  we do Logic, and we come to a contradiction of ideas. All of Logic has to stop, and fix the problem for Logic has to be contradiction free, at all times.

But there is another form of opposition somewhat different than contradiction and called Contrary. It uses "not" in the Equal-Not connector.

Many of us have met a person in life who seems to be contrary to much of what we say or do. We call such a person a contrarian.

For example: 99% of scientists and news reporters say this idea (c) Climate Change of rising heat and temperature is caused by the burning of fossil fuels emitting greenhouse gases like CO2.

AP has this idea (d) Climate Change of rising heat and temperature is caused due to the fact the Sun shines from Faraday law-structure, where every atom inside the Sun is producing new electricity like a power station and since the Sun increases its total number of atoms each year, that each year we have rising heat and temperature, and the fossil fuel burning is only a minor contribution to yearly rising heat and temperature.

Now, examine ideas (c) and (d). They are not contradictory. But they have some opposition between one another.

In Logic we say that (d) is Contrary to (c). Or, you can say that (c) is contrary to (d).

And the subject of Contradiction versus Contrary, has a test for contrary. When we have a Contradiction in Logic, it is all out red alert and we stop everything to fix the problem.

When we have Contrary in Logic, it is not so much a emergency. We can relax a bit and sort things out. 

The Test of Contrary is that when you have ideas like (c) and (d) that not Both can be true. One of them has to be false, or both can be false. But never can both be true.

I myself like to think of Contradiction as having a 100% Not involved. Whereas, Contrary has some percentage between 1 and 99% of a Not involved.

Think of Contrary as arguing over the details, a percentage of the details.

Think of Contradiction as total opposition and no arguing but fighting.

AP

[Archimedes Plutonium]

I surely am glad, I reversed my mind and now wrote High School Logic textbook rather than waiting much later, because in writing the High School Logic textbook, I uncovered several important features which is crucial to Logic, which I missed in writing the other 4 logic textbooks. For example, not only the 3D Calculus insights but now the fact that Kinetic Energy of physics is tied directly to integral in 2D Calculus.

And also, the idea that Reductio ad Absurdum does not exist in Logic when you have the true truth tables of If-->Then, but a replacement must exist in Logic once we throw out Reductio ad Absurdum. A "Proof by Impossibility" must exist when we no longer can argue from "Suppose A,,,, run into a contradiction,,, proves not A is true". We can no longer use that method of proof, some call it proof by contradiction.

But, there exists a proof method which I call Proof by Impossibility. And I need it in this circumstance of proving so many Corollaries off of the Fundamental Theorem of Calculus in 2D.

One of these corollaries is a proof that true numbers of mathematics have to be discrete. Reason: you cannot have a Fundamental Theorem of Calculus if the numbers are a continuum. Proof: Impossible to connect previous coordinate point of the derivative to the next coordinate point if there is no immediate successor but an infinity continuum.

Another corollary is a proof that the only valid function of mathematics is a polynomial and where a positive Decimal Grid Number is always on the right side of the equation, all alone, at all times. Another proof by Impossibility for unless that were true, we have no Fundamental Theorem of Calculus in 2D. Proof: Only polynomials have Straight Lines in geometry, for trigonometry, exponential, logarithmic, hyperbolic have no straight lines. And the derivative is always a straight line segment from previous point in graph to next point in graph. Only Polynomials have the straight line equation Y= mx + B.

And as for a Nonzero Decimal Grid Number always on the rightside of the equation all alone at all times. Proof: first we recognize that zero is different from all other numbers as representative of physically nothing whereas a number respresents something. When we have a physical environment and a equation to represent that environment, the only equation possible for a zero all alone on rightside of equation is Nothing = Nothing. We cannot have ludicrous math of ((4 - 4)/2) x (2) =0. No, whenever a zero is all alone on the rightside of the equation, the leftside must be 0 = 0. Algebra for a millenium has played games of folly when they put 0 all alone on the rightside of the equation.

Follow-up: the equation of the straightline is Y= mx + B, and what if we had 0 = mx + B is a folly delusion.

Proof: again, Fundamental Theorem of Calculus in 2D cannot exist if math had a valid function that was not a polynomial and allowed 0 all alone on the rightside of the equation.

Proof Method by Impossibility
----------------------------

Where in logic does this method come from??

I suspect it comes from the If--> Then connector which is division in math. And we all know that it is impossible to have division when the divisor is zero. The truth table of If-->Then has only one circumstance where it is true of a statement. That one circumstance is when If A then B where both A and B are true ideas. Sort of a uniqueness characteristic of truth of a statement.

So when the Fundamental Theorem of Calculus is true, proven true, and it requires certain conditions to be met--- numbers be discrete, functions be polynomials, a positive nonzero decimal grid number always on the rightside of the equation all alone. If any of those conditions are not met,then it is impossible to have a calculus.

In Old Math, a majority of their proofs were done by Reductio ad Absurdum and the reason for that is the method itself is a fake method. It is a pseudoproof. You suppose something true--- Wiles, Suppose FLT has solutions in exponent 3 ,,,, Wiles runs into a contradiction,,,, Wiles et al think they proved FLT has no solutions in exponent 3. That is not a valid proof in math, but a con job.

AP on FLT in exponent 3,,,, AP says 2+2 = 2x2= 2^2 is the reason we have solutions in exponent 2, but no number exists wherein N+ N+ N = NxNxN = N^3, thus, Impossible to construct a integer solution for A^3 + B^3 = C^3 in integers.

There is a great Direct link between Physics and Math
----------------------------------------------------------------

How does physics get the (1/2) in Kinetic Energy KE = (1/2) Mass x Velocity^2. Answer: it gets it from Math Integral being area under the derivative.

AP, King of Science

[Archimedes Plutonium]

So the concept of Contradiction is one of a 100% reversal of a previous idea-- not that idea. Yet the concept of Contrary is a percentage of difference between two ideas, being picky over details, and it can be picky over a few little things or over a large detail but not a complete 100% difference.

As a reminder to myself in the Introduction to Logic textbook where I do truth tables and were the OR connector has a percentage of truth, we run into this idea of percentage variance again, in OR and in the Not in Equal-Not.

As for Equal, well that is usually defined to be identical equal, completely the same.

In math though, there is a concept of Equivalence which is not identical sameness. For example 1/2 is equivalent to 3/6. They are not identical but equivalent. A pie cut into 2 portions is equivalent to a pie cut into 6 portions.

I usually think of fractions like 1/2, 2/4, 3/6 as a person who is too lazy to finish the division, too lazy to reduce to lowest terms.

Another concept that is kind of like "equal" is in geometry of concept of "similar". For example, a right triangle of 3,4,5 side is identical to another right triangle of 3,4,5 but is similar to a right triangle that is 6, 8, 10. They both are right triangles with the same angles, yet differ in size.

Reminder:: Talk about percentage variance of contrary in Introductory Logic. And talk about equal, equivalence, similar in Introductory Logic.

AP

[Archimedes Plutonium]

Alright, I am going to try to do something that is hard for me to do. Drop most everything and focus just on this book to get it done, or a state of near completion.

The summer months are getting too hard on me in heat and I need to finish this book by end of August.

So, do as I must---- focus focus focus.

Be sure to include the idea in first chapter that Logic follows much in the line of mathematics. We learn numbers and geometry figures. In Logic we learn ideas are what numbers are in math or what geometry figures are in math. And we learn in math the operators to play around with numbers and geometry figures--- add, subtract, multiply, divide, derivative, integral.

Likewise, Logic calls them connectors instead of operators. We use 6 connectors in Logic to play around with ideas.

So in teaching Logic, the best way to teach it is to recast mathematics and replace numbers with ideas, and replace the operators with connectors-- AND, OR, Equal-Not, If-->then, existential quantifier, universal quantifier. It is a shame that I know of no High School that teaches Logic, and I know of no college or university that requires Logic for a degree. A shame because Logic is the science of thinking straight, clear and correctly. And, is that not the whole point of a University education?????

AP, King of Science

[Archimedes Plutonium]

So AND is add, but there are many other words to describe AND, such as join, but, yet, also, still, although, however, moreover, nevertheless, conjunction. Also the comma or semicolon is AND.

OR is subtract, remove, but there are several other words to describe OR, such as alternatively, either-or, disjunction.

Equal-Not is multiply but is seldom called multiply. Other words for "equal" is same, identical, equivalent. Other words for "not" are negate, "it is false that".

If-->then is division and there are many words to describe If-->them, such as "hypothetical", conditional, implies, "antecedent-consequence"

Source for some of the above-- Copi, Introduction to Logic.

[Archimedes Plutonium]

The Existential quantifier and Universal quantifier.

The Existential quantifier of logic is "There exists" and comes from division as the calculus derivitive dy/dx. The change in y value divided by the change in x-value. 

The Universal quantifier is "For all" or "For every". It comes from multiplication for in calculus it is the integral as a rectangle underneath the function graph.

AP

[Archimedes Plutonium]

Mathematics is the science of numbers and geometry figures, that uses 6 operators to understand math. Those 6 are add, subtract, multiply, divide, derivative, integral.

Logic is the same as math once we replace numbers and geometry figures with Ideas, and we transform these ideas by using 6 connectors. Those 6 respectfully following the order of math operators are AND, OR, Equal-Not, If-->then, existential quantifier, universal quantifier.

To properly teach Logic, one must do a bit of calculus for derivative and integral.

AP

[Archimedes Plutonium]

So why does math have 6 operators to cover all of math. Here we use Logic to explain this question. For everything that is in math has to be in Logic also. For Math is about numbers and geometry figures, while Logic is about Ideas and numbers are ideas and geometry figures are Ideas.

We have in math add, subtract, multiply and divide and in Logic we have that corresponds to add is AND; corresponds to subtract we have OR; corresponds to multiply we have Equal-Not; corresponds to divide we have If-->Then.

Then we have in math, the calculus with its two operators of derivative and integral where the derivative is divide in dy/dx the change in y-value divided by the change in x-value of a graph of a function. So why do we need another division when we already have a division in If-->Then?? We need this new division because it is a geometry division.

In Numbers mathematics we need 4 operators on a straight line. But once we get to 2 dimension of the plane we need calculus where dy/dx, derivative is slope. We do not have slope in 1 dimension of a line. And in 1D of a straight line we do not have area for multiply, and we need area in 2D and 3D geometry. This is why the integral of math calculus is required and is the area under the function graph, a rectangle in 2D.

The 6 connectors of Logic are AND for add on a straight line 1D. The OR is subtract on a straight line 1D. The Equal-Not is Multiply on a straight line 1D. The If-->Then is division on a straight line in 1D. But we need connectors in 2D and 3D and that is why we have Existential quantifier as derivative in 2D and 3D as division dy/dx. And that is why we have integral as Universal quantifier in 2D and 3D as multiplication in geometry as area of rectangle in 2D, and as volume in 3D.

Numbers alone in math are all in 1D, but geometry included forces 2D and 3D and so we need 2 more operators. Numbers alone would be 1D and that would stay and remain as length. 2 units add 3 units is 5 units length on straight line. 3 units subtract 2 units is 1 unit length. 2 units multiply 3 units is 6 unit length on straight line in 1D. While 6 units divide 3 units is 2 unit lengths on straight line in 1D.

Calculus of math is formed because of multiplication and division in 2D and in 3D. In 2D, we need Area which is "squares" small tiny unit squares. And so the integral as Universal quantifier is how many unit squares in a rectangle. When we have 2 unit squares by 3 unit squares in a rectangle we have 6 unit squares altogether and that is the integral and that is the Universal quantifier.

When we have function graph, we have a slope as the function goes from x-value to y-value and then the next coordinate point of the graph. This slope is speed in motion and is the Existential quantifier. The first coordinate point forces the existence of the next coordinate point. This slope is the geometry of division dy/dx.

AP

[Archimedes Plutonium]

If-->Then connector, the material-conditional, but I like to think of it as the "move-into" connector. It is division. Examples: If it rains then my clothes get wet.

If the Sun shines from Faraday law, then every year gets more hot and hot.

If the Sun shines from Faraday law, then the Sun has moved into Red Giant Initiation Phase.

If the Sun shines from Faraday law and not from fusion, then the polar regions of Earth warm up faster than then the other regions.

If the Sun shines from Faraday law and is in Red Giant Phase, then all of life on Earth will go extinct and into oblivion unless we make a new home on Europa.

On Thursday, July 2, 2026 at 4:50:35 PM UTC-5 Archimedes Plutonium wrote:

So AND is add, but there are many other words to describe AND, such as join, but, yet, also, still, although, however, moreover, nevertheless, conjunction. Also the comma or semicolon is AND.

OR is subtract, remove, but there are several other words to describe OR, such as alternatively, either-or, disjunction.

Equal-Not is multiply but is seldom called multiply. Other words for "equal" is same, identical, equivalent. Other words for "not" are negate, "it is false that".

If-->then is division and there are many words to describe If-->then, such as "hypothetical", conditional, implies, "antecedent-consequence"

[Archimedes Plutonium]

On Saturday, July 4, 2026 at 5:03:07 PM UTC-5 Archimedes Plutonium wrote:

If-->Then connector, the material-conditional, but I like to think of it as the "move-into" connector. It is division. Examples: If it rains then my clothes get wet.

If the Sun shines from Faraday law, then every year gets more hot and hot.

If the Sun shines from Faraday law, then the Sun has moved into Red Giant Initiation Phase.

If the Sun shines from Faraday law and not from fusion, then the polar regions of Earth warm up faster than other regions.

If the Sun shines from Faraday law and is in Red Giant Phase, then all of life on Earth will go extinct and into oblivion unless we make a new home on Europa.

On Thursday, July 2, 2026 at 4:50:35 PM UTC-5 Archimedes Plutonium wrote:

So AND is add, but there are many other words to describe AND, such as join, but, yet, also, still, although, however, moreover, nevertheless, conjunction. Also the comma or semicolon is AND.

OR is subtract, remove, but there are several other words to describe OR, such as alternatively, either-or, disjunction.

Equal-Not is multiply but is seldom called multiply. Other words for "equal" is same, identical, equivalent. Other words for "not" are negate, "it is false that".

If-->then is division and there are many words to describe If-->then, such as "hypothetical", conditional, implies, "antecedent-consequence"

Some other equivalent (a form of equal) words that are If-->then

1) if...then

2) because

3) reason

4) implies

5) moves into

6) hypothetically

7) conditionally

8) consequently

Homework assignment for If-->Then. Using ideas or news from Science in this homework.

Write 5 sentences using "If.... then".

Write 1 sentence using (2) through (8) above.

For example: (2) because

(2) Old Chemistry is a failure science because it was so dumb as it thought the electron of atoms was the 0.5MeV particle when in fact it was the muon as electron of atoms.

(3) New Chemistry is true, reason being that all elementary particles have a task to do, not just sit around in a nucleus as a proton or neutron ball or float around the outside as a tiny ball.

(4) The Earth cores as a electric dynamo motors, implies continental-drift is a result of the vibrations from the motors.

(5) Humans evolved by throwing rocks moves into they had to become bipedal to free up the arms for throwing.

(6) Hypothetically, Sun gone Red Giant, means Earth will be swallowed up by the Sun in the near future.

(7) The saber-toothed tiger is genuine on the condition that the DNA testing of its saber teeth are not walrus tusks.

8) Humanity and much of life on Earth moves out to Europa as a consequence of Sun gone Red Giant.

AP

[Archimedes Plutonium]

Assign additional homework for next day.

Take your examples of (2) through (8) and convert them into a If...Then statement.

Example of mine shown below.

_If_ Old Chemistry thinks the electron of atoms is the 0.5MeV particle when in fact it was the muon as electron of atoms, _then_, Old Chemistry is a failure science. 

(3) New Chemistry is true, reason being that all elementary particles have a task to do, not just sit around in a nucleus as a proton or neutron ball or float around the outside as a tiny ball.

_If_ all of elementary particles have a task to do, not just sit around in a nucleus as a proton or neutron ball or float around outside as a tiny ball, _then_ New Chemistry is true.

(4) The Earth cores as a electric dynamo motors, implies continental-drift is a result of the vibrations from the motors.

_If_ the Earth cores are a motor electric dynamo, _then_ the continental drift is a result of the vibrations from this motor. 

(5) Humans evolved by throwing rocks moves into they had to become bipedal to free up the arms for throwing.

_If_ Humans evolved from throwing rocks, _then_ they became bipedal to free up the arms for throwing. 

(6) Hypothetically, Sun gone Red Giant, means Earth will be swallowed up by the Sun in the near future.

_If_ Sun goes Red Giant, _then_ Earth will be swallowed up by the Sun in the near future.

 

(7) The saber-toothed tiger is genuine on the condition that the DNA testing of its saber teeth are not walrus tusks.

_If_ the saber-toothed tiger is genuine, _then_ DNA testing of the saber teeth will not be walrus tusks.

8) Humanity and much of life on Earth moves out to Europa as a consequence of Sun gone Red Giant.

_If_ the Sun goes Red Giant, _then_, humanity and much of life on Earth moves out to Europa.

AP

[Archimedes Plutonium]

Now the If--> Then connector of Logic is one of the most important connectors because of its relationship to doing Science. When we hypothesize into the future, guess into the future we are using a If--> Then connector.

As I said earlier, If-->then is division, but also the Existential Quantifier which we cover next is division a sort of division in geometry as the derivative of calculus dy/dx.

If-->then is called the hypothetical as a moving of If into a  Then, a antecent into a consequent. Much like the slope of a line going from one point to the next point in the future.

Prediction is the heart of science and the If-->Then connector is the prediction connector. Example: If you touch a ongoing stovetop, then you get a burnt finger.

Next, we do the geometry division which is the Existential quantifier, the derivative of mathematics.

AP

[Archimedes Plutonium]

Existential quantifier

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

This is again division, only it is geometry division and comes from the calculus derivative. We learned that the derivative is dy/dx and is the straightline segment that goes from (x_1, y_1) to the next point of the function graph (x_2, y_2). Another example of "move into". If (x_1, y_1) Then, comes (x_2, y_2) in the function graph.

As the famous literature saying from Shakespeare's Hamlet "To be, or not to be, that is the question."

For Science, to exist or not exist is an important question. For if something exists, then we explore it. If it does not exist, no point in wasting time on it.

AP

[Archimedes Plutonium]

So let me write 5 examples of the Existential quantifier. We prefer to write them in science as that way we can have others check up on our work. If our example is ordinary daily living we cannot have a peer review or checkup.

Daily living example: I have a female cat who gave birth to 3 kittens and now 4 cats exist in my garage.

You see, the trouble with Daily living examples is that others have no easy way of checking up on the facts.

Science example: The 3 horned dinosaur called Triceratops existed.

You see, plenty of fossils have been found and displayed in museums to verify and peer review this claim.

Five Examples of Existential quantifier

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

(1) Communism as a philosophy, written up in Kaplan's book "The New World of Philosophy", 1961, _exists_ as an idea only, but _never existed_ as a practicing government. But rather instead, dictatorship governments _exist_ that call themselves "communism". Dictatorships that label themselves as "communist" is a shield to hide themselves from reality-- they are a brutal dictator, not communism.

(2) A strong magnetic field for Earth _exists_, and is easily seen by a compass needle pointing North in the northern hemisphere.

(3) Black holes do _not exist_ and have _never existed_, because what truly exists is the Pauli Exclusion Principle. The Pauli Exclusion Principle says that matter cannot be squeezed together into one another, which is the exact opposite of the idea of a black hole in physics.

(4) Did the Saber Tooth Tiger _exist_?? Or is that a walrus tusk glued and screwed onto the upper jaw of a normal tiger?? So far, paleontologists have been too lazy to DNA test saber tooth tigers to find out the truth.

(5) The Magnetic Monopole _exists_ despite the error and mistake in Old Physics Maxwell Equations. Gauss's law of electricity is true for it is the Coulomb law, but the Gauss law of magnetism of Maxwell Equations is false, for the magnetic monopole _exists_ and is the 0.5MeV particle, while the muon is the true electron of Atoms. 

AP, King of Science

[Archimedes Plutonium]

Homework: Write up 5 examples of the Existential quantifier using Science as the context. Try to pick 5 different sciences.

Although I have Existential quantifier next to last in this list of 6 connectors, it actually is the start as number 1 connector in Logic.

This book lists the connectors as this.

1. AND

2. OR

3. Equal-Not

4. If-->then

5. Existential quantifier

6. Universal quantifier

And the reason I list them in that order is because they follow the order of the 6 math operators which they correspondingly resemble.

1. Add as AND

2. Subtract as OR

3. Multiply as Equal-Not

4. Divide as If-->then

5. Derivative as Existential quantifier

6. Integral as Universal quantifier

We teach math operators in that order in our education school system because it is a natural order of going from the most simple to the more complicated and complex.

But the order for Logic really should be that of this.

1. Existential quantifier

2. Equal-Not

3. AND

4. OR

5. If-->then

6. Universal quantifier

What is that the best logical order to teach the 6 connectors???

Well the reason it is the best logical order to teach those 6 connectors is because the first thing we want to know about "something" is whether it exists or does not exist. Kind of silly and a bit crazy to spend time on something that is fictional, science fiction. And, that order as given above, the Logic Order closely follows what is called the Scientific Method.

The Scientific Method

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

The Scientific Method has 6 steps.

Step 1-- Make observations and ask many questions.

Step 2-- Research the subject matter and Review the literature on the subject.

Step 3-- Conduct Experiments pertaining to your hypothesis.

Step 4-- Collect data from the experiment/s and analyze the data.

Step 5-- Look for patterns on the data of what you think is going on.

Step 6-- Draw conclusions.

Now I am going to interpret those 6 steps of the Scientific Method and then give them a term name, afterwords. The reason I do this is to form a well-defined list of terms most often used in science, logic and math. Logic is precision and accuracy and you do not have that if the words you use are not well-defined.

I repeat again the 6 connectors of Logic will be discussed in detail in later chapters, but I introduce the 6 here and now to well-define science, logic, math terms. The 6 connectors in order are Existential quantifier, Not-Equal, AND, OR, If-->Then, Universal quantifier.

From the Scientific Method above
------------------------------------

Step 1-- Does something exist-- Existential quantifier --- call it a statement.
Step 2-- Research-- Not- equal connector --- call it a concept or premiss.
Step 3-- Experiments -- AND connector --- call it a hypothesis or connector structure. AND is like math add and you build a structure of ideas.
Step 4-- Analyze data and deciding experiment-- OR connector --- call it theory or principle or reasoning. The idea is that you pare-away ideas that are not good enough to become universal for a theory and principle has to be universal.

Step 5-- Patterns noticed --- If-->Then connector --- call it a theory or equalities  of what you think is going on and how it works.
Step 6-- Draw conclusion -- Universal quantifier ---  call it Universal Axiom or law-structure.

So for **Science** we have Statement, Concept, Hypothesis, Law-structure, Theory, Universal Axiom.

For **Logic** we have Statement, Premiss, Connector-structure, Reasoning, Equalities, Law-structure.

For **Math** we have Statement, Axiom, Operator-structure, Proof, Theorem, Theory

[Archimedes Plutonium]

Now I am thinking of 4 easy simple experiments for students to pick one and perform at home for which they see and understand the Scientific Method above. Chose from one of these 4 Experiments and write a paper on how the experiment fits the 6 Steps of the Scientific Method.

Perhaps we do the expeiments in class for some apparatuses are hard to obtain.

1) Faraday law-structure experiment where electricity is produced by thrusting bar magnet.

2) Magnets in a box experiment where it always ends up the magnets are attraction, never repelling.

3) Circle drawing Experiment by straight lines.

4) Slant cut of Cone Experiment is Oval, not ellipse.

[Archimedes Plutonium]

I am going to add a 5th Experiment to test the Scientific Method. And for homework have the students do all five, one at a time.

On Tuesday, July 7, 2026 at 3:44:02 AM UTC-5 Archimedes Plutonium wrote:

Perhaps we do the experiments in class for some apparatuses are hard to obtain.

1) Faraday law-structure experiment where electricity is produced by thrusting bar magnet.

2) Magnets in a box experiment where it always ends up the magnets are attraction, never repelling.

3) Iron Filings in a case or on top of a piece of stiff paper and a bar magnet causing the Magnetic Lines of Force 

4) Circle drawing Experiment by straight lines.

5) Slant cut of Cone Experiment is Oval, not ellipse.

Yes, well, all the three physics experiments will be done in class. The 2 math experiments can be done at home as well as in class.

For homework, each night we write a one page paper on how the experiment follows the Scientific Method.

AP 

[Archimedes Plutonium]

This is an excellent tie in to the Universal quantifier.

I go from Existential quantifier to that of the Scientific Method.

I correct the Scientific Method by placing the steps in a correct order as per the Logic connectors. 

Then I go directly into the Universal quantifier. The step before the Universal quantifier is the IF--> Then connector.

AP

[Archimedes Plutonium]

Science-- has a method called the Scientific Method. It is the gain of knowledge of the world by observation and experiments and careful reasoned logical ideas. Science uses math for precision and accuracy. Science uses Logic for precision and accuracy of ideas. Science is the only knowledge that can predict the future by the patterns of the law-structures such as the Atomic theory and electromagnetism forces. For example, watching Artemis 2 astronauts recently go around the Moon is predicting the future of the spacecraft, knowing the physics. Physics is the top science for it has the most experiments and law-structures of electromagnetism.

Logic is another science but much lower of importance than physics. Logic is the science of precise and accurate Ideas, while math is the science of numbers and geometry figures, Logic is the science of Ideas.

Logic is the science of Ideas, while math is the science of numbers and geometry figures.

Math is the lowest of sciences for it is that of precision and accuracy when using quantity, measure, formulas and geometry. Math actually is the easiest of the sciences and not only the lowest of the sciences. The only reason that math is difficult today as of May 2026, is that the math community uses the fake numbers of Reals and uses a fake calculus. In New Math of AP, all calculus textbooks can be reduced to 300 pages instead of the recent Stewart textbook, CALCULUS, his 5th edition, 2003, of over 1168 pages. The reason he needs so many pages is because he is in "fake calculus" that uses Reals, never understanding that Polynomials are the only valid function and thinks the derivative is a tangent line to the function graph at a point. No wonder Stewart could never do a geometry proof of Fundamental Theorem of Calculus, mired down in all those phony ideas.

Logic has had its share of horrible mistakes, in fact, the AP textbooks on Logic are the world's first logic textbooks that has all the 4 simple connectors --- AND, OR, Equal-Not, If-->then; and the 6 crucial Logic connectors taught correctly, I am speaking of Existential quantifier, Equal-Not, AND, OR, If-->then, Universal Quantifier in that sequential order. 

It is a sad and horrible reflection that not one single logician in modern times had AND, OR, Not-Equal, If-->then correct. Some prefer to write Equal-Not and I prefer to write it Not-Equal, it is your choice which you like.

In fact so ludicrous is Old Logic that they thought OR is addition as can be seen inside the front cover of Copi's logic textbook, 4th edition, 1972 where he writes
as Rules of Inference
9. Addition (Add)
p
therefore p or q

That entails... that means that Copi thought that AND is subtraction.

Example: Imagine that if I said to you "Tonight for dinner I will eat spaghetti and tonight for drink, I will drink orange juice. Question to reader, does the AND in that sentence sound like subtraction or removal to you? No, the AND sounds more like addition.

No wonder that people in math, especially math professors could never do a geometry proof of Fundamental Theorem of Calculus nor could tell the difference between a ellipse and a oval, because, well, they had no logical mind to even understand that AND has to be addition and OR has to be subtraction as remove. Remove either p or remove q.

Example: For dinner tonight I will eat spaghetti, or, I will eat a bacon, cucumber, tomato sandwich. Reader, does that sound like the OR is addition? No, it sounds like the OR is remove one of the items; subtract one of the items.

Logic is the science that helps you Think Straight and Think Clearly, and woe to any scientist who is bad in logic. But woe to any person who is not logical for life will have many stumbling blocks and losses to those that are illogical.


This is why I insist in College and University for all science majors to have 2 years of logic study, mandatory.

[Archimedes Plutonium]

I have forgotten whether I have earlier made this a homework assignment in High School. If not, well fit it into a appropriate spot.

2-Cannot understand slant cut of cylinder is truly ellipse but in cone it is oval? Do they have cataracts in their eyes or are they simpleton fools failures of math like you, Steve Huffman is a sci.math math failure???

2--- quoting Wikipedia---

--- end quoting Wikipedia---

2-Huffman's broadcast on math lunatics starting with Terry Tao UCLA, unable to admit slant cut of cone is Oval, never ellipse for it appears everything Terry does in math is a memorization of a Wikipedia page.

2--- quoting Wikipedia on Conic Section, for Jimmy Wales with his fascist editors never allow the truth of science into any of their entries of science in Wikipedia, not even a simple geometry of slant cut of cone is Oval, never ellipse--- 

--- end quoting Wikipedia on Conic Section---  

2--- quoting Wikipedia---

--- end quoting Wikipedia---

[Archimedes Plutonium]

The Scientific Method is crucial knowledge in all 5 of my Logic textbooks. And one of its features is that it gives the sequence of Logic connectors its ORDER.

It starts with the Existential quantifier, then moves to the Equal-Not connector, then AND which can entail a full blown experiment, then moves to OR connector, where OR can be a deciding-experiment, then moves to IF-->Then forming a rule maybe a law-structure of science and finally ending with Universal Quantifier.

And this is common plain sense, in the Scientific Method is Logic in practice at arriving at the truth in the world we live in.

AP, King of Science

[Archimedes Plutonium]

Old Math failed at the starting line of Geometry, with their fatally flawed Axiom or Postulate of that of "Point".

No wonder Old Math could never do a geometry proof of Fundamental Theorem of Calculus.

No wonder that Physics started the truth of geometry in year 1900 with Max Planck kicking off the century with Quantum Mechanics saying physics is discrete, yet the fools and cranks of math still in their cesspool of continuums.

I shall go through all the math history books here at home that talks about the First Axiom of geometry.

Old Math:: A point has no length, no width, no depth.

New Math:: A point must have at minimum a length of 1*10^-604 length and that same amount for width and depth. Note--- at minimum but in most circumstances it can have larger proportions.

ANALYSIS

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

We can analyze the fatal flaw of Old Math's axiom on what a point is by pure Logic.

A Point is something, otherwise it is nothing.

If a point has no length, no width, no depth, then it has 0 length, 0 width, 0 depth.

Hence a point is nothing for zero is nothing.

But a point is something and not nothing.

Therefore, A point must have some tiny finite length, finite width, finite depth. The minimum amount of metric for a point would be the Infinity Borderline and the induction element is 1*10^-604.

Common Sense Explanation

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

Physics is the material world science. The smallest thing in physics would be a point of math. Atoms are too big to be points, for they have subatomic particles. The smallest subatomic particle is the magnetic monopole of the 0.5MeV particle. Hence, the Point in Physics is the dimension of the magnetic monopole. 

Note: Old Physics and Old Chemistry were confused for the 0.5MeV particle is not the electron of Atoms, but rather instead the muon the 105 MeV particle is the true electron of Atoms.

The failure of Old Math to have a valid definition or postulate or axiom of "Point" contributed to the gargantuan failure of Old Math to understand what the Calculus was, and the derivative as part of the function graph, where the true numbers of mathematics have to be discrete with empty space in between one point and the next point.

Now, with Point having a tiny tiny finite size, we can backtrack and thus make Exacting Precision in Geometry figures such as a Circle is at minimum a 96-Regular Polygon when mapped on a Decimal Grid system.

AP, King of Science

I need to use these ideas in my Logic textbooks.

[Archimedes Plutonium]

Now I am going to go through a list of books that present Euclid's axiom of "point". For it is a Flawed Axiom that is self-contradictory.

GEOMETRY, 2nd edition Harold Jacobs, 1987, page 50 starts to talk about "points". Jacobs is following a practice that I suspect is dubious and flawed in that he insists on not defining "point".

Which reminds me of Edwin Moise "Elementary Geometry from an Advanced Standpoint" 3rd edition, 1990, pages, 43, 44, who does not define point and thinks that the axioms will specify what a "point" is. I am dubious of this claim.

Morris Kline, "Mathematical Thought From Ancient to Modern Times" volume 1, 1972, page 58, and where Kline writes the Euclid Elements definition of "point"  as "1. A point is that which has no part." "2. A line is breadthless length."

AP writes: this caught me be surprise for I had remembered the Euclid definition as a Point has no length, no width, no depth. Could it be that "no parts" is the same as "no length, no width, no depth" ?  No, I beg to differ that those two ideas are not the same. So I looked through other books to see if that was truly Euclid's definition of "point". So I go to Britannica-Great Books to see if "no parts" was Euclid's axiom.

Britannica "Great Books of the Western World" 1952, page 1, Book One Euclid Definitions 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself.

AP writes:: I see room for a-lot of mistakes there, especially the difference between straight-line and curved line. For AP, no continuum curves exist but rather instead, a large collection of tiny fine straight line segments all connected together to appear like a curve.

But let me get back to this idea of Point as having "no parts" and Point as having "no length, no width, no depth" to me those two are different definitions. And that the concept of Point from Euclid to 2026 AP was fatally flawed.

As I wrote earlier the true definition of Point must involve the infinity borderline which by tractrix computation is 1*10^-604. Hence, all points have at minimum 1*10^-604 length, and that amount of width and that amount of depth. When working in 10 Grid and borrowing up to 10000 Grid then the length, width and depth of a point can be 1*10^-4 long, wide, deep.

Modern day math with Quantum Mechanics Physics updates the definition of Point so that it is no longer a Self Contradictory definition.

A Point is Something. If it has no parts, or if it is metrically described as no length, no width, no depth means it is 0 long, 0 wide, 0 deep and 0 is Nothing. But a Point must be Something. Hence we have a self contradiction here. Proving that a Point must have some tiny finite metric. And this makes absolutely Common Sense for we know the length of a line distance is the summation of all its points. If points are 0, then all lines are 0 length.

AP, King of Science

[Archimedes Plutonium]

All of Logic requires WELL DEFINED Definitions

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

Two examples of Poor definitions-- physics -- "charge" and maths "point".

Two examples of Well Defined definitions-- biology's "DNA" and maths AP geometry proof of Fundamental Theorem of Calculus.

AP, King of Science

[Archimedes Plutonium]

On Sunday, July 12, 2026 at 12:02:38 AM UTC-5 Archimedes Plutonium wrote:

Now I am going to go through a list of books that present Euclid's axiom of "point". For it is a Flawed Axiom that is self-contradictory.

GEOMETRY, 2nd edition Harold Jacobs, 1987, page 50 starts to talk about "points". Jacobs is following a practice that I suspect is dubious and flawed in that he insists on not defining "point".

Which reminds me of Edwin Moise "Elementary Geometry from an Advanced Standpoint" 3rd edition, 1990, pages, 43, 44, who does not define point and thinks that the axioms will specify what a "point" is. I am dubious of this claim.

Clayton Dodge "Numbers & Mathematics" 2nd edition, 1975, pages 26 and 27 says:

"It follows that not all terms can be explicitly defined, for we do not have an endless supply of terms upon which to draw. And, of course, it is unthinkable to commit the "sin of circularity" by defining point in terms of line (say, as the intersection of two lines), and then turning around and defining line in terms of point (as a special collection of points)."

AP writes:: You see, Dodge, Moise, Jacobs would have immensely benefited if in college or university they had studied Logic for 2 years to know that what Logic says about Well Defining something, instead of leaving up to their notions.

In Logic, you have a choice--- you can well define "point" from other parameters. Or, you can use "point" in a axiom-postulate that describes what a point is. Either one you can use.

Continuing with Dodge : "How, then, are the primitive terms to be given precise meanings? By the postulates. These assumed statements involve the primitive terms and state their properties implicitly. The statement, 

two points determine exactly one line, 

tells something about points and lines."

AP writes:: Sorry to say, Euclid in Ancient Greek times had a better grasp of math geometry and of Logic than does Dodge, Moise, Jacobs.

For Euclid recognized that math proof is composed of (1) well defined terms (2) axiom-postulates (3) theorems as proofs using (1), (2).

Let me repeat what Euclid himself in Ancient Greek times does to "point". He defines "point" and defines "line".

Britannica "Great Books of the Western World" 1952, page 1, Book One Euclid Definitions 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself.

What is needed is Physics with physical reality. Physics has Distance measured by a metric most often in meters.

So when Euclid says a point has no part, can be interpreted as a point has no metric of length, width, depth and when Euclid says a line is breadthless length is the same as saying in modern times that a line has a metric length but no width and no depth.

So, well, how does Dodge solve his problem of telling us what "point" is??? He thinks the postulates-axioms will reveal what a "point" is. Dodge says "two points determine exactly one line". Which is silly. If one is looking at two lions determining a lion cub, then the lion is a point and the cub is a line.

No, what Dodge, Moise, Jacobs fail in realizing is that Physics is a Meta set above math geometry and which can easily provide a definition of point. Once a definition of point and line is given then axioms of point and line can further detail what point and line are.

AP Definition of Point: A point has a minimal length, minimal width and minimal depth given a specific Decimal Grid System. The smallest metric is 1*10^-604 because the infinity borderline itself is 1*10^604. In other words, a Point in geometry is an entity that has the smallest finite metrics, otherwise there is empty space at that location.

You see, Dodge, Moise, Jacobs are living with a self contradiction, on one hand they say a point is Something, but a Something that is Nothing.

AP, King of Science

[Archimedes Plutonium]

One of my best history of math books is Cajori, "A History of Mathematics", 1991. 

I am curious to see if Cajori adds some new light or new angle on well-defined concept of "point".

--- quoting Cajori, page 26---

The Pythagoreans called a point "unity in position," but this is a statement of a philosophical theory rather than a definition. Plato objected to calling a point a "geometrical fiction." He defined a point as the "beginning of a line" or as  "an indivisible line,"  and a line as "length without breadth." He called the point, line, surface, the "boundaries" of the line, surface, solid, respectively. Many of the definitions in Euclid are to be ascribed to the Platonic school.

--- end quoting Cajori---

AP writes:: Keep in mind it was Plato with his many students who gave a proof that the 5 Platonic Solids are the only existing regular polyhedron. You need immense well defined definitions and axioms to pull off a proof like that.

I was looking for when the first Graph paper was in use in the world. Wikipedia says the Metropolitan Museum of Art owns a pattern book dated to 1596 in which each page bears a grid printed with a woodblock.

The first commercial "coordinate paper" is attributed to Dr. Buxton of England.

I would have guessed Rene Descartes (1596-1650) would have used graph paper.

Cajori mentions Karl von Staudt (1798-1867) as graphic statics.

So, I wonder if Descartes had graph paper in his research?

AP

[Archimedes Plutonium]

Time to write up this book. I have only August left.

Need to change the title to say Logic for High School and 1st Year College// Logic science by Archimedes Plutonium

[Archimedes Plutonium]

Need to add a "charge" definition from Berkeley Physics Course -- volume 2 "Electricity and Magnetism" Purcell, 1965. 

AP writes: I like it when "charge" is the first topic on first page for "Electricity and Magnetism". However, I do feel this book is --- too difficult for most students. In college especially, it is rare that teachers connect with their students at their intellectual maturity level. In fact, it is rare that college professors take courses to help them teach.

--- quoting Purcell, Electricity and Magnetism, in parts pages 2, 3---

Electricity appeared go its early investigators as an extraordinary phenomenon. To draw from bodies the "subtle fire",  as it was sometimes called...

... We know now that electrical forces largely determine the physical and chemical properties of matter over the whole range from atom to living cell.

We assume the reader has some acquaintance with the elementary facts of electricity. We are not going to review all the experiments by which the existence of electric charge was demonstrated or all the evidence for the electrical constitution of matter. On the other hand, we do want to look carefully at the experimental foundations of the basic laws on which all else depends. In this chapter we shall study the physics of stationary electric charges-- electrostatics.

Certainly one fundamental property of electric charge is its existence in the two varieties that were long ago named positive and negative. The observed fact is that all charged particles can be divided into two classes such that all members of one class repel each other, while attracting members of the other class. If two small electrically charged bodies A and B, some distance apart, repel one another, and if A attracts some third electrified body C, then we always find that B attracts C. Why this universal law prevails we cannot say for sure. But today physicists tend to regard positive and negative charge as, fundamentally, opposite manifestations of one quality, much as "right" and "left" are opposite manifestations of "handedness." Indeed, the question of symmetry involved in right and left seems to be intimately related to this duality of electric charge, and to another fundamental symmetry, the two directions of time. Elementary particle physics is throwing some light on these questions.

--- end quoting Purcell, Electricity and Magnetism, in parts pages 2, 3---

AP writes:: Purcell missed one thing that is important--- he needed to name a physical particle that is the charge itself. You cannot have "charge" that is ephemeral, not distinctive, not a particle. EM waves are also particles of photons. We cannot have "charge" outside of being some wave of EM or photon. Charge ends up being a particle that is the 0.5MeV and lies between hard X-rays and Gamma rays.