Re: #366, 366th science book by AP

[Archimedes Plutonium]

On Saturday, February 14, 2026 at 10:50:59 AM UTC-6 Archimedes Plutonium wrote to Plutonium Atom Universe newsgroup:

Let me expand on Feynman's definition of science where he emphasizes the huge role of Experiment. And although throughout his life, Feynman, especially his book "The Character of Physical Law" 1964, does not really go into details of a good example of where Experimentation is the heart of science.

So in my long history of doing science, I found the very best example of showing where the Experiment--- as the heart of science-- what it means. It follows the Feynman "The Character of Physical Law" where he talks about gravity on page 4 and gives its formula as this.

F= G (M*M') / r^2

I am going to doctor his formula to make it more clear.

Force = (G a constant) ( Mass_1 times Mass_2) / (separation distance of masses)^2

Now, in my life, the very best experiment I have seen concerning the verification of the Universal law-structure of gravity as displayed above, comes from a rather unusual and outstanding source of a brilliant lady mathematician and physicist of France--- Madame Emilie du Chatelet 1706-1749 a friend of the famous writer Voltaire.

Women are seldom given credit in the world of science as we well know the genius of another French-Polish lady scientist-- Madame Curie. Well, Emilie was another genius scientist. For Emilie, stood head to head alongside Newton, Leibniz, Gravesande, and even showing where Newton was wrong on his own Universal Law-structure of Gravity, the gravity principle.

Emilie could be considered as a science-reporter-journalist of her time, translating and adding on to the explanation of Newton's Law-structure of Universal Gravity.

And the reason I pick the Law-structure of Gravity is to later on use this example to clarify UG, Universal Generalization and then UI, Universal Instantiation, and then EG, Existential Generalization and then EI, Existential Instantiation, __clarify Logic of Existential and Universal quantifiers____.

So the brilliant experiment that Emilie set up, incomparable to this day, in my humble opinion, and which I want to set up in this Logic classroom, is totally brilliant. If not able to set up then, at least, watch the PBS NOVA film.

What she did can be seen in a NOVA TV show, where she takes a large ball on a stand with clay underneath and lets the ball drop from 2 different heights. If Gravity is inverse square, the hole impression made by the ball falling is going to be 4 times bigger because of inverse square in denominator.

--- quoting NOVA on the lady physicist Emilie Du Chatelet with some AP edits---

NARRATOR: Emilie du Châtelet would have a huge effect on physics in her tragically short lifetime. Unheard of, for a woman of her time, she would publish many scientific works, including a translation of Sir Isaac Newton's Principia, the greatest treatise on motion ever written. Du Châtelet's translation is still the standard text in France today.

TUTOR (Dramatization): Musa, mihi causas memora?

CHARLES: Muse, my memory causes...?

EMILIE DU CHÂTELET: "O Muse. The causes and the crimes relate; what goddess was provok'd, and whence her hate; For what offence the Queen of Heav'n began to persecute so brave, so just a man."

EMILIE'S FATHER (Dramatization): Do not be cross with your sister because she persecutes many a just man. Only the other night Emilie silenced the Duc du Luynes when she divided a ridiculously long number in her head in a matter of seconds. You should have seen the incredulity on their faces when they realized Emilie was correct.

CHARLES: Was it my sister's astounding intelligence or her boundless beauty that made their mouths gape, I wonder?

EMILIE'S FATHER: Ah well, yes, you have a point, Monsieur.

EMILIE DU CHÂTELET: Messieurs, I thank you for your kindness. I fear, however, that my wit is only a curiosity to others. If only my mind was permitted opportunity.

EMILIE'S FATHER: My dearest, Emilie. You are blessed with intellect and courage. Use them both and the world will fall at your feet.

JUDITH ZINSSER (Du Châtelet Biographer): In one sense, she is a woman utterly out of her true time and place. She is a philosopher, a scientist, a mathematician, a linguist. She demands a freedom that women didn't begin to enjoy until over 150 years later, a freedom to study science, to write about it and to be published.

NARRATOR: Du Châtelet married a general in the French army at age nineteen and had three children. She ran a busy household, all the while pursuing her passion for science. She was 23 when she discovered advanced mathematics. She enthusiastically took lessons from one of the greatest mathematicians of the day, Pierre de Maupertuis. He was an expert on Newton, and she was his eager young student. It seems they had a brief affair. But then he set off on a Polar Expedition.

Du Châtelet then fell passionately in love with Voltaire, France's greatest poet. A fierce critic of the King and the Catholic Church, Voltaire had been in prison twice and exiled to England, where he became enthralled by the ideas of Newton. Back in France, it wasn't long before he again insulted the King. Du Châtelet hid him in her country home.

CHARACTER (Dramatization): The poor little creature is devoted to him.

NARRATOR: Isolated far from Paris, Du Châtelet and Voltaire turned her chateau into a palace of learning and culture—complete with its own tiny theatre—and all with the apparent blessing of her husband.

PATRICIA FARA: There is a great deal of myth surrounding Du Châtelet and her love life. And most of it is very exaggerated. But her husband did accept Voltaire into his household, and he often went to Paris on behalf of Voltaire. He went to his publisher to plead Voltaires' case, to keep Voltaire out of jail. And it is also true that Emilie Du Châtelet did have several affairs of a fleeting nature.

JUDITH ZINSSER: She created an institution to rival that of France's Royal Academy of Sciences. Many of the great philosophers, poets and scientists of the day visited.

EMILIE DU CHÂTELET: Ah, Monsieur you are young. I hope that soon you will judge me for my own merits or lack of them, but do not look upon me as an appendage to this great general or that renowned scholar. I am, in my own right, a whole person, responsible to myself alone for all that I am, all that I say, all that I do.

NARRATOR: Du Châtelet learned from the brilliant men around her, but she quickly developed ideas of her own. Much to the horror of her mentors, she even dared to suspect that there was a flaw in the great Sir Isaac Newton's thinking.

Newton stated that the energy of an object, the force with which it collided with another object, could very simply be accounted for by its mass times its velocity. In correspondence with scientists in Germany, Du Châtelet learned of another view, that of Gottfried Leibniz. He proposed that moving objects had a kind of inner spirit. He called it "vis viva," Latin for "living force." Many discounted his ideas, but Leibniz was convinced that the energy of an object was made up of its mass times its velocity, squared.

DAVID BODANIS: Taking the square of something is an ancient procedure. If you say a garden is "four square," you mean that it might be built up by four slabs along one edge and four along the other so the total number of paving slabs is four times four, is 16. If the garden is eight square, eight by eight, well eight squared is 64, it'll have 64 slabs in it. This huge multiplication, this building up by squares is something you'd find in nature all the time.

FRANCOIS-MARIE AROUET DE VOLTAIRE: Emilie, Emilie, you are being absurd. Why ascribe to an object a vague and immeasurable force like vis viva? It is a return to the old ways. It is the occult.

EMILIE DU CHÂTELET: When movement commences, you say it is true that a force is produced which did not exist until now. Think of our bodies, to have free will we must be free to initiate motion. So, all Leibniz is asking is, "Where does all this force come from?"

FRANCOIS-MARIE AROUET DE VOLTAIRE: In your case, my dear, the force, I'm sure, is primeval.

EMILIE DU CHÂTELET: Aaah, you're infuriating. You hide behind wit and sarcasm. You only think you understand Newton. You are incapable of understanding Leibniz. You are a provocateur. Everything you do is about something else and makes trouble for you. Criticize this, denounce that. Are you capable of discovering something of your own?

FRANCOIS-MARIE AROUET DE VOLTAIRE: I discovered you.

NARRATOR: Despite the overwhelming support for Newton, Du Châtelet did not waver in her belief. Eventually, she came across an experiment performed by a Dutch scientist, Willem's Gravesande that would prove her point.

EMILIE DU CHÂTELET: Gravesande, in Leiden, has been dropping lead balls into a pan of clay.

FRANCOIS-MARIE AROUET DE VOLTAIRE: Dropping lead balls into clay? How very imaginative.

EMILIE DU CHÂTELET: Using Newton's formulas, Monsieur Voltaire, he then drops a second ball from a higher height, calculated to exactly double the speed of the first ball on impact.

So, Messieurs, care for a little wager? Newton tells us that by doubling the speed of the ball, we will double the distance it travels into the clay. Leibniz asks us to square that speed. If he is correct the ball will travel not two, but four times as far. So who is correct?

AP writes: Newton was thinking of momentum as mass times velocity, while Leibniz was thinking of kinetic energy which is mass times velocity^2. And Emilie was going to show that Leibniz was correct while Newton was wrong. Most of us learned that there was a great fight between Newton and Leibniz on who discovered Calculus, but there also was this great fight between these two on momentum versus kinetic energy.

PIERRE LOUIS DE MAUPERTUIS (Dramatization): Messieurs, I feel Mister Newton's reputation dwindling, ever so slightly.

FRANCOIS-MARIE AROUET DE VOLTAIRE: Oh, Maupertuis, do not succumb to her. There is no earthly reason to ascribe hidden forces to this Dutchman's lead balls.

EMILIE DU CHÂTELET: Well, the ball travels four times further.

DAVID BODANIS: Turns out Leibniz is the one who is right. It's the best way to express the energy of a moving object. If you drive a car at twenty kilometers an hour, it takes a certain distance to stop if you slam on the breaks. If you're going three times as fast, you are going sixty kilometers an hour, it won't take you three times as long to stop, it'll take you nine times as long to stop.

AP writes: Yes, the car stopping as momentum= mass times velocity, or stopping as kinetic energy = mass times velocity^2.

PIERRE LOUIS DE MAUPERTUIS: Oh. Well, it does seem worth consideration.

FRANCESCO ALGAROTTI (Dramatization): Perhaps we might look over his calculations?

EMILIE DU CHÂTELET: I have already checked his figures. I am sure Leibniz is correct on this point. I intend to include a section on this matter in my book.

PIERRE LOUIS DE MAUPERTUIS: Really? Do be careful, Madame. Do you think the Academy is ready for such an opinion?

FRANCOIS-MARIE AROUET DE VOLTAIRE: Quite, quite. We really should be careful...

EMILIE DU CHÂTELET: "We?"

I see no reason to delay. There is no right time for the truth.

JUDITH ZINSSER: Emilie du Châtelet published her Institutions of Physics in 1740, and it provoked great controversy. Voltaire wrote that "she was a great man whose only fault was being a woman." In her day that was a great compliment.
--- end quoting PBS NOVA on Madame Chatelet proof of Universal law-structure of Gravity via Experiment with some AP edits---

AP writes: So, this experiment, balls dropped into clay, allows me to explain EI then EG then UI then UG. Experiment connects Existence with Universal Law-structures of Science. But, let me save that discussion for later.

Neanderthals had a fire making industry of flint striking iron pyrite.
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Well, I have always considered the Neanderthals smart, for,..... how could they have gotten across the English channel in the first place, from mainland Europe to the British Isles.

On the subject of "What is science?", fortuitously, I am helped by a recent report out of England of the very first evidence of Neanderthals, some 400,000 years ago used flint and iron pyrite to make fires.

It is safe to say that Science starts in history here, 400,000 years ago, as the Experiment that keeps making fire.

Homework: look up this report of Neanderthals making fire with flint striking iron pyrite and write a one page paper on the details of this report.

Archimedes Plutonium Jan 8, 2026, 8:28:31 PM to Plutonium Atom Universe newsgroup.

Question:: Iron pyrite struck by flint as in the BBC show, showing sparks flying everywhere, just profuse sparks flying everywhere.

My question is, is that a demonstration of the Faraday law-structure??? Does the Faraday law-structure have any role in sparks flying when flint strikes iron pyrite??????

The Faraday Law-structure for students who never saw that demonstration is where a magnet thrust through a coil of copper wire, produces electric current.

Faraday Law-structure Experiment in classroom
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Teacher Assignment:: set up the Faraday Law-structure experiment in the classroom so that all the students can eye witness this law-structure and allow some students to do the thrusting. You will need a copper coil connected to a Galvanometer and have a permanent bar magnet.

Lesson in classroom: Electric currents in Nature, come from thrusting magnetism through a current carrying material, a conductor like copper.


3) Logic is the "Scientific Method".



Archimedes Plutonium Jan 9, 2026, 4:03:31 AM to Plutonium Atom Universe newsgroup.

Alright, well some of the terms used by NOAA are different from my terms established earlier, and this is what I mean when I say Old Logic, Old Physics, Old Science never got their house of terms in order.

Scientific method from NOAA
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Step 1: Wonder

Observe something and wonder what is happening or why or how it happens. Ask a question or make a statement that can be tested by an experiment. This statement is called a hypothesis and defines the purpose of your experiment.
 

Step 2: Define

Define your variables (parts of your experiment that will change) and your controls (parts of your experiment that will not change).


Step 3: Review

Find out everything you can about what people have already said or written about the subject. Even if someone else has already done an experiment and reported results, you can still repeat their experiment or devise one of your own to verify (or refute) the results of the previous experimenter.

Step 4: Design

Design an experiment to test the hypothesis.

Make a step-by-step list of what you are going to do.

Step 5: Experiment

Do the experiment and carefully record the data.

Step 6: Analyze

Process and analyze your data. Do any calculations needed or draw graphs to help you make sense of the data.

Step 7: Conclude

Draw conclusions and write a report.

Did the data you collected support your hypothesis or not? Is there any reason to think there might be errors in your results? If the data did not support your hypothesis, what are some other hypotheses you might test to explain your initial observations? What further research could you or someone else do to verify your results?

Here again, well, NASA is all over the map when it comes to terms of science.

Scientific Method from NASA
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1. Form a hypothesis (a statement that an experiment can test)
2. Make observations (conduct experiments and gather data)
3. Analyze and interpret the data
4. Draw conclusions
5. Publish results that can be validated with further experiments


Scientific Method from various sources on Internet
-----------------------------------------------------------------

Step 1-- Make observations and ask many questions.

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

Step 3-- Formulate a Hypothesis of what you think is going on.

Step 4-- Conduct Experiments pertaining to your hypothesis.

Step 5-- Collect data from the experiment/s and analyze the data.

Step 6-- Draw conclusions.

Step 7-- Publish the results.

AP writes: Step 7 should be a part of step 6 where we say Draw conclusions and publish the results. Publishing is important for it forces others to repeat the experiment and repeatability is a major factor of true science experiments.

On Friday, January 9, 2026 at 4:03:31 AM UTC-6 Archimedes Plutonium wrote:
(snipped)
Alright, I am going to try to find an appropriate experiment to use the Scientific Method upon for 1st year logic in college or university.

1st Astronomy experiment: Observe the North Star and watch how the Big Dipper slowly rotates in the night sky. Focus on proving the Earth is Round and rotates on its axis.

2nd Astronomy experiment: Observe a few Blue Stars and Red Giant Stars and correlate their masses. Focus on whether stars shine from fusion or shine from Faraday law-structure.

1st Biology experiment:  Try imitating Smilodon, the saber toothed tiger by placing two fingers of left or right hand on your canine teeth, while with your free hand, try eating something, like a banana or apple. Look at the ridiculousness of David Attenborough and the entire paleontology community in getting you to museums and having interest in science because they glued on walrus tusks to the upper jaw of a fossilized cat. Focus on Darwin evolution that saber teeth would get in the way and interfere with everything the cat wanted to do in life if it had those massive saber teeth.

2nd Biology experiment: The Pterosaur dinosaurs which supposedly flew in the air yet had large bills and long necks and some weighed as much as a giraffe, yet grown up scientists say they flew in the air. Focus on aeronautical engineering that weight limits the ability to fly, and that rather than fly, these animals used their appendages as sails to boat oar sail on top of water.

3rd Biology experiment: The reflection of light that points out Superdeterminism.

1st Physics experiment: Take a bungee-cord fastened to fixed structure and see if you can get a sinusoid wave. Focus on whether physics has waves or rather, instead wires.

2nd Physics experiment: When Sun shines from Faraday law-structure, not fusion, then global warming accelerates year after year. Investigate how acceleration of global warming points to Faraday law-structure, not fusion for how the Sun and stars shine.

1st Chemistry experiment: Investigate H2 a molecule or a atom? All atoms require to have a neutron to store the energy of the muon thrusting through proton torus producing electric current in the Faraday law-structure. So in H2, is one of the protons acting like a neutron and not a proton.

Archimedes Plutonium Jan 9, 2026, 9:06:07 PM to Plutonium Atom Universe newsgroup.

I am going to have to narrow this list down.

1st Astronomy experiment: Observe the North Star and watch how the Big Dipper slowly rotates in the night sky. Focus on proving the Earth is Round and rotates on its axis.

The light pollution near cities is so bad that you cannot even see the constellations, so I may have to switch to the Moon.

2nd Astronomy experiment: Observe a few Blue Stars and Red Giant Stars and correlate their masses. Focus on whether stars shine from fusion or shine from Faraday law-structure.

Here the student is going to have to find out how astronomers determine the mass of a distant star. That maybe too much.
 
1st Biology experiment:  Try imitating Smilodon, the saber toothed tiger by placing two fingers of left or right hand on your canine teeth, while with your free hand, try eating something, like a banana or apple. Look at the ridiculousness of David Attenborough and the entire paleontology community in getting you to museums and having interest in science because they glued on walrus tusks to the upper jaw of a fossilized cat. Focus on Darwin evolution that saber teeth would get in the way and interfere with everything the cat wanted to do in life if it had those massive saber teeth.

This is a good experiment, for biology has little math and easier for students to comprehend.

2nd Biology experiment: The Pterosaur dinosaurs which supposedly flew in the air yet had large bills and long necks and some weighed as much as a giraffe, yet grown up scientists say they flew in the air. Focus on aeronautical engineering that weight limits the ability to fly, and that rather than fly, these animals used their appendages as sails to boat oar sail on top of water.

This is a good experiment. Too much of paleontology is con-art anti-science.
 
3rd Biology experiment: The reflection of light that points out Superdeterminism.

This is a nice experiment and would have the student dive into what is called Quantum Entanglement and the famous engineer John Bell who proved Einstein was wrong.
 
1st Physics experiment: Take a bungee-cord fastened to fixed structure and see if you can get a sinusoid wave. Focus on whether physics has waves or rather, instead wires.

I myself am doing this experiment, in showing that the Sinusoid wave is nonexistent and only appears on screens of appliances but not in physical reality. And where Light is not a wave but rather a Wire.

2nd Physics experiment: When Sun shines from Faraday law-structure, not fusion, then global warming accelerates year after year. Investigate how acceleration of global warming points to Faraday law-structure, not fusion for how the Sun and stars shine.

This is a good experiment for students for the most part are charged up about the climate and their future.

And recently in the news of February 2026 is the Thwaites glacier in Antarctica is accelerating melting. Probably because the Sun does not shine from fusion but from Faraday law-structure. To be sure, fossil fuel burning adds to the melt but the majority of Global Warming is due to Sun Gone Red Giant Phase.

1st Chemistry experiment: Investigate H2 a molecule or a atom? All atoms require to have a neutron to store the energy of the muon thrusting through proton torus producing electric current in the Faraday law-structure. So in H2, is one of the protons acting like a neutron and not a proton.

No, I better drop this chemistry experiment as too detailed in water electrolysis as a proof. Not all students in college take chemistry. And it maybe dangerous working with hydrogen and oxygen.

Archimedes Plutonium Jan 9, 2026, 9:36:41 PM to Plutonium Atom Universe newsgroup.

1st Astronomy experiment: Observe the North Star and watch how the Big Dipper slowly rotates in the night sky. Focus on proving the Earth is Round and rotates on its axis.

The light pollution near cities is so bad that you cannot even see the constellations, so I may have to switch to the Moon.

I have been working on astronomy in my Advanced Logic textbook, proving Earth is round, not flat. And proving Earth revolves around Sun and not the other way of geocentric system.

Of course when sailors sailed around Earth is demonstrable proof Earth is Round.

But, and however, the proof that Earth goes around the Sun, and not that Sun goes around Earth is best proved by what??? Is the Venus observations the best proof.

I am thinking we can make One Logical Argument that proves Earth is Round, and it spins on its axis with a tilt, and it goes around the Sun, not vice versa. All proven by the idea of the size of the Sun and its distance away from us and the speed of Sun and Earth in space.

I am saying speed and distance alone should prove these items. Only the proof is a Occam's Razor of easiest explanation is the true explanation.
1) Earth is Round
2) Earth spins on a tilted axis
3) Earth goes around Sun, not vice versa


4) Logic is the Science of Ideas.



We all know what mathematics is, for it involves playing around with numbers or playing around with geometry figures. Is there an easy definition of Logic?? Yes, of course, instead of playing around with numbers in math, we play around with "ideas". Just substitute ideas for numbers, and there you have Logic in its most simple form. However, Logic is bigger than math and Logic contains everything that is in mathematics, for everything inside of math must be inside of Logic, just as everything of chemistry, must be inside of physics. Math is a compartment of Logic.

The reason I bring this up about mathematics for Logic is somewhat similar to mathematics, only, instead of numbers and geometry figures to play with, we play with "ideas". Instead of numbers we manipulate ideas. Instead of geometry figures, we play with ideas in logic.

And in this book of logic, the best way to teach logic is to use what you already learned about math numbers-algebra and geometry figures.

Instead of 2 + 3 of math we play with two ideas P, Q as in P AND Q of logic. Instead of 5 - 2 of math, we play with two ideas R,S as in R OR S of logic. We call the ideas, P,Q,R,S as "statements of ideas". The statement P of Logic could be any idea, such as P = The winter is cold with the strawberries covered in pine needles to keep them alive. Such as Q = Some plants are kept inside the house near a window for survival from the winter.

Instead of numbers and geometry figures as the objects of study, in logic, we study statements of ideas and how we manipulate them via connectors such as AND, OR, Equal-Not, If-->Then.

Homework: There are 26 letters in the alphabet from A to Z, for 13 of those letters make up true statements of ideas about plants in Nature. For another 13 letters make up true statements of ideas about animals in Nature.

For example:
A= I have burr oak, Quercus macrocarpa, some over 100 years old growing on my property.

B= Both red squirrel and gray squirrel live in the burr oak and harvest their acorns.



5) Math is the science of numbers-algebra and geometry figures.



We learn mathematics in grade-school with arithmetic, adding, subtracting, multiplying and dividing numbers. Later we learn equations and using algebra to find number unknowns. Later we learn geometry figures, telling them apart. Finding their areas. So math is both numbers and geometry.

These operators in mathematics of add, subtract, multiply, and divide show up in Logic, only they are called "connectors of ideas".

In Logic there are what is called the 4 simple connectors of AND, OR, Equal-Not, and If-->Then. Can you guess which connector resembles which operator of math??

If you guessed that AND is add, and OR is subtract, and Equal-Not is multiply, and If-->Then is divide, you guessed correctly.

But there are 2 more operators in mathematics that are far more complicated and very important. These two come from calculus, the math of motion, geometry graph and energy. The two are called derivative and integral. The derivative is rate of change, like that of speed or velocity and is a sort of division of dy/dx, which in High School you learned as slope of the line. The change in the y component divided by the change in the x component is the derivative.

The other calculus operator is the integral and is the area under the graph of the function over an interval.

The reason I bring up the derivative and integral is because the Existential quantifier of Logic is the derivative of calculus and the Universal quantifier of Logic happens to be the integral of calculus. So I am going to have to teach students a little bit about calculus in order to teach Logic. I am going to bend over backwards to make it as easy as possible for many students are frightened by calculus.

So we end up with 6 operators of mathematics, add, subtract, multiply, divide, derivative and integral. And so Logic ends up with 6 connectors that resemble these 6 operators in the same order listed: AND for add, OR for subtract, Equal-Not for multiply, If-->Then for divide, Existential quantifier for derivative, and Universal quantifier for integral.

The easiest way of teaching logic is to use math as a guide, a template.


6) Both Logic and Math are precision sciences; math uses sigma-error and logic uses truth-tables.


Math is precise, and that is why the other sciences like using math for its utmost precision. But that precision is not always 100%, especially using numbers of measurements in experiments. And what math does in those circumstances is what is called Sigma Error. For example, the true value of the neutron is actually 945MeV but when physicists go to measure the neutron rest-mass, they find it to be 940MeV. So how much is the measurement off of actual? So we divide 940 into 945, dividing the smaller into the bigger number 945/940 and get 1.005 which means what?? Percentage is number in 100. And 1% is 1 out of 100. 10% means 10 out of 100. 50% means 50 out of 100. When we have a number 1.005, the 1 means 100 out of 100. So we move two decimal places to the right and end up with 0.5%. The number 945 is 0.5% more than the number 940. What is 0.5% of 945? 945 x 0.005 = 4.725, close enough to 5. If we subtract 940 from 945 we have 5 even.

0.5% as in percentage we go 2 places to the right of the decimal point. To check to see if that is correct we simply use a quick logic analysis multiply 945 by 0.5 and see if it comes close to 940.

Faster yet, we see if 1% of 940 gives us approximately a number when added to 940 is near 945.

I write this in detail because in sci.math Usenet where I spent enormous time from 1993 until Google exited, there are many college graduates who never learned what percentages mean, even this so called engineer from Rensselaer Polytech, as he mistakenly fumbles and makes error, after gross error.

Posted in sci.math Usenet (misspellings not corrected) and where Kibo Parry often used fake names. In my opinion, he is a good example of a big mouth with little logical brains and where this textbook would have benefitted him while in University.

May 26, 1989, 10:35:51 AM kibo parry wrote:

What's the largest prime currenlty known? (All the information
I could dig up here was either fairly old or contradictory...)

james "kibo" parry | Some days you just can't get rid of a bomb.
kibo%pawl.r...@itsgw.rpi.edu (internet)
userfe0n@rpitsmts (bitnet) | Anything I say represents the opinion of
kibo%mts.r...@itsgw.rpi.edu | myself and not this computer.
>
> Kibo Parry Moroney Volney wrote:
> On Wednesday, December 6, 2017 at 12:30:22 AM UTC-6, Michael Moroney wrote:
> > Silly boy, that's off by more than 12.6 MeV, or 12% of the mass of a muon.
> > Hardly "exactly" 9 muons.
> > Wednesday, December 6, 2017 at 9:52:21 AM UTC-6, Michael Moroney wrote:
> > Or, 938.2720813/105.6583745 = 8.88024338572. A proton is about the mass
> > of 8.88 muons, not 9. About 12% short.
>

Homework:: You may have to review percentages, so you do not make the same mistakes as engineer graduate Kibo Parry of Rensselaer Polytech.

Use the 1% check to know if you are in the range of the correct answer.

The Proton with a Muon inside the Proton actual rest-mass is 945 but Old Physics via experiments says it is 938MeV as listed in Halliday & Resnick and in Wikipedia.

Compute the sigma-error of Proton + Muon between 938 and its actual true rest-mass of 945. Show your work and show your check.

Compute the sigma-error of the Muon rest mass when it is actually 105MeV , yet Old Physics says it is 105.6 MeV. Show your work and show your check.

Compute the sigma-error of the magnetic monopole which is exactly 0.5MeV, yet Old Physics mistakenly thought this particle was the Atom's electron, it is not; but rather the muon is the atom's electron. Old Physics measures this particle as 0.51MeV. What is the sigma error between 0.5 and 0.51? Show your work and show your check.

I was a High School teacher myself and percentages was an extremely difficult concept for teenagers to learn. I advise them to use a Check on Range to know if they have the correct answer. You cannot be a good scientist or engineer, if your mind cannot handle percentages correctly.

Math is used throughout the sciences, more for physics, less for biology and the reason for its use is because math offers precision. Math is a language of precision. Logic is also a precision language as it tries to steer away from false ideas, false conclusions, vague ideas.

Logic uses Truth-Tables for precision. Math and Physics use Sigma-error for precision.

Logic seeks for you to Think Straight, Think Clearly, and Think Truthfully and uses Truth Tables to help you think the best.

Modeling Logic after Math in order to understand Logic better.
---------------------------------------------------------------------------------

Much of Logic has the same structure as mathematics, both have 6 connectors which math calls operators. Both seek precision. Only real difference is that Logic uses "Ideas" while math uses numbers or geometry figures to work on.

Ideas of course are bigger than is a number or a geometry figure and so we can readily see that Logic is the larger set and has all of mathematics inside of Logic.

Math is a subset of Physics supplying physics with correct and precise number quantity, size and geometry shapes and figures. While Logic supplies physics and the other sciences with precise ideas written as statements p,q,r,s, etc etc as those ideas. Both math and logic are precision languages.

Math numbers are quantity, size and amount, and math geometry is shape and describing space. Logic is the correct ideas and manipulation of those ideas of clear thinking and straight thinking whose truth value comes directly from the best science on the subject. Both math and logic are languages that describe Physics and all the other sciences.

Where to start in Math if this was a Math story?? Well, we could start with the true numbers of mathematics, the Decimal Grid Numbers and the smallest grid is the 10 Grid which has 100 members not counting 0. This set is 0, .1, .2, .3, ... , 9.8. 9.9, 10.0. And then start with addition then subtraction.

If Math starts that way, how should the language of Logic start? What is the parallel of Numbers in Logic?

Here the parallel are statements, which contain thoughts and ideas and are written in Logic as "p", "q", "r", "s", "t" etc. And the truth value of each of those statements comes from the best science of the time on the topic of that statement in question.

An example of statements is the famous Aristotle syllogism attributed to Aristotle.

All men are mortal.
Socrates is a man.
Therefore,
Socrates is mortal.

This can be rewritten as

p = q
S = p
therefore S = q

So for Math we have numbers, for Logic we have idea-statements p, q, r, s, t, etc.

Once we have numbers in math (and later have geometry figures), we then move on to that of operations on numbers. No point in creating numbers that sit around and do nothing. No, we want to operate and use numbers to figure out the world we live in.

And of course math operators have 4 simple operators which most students will guess or know what they are--- Add, Subtract, Multiply and Divide. In a very real sense, multiply is rapid add, while divide is rapid subtract.

Logic also has 4 Simple connectors on statements p, q, r, s, t, etc. and they are AND, OR, Equal-Not, If-->Then. Instead of calling them operators in logic, we call them connectors. But math has 2 more operators from calculus as derivative and integral.

Math has two more operators from calculus called Differentiation (the derivative) and Integration (the integral). Logic has two more connectors of quantifiers called the Existential quantifier and the Universal quantifier, related to derivative and integral, respectively.

One branch of Logic models math so much it is given a special name of "Symbolic Logic" where ideas are pushed around in arguments and all you see are symbols, much like a math computation where you see symbols from one step to the next step.



7) A lesson on Calculus derivative and integral in order to understand Existential and Universal quantifiers.



Again, I have to start over, for logic is my most painstaking textbooks. I would have thought it would be physics to give me the most problems. Turns out, it is Logic. And a simple reason is that Logic needs all things in proper place, proper order.

I thought I could do Elementary Logic with a brief mention of calculus derivative and integral, considering the fact that the Existential and Universal quantifiers are bound up with calculus.

For the past several days I have been debating this issue and finally came to the conclusion that you can not do Logic adequately without some calculus derivative and integral development. Just as AND is Add in math, the Existential quantifier is the derivative of calculus and the Universal quantifier is the integral of calculus.

So what I have to do now is make a Lesson on calculus, introducing the college or university student in knowing the relevant parts of Calculus in a story book telling of this calculus. Not to be scared or fearful, but a simple story book telling.

And this will be, obviously, the first Logic textbooks that views and sees Calculus essential in telling the true story of "What Logic Is?"

There is no doubt in my mind that without telling parts of calculus, I fail to instruct students, nay, humanity of "What is Logic?"

Students in High School are taught the function and the graph and so I can lean on, and be helped by that instruction.

So I am struck at this moment in time on what a marvelous idea I discovered about Logic, true Logic. A education and understanding of Logic is never complete unless you know the derivative and integral of Calculus.

Now the naysayers and old fashioned, conservative, or dull witted person of math and science would, like always rebuke and rebuff that claim. For all they know is defend the indefensible of what they have as a "memorization education". If it is in a textbook in school that is ultimate knowledge and nothing can replace or displace it in those weak minds. They can never admit slant cut of cone is oval, never ellipse. And would fiercely attack an idea that Logic as a science is incomplete unless it has math calculus, derivative and integral represented as Existential and Universal quantifiers.

So I was a High School teacher of math at one time in my life and learned well, how to make ideas simple for young people to learn. Here I need to teach in about 10 or 20 pages the rudimentary calculus to have students understand how Existential quantifier is the derivative and integral is the Universal quantifier.

But can I make a proof argument that Logic without calculus discussion, is flawed Logic????

Can I somehow come up with a viable argument that missing the derivative and integral in Logic class is like missing electricity and magnetism in physics classes????

I think I can in the very idea, that the definition of function has a Unique Y value given a x value--- which--- entails or causes existence. And the fact that the function definition requires every, or All x values have a unique y value entails a universal.

Law-structures of physics, chemistry, biology, are they the Universal quantifier? By all means. For the function in math requires All x values which determine a unique y-value.

So what I am saying here is that __no Logic textbook is complete__ unless it has a discussion of Calculus derivative and integral which forms the structure of Existential and Universal quantifiers.

There is another feature of derivative and integral that makes the case of a proof that                       they are Existential and Universal quantifiers. Here I speak of the idea that derivative is inverse integral returning the integral back to the starting function and that integral is inverse of derivative returning back to starting function.

When you take calculus in college or university you will learn that Polynomial functions are the easiest in the world as functions, to get a derivative or integral from. For all they are is a addition or subtraction of 1 from exponent of starting function, remembering to include a constant.

Let me show you this pattern on the Identity function, which is often written as Y = x. That is the starting function. Now the derivative of Y = x, which is a polynomial by the way. The derivative of Y = x is where x is x^1 and the constant of this x^1 is that I place the 1 in front of x^1 having 1(x^1) and if I subtract 1 from the exponent I get 1(x^(1-1)) which is 1(x^0). The x^0 turns out to have a value of 1 itself, so I end up for a derivative of x as being (1)(1)= 1.

Now that maybe confusing and scary to students upon first seeing that, so let us try another function of Y= x^2. When you see x^2 means nothing more than x times x. If you see x^3 means x times x times x. But now, what is the derivative of x^2. We follow the same procedure. We throw the 2 out in front and subtract 1 from exponent as that of 2(x^(2-1) and end up with 2x. The derivative of Y=x^2 becomes 2x and it is a function also denoted as Y=2x.


A graph of the function Y= x^2
-----------------------------------

Here is a picture diagram of what was talked about above on calculus.

The function x^2 -> Y looks like this in Integer Grid:

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

Cell or partition, some prefer the name cell, others prefer the name partition from 0 to 1 is a pure right-triangle sides 1 by 1 with hypotenuse sqrt2.
Cell from 1 to 2 is a right-angled-trapezoid (picketfence) and the picketfence is composed of a square with a right-triangle atop the square.
Cell from 2 to 3 is a right-angled-trapezoid (picketfence) and the picketfence is composed of a rectangle with a right-triangle atop the rectangle.

Homework: See if you can solve what the derivative of the function Y= x^3 is following the above pattern.

Now the Integral of calculus is almost the same method and pattern only it is a addition of 1 to the exponent. We started with Y=x function and what is the integral of that? Y=x is the same as Y=x^1. The rule is that you add 1 to exponent which is x^(1+1)and you throw in front of x^(1+1) a reciprocal of (1+1). Reciprocal means 1 over (1+1) and that is 1/2. So we have as the integral of Y=x to be Y= (1/2)x^2.

Another integral example is Y=x^2. So let us add one to the exponent and we have (2+1) and the constant thrown in front is 1/(2+1) = 1/3. So the integral of Y=x^2 is that of (1/3)x^3.

Homework: See if you can solve for the integral of the function Y= x^4 by following the above instructions.

If you go on to take Calculus in college or university you will learn a marvelous theorem called the Fundamental Theorem of Calculus which merely means that derivative and integral are inverses of one another. Inverse means one undoes the other.

In my example above of Y=x^2, we found the derivative is 2x and the integral is (1/3)x^3.
If we take the derivative of (1/3)x^3 we throw the 3 out in front and subtract 1 from exponent. That looks like this (3)(1/3)x^(3-1) and that reduces to x^2, our original starting function Y=x^2. But now, starting with the derivative 2x, what is the integral of 2x???? Here again we follow the same rules we add 1 to exponent and throw out a constant that is a reciprocal of the new exponent which looks like this (1/2)(2)x^2. Reducing that and again we have the starting function Y=x^2. This is the meaning of the Fundamental Theorem of Calculus, that the derivative undoes (inverse) the integral and the integral undoes (inverse) the derivative to establish the original starting function.

Homework: With the instructions given above on Fundamental Theorem of Calculus, your starting function is Y=x^3, see if your derivative undoes the integral and see if your integral undoes the derivative by returning to your original function Y=x^3.

Unfortunately as of this writing in February 2026, there exists no calculus textbook in the world at present who is teaching calculus correctly. I say that because the only valid functions in all of mathematics are the Polynomial functions, the easiest functions to find the derivative and integral. But that should not be strange because as of this writing, no textbook in Logic is correct for they all have the 4 simple logic connectors in awful error, the AND, OR, Equal-Not, If-->Then.

We will have to show that the Existential quantifier is the inverse of Universal quantifier and that Universal is inverse Existential. And that is easily done by showing the calculus is derivative inverse to integral.

The If-->then connector (detailed later in this book) or what I call "Move into" connector is easily seen as the derivative of a function as it moves from one cell to the next all along the x-axis to pair up a x-value to a unique Y-value. We study the If-->Then later but I wanted to introduce it to you now.

No, it is impossible to describe Existential with Universal quantifiers without bringing in the concept of calculus derivative and integral.

So, is there a Fundamental Principle of Logic, like there is in Calculus, for Calculus has a Fundamental Theorem which has as a statement-- the derivative is inverse the integral and the integral is inverse the derivative? Something along the lines of saying the Existential Quantifier is inverse to the Universal Quantifier, that would be a resounding proof that Logic without a discussion of calculus is half-baked logic.

Is not the UI, UG, EI, EG, Universal Instantiation, Universal Generalization, Existential Instantiation, Existential Generalization, much the same as the math teacher talking about Function then Function derivative, then Function integral, going from one to the other.

Take the function Y = x^2, its derivative is 2x, and its integral is (1/3)x^3. Taking the derivative of (1/3)x^3 leads me back to x^2, and taking the integral of 2x leads me to (1/2)(2)x^(1+1) = x^2. One undoes the other.

Taking the Derivative of Universal Quantifier ends up with Existence. Taking the Integral of Existence ends up as Universal.

What I am saying here, is the relationship of Logic's quantifiers out of necessity needs the calculus derivative and integral.

Just as AND is related to Equal-Not (multiply is rapid addition) and that OR is related to If-->Then (division is rapid subtraction). UG is rapid UI, EG is rapid EI.

Plus, AND is reverse of OR (add reverse of subtract). and Equal-Not the reverse of If-->Then (multiply the reverse of division).

What I am doing here is saying that a Logic textbook without involving math Calculus is missing a huge chunk of the science of Logic, itself. Is incomplete as Logic.

Calculus is motion and change, and Logic to be complete cannot have missing "motion and change".

Without Calculus involved in Logic, it is a premature and soiled and stained logic.

For Calculus is Motion and Change in the world. A logic without motion and change is a logic that is pathetically poor.

The Universal Generalization UG, the Universal Instantiation UI, the Existential Generalization EG, and the Existential Instantiation EI are elements of calculus, the Motion and Change needed to make Logic complete.

I have shown earlier that the Scientific Method starts with "There exists" which is EI moving over to EG through experiments, moving over to UI and finally being titled a Law-structure of Science in UG.

The force of gravity by Newton went through this process G = constant Mass_1 x Mass_2 / distance^2. Starting out with "There exists" moving into Generalized Existence in multiple experiments moving into Universal Instantiation leading to the recognition of Universal Generalization.

Much the same as a Function graph of the Identity function in Calculus.

Y = x, the identity function looking like this in 1st quadrant only.

|   /
| /_____

If we take the 10 Number Grid and only the integers, 0, 1, ...9, 10.

There exists 0 and the function links x= 0 with a unique y, being y=0. Next, x=1, links uniquely to y= 1.

x         y
0         0
1         1
2         2
3         3
4         4
5         5
6         6
7         7
8         8
9         9
10       10

The x number exists and moves uniquely into a y-value number. This is the definition of Function. For each x number must be 1 and only 1 y-value number. When you hear "1 and only 1" is the same as saying "unique". The motion in drawing the graph, starting at the origin (0,0) onto x=1 then x=2 and all the integers in 10 Grid.

And the derivative of Y = x, the identity function is dy/dx which means the change in y-value divided by change in x-value. As the function moves from 0 to 1 we have change in y as 1-0 divided by change in x 1-0 and so we have (1-0) / (1-0) = 1/1 = 1. Next the function moves from 1 to 2 and here again we have a dy/dx as (2-1)/(2-1) =1/1 =1.

What is the integral from x=0 to x=1??? That is the area under the function graph which is a right triangle.

A right triangle whose two legs are 1 and so its area is 1/2 base times height as that of (1/2) (1) (1) =0.5.

What is the integral from x=0 to x=9???? The base of right triangle is 9, the height is 9, the area is 1/2 base times height = (1/2) 9 x 9 =40.5.

You saw earlier that the integral of Y=x was that of (1/2)x^2 and that is the same formula geometry uses for the area of a right triangle.



And, no wonder math professors could never understand the truth about calculus derivative for it is the line segment that reaches for the next point of the original function graph and not the silly and absurd idea of Old Math-Old Calculus, full of error that it is a tangent line as pictured in Wikipedia. If a math professor cannot understand slant cut of cone is oval, not ellipse as a reasoned symmetry argument, why on Earth would anyone expect a math professor, he/she to have a correct explanation of the calculus and its derivative?

This is why the derivative is predictive of the future, the dy/dx is part of the original function graph. This is why the If-->then connector of logic is division just as dy/dx is division and I prefer to call If--> then as a concept of "move into". _If_ the Sun shines from Faraday law-structure, _then_ the Sun has gone Red Giant phase. Meaning the Sun moved into being a Red Giant star.

In the below picture from Wikipedia is a false statement for the derivative is not a tangent to the original function graph, rather instead, the derivative connects the previous (x,y) coordinate point to the very next (x,y) coordinate point of the original function graph.
--- quoting Wikipedia on the derivative---

Differential calculus

Main article: Differential calculus

Tangent line at (x₀, f(x₀)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The p

--- end quoting Wikipedia on the derivative---

ppp

[Archimedes Plutonium]

Before, I had thought of including this calculus of the truth about the Derivative in my Intermediate Logic textbook, thinking it would be too complicated and complex. But on second thought, calculus is usually taught to freshman in college and university, so why not teach this true derivative in Logic to Freshman in hopes that math professors will see the errors of their ways and fix their torture chamber calculus classrooms. Now I see the advantage of including it in my Elementary Logic textbook for Freshman in college or university. At UC in 1968-1972, I took calculus in 1968 as a Freshman. But it would have been far far more wonderful for me and all the other students to have learned True Calculus, and not the fake calculus taught by math professors who never realized ____calculus is geometry____ and thus--- the Fundamental Theorem of Calculus, FTC needs a geometry proof.

Because once a math professor has to make a geometry proof of FTC, then she/he starts to realize that Reals are fake numbers. That math has no continuum realizing that physics has no continuum because of quantum mechanics. Realizes that numbers, true numbers of mathematics are the Decimal Grid Numbers that have holes or gaps of empty space from one number to the next number. That the negative numbers are fakery and that the graph of numbers is in 1st quadrant only. That the only valid functions of math are Polynomials and if not a polynomial, you have to convert it to a polynomial over an interval. And that math Calculus is very very easy of the derivative and integral are merely add or subtract 1 from the polynomial exponent which is called the "power formula" for polynomials.

Note: the teacher of this course in logic is going to have to be familiar with the Power formula of calculus, taught in 1st year of college. The power formula is essentially the heart of Calculus, considering the only valid functions of math are polynomials and it makes calculus almost as easy as add, subtract, multiply, divide.

Have the teacher do this lesson in class on the blackboard or overhead projector. Sorry, but calculus of math is indispensable in teaching logic.

Power Formula of Calculus
-------------------------------------

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, the integral is addition.

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

Prefix-Area in calculus such as the 1/3 in 1/3*x^3 in integration of x^2 -> Y

Homework Exercises:

Derivatives using the Power formula of
3x -> Y
3x^2 -> Y
3x^3 -> Y
1/2x^4 -> Y
1/2x^6 -> Y
10x^8 -> Y

Integrals using the Power formula of
3x -> Y
3x^2 -> Y
3x^3 -> Y
1/2x^4 -> Y
1/2x^6 -> Y
10x^8 -> Y

So, yes, I should include this calculus in Elementary Logic because many students will be taking Elementary Logic as Freshman in college and university, and as the student is tortured in calculus class with nonsense of derivative as tangent and thousands of different functions, meanwhile, over in Logic class we teach you the true calculus that calculus is super easy with just add or subtract 1 from exponent, no torture chamber by crank-crackpot math professors who cannot even admit the slant cut of cone is Oval, never ellipse.

I hope students clash with the math professor and his/her torture chambers of calculus.


Homework: Do the assignment below of drawing the trapezoid and exposing the fact that when the true numbers of mathematics have holes and gaps in between one number and the next number-- not a continuum-- then the calculus derivative is not a tangent line to a point of the graph but that it spans to be the next point on the original function graph.

Now the teacher of this logic class will have to know well the calculus of math and to lecture on this chapter with students taking notes. Many items I have omitted such as the "power rule of polynomials". So the chore of the teacher is to clarify to the students. Many will have seen the derivative in math class, but many will not and so the teacher has to fill the gap.

--- quoting in part from my textbook
TEACHING TRUE MATHEMATICS: Volume 2 for ages 5 to 18, math textbook series, book 2 ---

Fundamental Theorem of Calculus, Picture

Draw a trapezoid so it is easier to explain this Calculus. This theorem is the essence of Calculus, so we want to understand it easily.

Draw the trapezoid  (0,0) (10,0) (10, 20) (0, 10)

Now look at the coordinate point (5,10) and mark it with a "m" meaning midpoint.

For it is the midpoint of the line segment that goes from (0,10) to (10,20) a slanted line segment.

Draw it in and draw the whole trapezoid.

Now the derivative in Calculus is this rooftop of a slanted line segment that goes from (0,10) to (10,20).

Here is a picture of what you have

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__
|         |
|         |
|         |
---------

That you can cut a right triangle from the midpoint

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

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

        B
        /|
      /  |
 m /----|
  /      |
|A      |
|____|

Or, you can start with that trapezoid and swivel the right triangle downwards to make the rectangle


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


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.

--- end quoting from my Teaching True Mathematics ---

Alright for an Exercise and then a Homework assignment of huge importance. We are going to prove that modern day math professors of Calculus are crank-crackpots when they believe the numbers of mathematics are the Reals as a continuum and that the derivative is a tangent line to the original function graph. Yes, we are going to prove math professors, not only are silly when it comes to slant cut of cone but crank crackpots when it comes to teaching true calculus. And let me use the glossary of terms for Mathematics given earlier.

Terms for math
--------------
Statement
Axiom (postulate)
Operator-structure
Proof
Theorem (and its corollaries)
Theory

We start with the Statement. And the statement is this.

Statement: Math professors who teach Calculus that the derivative is a tangent line to a point on the function graph (see Wikipedia diagram) are crank-crackpots of calculus.

We start with that as Statement. By the time we reach Theorem with Corollaries, the statement of the theorem will be far different. Something along the lines of this--- The derivative dy/dx of Calculus is a straight-line segment that joins the previous x-value and y-value of the original function to the successor x-value and y-value of the function. Geometrically it means the derivative is part of the original function itself and not some alien tangent line that touches the original function at a unique point.

I am showing the reader this because math professors since Newton and Leibniz when it comes to calculus have been nothing but crank-crackpots when understanding Calculus when they think the derivative is a tangent line to the original function graph. No, the derivative is a straight line segment that connects-up with the very next successor point of the x value and y value and forms the original function graph.

This is what I mean when scientists are not required to take 2 years of logic in college or university, for they come out as mostly kooks of science in their thinking. At least, when scientists take 2 years of University or College logic, at least they have a good chance of doing correct and proper science.

It is obvious to anyone, even those that hate math and especially Calculus, obvious that calculus is geometry. And yet in the late 17th century when Newton and Leibniz discovered calculus, they too realized calculus was geometry and they realized the Fundamental Theorem of Calculus, FTC, yet never ever proved this Fundamental Theorem of Calculus with a ____geometry proof_____. In fact no-one in math ever gave a valid proof of FTC, certainly impossible with the silly "limit analysis". And all the generations of math professors after Newton and Leibniz never had a geometry proof of FTC. They had some mindless analysis contraption called "limits" which is a bogus concept.

We have an original function Y--> x^2. By the way I write a function without a equal sign, but instead a arrow. For a function is a mapping of every x point linked up with a unique y-point and this is not equality, so I use an arrow. Now the function Y--> x^2 can be written as x^2 --> Y. The function plots coordinate points of a graph such as this integer table is trying to display.

x-value  --> y-value for x^2
0            -->   0 which is coordinate point (0,0)
1            -->   1 which forms (1,1) in the graphing
2           -->    4 which forms (2,4) in the graphing
3           -->    9 which forms (3,9) in the graphing

Now that table is just for a few points, but in a function, every x point must be represented with a unique y point forming a coordinate point. The Decimal 10 Grid of numbers has exactly 100 numbers not counting 0 on the x and on the y axes. Realizing there is a hole or gap in between 0 and the next number 0.1, another hole or gap between 0.1 and the next number 0.2 and this goes on all the way to the last two numbers in 10 Grid of 9.9 then 10.

In this diagram of the original function Y--> x^2 where derivative is 2x and integral is (1/3)x^3, AP draws the derivative as being a straight line segment from x= 1 to x= 1.1 in decimal-10-Grid, and showing that the derivative forms and produces the original function starting at x=1 and what x=1.1 has to be.

In other words, the derivative, fetches the future. The derivative at point x=1 will fetch the next original function point of coordinates (1.1, 1.21) and not the mindless tangent at the original function graph that crank-crackpot math professor teaches.

So looking at the function x^2 -> Y and we make a table in Decimal 10 Grid. Our table below is just a few points, and not all of the 100, or 101 counting 0 points.

x^2 -> Y
x       y
0      0
.5     .25
1      1
1.1   1.21
2.5   6.25
3      9

So we plug into the x^2 all the 10 Grid values on the x-axis and start making a table. But we are only interested in going from x=1 to x=1.1, because in 10 Grid no numbers exist between 1 and 1.1, or between 1.1 and 1.2. Grids are where there is ___no continuum___. Just like Quantum Mechanics in the year 1900 when Max Planck said physics is discrete with no continuum. But math professors were too dumb to ever entertain the idea, that math has to be discrete with no continuum if physics is discrete with no continuum.

Then, we graph our table of only the interval 1 to 1.1 for we are only concerned with this interval and I call this a cell, to see if the derivative is connecting up with the 1.1 as the successor number of 1.

y-axis
^
|
|
|
                             

                           |
                           |
                       /   |
                           |
                    /     |
                          |        
1.21  1.21/ |    |

                 |        |
           /     |        |
                 |        |
1    /  |1    |        |
   /     |       |        |
------------------------------------------------> x-axis

         1       1.1      

Now we are going to have to borrow from the 100 Decimal Grid the number 1.05 for in 10 Decimal Grid exists only empty space between 1. and 1.1. The 10 Decimal Grid is separated by increments of 0.1, in other words, holes of 0.1 from one number to the next number, while the 100 Decimal Grid is separated in increments of 0.01.

What is 1.05^2 for our original function graph is Y--> x^2. It is 1.1025.

We can picture this midpoint of 1.05 between x =1 and x=1.1 as a rectangle like this, which I call a cell.

____
|     |
|     |
------

Whose base is 1.1 subtract 1 equals 0.1. Whose height is 1.1025. The area of this cell would then be 0.1 x 1.1025 =0.11 approximately. The integral of Y-->x^2 using power rule is (1/3) x^3 and from the interval 0 to 1.1 is area of (1/3)1.331=0.444, while for area of 0 to 1 is (1/3)1 =0.333. If we subtract we have approximately, 0.11 a match to the area inside the cell.

Now the question, the big question is can we go down that cell from 1 to 1.1 with 1.05 as midpoint and find a right triangle to carve out and would, when pivoted onto the midpoint, end up landing at the coordinate point of (1.1, 1.21)??? In other words, the derivative is Not a tangent line to original function graph but is in fact a straight line segment that actually determines and connects up with the next point of the original function graph.

A week's homework assignment: This is a week long assignment. Go through two functions, Y--> 2x and Y--3x^2 using the power-rule make a table of coordinate point of x= 1 and x= 1.1 in Decimal 10 Grid. Plot your table similar to my plot above. Find the derivative and integral of these two polynomial functions. And show that the derivative connects the point (1,?) with (1.1, ?) for Y--> 2x and also for (1,?) with (1.1,?) for Y--> 3x^2. You will need the midpoint of 1 and 1.1. And sketch the right-triangle to lift up and pivot on the midpoint forming a trapezoid from rectangle. There, you have started a proof that says all mathematicians from Newton and Leibniz were wrong when they said a derivative is a tangent line to function graph.


So what failed between 17th century and 21st century? What failed most of all, is teachers do not teach Logic in College and University and the scientist has little to no logical brains. What Steve Huffman of the Reddit platform calls Lunatics. And why Australia, and now UK, France and many other countries by 2026 are banning these platforms for youngsters as these are brainwash ignorant platforms ruining the minds of our young students.


Steve Huffman lists science lunatics:
Reddit (symbol) r/math, 3 years ago Genius meets Lunatic: 1994 discussion between Terry Tao and Ludwig Plutonium
I remember Archimedes Plutonium and sci.math. He calculated the chromatic number of the plane: and it is 1 (color everything
..Is this crank...

Univ Virginia math dept: Peter Abramenko, Julie Bergner, Mikhail Ershov, Jeffrey Holt, John Imbrie, Thomas Koberda, Slava Krushkal, Thomas Mark, Jennifer Morse, Ken Ono, Andrei Rapinchuk, Christian Reidys, Jim Rolf, Charles Dunki, Ira Herbst, James Howland, Craig Huneke, Thomas Kriete, Nicholas Kuhn, Irena Lasiecka, Barbara MacCluer, Kevin McCrimmon, Karen Parshall, Loren Pitt, Donald Ramirez, James Rovnyak, Leonard Scott, Lawrence Thomas, Roberto Triggiani, Harold Ward

Steve Huffman on math lunatics Terence Tao,
Steve Huffman University of Virginia,
Reddit (symbol) An other Archimedes Plutonium rant about irrational numbers  Reddit · r/badmathematics 10+ comments · 6 years ago An other Archimedes Plutonium rant about irrational numbers ... In Grid Systems, you are exact only to the Grid, and forget about the beyond. "The ...

UCLA chancellor: Gene D. Block (biology)

UCLA Physics dept
Ernest Abers, Elihu Abrahams, Katsushi Arisaka, Michalis Bachtis
Eric Becklin, Zvi Bern, Rubin Braunstein, Stuart Brown, Robijn Bruinsma
Charles Buchanan, Wesley Campbell, Troy Carter, Sudip Chakravarty
W. Gilbert Clark, John Cornwall, Robert Cousins, Eric D'Hoker
Robert Finkelstein, Christian Fronsdal, Walter Gekelman, Graciela Gelmini
George Gruner, Michael Gutperle, Brad Hansen, Jay Hauser, Karoly Holczer
Huan Huang, Eric Hudson, George Igo, Per Kraus, Alexander Kusenko
Thomas Mason, George Morales, Warren Mori, Steven Moszkowski
Christoph Niemann, Kumar Patel, Roberto Peccei, Claudio Pellegrini
Seth Putterman, B. Regan, James Rosenzweig, Joseph Rudnick
David Saltzberg, William Slater, Reiner Stenzel, Terry Tomboulis, Jean Turner
Willard Libby (chem), Julian Schwinger (physics), Paul Boyer (chem), Andrea Ghez, James Fraser Stoddart (chem), Louis Ignarro (physio-medic)

UCLA math dept.

Donald Babbitt, Kirby Baker, Andrea Bertozzi, Mario Bonk, Lennart Carleson, Tony F-C Chan, Shiu-Yuen Cheng, Robert Edwards, Gregory Eskin, Hector Fattorini, Thomas Ferguson, Theodore Gamelin, John Garnett, David Gillman, Mark Green, Nathaniel Grossman, Alfred Hales, Robert Jennrich, Paul Johnson, Alan Laub, Thomas Liggett, Donald Martin, Sidney Port, James Ralston, Paul Roberts, Bruce Rothschild, Murray Schacher, Roberto Schonmann, Masamichi Takesaki, Terence Tao, Veeravalli Varadarajan, James White, Donald Ylvisaker



8) What is truth and falsehood and partial-truth in truth-tables of Logic.



Archimedes Plutonium Jan 12, 2026, 1:14:08 AM to Plutonium Atom Universe newsgroup.

Alright I better move along here. Hopefully the chapters will be short and simple.

8) What is truth and falsehood and partial-truth in truth-tables of Logic?

Here Logic takes a path never traveled before, where statements of ideas have partial truth value, not just all true or all false, but intermediate values. I was forced to do this because of the OR connector in solving for Deciding Experiments. Where two competing statements vie for the truth, one has some falsity however slight, mixed in with truth.

I am working on one such problem of a Deciding Experiment now, involving whether the Light-Photon is a wave or rather, what I suspect a wire. But both have truth value but one has falsity contained within.

So the Truth table of OR connector needs partial truth values. Later in this book we study OR in details, but for now I want to slowly introduce you to OR.

Here are some statements of ideas in Logic and here I have opined on how much truth value they have. Where falsehood and chatterbox gibberish has a value of 0, and where all truth is valued at 1, while a statement with partial truth is given a fraction value between 0 and 1.

A) Slant cut of cone is ellipse. [value of 0, since a cone has 1 axis of symmetry and ellipse has 2 axes of symmetry]
A') Slant cut of cone is oval.   [value of 1]

B) Smilodon is a saber toothed tiger. [value of 1/10 until DNA testing is done]
B') Smilodon is a normal cat with normal teeth but museums like gluing on walrus tusks to increase ticket sales. [value of 1/2, until DNA testing of tooth versus cat upper jaw takes place]

C) Pterosaurs the size of a giraffe flew in the air. [value of 0, as those appendages used as row-boat, oar, sail, and not to fly in the air]
C') Pterosaurs the size of a giraffe used its appendages to oar-row-boat-sail on the Cretaceous seas to catch fish. [value of 1, for aeronautical engineers all tell us such a weight cannot fly]

D) Darwin Evolution is how species evolve. [value of 1/2, for it is useful in preliminary explanations]
D') Darwin Evolution is a rule, not a theory, for Superdeterminism is how species evolve. [value of 1, for physics experiments in quantum entanglement keep coming in positively true]

E) Atoms have a nucleus of protons and neutrons. [value 0, because the bounce back alpha particles exited at a faster speed than they entered]
E') Atoms have no nucleus, rather instead, they have a proton torus in the middle of the atom. [value 1, because the elementary particles of physics must have a geometry in order to do the law-structures of electromagnetism, and not be silly balls, for the Faraday law-structure needs a torus]

F) Convection currents cause Continental Drift. [value 1/10 because observations are showing drift in one direction one day while reverse direction the next day]
F') Vibrations caused by the 2 cores of Earth cause Continental Drift. [value 1, because Earth cores are a dynamo and that causes vibrations]

G) Light Photons are sinusoidal waves. [value 1/100, because only a semicircle geometry can give a wave to a speed of light which is a constant speed]
G') Light Photons are wires, not waves. [value 9/10, because the Electric field in physics must deal with some form of conduction, and conduction exists only in wires]

H) Sun and stars shine from Faraday law-structure of every atom inside that star. [value of 1, for when a atom has proton as torus and muon inside proton torus produces electricity energy]
H') Sun and stars shine from fusion. [value of 0.05, because fusion events are rare in stars]

I) Global warming is caused by the Sun gone Red Giant phase. [value 0.95 because Faraday law-structure in atoms creates electrical energy and causes 95% of sunshine]
I') Global warming is caused by the greenhouse gas effect from burning fossil fuels. [value 0.05 because 5% of sunshine is caused by fusion]

J) Russia taking over Ukraine will make Russia surpass USA as superpower. [value 9/10 because Russia then is the largest land mass country with the most resources]
J') Russia not taking over Ukraine keeps USA #1 superpower, China #2, and Russia #3.
[value of 6/10, because the science and engineering of NATO is vastly better than that of China+Russia]

Homework assignment: Read and ponder these above 10 pairs of statements. Then make up your own list of 10 pairs of statements and assign a number value according to your opinion and state a "because".
1 for a statement that is all true.
0 for a statement that is all false.
A decimal fraction between 0 and 1 if the statement has some truth value.

Example:
K) Earth is Round and the Sun revolves around Earth. [value 0.5 because P=Earth is Round is true]
K') Earth is Round and the Earth revolves around the Sun. [value 1, because P=Earth is Round and Q=Earth revolves around the Sun, are both true]

I would give K a 0.5 (which is 5/10 = 1/2) truth value because the Earth is Round not flat, but the Earth revolves around the Sun. I would give K' a value of 1 for it is all true. But some may want to argue that Earth is not completely Round but oblate and give it a 0.9 truth value. Or argue that for the Sun to revolve around Earth is so preposterous and mind-boggling as to where to put the Sun orbit, that its value should not be 0.5 but rather 1/100.

The assignment of truth value is often what you opine it is, based on the best science evidence available to you. But it is important to recognize that a statement is only partially true.




9) The Existential quantifier.



Archimedes Plutonium Jan 16, 2026, 8:47:20 PM to Plutonium Atom Universe newsgroup.

9) The Existential quantifier.

Logic is about ideas and whether the ideas are good or bad and determining if the ideas are good or bad. In arithmetic we play around with numbers, in logic we play around with ideas. And one of the first things we do with ideas when presented to us, is question on whether the objects in the idea have existence, or, are simply fiction and have no existence.

Example: "Beware the Jabberwock, my son! The jaws that bite, the claws that catch! Beware the Jubjub bird, and shun the frumious Bandersnatch!"

This is a famous Lewis Carroll poem. Fun to read and think about, but is just nonsense with no existence in object reality. Logic uses science to determine a truth value which can be fully true, or partially true or all false.

Archimedes Plutonium Jan 17, 2026, 12:01:49 AM to Plutonium Atom Universe newsgroup.

Example: All drama TV shows are acting and a form of entertainment but not existing in reality. And woe to those who think they can form good ideas based on watching a TV drama show.

AP opines: Fiction shows like fiction books are stories not existing as real, but the book is real. A truth value of 100% false.

Example: All fiction and science fiction books, and woe to those who think they can form good ideas based on ideas in fiction novels.

AP opines: Same as drama TV shows.

Example: Did the Saber tooth tiger exist or were they glued on walrus tusks to upper cat jaw?

AP opines: We get the best biology scientist and see if ever there was a intact upper jaw with a saber tooth attached, as far as I know, never has one like that been found, so I opine the truth to be a tiger with normal teeth and the museums are gluing on walrus tusks found at the dig site, so I opine 90% fake saber tooth tigers. A DNA analysis should be conducted that may prove AP wrong or prove 100% fake.

Example: Does the slant cut of cone exist as a ellipse when a ellipse requires 2 axes of symmetry yet a cone has only one?

AP knows that by symmetry analysis, no slant cut in right circular cone is a ellipse, for it is a oval.

Example: Did the Moon exist as a satellite of Earth when the dinosaurs lived on both Antarctica and the Arctic circle?

AP opines: Using Occam's razor Law-structure, the timing of the rise of flowering plants with a meteor crash that killed dinosaurs points to a 100% truth value that the Moon was not a satellite of Earth until 90 million years ago.

Example: Can you have a black hole or can you have the Big Bang come into existence when you have a Pauli Exclusion Principle of physics?

AP knows that Pauli Exclusion Principle of quantum mechanics does not allow for the existence of black holes, so black holes and Big Bang are 100% false.

Example: Can you have Reals as numbers in math (Reals as a continuum) exist when physics in 1900 says all of physics is discrete with quantum mechanics. Both cannot be correct, and one is hugely mistaken.

AP knows that calculus must exist and in order for it to exist, the numbers of mathematics need to be discrete. So, Reals are 100%.

Example: Can Light be a Wave when waves require a medium to exist for the "wave to wave in", and yet Space has no medium? Is Light a Wire instead of a wave?

AP opines: Logic is a science of precision just as math is one also. A concept of wave needs a medium to wave in. So I suspect there is a ill-defined notion of a wave and where the concept of "wire" makes more sense. I would give this a truth value of 70% true that the Light ray is a wire, not a wave

The start of Logical thought often starts with the question of does this idea exist in reality or is it imagination run amok?

Really, not much use in spending a-lot of time on nonsense or fiction.

Does the object in the idea have Existence or is it Not-Existing, nonexistence. This leads us into the next operator of Logic which is Not-Equal. Not combined with Equal.

Archimedes Plutonium Jan 17, 2026, 12:26:50 AM to Plutonium Atom Universe newsgroup.

Let me not forget this important example of Infinity Borderline.

Example: Can you even have a concept of finite versus infinity without a concept of a borderline between the two??

AP knows that you cannot have a concept of finite versus infinity if there is no borderline separating out the two concepts. This is a 100% true value.



10) The 6 connectors of Logic resemble math 6 operators.



I am using mathematics to guide me on the 6 simple connectors. And as it just so happens mathematics has 6 basic operators which in grade school we learned first to add, then subtract, then multiply and then divide. Later, in college, usually, 1st year of college we learned two new operators of derivative called differentiation and integral called integration, both form the calculus. Some readers may not be familiar with calculus, and it is my hope that the student takes Calculus along with this logic textbook in college or university. That is fitting because much of Logic is a calculus of ideas rather than numbers and graphs. On the other hand, most calculus textbooks by 2026 are wrong and muddle-headed about the derivative and no-one in the math community by 2026 can give a geometry proof of the Fundamental Theorem of Calculus, in addition to using the wrong numbers, and so this textbook may help alert students to better to take this textbook first before taking calculus.

It is a double whammy, severe blow and setback to students across the world that there does not exist a logic textbook as of 2026 free of error in all 4 simple connectors--- AND, OR, Equal-Not, If-->Then. And there exists no calculus textbook which has the correct ideas of equation, what is true numbers, what is calculus derivative and integral. So students all over the world are taught nonsense of what is logic and what is calculus.

So mathematics has 6 simple operators. And given in order where Mathematical Induction starts mathematics. Mathematical Induction is a proof method, and gives all the Counting Numbers for that method is based on the idea of adding 1. Start with 0 and add 1 gives me 1, add another 1 to 1 gives me 2, add another 1 to 2 gives me 3, keep on going gives me all the positive Counting Numbers. That is how math starts by successively adding of 1 to get a new number.

Math 6 simple operators
-------------------------------

1) Add
2) Subtract (which should have been given the better name of "remove").
3) Multiply
4) Divide
5) Derivative
6) Integral

The above order of the 6 simple math operators is what the order they are taught in school and is a reasonable order. Add is likely the easiest concept of the 6 shown.

But surprisingly the order to teach Logic connectors starts with a complex concept of Existence. And this makes logical sense in that it is silly to argue over something that is nonexistent.

Logic has 6 connectors.

1) AND
2) OR
3) Not-Equal (two binaries combined to make 4 rows in a truth table)
4) If-->Then known as the material conditional, or the implication as "implies" but my favorite name is "moves into" because of calculus function and derivative is a move into.
5) Existential quantifier, because the derivative of a function moves into the next coordinate point.
6) Universal quantifier, known as "For every" or "all" but the best concept is the universal law-structures of physics such as Ampere law-structure, or Faraday law-structure or New Ohm's law-structure.

So, I listed the math operators and then the Logic connectors.

But in logic, we like to have things in order. Not enough to just list the connectors but to list them in order such that the most primitive connector is first and the last one the most complicated, needing the others to describe it.

In mathematics we can start with add first because of Mathematical Induction, given 0 and 1, add 1 to obtain 2, then add 1 to 2 to get to 3, and so on. By doing this we have all the counting numbers of mathematics. And a proof using Mathematical Induction if true for 0, 1, 2, 3 then suppose true for n, and if you can show that it is true for n+1, means your statement is true for all the counting numbers. This works by the reasoning that "n" is any counting number. So if you assume "n" and can show true for "n+1", you have proven true for all counting numbers. Now some students have trouble with Mathematical Induction when I went to college and learning this method of proof. And what seems to be the snag or hold-up in young students mind is that the "n" and then
"n+1" are numbers in general. The n can be any number. Some students have a difficult time of conceptualizing that "n" is any number. And teachers help explain Mathematical Induction by referring to dominoes falling. Dominoes ||||||||||||, if the first falls and a far off domino call it "n" and if "n+1" means all the dominoes had fallen.

Earlier in this textbook I listed the major terms of mathematics and theory was the last one of six terms listed. Mathematical Induction is a theory of mathematics for All True Numbers of mathematics are created by Math Induction. The Natural Numbers are math induction using 1 as inductor; the Decimal 10 Grid is math induction using 0.1 as inductor; the 100 Grid is math induction using 0.01 as inductor, etc.

This idea is worth great attention. The terms of mathematics as given earlier is this.

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

Which of those 6 terms would we classify the method of Mathematical Induction? Given any "n" assumed true and if "n+1" is shown to be true, according to Math Induction implies n is the infinite set of counting numbers.

Is Mathematical Induction a axiom?? Is it a operator-structure?? Is it a proof?? Is it a theorem?? Or is it a Theory??.

Before we find out the truth let us go back in math history to a famous mathematician who sized up what Mathematical Induction means. His name was Kronecker.

--- quoting Wikipedia---
free encyclopedia
Leopold Kronecker

Kronecker in 1865
Born 7 December 1823
Liegnitz, Province of Silesia, Prussia
Died 29 December 1891 (aged 68)
Berlin, German Empire
Citizenship Prussian
Alma mater University of Berlin
Known for
Arithmetization of analysis
Kronecker delta
Kronecker foliations
Kronecker limit formula
Kronecker symbol
Kronecker product
Kronecker quiver
Kronecker substitution
Kronecker's congruence
Kronecker's Jugendtraum
Kronecker's lemma
Kronecker's theorem
Kronecker–Capelli theorem
Kronecker–Weber theorem
Hermite–Kronecker–Brioschi characterization
Awards ForMemRS (1884)
Scientific career
Fields
Mathematics
Logic
Institutions
Berlin Academy
University of Berlin

Leopold Kronecker (German: [ˈkʁoːnɛkɐ]; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker[1] as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man").[2] Kronecker was a student and life-long friend of Ernst Kummer.
--- end quoting Wikipedia---

What Kronecker suggested and implied with his famous statement that the Natural Numbers, the Counting Numbers 0, 1, 2, 3, to infinity were something supremely special. And thus, Kronecker thought of Mathematical Induction as being a axiom-- something that cannot be proven true and that we accept it as "God-given" common-sense true, just like all the other axioms such as 2 points determine a straight-line-segment.

His biography suggests Kronecker knew Logic, however, AP thinks Kronecker made a mistake here on the Natural Numbers 0, 1, 2, 3, 4, .... to infinity. In that Mathematical Induction is easily proven true as theorems and a collection of theorems would make the method of Mathematical Induction be not a axiom, not a theorem but a Theory of Mathematics. That the method of Mathematical Induction is a Theory of mathematics.

Reasoning: The reasoning is that we can prove the Counting Numbers, starting at 0 is amenable to a proof given the axioms of math that 2 points determine a unique straight line segment interval and gives us a unique distance between point 0 on the x-axis and point 1 on the x-axis. By using this unique distance we lay out a further distance to point 2 as 1+1=2. Then we lay out still a further distance of 2+1 = 3.

In this sense we have built Mathematical Induction. That proof would be a Theorem, one theorem.

But now, we say our length is going to be 1/10 instead of 1. 1/10 is of course 0.1 and starting at 0 our next point using a 0.1 length is the point 0.1, next we have 0.1+0.1 = 0.2 and as we increasingly add 0.1 and reach 10 we have the entire Decimal 10 Grid of numbers. This 10 Grid is another theorem built by Mathematical Induction.

Next we use 0.01 to build the Decimal 100 Grid and that is another new theorem in math using mathematical induction method. Further we build the 1000 Grid with inductor element 0.001, then the 10^4 Grid, then the 10^5 Grid and on to infinity.

A "theory" in mathematics is a collection of theorems. And that is what the Method of Mathematical Induction is, for it is more than a axiom, but rather it is a "theory" of mathematics.

What Kronecker observed was that the Counting Numbers seemed to be special, but Kronecker failed to realize that the Reals of Mathematics were fake numbers. For the Reals are a continuum and not until the year 1900 when Max Planck came in physics to announce the birth of quantum mechanics physics that physics is discrete numbers only and no continuum, will this become apparent to mathematics only by AP starting in 1991 and ending up here in 2026, that the true numbers of mathematics have to all be Countable by Mathematical Induction.

So mathematics starts with add and ends up last with integration as the most complicated which in mathematics is Integration of calculus.

As for the ordering in Logic, it is a little different from math which has add as first.
Logic is a bit different. We first have to know what we talk about exists or does not exist and use the Existential quantifier. Next we use the Not along with Equal sign for truth tables, and so we need a connector for Not with Equality which is presented in a combined connector of Not-Equal, some prefer to call it Equal-Not.

Makes sense, that as we start Logic with the question of Existence, naturally that gives rise to whether the physical object Exists or does Not-Exist.

No use in arguing over things that do not exist. And just as a fiction story is fiction and non-existing, no use in arguing for truth.

For Logic seeks Truth and truth comes from the sciences. Science overhangs all of Logic as the bearers of truth, and, if partially true a dT value, or if not true at all then it has a 0-value for nonsense.

The Existential quantifier is the existence in Logic and for math that would be the derivative of calculus in First Quadrant Only, starting with origin at coordinate point (0,0,0) in 3rd dimension or (0,0) in 2nd dimension. In true math and true logic, no negative numbers exist and Logic explains why no negative numbers exist in this world.

Common sense why negative numbers do not exist. What is a negative-person if -1 is the negative of 1? What is a negative-statue of liberty? What is a negative-car? The point is. Existence is a positive reality. Negative reality does not exist. Archimedes Plutonium is a person, and what would a negative Archimedes Plutonium be?? Another example is a yard or lawn that has 3 dogs in it. What if we subtract 4 dogs? Does that mean we have a 1 negative dog in the yard? You cannot remove (subtract) more than what exists in reality.

When we graph in mathematics such as a function. The derivative is a motion starting from 0 and taking in all the positive decimal grid numbers as it moves from left to right in the first quadrant only. The derivative is existence of one coordinate point to the next coordinate point. A graph was shown in this book on calculus derivative motion.

So for logic we have to study the 6 connectors in an order from simple to most complicated.

Correct order of the 6 simple connectors of Logic.
-------------------------------------------------------------

1) Existential quantifier
2) Not-Equal
3) AND
4) OR
5) If--> Then
6) Universal quantifier

The truth-tables come in for the connectors that are ---not quantifiers---. We do not have truth tables on Existential and Universal quantification. We have truth-tables on Not-Equal, AND, OR, If-->Then.

And those are arranged in order for Not-Equal truth table is TTTT, while AND is TTTF, and OR is FTTF, and If-->Then is TFUU where U means unknown, undefined. Of course T means true and F means false.

You can see a pattern in the truth-table from that of 4 trues, then 3 trues, then 2 trues, then 1 true.



11) The Not-Equal connector.



Here is an awfully interesting problem that Logic must sort out. Do I call it the Equal-Not connector or call it the Not-Equal connector? At first I called it the Equal-Not, thinking Logic needed equality before it needed to have "not". But then I realized that "not" needs to come before equality in that of Exists is the first connector of Logic for no reason to talk at length and argue over about nonexistent objects, and also, the fact that Not Exists for contradictions stops all logic. All of logic comes to a grinding halt if a contradiction arises and Logic then has to sort out the contradiction by using the best science of the times in order to sort out the contradiction and then go forward. So that answers the question of Not-Equal versus Equal-Not. Existence first, Not second and then Equal third. But we have to combine Not with Equal as one connector. But that does not stop us from calling it Equal-Not or Not-Equal. Call it either one as you wish for often I call it Equal-Not when talking of only the 4 simple connectors but call it Not-Equal when speaking of the Existential and Universal quantifiers.

Order is essential in Logical truth, as essential as truth-value is essential.

Logic has to put things in Order. And the first connector we deal with is the Existence quantifier. No use in having an argument over something that does not exist or is idle chitter-chatter. So first comes existence. Then comes the need for "not" as in "does not exist" and then comes equality. Does it exist or does it Not-exist is the question of logic order.

Apparently I need to call it the Not-Equal connector, because Existence is the first question in Logic and to be involved with existence we are asking if it "does not exist". The issue of "sameness" in equality is remote when concerned over existence. In existence the issue is "does it exist" or "does it not exist". Like in the famous Shakespeare play of Hamlet-- "To be or not to be, that is the question?"

Let me reiterate the Not-Equal connector of Logic and why we start with Existential quantifier, next we introduce the Not-Equal connector in Logic. We start with existence for there is no point in making a Logic Argument over something that does not exist, is fictional, is imaginary. In our modern day TV world of drama and fiction, plays games on our minds, that if not careful, some people actually think the shows they watch have some reality. And because it is fiction and drama and does not exist we sometimes have to step back and remind ourselves that we are probably wasting the time of our life.

Then in modern society we have the Internet loaded with falsehoods and have to navigate a mountain of falsehoods.

Truth and reality and what exists is given by the best available sciences of the time pertaining to the subject or topic on hand. We have to have "Not Exist" of Logic. This causes and forces us to consider the next connector after Existential quantifier is the Not-Equal connector.

But there is a huge problem with the Not-Equal connector compared to AND, compared to OR, compared to If-->Then for they are 4 rowed truth tables, while Not is a binary two row truth table, same with Equal is a binary two row truth table. No worries, for to make Not-Equal into being 4 rowed truth-table we simply combine the two together making 4 rows.

And, this makes common-sense on another level. Is the statements P, Q, are they equal the same or not-equal. So we use "not" for exist or not-exist, and now we use "not" for "is equal" or is "not equal".

We start with Existence and then move to Not-Equal because all the other connectors need the concepts of existence, not and equal. Not-Equal is multiplication in mathematics and in geometry particularly, is area as length times width, and is all of Space as volume is all of space in multiplication. Volume as you remember is length times width times depth.

So our truth table of Logic for Not-Equal is made from two binary tables of equal and then of not, combined to form a quaternary table.

Not-Equal truth-table:
p     q      
T     T        = (T = T) = T
T     not F  = (T = T) = T
F     not T  = (F = F) = T
F     F        = (F = F) = T

Suppose we substitute numbers for T=1 and F=0 to see if we get multiplication out of the Logic connector that is Not-Equal.

p                q      
1  equal to 1 is true       = 1
1  equal to 1 is true       = 1
0  equal to 0 is true       = 1
0  equal to 0 is true       = 1

And we see a quick way to validate if any truth-table of Logic is valid or invalid. We simply see if we can substitute numbers into Logic truth tables and what those numbers become. In the above, only the Math operator of Multiplication can deliver a 1x1 = 1 and a 0x0= 0. Only Multiplication can deliver 1 when we have 1x1 and only multiplication can deliver 0 when we have 0x0. Only multiplication can substitute for Equal in 1=1 and 0=0.

Note: I use equal equality throughout this book for I have the symbol of equivalence not available. Equivalence is more general than equality, and without loss of generality in this book, I simply use equality. Equivalence for those who did not major in math, is such as 1/2 is equivalent to 3/6 is equivalent to 5/10. You get the picture. Equality is identical, the same, while equivalent can be reduced to become equal. Equivalence occurs when people want to relax the strict concept of equality. Equal is identity the same. While equivalence is almost equal but shades of differences. And all the more reason that the True Numbers of Mathematics are __not the Reals___ but are the Decimal Grid Numbers where we do not have the problem of running into 5/10 = 1/2 =0.5. For there in Decimal Grid Numbers we see only decimal numbers and not get hung up over the fact someone has an unfinished division problem as a Rational Number. In New Math and New Logic, we can eliminate "equivalence and make it all be equality".

Because Equivalence can be reduced to Equality, we hence-forth avoid the concept of equivalence. All equivalence can be reduced to equality, so we make no more fuss over shades of equality.

That is an important data to know and I shall repeat it. In science and math we often run into the idea that there are several different notions of "equality" such as equivalence. And one would have thought that "equal" is enough, without having the world cluttered up with a similar notion as equivalence. For example, 3/9 is not the same as 1/3 until we reduce 3/9. If we take a cherry pie and cut it into 9 equal pieces is not the same as cutting that same cherry pie into 3 equal pieces. But, 3/9 is reduced to 1/3. So instead of dreaming up different notions of equal, we just say that if it can be reduced to equality, then Equality being the same is all the concept of "sameness" we ever need.

For comparison sake we show the AND connector truth-table which is the next chapter. Paying particular attention to the fact it has 4 rows and why we had to combine Not to Equal to convert the two 2 rows into being also 4 rows.

AND truth-table:
p     q      p AND q
T     T  = T
T     F  = T
F     T  = T
F     F  = F

And if we plug in arithmetic of T= 1 and F = 0 we see that AND is addition in arithmetic.

1     1  = 2
1     0  = 1
0     1  = 1
0     0  = 0

In New Logic we no longer define connectors by their truth-table, as we already see that Existential quantifier has no truth-table and that Not-Equal just barely has a truth-table considering we had to lump the two binaries together to form a 4 row table. So in New Logic what we do is rely on science, especially Physics on defining the connectors by a universal structure that defines them. I call this universal structure the connector-structure.

This is important, for we define Logic connectors by a connector-structure.
------------------------------------------------------------------

We define logic connectors by a structure, not by its truth table. Although the truth-table helps us to ascertain what the structure is.

In New Logic we define all 6 connectors by a structure governing the connectors. The same as in physics where the essential ideas and truths of physics are given in a "physical law-structure" such as Coulomb law-structure, Faraday law-structure, Ampere law-structure, New Ohm's law-structure.

Truth-tables do not define a logic connector, and this avoids the AND truth table seemingly to have a contradiction of "True AND False being True". This appears to be a contradiction, but since AND is defined as a structure, we avoid the seemingly contradiction.

For __Existential quantifier that structure__ of defining was this--- look in the most relevant recent science pertaining to the existence of something and see if the object exists in that science, plus, no logic argument can have a contradiction such as A exists and A does not exist. If a contradiction arises in Logic, all must come to a halt and consult the relevant science to overcome the contradiction.

For the definition of __Not-Equal as a connector-structure__ we say this. Not-Equal is equality of identical sameness and the Not portion is a reversal of a statement. Keep in mind, Not is bound together with Equal and is inseparable from equal.

Philosophy warning for Not-Equal, which we have to add to the discussion. A major problem of Old Logic was the recurrent mistake of thinking ideas were tagged with negative numbers as being opposite of the true idea. For example: "Earth has one satellite called the Moon". The Not or negative of that statement is : "It is not the case that Earth has one satellite called the Moon".

So, does that mean Earth has 2 satellites or 3 or more, or perhaps no satellites at all. So in Old Logic there was obfuscation surrounding the Not connector and the philosophical idea that the negation of a true statement can have multiple or even an infinity of Not ideas.

While, in New Logic, there are only two values in truth tables-- a value of 1 for all truth and a fraction of 1 greater than 0. New Logic has truth values of 1 and a positive number value greater than 0; and, where all false or meaningless statements and chitter-chatter nonsense has a value of 0. New Logic truth values range from 0 to 1 with partial truth values in between. False is 0, and full truth is 1 while some values are dT a fractional truth.

So, when a Logician examines "It is not the case that Earth has one satellite called the Moon". The New Logic logician simply throws out the statement as meaningless nonsense with 0 value and be done with it, for he/she has looked up the science and wastes no more time on it. While the Old Logic logician spends hours upon hours mulling over the statement and wasting more time, and further, using the worthless statement in more argumentation. Does it have 0 moons, does it have 2 moons, does it have 3 moons.

In New Logic an idea in statements of p,q,r,s,t etc that is false from science, is thrown out. And logic only retains true ideas supported by science and manipulates those true ideas to make new true ideas. These True statements are given a name and called a "Premiss". Statements can be true or false or partially true dT, and statements can be compounded with the connectors. Premisses are individual statements or compounded statements, but, unlike statements, all premisses have to be "true" or dT partial true.

Further example. I love the old Irish saying : "If it works, do not be fixing it."

The Not or negation of that statement would be "It is not the case that if it works, do not be fixing it." Some would prefer to say it as this "If it works, do be fixing it". Here philosophers and Old Logic logicians would step in and say it is a worthwhile statement. While New Logic logicians would point to science and say, if you take apart something that works, the probability chances are risky that once reassembled it no longer works, or works as well as before. And look closely at that negation for it suggests a spectrum of benefits will accrue someone who takes apart a machine that is working. An infinity of negative number benefits from taking apart a working machine. While New Logic logician simply would say there is 0 value in taking apart a working machine is foolish for you risk making it be non-working.

To a large degree the concept of Not is a reversal connector, a contrary statement from the original statement. It reverses true statements into becoming 0 value statements. But in many cases, the Not reverses a 0 value statement into a true statement. So here is a major difference between New Logic and Old Logic. The "Not" connector in Not-Equal does not necessarily convert a 0 value statement (false statement in Old Logic) to a true statement. To the contrary, the Not connector often leaves a 0 value statement -- a false statement remain to be of 0 or nonsense value. And the Not connector can leave a 1 value statement of true and the not statement remain true. The reason being is seen in the truth tables above where we manipulate two rows to force a table of TTTT.

Example: P= Ships are made out of paper. The not-P would be "Ships are not made out of paper." P is false but not-P is true.

Another example: P= Ships are made out of wood. The not-P would be "Ships are not made out of wood." P is true and also not-P is true, for some ships are made of wood and some are made of steel.

Another example: Of where P is true and not-P is true, even though contrary. Here I am going to apply biology statements. P= Viruses are living organisms. The not-P= Viruses are not living organisms. Analysis: throughout this textbook I have been harping the idea that truth is determined by the best available science on the topic. The science of biology claims that Viruses are living in the fact that they have DNA and hijack other cells into making more virus DNA. But some biologists reckon that Viruses are not living because they lack cellular structure and independent metabolism to make energy. So in a sense, the science of biology has come to a standstill saying P is true and not-P is also true.

Again, this reflects back to the Not-Equal truth table of TTTT, where the Not turns a P and a not-P to both be true.

Another example: this time where P is false and not-P is also false. P = Atoms have a nucleus, a center with all the protons and neutrons are clustered into a ball. The not-P= Atoms have no nucleus, a center with all the protons and neutrons are clustered into a ball. Both P and not-P are false according to the best science on the topic. Atoms have a Proton torus surrounded by neutrons as parallel plate capacitors. Yet, the center of Atoms would be these neutrons as parallel plates.

Another example: another P is false and not-P is false. P= If intelligence requires self-awareness, then no AI exists. The not-P= It is not the case that if intelligence requires self-awareness, then no AI exists. Both statements are false because the best available science says that the threshold for being "intelligent" is to have self-awareness. All plants and animals have self-awareness. No computer machine to date has self-awareness as measured by the ability of the machine to turn itself on, or off from its own volition.

About the concept of the Contradiction
---------------------------------------------------

Contradiction in Logic and science is defined from the Not-Equal connector. Throughout this book I have been harping of the fact that when logic arguments run into a Contradiction, all things must stop and sort the contradiction out, using the best available science data and facts.

A= A, and A does not equal not-A, and A does not equal B, C or any thing else but A.

Logic starts with Existence and the Existential quantifier. Logic then moves into the Not-Equal connector. Does it exist or does it not exist. And while inside of the Not-Equal connector, the Contradiction concept itself is defined.

Not-Equal truth-table:
p     q      
T     T        = T
T     not F  = T
F     not T  = T
F     F        = T

The Not-Equal truth-table defines the concept of Contradiction as shown in second and third row as we turn the F into "not F" and the T into "not T" as that which A exists plus A does not exist. And more generally A multiply not-A.

Mathematics as a science never comes to the contradiction concept until math comes to division. There, at division, mathematics sees that division by 0, tears up all of mathematics. If we allow 1/0 to equal something, then we destroy all of mathematics for then we have 1=2, or 3= 11, or 0 = 1 all because we allow division by 0. We cannot have division by 0 for we lose all of mathematics. We cannot have a Contradiction in Logic for that tears up all of Logic.

The structure of Not-Equal is that equality is sameness, A= A, plus, Logic itself cannot have any contradictions where A = not-A. If a contradiction arises in Logic or a argument of Logic, all must stop and come to a grinding halt and only resume by correcting the contradiction.



12) The AND connector.



So, we define and describe the connectors of Logic, not by their truth tables but as a structure of Logic, much like physics is a collection of Law-structures, the Faraday law-structure, the law-structure of universal gravity and others. We define AND connector not as a truth table of TTTF, but as a structure that says in a string of ideas, statements of ideas p,q,r,s,t etc connected through AND, if one of the ideas is true, the entire string is true. Why define by structure instead of the Truth-table is evident in AND, in that a table cannot express the limiting idea that what if P AND Q are two contradictory statements. P = Earth is flat while Q = Earth is not flat. So we have P AND Q as true if we relied only on truth tables of TTTF for AND. But when we write AND as a connector-structure of logic, we state in the structure that AND cannot contain two contradictory statements and we have to stop the logic work and resolve the issue of contradiction.

The truth-table of OR is riddled through with a strange truth value of a partial-truth in order for science to argue in a Deciding Experiment, which of statements P OR Q is the true statement, and the other partially true. By using truth-tables as defining the connector is just inadequate and we have to resort to a structure of AND and a structure of OR.

And the connector If --> Then is riddled full of strange things such as the U for undecided or unknown along with T for true and F for false or gibberish. So writing the definition of If-->Then, as a structure opens up and reveals much more about the connector then if we accepted the truth-table for If-->Then as its definition.

Structures express more details of the connectors than just plain using the truth table.

So we define connectors of true logic, not by a truth-table but by connector-structures, same as in science, for science is defined by their universal law-structure, much like the law-structures of physics. For example the law-structures of electromagnetism-- Coulomb, New Ohm's, Faraday, Lenz, Ampere.

The Existential quantifier is defined by structure as something exists due to the available best science on the subject showing the object exists, plus, you cannot have A exists and A does not exist for that is a contradiction and Logic comes to a grinding halt to straighten-out the contradiction before continuing further.

The Not-Equal connector is defined by structure as "Not" is the reversal of a statement while "Equality" is identical sameness. Keep in mind, Not is bound together with Equal and unable to be a separate concept in itself.

Example: "Plants are species that live on CO2 while animals are species that do not live on CO2."

Explanation: All plants share the sameness of living on CO2, while animals do not share a sameness with the breathing in of CO2 to live on that gas molecule. This example shows how "equal" is bound up with "not".

We now define the AND connector, not with truth-table but with a connector-structure saying that AND connector is one of add or join two or more statements of ideas together. And the structure that defines AND is that within a string of statements joined by AND that at __least one of the statements has a full true value__ ascertained from science, and where all the other statements in the string can be 0-valued out right false or mere worthless chitter-chatter, ___except a contradiction___, but the overall chain of statements is thus true. By full truth value the AND connector is not true if only a partial true value of dT. If a contradiction occurs in a string of statements, then all stops and until the contradiction is excised out, and then does logic continue further.

That means a string of statements, p,q,r,s,t,u,v connected by AND can be true if just one of the statements is fully true, and the rest be worthless nonsense, chitter-chatter and outright false. However, beware, there cannot be a contradiction of say v and not-v in the string. The expression given of AND is "we do not throw the baby out in the bathe water".

If one wanted to give a truth-table of AND it would look like this.

New Logic
AND truth table
p     q     p AND q
T     T  = T
T     F  = T
F     T  = T
F     F  = F

And with modern day computers needing to do arithmetic Add, they have their software make addition with a truth table of TTTF.

And if we substitute T with 1 and F with 0 we see again that AND is add of arithmetic.

p     q     p AND q
1     1  = 2
1     0  = 1
0     1  = 1
0     0  = 0

Now the AND connector of Logic has several replacement terms in English as being "but", "yet", "also", "still", "although", "however", "moreover", "nevertheless", even the comma and semicolon are AND replacements (source: Copi on conjunction).

The AND connector of Logic. To my mind the easiest connector for it is simply add of arithmetic. In fact, we can replace the word "and" with that of "add".

However, AND does get confusing or distracting in arguments because T AND F or F AND T both result in a true overall statement, yet it contains a falsehood or gibberish.

Homework: Examine these AND connected statements and pick which is the true statement and the other a false or gibberish statement.

Note: sometimes we use other words that mean AND, such as "both" sometimes "because".

1) The Earth is flat and it rotates on an axis.
2) The Winter solstice is 21 December this year and it is the first day of Winter.
3) The Big Dipper points to the North Star, Polaris, and Polaris is in Cassiopeia constellation.
4) Higgly piggly, the cow jumped over the moon, because the Moon arrived to Earth to be a satellite only 90 million years ago.
5) The Sun has gone Red Giant and so Santa will be late for Christmas.

Archimedes Plutonium Jan 20, 2026, 11:57:06 PM to Plutonium Atom Universe newsgroup.

So here we see that the AND connector in Logic is similar to the Add operator of mathematics. But carefully notice that in a logic arguments, statements using the AND connector can have falsehoods and gibberish nonsense in addition to a fully true and worthwhile statement.

This is often seen in the case of mathematics proofs. Where a proof is given but carries a-lot of side-show nonsense, even a falsehood here and there. And the way many mathematicians react to the nonsense or falsehoods, they eventually trim out the nonsense and throw-out the falsehoods. But still leaving behind a valid proof.

But, even if they did not, the proof is still valid with or without the falsehoods and gibberish nonsense.

Now, one has to ask the question why on Earth would Logic be a systematic science of ideas and need a connector such as AND that can carry around falsehoods and gibberish nonsense in arguments of Logic?? What is the need for this transport of 0 valued falsity in arguments or partially true dT statements?? Here I look to the Scientific Method for an answer. What is the function of carrying extra baggage in an argument, some outright false, some gibberish? Well in taking baggage on a expedition, we often take more than needed, just in case. And it is this idea of utility. The AND operator is a utility operator, just in case an idea needs support, like supporting evidence. We can think of the AND carrying extra baggage as carrying extra hypotheses of science, where the first hypothesis is wrong so we go to a second hypothesis.


13) The OR connector.



Alright, I am up to OR connector but need a vast overhaul of OR in order to make its truth table align with mathematics arithmetic.

The Truth table representing subtraction or Remove, for the term "remove" would have been a far better name for subtraction and could have made silly math professors realize that negative numbers are crank-crackpot ideas, because removing more than what is available to remove is insanity. Remove is what the OR connector is all about. As we saw before, AND is add or join together. So Logic would need a connector of Remove or subtract.

In the OR connector we have the Deciding Experiment to take into account, where two statements are competing for the truth, one statement is all true while the other statement has a fractional truth, a fraction between 0 and 1 such as being 1/4 true or 1/2 true. And science then conducts experiments to see which of the two competing statements is the full truth of 1 and the other a partial fraction of the truth say 2/3 true.

New Logic OR (exclusive)
p     q      p or q
____________
T    T        F
T    dT       T
dT   T        T
F    F        F

Math validation of correctness
p     q               p or q
____________
1    1               0    so in this row we can see 1 - 1 =0
1    fraction     1    in this row we see 1 remove fraction =1  
fraction    1     1    in this row we can say remove fraction leaving us with 1
0    0               0    in this row we can say 0-0 =0, alternatively we can say remove p leaving q, or remove q leaving p

So we have 4 possibilities.

1) Remove P keeping Q
2) Remove Q keeping P
3) Subtract P from Q provided Q is larger
4) Subtract Q from P provided P is larger

As I write the connector-structure of OR, the structure must consider the 4 possibilities of Remove (subtract).

This is why I use Mathematics to guide me in the True Logic connectors. You see the third row above of F T then T is  (fraction -1) = 1 is not allowed in arithmetic by the axiom that you cannot subtract more than what is available.

And this is why the Truth Tables are not the correct definition of any of the connectors but has to be a structure that states-- in statement form-- the correct definition of any of the Logic connectors.

The definition of all 6 Connectors of Logic is best served by a written statement or statements as a structure of Logic. The Truth-tables ___cannot____ properly define the 6 connectors for they leave too much out of the meaning of the 6 connectors. Written structures, like the written law-structures of Electromagnetism in Physics best describes the phenomenon that is being defined. Truth tables in Logic are only a shadowy glimpse of what the definition may be, but cannot adequately define Existential quantifier, Not-Equal, AND, OR, IF-->Then, Universal quantifier.


Alright, I am up to OR connector but need a vast overhaul of OR in order to make its truth table align with mathematics arithmetic.

The Truth table representing subtraction or Remove for OR.

New Logic OR (exclusive)
p     q      p or q
____________
T    T        F
T    dT        T
dT    T        T
F    F        F

Where the dT represents partial true value of a fraction value between 0 and 1.

So, OR cannot be defined from truth table but must be defined by a statement summary, just like defining the Faraday law-structure as--- thrust a bar magnet through a copper coil connected in circuit with a Galvanometer and watch for the reading of electric current produced by the thrusting bar magnet.

That is the OR truth table, but it does not give you information on how it is formed. For information we go to math arithmetic of subtraction which is better called Remove.

AND connector in previous chapter is addition and joining together of ideas, while OR should be the reverse of joining together but removing.

Archimedes Plutonium Jan 21, 2026, 12:12:33 AM to Plutonium Atom Universe newsgroup.

13) The OR connector of Logic
------------------------------------------------

AND or add connector was rather simple, but now we come to the OR connector which is far more complex and complicated.

OR is sometimes stated as "Either....or" another replacement is "alternatively". OR is the opposite of AND where we add, but in OR we remove, we subtract.

Physics is famous for experiments that tell us the truth of the world in Law-structures of physics or law-structures of any science. Often, a science has to decide on which of two statements is the true statement and which is wrong and discarded.

Example. The Earth is flat or, the Earth is Round.

Here we have two statements "P= The Earth is flat." With "Q= The Earth is Round" and it is the job of science to make experiments and decide which is true P or Q, whichever is true, we discard the false one.

OR as a connector does not tolerate falsehoods in deciding experiments and so its truth table is very different from AND in that it has partial truths written as dT. When you see the symbol dT, it means the statement has a fractional truth value but parts of the statement are false.

New Logic OR (exclusive)
p     q      p or q
____________
T    T          F
T    dT        T
dT    T        T
F    F          F

Math validation of correctness where T is valued at 1, F at 0, and dT a fraction of 1.
p     q      p or q
____________
1    1          0
1    1/10    1 once we remove 1/10
1/10    1     1 once we remove 1/10
0    0          0

So we have 4 possibilities.

1) Remove P keeping Q
2) Remove Q keeping P
3) Subtract P from Q provided Q is larger or the same size
4) Subtract Q from P provided P is larger or the same size


New Logic OR (exclusive)
p     q      p or q
____________
T    T           F
T    dT        T
dT    T        T
F    F          F

Examples of 2nd row and their deciding experiments.

1) Either the Sun shines 90% from Faraday law-structure or the Sun shines 100% from fusion of light elements forming heavier elements.

Deciding Experiment: It is found that a muon of 105MeV is inside a proton torus of hydrogen of 840MeV. Why would that be a deciding experiment?? Because a muon thrusting through a 840 windings of 1 MeV each is the Faraday law-structure producing electrical energy.

2) Earth has earthquakes caused by the rattling and vibration of the two inner cores, or, Earth has earthquakes due to the motion of convection currents in mantle and crust.

Deciding Experiment: If Convection currents are the cause, earthquakes would not move in one direction then back up and move in the opposite direction. But if it is the vibrations of electric motors that the cores are, then the motion is forward then backward.

Examples for 3rd row and their deciding experiments.

3) Either Smilodon, the saber tooth tiger really did grow large canine teeth, or, they are walrus tusks glued on by paleontologists and museums.

Deciding Experiment:: DNA test all museum specimens of Smilodon of the jaw and of the canines, if found to be cat DNA or walrus tusk DNA.

4) Either the chemical formula for Water is H2O, or, the true chemical formula for Water is H4O.

Deciding Experiment:: Insist that lazy chemists and physicists stop their water electrolysis experiment when they check for volume of hydrogen compared to oxygen, then hop skip jump back to the lounge for cake coffee and donuts, but rather, insist they get out the micro quartz balance and actually weigh the mass of the hydrogen as compared to oxygen. What prompts AP to do this extra work, is that the Atom needs all three of muon, proton and at least one neutron to store the energy created by Faraday law-structure as the muon thrusts through the proton torus of hydrogen. No atom can exist without some form of neutron. Chemists and physicists have been exceptionally lazy and ignorant in weighing the results of Water Electrolysis.

Archimedes Plutonium Jan 21, 2026, 5:56:09 PM to Plutonium Atom Universe newsgroup.
On Wednesday, January 21, 2026 at 4:21:43 AM UTC-6 Archimedes Plutonium wrote:

4) Either the chemical formula for Water is H2O, or, the true chemical formula for Water is H4O.

Deciding Experiment:: Insist that lazy chemists and physicists stop their water electrolysis experiment when they check for volume of hydrogen compared to oxygen, then hop skip jump back to the lounge for cake coffee and donuts, but rather, insist they get out the micro quartz balance and actually weigh the mass of the hydrogen as compared to oxygen. What prompts AP to do this extra work, is that the Atom needs all three of muon, proton and at least one neutron to store the energy created by Faraday law-structure as the muon thrusts through the proton torus of hydrogen. No atom can exist without some form of neutron. Chemists and physicists have been exceptionally lazy and ignorant in weighing the results of Water Electrolysis.

The message I was getting across in Water Electrolysis is that a hydrogen atom with only a proton and muon inside is not a Atom at all, but a subatomic particle. All Atoms need some form of a neutron. It does not have to be a fully grown neutron of 945MeV but a partially grown Neutron to store the electrical energy giving off by the Faraday law-structure of muon thrusting through proton torus.

Archimedes Plutonium Jan 8, 2026, 2:49:14 AM to Plutonium Atom Universe newsgroup.

I had to modify drastically, the OR connector, where truth has a Range of values. Where full truth T has a value of 1, and full false has a value of 0, but you can have truth values between 0 and 1, and those variable values need to show up as "dT" in OR truth table.

P     Q        P OR Q  where dT stands for a statement that has a partial truth
T      T            F
T      dT          T
dT      T          T
F      F            F

Example: P = The Sun is a star. Is a true statement with value 1. Q = The Sun is a planet. Is a false statement with value 0.

R = The Sun is a blue star. Is a statement that has a fractional truth value, for the Sun is a star but it is not a blue star. So we say the truth value of R is intermediate between 0 and 1 and give it a truth value of 0.25.

The greatest use of OR in logic is to evaluate Deciding Experiments of Physics. Is the Light-photon a Light Wave or as AP thinks, a Light Wire.

So in Logic we have the statement Either the Light-Photon is a Light-Wave, OR, it is a Light-Wire. From there, that argument proceeds and one of them will win, the other will lose. Both have a truth value greater than 0.

Archimedes Plutonium Jan 8, 2026, 8:15:21 PM to Plutonium Atom Universe newsgroup.

A logic textbook is the very hardest science book to write. I should know, finding out from experience, for now I start over again all 4 my of logic textbooks because of "dT".

Now some may question if the dT variable truth, part true and part false is needed in any of the other 5 connectors of Logic, other than the OR connector. And here is the beauty of modeling Logic after mathematics. In mathematics, as we substitute 1 for True and 0 for false in the connectors, the only time in which this substitution breaks apart is in the OR connector, and demanding for us to craft and create the dT variable truth value. You will notice in the next chapter on If-->Then we introduce a new parameter of U in the truth tables meaning uncertain or undefined. The U is different from T as 1, and F as 0, and dT as partial truth. The U also comes from mathematics in the knowledge that we cannot divide by 0. But that is not solved by using a dT in the If-->Then. The use of dT in OR arose because we cannot mathematically take 0 -1 = -1. The dT arose from that impossible math arithmetic of getting negative numbers. And although some will complain that 1/10 subtract 1 is still a negative number, in the structure of OR will stipulate that the subtract or removal of (1/10) - 1 is such that you remove 1/10 altogether leaving only 1. When we look at the second row instead of the third row of OR, we have 1 - 1/10 and here the subtraction is not ending up as 9/10, no, it ends up as removal of 1/10 altogether and leaving behind 1.

So in summary, the dT special variable comes into Logic only in the OR connector, only 1 of the 6 connectors. But, and however, we can use a partial truth dT in any connector we want, we need not confine dT to just the OR connector.

Before I leave the OR connector, I must say something about the Commutative property in math versus logic, especially OR. Commutative simply means the order in which you do the operator or the connector does not matter. For example 5+10 is the same as 10+5. Logic example: For drink we have orange juice or cola is the same as "for drink we have cola or orange juice".

OR seems to appear commutative, but is it really???

Painstakingly I found out that OR is noncommutative and nonassociative. When we replace "remove" into the example "For drink we have orange juice or remove cola" is not the same as "for drink we have cola or remove orange juice". You see in the OR connector you remove either P or Q and thus the removal is not commutative which agrees with math subtraction that 10 -6 is not the same as 6-10.

A vacillating mistake of mine in early 2026 was that I said OR was commutative because of its truth table where T OR F is the same as F OR T. But then later I switched by citing that math subtraction was Not Commutative where for example 5 -3 is not the same as 3-5. So OR was not commutative.  So what is the logical truth, for it cannot be a contradiction--- OR is commutative and OR is not commutative. Which is it?? Finally it is answered by seeing that OR is the same as substitution of "remove" and this full agreement with mathematics subtraction.

OR is Not-commutative and Not-associative.

But there is far more going on here with subtraction in math. In Old Math they had fake things like negative numbers and they forgot an important axiom-- You cannot subtract more than what is available to subtract. So that we can have 5-3 but we cannot venture into 3-5.

Yes, if you hate and despise revising and editing, never ever write a logic textbook.

[Archimedes Plutonium]

Logic textbooks are so difficult. I have to start all over again, and get the Counterexample in the correct spot.