Update: 20210721 04:55 AM 0400
Boolinthought.htm
by U Kyaw Tun (UKT) (M.S., I.P.C., USA),
Daw Khin Wutyi, Daw Thuzar Myint, Daw Zinthiri Han
and staff of Tun Institute of Learning (TIL).
Not for sale. No copyright. Free for everyone.
Prepared for students and staff of TIL
Research Station, Yangon, MYANMAR

http://www.tuninst.net ,
www.romabama.blogspot.com
The ability to communicate with each other of the species by use of Language involving syntax is a human achievement. We Communicate about the worldly affairs and also about philosophical questions. Another human achievement is the ability to count  the basis of Mathematics. Without Mathematics our communication on Science becomes very unreliable.
 UKT 210628 :
If you are a Christian, Hindu, Jew, or Muslim, you would be accepting that there is a Creator and of course an "afterlife" in Heaven or Hell. However, if willed by the Creator you will have another lifetime on what we call Earth or World as a human who may be mad, an idiot, a philosopher, or even buddha (the wisest of all beings). Worse, you may be born an animal or an insect. Buddhists too believe in an afterlife: the religionists also stipulated other unseen beings which my scientific training would not permit me to believe.
Are these the Right View, you might be wondering? As a Buddhist, I must know: because the Fourth Law of Truth emphasises that a Buddhist must hold the Right View.
Studying the Questions and Answers of the Sixth Buddhist Council, Rangoon, 1954, has increased my knowledge of Theravada Buddhism. In one of the sessions {miz~Zi.ma. ni.kaaý} (vol 1), in one sermon named {a.pûN~Na.ka. þoat~tän} we find a partial list of nonBuddhist views, and Buddha's answer to them. First, Buddha lists the various views as follows:
1. ,
2. ,
3. ,
4. ,
5. ,
 etc.
See Wikipedia  https://en.wikipedia.org/wiki/Indian_philosophy 210721
"Indian philosophy refers to philosophical traditions of the Indian subcontinent. A traditional classification divides Skt: «āstika» (orthodox), Pali: {ût~ti.ka. a.yu}, and «nāstika» (heterodox) {nût~ti.ka. a.yu} schools of philosophy. [The division depends] on one of three alternate criteria: whether it believes the Védas [ since the Védas {wéda}. have been rewritten at least 3 times, you have to specify which layer you are talking about] as source of knowledge; whether the school believes in the premises of Brahman and Atman ; and whether the school believes in afterlife and the Devas {déwa.}. (ref. 1 , 2 , 3)
There are six major schools of orthodox (vedic) Indian philosophy  Nyaya, Vaisheshika, Samkhya, Yoga, Mīmāṃsā, and Védanta, and five major heterodox (srmanic) schools  Jain, Buddhist, Ajivika, Ajñana, and Charvaka . However,there are other methods of classification: Vidvaranva for instance identifies sixteen schools of Indian philosophy by including those that belong to the Śaiva ad Raseśvara traditions. (ref. 4, 5).
Buddha sums them as either as "negative" {nût~ti.ka. a.yu} and "positive" {ût~ti.ka. a.yu}. He doesn't say anything about his Anatta {a.nût~ta.} view, in which he questions the "existence" of nonchanging, everlasting Self or Atta or Ātman आत्मन् which is related to the CreatorcumUniversal Judge .
Buddha's advice to those who do have either "negative" {nût~ti.ka. a.yu} or "positive" {ût~ti.ka. a.yu}, to side with {ût~ti.ka. a.yu}  for even if it were false, they have nothing to lose. On the other hand if they were to side with {nût~ti.ka. a.yu}, they would lose everything if the positive view proved to be true. This view of the Buddha is very pragmatic.
Now, we must ask if Buddha's view of Anatta {a.nût~ta.} were wrong, what then? It seems that Buddha's view on CreatorcumUniversal Judge is also pragmatic. According to Buddha, everyone, you and I, is responsible for our actions. You must not lay the blame or praise on the CreatorcumUniversal Judge. There is none to save you from your Sins. It is just a waste of time to pin your hope on God. I came to this view after listening and reading {a.pûN~Na.ka. þoat~tän} .
Let's now see what George Boole, a mathematician, said
in 1853, in his An Investigation of the Laws of
Thought p003. See TIL HDPDF and SDPDF libraries

GBooleLawsOfThought<Ô> /
Bkp<Ô> (link chk 210628)
"The general laws of Nature are not, for the most part, immediate objects of perception. They are either inductive inferences from a large body of facts, the common truth in which they express, or, in their origin at least, physical hypotheses of a causal nature serving to explain phenomena with undeviating precision, and to enable us to predict new combinations of them. They are in all cases, and in the strictest sense of the term, probable conclusions, approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience."
UKT 210629: The above excerpt shows Boole's writing style is not easy to follow. Yet, I've to try.
What do we understand by "a large body of facts"? Scientific facts are borne out of physical experiments. If a current Scientific Theory can no longer explain observations of new physical experiments, the current theory must be modified or rejected. Here, what I have in mind is the birth of Quantum Theory. Read an interesting article in Scientific American by Scott Bembenek on March 27, 2018 :  https://blogs.scientificamerican.com/observations/einsteinandthequantum/ 210629
"By 1926, Albert Einstein had become completely unforgiving of quantum mechanics' probabilistic interpretation of the universe and would step away from it forever. In Einstein's mind, the universe must ultimately obey laws of physics that are fundamentally deterministic, and with respect to this, he would be uncompromising. Einstein made this most clear in response to a letter Marx Born (18821970) had written to him when he said: 'Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One [Christian God]. I am at all events convinced the He [God] does not play dice.' "
What would Gautama Buddha (Siddhartha Gautama, fl. bet. 6th and 4th century BC), whose First Four Laws, and Anatta Principle are the earliest scientific observations in the history of mankind, would say? I am not the one to answer my question, which means, I will have to learn more abut Buddhism from authentic sources such as the Questions and Answers of the Sixth Buddhist Council . But first, let's see what Wikipedia has to say of Boolian's views on Thought:
From: https://en.wikipedia.org/wiki/The_Laws_of_Thought 210628
An Investigation of the Law of Thought on Which are founded the Mathematical Theories of Logic and Probabilities , George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathematics at what was then Queen's College, Cork (now University College), in Ireland.
The historian of logic John Corcoran wrote an accessible introduction to Laws of Thought [ref 1] and a point by point comparison of Prior Analytics and Law of Thought. [ref 2]. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. [Aristotle: 384322 BC] Boole's goals were "to go under, over and beyond" Aristotle's logic by:
1. Providing it with mathematical foundations involving equations;
2. Extending the class of problems it could treat from assessing validity to solving equations, and;
3. Expanding the range of applications it could handle
 e.g. from propositions having only two terms to those having arbitrarily many.
More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. [UKT ¶]
First, in the realm of foundations, Boole reduced the four propositional forms of Aristotle's logic to formulas in the form of equations  by itself a revolutionary idea. [UKT ¶]
Second, in the realm of logic's problems, Boole's addition of equation solving to logic  another revolutionary idea  involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. [UKT ¶]
Third, in the realm of applications, Boole's system could handle multiterm propositions and arguments whereas Aristotle could handle only twotermed subjectpredicate propositions and arguments. For example, Aristotle's system could not deduce
"No quadrangle that is a square is a rectangle that is a rhombus"
from
"No square that is quadrangle is a rhombus that is a rectangle"
or from
"No rhombus that is a rectangle is a square that is a quadrangle".
Boole's work founded the discipline of algebraic logic. It is often, but mistakenly, credited as being the source of what we know today as Boolean algebra. In fact, however, Boole's algebra differs from modern Boolean algebra: in Boole's algebra A+B cannot be interpreted by set union, due to the permissibility of uninterpretable terms in Boole's calculus. Therefore, algebras on Boole's account cannot be interpreted by sets under the operations of union, intersection and complement, as is the case with modern Boolean algebra. The task of developing the modern account of Boolean algebra fell to Boole's successors in the tradition of algebraic logic (Jevons 1869, Peirce 1880, Jevons 1890, Schröder 1890, Huntington 1904).
UKT 210720: Now let's see what George Boole has to say.
[p001]
1. The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; [UKT ¶]
UKT 210720: Being a Theravada Buddhist, I'll compare "those operations of the mind" to what is popularly known in BurMyan as Nicsaclé Picsie {nhichswè.lé: pic~sæÑ:}, and in PalMyan as {pûT~HTaan: pic~sa.yoad~déþa.}. We usually write the Pali version in 12 lines of 2 entries each, arriving at 24 items, hence {nhichswè.lé:}
line01:
Please note, I'm not a Pali scholar, nor a Mathematician. I'm just trying to make those Buddhists who are Mathematicians, and those Mathematicians who are scholars  who may or may not be Buddhists, or who either have a religion or not, to become interested in Buddhism as a science. As reference, see (p109 of my copy), in Questions and Answers of the Sixth Buddhist Council.
to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; [UKT ¶]
and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.
2. That this design is not altogether a novel one it is almost needless to remark, and it is well known that to its two main practical divisions of Logic and Probabilities a very considerable share of the attention of philosophers has been directed. [UKT ¶]
In its ancient and scholastic form, indeed, ... [Boole mentions that Logic owes its origin to Greeks  to which I'm sure Easterners, Chinese and Indians in particular, will disagree]. ... The history of the theory of Probabilities, on the other hand, has presented far more of that character of steady growth which belongs to science. ...
[p003]
The general laws of Nature are not, for the most part, immediate objects of perception. [UKT ¶]
They are either inductive inferences from a large body of facts, the common truth in which they express, or, in their origin at least, physical hypotheses of a causal nature serving to explain phenomena with undeviating precision, and to enable us to predict new combinations of them. They are in all cases, and in the strictest sense of the term, probable conclusions, approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience. [UKT ¶]
But of the character of probability, in the strict and proper sense of that term, they are never wholly divested. On the other hand, the knowledge of the laws of the mind does not require as its basis any extensive collection of observations. [UKT ¶]
UKT 210720: Why should be Laws of Nature, and Laws of Mind be different? Laws of Mind should also be confirmed by repetition of instances, and so by checking with other instances.
The general truth is seen in the particular instance, and it is not confirmed by the repetition of instances. We may illustrate this position by an obvious example.
It may be a question whether that formula of reasoning, which is called the dictum of Aristotle, de omni et nullo, expresses a primary law of human reasoning or not; but it is no question that it expresses a general truth in Logic. ... ... ...
UKT 210720: Aristotle's dictum de omni et nullo expresses a general truth in Logic  maybe. We are interested in Laws of human reasoning. But first for those who are not rigorously trained in Logic should look into: A Brief History of Natural Logic , by Johan van Benthem, ILLC Amsterdam & Stanford, http://staff.science.uva.nl
 JvanBenthemBriefHistLogic<Ô> / Bkp<Ô> (link chk 210721)
"Abstract : This paper is a very brief history of natural logic at the interface of logic, linguistics, and nowadays also other disciplines. It merely summarizes some facts that deserve to be common knowledge."
... ... ...
UKT 210720: I don't know what Boole exactly means by "Signs". Are they individual words and phrases of a language? Let's see:
Chapter 02. Of Signs in general, and of the Signs appropriate to the science of Logic in particular; also to the Laws to which that class of Signs are subject.
[p017]
1. That Language is an instrument of human reason, and not merely a medium for the expression of thought, is a truth generally admitted. It is proposed in this chapter to inquire what it is that renders Language thus subservient to the most important of our intellectual faculties. In the various steps of this inquiry we shall be led to consider the constitution of Language, considered as a system adapted to an end or purpose; to investigate its elements; to seek to determine their mutual relation and dependence; and to inquire in what manner they contribute to the attainment of the end to which, as coordinate parts of a system, they have respect.
UKT: Language is not just a message from one person to another. It can be just "where is my dinner", or a deep thought on a subject such as "where do I come from". When a young child asks that question, the mother would say, "A big bird brought you to me".
In proceeding to these inquiries, it will not be necessary to enter into the discussion of that famous question of the schools, whether Language is to be regarded as an essential instrument of reasoning, or whether, on the other hand, it is possible for us to reason without its aid. [UKT ¶]
UKT: Whenever I think of a subject, I always talk to myself  either in English sentences or Burmese sentences, or mixed. Words are formed in my mind, so when I write these thoughts down, they are already in sentence, and phrases, with Subjects and Predicates.
[p018]
2. The elements of which all language consists are signs or symbols. Words are signs. Sometimes they are said to represent things; sometimes the operations by which the mind combines together the simple notions of things into complex conceptions; [UKT ¶]
[Language expressed in words] .... express the relations of action, passion, or mere quality, which we perceive to exist among the objects of our experience; sometimes the emotions of the perceiving mind.
• Mathematics of the Law of Karma
: A mathematical analysis of Law of Karma. Read: Mathematical Proof of the Law of Karma,
by Jargal Dorj , in American Journal of Applied Mathematics. Vol. 2, No.
4, 2014, pp. 111126, in TIL HDPDF and SDPDF libraries:
 JDorjMathLawKarma<Ô>
/ Bkp<Ô> (link
chk 200301)
Abstract: "... The existence of the Law of Karma will be proved and verified in
this article using the mathematical Set Theory. The incomprehension of the
“Self” and its emptiness is described in the Buddhist teachings as ignorant.
Herewith we shall explain the theory of the “Self’ and its emptiness founded on
the possession of the body and mind using the mathematical Set Theory. By
reading this article the reader will comprehend the “Self” and its emptiness and
overcome this ignorance.
Dolphins may be maths geniuses , Jennifer Viegas , Discovery News, 2012Jul18
 https://www.abc.net.au/science/articles/2012/07/18/3548573.htm 201030
Dolphins may use complex nonlinear maths when hunting, according to a new study that suggests they could be far more skilled than was ever thought possible before.
Inspiration for the new study, published in the latest Proceedings of the Royal Society A, came after lead author Tim Leighton watched an episode of the Discovery Channel's Blue Planet series and saw dolphins blowing multiple tiny bubbles around prey as they hunted.
"I immediately got hooked, because I knew that no manmade sonar would be able to operate in such bubble water," says Leighton, a professor of ultrasonics and underwater acoustics at the University of Southampton.
"These dolphins were either 'blinding' their most spectacular sensory apparatus when hunting  which would be odd, though they still have sight to reply on  or they have a sonar that can do what human sonar cannot …Perhaps they have something amazing," he adds.
Leighton and colleagues Paul White and student Gim Hwa Chua set out to determine what the amazing ability might be.
They started by modelling the types of echolocation pulses that dolphins emit. The researchers processed them using nonlinear mathematics instead of the standard way of processing sonar returns. The technique worked, and could explain how dolphins achieve hunting success with bubbles.
The math involved is complex. Essentially it relies upon sending out pulses that
vary in amplitude. The first may have a value of 1 while the second is 1/3 that
amplitude.
"So, provided the dolphin remembers what the ratios of the two pulses were, and
can multiply the second echo by that and add the echoes together, it can make
the fish 'visible' to its sonar," says Leighton. "This is detection
enhancement."
But that's not all. There must be a second stage to the hunt.
"Bubbles cause false alarms because they scatter strongly and a dolphin cannot
afford to waste its energy chasing false alarms while the real fish escape,"
explains Leighton.
The second stage then involves subtracting the echoes from one another, ensuring the echo of the second pulse is first multiplied by three. The process, in short, therefore first entails making the fish visible to sonar by addition. The fish is then made invisible by subtraction to confirm it is a true target.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
"Until measurements are taken of wild dolphin sonar as they hunt in bubbly water, these questions will remain unanswered," says Leighton. "What we have shown is that it is not impossible to distinguish targets in bubbly water using the same sort of pulses that dolphins use."
UKT: More in the article.
Nonlinear system , https://en.wikipedia.org/wiki/Nonlinear_system 201020
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.^{[1]}^{[2]} Nonlinear problems are of interest to engineers, biologists,^{[3]}^{[4]}^{[5]} physicists,^{[6]}^{[7]} mathematicians, and many other scientists because most systems are inherently nonlinear in nature.^{[8]} Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Dolphins may be maths geniuses , Jennifer Viegas , Discovery News, 2012Jul18
 https://www.abc.net.au/science/articles/2012/07/18/3548573.htm 201030
Dolphins may use complex nonlinear maths when hunting, according to a new study that suggests they could be far more skilled than was ever thought possible before.
Inspiration for the new study, published in the latest Proceedings of the Royal Society A, came after lead author Tim Leighton watched an episode of the Discovery Channel's Blue Planet series and saw dolphins blowing multiple tiny bubbles around prey as they hunted.
"I immediately got hooked, because I knew that no manmade sonar would be able to operate in such bubble water," says Leighton, a professor of ultrasonics and underwater acoustics at the University of Southampton.
"These dolphins were either 'blinding' their most spectacular sensory apparatus when hunting  which would be odd, though they still have sight to reply on  or they have a sonar that can do what human sonar cannot …Perhaps they have something amazing," he adds.
Leighton and colleagues Paul White and student Gim Hwa Chua set out to determine what the amazing ability might be.
They started by modelling the types of echolocation pulses that dolphins emit. The researchers processed them using nonlinear mathematics instead of the standard way of processing sonar returns. The technique worked, and could explain how dolphins achieve hunting success with bubbles.
The math involved is complex. Essentially it relies upon sending out pulses that
vary in amplitude. The first may have a value of 1 while the second is 1/3 that
amplitude.
"So, provided the dolphin remembers what the ratios of the two pulses were, and
can multiply the second echo by that and add the echoes together, it can make
the fish 'visible' to its sonar," says Leighton. "This is detection
enhancement."
But that's not all. There must be a second stage to the hunt.
"Bubbles cause false alarms because they scatter strongly and a dolphin cannot
afford to waste its energy chasing false alarms while the real fish escape,"
explains Leighton.
The second stage then involves subtracting the echoes from one another, ensuring the echo of the second pulse is first multiplied by three. The process, in short, therefore first entails making the fish visible to sonar by addition. The fish is then made invisible by subtraction to confirm it is a true target.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
"Until measurements are taken of wild dolphin sonar as they hunt in bubbly water, these questions will remain unanswered," says Leighton. "What we have shown is that it is not impossible to distinguish targets in bubbly water using the same sort of pulses that dolphins use."
UKT: More in the article.
Nonlinear system , https://en.wikipedia.org/wiki/Nonlinear_system 201020
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.^{[1]}^{[2]} Nonlinear problems are of interest to engineers, biologists,^{[3]}^{[4]}^{[5]} physicists,^{[6]}^{[7]} mathematicians, and many other scientists because most systems are inherently nonlinear in nature.^{[8]} Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.