An Interview With My Masters: Yes, No, Both, Neither
From the teachings of Nabir
Nabir was my spiritual teacher and mentor concerning the ways of the world. The things I learned from him, over half a century, have been the foundation for my meaningful, and very successful life.
Here is an introduction to Nabir and his teachings.
Here is Part 1 to this thread.
In this talk, Nabir spoke about the concept of magnification in the seeking of truth. He explained that what is often true at one level is not so at another. He specically stated that this process is different than manipulating, or “fudging facts”. This distinction concerning multiple levels of truth has a basis in logic.
This is the seond part of a specific Q & A session on this issue. Nabir’s presentation really got us thinking, and the Q & A session (This is the second part) on this particular night was lively and had a lovely, Zen, counter-intuitive sensibility about it.
Here is the Q & A session that followed that talk.
Q. What you have said here makes sense. How would you explain this to a more advanced student? Someone with some understanding of mathematics, or any of the formal sciences under their belt?
A. Let’s take a shot at this. I will avoid using technical terms here, but you may still need to parse it down word by word to make sense of it all. It will be worth it.
Multi-valued logics refer to logics defined by value matrices (“truth tables” in propositional logic). Some paraconsistent logics are multi-valued, and some multi-valued logics are paraconsistent. It depends on how the logics are defined. A paraconsistent multi-valued logic might define negation as {0,1,2,3; 1,0,3,2}. So the 0,1 pair behave exactly as they do in the binary logic, and so do the 2,3 pair. Each might go on to be defined with their own operators for IF, AND, and OR which are equivalent to their binary counterpart. That would be a good start to defining a basic paraconsistent logic using value matrices.
Q. So this is why a person might ask you a question, and the answer will be “Yes”, “No”, “both”, “neither”?
A. Yes! The answer given will depend on whether you need to use quantification or qualification for your paraconsistent application. Here, you’re paraconsistent logic will need non-valued operators. For those of you who worship at the alter of linear thinking, keep in mind that here quantified and modal logics cannot (easily) be defined by value matrices. It can occasionally but it’s not standard practice, among engineers or computer scientists, probably because it becomes very cumbersome manually.
Q. But it can be done?
A. Yes. Technically, however, if we are defining something computationally, it’s ultimately being defined by some set of value pair operations. So one might argue that all (standard) logics are valued, and even binary (aside from quantum logics and quantum computing, which is essentially multi-valued).
Q. Which are more effective, paraconsistent logics or multi-valued logics?
A. It depends whether you are philosopher or a scientist. When I gather students and peers and ask them this question, I get different answers. A mathematical logician may answer with a brief introduction, including a couple of definitions. For the mathematical logician, this is just boring routine stuff, something you need to go through before you write down the theorem and gets to the interesting part, the neat techniques they may have invented to prove it.
However, as soon as the definitions are shown, the mathematical logician may feel they have sufficiently answered the question. Now the philosophers will jump in. They will want to discuss whether this is the “right” definition. For them, the definition is supposed to clarify what you are studying; the definition itself should capture some underlying basic truth. The mathematical logician just doesn’t care about that. He or she will rather be thinking something along the lines of “Clearly it is the right definition, because that is the definition that lets us prove this extremely cool theorem that I haven’t even gotten to write down yet! Stop philosophizing and let me get on with it!”
Philosophical logic can also be rigorous and mathematical, but can also appeal to some rather less rigorous notions that, nevertheless, have some validity in some (typically philosophical) domain of discourse. A classic domain here is the question of “existence”. This including perhaps the most well-known philosophical quote of all by René Descartes’, a great mathematical logician, and philosopher.
“Cogito ergo sum — I think, therefore I am”
Both philosophical and mathematical logic have several branches, that differ quite a bit from each other. If you want to go deeper a good example of philosophical logic is Intuitionistic type theory. For mathematical logic Google “Forcing (mathematics)”.
I did not intend to put down philosophers here. I was describing the experience of how they may interest concerning this question. Both approaches are valuable. I also think it is difficult to do both approaches at the same time, so having two separate disciplines is important.
Q. I can meditate study Zen, and talk about philosophy and metaphics. What I want to do hear, as a critical thinker, is what is hardcore, not spiritual-new age philosophy. In other words, what mathematical logic about?
A. O.K. then. There are several things that will need to be discussed here then. Firstly, I shall skip out the standard naive notion that logic has to do with “correct ways of reasoning”. Logic is a wide field of knowledge. For instance, it is essential for the understanding of how scientific theories can be stated, and I shall fix this point; others can be cited as well.
To do what you are asking here we need to understand how any idea can be schematized within a mathematical framework. And the choice of this framework, and it is a choice, may influence the theory itself.
Secondly, we usually assume that, at least in principle, theories can be reduced to a system of basic truths (axioms). That means that we put in advance the concepts we assume to be considered and the principles we admit are to be taken as”So”. This presupposes a logic, for we (surely) aim at to make inferences. If you want to develop or understand the language for expressing our theories, you would need to learn about standard set theory. Of course, Category theory and higher order logic could be used as well, with certain restrictions (there are always some restrictions!)
Q. So what is he difference between what a theory is and a statement of fact?
A. A theory is a set of formulas in the language of set theory, which constitute its mathematical counterpart, plus a domain of application, or set of possible domains of applications, plus a collection of rules which relate the terms of the theory with the elements of the models.
Q. Which are more dominant today. Philosophical logicians or Mathematical logicians?
A. In daily life Mathematical logicians? People introspect less and less, and AI is definitely tied to Mathematical logic.
Alan Turing, Alonzo Church, and Stephen Kleene started grappling with the idea of computation in the 1930s. And their work in developing models of computation such as the Turing Machine, the Lambda Calculus, and recursive functions heavily influenced the design of early computers and programming languages.
For example, Lisp (programming language), a language that has heavily influenced many of today’s most popular programming languages, was itself inspired by Church’s lambda calculus. Interesting many of these programmers were student of Zen and D.T. Suzuki. These scientists were the precursors to today’s computer scientists. Today, many compilers and interpreters are designed as finite automata that recognize context-free grammars, two more concepts from the subfield of computability theory. In addition to computability and recursion theory, mathematical logic also entails other areas such as set theory, model theory, and proof theory which have their own useful applications, especially in applied game theory.
Q. What is the difference between mathematical logic and mathematical philosophy?
Mathematical logic. Much of logic is mathematical. That includes all of symbolic logic, also called formal logic. But, all logic isn’t mathematical. For instance, analytical reasoning analyzes arguments that may include more than just logic.
Q. Where is the boundary?
A. It isn’t clear. George Boole explored these ideas and changed the world by doing so. He was able to classify thought and codify it using algebraic language. Boole invented a new kind of mathematics, called Boolean algebra. A century later it would provide an ideal foundation for designing the electronic structure of computers, and for manipulating information within computers.
Q. Why is Boole of interest to you?
A. I often consult with clients on futurism and predictive probabilities. As a forecaster Boole’s treatment of probability is an invaluabel mathematical analysis of inductive logic — Making decision when you have a limited amount of factual data available. In a sense, all of probability and statistics can be thought of as such a mathematical analysis.
Mathematical logic starts with symbolic logic and then asks questions about logical theories. It uses mathematics to answer those questions.
Q. What is symbolic logic?
A. In symbolic logic, a letter such as p stands for an entire statement. It may, for example, represent the statement, “A triangle has three sides.” In algebra, the plus sign joins two numbers to form a third number. In symbolic logic, a sign such as V connects two statements to form a third statement. In game theory a sign such “n” is often used in this way. In game theory, there may be any number of players in a game. Think of poker. An n-player game is a game which is well defined for any specific number of players beyong just two. “N” is usually used in contrast to standard 2-player games that are only specified for two players.
Q. Can you speak more about this concept of Mathematical philosophy?
A. The term appears in the title of Bertrand Russell’s 1919 book Introduction to Mathematical Philosophy. The content of that book consists of mathematics and what we now call mathematical logic. Mathematical philosophy has broadened since then.
Just to review a bit of what I have already said. Mathematical philosophy is the study of philosophical concepts by means of mathematics. It would include mathematical logic, but it more than just that. It shouldn’t be the philosophy of mathematics; that’s philosophy that deals with mathematics.
An example is George David Birkhoff’s theory of aesthetics. This would be an example of mathematical philosophy.
Q. It would seem that the basic idea of logic would seem, well, logical?
A. For a while that was actually so. For a while mathematical philosophers thought logic, their tool, was the foundation of mathematics. It was like thinking a hammer/saw were the foundation of a house. But that isn’t so it? Mathematical philosophy is actually more than what formal logic can tell us about it.
Q. If I wanted to go down that rabbit hole what would we call this.
A. Mathematical logic can analyze itself and this reflection is called metalogic.
Q. So what makes mathematical philosophy, similar or different than any other type of philosophy?
A. Like any philosophy, mathematical philosophy studies the existential status of definable objects. We start with a number. Let’s say “one person”. Then we begin asking queton such as are numbers real? Are groups? Are Categories? Then we can ask “what is mathematical truth? Or, What is intuition? What is a proof? Maybe math is just a huge is just one big thing with different names used to describe the same thing.
With this Nabir smiled, arose, and the class ended.
I want to acknowledge Mian Ahsan, Thom Grunauer, Quinn Rusnell, Mian Ahsan, Anders Ahlgren, Alan Bustany, Mian Ahsan, Décio Krause, Arkajit Dey, David Joyce, Frank P Mora, David Joyce whose informative posts on various online forums got me motivated to explore this subject in greater depth.
Here is a Medium story on this subject @siosond
Here is one from the archives @LewisCoaches
Tips on A Game Theory Strategy: https://readmedium.com/tips-on-a-game-theory-strategy-battle-of-the-sexes-cfff5c9a9fac
To learn more about studying “predictions” directly with me, just email me at [email protected]. I will respond personally.
Author: Lewis Harrison is a Manifestation Coach, and a professional blog and copywriter.
He is the creator of the Ask Lewis Mentoring Method as well as HAGT — Harrison’s Applied Game Theory. He is the Executive Director of the International Association of Healing Professionals an educational organization that offers programs around the world in Intentional Living. He is also Independent Scholar, with a passion for knowledge, personal development, self-improvement, creativity, innovation, and problem-solving. You can read all of his Medium stories at [email protected].
For a decade, Lewis was the host of a humor-based Q & A talk show on NPR (National Public Radio) affiliated WIOX FM in NY.
To learn more about studying with me, email me at [email protected]
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