Alonzo Church: The Great Mathematician of Modern Day Logic and Frege–Church ontology by Andrew M. K. Nassief
Introduction:
On June 3rd, 1903, one of our era’s greatest mathematicians was born. His name was
Alonzo Church. He had many great contributions to mathematics and logic. (INTRODUCTION Alonzo Church: Life and Work, 2020) Amongst these contributions included the Frege–Church ontology. This ontology is one of modal logic in representation of existence and paradoxical theory. Truth contexts and the ordinal language of whether something is a paradox or not is quite important. Not only is this research affiliated with traditional logical reasoning, but the ability to tie mathematical representations with paradoxes have huge implications in the world of computer science and computational physics.
Some History:
Alonzo Church was a mathematical logician, who has many works attributed to his name.
(Alonzo Church, 2020) Among those works include Church’s Theorem in which he looked at undecidability in first-order logic for variables, as well as many other contributions in the sciences. Dr. Church had a passion as well for symbolic logic which was at its infancy at the time, as an emerging field in mathematics. His contributions to mathematics helped start the computing revolution. He had many students including notably C. Anthony Anderson, Norman
Shapiro, Stephen Cole Kleene, and Turing. Norman Shapiro co-invented the ABEL
programming language, Stephen Cole Kleene was a founder in the field of Recursion Theory, and Turing is believed to have created the Turing test and helped shape the way as well for modern day cryptography. Without the works of Alonzo Church, the world would likely not be the same.
In many of the areas that Alonzo worked, he has been a distinguished scholar and researcher. His ideas on calculability, acceptability, and complexity likely was a precursor to many of the modern developments surrounding computational hardness. It also likely played a large role in inspiring code breakers such as Alan Turing. Dr. Church compiled a monograph titled, “The Calculi of Lambdaconversion” which was one of the factors to help form the field of λ-calculus, and was an inspiration for programming languages such as LISP. The great mathematician John Von Neumann, (one of the greatest mathematicians of our time), encouraged
Turing to complete a PhD under Church. This was as he noticed that Turing has great respect for
Church’s work and Turing was also inspired by that monograph.
Alonzo Church contributed greatly to set theory in the fields of propositional calculus.
His PhD work even talked about the axiom of choice, and likely it served as a precursor to the
Frege–Church ontology. Some people even believe that universal choice has a lot to do with free will, and concepts in human nature can be portrayed by mathematics. Alonzo and other mathematicians may have indirectly inspired what is modern day sociophysics and even the works of people like John Forbes Nash Jr.. John Nash also was a prominent Princeton University alumni and a polarizing figure in mathematics. (John Forbes Nash Jr., 2020) Because of people like Dr. Church, Dr. Nash and many others, I want to say that mathematics will never be the same. However, mathematics always was the same, the laws of logic and nature are absolute.
Hence, it is more accurate to say the field of mathematics will never be the same. These works lay as groundwork for foundations for future mathematicians, physicists, programmers, and
Quantum engineers.
In regards to the nature of certainty, certainty is very important in the field of mathematics. Math in its purest form is derived from logic. One of the subjects of nature is certainty. Once one is able to numerically represent certainty or a said choice being made, this can be the start of a logical proof. This is also important in the field of computer science. When one communicates to a computer, they are doing so with the implications of logic. Certainty isn’t just tied to logic, but a recurring theme in Theoretical and Quantum Physics. This makes sense on the basis that the laws of nature are laws of nature. Deriving an example of certainty, let us look at the following: If avb =b then bva= a. This is showing a basic duality equation as an if then. The then is always certain to be true. Logic for it to be considered permissible is absolute.
One can also notice how statements in programming such as if, than, or else statements have been derived from certainty. Schrödinger's cat is another prime example of certainty in a paradoxical setting. The idea that a cat could be thought of as alive or dead or a position can be thought of as at both 0 or 1, is a logical implication of paradoxical certainty. (Schrödinger's cat, 2020) Some paradoxes hence are found to be true, and the representation of uncertainty may actually be viewed as certainty when looked at in the eyes of absolute vs. relative truth. When a paradox is deemed as absolute, it is really true since all logic is absolute, hence it is truth. If a paradox was relative, then hence it would be false.
Given the implications and importance of the Frege–Church ontology, it is perhaps one of Church’s most underrated works. When looking at certainty, one of the biggest implications of
importance is state of being. The fact that something is in the state of existence dictates an identity. The qualifier symbol for there is, is ∃. (Frege's Ontology: Being, Existence, and Truth,
2020) Existence and truth directly derives important findings in figuring out simple truths or stating statements of truth.
The concept of truth and what is truth have been there since the beginning of time.
Questions have been asked from prophets to ancient philosophers until now. However, mathematicians have only recently started numerically representing truth in a way that makes it applicable to the computational sciences. In programming, truth and certainty are core values into how computers interpret what to do. This applies from basic things such as data parsing, grammar, and interpreting statements. Even with things such as Regex (Regular Expressions) or running recursion usually revolves around some sort of rational argument or statement. If char=is.name than interest is.name to tbl(x). Whether it is adding names to tables, or running some sort of function that recursively goes through a sorting list, pure logic in terms of certainty is involved.
The next important thing to look at is also complexity. Complexity impacts a variety of different fields including cryptography. How hard is it to break or decrypt a certain cypher?
Complexity is also tied to concepts such as the Halting Problem. The theoretical halting problem is actually tied to Alan Turing, a doctoral student of Church’s. Its argument is that of decidability. How one measures when a theoretical program will halt or infinity runs? (Halting problem, 2020) So far in computational theory and turing-completeness proofs, this is computationally undecidable or unsolved, in terms of lots of the research that has been put out to the general public.
The concept of complexity in computer science, is also utilized in modern day technologies as well. An example of this is in cryptocurrency mining, where many blockchain networks with a PoW “Proof of Work” algorithm have an estimated time for finishing the next block. Infact, cryptocurrency mining in essence is a form of cryptography and cryptographic hashing. Cryptography in terms of computational hardness will exist in the post-classical computing world as well. Many Quantum computers will derive Quantum encryption algorithms.
This makes sense because this is an example of yet another form of advanced technology that still needs to rely on logic and the “laws of nature”.
If you think about it, Alonzo Church and many other mathematicians are storytellers. The stories they tell are explaining already existing concepts derived in nature. Logic in the purest form is on finding truth. Truth and these applicable patterns are what technically ties everything together. We talked about everything such as determinability, truth, and even computers and technology. These things are all tied to formalities together with logic. The only thing theoretically that doesn’t need to apply to logic is what exists outside of space, time and nature itself. That means even random human decision making can be explained through terms of certainty, uncertainty, and “x and y”, etc. Game Theory when describing patterns or sociophysics describing human decision making and mathematical implications of decision making, are all logic. If someone makes an illogical or irrational decision, the fact that they made a decision and x happened can be defined as a logical statement. Even if someone tells a lie, the fact that what he said is a lie is an absolute truth though his said statement is false. Things are described as that of patterns. We as humans even act as variables to a variety of things when deriving certain equations or conclusions. Human based social behavior such as how they react to certain
sentiment in stocks, or how emotional someone gets can be even described in terms of numerical values. Not all statements are true, and all truth is absolute, but the way to describe all statements can be absolute. Nature in essence, whether looking at theorems or many other forms, all show numerous examples of math at work.
References:
1. Web.archive.org. 2020. INTRODUCTION Alonzo Church: Life And Work. [online] Available at: