Mathematics of Philosophy — An Introduction
Philosophy and mathematics are considered two different fields of study, but from ancient times, many philosophers and mathematicians have attempted to see the relationship between both. Their effort helped humanity to see a more clear picture of reality.
Zeno’s paradoxes were constructed with the philosophical purpose of demonstrating that all change is illusory, but raise important and challenging questions about geometry, measure, limits, and the nature of infinity.
Gottlob Frege’s interest in the nature of mathematical truth prompted his invention of modern mathematical logic, a subject of central importance in understanding the nature of proof and of computation more generally.
David Hilbert’s philosophical view that mathematical truth reduces to provability prompted mathematical work of great profundity, notably Gödel’s incompleteness theorems and Alan Turing’s related work on computability. Concerns about the nature of numbers, sets, and the paradoxes of set theory, have animated the work of both philosophers and mathematicians.
Table of content:
1. Preliminaries
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|_a. Why anything is important to you?
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|_b. Basic terminologies
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|_c. Introduction to some well known school of thoughts
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|_d. Required mathematical concepts(Vector,Basis,and Vector Space)2. Philosophy of Maths and Maths of philosophy.
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|_a. Nomenclature and Notations
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|_b. Assumptions
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|_c. Theory
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|_d. Use Cases
In the end we will try to see the mathematical relation between Philosophy and Mathematics.
Let’s start our discussion with Preliminaries:
Why anything is important?
Our brain makes connections between neurons. Connections between neurons define an individual. An Individual has some value for everything. His values determine the importance of something for an individual. Values are created by the environment, more precisely, how the environment shapes the neuron connection inside the brain. Now, suppose something is essential for an individual. In that case, it means the individual has neuron connections that contain the value regarding the same. Suppose any factor in its environment tries to alter those connections. In that case, the individual will resist because he has lived a part of his life with those values, and sudden changes will make their past lifeless meaningful.
We will know about value in more detail in next section.
Basic Terminologies:
Values: Values are the individual principles or quality that guide judgement and behaviour of a person or group. Attitude: Attitudes are the feelings or beliefs/values that determine the behaviour of an individual.Moral Values: When values are seen from the lens of right or wrong on individual level those values are known as moral values.Ethical values: When values are seen from the lens of right or wrong on Societal level those values are known as Ethical values.
The diagram above explains how values lie at the core of Attitude(our behavior), Morality, and Ethics.
Now Let’s discuss some existing schools of thought to get familiar with philosophical ideas.
Introduction to some well known school of thoughts-
Nihilism:
Nihilism is a philosophy or family of views that reject general or fundamental aspects of human existence, such as objective truth, knowledge, morality, values, or meaning.
Although there is more than one form of nihilism, they all discuss the human condition and its existence. Here are a few underlying principles of nihilism:
- Existence is useless. A nihilist believes there is no purpose to having values or beliefs because everything in existence is unfounded.
- There is no truth. Everything is unfounded and useless, including the truth, so there are no reasons to uphold moral principles for your own sake or the sake of anyone else.
- Everything is meaningless. Active nihilism says that since there is nothing and nothing we do matters, all things are therefore meaningless, including the meaning of life.
German philosopher Friedrich Nietzsche put forth many concepts related to nihilism and have exerted an enormous influence on Western philosophy and intellectual history.
Existentialism:
The central proposition of existentialism is that “existence precedes essence.”
which means-
- We have no predetermined nature or essence that controls what we are, what we do, or what is valuable for us.
- We are radically free to act independently of determination by outside influence.
- We create our human nature through these free choices.
- We also create our values through these choices.
“We create our own nature”: We are thrown into existence first without a predetermined nature, and only later do we construct our nature or essence through our actions.
Jean-Paul Satire is one of the best-known philosophers of the 20th century also regarded as the father of existentialism.
Absurdism:
Kierkegaard has first talked about the absurd, and Albert Camus fully developed the philosophy of Absurdism in his book The Myth of Sisyphus. The Absurd is the collision between the inherent human hunger for meaning and the impossibility of satisfying this drive in a meaningless world. Camus says we have three options in facing the Absurd:
- Commit Suicide: Camus found that there is no meaning in death, so this option is not better than before.
- Commit Philosophical suicide: Meaning is to take a leap of faith and believe in some meaning (like Christianity. Buddhism, Marxism, existentialism).
- Absurdism: The third option is Absurdism. Absurdism is the rebellion against the Absurd. It is to refuse to give in and create meaning. For Camus, Absurdism means holding the space of the absurd, staring into its face and rebelling against it, and out of this rebellion, letting our freedom and passion flow.
He identifies three characteristics of the absurd life:
1. Revolt: We must not accept any answer or reconciliation in our struggle.
2. Freedom: We can think and behave as we choose.
3. Passion: We must pursue a life of rich and diverse experiences.
if you want to know the myth of Sisyphus then checkout this TED-Talk
Hedonism:
According to hedonism, the meaning of life is to minimize pain and maximize pleasure while we are alive.
As a theory of value, hedonism states that all and only pleasure is intrinsically valuable and all and only pain is intrinsically not valuable.
The earliest written record of hedonism comes from the Charvaka. Charvaka belongs to Heterodox(Nastika) schools of Indian philosophy. Persisted for two thousand years (from about 600 B.C.E.), original works have been missing, but we find its ideas by criticism of other Indian schools of philosophy.
Most notably, the Charvaka advocated scepticism and Hedonistic Egoism — that the right action is the one that brings the actor the most net pleasure. The Charvaka acknowledged that some pain often accompanied, or was later caused by, sensual pleasure, but that pleasure was worth it.
Let’s understand/refresh some essential mathematical topics such as vectors, basis, and vector space.
Required Mathematical Concepts:
Vector: A Quantity that has size and direction.
A vector space is a set of objects called vectors, which may be added together and multiplied by numbers called scalars. These operation satisfies some axioms.
A vector space can have several bases; however, all the bases have the same number of elements, called the dimension of the vector space.
Here we can see a vector F in 3-Dimensional vector space.
Basis:
Any set of linearly independent vectors which spans the whole space. In above figure set {i,j,k} can be considered as basis for 3 dimensional vector space.
In other words, a basis is a linearly independent spanning set.
Philosophy of Maths and Maths of Philosophy:
The philosophy of mathematics is the branch of philosophy that studies mathematics’s assumptions, foundations, and implications. It aims to understand the nature and methods of mathematics and find out the place of mathematics in people’s lives.
On the other hand, mathematical philosophy( Maths of Philosophy) is an attempt to find hidden mathematical structures in philosophy.
We will be focusing on Mathematics of Philosophy in this article.
The idea of Mathematical philosophy:
The idea is to get a picture that shows the hidden mathematical structure in philosophy.
Nomenclature and Notations:
Assumptions to the theory of Mathematical philosophy:
- The philosophy of an individual can be described as a philosophy vector.
- The projection of a philosophy vector on a basis represents the probability of an individual to have certain philosophical orientation.
- Each school of thought forms a basis.
- The contribution of basis can be calculated after observing an individual’s behaviour in a certain period (for simplicity, we take observation period 24hr.).
- For a day philosophy vector remains constant. Changes will happen in the meantime but will be observed after a day.
- A function H(time, environment) constantly acts on the philosophy vector and determines its evolution with time.
- Function H depends on time and environment.
- The sum of every school of philosophy’s/basis contribution to an individual(i.e., the sum of the dot product of philosophy vector ϕ of an individual with every basis x) is 1.
Theory:
A vector can describe the philosophy vector of an individual in n- Dimensional vector space, where n is the number of schools of thoughts taken into account. In reality, n will be infinity, but we take it as a finite number for simplicity.
The evolution of the Philosophy vector will be determined by an operator H.
H is the combination of many instantaneous h operators that accounts for instantaneous changes in the philosophy vector. So, H is nothing but a combination of all the attempts in a given time. For now, we are assuming given time to be a day.
here philosophy vector for today is ϕ(t) and ϕ(t+1) for tomorrow.
We can derive a quantity which will be directly associated with how likely an individual’s philosophy can be changed towards a particular basis in a finite time. I will discuss in more details in near future article. For now here are some potential applications of the above discussed theory —
Use Cases:
- Human behaviour prediction in future.
- Estimation of how much likely one’s philosophy can change.
There is much to cover, but I will leave it as an introduction here. This is a new approach toward thinking philosophy as a mathematical object. This article was made possible after discussing in length with my friends (Devang Jain and Anurag Gupta). There is much to explore and derive so, your suggestions regarding the Theory of Mathematical Philosophy will be highly appreciated!
References:
- https://youtu.be/J0aX8QMkFAI
- The Myth of Sisyphus — Albert Camus
- https://youtu.be/ozwodzD5bJM
- The Lexicon- Chronicle
- https://www.britannica.com/
- MAPhilosophy32.pdf (patnauniversity.ac.in)
- What Is Nihilism? Definition and Concepts of the Philosophy — 2021 — MasterClass
- The Myth of Sisyphus: Summary | SparkNotes
