An Introduction to Philosophical Thinking, Part III

Content notes: This series is intentionally opinionated and exploratory in nature. It is not intended as an academic treatment of philosophy, nor as a rigorous defence of particular positions. Instead, it serves as an accessible introduction to a handful of philosophical ideas and ways of thinking, aimed at casual readers who are curious about philosophy but may not wish to engage with formal proofs, technical debates, or scholarly conventions.

In Part I we examined how philosophical thinking methods and practices can be useful. Part II offered a brief look at the differences between maths and science. In this part, philosophy will be considered alongside both.

Why do we need philosophy?

Most people fail to explain why philosophy is needed at all, even though many agree that the answer is, or at least should be, obvious. Let us begin by offering some possible answers.

One possible answer is that philosophy is needed because there are questions that are not readily answered—or cannot be answered at all—using empirical data or formal systems alone. These include fundamental questions about knowledge, existence, and reasoning itself.

A broader definition of philosophy could describe it as a pursuit of meaning and understanding, with an emphasis on clarification and conceptual analysis rather than testable hypotheses.

Another view can be gained by comparing philosophy with maths and science, to see whether this sheds any light on what philosophy could mean.

Philosophy of Mathematics

Both maths and formal logic begin with explicit axioms. These axioms are not meant to discover or guarantee truth about the world. They are accepted without proof, as long as they are internally consistent.

In other words, axioms in maths and logic exist to preserve the validity of formal proofs. This is why division by zero is not allowed—not because it reflects a constraint of the universe, but because allowing it introduces contradictions within the system. Without this restriction, mathematics would cease to function predictably.

In contrast, although philosophy can and does make use of axioms, philosophical inquiry is often pre-axiomatic, since it tends to question whether axioms themselves make sense.

Philosophy of Science

Unlike maths, science relies on epistemic tools such as observation and experimentation, with falsification as a central aim. As a result, scientific theories remain provisional and are never absolutely certain.

Scientific endeavours often begin by observing regularities: the apple always drops. From this, one may ask whether a gravitational force is at work.

Philosophy of Philosophy

Metaphilosophy—or the philosophy of philosophy—is distinctive in that philosophy alone turns its methods back onto itself. It is philosophy that examines and revises its own definition.

Philosophy, in this sense, becomes the foundation of other forms of inquiry.

Let me explain.

When we begin to think about a scientific theory, we do not begin with formal axioms as we do in maths. Nevertheless, scientific reasoning still relies on certain unproven philosophical assumptions a.k.a. axioms. As mentioned earlier, axioms do not need to be proven.

For example, when we observe an apple dropping from a tree, we implicitly rely on assumptions such as:

1. The external world exists and it is not just a dream.
2. Our senses are generally reliable and what we feel and observe is not filled with errors.
3. The laws of logic apply to the universe so the apple cannot go up and down at the same time.

These are only a few examples. As you can see, such axioms must be taken as true; otherwise, no meaningful theory could ever be discussed. In this sense, philosophy provides the ground upon which other intellectual practices stand.

The axioms mentioned—along with many others—are philosophical beliefs that make knowledge possible. Through philosophy, we establish the “rules of the game” to enable other practices to play the game.

Aristotle called this the “First Principles.” He argued that every field of knowledge must start with axioms that cannot be proven but are self-evidently true. He called philosophy “First Philosophy” because it examines those very starting points.

He distinguished between two categories: Common Principles and Proper Principles.

Common Principles are axioms that apply universally. The most well-known is the Law of Non-Contradiction (LNC): A cannot be both B and not-B at the same time. This principle cannot be proven, since any attempt at proof would already presuppose it.

Proper Principles are definitions specific to a given field. In geometry, this includes the definition of a line; in biology, the definition of what counts as a living thing.

Aristotle argued that attempting to prove every statement leads to an infinite regress. Each proof would require another proof, without end. To have any knowledge at all, inquiry must eventually rest on a First Principle that is indemonstrable—known not through proof, but through direct understanding.

This is why I believe philosophy often has to establish its own beginnings.

For this reason, Aristotle called philosophy “the Queen of the Sciences.” A biologist accepts that life exists and proceeds to study it. A physicist accepts that matter exists. Only the philosopher pauses to ask what it even means to exist.

In other words, philosophy operates at a different level of inquiry. It does not only ask questions such as “what is X?” but also asks what it means to ask a question at all, and what can or cannot be answered.

Starting from such beginnings is equivalent to formulating hypotheses. This is often called deduction and its purpose is to determine whether a starting axiom leads to contradiction, in a way that parallels how science uses hypotheses to generate predictions.

Another useful method is what might be described as “working backwards.” Its technical name is induction, and it resembles how science observes a falling apple and infers the law of gravity.

An example of induction in philosophy can be found in ethics, where one might begin with a specific moral intuition—such as “killing is wrong”—and try to work backward to see whether a universal rule can be derived from it.

In Part IV, we take a deeper look at philosophy and a few mindsets that could help in understanding it better.