When we do science, maths or anything, we always assume something to be true without arguing back. Is this always true? Let us dive deeper and find out!

The Nature of Proof

A proof is a logical argument that shows a conclusion follows from a set of premises. But notice: even the process of reasoning itself depends on some unspoken rules. For example:

1. Rules of Inference: Rules like modus ponens (if “A” is true and “A implies B” is true, then “B” is true) are taken as given in any logical system. These rules are not proven within the system; they’re assumed to be valid.

2. Logical Framework Whether we’re working in classical logic, intuitionistic logic, or some other system, we start by assuming certain properties (like the law of non-contradiction or, in classical logic, the law of excluded middle). These aren’t proven—they’re the groundwork that makes proof possible.

Tautologies: The Exception

The only things that might be considered “provable without additional assumptions” are tautologies. Tautologies are statements that are true solely by virtue of their logical form. For example: “Either it is raining, or it is not raining.” This is true regardless of the weather because it follows the law of the excluded middle, a basic rule of classical logic.

But again, in quantum physics, there are instances we are not sure about the exact state of something. Cat is either alive, or dead or both until someone observes it.

Tautologies are not informative about the world; they simply reflect the structure of our logical system.

No Proof from Nothing =======

If you try to prove something without accepting any starting assumptions at all, then you have no foundation upon which to build a proof. In other words:

• Without axioms, there is no baseline Even if we think of some “self-evident” truths as not needing justification, choosing them is already an act of acceptance. You must decide what reasoning rules or definitions you trust, even if they seem so basic they go unnoticed.

• Circularity and the Need for Foundations Philosophers like Descartes attempted to find an andubitable foundation for knowledge (e.g., “I think, therefore I am”). Yet, even these foundational statements are accepted as starting points, not proven from nothing.

Summary and Implications ========

• Every proof has assumptions In any logical or mathematical system, proofs are built upon axioms or starting rules. You cannot escape this unless you restrict yourself to proving tautologies.

• The “Nothing” Case is Trivial If you disallow any unconditional assumptions, the only “proofs” you can muster are those that are true by definition, statements that don’t tell you anything new about the world.

• Reflection on Knowledge This insight is one reason why epistemolgists study the justification of our most basic beliefs and why mathematicians choose axiomatic systems (like ZFC in set theory). We must decide which assumptions are acceptable if we want to build any knowledge at all.

In short, without accepting some underlying assumptions (even if they seem trivial or self-evident), we cannot prove anything of substance. The only “proofs” available in an assumption-free context are the tautologies that are built into the very language of logic. This is both a limitation and a foundational truth about reasoning itself.