On axioms and assumptions. How changing them can change the world.

26 May, 2025

What are axioms?

Axioms, especially in math and sciences, are statements we regard as foundational assumptions accepted without the need to prove them. They serve as starting points to deduce other truths within a formal system. Coming together, they form the foundations of different fields. We can derive numerous logical assumptions based on the axioms. They have specific logical properties which enable us to derive lemmas and theorems from them. Thanks to them, we can construct whole domains based on them.

While doing that, one must remember the insight from Gödel's incompleteness and completeness theorems. He showed that in any consistent system capable of expressing basic arithmetic, there are true statements that cannot be proved within the system. This means that such systems cannot be both complete and consistent. One could say that axioms are assumptions we take as true to proceed further in a given field. 

There are different kinds of axioms. I am interested in the division between logical and non-logical axioms.

Logical axioms are, in general, valid within a logical system. It means that they are regarded as such in different kinds of settings and not only in a specific domain. An example would be the following sentence: if A and B are true, then A is true. It would be hard to disprove this. 

The other variety of axioms are the non-logical ones. They are used to build insights, although their truthfulness is limited to a particular domain. We assume them to be true to proceed in a given field. There are multiple examples of them in economics, mathematics, and physics. Think about the assumption that prices are positive in economics or that classical physics is assumed to hold. 

Where did they come from?

Two figures that come to mind when thinking about the beginnings of axioms and logic in general are Aristotle (388 - 322 BC) and Euclid (325 - 265 BC). In his work "Posterior Analytics", Aristotle establishes the foundations of scientific knowledge. He makes a distinction similar to the one I outlined before about logical and non-logical axioms. He explains a rigid system, according to which we could deduce true statements, based on previous assumptions. A famous example of Socrates being mortal comes to mind¹.

Euclid came up with his "Elements of Geometry", which marked the beginning of geometry as a rigid science as we know it today. In it, he came up with a definition like the one of a point. As defined by Euclid, a point is that what has no part. He also formalized postulates like that you can draw only one straight line through two points. They came together to establish what is now called Euclidean geometry. Although a contemporary mathematician would describe his axioms as lacking some rigor, his achievement was extraordinary.

Are axioms always true?

Given axioms are always true, and we do not require proof for them, one could say that they always act accordingly. Interestingly enough, history shows us that it is not always the case.

There exist examples of statements we regarded as axioms for centuries, but some people proved that it might not necessarily be so. A famous one is the case of one of the axioms proposed by Euclid in his work I mentioned before. Specifically, it was the Parallel Postulate that people challenged. The postulate goes as follows; If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

As this postulate was picked up and challenged, it gave rise to a new domain, now known as non-Euclidean geometrics. This in turn revolutionized multiple other fields. Einstein's General Relativity uses non-Euclidean geometry to describe spacetime.

Ordinary modern logic shows us that a statement is an axiom of a given system. That is to say that we can construct a system with an axiom, which is true in some other domain, which will be false in our domain. This is the case of the Parallel Postulate in Euclidean geometry vs non-Euclidean geometries. When we talk about axioms, we should refer to specific systems within which they are true, or at least it should be implied.

How malleable axioms help us navigate different fields

The most fascinating thing about axioms is that the system we use them in defines whether they are true. Therefore, we can define a system in which an axiom, like prices are always positive, is unconditionally true. This realization leads us to being able to define a system, whose foundations, despite not being true in general, are helpful to think about in particular domains. This gives us a valuable tool we can employ when wandering in new domains.

When we try to model the complex world we live in without simplified models, we need to make assumptions or define axioms. But that doesn't stop us from making the models useful. Some settings can help us define models to apply in the real world. An example of this is quantum physics. Quantum physics is a field which has multiple counterintuitive axioms at its core. Think about superposition. That is essentially existing in various states at the same time until someone takes a measure. Despite this, it is tremendously helpful and can be applied in many subject areas.

Another interesting field is game theory. It is defined as a branch of (applied) mathematics. Despite its definition as a it is, I can't stop but see more resemblances of puzzles or board games, rather than linear algebra or calculus. That is surely a byproduct of applying many assumptions, that limit the math part by a lot. Despite the limitations and the apparent differences in notation, this is still useful and can be of plenty of help to scientists alike.

What would happen if we were to change our axioms?

We commonly use the decimal (base-10) math system. What would happen if we were to suddenly change, from 10 digits, so from 0-9, to say 7, that is 0-6? Nothing would change, except the look. This is understood by noticing that the two numbers would still represent the same quantity even though they would be written differently. We could change certain axioms without changing too much except the looks. But as we have seen with the removal of other axioms, there arise completely different domains, as with Euclidean and non-Euclidean geometrics. The choice of axioms is fundamental to the undertakings within various fields. Choosing the appropriate axioms can be tremendously powerful in making progress within specific fields.

Knowing the limitations of axioms

While seeing the importance and power of appropriate axioms, it is essential to keep in mind their limitations. Some things, even though true in some systems, do not transfer to the real world. It is not possible to divide a ball into 5 pieces and then join them together to create two balls, of the size equivalent to the previous ball, like the Banach-Tarski paradox suggests. Their counterintuitive result applies to purely mathematical domains and should not be treated with respect to physical dimensions. Nevertheless, there are examples of axioms we can get more use of, such as the ones in game theory or quantum physics.

Gödel's incompleteness and completeness theorems are some of the best expressions of the limitations of systems that we create with axioms. In his works, he shows that there are fundamental obstacles that are impossible to overcome within a defined system. This is represented by the fact that never can every axiom within a system be appropriately defined.

Final words

Considering all the above, it is possible to see a glimpse of how changing axioms can lead to discovering huge things. This might be immensely useful in domains that we only start to investigate deeper.