Can 1 1 3 if We Change the Axioms of Math? Exploring a Different Mathematical Universe

When we consider the truth of mathematical statements, such as 1 1 2, we often think of them as universal and absolute. However, in reality, these truths are derived from the axioms of mathematics, which are the foundational rules and assumptions that define the structure of mathematical systems.

Changing Axioms

Axiomatic systems are the backbone of mathematics. Different sets of axioms can lead to different mathematical structures. For example, in modular arithmetic, 1 1 is congruent to 0 (mod 2). This is a result of the specific axioms and definitions that govern this system. Similarly, if we were to redefine the axioms and operations, we could construct a system where 1 1 3.

Axiomatic Systems

Mathematics is built on axiomatic systems. These systems consist of a set of basic statements (axioms) that are considered to be self-evident or true without proof. From these axioms, more complex mathematical structures and theorems are derived. For example, the Peano axioms define the properties of the natural numbers, including addition and multiplication, which are the foundation for many areas of mathematics.

Non-standard Arithmetic

One can also explore non-standard arithmetic, where the definitions or operations of numbers are changed. In this context, it is possible to create a system where 1 1 3. This could be achieved by defining new axioms and operations that do not conform to the standard rules. For instance, if we redefine the operator “ ” to behave differently, we can make 1 1 3 a valid statement.

Freedom in Definitions

Mathematicians have the freedom to choose axioms and definitions that suit specific needs or to explore new mathematical concepts. This freedom is particularly evident in fields like abstract algebra, where mathematical structures are defined by their operations and properties. For example, in group theory, we define groups based on certain axioms and operations, which can lead to a wide range of mathematical structures.

Consistency and Coherence

While it is fascinating to explore new mathematical systems, it is crucial that any such system be consistent and logically coherent. A system without contradictions is necessary for it to be useful and meaningful. This is why mathematicians rigorously check the consistency of their systems before adopting them.

Conclusion: While 1 1 3 is not true in standard arithmetic, it can be made true in a different mathematical framework by carefully choosing and defining new axioms and operations. This flexibility is a fundamental aspect of mathematical exploration and theory development. Mathematics, in a sense, is a product of our way of seeing the world. Galileo's claim that the world is written in a mathematical language suggests that mathematics is our best tool for explaining our surroundings and predicting future outcomes. However, the nature of such a mathematics depends on our perceptions and the axioms we choose. A different perspective or a different set of axioms could lead to a different mathematics that still explains the world in a coherent and meaningful way.

So, to directly answer the question: Yes, you can create an operation where one plus one is three. For instance, you could change the names, redefine the symbols, or introduce new operations that adhere to a different set of axioms. The possibilities are vast and only limited by the creativity and logical consistency of the mathematician.