Questions to consider

• Is maths an invention or a discovery?
• It can be considered a discovery, as mathematical laws and formulas hold even when unknown to humans and independent of who is doing the mathematical operation, so it is independent of humans. (This is called the platonic view of maths)
• But it can also be considered an invention as the concepts, laws and definitions used have all been formulated and drawn up by human hand. It can be argued that mathematics exist solely in the human mind, opening up a plethora of more questions, such as “why don’t different mathematicians invent different maths?” and “if it’s invented like art, how can there be a mathematical right or wrong?”

Further Questions that were posed but not discussed

• Is mathematics a universal language?
• What does it mean to get the wrong answer in mathematics?
• What is the difference between maths and natural sciences?
• What is the difference between maths and informatics?
   

Math as a creative art

For certain questions, where the answer is complex and not intuitive to solve, maths become a sort of creative art, where finding an original, neat solution is the creative process, with each “artist” finding their own way of proving/solving something.

Axioms, Proofs & Paradoxes: The structure of math

Although Math is the only field where each statement is rigorously proven within the given system, this given system itself has to be assumed and cannot be proven.

The system of maths is structured using the following elements:

• Axiom: a proposition regarded as self-evidently true without proof. This is the fundament of math. All other mathematical statements can be reduced to a set of axioms.
• Definition: assigns properties to some sort of mathematical object. (E.g: “a line is a breadthless length”)
• Proposition: A hypothesis consistent with known data which has been neither verified nor shown to be false. The next step is to prove it.
• Proof: A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition.
• Theorem: A mathematical statement that has been proven is called a Theorem.
• Lemma: A minor theorem, which often helps to prove a “bigger” theorem.

Pure and applied mathematics

Mathematics has developed 2 distinct branches. Pure mathematics which is used for its own sake, and applied mathematics which is studied for its usefulness.

Pure Mathematicians find mathematical problems interesting for their own sake, while applied mathematicians seek mathematical objects such as algorithms or equations that can be applied to predict what will happen if we follow a particular course of action, such as building a bridge a particular way.

Axioms

Mathematics is based on axioms. There are various sets of axioms, such as those written up by Euclid in which the shortest distance between two points is a straight line, which can be used depending on the mathematicians intentions of studying a certain thing (there is, for example, non-Euclidean geometry on curved planes where the previous example no longer holds).

Mathematics has, in some sense, been a search for the smallest possible set of consistent axioms without paradoxes. Pure mathematics is concerned with absolute truth in the sense of creating a self-consistent structure of thinking.

Applied mathematics, by contrast, is judged by its ability to predict the future and not necessarily by its self-consistency. It’s about getting a model, using it to predict, and then improving it to predict more accurately.

Proofs

In Mathematics, proof is an argument which has no doubt whatsoever, in contrast to the everyday term which generally means “proof beyond reasonable doubt” and can be refuted. Once a statement has been proven, it becomes a theorem accepted as true, which can then be used to support new theorems, thereby building a structure we call “mathematics”. The foundation of this structure are the axioms.

There are a variety of methods of proof, these include:

• Rules of inference, generally reliant on the idea of “implication”. There are 4 main types:

• The rule of detachment: if a is true and a -> b (a implies b), then we can infer that b is true. A and b are propositions
• The rule of syllogism: if a -> b is true and b -> c, then we can conclude that a -> c is true.
• The rule of equivalence: At any stage in an argument we can replace any statement with an equivalent statement. (E.g: “x is even” can be replaced with “x is divisible by 2”)
• The rule of substitution: if we have a true statement about all the elements of a set, then that statement is true for any individual member of the set.
• Proof by exhaustion, the process of considering every single possibility to check whether the statement to be proven always holds. Very cumbersome for large sets of possibilities.
• Direct Proof, proving something by simply showing it to be true.
• Proof by contradiction, assuming that the statement to be proven is false and then showing that this assumption leads to a contradiction, therefore necessitating the correctness of the statement to be proven.

Paradoxes

A paradox is a statement that appears to be sound but has a hidden error, an inconsistency. An example of this is is the following:

X = 1

X² — 1 = X – 1

(X + 1)( X — 1) = (X — 1)

X + 1 = 1

2 = 1

While every solitary step appears correct (1² = 1, factorising, dividing by like terms, etc.), the result is a contradiction. This means that the set of axioms we used here is insufficient and inconsistent. IN order to fix this paradox, we add the axiom to never divide by a quantity that is, or will become, zero. In our example, we divided by (X — 1), which, given that X = 1, is now forbidden. So, the paradox is solved.

Gödels incompleteness theorem​​​​​​​

One of the most groundbreaking proofs in the history of mathematics is that of Gödels theorem. It states that if a set of axioms is consistent, then it is incomplete. And that, no set of axioms can prove its own consistency.

In other words, math itself makes the dream of pure mathematicians, of a complete and consistent set of axioms, impossible.