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### So what is maths...?

Mathematical 'truth' is considered irrefutable to some, but why is this the case? It is quite remarkable how we can seemingly claim something with such a high degree of certainty within mathematics. Mathematics seems to embody principles and assumptions which are universally valid. This is quite unique compared with other areas of knowledge.

Perhaps this is due to the fact that mathematics is heavily based on reason. By creating its own language of symbols, mathematics also aims to reduce cultural or contextual influences in the creation of knowledge. In that sense, it may comes as no surprise that mathematicians across the globe readily agree about the validity of things such as  geometry. However, to say that mathematics is completely removed from human experience would perhaps be too hasty.

In fact, interestingly, mathematics has been used to prove what some people feel intuitively. Genuine new knowledge in mathematics is often the product of imagination rather than merely following the rules of reason. Things that are very much part of our human experience and intuition, such as concepts like beauty, can sometimes be explained through mathematics.

The infamous 'golden ratio' calculation, for example, can be found in nature. This calculation also illustrates how facial symmetry and harmony in things like architecture are linked to the concept of beauty. Links between mathematics and other areas of knowledge such as the arts (where beauty and aesthetics play a role) can lead to interesting knowledge questions.

Sometimes, we use mathematics to offer "proof" and produce knowledge in other areas of knowledge. The applications of mathematical knowledge are not confined to its own discipline. In fact, we like to use mathematics to add value to knowledge in other areas of knowledge such as the natural sciences. We also like to use mathematical calculations or mathematical language to explain behaviour in the human sciences.

This utility of mathematics seemingly enhances the credibility of the knowledge it produces. However, we could wonder how useful it actually is to explain, let's say, human behaviour in mathematical terms. Are there circumstances in which applying mathematical knowledge to other areas of knowledge is not useful? The notion of the applicability of mathematics in the world around us leads to one of the most fundamental philosophical questions about the nature of mathematics.

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Philosophers have debated for centuries whether mathematics is discovered or invented. Formalists believe that mathematics has more similarities with a kind of game, which does not need to be reflected by the outer world. Platonists, however, believe that mathematical concepts exist independent of human understanding. The relationship between maths and the world around us is an important one.

Other areas of knowledge, such as the natural sciences, are very much dependent on what they observe in the natural world. When we study mathematics, it is as if we enter into its own world, which is removed from what we can or should observe. Although the initial principles of mathematics may be based on what is present around us, we build much mathematical knowledge by following mathematical rules that are independent of the natural world. It is true that most mathematical knowledge will find real-life applications eventually.

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Yet, mathematics has its own methodology, which is very different from, let's say, the scientific method. Mathematics is often very abstract and far removed from every day life. In this sense, it is perhaps no surprise that many ancient mathematicians were also philosophers. these philosophers were very much concerned with the complex relationship between what we can(not) observe around us and what actually is "out there".

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What links can be made between maths and human sciences, between maths and history?  Watch the TEDtalk and note down those ideas that help explain these links..

How can maths work alongside science?

Watch the mini-lecture left and note down the links between the two

To what extent has maths been invented or discovered - watch the video on the left to help explain the debate..?

### So, is maths perfect? Is it 100%?... Is it the'Knowledge' we've been looking for?....

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.

Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.

But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.

His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

In the 89 years since Gödel’s discovery, mathematicians have stumbled upon just the kinds of unanswerable questions his theorems foretold. For example, Gödel himself helped establish that the continuum hypothesis, which concerns the sizes of infinity, is undecidable, as is the halting problem, which asks whether a computer program fed with a random input will run forever or eventually halt. Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality.

0:30 What is a math system?

0:47 Axioms

1:09 Theorems

1:42 Sufficiently Expressive

2:10 Completeness

2:32 True and its opposite are incompatible.

3:04 Inconsistent System

3:31 Unicorns exist

4:16 Principal of Explosion

4:36 Godel's 1st Incompleteness Theorm

5:36 Godel's 2nd Incompleteness Theorm

6:31 Beauty of Incompleteness

### After all this, what does Maths mean to you?

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