The validity of mathematics as an area of knowledge has been founded on its ability to rely on basic assumptions and universally accepted definitions.

Mathematicians often use aspects such as deductive reasoning and a system of axioms to prove mathematical truths or theorems.

Based on this factor, students can easily answer any knowledge questions that arise with a degree of certainty that is unmatched in other fields.

This makes it a perfect area of knowledge to explore various knowledge issues.

One of the main reasons why mathematics embodies assumptions that are universally valid is the fact it based on reason. Additionally, as it relies on its own language of symbols, it is able to reduce any contextual or cultural influences when it comes to knowledge production.

Despite the strict confines associated with its methodology, mathematics has also been regarded as an enormously creative subject. This is highlighted by the extent to which it asks its practitioners to rely on ways of knowing such as their imagination.

In addition, while pure mathematics does not rely on the application of sense perception during the process of inquiry, the application of the resultant knowledge relies on techniques used in other areas of knowledge.

This explains why most of the knowledge questions that arise in fields such as the arts and the natural sciences are underpinned by mathematical theories.

For instance, the arts possess a close link with mathematics due to the fact that formal requirements for symmetry or harmony rely on mathematical structures.

## Examples and Explanations

### Example #1

Different guidelines on how to structure a tok essay have shown that some areas of knowledge easily resolve disagreements when compared to others.

Mathematics is one of the fields whereby experts rely on methodologies that are geared towards the development of a consensus.

As highlighted in the theory of knowledge, IB Diploma Program, mistakes often occur in mathematics. However, the experts involved often use ways of knowing such as intuition and reason to spot these mistakes and dispel any disagreements that arise from them.

It can, therefore, be surmised that consensus in mathematics is not achieved through coercion. Mathematicians recognize that there is a shared genuine conviction adopted by the individual researchers which allow them to examine the alternating sides of a disagreement. This is mainly because there exists a social standard of what experts regard as proof.

A tok real-life example that illustrates this claim is the assertion by Edward Nelson in 2011 that the Peano Arithmetic was essentially inconsistent.

The professor of mathematics from Princeton University argued that he had found proof that showed this inconsistency. The mathematical community quickly dispelled the disagreements that begun to rise within the field by finding a loophole in Nelson’s argument.

This example shows that the internalization of mathematical knowledge contrasts with other areas of knowledge because different opinions often lead to a consensus.

Alternatively, one can argue that giving equal treatment to the contrasting sides in mathematics does not necessarily result in the production of knowledge or new insights.

One of the main reasons that mathematicians appeal to mutually inconsistent propositions during the analysis of knowledge claims is to ensure that they can establish a sound position.

However, based on aspects such as analogical reasoning, different points of view can be perceived as accurate due to factors like confirmation bias. An instance that exemplifies this argument in the continuous disagreement over the axiom of foundation.

This specific theory has been cited as one of the fundamental backbones to the understanding of the set-theoretic universe.

Despite this fact, there are advocates and detractors that conflict on the validity of the Foundation. However, due to the fact that there is a consensus that this foundation has no significant impact on other parts of this field, this disagreement has not been dispelled.

### Example #2

“Reliable knowledge can lack certainty.” Explore this claim with reference to two areas of knowledge.

The maverick tradition within this area of knowledge has continuously rejected the claim that the universal truth of mathematical knowledge exists.

Mathematicians who support this idea have argued that no mathematical knowledge can be considered valid for all knowers.

As a result, while the concept of certainty in mathematics is acknowledged, it is limited to human knowledge. This essentially means that the trajectory of knowledge production within this area of knowledge follows a path that is gradually refined and developed. During this process, the certainty present is increased.

This assertion justifies the claim that reliable knowledge within mathematics can possess some form of uncertainty. This is evident from the mathematical proofs that have been appropriated by this knowledge community such as the infinite number of primes and the irrationality of root 2.

This knowledge has been considered reliable despite lacking certainty in the past millennium.

To counter this claim, one can also argue that the main defining characteristic of this area of knowledge is its ability to generate formulas that are known with certainty.

Some of the accepted mathematical truths are considered irrefutable due to their level of certainty. In reference to this tok prescribed title, mathematics is unique in that it embodies assumptions and principles that are universally accepted. This can be vindicated by the experts’ reliance on reason.

Therefore, any knowledge that is determined to be reliable has to possess a high level of certainty.

For example, geometry has been readily accepted in all parts of the world due to the certainly of the language and symbols involved.