Consensus and disagreement are both stages necessary in the production of knowledge, but depending on the area of knowledge discussed, the extent to which they rely on consensus or disagreement may differ.

One area of knowledge in which this is the case is that of mathematics, the ultimate goal of which is to arrive at logically valid statements that are objectively true. If this is the case, then the end product must involve a large extent of consensus regarding the theories that are proven. When Professor Thomas Hales, from the University of Pittsburgh, Pennsylvania first proposed a proof to Johannes Kepler’s theory that the most efficient way to stack spheres such as oranges and melons was in a pyramid formation in 1998, he created a three-hundred-page proof, which took 12 mathematicians four years to review and check for the errors. In this case, 12 mathematicians were able to form consensus with each other to the idea Thomas Hales’ proof was right. Such peer assessment process is widespread among the mathematicians because maths is an area of knowledge where most of its rules and patterns come from deductive reasoning, under the bases of axioms and logic. Mathematicians were able to Thomas Hales’ work because mathematicians have a general agreement about the logical deductive reasoning behind the well-established mathematical language system.

There are also cases where mathematicians face the challenge. When the reviewers find an error in the mathematical proof, this error would likely to influence the overall validity of the theory proposed. A famous mathematician, Nelson, claimed to give proof to an argument, where “Peano Arithmetic is inconsistent.” Two other famous mathematicians, Tao and Tausk, was able to spot a mathematical error in Nelson’s paper, and they spelled out. Nelson’s reaction was: “Ah, you’re right. So I have not proven that Peano Arithmetics is inconsistent”. That is the end of the story. There were no fight, or any disagreement, a formation of alternative schools of thoughts, and no playing with how to interpret this or that claim. Whenever the logic is proven wrong, mathematicians would just come to a clear consensus. It is difficult to disagree in math because when a claim contradicts to a certain axiom that was proven to be true, it is very likely that the claim is wrong with regards to its fundamental knowledge. To develop mathematical knowledge, mathematicians must share absolute consensus of the basic self-evident knowledge, so that new knowledge will stand.

Not only to the production of the mathematical knowledge, but consensus in math is also required for regular uses/applications of math to solve real-life situation problems, and for students to learn math and participate in exams. Mathematical knowledge is supposed to be built on top of these axioms and to form consistencies with each other. That’s why often in math exams, there is only one exact correct answer in the mark scheme, unlike other subjects such as human sciences, where the cause of to a specific social phenomenon can be answered in different ways, based on varying perspectives.

However, that is not to say that no disagreement is available in studies of math. In 2012, Shinichi Mochizuki at Kyoto University in Japan produced proof of a long-standing problem called the ABC conjecture. The ABC conjecture was first proposed in the 1980s and concerns a fundamental property of numbers, based on the simple equation a + b = c. For a long time, mathematicians believe that the conjecture was true, but nobody had ever been able to prove it.

Mochizuki developed a whole new type of mathematics called inter-universal Teichmüller theory (IUT) and claimed to have proved the theory. Peter Sarnak seems to believe that it isn’t, and Terry Tao seems to believe that it is, and I don’t think either holds an unpopular opinion. As for His proof, it is 500 pages and baffled almost everyone who read it.

This is because, in order to ensure that proofs are mathematically valid, it must be examined rigorously by others, in the review process. If his students never challenged Aristotle about the idea of all numbers being rational, the cult of Aristotle would never be corrected, and hence irrational numbers will not be discovered. In modern times, the review process is also very important in that it allows for logical leaps to be discovered, Andrew Wiles’ proof of Fermat’s last theorem. For instance, was initially invalid, but would not have been discovered without any disagreement. It is, therefore, the disagreement that drives conclusions that can be agreed upon, for consensus to form.

Disagreement, however, is relatively rarer due to a combination of two major factors. Firstly, as mentioned earlier, to disagree a mathematical prove means to disagree it’s foundation axioms. However, in the modern society, there is not much room for this kind of disagreement because most of us accept, and uses the Zermelo-Fraenkel Set theory. Mathematicians seem perfectly content to use ZFC as the foundational axioms of mathematics. Secondly, the field of mathematics is much larger than it used to be. Thus it is much more difficult for any interpersonal debates and arguments to take place.

Another area of knowledge that requires consensus and disagreement to its knowledge production is that of natural science. Knowledge in the natural science is produced in its distinctive way: by believing that theories should be based on our observations of the world rather than on intuition, faith, reasoning, or appeals to authority. It is commonly known by the name “scientific method”, and is heavily influenced by Galileo. Galileo overturned the Aristotelian worldview, according to which all the phenomenon that occurs in this world can be explained by science (). The Galileo’s scientific method is divided into three general stages. Intuition, demonstration, and experiment. The modern scientific method evolved around it, but both of them starts with empirical, organized, systematic observations.

In the view of observationalist-inductionism, scientists or people can induce and predict the outcome of an experiment or a natural phenomenon based on their previous experiences of observing it, as the phenomenon has always occurred in the past the same way. Therefore, after systematic observation, scientists can make a hypothesis based on the foundation of the publically available knowledge, which refers to the knowledge that is proved in the past, under the consensus to the means and application of math and it the deductive logic. However, this raises problems. There is no absolute certainty in the natural world. For instance, the gravitational field strength is not always the same anywhere on earth. But we still accept it and continues to apply these existing public knowledge into further experiments and studies is to show that the purpose of science is not to proof or show whether or not things are true, but rather that things are false. When the hypothesis can still stand after rigorous tests and debates over a period, it is then considered as a theory and is being applied by other scientists or people as a publically available knowledge.

However, there is a general lack of demand in the consensus among all the theories in the realm of natural science besides the use of mathematical language. The scientific method carries the aim to find knowledge that is not false or to find knowledge that false so that we can get closer to the truth. This objective may be achieved through the peer assessing stage after the theory is being tested by the researchers, and is evaluated by other scientists with the same amount of information as the researchers who experimented. Therefore, the disagreement may play a significant role to distinguish between a false theory and a not false theory. However, it is still arguable that peer assessing is just a process for the other scientists to make sure that all of the calculations and formulas in the paper are used accurately. These scientists are not judging on the researchers’ hypothesis itself; they are just checking the researchers’ mathematical language. It is like when the English teacher checks the grammar of an essay without really giving an opinion on the topic of that essay.

Even though the publically available theories of the natural science are taught, learned, and used as the robust scientific knowledge that has survived rigorous testings and assessments, there are still examples of scientific theories being contradicting to one another. The general theory of relativity and quantum mechanics are fundamentally different theories that have different formulations. In a brief description, quantum mechanics has a fairly antagonistic relationship with general relativity (World Science). In Einstien’s relativity, he envisions that space is curved, and it is nice and gently curved. Things are continuous and deterministic, meaning that every cause matches up to a specific, local effect (Powell). In quantum mechanics, due to its uncertainty principle, events produced by the interaction of subatomic particles happen in jumps (quantum leaps), with probabilistic rather than definite outcomes. In other words, things are uncertain because they fluctuate and move chaotically. This fluctuating uncertain and messy behavior in the macro realm conflicts with the gentle behavior of the general relativity macro realm. These two leading physics theories do not come together in a harmonious manner in the conventional way of looking at things. Since virtually, everything we know about the laws of physics falls into one of two piles; these fundamental piles conflict with one another, disagreements between theories thus become unavoidable. Scientists, especially physicists nowadays are concentrating their focus on solving this issue, and are anticipating to the theoretically potential discovery of the “theory of Everything,” in which all physics theories can cohere.

Mathematics is the study of abstract models by specific/individual rules/axioms.Science is to formulate the model of real-world phenomena. These two areas of knowledge overlap, like science, uses an abstract Mathematical model if it fits any real-world events such as the application of Fourier series in Music. Due to the reliance on mathematics, these two subject areas share similar consensus towards to use of mathematics, but due to the different methodologies and aim, disagreements exist for various reasons. In math, the consensus is formed by the reliance and acceptant on the Zermelo-Fraenkel Set theory. It is challenging to disclaim a mathematical theory due to its deductive nature and logical basis. Therefore it is a much-preferred tool for scientists to represent or proof their scientific observations and results. Disagreements are rare in math, besides spotting out errors in the assessment process, which are usually accepted by the person who made such error afterward, like in the example of Thomas Halees. Science, on the other hand, there is a consensus towards the use of mathematics proof, since math is used to examine the logic behind the proposed theory. Due to this factor, scientists may come up with new “theories,” as long as their proposal has been mathematically proved and by reviewers. However, disagreement in the natural science exists with due to the availability of the inconsistent and contradictory theories. These are prior proposed theories that have been rigorously assessed mathematically, and the reasons for which they share contradictory relationship is still unknown.