Uniformities are an aspect within disciplines and areas of knowledge that stay constant throughout. They are also closely related to axioms, this is due to how they both can be used for assuming that something stays constant within thier respective disciplines. Axioms are usually used as a foundation for a discipline or area of knowledge to build off of. That does not necessarily mean that they were the beginnings of that area of knowledge, to make a uniformity, there has to be knowledge of the subject present, or else what uniformity can you make if you know nothing, It should be argued that this claim is to be agreed to an extent.
To the extent that you know that uniformities are necessary for delving into the more complex areas of that area of knowledge, however, you know that they are not necessary for developing the most basic knowledge in that area of knowledge. This claim will be studied by looking into mathematics and the sciences as areas of knowledge. The field of mathematics is built upon logic and some intuition, however there was a point in the early to mid 20th century when this was under question. It was also a time of great mathematical progress, one of those great accomplishments was Kurt Godels incompleteness theorem. Godel published two incompleteness theorems, the first one addresses the inherent limitations that are within every formal axiomatic system.
While the second one addresses the fact that none of these systems can prove their own consistency (assuming that they are consistent to begin with). To fully understand Godels incompleteness theorems, you first must know what a “formal system” is. Basically, a formal system is a system of axioms that has rules of deduction, this allows you to generate more theories based off of this system. So Godel is saying that each of these systems has natural limitations within the system, that everything cannot be proven or disproven. The second states that these systems of axioms are unable to prove their own consistency, assuming that they are indeed consistent in the first place. This tells us that there can be no such thing as a uniformity or a universal axiom, because it will not be able to prove or disprove all questions that it encompasses. Showing us that knowledge must be able to be built without the assumption of the existence of uniformities, because, according to these theorems, there is no such thing as a uniformity.
On the other hand is the axiom that is mathematical induction. Mathematical induction has beend used since 370 BC by Plato, however this hypothesis was never actually stated until 1665 by Pascal, despite this the modern system that we now know didnt come until the 19th century with Giuseppe Peano, George Boole, Charles Sanders Peirce and Richard Dedekind. Mathematical induction is the proof that is used to prove a statement for any set, such as the whole, natural and integer number sets. This means that you are able to “close” any operation under any set, and if not, another set is developed. To determine if something is closed, you must perform the operation with two numbers from that set, you do this for the numbers you chose and eachnumber after that, or n and n+1. If there no exceptions then that operation is closed under that set, as previously stated, if there is then a new set might need to be developed. Such as Subtraction is not closed under the whole number set, so the integer set was developed to satisfy this.
Or how exponentiation is not closed under the real number set, because you can’t take the even root of a negative number, so the imaginary and complex number sets were developed to satisfy this. So mathematical induction gives us the ability to “close” or prove any statement about a number set. Mathematical induction shows us that it is necessary to assume that there are uniformities, this is due to the fact that itself is a uniformity within mathematics. It also leads us to believe the point that uniformities are necessary for the basis of knowledge, because, in this specific case, mathematical induction tells us when our number system is in need of expansion or improvement, and without it we would not have the knowledge that it is in need of improvement.
The sciences also have forms of axiomatic assumptions throughout there respectful disciplines. The most basic of these are axiomatic assumptions which are necessary for the scientific method to be justifiable. These are: 1. There is an objective reality shared by all rational observers. 2. This objective reality is governed by natural laws.
3. The laws that govern this objective reality are able to be determined using systematic observation and experimentation. These three assumptions are necessary in order for the scientific method to be used, this is because the scientific method is used to collect more knowledge on something in this objective reality, if this objective reality was not shared by all observers, or was not governed by natural laws, then there would be no way of gathering knowledge. However Paul Feyerabend and his ideology of epistemological anarchism, would argue that there are no methodological rules or assumptions that are able to govern all of science, such as the scientific method and it’s basic assumptions. Epistemological anarchism is the ideology that there are no exception free rules that govern the progress of science or knowledge on it’s own. It also states that the idea of science being controlled by these universal rules is unrealistic and is actually detrimental to science as a whole. Showing us that if we do assume that there are uniformities that govern such areas of knowledge, it would actually harm the advancement of knowledge in this area of knowledge.
Which means that science would have had to build its knowledge without the use of an axiomatic foundation. On the other hand is the ideology of Hugh G. Gauch Jr.
which states these axiomatic assumptions are not only necessary, but demanded by the sciences. Gauch states that “science presupposes that the physical world is orderly and comprehensible.”, which implies that there must be an assumption of the uniformities that are necessary to justify the scientific method. An ideology that shares this belief is uniformitarianism, outside of geology, it is the belief that all of the natural laws that exist and operate now, have always been operating throughout the universe, and that they are applied everywhere. This belief tells us that there are uniformities throughout science, but they also tell us that they have always existed.
It gives us a foundation which science uses to build from, and expand, which implies that without this uniformity there would be no knowledge, because there would be no foundation for science to grow from. In the end, Feyerabends epistemological anarchism shows us that if we do assume that uniformities exist and that they are the basis of knowledge, it actually harms the advancement of science, while uniformitarianism and Gauch state that there must be an axiomatic foundation for science to base it’s knowledge off of.In mathematics, mathematical induction shows us that there are uniformities within mathematics to base its knowledge off of, and so does the ideology of uniformitarianism. The ideology of Feyerabend and Godels incompleteness theorem shows us that these are not necessarily uniformities, but points in which to build knowledge off of. In the end this claim should be agreed with to an extent, to an extent of which we know that uniformities are points in which we can build knowledge off of, as previously mentioned, but they are not a necessity for the building of knowledge.