My interpretation of this prompt is that without a basis of knowledge stemming from widely accepted facts, there can be no new knowledge that can be built upon it. Prior to answering this question, however, certain ambiguous terms must be defined. Firstly, assumptions don't necessarily require proofs as they are a conscious or subconscious part of human nature. Secondly, for the sole purpose of this essay, an assumption is that the same natural laws in the universe have always operated in the universe in the past and apply everywhere. This uniformity of nature is the principle that the course of nature continues uniformly the same. As well, truths are justified, true beliefs. According to Plato, knowledge is truth. Within this, both shared and personal knowledge come into play. Shared knowledge is widely accepted by a group of people whereas personal knowledge about a certain fact is gained through first-hand observation or experience. I therefore believe that uniformities are needed to build further knowledge in both the natural sciences and mathematic areas of knowledge. From this, the knowledge question is formed: To what extent does reason verify the validity of both shared and personal knowledge in mathematics and natural sciences?
To build new knowledge in the natural sciences, uniformities are needed. As scientist Hans Christian Oersted once said, uniformities are a necessary aspect in the sciences and are understood by all. This understanding rapidly leads to a secondary knowledge question, which is “How are languages and reason used to justify shared knowledge in the Natural Sciences?" Firstly, reason adds validity to our shared knowledge. Then, language spreads the knowledge and leads to assumptions of uniformities in our shared knowledge. Lastly, reason and language either work for or against an argument against accepted shared knowledge.
In the sciences, predictive statements about nature are used; a hypothesis incorporates language to properly articulate the scientist's predictions about nature and the uniformity which is needed to communicate with others. To reach accurate conclusions in the sciences, reason and language must therefore be used. Reason is used to make sense of scientific information in order to properly discover and eventually apply uniformities. It is used to justify the existence of uniformities by allowing the reader to formulate a personal opinion and validate the scientist's beliefs. If the hypothesis is generated through induction, then its prediction of the future is based on observations of the past.
The hypothesis is a predictive statement about nature that is based on uniformities. Therefore, science requires the uniformity of nature and is based on certain assumptions in natural causality and uniformities in space and time. Ultimately, particularly in the natural sciences, it is assumed that all human interpret certain events from a similar perspective, allowing the world to consider scientists’ observations as reliable. A perfect example of this is Mendel's first law, which assumed the law of uniformity in order to gain knowledge about genes. He assumed that all pea plants were physically and genetically identical, hence why they should act the same; however, he assumed that the first 100 pea plants would act the same as the last 100. In this case, Mendel properly understood (and, hence, relied) on both the law of uniformity and on old theories to build further knowledge; he used deductive reasoning based on past events. If he had not taken this into account, the Genetics Law would not exist. Consequently, DNA and genetic factors would be unpredictable and there would be vague explanations about certain diseases involving alleles. Another perfect example is that of correlational research in the natural sciences. This is a historical approach to studying life, in which each event can stem from a natural cause. In this case, uniformities are used to explain the relationship between certain diseases and triggering factors. Notably, cigarette cancer is linked to approximately 90% of lung cancers in the United States. In this case, the correlation and causation scenario comes into play as doctors and patients alike are well aware of the effects of smoking; the patient knows that smoking might kill him due to his reasoning and language skills acquired by reading articles online which confirms what others around him say. Ultimately, it is safe to say that both reason and language are key determinants of new scientific knowledge and help the scientist form inductive conclusions.
From this, another point worth addressing is that of science being based on reasoning. Throughout the development of certain scientific theories, (which are generalizations created from many observations), inductive reasoning is used. For instance, the cell theory, stating that all living things are made of one or more cells, arises from various observations indicating a cellular basis for life. It becomes clear that science is also based on deductive reasoning. While formulating a hypothesis, certain well-supported generalizations (theories) are taken into account. In this case, the new idea cannot be justified in any way and its conclusions are therefore drawn by logical deductions. So, if a new organism is discovered, it would likely be assumed that it is composed of cells, based on the cell theory.
Generally, in order for new scientific discoveries to be made, a strong base of previous research and theories must be present to properly build upon certain topics. One must rely on other scientists’ findings while assuming certain universal, commonly-accepted theories to further discover information.
Uniformities, which are justified by reason, are needed to verify the validity of knowledge in mathematics, which is often based on the assumptions that we make. Certain mathematicians have formed the "basis" for mathematics: common math rules that are applied by everyone worldwide. For instance, it is common knowledge the 2+2=4, and this uniformity in knowledge is what allows further algebra and calculus to exist. A famous mathematician, Euclid, is known to have organized geometry into one area of knowledge. His model of reasoning allowed him to form the shared knowledge based on axioms, deductive reasoning and theorems. According to Lagemaat, axioms are self-evident truths that provide a foundation for mathematical knowledge. From this, Euclid formed assumptions that are accepted as uniformities worldwide. Using five axioms and deductive reasoning, Euclid came up with the following simple theorems: Lines perpendicular to the same line are parallel, two straight lines do not enclose an area, the sum of the angles of a triangle is 180 degrees and the angles on a straight line sum to 180 degrees. Euclid stated certain axioms and hoped that the rest of the population would follow them as well in order to discover new information and make proofs. Nowadays, these assumptions are commonly used because each individual uses his reasoning to conclude that Euclid was, indeed, right as no contradictions have arised.
A second example worth addressing is that of basic shapes. From a young age, students are taught how to properly draw circles and rectangles and calculate volumes (Vprism = Areabase · Heightprism), however, when looking around in nature, most solids will not have perfect shapes nor volumes that can be calculated using the formula above. Volumes of certain objects are often needed and therefore, it is assumed that the shapes are “perfect” In actual mathematics, actual circles will never be seen because nothing is perfectly circular in the universe. For instance, it is assumed that the planet is perfectly circular and doesn’t have a tilt. However, in order for this to be true, there wouldn’t be any seasons; the earth’s tilt is what leads to seasons. To simplify everyone’s lives, an assumption of certain shapes is made and is uniformly accepted throughout the world. Ultimately, while it is assumed that the earth is round to simplify society’s understanding, certain mathematicians have indeed acknowledged that earth isn’t perfectly round due to the season changes.
Some, however, argue that certain commonly accepted algebraic uniformities aren't reliable because the next number might disprove the uniformity and assumption. Reason, therefore, might not always justify the claim. Inductive reasoning is helpful but cannot give us certainty. Goldbach’s conjecture (in which every even number is the sum of two primes) is correct through hundreds, thousands, and even ten thousands. However, the next number could disprove a conjecture, even if it is proven millions of times. Some people therefore reason against the common knowledge and form their own personal knowledge based on their emotions and past experiences. For instance, as addressed in TOK, the curious Georg Friedrich Bernard Riemann came up with the idea to replace some of Euclid’s axioms with their opposites. In this case, Riemann’s axioms differed from Euclid’s as followed: A. Two points may determine more than one line (instead of axiom 1). B. All lines are finite in length but endless, like circles (instead of axiom 2); C. There are no parallel lines (instead of axiom 5).
Among the theorems that can be deduced from these axioms are: 1.All perpendiculars to a straight line meet at one point. 2.Two straight lines enclose an area. 3.The sum of the angles of any triangle is greater than 180 degrees.
Along these same lines, Gödel’s incompleteness Theorem states that we cannot prove that a formal mathematical system is free from contradiction.
To conclude, it is safe to say that in order for new information to be discovered, certain uniformities must be agreed upon by the rest of society; everyone must be on the same page. Without this, we will always wind up back at Point A and will be unable to advance further. In order to advance, one must use his reason and language to properly understand the solid foundation set by society.