Knowledge and making it easier to apply to related

Knowledge are the justified belief, skills or information acquired through experiments, processing data and other related evidence. Knowledge can be personal or shared and doesn’t necessarily have to be correct, as the evidence acquired can be false. Hence, most individuals carefully assess knowledge they obtain by looking at it from different perspectives. According to Goethe, through acquiring more knowledge doubt, the lack of certainty, in that topic increases. However, can this also be applied to mathematics and natural science?Confidence in this case is the belief of certainty in a specific fact or piece of information. Little knowledge refers to having a limited set of evidence or facts to evaluate available, therefore making their assessment fairly easy as they encompass only one specific perspective. The title tries to suggest that the more knowledge we have available to assess, the more perspectives we need to take into consideration. Thus, our initial perspective may seem odd and we doubt it. Sometimes we may even find the initial knowledge that we have to be useless or false. However, I disagree with the title.In natural science, new or additional knowledge arises through paradigm shifts, falsifications or simply the suggestion of new theories in par with data from observations and experiments. The theories and shifts in knowledge usually come with a new set of evidence. The more we have available, or the more conclusions we can draw from a theory, the better we perceive it to be, increasing our knowledge and confidence in the subject and making it easier to apply to related situations. An example for this would be science education in high schools. Students are taught Bohr’s, Dalton’s and Rutherford’s atomic models of the electron configuration. These models arose from one another, Bohr’s being the latest. However, through new research, those models were deemed inaccurate, as electrons never stay in a particular pattern around the nucleus, but can be expected in orbitals around the nucleus (Murphy, Horner). This shows that students are taught the basics, starting with easy comprehendible scientific models, in order to understand the thinking involved behind them. They are a prerequisite that needs to be understood before learning the orbital model, as their straightforward nature allows to visualize the concept.  Furthermore, those prior models underwent shifts in knowledge to arrive to a newer complex model which is understood to be the universal truth, so teaching previous models allows to visualize the historical development in scientific knowledge. Having all this information and evidence strengthens the student’s knowledge of the concept of electron configuration, and students are more confident in explaining ideas which consequently arise. Following experiments and investigations will deepen this understanding and more observations can be included based on this prior knowledge learned from facts. The confidence is backed up by gaining knowledge in a correct way, by explaining easier topics first and then going deeper, instead of limiting knowledge by only providing the complicated facts. Yet, for each piece of information we evaluate its strength by looking at the existing evidence or beliefs this information implies. As such, we can have doubt in newly arising knowledge or information, as we are not confident in the evidence available. We may feel uncomfortable with the new set of beliefs we have to incorporate into our knowing. Additionally, even though it tries to be more reliable, new evidence from experiments comes with more room for uncertainty and error, increasing disbelief that it is accurate enough for quantitative or qualitative explanation and further prediction. This phenomenon can be described by natural science, where newly arising climate models are at their limit. As they become more modern and try to be more accurate through increased precision and larger database, they also come with a greater room for uncertainty. Scientists found out that making calculations more realistic and accurate had a negative effect on predictions for the future. This can be attributed to the fact that errors which used to cancel before no longer cancel, an increase in outliers or overfitting the data, meaning that the data is so accurately spread out that a slight change in distribution will completely fail its predictive power. Climate scientists therefore claim that sometimes more simple and crude models are more useful to provide knowledge rather than ones trying to encompass multiple observations at once (Jogalekar).Because mathematics are an area of knowledge based entirely on reasoning and logic, it seems to be certain and timeless. A statement in mathematics is only regarded as true if it is fully supported by quantifiable evidence, therefore confidence is limitless as there is full proof and support for a theory.As John Polkinghorne said, “Mathematics is the abstract key which turns the lock of the universe”. It seeks to explain concepts which shape reality, therefore absolutely increasing our knowledge in a certain area or subject. Mathematics such as fractions, numbers or accurate measurements quantify observations, and therefore increase confidence in a theory or hypothesis by either supporting or disproving it. An example would be the discovery of the number zero. When looking at past mathematicians, their intent of describing a phenomenon led to a complete transformation of the world by increasing understanding of concepts and ideas that shape reality . For example, number systems had been used ever since the existence of the homo sapiens. Babylonians used number systems that led up to the number 60. Sumerians adopted a system which used spaces in order to quantify things. However, it wasn’t until the seventh century A.D. when an Indian mathematician showed the number zero, or ‘sunya’. The number had technically always existed, but it wasn’t known before. Mathematics was a concept shaped around the number zero, and there was room for doubt or wondering how to quantify the word “nothing”, and whether there is such thing. Its use spread from India, to the Middle East to Europe, and has been the foundation of every single mathematical formula ever since (Guha).The description of such a small number led to infinite understanding of the world and transformed the way mathematics has been used since then. It transformed our quantitative thinking and strengthened our confidence in applying numbers and numerical systems.Investigating mathematics from another perspective, individuals may show doubt in the statement that mathematics is always certain, and more of this certain knowledge improves understanding of the concepts. Claiming so implies the generalization that more knowledge in it increases confidence in mathematics for everybody. However, this can be disproved by the IB syllabus. For students who show good understanding of mathematics and can apply it to real life situations easily, classes such as Higher Level Mathematics or Further Mathematics have been created. These concepts seek to find the origin of a mathematical formula and how it has shifted understanding of the concept at hand. These classes provide students with a greater set of mathematical knowledge, and seek to challenge the thinking involved. However, for less confident students who get confused by complex mathematics a class called Mathematical Studies has been created. It deals with the basic definitions and formulas of a mathematical area but does not go deep into its background. An example would be studying the topic trigonometry. In Higher Level Mathematics, students study trigonometry also taking into consideration inverse trigonometric functions or compound angle identities. Comparatively, mathematics SL covers most of the trigonometry syllabus, omitting the previously mentioned components, and mathematical studies only covers basic trigonometry such as the angle rules and trigonometric identities. The understanding of maths is reflected in a student’s grades. Some students in my school had to drop down one mathematical level because they were risking failing conditions in their course. The depth and variety of information in these harder classes were more difficult to process, and therefore students were struggling with understanding. However, after commencing a lower level course with less in depth mathematical analysis, their grades went up and they were able to pass. This shows that more knowledge may increase confusion and therefore doubt in a mathematical system or formula.To sum up, the statement that more knowledge increases doubt cannot be applied fully to mathematics or science. An increase in knowledge in these two areas increases confidence in the subject as theories can be interconnected and formulas or other tools can be applied to describe or interpret a phenomenon. Limitation or lack of knowledge leaves room for doubt or uncertainty and the desire to study more.Although knowledge in these two AoKs has to be supported by evidence to be accepted as the general truth, there are limitations to both mathematical and scientific knowledge. Specifically in science, some knowledge comes with a level of arbitrary acceptance, as the AoK cannot be the universal truth but is rather an approximation of truth. At some point, it is impossible to gather even more evidence through observation or reasoning and therefore a theory needs to be regarded as “true”. Therefore, individuals must see if they can apply those approximate results to other observed phenomena and, if it is still numerically and reasonably valuable, they can include this new observation or knowledge into one larger part of the understanding of reality. In some, this arbitrary acceptance may increase confidence, in others, this may increase doubt by wondering “what if there is more evidence that we just have not found yet?”.Additionally, there will always be individuals doubting scientific theories, no matter how much evidence or knowledge they are presented. To some extent, doubt and confidence are very subjective, vary across the board and therefore for some people increasing knowledge may deepen understanding, while for others confusion arises.

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