Paper Sample: Intuition in Maths and the Role That It Plays in This Field

Intuition can be perceived as an important element about the decision-making process that either agrees to or neglects an underlying concept CITATION Leo99 \l 1033 (Burton). However, the fact that most people doubt the aspect of intuition in mathematics is not peculiar. Mathematics is believed to be founded on facts. Thus nothing can go wrong. Nonetheless, the application of intuition aspect cuts across all fields CITATION Joh61 \l 1033 (Kemeny). The aspect can be applied in most if not all domains and fields; including mathematics. Consequently, it is imperative to question the extent in which mathematical intuition differs from mathematical proof or calculation.

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Real Life Situation

According to the birthday paradox, as explained in the Business Insider by Sara Silverstein, two out of twenty-three people in a room share a birthday CITATION Sar14 \l 1033 (Silverstein). In an analytical perspective, assuming that a birthday is 12th March, the probability of two people in the same room (a room with 23 people) to share a birthday is 0.27, with the probability of any of the 22 people sharing the birthday being 6%. Nonetheless, there are 253 pairs among the 23 people. Therefore, the likelihood of getting one match becomes 51% which is a better figure than 50%. Silverstein also uses the same principle in a room with 118 people whereby there is a probability of two individuals having the same last four digits in their social security number. The principle is that the math found in the birthday paradox can be used to explain unusual coincidences.

Main Knowledge question

The birthday paradox represents some of the situations in real life that use mathematical intuition. Therefore, to understand the concept better, one of the primary questions to ponder upon is; to what extent does mathematical intuition differ from mathematical proof/calculation?

The link between Real Life Situation and the Main Knowledge Question

The main knowledge question analyzes the extent to which mathematical intuition is different from the actual calculation. For the intuition aspect, one can easily pinpoint a problem given their extensive experience in a particular field. Taking a gander at the paradox, the aspect of probability is introduced whereby by applying certain principles; one can easily deduce that a certain figure exists in a particular situation. For instance, Silverstein explains that two individuals in a room of twenty-three people have 0.27 chances of sharing a birthday. Therefore, synonymous to intuition, when one is in a crowd, the paradox may be applied with regards to a particular assumption. However, calculations have to be effected to prove that the stipulated assumption is true. Possibly, when other figures are incorporated, there might be a slight difference in the answers and therefore showing a variation between the intuition and the calculations. In synopsis, the RLS fits the MKQ as one would first use intuition but after comprehension of the real answer mathematical proof becomes imperative. Successively, the Birthday Paradox is a whole mathematical counterintuitive problem.

MKQ Breakdown

Succinctly, there is more to mathematics than calculations and evidence. Mathematics, therefore, encompasses a blend of natural intuitive aspects, formal aspects, as well as precise foundations in underlying fields. The transitions and interdependence of these key components can often result into contentious debates. Intuition developed through experience in a field, coupled up with critical thinking, helps mathematicians to acquire a precise discipline that can prove immensely valuable in mitigating common errors and misconceptions. Our underlying presentation to a thought shapes our intuition. Furthermore, our intuition affects the amount we appreciate a subject.

The MKQ focuses on the extent to which mathematical intuition differs from mathematical proof. Therefore, the analysis brings into perspective the relationship between mathematical intuition and mathematical proof. The relationship is often depicted when proving mathematical theorems whereby both intuition and formal manipulation have to be employed. Intuition, in this case, entails analyzing whether a theorem is reasonable such that facts are kept together. The difference occurs when, even though intuition shows that a theory is true, it 's hard to show how the final position was obtained.

Suggestion

A plausible example is using the prime number theorem whereby, prime numbers that lie in the range of {[N, N+\ D]} consist of odd numbers that are randomly selected such that they have the required density. The prime numbers found in the interval for {\ D} is approximately {\ D /\ln N}. The difference emanated in proving the theorem or rather the intuition by the actual calculation.

First MKQ Breakdown

John Von Neumann stated that In mathematics, you do not understand things. You just get used to them.

What is intuition in maths?

Intuition in mathematics entails assessing whether a theorem is reasonable such that its components can be easily figured out. The lemma example illustrated in the backdrop above depicts the real analysis expert who managed to point out flaws in a new theorem, owing to his experience. In a more practical perspective, when a series that contains only prime numbers and one is asked to fill the remaining gaps, the knowledge that one has on prime numbers makes it easier to predict the next numbers. Another example relates to geometry whereby n-dimensional figures have distinct properties with regards to angles and sides such that m-dimensional figures cannot exhibit similar features. Additionally, in an isosceles triangle, assuming that one of its sides 450, because of the intuitive aspect that isosceles triangles have two equal sides and two equal angles, it means that one angle has 450, while the other one 900 in order to add up to 1800 (all angles in a triangle add up to 1800).

What is mathematical proof?

The concept of mathematical proof primarily endeavors to provide succinct and thorough evidence. As a result, the proof ultimately affects the comprehension of something is it a theorem, an idea, or an assertion with complete assurance. Mathematical proof is founded on the degree to which complete mathematical certainty may be attained. Despite the fact that mathematics encompasses an array of geometrical clarifications, intuitive comprehension is imperative. All proofs and first rule closures work to legitimize just on the premise of natural comprehension and intuition.

Henri Poincare (1854-1912) asserted that successful mathematicians are guided as much by the intuition of magnificence as by mechanical estimation.

Cultural Viewpoints

Mathematics as a comprehensive domain has had a huge influence in the improvement and cultural development over the past as well as in the present day. For instance, historic cultural pillars such as the pyramids in Egypt were arguably constructed by mathematically-conscious architects. Correspondingly, mathematics can be traced back in history through ancient Chinese and Indian cultures. Although mathematical proofs may vary across different cultural perspectives, its place and position in these cultural viewpoints are undeniable.

Another RLS: The Secretary Problem

The secretary problem is widely adopted in applied probability and statistics as it utilizes the optimal stopping theory. Ideally, the secretary is founded on the constructs of envisioning a manager seeking the best secretary from a group of n interested candidates. These n candidates are therefore screened randomly, with a decision being arrived at after every screening process; it is the end of the road for rejected candidate as their status cannot be reviewed in future. As the manager successively ranks these candidates after every interview, he does so without the knowledge of the competence of the pending candidates. The underlying principle is the ideal technique to boost the likelihood of selecting the best candidate. The theorem recommends continually dismissing the initial n/e candidates after the screening process; where e is the base of the natural logarithm and settling on the next candidate who proves to be better. In synopsis, the problem chooses the best secretary 37% of the time.

Conclusion

The backdrop above defines intuition in maths and outlines the important role that it plays in this field. Be that as it may, numerous mathematicians and philosophers often conflict on the concept of intuition with reason and logic, a viewpoint that has progressively found its way into the present day mathematical intuition conceptualization. Intuition is primarily the process of comprehending proofs and reasoning CITATION JHa54 \l 1033 (Hadamard). Progressively, intuition is embedded in various mathematical concepts, forming a common tool that individuals can dependably employ to draw to a conclusion. Intuition also forms a dependable decision-making construct. It is the plausible gut feeling. The real life situation illustrated above therefore supports the viewpoint that reason and logic are more effective in mathematics. Subsequently, mathematical reasoning endeavors to establish underlying reason in maths, bearing it in mind that intuition is the chief driver that supports reason. Thus, mathematics is a comprehensive domain that incorporates intuition, logic, and discovery. Ultimately, the backdrop provides sufficient basis to draw the conclusion that there are solutions to math. The conventional concept that there are no solutions to math is fallacious. Every mathematical equation is calculated on the fundamental assumption that there is a solution, but in fact, after calculation, there may be no solution. By so doing, the differences and similarities between reasoning and intuition become apparent. Finally, intuition not only saves time but also helps individuals to arrive at a solution to prevailing problems.

Works Cited

BIBLIOGRAPHY Alibert, Daniel and Thomas Michael. "Research on Mathematical Proof." Advanced Mathematical Thinking (2002): 215-230. Web.

Burton, Leone. "Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education?" For the Learning of Mathematics (1999): 27-32. Print.

Hadamard, J. An essay on the psychology of invention in the mathematical field. New York: Dover Publishing Inc, 1954. Print.

Kemeny, John. "Rigor vs. intuition in mathematics." The Mathematics Teacher (1961): 66-74. Web.

MacDonald, I. D. "Insight and intuition in mathematics." Educational Studies in Mathematics (1978): 411420. Web.

Silverstein, Sara. 2 Out Of 23 People In A Room Have The Same Birthday Here's Why. 31 October 2014. Web. 02 October 2016.