Yes, mathematics is useful to a variety of people without knowing the proofs as it helps in problem-solving, decision-making, analysis, and critical thinking.
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Mathematics has always been regarded as an instrumental tool in various fields of life, such as science, engineering, economics, and finance, to name a few. From the simple calculation of everyday life to advanced scientific discoveries, math is everywhere. Fortunately, using algebra, geometry, calculus, and other mathematical concepts do not necessarily require one to know all the nitty-gritty details about every theorem and proof from their respective books. People who do not know how to prove theorems, still find math very pertinent to their life, passions, and career.
In fact, using math to solve problems, analyze situations, and make informed decisions is the crucial aspect, mastering the proof is merely a part of the process. As scientifically expressed by Carl Friedrich Gauss, “Mathematics is the queen of sciences and arithmetic the queen of mathematics.” In essence, mathematics is not only about finding the correct answer to a problem but also the process used to reach that solution is crucial. Hence, regardless of one’s mathematical proficiency, knowing the fundamental structure and foundation of math is of utmost importance.
Interesting facts about the utility of math without proofs include the following:
Despite being a theoretical physicist, Albert Einstein was also skilled in mathematics and has contributed remarkably to calculus, geometry, and algebra.
Math is very much relevant to art as well. Fractals, for instance, are known for the beautiful design patterns that are an extension to the concept of self-similar geometric shapes with intricate structures.
The International Mathematical Olympiad is arguably the most famous math competition globally and is held annually. It is sponsored by the International Mathematical Union and is considered to be the most prestigious academic competition for high school students.
Apple co-founder Steve Jobs once said, “I think everybody in this country should learn how to program a computer because it teaches you how to think.” This holds true not only for programming but also for math.
Mathematics has applications across environments such as medicine, finance, architecture, sports, and other fields apart from science and engineering.
Here’s an example of a table demonstrating the utility of math without proofs:
|Field||Utility of Math|
|Engineering||Engineers use math to design, test, develop machines, and other equipment.|
|Business and finance||Math is critical in understanding and analyzing data, making informed decisions, and performing financial analysis.|
|Art||Fractals, geometry, and other mathematical concepts are the fundamental structures of modern art.|
|Sports||Statistical modeling and analysis, geometry, and calculus are essential in sports, analyzing performance, evaluating risk, and making strategies.|
|Medicine||Medical image processing, algorithm design, and statistical analysis all have their basis in math.|
In Conclusion, Mathematics is a versatile tool used to solve real-world problems. The ability to prove a theorem is not the only parameter to gauge mathematical proficiency. In addition to memorizing theorems, understanding the logic, and reasoning ability is also essential. As Richard Feynman, a Nobel Prize-winning theoretical physicist, rightly stated, “I learned very early the difference between knowing the name of something and knowing something.”
See a video about the subject.
In the YouTube video “Anyone Can Be a Math Person Once They Know the Best Learning Techniques | Po-Shen Loh | Big Think”, Po-Shen Loh argues that anyone can understand mathematics if they focus on the principles of reasoning and learn at their own pace. He believes that this would make mathematics the easiest subject to understand.
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Proofs are important because proofs are just understanding how we know that something is true. This is what mathematics is all about!
What if all you care about is using the results of mathematics: you don’t particularly care about why it works, just how to use it. Should you still care about proofs?
I am teaching infinite series in calculus at the moment, so I will use an example from this subject to illustrate some reasons why.
One result which is important in the study of infinite series is that if a and r are two real numbers then:
a+ar+ar2+ar3+ar4+⋯=a1−r if −1<r<1.