Some people study mathematics for its own sake: they find maths beautiful and they study maths to enjoy its beauty. They are like people who eat broccoli, not because it is healthy. They eat it just because they enjoy the taste.
But even if you are not a maths enthusiast you may still have reason to bother to study maths. First and foremost, mathematics is a means of understanding the world: "The book of nature," as Galileo famously said, "is written in the language of mathematics" and the key to a successful and meaningful life is understanding the world you live in.
You may wonder how on earth mathematics can be relevant to understanding the world. How you may wonder, mathematics can explain things? When we think about mathematical explanation, we should clearly distinguish it from explanation by means of a scientific law. True, scientific laws can best be expressed in the language of mathematics but when we explain things by means of a law, it is the law that explains not the maths. The maths works merely as a medium for the formulation of the law.
However, it happens at times that our understanding of a phenomenon is not complete unless we resort to a mathematical truth and here is where mathematics plays an explanatory role. Here is a nice example.
Hive-bee honeycombs have a hexagonal structure. Part of the explanation depends on evolutionary laws. Bees that use less wax and thus spend less energy have a better chance at evolving via natural selection. The explanation is completed, however, by pointing out the mathematical fact that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Thus, the hexagonal tiling is optimal with respect to dividing the plane into equal areas and minimizing the perimeter. The explanation of the biological fact seems to depend essentially on a mathematical fact!
But secondly, even if the maths you study has no direct relevance to your life and world, it is still worthwhile because it helps your mind develop problem-solving and thinking skills-skills that you need in dealing with the world and with other fellow human beings.
Mathematics, as practiced in our primary and high schools, is mostly, if not wholly, about following rules and applying them to particular problems. This, however, is not how mathematics was historically practised. Historically, mathematics was practised as both a pure and an applied science. In the pure realm, people were concerned with discovering mathematical truths. In the applied realm, people engaged themselves in the quite distinct activity of using mathematical truths as rules to solve particular problems. The two realms are distinct in the following sense: you may quite successfully apply a mathematical rule without ever understanding the mathematical truth that it represents.
The really existing educational systems around the world gravitate towards applied maths for obvious reasons: they are planned to serve the purposes of corporations by training people who can apply maths to problems in business and industry. There are reasons, however, to believe that our educational systems should not be like that. After all, the sole and primary purpose of education is not to train rule-followers. Its primary purpose should rather be to train creative thinkers who can discover new mathematical truths and challenge well-established rules. This is why pure maths should also be an integral part of mathematical education. Students should learn maths not only as a rule-following exercise, but also as an exercise in creative thinking for the purpose of discovering and understanding mathematical truths.
Mathematics is hard, true, but let it bring to you the joy of understanding and clear thinking and the hardship of learning it will go away!
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