Crash course in mathematics: Lesson 1 - Mathematical thinking

This lesson is about mathematical thinking. If it is the first lesson of this crash course you encounter, or you missed some lessons, please go back to the beginning and follow the course trough, by using the framework and index, at the end of the introduction

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Let’s say you are a student in some university in the US. It's the summer vacation and you want make some extra money, so you go and play music in the railway station. People go by and throw small change into your hat. At the end of the day, you take a whole bunch of dimes and go to the railway station’s cafeteria to buy a cup of coffee. The coffee costs four dollars and you want to pay it all in dimes. Now the question is, how do you count the coins?

Do you count, “Ten, twenty, thirty… one dollar, ten, twenty, thirty… two dollars…” and so on? Or do you count, “one, two, three … forty”?

If you think like a mathematician, you probably count the dimes in the second manner. You know you need forty dimes to pay four dollars because you made a calculation. You took few things that you know already and combined them together to reach a new conclusion. Of course, you may have done that some time ago, and now you simply remember that you have to pay forty dimes for your coffee, but it is easier for you to remember that because you understand the reasoning behind the fact, that if your coffee costs four dollars then you can pay that with forty dime coins. And even if you have a very poor memory, you can write it down and use the note to remind you, because as long as there will be ten dimes in a dollar, and the price of the cup of coffee won't change, it will remain true.

As we will see in this course, mathematical thinking is based on simply repeating this inference process again and again. We will see that all is needed, in order to explore even the most complex mathematical concepts and ideas, is a very simple skill set and the ability to use it repeatedly and consistently to solve problems. The genius of great mathematicians, like Gauss, Riemann or Nash, is that they can take a very difficult problem and see how it can be solved mathematically, in the most elegant way. But anyone can use mathematical thinking to solve everyday problems, in a way that saves allot of physical and mental effort.

You may think that mathematical thinking is only good for situations where you have to deal with numbers or geometry, etc. But that is not true. As we will see, mathematical thinking may be useful in a surprisingly large variety of cases, and most importantly, it can always be used for one thing at least, and that is to help you make assessments, and especially to assess how much you should be skeptic about ideas and conjectures. And this is what the next lesson is going to be about.