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What the hell is mathematics anyway?

Education methods

Last week I mentioned how one friend’s response to “what is an education?” included that you should be able to “do maths”. It got me thinking, what does that even mean to “do maths”? I mean, what is maths anyway?

What is maths counting numbers home made math blog

A few days later, I was researching some quotes about mathematics when I stumbled upon this quote by Georg Cantor:

“The essence of mathematics lies in its freedom.”

Who was this man? And why should I be interested in his opinion on mathematics? After all you can quote anyone on the internet! So I went looking for information on Georg Cantor.

Turns out he was a pure mathematician, in the 19th Century, who did a lot of work looking at infinity. Infinity is a concept that you and I take for granted, but it actually has a strong influence on our philosophy and spirituality… the existence of our concept of infinity plays a role in our understanding of God, of the Universe, and of time… and equally our belief in the converse “zero” also plays a role. If you think about it, zero is a human construct; is there ever actually nothing? If zero is just an arbitrary point, then our question of “what happened before…creation/the big bang/the universe?” (whatever your belief system) becomes nonsensical, because zero does not exist. Do you see where I’m going here my friends? We have actually just started talking about pure mathematics… we are “doing maths” by posing these questions.

Let’s go back a little bit.

There are 2 types of mathematics. The line between them blurs, but if we are going to define them…

Pure mathematics is the theory of mathematics. Pure mathematicians work in the abstract, using mathematics to prove theorems.

Applied mathematics is the application of those theories to solve real world problems.

(Mathematicians argue over these definitions but this was the simplest, clear way I can define them, see this article and here for more info)

For the most part, in schools, and most homeschool mathematics courses, we use applied mathematics, and for good reason. Applied mathematics is very useful in our day to day. It helps us determine if a bridge will stand, give accurate medicinal dosages, record data, and successfully operate in our capitalist society. 

But mathematics is actually more than just a useful operational tool. Let’s go back to my mate Georg Cantor… ok further back, let’s consider Galileo.

In the 17th Century Galileo pointed out that a circle must be made up of infinite points, but if you had a larger circle that circle would have a larger number of infinite points…. hmmm a larger infinity, does that even work? (And does it really matter?) Galileo settled by saying that terms such as equal to, greater than and less than could only be applied to finite number sets (groups of numbers that could be counted). Cantor was not happy with this solution, he really wanted to get to the bottom of what appeared to be different sizes of infinity. Using mathematical theory and rational extrapolation (which you can read all about here) “he showed that there may be infinitely many sets of infinite numbers - an infinity of infinities - some bigger than others, a concept which clearly has philosophical, as well as just mathematical, significance.”(source) He showed that this philosophical idea of infinite infinities actually can be proved, calculated, i.e. it makes sense, mathematically.

Now, I’m going to say this:

If we believe that our children's Spiritual development is equally as valuable as their “success” in the capitalist society, then why are we not exposing them to more pure mathematics? 

Consider this description of a Waldorf mathematics lesson from rootparenting:

“Waldorf starts off the introduction to math by asking a seemingly simple question, “What is the largest number in the universe?”. My son (aged 5) came home from school and asked me the same question. I answered “Well, erhmmm, it’s infinity.”. He said “No, one is the biggest because I am one.”. Other responses discussed in class are “One is the biggest because without it there isn’t any 2, or 3, or even a million.” “One is the biggest because everything there is is in one Universe.” “One is the biggest because it can be any number it wants.” All sorts of philosophical and mathematical truths become evident through just this “one” discussion. This gets them thinking in a whole new way about numbers, and how they relate to us and the world. Eventually the children arrive at “I am one!”, they see how their bodies are shaped like the number one, they relate themselves to the vastness of the Universe, and realize at that point that they are co-creators.”

If this all feels a bit hippie and overwhelming, how about starting with the simple premise that we can question mathematics.

In the home made math "Exploring Number in Charlotte's Web" unit, there is an activity on negative numbers inspired by the quote “what do you mean less than nothing?”. I was a bit nervous about including it in the unit for younger children at first. Now that I’ve been thinking about pure mathematics I feel justified. Let’s expose our kids to these ideas, let’s encourage them to think outside the applications of mathematics, to the implications of mathematical statements.

“In mathematics the art of proposing a question must be held of higher value than solving it.”
Georg Cantor

Xx Becky

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