Maths: Conjectures and Theorems

1. What is the difference between a conjecture and a theorem?

A conjecture is equivalent to a hypothesis in natural sciences: it is an educated guess by mathematicians on a subject in math, and is open to debate. Conjectures can be disproven by others. Conjectures are not proven: once they are proven, they become theorems.

A theorem is a further step to a conjecture. It is a conjecture that is demonstrated and proven. In math, once something is proven, it is forever true. For example, the Pythagoras theorem, where the square of two sides of a right-angled triangle will always equal the square of the hypotenuse, will always be true no matter what is happening to the world. 

2. In the video Eduardo Saenz de Cabezon uses the example of people being surprised that folding a normal piece of paper 50 times, will reach a thickness as high as the sun. He challenges us to ‘do the math’ and see that he is correct. What do you think meant when he said that Maths dominates intuition and tames creativity? Do you agree with this?

What Eduardo Saenz de Cabezon was saying is that Mathematical logic can sometimes bring us really far to places that people have never dreamed of, especially when it draws us away from all other factors in reality that might contradict with a mathematical idea.

Implicitly, Cabezon is saying that our intuition is limited by reality, and this is quite true, because intuition is mostly based on one’s experience and extending it further with ideas. Math is extending reality: except unlike intuition, it brings our daily problems away from the context of reality and puts it in a field purely based on reason. Unlike intuition, the experiences supporting knowledge in Math is conscious. In the example, math takes our observation of folding paper and extends it by asking us what happens when we fold it 50 times?

But can it be said that the maths dominate intuition? Even though maths can be proven true, and that truth stays eternal, but math is confined to mainly its own area of knowledge. However, intuition is a way of knowing: it can be used in any area of knowledge. We might even find a new area of knowledge through intuition in the future!

Furthermore, there are parts of math that require intuition, so intuition might dominate math as some parts of math might not exist without it. This occurs especially in pattern recognition, where it just naturally comes to you.

3. Saenz de Cabezon claims that the truths in maths are eternal. Do you think this gives maths a privileged position in TOK?

This question basically aims to answer the question do eternal truths have a privileged position in TOK? In a way, this does make math have a privileged position as math is the only area of knowledge where truths are eternal. It is a special characteristic of knowledge that I believe should be regarded higher than knowledge in other AOKs.

Yet, there is the other side of the argument that math is only 1 area of knowledge out of the other 8. It cannot fully represent other AOKs. Even though the AOKs overlap, such as maths playing a huge role in the foundation of physics, but for example, math cannot explain the concept of why gravity exists. Math can be used to determine the magnitude of gravity on an object or explain how the magnitude got its number, but math cannot explain why there is the gravitational force between two objects: that knowledge rests in physics, or the natural science AOK.

4. List any of the knowledge questions related to maths that came out of your discussion in class.

  • What makes math the area of knowledge where something holds true forever once it is proven?
  • Was math invented or discovered?
  • How can one be sure that knowledge in math holds true forever?
  • Why is 1 + 1 = 2?
  • Can theorems be disproven?

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