What Could Be Meant by Generalization in Maths?
Insights for ToK Essay 2
The idea of generalization in AoK mathematics has certainly become more conspicuous since Theory of Knowledge (ToK) Essay 2 was published a couple of weeks ago. So today, we look at what could be meant by "generalisation” (I’m going to use the British spelling because I’m British) in maths? In essence, generalisation in this discipline involves applying mathematical knowledge, initially developed to solve a specific problem, to a broader range of issues. It could also involve understanding new cause-and-effect relationships using mathematical concepts, or principles, derived from previously studied cause-and-effect relationships.
In ToK Essay 2, the notion of generalisation can give rise to discussion about the nature and scope of mathematical knowledge. Does generalisation in maths lead to more 'true' or 'universal' forms of understanding, as opposed to the limited scope offered by specialisation? What are the ethical considerations, if any, when applying generalised mathematical models to real-world scenarios?
Generalisation also plays a role in understanding newly observed cause-and-effect relationships. For instance, the principles behind the spread of disease could be mathematically modelled, drawing from prior models related to diffusion or information dissemination. By leveraging existing mathematical frameworks, researchers can quickly make sense of new phenomena, making generalisation a powerful tool for both scientific inquiry and problem-solving.
As such, we start to see that generalisation in maths serves as a bridge between specific mathematical problems and broader applications. It allows mathematicians (and scientists amongst others) to extrapolate from known situations to solve new, unexplored problems. In ToK Essay 2 you can discuss the challenges and limitations of this approach alongside the value of generalisation in expanding the scope and applicability of mathematical knowledge. It not only aids in the advancement of the field but also enhances our understanding of the world through the universality of mathematics.
Find out more about this essay title in the overview discussion with Gareth Stevens.
This is just a start of the type of overview that you can find in our ToK Essay 2 Guidance Notes, in these notes we get into what the opposing demands might be for specialised and generalised knowledge, and how we could reconcile these demands. This year we have two versions of the notes:
The Foundation Notes fully unpack the title, explore different ways to approach the concepts in the title, and explain a number of knowledge arguments that could be used. These notes are 4,000-5,000 words.
The Complete Guide has all of the same content as The Foundation Notes, and in addition has fully explained real life examples to illustrate each knowledge argument.The Complete Guide also has evaluation points and implications for each knowledge argument. These notes are 8,000-11,000 words.
You can find essay guidance notes for all of the essay titles at this link.
Stay Toktastic my friends,
Daniel,
Bangkok, September 2023
Detailed guidance video for Essay #2 May 24
Initial overview thoughts with Gareth Stevens for Essay 2 May 24