Summary, Terminology and Practice (DP IB Theory of Knowledge): Revision Note

Scope

Mathematical knowledge aims for generality by using abstraction and proof.
 
Mathematics can feel certain within its system, but application to the real world has limits.
 
Mathematics can extend knowledge by making relationships visible that are hard to see directly.

1) A mathematician proves a theorem about prime numbers that applies to all primes, including ones too large to check individually. The scope of the claim is universal because it is justified by deduction, not by testing many cases.
 

2) A scientist uses a mathematical model to predict the spread of a disease in a population. The results seem general and precise, but the scope is limited by assumptions about behaviour, measurement accuracy and what variables the model includes.

To what extent does mathematical generality depend on the definitions and axioms chosen?
 
How far can mathematical certainty be transferred to knowledge claims about the real world?
 

Does mathematics reveal relationships that exist independently, or does it create them through formalisation?

Perspectives

Different mathematical frameworks can lead to different valid conclusions about the same idea.
 
Mathematical work is shaped by human choices about what problems are worth solving.
 
Mathematical interpretation matters when applying abstract structures to real situations.

1) In Euclidean geometry, the angles in a triangle add to 180°, and this can be proven from the axioms. In non-Euclidean geometry, a different set of axioms changes what is true, showing that “truth” depends on the chosen system.
 
2) Two statisticians analyse the same set of exam scores, but one reports the mean while the other reports the median. Both methods are valid, but they create different impressions of “typical performance”, shaping how the knowledge claim is interpreted.

In what ways do different mathematical systems change what counts as a true claim?
 

How do social and practical priorities shape what is treated as important mathematical knowledge?
 
How does perspective influence which mathematical model is accepted as the best explanation?

Methods and tools

Proof is the central method for justifying mathematical knowledge claims.
 
Deductive reasoning produces certainty if premises and logic are valid.
 
Mathematical tools allow the representation and manipulation of abstract ideas.

1) A student notices a pattern from several examples and makes a conjecture that it always holds. The claim only becomes mathematical knowledge once a proof is produced, showing that evidence can guide discovery, but proof provides justification.
 
2) An economist uses regression software to model the relationship between unemployment and inflation. The tool produces a neat equation, but the knowledge claim depends on modelling choices like which variables are included and whether the assumptions of the method are met.

Why is proof treated as a stronger justification than evidence in mathematics?
 
What makes a deductive argument valid, and how can validity be checked?
 
How do mathematical tools and notation shape the kinds of knowledge claims that can be made?

Ethics

Mathematical outputs can appear objective, creating strong trust in numbers.
 
Ethical issues arise when mathematical results are communicated without uncertainty.
 
Ethical use of mathematics requires transparency about assumptions and limitations.

1) A news report uses a graph with a shortened axis to show that prices have risen sharply, even though the change is small. The mathematics may be correct, but the presentation makes the claim more persuasive than the evidence supports.

 
2) A company uses an algorithm to rank job applicants based on historical hiring data and calls the results objective because they are statistical. If the data reflects past bias, the mathematical method can reproduce unfair outcomes while appearing neutral and reliable.

Why do numerical claims often seem more trustworthy than qualitative ones?
 
What responsibilities do knowers have when presenting statistical claims to an audience?
 
How can mathematical knowledge be used responsibly when it affects real-world choices?

Proof

A conclusive deduction from axioms that leaves no room for doubt or argument 

Formal system

In mathematics, a system used to deduce theorems from axioms according to a set of logical rules 

Golden ratio

When a whole is divided so that the ratio of the whole to the big part is the same ratio as the big part to the small part. The ratio is 1:1.618

Logicism

The theory that mathematics can be derived from logic, without the need for any specifically mathematical concepts

Outlier

a value or datum very different from others