TOK food for thought - Mathematics
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Ms Ortiz


For your consideration: What do you think? Is Mathematics discovered or invented?

In Year 12, our foray into Mathematics as an Area of Knowledge often begins with discussing the true nature of mathematical knowledge and debating two commonly held views:

  • as Mathematics discovered (the Platonists’ view)
  • or it was invented (the Formalists’ view)

  Which view do you side with? See below for more info!

According to the Platonists’ view, mathematical objects are real, and their existence is objective and independent of our knowledge of them. They exist outside the space and time of physical existence, they were not created, and they will not change or disappear. 

Any meaningful question about a mathematical object has a definite answer, whether we are able to determine it or not. According to Platonism, a mathematician is an empirical scientist like a geologist; he cannot invent anything, because it is all there already. All he can do is discover.  

On the other hand, the Formalists define mathematics as a game of logical deduction: either we have a proof derived from logic or we have nothing.

And just as any logical statement starts with a premise, a mathematical proof must always start with “assumptions” or “axioms”. Where pure mathematics is concerned, the interpretations we give to the axioms are actually irrelevant. We’re concerned only with valid logical deductions from them.

We can’t state that a theorem is true, any more than one can assert that the axioms are true, the statements of mathematical theorems can be said to be arbitrary, depending on the rules we assign the ‘game’.

On the other hand, according to the Formalists, they are also free of any possible doubt or error, simply because the process of logical deduction and rigorous proof leaves no gaps or loopholes.  


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