Opinion – Marcelo Viana: Is mathematics deductive or inductive?

Philosophers consider two main types of reasoning: deduction and induction. The two can be compared as follows (very simplified!).

The deduction starts from general statements, the premises, and, through logical rules, arrives at more specific statements, called conclusions. A simple example, attributed to Aristotle (384 BC-322 BC): Every man is mortal. Socrates is a man. Therefore, Socrates is mortal”.

Induction goes in the opposite direction, starting from particular cases to arrive at general rules. For example, noting that known living things are made up of cells, we assume that the same holds true for all living things.

Induction is the foundation of experimental science. But it can fail: knowing that the chickens in my backyard have red plumage does not guarantee that every chicken in the world is red. The deduction, on the other hand, is rigorous: if the premises are true, the conclusion is necessarily true. But that’s because this one is contained in those, not bringing new information.

And the math, how is it in this discussion?

Mathematical reasoning is deductive in nature: it starts from certain statements, the axioms, and, through well-defined logical steps, arrives at new statements, called theorems. That’s what makes math a rigorous science.

But mathematical reasoning also has a remarkable ability to produce new knowledge. Fermat’s theorem states that the equation xⁿ+ andⁿ= zⁿ it has no positive integer x, y, z solutions if the exponent n is greater than 2. It is deduced from the axioms of algebra, but it is certainly not contained in those axioms. It’s as if the theorem was created (or discovered?) in the act of deduction.

What is special about mathematical reasoning that makes it both rigorous, like classical deduction, and creative, like experimental induction?

The answer lies in mathematical induction, the principle that if a given statement is true for the number 1, and if being true for a given integer N entails that it is also true for N+1, then that statement is true for all positive integers.

This remarkable law, which has no counterpart in experimental science, allows mathematics to pass from finite to infinity in a way that is both fruitful and rigorous. I will comment next week on its origins.