There is a standard example people use when they want to show what bad scientific reasoning looks like:

The sun has risen every day so far. Therefore, the sun will rise tomorrow.

This is usually called induction. We observe many instances of the same event, notice a pattern, and extend that pattern into the next case. The sun rose yesterday. It rose the day before. It has risen every day we remember. So we conclude that it will rise tomorrow too.

A physicist will be suspicious of this. Not because the conclusion is false, but because the reasoning is weak. The fact that something has happened many times does not, by itself, explain why it must continue happening. Repetition is not explanation. A long list of observations is not yet a theory.

This is one reason induction has a bad reputation in physics. It can look like a shortcut from pattern to truth.

But this is not what mathematicians usually mean when they use induction in a proof.

In mathematics, induction is not the claim that something happened many times, so it will keep happening. Mathematical induction is closer to this: if a world of objects is generated by some rules, and if a property is preserved by those rules, then every object generated by those rules has that property.

The important question is not simply whether we have seen many cases. The important question is: what licenses the step from the cases we know to the cases we have not yet written down?

In the sun example, repetition alone is doing the work. The sun has risen many times, so we expect it to rise again. But repetition alone does not tell us what is going on. Physics wants a mechanism. It wants a model. The earth rotates, and the apparent rising of the sun is explained by that rotation. The prediction becomes stronger because it is not merely a record of past sunrises. It is connected to an explanation of why the phenomenon happens.

Mathematical induction is different. The inductive step is licensed by the way the objects are made.

For example, the natural numbers are generated from a starting point and a successor operation. Start with $1$. Then apply successor to get the next number, and keep going. If we prove that a property $P$ holds for $1$, and if we prove that whenever $P$ holds for $n$, it also holds for $n+1$, then $P$ holds for every natural number.

This is not guessing from repeated observation. We are not saying:

I checked $1$, $2$, $3$, $4$, and $5$, so the result is probably true for all numbers.

We are saying something stronger. We have covered the beginning, and we have covered the rule that makes the next object. Since every natural number is reached by repeatedly using that rule, the proof reaches every natural number.

The same idea appears in structural induction. If trees are generated by rules such as

\[T ::= \operatorname{Leaf} \mid \operatorname{Node}(T_1, T_2),\]

then to prove something about all trees, we prove it for leaves and prove that the node constructor preserves the property. The proof works because every tree is made by those rules.

I wrote more about this in Induction is about how things are made.

So the disagreement is partly a confusion of names. The induction physicists distrust is empirical induction: seeing a pattern and projecting it forward. The induction mathematicians use is proof over generated objects: showing that every way of making an object preserves the property we care about.

The physicist is right to reject repetition as a foundation for certainty. The mathematician is right to use induction when the domain has a generative structure and the proof follows that structure.

The word is the same. The justification is not.