One out of every seven mathematicians is a philosopher, and one out of every nine philosophers is a mathematician. Are there more philosophers or mathematicians?
1. Let \( m \) represent the number of mathematicians.
2. Let \( p \) represent the number of philosophers.
3. According to the problem, one out of every seven mathematicians is a philosopher. This can be written as:
   \[
   \frac{m}{7} = \text{number of mathematicians who are philosophers}
   \]
4. Similarly, one out of every nine philosophers is a mathematician. This can be written as:
   \[
   \frac{p}{9} = \text{number of philosophers who are mathematicians}
   \]
5. Since the number of people who are both mathematicians and philosophers must be the same, we set the two expressions equal to each other:
   \[
   \frac{m}{7} = \frac{p}{9}
   \]
6. To find the relationship between \( m \) and \( p \), we solve for \( p \):
   \[
   \frac{m}{7} = \frac{p}{9} \implies 9m = 7p \implies p = \frac{9m}{7}
   \]
7. Since \( \frac{9m}{7} > m \), it follows that \( p > m \).

Conclusion:
There are more philosophers than mathematicians.

The final answer is \( \boxed{ p > m } \).