Examine the logical weaknesses in modern pure mathematics through this 27-minute video lecture from the MathFoundations series. Explore five key problems affecting the field, including inconsistent rigor, problematic definitions, and reliance on axioms. Distinguish between problematic and non-problematic areas in mathematics, and compare applied and pure mathematics approaches. Analyze concepts that are clearly defined versus those that are not, and consider three consequences of these logical weaknesses. Conclude by reviewing four aims for addressing these issues in subsequent videos, inviting further discussion on these contentious yet crucial topics for the foundations of mathematics.