Mathematics is the foundation of all engineering courses, because it provides an accurate, logical and abstract way of thinking, which is very important for understanding and solving engineering problems. The following are the main reasons:

1. Modeling: In engineering, we need to model real-world problems. This requires us to use mathematics to describe problems, such as algebraic equations to describe physical phenomena, or calculus to describe the behavior of the system.

2. Analysis: Mathematics provides a powerful tool, which can be used to analyze and understand complex systems. For example, we can use linear algebra to analyze circuits, or use differential equations to analyze fluid dynamics.

3. Optimization: In engineering, we often need to find the optimal solution. This requires us to apply the optimization theory in mathematics, such as linear programming or nonlinear optimization.

4. Prediction: Mathematics can help us predict the future. For example, we can use statistics and probability theory to predict the market demand of products, or use numerical simulation to predict the structural performance of buildings.

5. Verification: In engineering, we need to verify whether our design and solution are effective. This requires us to use mathematics for strict proof and verification.

Therefore, mathematics plays a vital role in both the design stage and the implementation stage. Without mathematics, we can't understand and solve engineering problems. This is why mathematics is the foundation of all engineering courses.