Physicists do study mathematics rigorously, but their approach to mathematics differs from that of mathematicians. Physics is primarily concerned with understanding the physical world and developing mathematical models to describe and predict its behavior. Therefore, physicists often focus on using mathematics as a practical tool to solve specific problems rather than exploring mathematical concepts for their own sake.
While physicists may not delve into the depths of pure mathematics in the same way as mathematicians, they still require a strong mathematical foundation to formulate and solve problems in physics. Physics relies on mathematical techniques such as calculus, differential equations, linear algebra, and statistics. Physicists need to understand these mathematical tools and their applications thoroughly to address complex physical phenomena.
That being said, collaborations between physicists and mathematicians are not uncommon. Physicists may work closely with mathematicians when they encounter mathematical challenges beyond their expertise. Mathematicians can provide insights, develop new mathematical techniques, and prove theorems that help physicists advance their understanding of the physical world.
As for making breakthroughs, while a deep understanding of mathematics is crucial in many areas of physics, it is not the sole factor. Creativity, intuition, experimental observations, and interdisciplinary thinking also play significant roles. Breakthroughs can come from exploring new experimental techniques, formulating novel hypotheses, or making unexpected connections between different areas of physics. While a strong mathematical background is necessary, breakthroughs in physics often require a combination of mathematical expertise and innovative thinking.