I think they make it normal and you can't spot a mathematician or a physicist out of a certain social group until they tell you who they are.
But to be precise enough, they are completely different and think far from normal when doing their specific job.
An easy-going engineer might assume that e = π =3. in emergent situations whereas a mathematician can only be satisfied with precision to the nearest 10th digit.
A mathematician would say that the yield of mathematical equations and reality or what we observe in nature may be two different things, but a faithful physicist insists that they should be one and the same.
We cannot judge man for what he is. We can only judge the results that flow from what he does.
I can just tell a funny story, a mathematician friend told me about the intertwining between math and physics:
Two friends, one a mathematician and the other a physicist, went to a conference and slept in the same room in a high-rise hotel.
Late at night, a fire broke out in their room and the mathematician was the first to wake up to turn on the water tap and write in his papers:
We have fire and water,
the solution exists,
then he went back to sleep.
When the fire escalated, the physicist woke up and began to plan and cooperate with his friend and the right tools to transfer water to extinguish the fire.
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