I have asked this on SE before, But i DO hope that you kindly give me more insight about this question since it's very important to me
Having come from engineering background, I am kinda obsessed with the meaning with things, like the physical meaning of derivatives or the meaning of the nonlinear growth of functions.
but recently i fell in love with the beauty of the mathematical patterns, and i decided to do a master degree in pure mathematics. hopefully in analysis. I recently appreciated more the value of rigour and the structure of arguments.
I had that conversation with one of my great professors that has an applied mathematics background and his opinion was that pure mathematicians don't usually understand the "meaning" of the objects they're working with and they so often work with objects that don't exist in the physical world. I didn't totally agreed with that but i wondered.
still when i study i use my old way of thinking and wonder all the time about the meaning.
Should i instead pay more attention to understand how the mathematical objects "consist of" and how they "work" instead of how they "mean"? and use mathematical objects to build up new ones? and care more about the patterns and how i can build new ones from the existing ones?
In other words, how far does the meaning of mathematical objects matter in doing research in pure mathematics? Is it more important to use mathematical objects and understand their mathematical structure and use them to develop new ideas?
I have no problem with that, i would enjoy mathematics more if i save this amount of energy spent in visualizing things and trying to understand how they physically mean. just wanted to make sure