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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

Thanks

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Of course mathematicians care about what things mean! Do you think we just blindly manipulate symbols on a page? We just have a broader understanding of the word "meaning" than an engineer or physicist. –  Andy Putman Feb 7 '13 at 21:21
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The insitence on the vague notion "meaning" and others is in my opinion problematic. I am sorry but I have no idea what is the idea behind 'how the mathematical objects "consist of" and how they "work" instead of how they "mean" ' I do not vote to close, perhaps this resonantes with somebody and they can tell you something. But to me this is not a real question, and perhaps you can clarify via means of a simple example to highlight the difference of 'consist of', 'work', and 'mean'. –  quid Feb 7 '13 at 21:22
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first posted (and closed) at math.stackexchange.com/questions/297111/… –  Will Jagy Feb 7 '13 at 22:16
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What is the physical meaning of a derivative, to you? It could be the rate of change of something, if your function is measured against a time variable. It could be the slope of a tangent line, if your function is a graph of one variable. It could be the best linear hyperplane approximation, if your function has a vector of inputs. The nice part about using the derivative abstractly is that it could have any of those physical meanings, or many others, but it doesn't have to. You are free to use any reasonable physical interpretation, or none at all. –  Zack Wolske Feb 7 '13 at 22:20
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I really wish experienced users of MO wouldn't answer questions that are clearly wrong for MO. –  Todd Trimble Feb 8 '13 at 1:59
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closed as not constructive by Andy Putman, Will Jagy, Nik Weaver, Daniel Litt, Martin Brandenburg Feb 7 '13 at 21:47

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2 Answers

To me, it is a bit like asking the meaning of music; it does not have to be "applied", but may exist and be beautiful in it self.

Now, beautiful mathematics is usually simpler to understand, than ugly mathematics, so sometimes, it might be necessary to do beautiful, non-meaningful things, to simplify mathematics which has a meaning. This was sort of the story when the complex numbers were invented; these were strange, and did not have a meaning, but simplified the mathematics. Now, we cannot live without them.

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Actually what pure mathematicians typically lack are the frameworks of intuitions that are common in end-user mathematical fields. Calling such intuitions "meanings" of the mathematics is some sort of fallacy, really. It is the stuff that really is not coded into the mathematical theory that is hard to get "if you only know the theory". Mathematicians have their own sorts of intuitions, also, an obvious example being "higher dimensions".

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