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Preferably, this book/set of resources would provide some history of the motivation that led to the development of the modern subject, how these motivations evolved with time, where the "state of the art" lies today, and what connections have been discovered (or tentatively proposed) between one subject and the others.

For example, the resource may provide a blurb on the development of the Chicago school of analysis. What motivated Anton Zygmund and his group early on? At the time, which fields in mathematics would their school have been associated with? How does their work connect with other subjects? What was the impact of their school? Which fields of mathematics is their school associated with today?

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This similar question has additional resources, aimed at graduate students / researchers... – Assad Ebrahim Feb 20 '14 at 11:53
up vote 6 down vote accepted

The Princeton Companion to Mathematics might be what you're looking for. It surveys an impressive number of fields in mathematics, including their historical and modern contexts, as well as describing basic definitions, theorems and open problems. And it gives mini-biographies of a whole load of important mathematicians. It's a very good reference.

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Wow, this does look exactly like what I was hoping for! Thank you! – user89 Jan 25 '14 at 6:50
@twirlobite: It's well worth the investment. I choked up the money to buy a copy a couple of years ago and never looked back. – Clive Newstead Jan 25 '14 at 6:51

It's unclear as to what exactly your question is, but if you want a comprehensive overview of modern mathematics, I'd suggest What is Mathematics? If you want a full history of all schools of modern mathematics down to developments at the individual level: good luck.

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Thank you, I will have a look at that resource. Do you have any suggestions for what clarifications I could make in order to improve my question? – user89 Jan 25 '14 at 6:49
The issue is not so much with your question as with what you're demanding. You are essentially asking for a commentary on mathematics or a book on the history of mathematics, which is a sparsely-populated genre, and most texts that fall into it are of questionable quality. Additionally, I'm afraid that your request may be far too broad: mathematics is a really big field, and a great number of people, even in modern history, have contributed significantly to it. A single book containing the information you request would easily span thousands of pages. – Newb Jan 25 '14 at 6:51

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