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Unfortunately my background's in engineering, so we've only been taught bits and pieces of math needed to be fluent in the science, but as I've started studying abstract algebra and real analysis, I've had a hard time seeing the relationships between the fields. For example, when I had to teach a survey course in physics, it was always useful to think in terms of fundamental forces and fundamental laws (conservation this and that) as a cornerstone despite whatever topic I was covering. But then I stumbled across a quote in a lecture on category theory that said "category theory is the abstract algebra of functions'.

So my question is,

What are some unifying concepts in mathematics that's always useful to keep in mind? Or is each field just a disparate branch built up from set theory and logic and should be thought of as entirely independent?

Thanks for insight


So I've been asked to reword the question because it was too broad. Please keep in mind I'm approaching from a natural science point of view, whereby abstract algebra and real analysis seem as disparate as chemistry and biology, which are treated as distinct disciplines in academia, but the former are all lumped together in one discipline called 'Math'. It's extremely easy to point to relevant foundational theory for any discipline, so I'm having a hard time understanding why my question is 'too broad' (unless it says something about the nature of math that needs to be posed in a philosophy forum?)

Here goes:

Have there been any concepts you found to be useful to keep in mind while studying any branch of mathematics, irrespective of the field? This must exist right, for a subject to be categorized under 'Mathematics'?

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closed as too broad by AlexR, Olivier B├ęgassat, Yiorgos S. Smyrlis, Your Ad Here, user86418 Feb 8 at 13:46

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

Perhaps you may find this MO topic interesting/related. –  Karl Kronenfeld Feb 8 at 11:54
Community wiki? –  Thomas Klimpel Feb 8 at 12:27
If I were to write an answer, it would go something like this: Mathematics is as much about teaching and communication of knowledge than about the knowledge itself. Mathematics should not to be confused with an IQ test. The aspect of creation of new domain specific languages should not be underestimated. Also take into account that the greeks used sketches as an integral part of the communication, and that mathematics might once again decide to use additional media as part of the language, in addition to pure words. Take the commutative diagrams of category theory as an example. –  Thomas Klimpel Feb 8 at 12:38