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Next year in my 3rd year at uni I've got to do an individual project and I was thinking of doing it on the study of metamathematics or mathematical logic (I'm not sure quite what the difference is as I have no experience in this area). What got me thinking of doing this for a project was my curiosity about the justification of mathematical definitions/conventions - for example the definition of a complex integral, and conventions such as 0!=1. The things we define arise in calculations and are used in theory and proofs, so they must be sensibly defined, and I'd love to discover the reasoning behind defining them.

Does anybody have any recommendations for books or online lectures to get stared on? My relevant knowledge so far is a basic grounding in abstract maths such as introduction to groups, rings, integer arithmetic, etc.

Thanks

MattBurrows
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  • I think you're going to be disappointed if you're looking for answers to questions like that in formal logic. Definitions are just definitions. The motivations behind them varies, but investigating mathematical motivation is outside of the purview of the field of mathematical logic. Those would make good questions to ask here, though. – Jack M May 21 '14 at 10:25
  • Is there a name for the study of definitions and their reasons? Its always puzzled me how some things which arise naturally in maths and science are just defined - when something new arises in maths, lets say when complex integration was first 'thought of' (i guess) , how does the person who first defines it know that it won't conflict with some other mathematical concept, or contradict itself somehow, somewhere further down the line? I hope i'm making sense... – MattBurrows May 21 '14 at 15:24
  • Not really as a "study" in its own right, no, but then I'm not entirely sure what you're talking about. I think you should ask this as a separate question. – Jack M May 21 '14 at 15:43
  • Not much to sink your teeth into, I would think. Maybe you really want to do something on the history of mathematics -- how certain concepts evolved over the millennia. – Dan Christensen May 22 '14 at 14:12
  • Hilbert and Ackermann's Foundations of Mathematical Logic is an excellent introduction :) – Yes Sep 03 '14 at 00:54
  • @Jack M: Before A. Einstein published his first paper on electrodynamics, if he told someone else that he was about to investigate the notion of simultaneity then he probably would get negative feedback. We know the importance of simultaneity because Einstein successfully convinced his contemporary physicists. Are you sure you can be for Einstein's ideas when he was a member of staff in patent office? Are you sure you can be for G. Cantor's set theory when he was vehemently attacked by Kronecker, one of the then greatest mathematicians? – Yes Sep 03 '14 at 01:04

2 Answers2

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Online you have Wiki, starting from Mathematical logic, but it is not easy to "design" a consistent path through all the articles available.

You can use SEP : articles are longer with detailed biblio and there are pointr to other entries.

You can start with :

Classical Logic

Model Theory

First-order Model Theory

Logical Consequence

Tarski's Truth Definitions

Second-order and Higher-order Logic

Try to read them and use Bibliography and Related Entries.

Community
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Mauro ALLEGRANZA
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There are some interesting formal notions of both Model Theoretic and Proof Theoretic definability. If this is something that you are interested in, then please let me know.

I think understanding how people create mathematical definitions in a non-formal setting is likely even more valuable. Occasionally, definitions are the beautiful result of one specifying exactly what he/she wants. Other times, definitions are almost randomly created.

Michael Wehar
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