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Hi all,

I am looking to do some linguistic analysis of informal proofs. Therefore I am on a search for a collection of entry level proofs written in a clear, uninvolved style. I have one recommendation for Hardy and Wright's "An Introduction to the Theory of Numbers," and was wondering if there is something else you may add to this.

Many thanks in advance, Nickolay

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This question should be community wiki in my opinion. –  Grétar Amazeen Oct 21 '10 at 16:34
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9 Answers

up vote 2 down vote accepted

You can try Aigner and Ziegler's book Proofs from the book

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I think you have to determine some categories, as like as number theory, combinatorics, geometry and etc. But I think this book is so interesting:

"Ingenuity in Mathematics" by Ross Hansberger

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M. Ganesalingam did a pretty interesting dissertation about linguistic analysis of math texts. I thought I read about it here on MO but I can't seem to find the pointer right now. Anyway, see:

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A very different class of examples are to be found at the Naproche project:

About Naproche: The Naproche project (Natural language Proof Checking) studies the semi-formal language of mathematics from a linguistic, philosophical and mathematical perspective. A central goal of Naproche is to develop a controlled natural language (CNL) for mathematical texts and adapted proof checking software which checks texts written in the CNL for syntactical and mathematical correctness.

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Here is a book that examines the structure of proofs in detail:

The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs

by Antonella Cupillari

It contains examples of entry level proofs of various types.

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The classic "Foundations of Analysis" by Edmund Landau. It pedantically and very carefully derives elementary properties of integers, rationals, etc., from Peano axioms. Or is it too formal?

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Beweis: klar. :-) –  Felipe Voloch Oct 22 '10 at 19:06
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Truly "entry level" ... How to Read and Do Proofs by D. Solow.

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Hello,

I cannot add a comment, so I must have to ask this as an answer. What exactly does linguistic analysis consist of? Are you really going to study the wording used, the vocabulary, the ontology, and the structuring of the lexical elements of the proof?

Are you going to analyze the symbols and formulae along with the textual words? Thanks for considering my question.

Another question: what if you did linguistic analysis on a proof that was incorrect? Is there any relation between

  • what the content of the proof is

  • what the correctness of the proof is

  • what the linguistic content and the syntactical form of the proof is

  • what the ontological underpinnings are of the vocabulary used in the proof

  • what symbology and representational schema are used in the formulae in the proof

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I admit to having some similar questions myself. For instance, if you're doing a linguistic analysis, why does it matter whether the proofs are "entry level" or not? Or are you thinking that proofs written for a less mathematically sophisticated audience will also be less sophisticated in a purely linguistic way? (It's not clear to me whether that's a reasonable assumption or not.) –  Pete L. Clark Oct 22 '10 at 3:41
    
I will be looking for a way to represent the proofs as a symbolic structures which would enable further processing. Formulae and symbols will be treated along with the "natural language bits". Ideally an identical proof step expressed as a formula and expressed in words should yield the same output structure. The long term goal is the interpretation of arbitrary proofs in expressions of Artemov's Justification Logic (see e.g. logic.uconn.edu/readings/Milnikel1.pdf). Checking the correctness of a proof is not a goal. –  Nickolay Kolev Oct 22 '10 at 16:37
    
A clarification on the requirement of "entry level": I would like to work with entry level proofs so as to concentrate on the linguistic means used in proofs while still being able to understand the proofs. I am mostly interested in the expressions used to denote premise->conclusion substructures. –  Nickolay Kolev Oct 22 '10 at 16:41
    
@Nickolay-Kolev, thanks for the clarifications. What sort of mapping are you using from textual words to symbology? (I guess I'm asking what dictionary you're using as the canonical mechanism for defining a string of letters unambiguously.) The same string of letters is the same word but can have very different meanings in different contexts, for example. –  PamNDRome Oct 23 '10 at 1:49
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Take a look at Lovász' paper, "Three short proofs in graph theory", Journal of Combinatorial Theory, Series B, vol. 19, 1975. Maybe those proofs are a little too involved for what you want, but they are worth checking out.

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