5
$\begingroup$

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

Improve this question
$\endgroup$
0

9 Answers 9

Reset to default
2
$\begingroup$

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

Improve this answer
$\endgroup$
1
$\begingroup$

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.

Improve this answer
$\endgroup$
1
$\begingroup$

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

Improve this answer
$\endgroup$
4
  • $\begingroup$ 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.) $\endgroup$
    – Pete L. Clark
    Commented Oct 22, 2010 at 3:41
  • $\begingroup$ 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. $\endgroup$
    – Nickolay Kolev
    Commented Oct 22, 2010 at 16:37
  • $\begingroup$ 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. $\endgroup$
    – Nickolay Kolev
    Commented Oct 22, 2010 at 16:41
  • $\begingroup$ @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. $\endgroup$
    – PamNDRome
    Commented Oct 23, 2010 at 1:49
1
$\begingroup$

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?

Improve this answer
$\endgroup$
1
  • 3
    $\begingroup$ Beweis: klar. :-) $\endgroup$
    – Felipe Voloch
    Commented Oct 22, 2010 at 19:06
1
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ I did my PhD in the Naproche project. For those interested in doing linguistic analysis of mathematical texts, I can recommend the PhD thesis of Mohan Ganesalingam, my own PhD thesis (icr.uni.lu/mcramer/downloads/doktor.pdf) and my paper "The Naproche system: Proof-checking mathematical texts in controlled natural language" (icr.uni.lu/mcramer/downloads/SDV.pdf). $\endgroup$
    – Marcos Cramer
    Commented Jan 8, 2018 at 13:48
1
$\begingroup$

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:

Improve this answer
$\endgroup$
1
$\begingroup$

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

Improve this answer
$\endgroup$
0
$\begingroup$

Truly "entry level" ... How to Read and Do Proofs by D. Solow.

Improve this answer
$\endgroup$
0
$\begingroup$

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.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .