Book Proof Research Articles

A Book Proof of the Middle Levels Theorem

We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the (2n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2n+1)$$\\end{document}-dimensional hypercube induced by all vertices with exactly n or n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} many 1s.

Takeuti's Well-ordering Proof

G. Genzten’s 1938 proof of the consistency of pure arithmetic was hailed as a success for finitism and constructivism, but his proof requires induction along ordinal notations in Cantor normal form up to the first epsilon number, ε0. This left the task of giving a finitisically acceptable proof of the well-ordering of those ordinal notations, without which Gentzen’s proof could hardly be seen as a success for finitism. In his seminal book Proof Theory G. Takeuti provides such a proof. After a brief philosophical introduction, we provide a reconstruction of Takeuti’s proof including corrections, comments, re-organization and notational adjustments for the sake of clarity. The result is a much longer, but much more tractable proof of the well-ordering of ordinal notations in Cantor normal form less than ε0, that nevertheless follows Takeuti’s strategy closely. We end with some more general comments about that proof strategy and the notion of accessibility more generally.

A One-Page Solution of a Problem of Erdős and Purdy

The following theorem was conjectured by Erdős and Purdy: Let P be a set of $$n>4$$ points in general position in the plane. Suppose that R is a set of points disjoint from P such that every line determined by P passes through a point in R. Then $$|R| \ge n$$ . In this paper we give a very elegant and elementary proof of this, being a very good candidate for the “book proof” of this conjecture.

My History: A Memoir of Growing Up

My History: A Memoir of Growing Up

Santilli’s Prime Chains: Pj +1=aPj ± b

Santilli’s prime chains: 1 j j P + = aP ± b , j=1,iŒ,k−1, (a,b)=1, 2|ab. If 1 1 n n n a − = Pλ iŒPλ , P1iŒ Pn|b, we have J2(ω)→∞ as ω→∞. There exist infinitely many primes P1 such that P2,iŒ, Pk are primes for arbitrary length k. It is the Book proof. This is a generalized Euclid-Euler proof for the existence of infinitely many primes. Therefore Euclid- Euler-Jiang theorem in the distribution of primes is advanced. It is the Book theorem.

Finding Proofs in Tarskian Geometry

We report on a project to use a theorem prover to find proofs of the theorems in Tarskian geometry. These theorems start with fundamental properties of betweenness, proceed through the derivations of several famous theorems due to Gupta and end with the derivation from Tarski’s axioms of Hilbert’s 1899 axioms for geometry. They include the four challenge problems left unsolved by Quaife, who two decades ago found some OTTER proofs in Tarskian geometry (solving challenges issued in Wos’s 1998 book). There are 212 theorems in this collection. We were able to find OTTER proofs of all these theorems. We developed a methodology for the automated preparation and checking of the input files for those theorems, to ensure that no human error has corrupted the formal development of an entire theory as embodied in two hundred input files and proofs. We distinguish between proofs that were found completely mechanically (without reference to the steps of a book proof) and proofs that were constructed by some technique that involved a human knowing the steps of a book proof. Proofs of length 40–100, roughly speaking, are difficult exercises for a human, and proofs of 100–250 steps belong in a Ph.D. thesis or publication. 29 of the proofs in our collection are longer than 40 steps, and ten are longer than 90 steps. We were able to derive completely mechanically all but 26 of the 183 theorems that have “short” proofs (40 or fewer deduction steps). We found proofs of the rest, as well as the 29 “hard” theorems, using a method that requires consulting the book proof at the outset. Our “subformula strategy” enabled us to prove four of the 29 hard theorems completely mechanically. These are Ph.D. level proofs, of length up to 108.

Proofs (Not) from The Book

We all know about The Book in which, according to Erdős, God keeps the most elegant proof of each mathematical result. It is interesting to speculate whether every theorem has a “Book proof” (say what about the Four Color Theorem or the Collatz 3n+1 Conjecture?) or whether a “Book proof” of a theorem is unique. I believe many mathematicians have their own private collections of proofs from The Book. In fact, such a collection, and a highly successful one, was published by M. Aigner and G. M. Ziegler [1] (see [2] for a complement). This article contains several proofs, not included in [1], that, in my opinion, are serious contenders for inclusion in The Book. Needless to say, the selection reflects my taste and mathematical interests. Most of the theorems discussed here are 100–150 years old, but these proofs are considerably newer. This is another issue to mull over: how long does it take for a Book proof of a theorem to emerge?

A tribute to Bryan Joseph Cremin (1929–2012) – an Irishman and a South African legend

Prof. Bryan Joseph Cremin, M.B., B.S., DRCOG, MCRA, FFR, FRCR, FRACR was from a different generation from my own, a radiologist of the analogue age. Many of you who were friends with him will wonder why I am writing this tribute, since I met him only three times. When Professor Cremin started at the Red Cross Children’s Hospital in Cape Town I had just been born into the world. However, I have lived with his presence daily for many years. As a green paediatric radiologist starting at the Red Cross Children’s Hospital in 1998 when he had already been retired since 1995 I encountered the legend of Professor Cremin – it awed me and inspired me. This giant of paediatric radiology left such a footprint that I feared to occupy his office, the holy place where he wrote and worked. There were stories of him everywhere. For example, it was said that he would not let anyone else reduce intussusceptions, insisting that he be called out personally, day or night, until he considered himself an expert. When I did eventually move into his office, I found not scents or traces of Professor Cremin, but instead an astonishingly large body of work. There were hundreds of articles he had written, book proofs, drawers and drawers of slides and lectures he had delivered. The majority of this had handwritten origins, typed pages and analogue photographs. None of Professor Cremin’s work had the luxury of digital photographs, electronic submissions and computerized versions that speed up our work today – it smelt of blood and sweat. However, it was in the cracks between the science that I glimpsed the life of the man beyond paediatric radiology – a picture of him in the British army in Malaya, an action photo running with rugby ball in hand, an image of a laughing Bryan with friends, an image of the orator. Who could fit this all into one life? The third and last time Professor Cremin visited me, he sat very close and whispered to me in a conspiratorial tone: BThey all need a photographer. The royal family needs one. So does every doctor here. Be their personal photographer and it will make you a great paediatric radiologist^. I think he underestimated what he did and the lasting imprint he would have on us. I look upwards to Professor Cremin as the South African figure with the greatest influence on the international paediatric radiology community to date. May his spirit live long. Pediatr Radiol (2012) 42:1024 DOI 10.1007/s00247-012-2445-8

Meta-Evaluation of Worksite Health Promotion Economic Return Studies: 2012 Update

Two earlier editions of The Art of Health Promotion reported formal meta-evaluations of economic return studies of worksite health promotion programs. These meta-evaluations were among the most popular articles in our publishing history because practitioners, scientists, and employers continue to seek evidence on the economic effectiveness of workplace health promotion. The 2003 report examined 42 peer-reviewed journal articles that met the inclusion criteria, and the 2005 report examined 56 peer-reviewed journal articles. This 2012 update examines 10 additional studies that met the same set of inclusion criteria; four of the weaker studies were dropped, resulting in a total of 62 studies in this report. The meta-evaluation method used here is the same as used in the earlier reports and has been adapted from use with other preventive health programs, providing an overall summary and individual look at the quality of the peer reviewed articles that comprise the current scientific evidence of economic return for worksite health promotion and wellness programs. The term ‘‘meta-evaluation’’ as used in this article is defined as the application of a systematic review process to a set of evaluation studies with a similar purpose in order to determine their respective quality and to summarize their primary findings. It applies the formal meta-evaluation review process and methodology developed and refined by Windsor and Orleans and further modified by Boyd and Windsor to studies of multi-component worksite health promotion programs as defined by Heaney. This report include excerpts from the book Proof Positive: An Analysis of the Cost-Effectiveness of Worksite Wellness, seventh edition, revised and expanded in December of 2011, which applies the same methodology. This edition of The Art of Health Promotion addresses the following topics:

Quantum Locality?

Robert Griffiths has recently addressed, within the framework of a 'consistent quantum theory' that he has developed, the issue of whether, as is often claimed, quantum mechanics entails a need for faster-than-light transfers of information over long distances. He argues that the putative proofs of this property that involve hidden variables include in their premises some essentially classical-physics-type assumptions that are fundamentally incompatible with the precepts of quantum physics. One cannot logically prove properties of a system by establishing, instead, properties of a system modified by adding properties alien to the original system. Hence Griffiths' rejection of hidden-variable-based proofs is logically warranted. Griffiths mentions the existence of a certain alternative proof that does not involve hidden variables, and that uses only macroscopically described observable properties. He notes that he had examined in his book proofs of this general kind, and concluded that they provide no evidence for nonlocal influences. But he did not examine the particular proof that he cites. An examination of that particular proof by the method specified by his 'consistent quantum theory' shows that the cited proof is valid within that restrictive version of quantum theory. An added section responds to Griffiths' reply, which cites general possibilities of ambiguities that make what is to be proved ill-defined, and hence render the pertinent 'consistent framework' ill defined. But the vagaries that he cites do not upset the proof in question, which, both by its physical formulation and by explicit identification, specify the framework to be used. Griffiths confirms the validity of the proof insofar as that framework is used. The section also shows, in response to Griffiths' challenge, why a putative proof of locality that he has described is flawed.

Would the Real Lakatos Please Stand Up

Although Imre Lakatos described the work published in his book Proofs and Refutations as a study of mathematical methodology, work which has been responded to and criticized by philosophers and historians of mathematics more on its own terms, a significant body of writing in the 30 years since its appearance has used it as a pertinent cognate text appropriable for school mathematics education. In this paper, we contrast the responses these two fields have generated, with an emphasis on that of mathematics education. Doing so offers a potentially salutary case study of how challenging and fraught it can be at times to undertake work at the nexus of history and philosophy on the one hand, while at the same time seeking to explore its possible relevance and significance for education. As our title suggests, we are concerned about the proliferating Lakatos personas that seem to exist, including a growing range of self-styled reform or progressive educational practices which get attributed to him.

The 99th Fibonacci Identity

We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences $G_n$ that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well. Among these, we prove $\sum_{k\geq 0}{n \choose k}G_{2k} = 5^n G_{2n}$ and $\sum_{k\geq 0}{n \choose k}G_{qk}(F_{q-2})^{n-k} = (F_q)^n G_{2n}$.

The irrationality of e4: a simple proof

The well-known simple proof that the number e is irrational consists of writing e as its exponential series, multiplying by a suitable factorial, r!, and deriving the contradiction that, if e were rational, there would be an integer lying strictly between 0 and 1. This proof is referred to as a ‘trick’ in the recent highly acclaimed book Proofs from THE BOOK [1, p. 27], where the surprising statement is made that ‘This trick … isn’t even good enough to prove that e2 is irrational’. In this article we show that there are nevertheless two similar simple proofs that e2 is irrational. The first proof is in fact well known [2, p. 12]. The second, though elementary, is less obvious and certainly not well known but, together with the idea in the first proof, it implies the even stronger property that e4 is irrational.

Countable Additivity and Subjective Probability

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti's central argument against countable additivity and provide a new Dutch book proof of the principle, To argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint.