A POSSIBLE RESOLUTION TO HILBERTâS FIRST PROBLEM BY APPLYING CANTORâS DIAGONA...ijscmcj
Â
We present herein a new approach to the Continuum hypothesis CH. We will employ string conditioning, a
technique that limits the range of a string over portions of its sub-domain for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantorâs Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
A POSSIBLE RESOLUTION TO HILBERTâS FIRST PROBLEM BY APPLYING CANTORâS DIAGONA...ijscmcj
Â
We present herein a new approach to the Continuum hypothesis CH. We will employ string conditioning, a technique that limits the range of a string over portions of its sub-domain for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantorâs Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively
Associate Professor Anita Wasilewska gave a lecture on "Descriptive Granularity" in the Distinguished Lecturer Series - Leon The Mathematician.
More Information available at:
http://dls.csd.auth.gr
A POSSIBLE RESOLUTION TO HILBERTâS FIRST PROBLEM BY APPLYING CANTORâS DIAGONA...ijscmcj
Â
We present herein a new approach to the Continuum hypothesis CH. We will employ string conditioning, a
technique that limits the range of a string over portions of its sub-domain for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantorâs Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
A POSSIBLE RESOLUTION TO HILBERTâS FIRST PROBLEM BY APPLYING CANTORâS DIAGONA...ijscmcj
Â
We present herein a new approach to the Continuum hypothesis CH. We will employ string conditioning, a technique that limits the range of a string over portions of its sub-domain for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantorâs Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively
Associate Professor Anita Wasilewska gave a lecture on "Descriptive Granularity" in the Distinguished Lecturer Series - Leon The Mathematician.
More Information available at:
http://dls.csd.auth.gr
Logic and mathematics history and overview for studentsBob Marcus
Â
Math and logic overview for students. Covers a wide range of topics including algorithms, proofs, probability, networks, number theory, statistics, causality, WolframAlpha, and Python programs.
Earlier a place value notation number system had evolved over a leng.pdfbrijmote
Â
Earlier a place value notation number system had evolved over a lengthy period with a number
base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to
be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied
from at least 1700 BC. Systems of linear equations were studied in the context of solving number
problems. Quadratic equations were also studied and these examples led to a type of numerical
algebra.
Geometric problems relating to similar figures, area and volume were also studied and values
obtained for ĂâŹ.
The Babylonian basis of mathematics was inherited by the Greeks and independent development
by the Greeks began from around 450 BC. Zeno of Elea\'s paradoxes led to the atomic theory of
Democritus. A more precise formulation of concepts led to the realisation that the rational
numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers
arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius.
Further mathematical discoveries were driven by the astronomy, for example the study of
trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress
continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This
work did not match the progress made by the Greeks but in addition to the Islamic progress, it
did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later
Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into
Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with
Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic
equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of
the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical
research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in
France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a
calculatory science with his discovery of logarithms. Cavalieri made progress towards the
calculus with his infinitesimal methods and Descartes added the power of algebraic methods to
geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the
mathematical study of probability. However the calculus was to be the topic of most significance
to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow,
developed the calculus into a tool to push forward the study of nature. His work contained a
wealth of new discoveries showing the interaction between mathemat.
Some Interesting Facts, Myths and History of Mathematicsinventionjournals
Â
This paper deals with primary concepts and fallacies of mathematics which many a times students and even teachers ignore. Also this paper comprises of history of mathematical symbols, notations and methods of calculating time. I have also included some ancient techniques of solving mathematical real time problems. This paper is a confluence of various traditional mathematical techniques and their implementation in modern mathematics.
Logic and mathematics history and overview for studentsBob Marcus
Â
Math and logic overview for students. Covers a wide range of topics including algorithms, proofs, probability, networks, number theory, statistics, causality, WolframAlpha, and Python programs.
Earlier a place value notation number system had evolved over a leng.pdfbrijmote
Â
Earlier a place value notation number system had evolved over a lengthy period with a number
base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to
be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied
from at least 1700 BC. Systems of linear equations were studied in the context of solving number
problems. Quadratic equations were also studied and these examples led to a type of numerical
algebra.
Geometric problems relating to similar figures, area and volume were also studied and values
obtained for ĂâŹ.
The Babylonian basis of mathematics was inherited by the Greeks and independent development
by the Greeks began from around 450 BC. Zeno of Elea\'s paradoxes led to the atomic theory of
Democritus. A more precise formulation of concepts led to the realisation that the rational
numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers
arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius.
Further mathematical discoveries were driven by the astronomy, for example the study of
trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress
continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This
work did not match the progress made by the Greeks but in addition to the Islamic progress, it
did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later
Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into
Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with
Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic
equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of
the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical
research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in
France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a
calculatory science with his discovery of logarithms. Cavalieri made progress towards the
calculus with his infinitesimal methods and Descartes added the power of algebraic methods to
geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the
mathematical study of probability. However the calculus was to be the topic of most significance
to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow,
developed the calculus into a tool to push forward the study of nature. His work contained a
wealth of new discoveries showing the interaction between mathemat.
Some Interesting Facts, Myths and History of Mathematicsinventionjournals
Â
This paper deals with primary concepts and fallacies of mathematics which many a times students and even teachers ignore. Also this paper comprises of history of mathematical symbols, notations and methods of calculating time. I have also included some ancient techniques of solving mathematical real time problems. This paper is a confluence of various traditional mathematical techniques and their implementation in modern mathematics.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
Â
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
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In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Operation âBlue Starâ is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
A Strategic Approach: GenAI in EducationPeter Windle
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
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Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Model Attribute Check Company Auto PropertyCeline George
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In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
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Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Hanâs Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insiderâs LMA Course, this piece examines the courseâs effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
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Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
1. arXiv:2110.09718v1
[math.GM]
19
Oct
2021
Update as on October 20, 2021.
Update as on October 20, 2021.
A pictorial proof of the Four Colour Theorem
Bhupinder Singh Anandâ
https://orcid.org/0000-0003-4290-9549
Abstract. We give a pictorial, and absurdly simple, proof that transparently illustrates why four colours suffice to chromatically
differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar
map. We show, moreover, why the proof cannot be expressed within classical graph theory.
Keywords. contiguous area, four colour theorem, planar map, simply connected.
2010 Mathematics Subject Classification. 05C15
DECLARATIONS ⢠Funding: Not applicable ⢠Conflicts of interest/Competing interests: Not applicable ⢠Availability of data and
material: Not applicable ⢠Code availability: Not applicable ⢠Authorsâ contributions: Not applicable
1. Introduction
Although the Four Colour Theorem 4CT is considered passeĚ (see §1.A.), we give a pictorial, and
absurdly simple, proof1 that transparently illustrates why four colours suffice to chromatically differ-
entiate any set of contiguous, simply connected and bounded, planar spaces; by showing that:
(1) If, for some natural numbers m, n, every planar map of less than m + n contiguous, simply
connected and bounded, areas can be 4-coloured;
(2) And, we assume (Hypothesis 1) that there is a sub-minimal 4-coloured planar map M, of m+n
such areas, where finitary creation of a specific, additional, contiguous, simply connected and
bounded, area C within M yields a minimal map H which entails that C require a 5th colour;
(3) Then Hypothesis 1 is false (by Theorem 2.1), since there can be no such sub-minimal 4-coloured
planar map M.
Moreover we show whyâchallenging deep-seated dogmas that seemingly yet await, even if not
actively seek, a mathematically âinsightfulâ, and philosophically âsatisfyingâ, proof of 4CT within
inherited paradigmsâthe pictorial proof cannot be expressed within classical graph theory.
1.A. A historical perspective
It would probably be a fair assessment that the mathematical significance of any new proof of the Four
Colour Theorem 4CT continues to be perceived as lying not in any ensuing theoretical or practical
utility of the Theorem per se, but in whether the proof can address the philosophically âunsatisfyingâ,
and occasionally âdespairingâ (see [Tym79]; [Sw80]; [Gnt08], [Cl01]), lack of mathematical âinsightâ,
âsimplicityâ, and âeleganceâ in currently known proofs of the Theorem (eg. [AH77], [AHK77], [RSST],
[Gnt08])âan insight and simplicity this investigation seeks in a pre-formal2 proof of 4CT.
For instance we noteâamongst othersâsome candid comments from Robertson, Sanders, Sey-
mour, and Thomasâs 1995-dated (apparently pre-publication) summary3 of their proof [RSST]:
â
# 1003, Lady Ratan Tower, Dainik Shivner Marg, Gandhinagar, Worli, Mumbai - 400 018, Maharashtra, India.
Email: bhup.anand@gmail.com. Mbl: +91 93225 91328. Tel: +91 (22) 2491 9821.
1
Extracted from [An21], §1.G: Evidence-based (pictorial), pre-formal, proofs of the Four Colour Theorem.
2
The need for distinguishing between belief-based âinformalâ, and evidence-based âpre-formalâ, reasoning is addressed
by Markus Pantsar in [Pan09]; see also [An21], §1.D.
3
See [RSSp]; also [Thm98], [Cl01], and the survey [Rgrs] by Leo Rogers.
2. 2 1. Introduction
2 1. Introduction
âWhy a new proof?
There are two reasons why the Appel-Haken proof is not completely satisfactory.
⢠Part of the Appel-Haken proof uses a computer, and cannot be verified by hand, and
⢠even the part that is supposedly hand-checkable is extraordinarily complicated and tedious,
and as far as we know, no one has verified it in its entirety.â . . . Robertson et al: [RSSp], Pre-publication.
âIt has been known since 1913 that every minimal counterexample to the Four Color Theorem is an
internally six-connected triangulation. In the second part of the proof, published in [4, p. 432], Robertson
et al. proved that at least one of the 633 configurations appears in every internally six-connected planar
triangulation. This condition is called âunavoidability,â and uses the discharging method, first suggested
by Heesch. Here, the proof differs from that of Appel and Haken in that it relies far less on computer
calculation. Nevertheless, parts of the proof still cannot be verified by a human. The search continues for
a computer-free proof of the Four Color Theorem.â . . . Brun: [Bru02], §1. Introduction (Article for undergraduates)
âThe four-colour problem had a long life before it eventually became the four-colour theorem. In 1852
Francis Gutherie (later Professor of Mathematics at the University of Cape Town) noticed that a map
of the counties of England could be coloured using only four colours. He wondered if four colours would
always suffice for any map. He, or his brother Frederick, proposed the problem to Augustus De Morgan
(see the box at the end of Section 3.5 in Chapter 3) who liked it and suggested it to other mathematicians.
Interest in the problem increased after Arthur Cayley presented it to the London Mathematical Society
in 1878 ([Cay79]). The next year Alfred Bray Kempe (a British lawyer) gave a proof of the conjecture.
His proof models the problem in terms of graphs and breaks it up into a number of necessary cases to be
checked. Another proof was given by Peter Tait in 1880. It seemed that the four-colour problem had been
settled in the affirmative.
However, in 1890 Percy John Heawood found that Kempeâs proof missed one crucial case, but that the
approach could still be used to prove that five colours are sufficient to colour any map. In the following
year Taitâs proof was also shown to be flawed, this time by Julius Petersen, after whom the Petersen graph
is named. The four-colour problem was therefore again open, and would remain so for the next 86 years.
In that time it attracted a lot of attention from professional mathematicians and good (and not so good)
amateurs alike. In the words of Underwood Dudley:
The four-color conjecture was easy to state and easy to understand, no large amount of tech-
nical mathematics is needed to attack it, and errors in proposed proofs are hard to see, even
for professionals; what an ideal combination to attract cranks!
The four-colour theorem was finally proved in 1976 by Kenneth Appel and Wolfgang Haken at the Uni-
versity of Illinois. They reduced the problem to a large number of cases, which were then checked by
computer. This was the first mathematical proof that needed computer assistance. In 1997 N. Robertson,
D.P. Sanders, P.D. Seymour and R. Thomas published a refinement of Appel [and] Hakenâs proof, which
reduces the number of necessary cases, but which still relies on computer assistance. The search is still on
for a short proof that does not require a computer.â . . . Conradie/Goranko: [CG15], §7.7.1, Graph Colourings, p.417.
âThe first example concerns a notorious problem within the philosophy of mathematics, namely the ac-
ceptability of computer-generated proofs or proofs that can only be checked by a computer; for instance
because it includes the verification of an excessively large set of cases. The text-book example of such
a mathematical result is the proof of the 4-colour theorem, which continues to preoccupy philosophers
of mathematics (Calude 2001). Here, we only need to note that the debate does not primarily concern
the correctness of the result, but rather its failure to adhere to the standard of surveyability to which
mathematical proofs should conform.â . . . Allo: [All17], Conclusion, p.562.
âBeing the first ever proof to be achieved with substantial help of a computer, it has raised questions to
what a proof really is. Many mathematicians remain sceptical about the nature of this proof due to the
involvement of a computer. With the possibility of a computing error, they do not feel comfortable relying
on a machine to do their work as they would be if it were a simple pen-and-paper proof.
The controversy lies not so much on whether or not the proof is valid but rather whether the proof is a
valid proof. To mathematicians, it is as important to understand why something is correct as it is finding
the solution. They hate that there is no way of knowing how a computer reasons. Since a computer runs
programs as they are fed into it, designed to tackle a problem in a particular way, it is likely they will
return what the programmer wants to find leaving out any other possible outcomes outside the bracket.
3. B S Anand, Four Colour Theorem 3
B S Anand, Four Colour Theorem 3
Many mathematicians continue to search for a better proof to the problem. They prefer to think that the
Four Colour problem has not been solved and that one day someone will come up with a simple completely
hand checkable proof to the problem.â . . . Nanjwenge: [Nnj18], Chapter 8, Discussion (Student Thesis).
âThe heavy reliance on computers in Appel and Hakenâs proof was immediately a topic of discussion and
concern in the mathematical community. The issue was the fact that no individual could check the proof;
of special concern was the reductibility [sic] part of the proof because the details were âhiddenâ inside the
computer. Though it isnât so much the validity of the result, but the understanding of the proof. Appel
himself commented: â. . . there were people who said, âThis is terrible mathematics, because mathematics
should be clean and elegant,â and I would agree. It would be nicer to have clean and elegant proofs.â See
page 222 of Wilson.â . . . Gardner: [Grd21], §11.1, Colourings of Planar Maps, pp.6-7 (Lecture notes).
2. A pictorial proof of the 4-Colour Theorem
â
â
â
â
â
C
Bn
Am
Minimal Planar Map H
Bn
Only the immediate portion of each area cn,1, cn,2, . . . , cn,r of Bn abutting C is indicated.
cn,i
Fig.1
We consider the surface of the hemisphere (minimal planar map H) in Fig.1 where:
1. Am denotes a region of m contiguous, simply connected and bounded, surface areas am,1, am,2,
. . . , am,m, none of which share a non-zero boundary segment with the contiguous, simply con-
nected, surface area C (as indicated by the red barrier which, however, is not to be treated as
a boundary of the region Am);
2. Bn denotes a region of n contiguous, simply connected and bounded, surface areas bn,1, bn,2,
. . . , bn,n, some of which, say cn,1, cn,2, . . . , cn,r, share at least one non-zero boundary segment of
cn,i with C; where, for each 1 ⤠i ⤠r, cn,i = bn,j for some 1 ⤠j ⤠n;
3. C is a single contiguous, simply connected and bounded, area created finitarily (i.e., not pos-
tulated) by sub-dividing and annexing one or more contiguous, simply connected, portions of
each area câ
n,i (defined in Hypothesis 1(b) below) in the region Bâ
n (defined in Hypothesis 1(b)).
Hypothesis 1. (Minimality Hypothesis) Since four colours suffice for maps with fewer than 5
regions, we assume the existence of some m, n, in a putatively minimal planar map H, which defines
a minimal configuration of the region {Am + Bn + C} where:
(a) any configuration of p contiguous, simply connected and bounded, areas can be 4-coloured if
p ⤠m + n, where p, m, n â N, and m + n ⼠5;
(b) any configuration of the m+n contiguous, simply connected and bounded, areas of the region, say
{Aâ
m + Bâ
n }, in a putative, sub-minimal, planar map M before the creation of Câconstructed
finitarily by sub-dividing and annexing some portions from each area, say câ
n,i, of Bâ
n in Mâcan
be 4-coloured;
4. 4 2. A pictorial proof of the 4-Colour Theorem
4 2. A pictorial proof of the 4-Colour Theorem
(c) the region {Am +Bn +C} in the planar map H is a specific configuration of m+n+1 contiguous,
simply connected and bounded, areas that cannot be 4-coloured (whence the area C necessarily
requires a 5th colour by the Minimality Hypothesis).
Theorem 2.1. (Four Colour Theorem) No planar map needs more than four colours.
â
â
â
â
E1
E1
D1
E1
Bn
Am
Planar Map Hâ˛
Bn
cn,i
Fig.2
Proof. If the area C of the minimal planar map H in Fig.1 is divided further (as indicated in Fig.2)
into two non-empty areas D1 and E1, where:
⢠D1 shares a non-zero boundary segment with only one of the areas cn,i; and
⢠D1 can be treated as an original area of câ
n,i in M (see Hypothesis 1(b)) that was annexed to
form part of C in H (in Fig.1);
then D1 can be absorbed back into cn,i without violating the Minimality Hypothesis. Moreover,
cn,i + D1 must share a non-zero boundary with E1 in HⲠif cn,i = bn,j for some 1 < j < n, and bn,j, C
are required to be differently coloured, in H.
Such a division, as illustrated in Fig.2, followed by re-absorption of D1 into câ
n,i (denoted, say, by
Bn + D1), would reduce the configuration HⲠin Fig.2 again to a minimal planar map, say H1 with a
configuration {Am + Bâ˛
n + E1}, where Bâ˛
n = (Bn + D1); which would in turn necessitate a 5th colour
for the area E1 â C by the Minimality Hypothesis.
Since we cannot, by reiteration, have a non-terminating sequence C â E1 â E2 â E3 â . . ., the
sequence must terminate in a non-empty area Ek of a minimal planar map, say Hk, for some finite
integer k; where Ek contains no area that is annexed from any of the areas of Bâ
n in M prior to the
formation of the minimal planar map H (in Fig.1).
Comment: Note that we cannot admit as a putative limit of C â E1 â E2 â E3 . . . the configuration
where all the câ
n,iâcorresponding to the abutting areas cn,i of C in the Minimal Planar Map Hâmeet
at a point in the putative, sub-minimal, planar map M, since any finitary (i.e., not postulated) creation
of C, begun by initially annexing a non-empty area of some câ
n,i at such an apex (corresponding to the
putative âfinally mergedâ area of the above non-terminating sequence C â E1 â E2 â E3 â . . .), would
require, at most, a 4th
but not a 5th
colour.
However, by Hypothesis 1(b), this contradicts the definition of the area C, in the minimal planar
map H (in Fig.1)âergo of the area Ek in the minimal planar map Hkâas formed finitarily by
sub-division and annexation of existing areas of Bâ
n in M.
Hence there can be no minimal planar map H which defines a minimal configuration such as the
region {Am + Bn + C} in Fig.1. The theorem follows.
5. B S Anand, Four Colour Theorem 5
B S Anand, Four Colour Theorem 5
We conclude by noting that, since classical graph theory4 represents non-empty areas as points
(vertices), and a non-zero boundary between two areas as a line (edge) joining two points (vertices),
it cannot express the proof of Theorem 2.1 graphically.
Reason: The proof of Theorem 2.1 appeals to finitarily distinguishable properties of a series
of, putatively minimal, planar maps H, H1, H2, . . ., created by a corresponding sequence of areas
C, E1, E2, . . . , where each area is finitarily created from, or as a proper subset of, some preceding
area/s in such a way that the graphs of H, H1, H2, . . . remain undistinguished.
References
[AH77] Kenneth Appel and Wolfgang Haken. 1977. Every planar map is four colorable. Part I: Discharging. Illinois
Journal of Mathematics, Volume 21, Issue 3 (1977), pp. 429-490, University of Illinois, Urbana-Champaign.
http://projecteuclid.org/download/pdf 1/euclid.ijm/1256049011
[AHK77] Kenneth Appel, Wolfgang Haken and John Koch. 1977. Every planar map is four colorable. Part II: Re-
ducibility. Illinois Journal of Mathematics, Volume 21, Issue 3 (1977), 491-567, University of Illinois, Urbana-
Champaign.
http://projecteuclid.org/download/pdf 1/euclid.ijm/1256049012
[All17] Patrick Allo. 2017. A Constructionist Philosophy of Logic. In Minds Machines, 27, 545â564 (2017).
https://doi.org/10.1007/s11023-017-9430-9
https://link.springer.com/article/10.1007/s11023-017-9430-9#Sec4
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4
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6. 6 2. A pictorial proof of the 4-Colour Theorem
6 2. A pictorial proof of the 4-Colour Theorem
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