I can disprove the 4-color map theorem

The four-color-map theorem, demonstrated here [1], is a theory that declares that "every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. Two regions are called adjacent if they share a border segment, not just a point." [2]

The rules for this debate are very simple.

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1) In round one, my opponent will ONLY accept the debate. The definition of the four-color map theorem cannot be altered; the reason I emphasize this is because I wish to stress "possible geographical map". I do not wish to use any other definition.

2) Semantics should not be used. My opponent may NOT define any terms in round 1, or any future round.

3) In round 2, I will present a picture I have created to "disprove" the theorem. In round 2, my opponent will attempt to point out flaws in my drawing and defend the theorem. Round 3 will be my only defense, and my opponent's closing remarks.

4) Please note- this debate allows for ONE full day to argue, and 6,000 characters to argue with. If you don't have the time, please do not accept- I don't want a forfeiture.

Good luck to my opponent.

~SOURCES~

[1] http://upload.wikimedia.org...

[2] http://www.wordiq.com...

As per my opponent's request, I will only accept the challenge in this round. I look forward to an interesting debate.

I thank my esteemable opponent for accepting this debate, and look forward to a good exchange.

Here [1] is a "possible geographical map" that I have drawn out, with the addition of two photos.

The two photos I used can be seen larger, here: [2], [3]

~~~ANALYSIS OF PHOTO~~~

Labled in the photo are 5 hypothetical countries; they are named "1", "2", "3", "4", and "5". The country that is in question is country "5", hence the lack of color (unless you consider white). What previously used color could be used to fill 5, without touching an adjacent nation/state? "5" borders all of the remaining nations, somewhere or other.

As a preemptive response, notice the two photos to the right.

We see Michigan. Michigan is a single state; it is also discontinuous. This makes country "5" legal, by the standards of the shape of a country. Meanwhile, the second photo shows the country of "Lesotho"; this nation is entirely engulfed by South Africa, leaving it with no bordering country except for South Africa, which goes all the way around it.

My country "5" is a hybrid of Michigan and Lesotho; it is discontinuous, and part of it is engulfed. For this reason, country "5" requires a fifth color, which I will offer to be burnt orange because I go to college at UT Austin.

I wish my opponent luck, and look forward to his response.

PS If the link to the picture does not work, (the rich text feature has been giving me this problem) I'll repost it in the comments section.

[1] http://i56.tinypic.com...

[2] http://www.infomi.com...

[3] http://www.thecommonwealth.org...

Refer back to the initial definition of the four color theorem: "every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. Two regions are called adjacent if they share a border segment, not just a point."

'Region' doesn't mean the same thing as 'nation.' #5 on Pro's map is one nation, but two different regions. While region was not specifically defined in the resolution, it is absurd to assume that it means the same thing as nation, especially since we are talking about a mathematical theorem. 'Four Color Theorem', referring to the mathematical problem doesn't necessarily have anything to do with actual maps or countries, as having a mathematical theorem dependent on whatever the current political boundaries would be absurd. [1]

Pro does mention "geographical map", rather than just map, but that doesn't change the definition of 'region' to 'country.'

[1] http://en.wikipedia.org...

I thank my opponent for his rapid reply.

I'll keep my argument short, for there isn't much to say here.

Con writes,

"'Region' doesn't mean the same thing as 'nation.' #5 on Pro's map is one nation, but two different regions. While region was not specifically defined in the resolution, it is absurd to assume that it means the same thing as nation, especially since we are talking about a mathematical theorem."

Refer back to my round one; even to the definition; even further, look to the title of the debate. The "4-color map theorem" is what is being debated. Regardless of whether or not it is called a nation or a region, my argument holds.

Essentially, my opponent HAD to break a rule to win. He argued from semantics, distinguishing nation from region and then arguing that regions MUST be continuous. He neglects the example of Michigan, that wins me the round given the argument provided by Con. To disambiguate the issue; regions and nations are non-distinct in this debate. I'm Pro my ability to create a 4-color map, not a random arrangement of shapes that are independent as soon as their borders end. The reason there is a distinction is because countries can be split and discontinuous, as opposed to polygons and geometric figures, which can not.

Had I not cared whether we were discussing mathematical "regions" or geographical "countries", I would not have so vehemently emphasized "possible geographical map" in R1, or used Michigan and Lesotho (real states/nations) for my examples.

Furthermore, it only makes sense realistically to assume that regions don't have to be coninuous for the sake of this debate given the fact that all maps keep discontinuous nations/states the same color anyways. E.g. Michigan, the US/Alaska/Hawaii, Pacific Islands, etc.

I urge a Pro vote; Con's argument was insufficient to negate the resolution.

I did not use semantics, which, by my opponent's definition, "My opponent may NOT define any terms in round 1, or any future round." I did not define region, I merely pointed out that Pro's unwritten assumption that region means the same thing as nation is wrong.

Of course, Pro may have meant countries when he said regions, but that doesn't change the fact that that isn't what the word regions means.

"He neglects the example of Michigan, that wins me the round given the argument provided by Con. To disambiguate the issue; regions and nations are non-distinct in this debate. I'm Pro my ability to create a 4-color map, not a random arrangement of shapes that are independent as soon as their borders end."
Pro's example means nothing, unless you first accept his assumption that regions means the same thing as countries or states.

"The reason there is a distinction is because countries can be split and discontinuous, as opposed to polygons and geometric figures, which can not."
There is a distinction in Pro's mind, but not in what was written during R1, which is what counts. He defined 'Four Color Theorem' to mean a specific thing, and failed to prove that he had disproved that specific thing. If he had said countries in his initial definition, which he could have, he would be right. But he didn't.

"Furthermore, it only makes sense realistically to assume that regions don't have to be coninuous for the sake of this debate given the fact that all maps keep discontinuous nations/states the same color anyways. E.g. Michigan, the US/Alaska/Hawaii, Pacific Islands, etc."
No, this assumption does not make sense. One cannot just assume that you meant something that you did not actually say. If Pro's intention was really to show your claim was true of countries, rather than regions, he could have simply said so. His actual intention was to be deliberately deceptive, by saying something that he knew was different from what he meant.

In conclusion, if Pro had meant countries when he said regions, he could have said so. He did not. Because he defined the four color theorem to mean a certain thing, his burden of proof was to show that he had a valid counter example to what he actually defined, not what he secretly meant by what the words in R1 said. He did not meet this burden of proof.