The Instigator
mostlogical
Pro (for)
Losing
7 Points
The Contender
Biodome
Con (against)
Winning
14 Points

"There aren't 360 Degrees in a Circle"

Do you like this debate?NoYes+1
Add this debate to Google Add this debate to Delicious Add this debate to FaceBook Add this debate to Digg  
Post Voting Period
The voting period for this debate has ended.
after 3 votes the winner is...
Biodome
Voting Style: Open Point System: Select Winner
Started: 6/6/2015 Category: Education
Updated: 1 year ago Status: Post Voting Period
Viewed: 2,364 times Debate No: 76235
Debate Rounds (5)
Comments (25)
Votes (3)

 

mostlogical

Pro

I agree with the statement: "there aren't 360 degrees in a circle".

It may be that we chose 360 because it's close to the number of days it takes Earth to orbit the Sun, but it's more likely because the Babylonians chose to divide a circle by 360 degrees. But the Babylonians couldn't even calculate Pi to more than 2 decimals places! Whatever the reason, it seems we have accepted a circle has 360 degrees on pure faith, as no-one seems to know why a circle has 360 degrees to this day.

My opponent must argue why there is in fact exactly 360 degrees in a circle, while I will prove that there are not 360 degrees in a circle.

Good luck Con
Biodome

Con

I would like to thank my opponent for initiating this debate. I will dedicate this round to my constructive case.


Definitions

Degree - (in Geometry) the 360th part of a complete angle or turn, often represented by the sign °, as in 45°, which is read as 45 degrees. [1]

also

Degree - π/180 radians. [2]

Radian - a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1. [3]

Circle - A circle is the set of points in a plane that are equidistant from a given point. [4] A circle can also be called a turn, revolution, complete rotation etc. [5]


Arguments

Note that I will assume the standard and fundamental axioms of Mathematics and Number Theory, such as the Peano axioms.


A. By definition of a degree, there are 360 degrees in a circle.

1) A degree is the 360th part of a of a circle (see Definitions above)
2) Therefore, there are 360 degrees in a circle.


B. Proof by a unit circle.

Radian

1) Take a circle with a radius of 1 (a unit circle)
2) By definition, an arc of length 1 of such a circle corresponds to an angle of 1 radian. (see Definitions above)
3) Observe that for any circle, its Circumference (let's call it C) is calculated by the formula C = 2πr [6]
4) By proportion, an arc of length C corresponds to an angle of C radians.
5) By substitution, an arc length of C corresponds to an angle of 2πr radians, which is simplified to 2π radians, as we have a unit circle.
6) By inverse geometry, this can be generalized to all circles [7]. Thus, all circles have an angle of 2π radians.
7) By definition and the relation of radians and degrees, it can be shown that 2π radians are equal to 360 degrees.
8) Therefore, all circles have an angle of 360 degrees.


My opponent has the Burden of Proof to show that both of these arguments are unsound.


Sources:

[1] http://dictionary.reference.com...
[2] http://physics.nist.gov...
[3] http://mathworld.wolfram.com...
[4] http://mathworld.wolfram.com...
[5] https://en.wikipedia.org...
[6] http://www.mathgoodies.com...
[7] https://en.wikipedia.org...

Debate Round No. 1
mostlogical

Pro

Thank you for providing definitions, however they are very specific. The Oxford dictionary defines a degree as a unit for measuring angles, which is a more accurate definition in my opinion.

Argument A

The definitions you have used assume it is impossible to calculate the number of degrees in a circle.

Circles are commonly divided into 360 degrees because the Babylonians did [1] but they couldn't have calculated how many degrees there are in a circle because they couldn't calculate Pi accurate to 2 decimal places.

It is possible to divide a circle by whatever we want to make useful angle measurements, 360 just happens to divide evenly by 2, 3, 4, 5, 6, 8, 9, 10) however a new unit should be used if there aren't 360 degrees in a circle, or if it is not known.

Mathematicians do not show angles in more than 1 decimal place, so this means they use a polygon with 647,640 degrees to make angle measurements. Therefore there is no need to "make up" a unit for a circle. Doing so hides the real internal angle of a circle.

Argument B

Saying that one degree is equal to 180/ pi does not prove there are 360 degrees in a circle, a circle can be divided any number of times. I could just as easily create my own unit and call it a 'klark' and say 1 degree equals 1.6... 'klarks'.

I will now calculate the number of degrees in a circle, to do this I will effectively place a regular polygon inside a circle and increase the number of sides.

16 sides:
360/16 = 22.5 degrees
There are 180 degrees in a triangle, so the edge angles add up to 180 - 22.5 = 157.5

1000 sides:
360/1000 = 0.36
There are 180 degrees in a triangle, so the edge angles add up to 180 - 0.36 = 179.64

3 million sides:
360/3 million = 0.00012
There are 180 degrees in a triangle, so the edge angles add up to 180 - 0.00012 = 179.99988

20 million sides:
360/20 million = 0.000018
There are 180 degrees in a triangle, so the edge angles add up to 180 - 0.000018 = 179.999982

If I continue increasing the number of sides of a polygon, each edge angle will become "flatter" i.e. closer to 180. But it will NEVER reach 180. By increasing the sides an infinite number of times I will create a polygon with one side i.e. a circle. This means a circle has 179.9... degrees.

This should be obvious really because a line has 180 degrees, and for it to form a circle it must bend by the tiniest amount.
Imagine if I make this curved line longer, it will still have 179.9... degrees, and if I made it so it almost came back on it's self it would still have the same number of degrees, and there is no reason to assume this wouldn't change if both ends met.

Thanks for reading
Biodome

Con

I thank my opponent for producing and argument. I will now defend my position and rebut my opponent's claims.


Definition of a Degree

My opponent contests the first definition of a Degree that I have provided, i.e.:

Degree - (in Geometry) the 360th part of a complete angle or turn, often represented by the sign °, as in 45°, which is read as 45 degrees.

He states that it is "very specific" and not as "accurate" as the Oxford Dictionary Definition.

I disagree with him. Firstly, we cannot use any definition of a degree, as the debate we are currently having focuses on Mathematics, specifically, Geometry. Therefore, there is a need of a specific definition, which is applicable to the field of Geometry, and which is widely accepted by mathematicians. It seems that most, if not all serious dictionaries and sources provide a geometrical definition of a degree, as the 360th part of a complete circle.

My opponent claims that The Oxford Dictionary defines a degree as "a unit for measuring angles". This is not true, as this is not the whole definition. The whole definition of a degree, provided by the Oxford Dictionary in the context of Geometry, goes like this:

Degree - unit of measurement of angles, one ninetieth of a right angle or the angle subtended by one three-hundred-and-sixtieth of the circumference of a circle. [1]

The library version of the Oxford Dictionary provides a definition as follows:

Degree - Geom. (Astron., Geogr., etc.) A unit of measurement of angles or circular arcs, being an angle equal to the 90th part of a right angle, or an arc equal to the 360th part of the circumference of a circle (which subtends this angle at the centre). [2]

Therefore, it seems that my opponent is simply ignoring an important part of the definition.


1. Defense of Argument A.

My opponent claims that "The definitions you have used assume it is impossible to calculate the number of degrees in a circle." I fail to see any kind of justification for this claim. The definition that I have provided clearly defines a degree as being the 1/360th part of the circle. Thus, it is trivially easy to calculate the amount of degrees in a circle.

By simple mathematics,
If 1/360 of a Circle corresponds to 1 degree,
Then 1 (full) Circle corresponds to 1 * 360 = 360 degrees.

My opponent also talks about the Babylonians, claiming that they could not have calculated the degrees in a circle without knowing the value of Pi. But my opponent fails to realize that Pi is not needed to calculate the angle of a circle, especially if one defines a degree as the Babylonians did, that is, the 1/360th part of a circle. The Babylonians would have perfectly been capable of calculating the amount of degrees in a circle, as I did above.

We know how many degrees there are in a circle, because we have defined a degree in this way.

There is no need for any polygons or complex measurements when the answer simply comes straight out of the definition of the word.


2. Defense of Argument B.

My opponent completely drops the argument. He says that my second definition of a degree, which relates degrees and radians, is insufficient proof for the conclusion that there are 360 degrees in a circle. He does not dispute any of my premises or calculations; he completely ignores my whole proof and effectively ignores the whole argument I have provided.

He then provides a completely irrelevant comment about the arbitrariness of a degree, claiming that "I could just as easily create my own unit and call it a 'klark' and say 1 degree equals 1.6... 'klarks'"

I do not dispute that. I completely agree that the definition of a degree, or, indeed, the definitions of all units in Mathematics are completely arbitrarily defined. However, they are defined nonetheless, and those definitions have been used extensively in mathematics, and they are accepted. If the mathematical community accepts your definition of a 'klark' I will see no problem in measuring angles in klarks.

We are not discussing whether the definition of a degree is arbitrary. We are discussing whether a degree, as it is defined, leads to the conclusion that there are 360 degrees in a circle. Both of my arguments, which support this conclusion, are still standing at this point.


3. My opponent's Argument.

My opponent provides an iterative process, showing how increasing the number of sides of a regular polygon increases its internal angle. Please remember that this debate is about the central angle of a circle. Therefore, whatever my opponent has written here is actually irrelevant to the debate. Nevertheless, I will go through his argument to clear any misconceptions.

He shows that a hexadecagon has an internal angle of 157.5 degrees, that a chiliagon has an angle of 0.36 degrees and so forth.

He makes the assumption that a circle is a polygon with infinite sides. He then states that by his proof, the internal angle of a circle is equal to 179.9... degrees.

Indeed, if one assumes a circle to be a polygon, the same rule applies. It can actually be expressed by the following formula:


My opponent seemingly arrived at the answer 179.9..., which he claims to be contradictory. But he failed to realize, that this is a repeating decimal, that is, 179.999..., which can be written as 179.(9), which is exactly equal to 180.

Thus, if a circle is defined as a polygon, it can be concluded that its internal angle is 180 degrees. The sum of its internal angles is equal to infinity. This is to be expected, as a circle has no corners, or infinite sides. However, my opponent is incorrect in assuming that such a circle would have no curvature. That would only be the case if the radius of such a circle would be infinitely large, but that would be a special case known as an Apeirogon. [3]

Anyway, as the proof goes, please remember that we are debating the central angle of a circle, not the internal one, as that's exactly the angle we mean when we say "360 degrees in a circle". I ask my opponent to refrain from conflating the terms.


The Debate so far

At the moment, I am still waiting for my opponent to refute my both arguments. I.e. he should show that the definition of a degree does not lead to the conclusion of a 360-degree circle. Also, he should point out flaws in my second argument, which he hasn't done yet.

Also, I would advice my opponent to produce a constructive case, which would be based on the central angle of a circle. Otherwise, my opponent might fail to meet his burden of proof.


Sources:

[1] http://www.oxforddictionaries.com...
[2] http://www.oed.com... (See 9th definition; Library access or subscription will be required)
[3] https://en.wikipedia.org...
Debate Round No. 2
mostlogical

Pro

The definition of a degree I gave was from the Oxford English "mini" dictionary, I apologise for not making this clear. The full definition is what you have said, that is a unit of measurement of angles, one ninetieth of a right angle or the angle subtended by one three-hundred-and-sixtieth of the circumference of a circle.

I do not question that 360 degrees are used to represent the number of degrees in a circle since there are full circle protractors which display 360 degrees.

Suppose I took a flat bit of wood with a length of 10cm, then added 15 markings spaced equally along the length and declared it has 15cm. How long is the ruler I made? I'm sure you will agree that it is 10cm long. It can only be 15cm if centimetres were not already a unit used for measuring distances. I'll give another example to make this point clear, if someone along time ago said 'the Sun is 360 miles away' and it was accepted because no-one thought it is possible to calculate, but then someone worked out that the Sun is 90 million miles away, we wouldn't just say that a mile is equal to 1 mile and also 250,000 miles. Someone would be wrong! It's the same with degrees.

In order to create an instrument for measuring angles you'd first need to define the number of degrees in a straight line. Once an angle has been defined it is possible to calculate the number of degrees in every polygon including a polygon with infinite sides i.e. a circle. A triangle has 180 degrees because as you can see from the diagram below the angles A, B, and C add up to 180 degrees. Adding a side to this polygon will give the next polygon (a square) 180 degrees, adding a side to a square to make a pentagon means I must add 180 degrees to 360 giving 540 degrees, and you get the pattern. Repeatedly adding 180 degrees in an attempt to calculate the internal angle in a circle will give an infinite sum of angles, but a circle only has one side and thus one angle, so it can't have an infinitely large angle. Using the method used in round 2 will give the correct answer, in my view and shows that the definition of a degree defined by the method does not lead to the conclusion of a 360 degree circle.


s://www.mathsisfun.com...; alt="Proof the angles in a triangle add to 180" />

Image source: https://www.mathsisfun.com...

You will notice that angles will be enclosed by an arch unless they are right angles, this doesn't prove there actually are 360 degrees in a circle, the only thing it proves is that we are taught to accept this idea.

The only calculations provided by my opponent in round 1 were conversions, which do not prove anything. He does use an equation to show that the internal angle of a circle is 180 degrees, not 179.9... degrees, but fails to show how 360 is derived.

Pi iself is not needed to calculate the angle of a circle, the method used is, which is why I believe the Babylonians made up the number of degrees in a circle.

I challenge my oppoent to prove there are 360 degrees in a circle without making any assumptions, accepting only that there are 180 degrees in a straight line. I also ask my opponent to prove that the central angle as he calls it is of a circle and not a square

Many thanks







Biodome

Con

I shall now go through my opponent's arguments and clarify my own.


1. The Arbitrary Nature of Definitions.

My opponent continues to provide examples on the arbitrariness of the definition of a Degree. For instance, he starts talking about alternate realities, where a degree, a centimeter or a mile is defined differently. However, I have addressed this argument in detail in my 2nd round. I can only reiterate what I said: the arbitrary nature of Mathematical definitions has nothing to do with the Resolution of this debate. We are talking about a specific definition of a Degree, and we are using this definition to discuss a specific geometric concept - a Circle. My opponent does not address the issue with those arguments. He completely ignores my rebuttal and goes on asserting the same thing. This is a logical fallacy, known as Argumentum ad nauseam [1] and is also a red herring. [2]


2. Misunderstanding of the Nature of an Angular Measurement.

My opponent claims that in order to measure an angle, you "need to define the number of degrees in a straight line". This is simply not true. A degree is an angular measurement, which looks as follows (it is represented by θ):



A measurement of a straight line is carried out completely differently (i.e. the distance between points A and B):



I feel that it is obvious, from a mathematical standpoint, that an angular measurement, and a distance measurement, are completely different terms. Somehow, my opponent is trying to conflate them here. He needs to provide justification for his claim that a Degree can be represented as a distance measurement on a straight line. As far as I know, this would be impossible by definition.


3. Internal Angle is not a Central Angle.

Again, my opponent completely drops my rebuttal in the 2nd Round, and continues to use Internal angles as arguments for his Resolution. As I have clearly stated in the previous round, we need to be talking about Central angles, as that is exactly what the wording of the resolution suggests.

My opponent is talking about the Internal angles of polygons, which look like this (i.e. angles A, B and C):


However, I have asked my opponent to provide an argument, using the Central angle (i.e. θ):


It seems that my opponent is trying to infer something about the Central angle of a Circle, using Polygons and their Internal angles as examples. While this is not necessarily a fallacious method, my opponent has not provided any mathematical justification for his inferences, especially since he has misunderstood the concept of a Central angle altogether.

My opponent also shows a misunderstanding of Polygons and Limits. He states that "a circle only has one side and thus one angle, so it can't have an infinitely large angle". His argument can be represented by a syllogism:

P1. A Circle has one Edge
P2. A Circle has one Internal Angle.
C1. Therefore, a Circle cannot have an infinitely large Angle.

All of the premises are demonstrably false.

Let's assume that a Circle is a Polygon. Then, it is a Polygon with infinite edges (or "sides"), as I have demonstrated with my Limit formula in the previous round. This follows from the fact that in order to construct a Circle via the Polygon iteration process, the amount of Edges has to be increased to infinity (while keeping the Perimeter constant). [3] Secondly, an Edge cannot be curved, as it is by definition a line segment, which is always straight. [4] Therefore, P1 is completely and utterly false.

The second premise of my opponent can be negated by observing that each Internal angle of a regular polygon corresponds to a Vertex. A Vertex is located at the intersection point of two Edges. [5] Thus, if we assume that a Circle is a regular polygon with infinite Edges, it follows that it has infinite vertices, and, therefore, infinite internal Angles.

Although my opponent's Conclusion never follows from his incorrect premises, it is mathematically true. Observe that for any regular n-sided polygon, its internal angle is calculated as follows [3]:



Thus, if we take a limit of this, as n-approaches infinity (which is essentially a Circle), then we see that the internal angle of a Circle has a measure of exactly 180 degrees, not infinity. Moreover, if we add up all these angles, we get infinity, because a circle has an infinite amount of angles.

However, any of this has nothing to do with the central angle, and thus my opponent's whole argument is negated.


4. Final Rebuttals.

My opponent asserts that "angles will be enclosed by an arch unless they are right angles", concluding that this somehow shows that "we are taught to accept this idea [of 360 degrees in a Circle]".

I fail to see any logical or mathematical coherence within this argument. I might just negate the whole premise by showing you a right angle that is enclosed by an arc:



Secondly, my opponent disputes my calculations in Round 1 by stating that my conversions "do not prove anything". I suppose he is talking about the conversion from Radians to Degrees, but I fail to see his justification for this assertion. All of my conversions adhered to the definitions of both Radians and Degrees. Therefore, they are consistent within Mathematics, and successfully show that there are 360 Degrees in a Circle. I challenge my opponent to point out a specific flaw in those calculations. He has to show that they are mathematically or logically inconsistent, but he hasn't done so yet.

Thirdly, my opponent says that I have demonstrated that "the internal angle of a circle is 180 degrees, not 179.9... degrees" but that I have failed to "show how 360 is derived".

To start with, I didn't show that "the internal angle of a circle is 180 degrees, not 179.9... degrees". I actually showed that the internal angle of a circle is both 180 and 179.999... degrees, which are mathematically equivalent expressions, thus, not contradictory. [6] The reason why I didn't derive any "360" from this, is because I have explicitly stated that the argument from Internal degrees had nothing to do with the Central angle of a circle. Therefore, my whole refutation was simply due to my mathematical interest in clearing up any confusion. It was never supposed to be an argument, because my opponent's argument was a fallacious red herring in the first place. I demonstrated the derivation of 360 degrees in my 1st Round arguments A and B, which my opponent has still not managed to refute mathematically or logically.

Fourthly, my opponent concedes that Pi is not needed to calculate the angle of a circle. Thus, I stand by my assertion that Babylonians could have used the same argument, which I used in my 1st Round (Argument A), since they were the ones who defined a Degree.

Finally, my opponent challenges me to prove that "there are 360 degrees in a circle without making any assumptions". I have done so in Round 1, with both argument A and B. I did not make any unnecessary assumptions, but I had to make assumptions of basic mathematical axioms, such as Peano axioms. I sincerely hope that my opponent will not dispute these, as this is, by nature, a mathematical debate, thus, fundamental axioms should be assumed.

I have demonstrated how, using the definition of a Degree, one may conclude that there are 360 degrees in a Circle (Round 1, Argument A). Also, I have demonstrated how, using the definition of a Radian, and the relation of Radians and Degrees, that there are 360 degrees in a Circle (Round 1, Argument B).


Conclusion so far

My opponent has failed to satisfactorily challenge any of my arguments. Therefore, I conclude that my opponent has still not met his burden of proof, while my arguments still stand strong and unrefuted.


Sources:

[1] http://rationalwiki.org...
[2] http://rationalwiki.org...
[3] https://en.wikipedia.org...
[4] https://en.wikipedia.org...
[5] https://en.wikipedia.org...
[6] https://en.wikipedia.org......
Debate Round No. 3
mostlogical

Pro

In order to make angle measurements it is essential to first define a straight angle, that is to say the total number of degrees on one side of a straight line. A straight angle has 180 degrees, and just by knowing this I have shown the interior angles of a triangle add up to 180 degrees. The central angle is the sum of angles on both sides of this line. We are taught to imagine the central angle takes the form of a circle, when in reality it can be any polygon. Since the angles add up to 360 degrees like a square it makes more sense to think of the central angle as a square.

To counter my arguments my opponent relies heavily on definitions, which he assumes to be true because the Babylonians assumed them to be true. He wishes for me to talk about central angles where it is easy to imagine a circle having a central angle of 360 degrees, especially when we are taught to do so. However, if we accept that a circle has 360 degrees then we must accept that there are 360 degrees in a pentagon, hexagon, septagon etc because they also have a central angle of 360. We wouldn't say a pentagon has 360 degrees like a hexagon, we would say a square has 360 degrees though.

The only way it can make sense for a circle to have 360 degrees is if:
a) circles do not have an internal angle, or
b) the internal angle of a circle is 360 degrees.

I have shown that circles do have an internal angle, which is equal to 179.9...

Despite definitions saying otherwise, a circle is a polygon with infinite sides. If this wasn't true we wouldn't be able to calculate Pi. A circle can have an infinite number of edges yet be curved, because infinite edges become smaller and smaller into points just like those that make up a straight line, which is probably why there are 180 degrees.

If I look at a regular polygon with 3 million sides, each interior angle equals 179.99988, meaning the sum of it's interior angles = (179.99988 x 3 million) = 539999640 degrees. Increasing the number of sides will increase the sum of interior angles. By increasing the number of sides I can establish the 4th decimal point of an infinite sided polygon, and the one after that and so on, each decimal point is found to be a 9 followed by another 9. When there is a flat angle 179.9...(or 180) and an infinite number of angles of 180 degrees, there are no other sides! This is why the sum of interior angles would be 179.9... or 180 degrees, and why a circle has 179.9... degrees, rather than 360 degrees.
Biodome

Con

A. Proof from a 180 degree angle

My opponent concedes that a straight angle has 180 degrees. Note that from this it is trivially easy to prove that there are 360 degrees in a circle.

I can produce an illustrated proof. Here is a straight angle of 180 degrees. My opponent agrees with this:


Let's now add two such angles together:

We obtain a full angle, which is a circle. And we can conclusively deduce the amount of degrees in this circle: 180 degrees + 180 degrees = 360 degrees.

Let's this be an additional, 3rd proof, which must be negated by my opponent.


B. Rebuttals


1. My opponent is ignoring my rebuttals in the previous round.

My opponent again conflates the angle terminology. Please note that I am not arguing about how many degrees there are in the internal angles of a circle. I have already discussed this in the previous round, and I still maintain that this is completely irrelevant to the resolution that we have. My opponent must produce an argument, which would support his claim that there are not 360 degrees in the central angle of a circle.

Frankly, I am tired of this back and forth arguing about internal angles. It is completely unnecessary and irrelevant, and I find it puzzling why my opponent completely ignores this fact. I will just refer to my arguments in the previous round, as they are sufficient to completely negate whatever my opponent stated in this current round.


2. Regular polygons have a central angle of 360 degrees as well.

My opponent seems to think that the circle cannot have a central angle of 360 degrees, because that would mean that other regular polygons would also have a central angle of 360 degrees. Quoting him, "We are taught to imagine the central angle takes the form of a circle, when in reality it can be any polygon"

This is indeed true. The central angles of other polygons may also be deduced similarly. They are equal to 360 degrees, because they can be subtended by complete arcs. Here are a few examples:

If we assume that a circle has a central angle of 360 degrees, which, I believe, I have successfully shown during this debate, then by simply superimposing other polygons onto that circle we can see that their central angles are also 360 degrees. The type of regular polygon itself doesn't matter - it can be a triangle, a square, a hexagon etc. If we take its central angle, it is still 360 degrees.

My opponent keeps referring to the internal angles of these polygons. Please note that, again, those angles are completely irrelevant and any arguments by my opponent discussing only these internal angles should be ignored.


3. My opponent misunderstands the nature of definitions.

My opponent (again) accuses me of relying on definitions, which he believes are questionable and arbitrary: "my opponent relies heavily on definitions, which he assumes to be true because the Babylonians assumed them to be true."

I must again reiterate my stance of this, which I have analyzed in detail in the 2nd Round, and which I also restated in my 3rd round. Definitions cannot be true or false. They are arbitrary constructs of Mathematics, which help us deduce real knowledge. A degree was defined as a 360th part of a complete circle, therefore a degree is the 360th part of the circle. This definition has been accepted by mathematicians, and it means exactly what it means. If my opponent prefers to use other units for angle measurement, he is free to do so: he may use radians, gradians, minutes, seconds, turns - whatever. But that doesn't actually make any difference to what a degree is, nor does it challenge any of my original arguments. I ask my opponent to acknowledge that the arbitrariness of a degree has no impact on this debate.


4. More bare assertions by my opponent.

My opponent asserts that:

The only way it can make sense for a circle to have 360 degrees is if:
a) circles do not have an internal angle, or
b) the internal angle of a circle is 360 degrees.

Note that the whole premise is false, because my opponent does not realize that central angles and internal angles are completely different concepts. I have shown that:

a) Circles have infinite internal angles of 180 degrees
b) Circles have a central angle of 360 degrees.

I must once more emphasize that my opponent's argument makes no mathematical or logical sense.


My opponent also asserts that the internal angles of a circle sum up to 180 degrees. He uses a very dubious and mathematically incoherent reasoning:

"When there is a flat angle 179.9...(or 180) and an infinite number of angles of 180 degrees, there are no other sides! This is why the sum of interior angles would be 179.9... or 180 degrees, and why a circle has 179.9... degrees, rather than 360 degrees."

I see no justification for these bare assertions. My opponent states that a circle has just only one side, and makes a false conclusion about internal angles of a circle.

This is problematic for several reasons:

a) This debate is not about the internal angle of a circle (yet another conflation of the terms Central angle and Internal angle).
b) My opponent completely ignores my proof in the last round, which shows that a circle has infinite sides (or edges).


Conclusion thus far

My opponent has virtually produced no arguments at all during this debate. All of his attacks were either based on:

a) Internal angles
b) The arbitrariness of definitions

I have shown that both (a) and (b) are completely irrelevant to this debate. He has not produced any reasons why a circle does not have a central angle of 360 degrees. Nor has he shown why my arguments fail.


My opponent still has the last round, and I will give him one more shot at producing a logically justified argument. For now, I believe that my case is clearly better here.
Debate Round No. 4
mostlogical

Pro

Thank you for providing images, (I will learn how to put them on one day). I have explained that the central angle is the sum of angles on both sides of a line, and because a straight angle has 180 degrees, the central angle has 2 x 180 = 360 degrees. I have never said that a circle cannot have a "central" angle of 360 degrees.

The number of degrees a polygon has is the sum of its interior angles. To calculate the number of degrees a polygon has (including a circle) we add all the interior angles together, e.g. a pentagon has 5 x 108 = 540 degrees. This method is used to define the number of degrees a shape has, strangely my opponent does not wish to talk about interior angles, believing them to be irreverent.

With an infinite sided polygon (a circle), one of it's infinite interior angles is 179.9... which mathematicians say is another way of saying 180 degrees. Multiplying 180 degrees by infinity or one (number of sides) will give 180 degrees. I understand that any number x infinity equals infinity however when you have a straight angle it doesn't matter how many times you multiply it by; it can't produce any more angles, as any additional sides will be parallel.

The central angle is not really a shape, we are simply taught to believe it is and that the shape is a circle. From an early age we draw a number of small arcs inside polygons which make up a circle, so we imagine a circle has 360 degrees. If the sum of the interior angles of a circle were 360 exactly like a square has 360 degrees then it would be correct to say a circle has 360 degrees, but the internal angle of a circle is 180 degrees. If I say 'the central angle "is" a square' that would make sense. But it wouldn't make much sense to say 'the central angle "is" a circle'. When someone says a circle has 360 degrees what they are saying is the central angle"is" a circle.

A circle is defined as being a 360th part of a circle. This definition has been used ever since Babylonians imagined the central angle to be a circle due to being unable to calculate the internal angle of a circle. Imagination doesn't make something a fact. If they imagined the central angle to be an octagon and chose to define an octagon as being a 360th part of an octagon it wouldn't mean an octagon has 360 degrees when they have defined a straight angle as having 180 degrees. So why would a circle have 360 degrees?

I understand the difference between central and interior angles, this should be evident when I calculated interior angles. It would be extremely difficult to calculate interior angles without any knowledge of central angles.

Infinite angles of 180 degrees means a shape has exactly one angle of 180 degrees,

Proving a circle has a central angle of 360 degrees or any shape for that matter does not prove it has 360 degrees. E.g. I could prove an octagon has a central angle of 360 degrees but it wouldn't prove an octagon has 360 degrees.

To sum up, the number of degrees a shape is said to have is determined by the sum of its interior angles, which in the case of a circle is 179.9... (or 180) degrees, and not 360 degrees. This conclusion has been reached by using mathematically coherent reasoning. I tank my opponent for having this debate, and ask voters to vote on this basis.
Biodome

Con

As this is the last round, I will try and summarize the debate.

What was the debate about?

The resolution of the debate, as well as the original argument that my opponent put in the 1st round clearly implies that we are discussing the degrees in the central angle of a circle. It makes no sense to assume that we are discussing the internal degrees of a circle because of two reasons:

1. The usage of the phrase "360 degrees in a Circle" is always in the context of the central angle.

When one says "There are 360 degrees in a Circle", or "There are X degrees in that particular arc" or "There are X degrees in that particular section of a Circle", one always implies that he is talking about the central angle. There is no reason to think that one is talking about internal angles. For instance, if you look up this phrase on Google, all of the sources, at least in the initial pages, will refer to the central angle. [1] The meaning of this phrase is simply accepted and consistently used throughout the context of Geometry and regular speech, i.e. "He did a 360 on his skateboard".

My opponent is simply using definitions and meanings which are completely unjustified. He has, during this whole debate, starting on Round 2, talked about internal angles, completely ignoring my rebuttals that internal angles are completely irrelevant to the debate. It is cheating and unjust behaviour if during the debate you are suddenly turning the topic upside-down without explicitly noting in Round 1 that you will be talking about internal angles.

I therefore ask voters to ignore my opponent's arguments, which were based on internal angles, because they are irrelevant, and my rebuttals were completely dropped and not once contested.

2. It is not obvious that Circles are regular polygons.

While a mathematician might certainly refer to a circle as a polygon, this characterization is by no means obvious, and not necessarily true, because it results in some geometrical paradoxes relating to curvature. One should by no means assume, finding a debate on circles, that this debate will require the assumption that a circle is a polygon, especially if that is not clearly stated in the 1st Round by the instigator.

I am aware that I did not contest this assumption, on the grounds that internal angles were irrelevant to the debate anyways. However, this is still another reason why this debate should have been about central angles, not internal angles. I therefore also ask the voters to take this into account.

---

These two reasons explain why the constant jumping between different definitions of the words in the Resolution, as well as conflation of terms was completely unjustified, non-obvious and a red herring.

Note that my opponent never provided any argumentation based on the central angle of a circle, which he should have, since that's what was expected of him during this debate.


Summary of Arguments

I have provided three constructive arguments during this debate:

1. Argument from the definition of a Degree.

2. Argument from the mathematical relation between Radians and Degrees.

3. Argument from a straight angle.

My opponent tried to contest Argument 1, however, he clearly failed in constructing a good rebuttal, because he was assuming that the arbitrary nature of a degree was somehow relevant to the debate. I have successfully shown that this is not the case. He still continued repeating different variations of the same argument, completely failing to address my rebuttals.

My opponent tried to contest Argument 2 by stating some vague observations about how my proof was unsound. I challenged him to provide clear and specific examples, showing where I went wrong. He didn't do that and completely dropped that argument.

My opponent conceded that a straight angle has 180 degrees. I used this fact in order to construct my third argument, where I simply added two straight angles to produce a full circle. My opponent never contested that argument.

To conclude, I would like to ask voters to recognize the fact that these arguments were dropped or never adequately contested during the course of this debate. Therefore, I would maintain that I have successfully defended my case, and that I therefore have met my Burden of Proof.


Last Round Rebuttals

My opponent completely ignored my concerns about the relevancy of internal angles to this debate, and he still chose to use the final round for the discussion of internal angles. Since he never actually addressed my earlier rebuttals, I will simply declare his arguments as yet another argumentum ad nauseam. [2] I could certainly go on showing how his thoughts about internal angles do not make sense in the context of circles. I could also show that his disbelief in the fact that "any number x infinity equals infinity" is unsound and that his use of Mathematics during this debate has been thoroughly and fundamentally flawed. However, since all of his arguments are simply based on internal angles, I will continue to ask voters to look through my earlier rebuttals as references, since they were never addressed.


Vote CON!


Sources:

[1] https://www.google.co.uk...
[2] http://rationalwiki.org...
Debate Round No. 5
25 comments have been posted on this debate. Showing 1 through 10 records.
Posted by mostlogical 1 year ago
mostlogical
thanks
Posted by whiteflame 1 year ago
whiteflame
Working on it, guys. I'll have a vote up over the next couple of days.
Posted by Raisor 1 year ago
Raisor
This debate turns on whether the number of degrees "in" a circle refers to the central angle or the sum of interior angles.

I found Pro's case very confusing, it was hard to understand the case he was making, and I still don't understand what he was trying to gt at with some of his arguments.

Con was consistently clear, and when Con finally understood that Pro was arguing about the sum of internal angles, not the central angle, he made a clear argument that the Resolution should be taken to refer to the central angle. Con's arguments are pretty straightforward- b definition the central angle of a circle is 360 degrees.

Pro should work on clearly communicating his arguments, I wasn't able to figure out your basic thesis until maybe the third or fourth round.

Clear Con win.
Posted by F-16_Fighting_Falcon 1 year ago
F-16_Fighting_Falcon
****************************************************************************************
>Reported vote: TheJuniorVaristyNovice // Mod action: NOT Removed<

Voted for Pro (Select Winner). Reasons for voting decision: none of con's arguments show how people came to the conclusion that we should have 360 degrees in a circle, he rather shows that according to currently accepted mathematical rules circles must have 360 degrees. Pro wanted con to show that there is purely mathematical and non arbitrary reason that a circle is considered to have 360 degrees. Con doesn't do this and pro let's it slide casually along the whole way so as to allow con to concede the very point of his argument, I would be surprised if this is not what pro wanted, and if it is not them I ask pro to message me with more information so that I can re-evaluate the round. But as of now the win is easily given to pro because con failed to tend his R1 duties and thus loses.

Report: A lot of arguments were not analyzed. Also he actually misrepresents my position.

[*Reason for non-removal*] (1) The voter explains why he felt that Pro's arguments affirmed the resolution and why Con's refutations were insufficient. (2) We don't remove votes because the debater has a substantial disagreement with the voter. (3) Also, consider this a pre-emptive warning that votes must be made based on the debate rounds and shouldn't be re-evaluated based on private messages.
*****************************************************************************************************
Posted by Biodome 1 year ago
Biodome
@TheJuniorVarsityNovice

I find your RFD inadequate. Could you please assess ALL arguments put forward by both sides? I did not understand why you think that my use of mathematical definitions fail to qualify as an argument. I.e. you have never thoroughly analyzed why my 3 arguments fail against Pro's rebuttals, especially after he conceded to using the same definitions as me (i.e. look at Argument 3). Are you suggesting that I should not have used the accepted definition of a degree? How else could I have defined that term? Could you elaborate on how I actually "failed to tend [my] R1 duties"?

Thanks in advance.
Posted by mostlogical 1 year ago
mostlogical
#Cottrill - a square has 360 degrees because you multiply one of it's interior angles by the number of sides. With an infinite sided polygon one of it's infinite interior angles is basically 180 degrees (see calculations), multiplying 180 degrees by infinity will give 180 degrees. You may disagree here, I understand that any number x infinity = infinity, but when you have a straight angle it doesn't matter how many times you multiply it by because it can't produce any more angles, as any additional sides will be parallel.
Posted by cottrill 1 year ago
cottrill
#mostlogical - If we are declaring that a shape "has" as many degrees as its internal angles (a square has 360, a triangle has 180), and we are declaring that a circle has and infinite number of internal angles, then it has to follow that a circle has an infinite number of degrees.

To put it as a syllogism:

Shapes have degrees equal to the sum of their internal angles.
Circles have an infinite number of internal angles.
Therefore circles have infinite degrees.
Posted by mostlogical 1 year ago
mostlogical
True, every polygon (including a circle) has a central angle of 360, but we wouldn't say an octagon has 360 degrees. We would say a square has 360 degrees because the sum of interior angles equals 360 degrees.

Some think that a circle only has a central angle because it is has curved lines, or because they like yourself think that there are an infinite number of internal angles and so think it's acceptable to say a circle has 360 degrees. While my maths hopefully shows there are an infinite number of internal angles, they are straight angles meaning a circle has 179.9... or 180 degrees.
Posted by cottrill 1 year ago
cottrill
#mostlogical - Don't let me distract you from the debate, but I was enjoying the back and forth we were having.

I think the issue with your argument boils down to your conflation of internal angles with central angle. A circle has an infinite number of internal angles, meaning that, yes, you are correct, there are not 360 degrees worth of internal angles. There is, however, 360 degrees worth of angle using the center of the circle as the point of origin.
Posted by Ragnar 1 year ago
Ragnar
@Mostlogical
This may seem strange, but I advise talking less in the comment section while the debate is ongoing. Voters can't give you any credit for anything you write in it, and it's easy to think you already made a point in an argument when in fact you only stated it in the comments.

Due to the nature of this debate, I also suggest including a link to the voting standards in your final round. If anyone disagrees with you, is a matter separate from how well you argued and supported your claim.
3 votes have been placed for this debate. Showing 1 through 3 records.
Vote Placed by whiteflame 1 year ago
whiteflame
mostlogicalBiodome
Who won the debate:-Vote Checkmark
Reasons for voting decision: Pro's arguments just continued to confuse me throughout the debate. Perhaps, if the debate was "there aren't 360 degrees total among the internal angles of a circle" he would have gotten somewhere. But that's not the topic. All Con had to show was that 360 degrees exists somewhere in the circle. He managed that. The definition of what is a degree is inherent to the resolution, so all I have to do is determine if Con has found a way that the circle represents 360 degrees. He did that via the central angle, something Con never contests. Even if I believe that the internal angles don't add up to 360 degrees, I don't need to. Con has satisfied his burdens in this debate by showing that the central angles = 360 degrees. The definitions don't help Pro, and neither does the lack of response to Con's proof. The line proof that Pro himself buttresses ends up just sealing the deal. Con has proven that there are 360 degrees in a circle, ergo that's where I vote.
Vote Placed by Raisor 1 year ago
Raisor
mostlogicalBiodome
Who won the debate:-Vote Checkmark
Reasons for voting decision: RFD in comments
Vote Placed by TheJuniorVarsityNovice 1 year ago
TheJuniorVarsityNovice
mostlogicalBiodome
Who won the debate:Vote Checkmark-
Reasons for voting decision: none of con's arguments show how people came to the conclusion that we should have 360 degrees in a circle, he rather shows that according to currently accepted mathematical rules circles must have 360 degrees. Pro wanted con to show that there is purely mathematical and non arbitrary reason that a circle is considered to have 360 degrees. Con doesn't do this and pro let's it slide casually along the whole way so as to allow con to concede the very point of his argument, I would be surprised if this is not what pro wanted, and if it is not them I ask pro to message me with more information so that I can re-evaluate the round. But as of now the win is easily given to pro because con failed to tend his R1 duties and thus loses.