Fifth Axiom (extra footage) - Numberphile 

Kram1032 playlist says "(more videos coming soon)"

KarstenOkk I know. I just hope it'll be a *lot* more :D

Kram1032 MOOAAARRRR!!!!!

MrCybin EXACTLY!

Haniff Din Geometric Algebra is a cool way to describe various geometric spaces, but it's in no way the only and certainly not "the true" mathematics. As powerful as Geometry is, it already has a ton of structure that could merely be unnecessary luggage for the description of some systems, overcomplicating things in the process, or it could downright be incompatible with others.

Frame of reference is key when dealing with this topics.

KillMeSeason Yes!

KillMeSeason I agree, and the topic is so interesting

He is very down to earth. Unlike some others featured on these videos sometimes

My favorite is still James grime, but that's never going to change. He hasn't been featured in any of the recent ones though

whydontiknowthat yeah the singing banana is great ;)

Fantastic reference. I love it.

Adrian VanRassel Go look up the "metric", this defines the space.

+Adrian VanRassel But Toph, you're blind.... You want us to write on earth??

Callme Enkay You're damn right I want it on earth!

StunFlash Idk why but it sounds reaaally strange when i red it.

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collopa1 ?

collopa1 go outside kid

maybe by intention? :P

Dixavd You know that graphics computation comes mainly from mathematics... Right? If you don't, then look up on google the source codes. They comply mainly of calculus and geometry.

gtheskater Thanks for the heads-up but I am aware of that (I'm a physics student with a potential career in game design). Regardless, thanks for reminding me (or others) anyway.

Oh right. Cheers for this non-aggressive denouement.

minch333 You the best thing about affine geometry, it's a Fine geometry. Sorry I had to. But more to the point, really what affine geometry gets rid of is the notation of an origin while maintaining the concept of the metric. So the third axiom is still in important. What is preserved in affine geometry is ratio of lengths.

minch333 So I thinking more about your statement. So really what the third axiom forces you to have is a metric. It defines a circle as the set points equal distance from a point. So in hyperbolic geometry, this is a hyperbola while in Euclidean geometry it's a circle like we think of it. So any geometry with a metric needs the third axiom, I think the geometry you are thinking of is projective geometry which is geometry without a distance. You can actually see this pretty easily. So if you have a circle in a plane, than project into a "3d" image it can look like an ellipse, but is equivalent to the original circle.

minch333 You not being mean, just mostly wrong. Though I was also mistaken about affine geometry as well. So projective geometry is where you ignore the third and fifth axioms. This is because the third axiom implies you have a metric, because if you have distance you have a metric, if you have a metric you define POset. If you include the fifth axiom you get affine geometry. Now where you are getting confused about projective geometry having a metric is there is a projective metric, but this is additionally structure added to a subset of the projective plane, that will allow you to define Hyperbolic, Euclidean, or Elliptic geometry. The same thing is possible in affine geometry(obviously), but you can only define Hyperbolic or Euclidean geometries(Though locally all theses geometries look Euclidean). So really Projective and Affine geometry aren't really geometries in the same way Hyperbolic, Euclidean, or Elliptic geometry are. Projective and Affine have less structure then the other 3, while Hyperbolic, Euclidean, or Elliptic geometry have the same amount of structure(i.e. they have lines, and planes, they are all metric spaces, and we can define what a parallel line is, and thus do parallel transport). So I would also say that the video misspoke when they said the got rid of the fifth axiom, rather they modified it.

Saki630 I did... The only thing I didn't understand is the relation between borders on euclidean geometry and hyperbolic geometry. In euclideans' geometry, the border is a line between two points, but in hyperbolic geometry, the border is a line where the two points are the same... I understand it, but WHY join the two points?

Models of hyperbolic plane shown in this video (Poincare disk model and half-plane model) are conformal, which means that two curves intersecting at angle alpha will be represented as curves intersecting at angle alpha. So, simply draw two hyperbolic straight lines, and if they appear to intersect at a right/acute/obtuse angle, they actually do. Look up tesselations of hyperbolic plane, you will see lots of acute angles there, and some obtuse and right ones too.

1. Polar coordinates, or a similar system 2. Normally, just in a polar-centric system 3. An Euclidean circle.

Yes now I see :-) It was hard to think about it, as i am used to the infinity on the x,y,z. But the 3rd one doesnt really makes sense, only if I merge two globes in 3D x,y,z coordinates, but there is no such a thing, so you cant do that. By the way, thanks for answering :-)

Sorry, I didn't see the plane. Yes, it wouldn't make a circle.

No, it states that a logical system cannot be self-consistent if it is complete. The converse statement is also true. Most mathematicians agree that our axioms are incomplete, not contradicting.

Ratstail91 I would gladly read a source for your claim that the universe is non-euclidian. I know that it is locally euclidian, but speaking globally about the universe would be pure speculation I guess.

Ratstail91 I like this definition, because in any other space (literally) that isnt true of triangles. I cosign this.

Ratstail91 But locally at least the universe appears to be flat.

dexter9313 Disclaimer: This post is very simplified - mostly for my own sake - and might contain inaccuracies. But numberphiles are usually good at correcting mistakes, so it will probably be sorted out swiftly. In general relativity, gravity is interpreted as the curvature of space time. Newton stated that the gravitational force between two bodies depends on the mass of said bodies, and the distance between their centers of mass. This works great for stuying things that have mass, and how they interact and change their trajectories through the universe. But then we find a problem. Light also changes its trajectory if it gets close enough to a body with great mass, even though the light itself doesn't have mass. The interaction of masses can't explain this. If massive objects instead curve space time, any trajectory can look curved if seen through the perspective of euclidian geometry, despite actually being the straightest possible path through a curved geometry.

I see, you were speaking about General Relativity. I, like Deuce1042, thought you were talking about the "flatness" of it, I mean the actual curvature of the Entire universe. (Is the universe the enveloppe of an hypersphere for example ?) On that question we have no answer yet, but we measure it flat locally, in the known universe.

fcolecumberri ... um... what's your point?

fcolecumberri we're all aware, see the little lock on the title? that lets you know it's private

fcolecumberri Yes. From my understanding Brady does that on purpose to not clog people's subscription boxes.

EGarrett01 thanks

Then you break all of real numbers as we know them. At least I think so.

Aeroscience "1+1=2" actually is not used as an axiom. The natural numbers are set up according to the "Peano axioms" (unless I'm mistaken -- my path diverged from math a few years ago), which mainly involve two objects, the first an object "0" designated to be a natural number, and the second a "successor function" S. The axioms delineate particular properties of these objects (e.g., there is no number whose successor is 0 (i.e., S(0) does not exist), meaning that 0 is the "smallest" natural number) so that the set of natural numbers emerges from them. Addition can be defined in Peano terms, and the symbols "1" and "2" are just names for S(0) and S(1)=S(S(0)). The oft-asked question "Why does 1+1=2?" is really a misguided one. Interpreting the question to mean "Why is it that 1+1 equals 2 instead of, say, 3?", then it's all just a matter of naming. "1", "2", and "3" are just names/symbols used to designate objects that exist as part of an abstract structure. The structure that is the natural numbers, together with the additional structure of the addition operation, have the property that there is an object, which we designate "1", that when added to itself coincides with the object designated "2". You could say "1+1=3", but that would only be true if all of the definitions of your names/symbols/operations accurately lined up with the underlying structure (in this case, the name "3" would refer to the object conventionally labelled "2"). As for "getting rid of" axioms, keep in mind that the system of numbers that we're all familiar with is merely a set of objects. There are no "foundational" axioms for math -- Russell and others tried, and Gödel proved it a fool's errand. Math is the abstract study of structure. It is the study of the interrelatedness of objects, whatever they may be, based on whatever properties are declared for them. "Getting rid of axioms" isn't, perhaps, the best phrasing, because it's not really a matter of throwing things away; the matter concerning Euclid's 5th is that, by including it in the system or not, different structures arise with properties that are different, and different in interesting and significant ways, and those differences point to a higher-level, more abstract structure involving the idea of curvature.

Aeroscience IDTIMWYTIM

Hmm, well in a binary field this is kind of true, 1+1=0.

James Owen Where are you getting that idea from? 1+1=10 in binary

I still don't understand though. Am I trying to visualize a 90 degree angle on a spherical plane? intersecting a spherical line? Is this hyperbolic world even applicable to our 3-dimensional world?

Truth is not exactly 'metaphysical', I do not think he was using that word correctly, but the (presumed) meaning of this phrase is. Truth can be analyzed at different layers of abstraction. So Truth, in a sense, is metatrue.

racoiaws I am curious about this as well. I have heard of and read about this before, and I have nothing against fun thought experiments, but I am unaware of any practical use. I would be very excited to find some though! =D

Flexi Co Physics, in particular cosmology and relativity.

racoiaws "The axioms surely aren't just arbitrarily being changed for the sake of a thought experiment, right?"Actually, largely, that's how it works :) Mathematicians try and change up various rules and explore what new relationships between abstract mathematical objects that would imply. Initially that produces nothing but a fun idea, something that might entertain you and your fellow mathematicians, but would seem utterly useless to an outsider. But soon as you find a real-world problem that can be studied using your new method and the tools you've developed, that's when the playful idea turns into gold. For example, non-euclidean geometry has vast applications in map making and even general physics. In fact, physical theories such as relativity are entirely expressed in terms of non-euclidean geometries. This then enables you to build super-high-precision clocks and systems such as GPS (which absolutely rely on these tools). So you see a playful, seemingly pointless idea can later turn out to be absolutely vital to technological progress.

Flexi Co Radionavigation is a big one, the old LORAN-C radio-compass system that ships used to use required hyperbolic geometry to produce the charts.

racoiaws Relativity uses hyperbolic functions, and as he was saying, we often deal with things that aren't flat like Euclid's plane. A hyperbolic plane is curved like a saddle. If you draw some straight lines on a piece of paper and then fold it into a saddle shape those lines won't look so straight anymore. As for what you said about axioms, you can get rid of any axiom as long as you can create a logical system without it. Axioms aren't fundamental, they're just assumed. You must have axioms, and there aren't a finite number of axioms, but there's nothing fundamental about the 5th axiom in particular.

-8 Nicholas Ivanovich Lobachevski is his name! -8

Crizzl I believe he said "any N greater than 6".

Phillip H I'm hearing a t in there but maybe I'm going insane.

I kind of hear it too, but it's probably safe to chalk it up to a slight stutter in the pronunciation of the word.

You're probably right. Your interpretation makes more sense as well.

Crizzl Sticks=11 obviously

***** *"In math you cant just do whatever you want."* Euhm, you can. *"Everything you want to do needs to consist with one another."* Sure, the episemological must be consistent with itself. *"The results from mathematics with just a few axioms are huge and indepenend of human creation. That's the beauty of mathematics."* That does not mean mathematics is not invented. Of course you will come to the same conclusions using the same rules.

TheLeftLibertarianAtheist I agree with MrLuchtverfrisser. The patterns, properties & structures math investigates are DISCOVERED. Our methods of understanding those patterns, properties & structures are INVENTED.

Kaelyn Willingham Eh, we invent the patterns...

Markus Uh, no we don't. Patterns are self-evident in nature. As well as in the mathematical universe. A prime example of this is the Fibonacci sequence. A formula for listing the terms of the Fibonacci sequence was created, but only after the underlying patters that led to the creation of that formula were discovered. Patterns exist independent of human existence. Just like science does. Those things aren't created, they just 'are'. Axioms are "self-evident truths". We claim them as true independent of proof. What math does is take those "self-evident truths" and find relationships. Those relationships are true independent of our observation. How we interpret & understand those relationships is what we create. So really, math is BOTH. It's discovered AND created. We discover the relationships between different sets of objects, and create interpretations of them. Those creations in turn lead to new discoveries.

No, these are axioms, they are talking about straight lines, and flat surfaces (planes). They do not preclude the existence of curves. Furthermore, there is no real difference between a curve and a straight line.

+Hawk SilverDragon Because the Universe may simply appear to be Euclidean, but in actuality be non-Euclidean. These are postulates. Which means they're simply accepted as true, but not necessarily provable. When Euclid made his postulates he was assumed to be correct simply because we couldn't comprehend of another way to look at the universe. Now with non-Euclidean geometry, we know there are other possibilities, but we still can't prove one over the other.

The postulate of Euclid is provable all we have to do is look at our houses, or our skyscraper buildings, or one of the best examples is the pyramids. if Euclid's geometry does not actually exist than neither would any of these structures, but they do! What does not actually exist is the Global reality that we are force fed as children as reality. It is a Santa Globe!

You're confused. Geometry and structures and lines exist. What's being postulated is how to describe parallel (and straight) lines. Your reality isn't at stake here, just your understanding of triangles on curved planes and thus the general equations that describe the universe.

You are contradicting yourself, if a plane can be considered a plane then it cannot be curved. IT MUST BE FLAT!

A plane can be curved through a 3-dimensional space, just not from the perspective of the plane itself. But you're aiming for low hanging fruit. Would you like me to call them curved surfaces or two-dimensional manifolds?

They are the shortest lines if you are only allowed to go on the surface of Earth (not trough Earth) -- they are straight in this sense.