Fifth Axiom (extra footage) - Numberphile
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- Опубліковано 10 кві 2015
- Some extra footage not used in our Fifth Axiom video with Dr Caleb Ashley.
Main video: • Ditching the Fifth Axi...
bit.ly/HyperbolicGeometry
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0:32 "We've done a violent act ... we've allowed numbers to be used as coordinates"
Did I hear that right? This is gold.
I hope there will be a lot more on non-euclidean geometry.
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.
From our perspective, the lines in hyperbolic geometry bend - but from the plane's perspective, the lines are straight - it's the plane itself that bends. It means that distance and direction aren't absolutes in hyperbolic geometry, and change with your frame of reference. A flat surface in hyperbolic geometry is concave in planar geometry.
Frame of reference is key when dealing with this topics.
I get that the N1 video is direct and clear and skips a lot of this extra N2 footage, but the way Dr Ashley talks about this is just incredible. Thanks for putting up the rest of this conversation. I hope to have the chance to hear more like this from him in the future.
All these circles make a square.
*All these circles make a square.*
*ALL THESE CIRCLES MAKE A SQUARE.*
Fantastic reference. I love it.
Nice video! Love this guy, more from him please!
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 ;)
Great presenter, hope you've got more from him.
I'd love to know the actual math behind translating euclidean geometry to hyperbolic geometry.
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!
This guy is great. I hope to see more contributions by him to the channel.
I was just looking at the suggestions below, and I realised that the greatest part of Numberphile (2) is not just the nice numbers/math(s) and explanation, but the huge variety of really interesting mathematicians.
bringing it to us smoothly
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
I hope there will be more videos of Dr Caleb Ahsley. He seems a bit nervous video (I suppose it's his first) and with routine and his very nice way of presentation there will be awesome videos.
I love the way he kind of chuckles when he says "we said parallel lines are lines that don't intersect...so they don't intersect" Like he knows deep down he is cheating the definition. I get the same kind of joy from thinking about stuff like this.
This video had me questioning everything I've ever learned in just the first 3 minutes - excellent!
4:23 I think people discovered the consequences of changing the Fifth Postulate before Lobachevsky and Bolyai, but they were seen as “absurd” (really just “counter-intuitive”), and that was sufficient excuse to not pursue the matter further. What L&B did was stick to the courage of their convictions, and say that, since alternatives to the Fifth Postulate did not create any (new) contradictions in the mathematical theory, they must be considered to be just as valid as Euclid’s version.
In other words, this was a change in the meaning of mathematics, from thinking about things that had to be “intuitive”, to things that had to be “logical”.
We really need a lot more on this.
This is a great topic. Brady, You have a gift for discovering the great professors to include in your movies.
When I get bored or stuck in my studies, I turn to Numberphile.
Would love a more technical video, maybe show how they continued to build the geometry from this point. Cool video anyway!
Great video. I've heard the term hyperbolic geometry before but I've never really understood it. I look forward to seeing the rest of the series.
This is great stuff Brady, keep it coming.
this video clarified more than the previous one
Oh wow, this really helped me to better understand the shape/geometry of the universe.
This guy is seriously good at these videos. More from him and more non-Euclidean geometry!
This is beautiful, thank you.
Thanks Brady, you are doing a wonderful Job.
This guy is really good at drawing spheres
This is I think one of the best explanations of this idea I have seen.
I'm supposed to be a smart person, I have just received my PhD in the biological sciences and I work on the frontiers of my field, yet I still struggle to wrap my head around this sort of thing. I spend some time here and there trying to better understand the upper levels of mathematics but ultimately I have to accept that I don't have the intellect required too truly understand it.
Videos from numberphile have helped me to comprehend a few things that were on the limits of my ability, I may never be able to grasp the upper tiers of maths and physics but at least with resources like this I can try to hold my own at a conference when a fellow biologist tries to show off their mathematical skills :)
Wonderfully, very well explained.
Thank you for making this video.
dr. ashley is amazing!!!!!!!!!
It slightly wrong to say that we for a long time beloved the earth was flat. Eratosthenes calculated the circumference about 200 years after Euclid's death...
His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis (again with remarkable accuracy).
He created the first map of the world, incorporating parallels and meridians.
Like the one drawn on the brown paper
More from this guy and on this topic please!
Excellent series! I hope to see more describing surfaces and different ways to tessellate.
My topology professor does his research on geometrizing the Hausdorff metric. That would make an interesting Numberphile video :)
For me this hyperbolic space seems like a hemisphrerical-shaped minigolf greens. If you hit he ball from the edge straight to the middle, the ball goes straight, but if you hit even a bit to the side, the ball moves in an arc.
Which also brings vectors into my mind, because in euclidean plain you can determine a line by giving two vectors: one to determine any point on the line, and one to determine the direction which the line goes. The length of the direction-vector doesn't matter because it doesn't matter how hard you "throw" the line to go, the direction is the same whatsoever.
But on the hyperbolic plane it seems for me that the length of the direction vector actually matters. In golf-example, it does matter how hard you hit the ball. If you hit the ball to the side with full force, it goes almost straight, but if you just gently pat it, it moves just an inch to the side.
The thing where I thnk this example fails is that every of those shortest routes leave the edge in perpendicular angle, where golfball needs to leave anything but perpendicular to get a curve.
I love non-euclidean geometry topics, please do some more if you can! Very smooth presentation too! Thanks!
MIND BLOWN !!!!!
this is totally new to me
keep the good stuff coming :D
5:33 I believe there is a society which has the toast “To mathematics! And may it be of no damn use to anybody!”
Of course, “pure” maths often turns out to have surprising applications...
Was inspired by Dr. Ashley's comments to look up William Thurston - died in 2012, unfortunately. But he was a really fascinating guy! A rare mathematical genius and also a skilled teacher/explainer. Was one of the Fields Medal winners who would drop by MathOverflow on occasion to answer people's questions. He enjoyed other intellectual pursuits and valued human creativity no matter what the subject area. He was even involved in designing clothes for a fashion show! Definitely not your typical math genius.
Can you please make a video about the Thurston geometrization?
Roger Penrose writes some very informative pages on basic hyperbolic geometry in 'The Road to Reality'. I highly recommend it.
I'm actually starting to 'get' this. Very cool stuff!
this. is. amazing!
I want to hear more from Dr. Ashley, I really like the way he presents the ideas.
just fantastic
Definitely want to see more along the liines of Episomology.
he keeps giving these sly smiles as if he can't believe it's all true. but he knows better, and knows hyperbolic geometry is a consistent and valid model. funny when the human mind's intuition and reason are at odds. I must admit I've struggled with this myself over hyperbolic geometry.
You should do a short video about how to calculate how far away the horizon is on different planets - could be interesting :)
brilliant!
please more videos on this
6:24 The other “axioms” are more about definitions of terms. If you change them, I guess you don’t end up with geometry at all. Instead, you probably create something more like topology.
mind implosion main sequence start 0:00
These axioms are never 'ignored', but their manifestly geometrical nature is only true in Euclidean space. Hyperbolic spaces do obey these axioms, albeit more abstractly. This leads to an interesting thought on what is mathematically real or not; Is a mathematical object real if it is represented physically, or is merely proven to emerge from the agreed upon axioms? I am believe the latter.
😊 thank you
"Train tracks would not work in this world". Therefore we are not in hyperbolic, non euclidien space. We exist within a euclidian plane. The material world is plane in a euclidian plane seen from all directions. Frames of reference.
What great thoughts on the foundations of mathematics
So if a circle is a staright line in the hyperbolic plane - is the (on paper) straight line he drew first actually a straight line in that plane?
i like this guy
How to identify which one is a straight line and whixh one 's not?? Btw beautiful video!
This blew my mind. I guess its used somewhere in general relativity, its not my field though. I wonder if theres a discusion about if our Universe is euclidian or non-euclidian in nature.
This is some heavy stuff. I highly doubt anyone who is not a mathematician could follow along.
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?
And I thought Tessellation was just a game option that makes my machine run like crap.
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.
Two notes:
1) "A parallel line is a line that does not intersect" This is not true. Two lines can not intersect in 3D space while not being parallel.
2) It seems parallelism fails at being an equivalence relation in hyperbolic space because if line A is parallel to line B and line B to C, it doesn't seem to imply that A is parallel to C in the example given.
Apropos "Traintracks would not work...": Does that imply that a moving body can measure it's velocity by measuring the force that pulls on points beside the trajectory of the center of mass? That would mean that a non euclidian geometry is not compatible with the principle of relativity and there is a stational reference system. That further means that either our universe has a curvature of zero or the principle of relativity doesn't hold. Is that correct?
'The unifying object in these different worlds are surfaces.' Anyone please care to explain what he meant by that very last line?
This thing he says about logic ties into what's called "Godel's Incompleteness Theorems". In a sense it says "Nothing is *perfectly* self-consistent".
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.
So "Truth" is only "Truth" when defined within parameters that make it so.
So if you define a plane as a curved object instead of a flat one,
define a straight line as one going across a plane at a consistent rate regardless of whether it curves or not,
and define a parallel line as two lines that don't cross, not as two lines with the same angle,
then you can make infinite "parallel, straight" lines that are not at the same angle and don't cross each other. Makes sense. If you redefine everything, you can get different results.
I think the speaker intentionally skirts around the concept of divine maths, because of the divine part. Math's territory generally does not include religion or spirituality. Some ancient cultures have explored these ideas, but many of them have faded.
I think the key here is in how the extrapolation process works, and the scales upon which it is built.
You were asking about whether the third axiom has ever been questioned and, forgive me if I'm wrong, but doesn't affine geometry kind of ignore it? In affine geometry, there's no real difference between and ellipse and a circle
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.
omg i really liked the video but the way he drew the equator and the meridians is just wrong: first of all the eqator and the meridians have to elipses that touch the contour of the spere. Also if you can see the eqator as an elipse the north and south pole can never be on the contour of the spere. Sry 4 bad english
Does anyone know how you would draw an acute or obtuse angle on a hyperbolic plane?
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.
Hold up let me get this straight... If you have two parallel 90 degree lines intersecting a single 90 degree line, somewhere up along the two lines they will intersect each other... as its stated in the postulate that any line that is less than 180 degrees will always intersect.... so the the big mystery is that anything under 180 degrees or over 180 degrees curves.... and that's what the problem is? considering also that a line within the Hyperbolic realm curves & further that maths and the universe works by scale if one wanted to enter the Hyperbolic state they would just need to scale themselves exponentially until all straight lines curved?
what is non euclidean geometry
another problem: only in flat-plane geometry do you get similar non-congruent shapes
OK!
So on that globe plain how are you going to identify points?
How do you draw and calcuate vectors?
Is the intersection of two globe plains a line or a circle?
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.
Triangles with more than 180º or they are not triangles?
I don't know why it seems to me they are mixing dimensions with that way of thinking.
I remember in like kindergarten or first grade I said to my teacher that drawing 2 parallel lines means they will intersect..I still don't know the meaning but at least I was right
These videos are very interesting, it would be nicer to watch if it wasn't overexposed all the time^^
are you aware that this video is hidden and only ones with the link can see it?, what I mean is that if you paste "Fifth Axiom (extra footage) - Numberphile" inside the search bar, it wont give you this video, maybe by accident
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
Is the main problem trying to explain non-Euclidian geometry, on a Euclidian surface?
At 5:10, describing truth as "metaphysical" is problematic. (If that's what he is getting at)
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.
-8 And who made me a big success and brought me wealth and fame? -8
-8 Nicholas Ivanovich Lobachevski is his name! -8
Here's an idea for defining non-euclidian space:
"Euclidean space is defined as a region of N-dimensional space where every possible triangle drawn within that space will have angles that total 180°. Non-euclidian space is any space which violates this rule."
Funny thing is, the universe is non-euclidian. I wonder if there's a way to isolate "space" like there is for isolating any form of energy within that space...
P.S. Can a torous be euclidian?
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.
But the triangles aren't 90 degrees when measured 2 dimensionaly. The two right angles at 2:20 are only 90 degrees if measured on a tangetial plane.
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?
What if we're in a 4d hyperbolic universe and traveling through time causes things to seperate just like the train tracks would, explaining dark energy, and we're all gunna die? I don't wanna be split open like train tracks.
Oh look - M C Escher!
What if we get rid of this axiom:
"Let 1+1=2"
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 found his explanation very 'surfacy'.
I can somewhat follow along with how this works, but what are the practical applications for the hyperbolic space?
The axioms surely aren't just arbitrarily being changed for the sake of a thought experiment, right?
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.
Somehow this feels like cheating. This is comparing 2D plane straight lines and 3D sphere straight lines translated to a 2D plane surface. I'm not sure if this comparison is valid as their conditions are very different. But I am no mathematician...
Did he say "any N greater than sticks"? If yes, does that mean any N greater than 2?
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
"there are subtle questions about this notion of math being connected fundamentally with proof, whereas science is connected to accuracy and observation, and truth is this subtle metaphysical thing that is in between or independent" the first four postulates are true in hyperbolic, euclidean, and spherical geometry. the fifth is true in euclidean but false in the other two. this means we can't prove the fifth postulate given only the first four (otherwise the fifth would be true in all three geometries). well in order to perfectly describe euclidean geometry, why not just add the fifth axiom or add other axioms consistent with the first four which imply the fifth?
godel's incompleteness theorem (*kind of) says that no matter how many axioms you add, there will always be something true about euclidean geometry that you won't be able to prove from your axioms. equivalently, there will always be a thing true in euclidean geometry which will be false in some other geometry that satisfies your new set of axioms. so maybe we won't be able to perfectly describe euclidean geometry with mathematics. but then this would seem to suggest that euclidean geometry is something independent of mathematics if we can't perfectly describe it with mathematics. i believe this is what he means by math being connected to proof and truth being independent.
*it's possible there might be a perfect description of euclidean geometry. however, in that case, one of two things will be true. either the description will not describe enough of arithmetic (not likely since arithmetic is pretty essential), or a computer won't be able to tell you what is an axiom and what isn't (i.e. you would not be able to write a program to do this)
I don't get why people don't get this. Maths is just a bunch of axioms and rules, and you can define anything in any way you like. If I felt like it, I could define a "cow" to be the the work required to lift me 1.86656478 meters in a universal gravity field of 0.00067464 m/s² divided by the mass of all bacteria in the world as 2015-01-01 19:46:74.535 UTC. But that's just a definition.
Folks need to give up the Platonic notion of math. Math is invented, not freaking "discovered."
*****
*"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.
👋 flerfs
I don't think this video really gets at what is basically going on here. Euclid made an assumption that he shouldn't have: That a "plane" is not curved. If a plane is in fact flat, then you get Euclidean geometry, which is everything we are familiar with. But nothing in the first four axioms actually REQUIRES the plane to be flat. It can be curved and those four axioms still hold. It is only the fifth axiom that changes based on the curvature of the surface. If the surface is curved in on itself--a sphere, then a "point" is actually two poles on the sphere, not one "point" at all. A "line" is any great circle around the sphere. In spherical space, there are NO parallel lines, because EVERY great circle intersects every other great circle on a sphere!
Hyperbolic space is a lot harder to picture, but if you think about spherical space for a minute, and how that changed the axioms, then you can imagine the kinds of things that might change in hyperbolic space.
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.
Longitudinal line are straight? So, if it's following the curvature of a sphere, how is it straight? You lost me.
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.
No offense to you sir, but I barely understood everything you said. I don't know why, I've dealt with positive and negative curvature before, but the way you word things... in one ear and out the other.
all these shapes can be created with Euclid's geometry so why make up an erroneous set of calculations.
+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?
Oops, polygons, not polynomials.