Would the circle problem be resolved if there were a postulate assuming lines and circles were continuous and contained all real numbers between their points?
I think a better resolution is the algebraic one. Basically, viewing lines and circles as just equations, an intersection becomes a solution to when you set these equations equal to each other. Then you can define the _constructable_ plane as the set of solutions to these equations. Solutions exist because of field theory. This leaves open the possibility for non-constructable points, such as pi or the cuberoot of 2.
I think there are a lot of ambiguous things that you're saying, I don't think the problem of circles not intersection in our modded "Rational coordinates only" plane simply by saying circles are continuous, in that plane they can't be, that is if you define circle to be the locus of points equidistant from a point, circles just won't be continuous curves in such a plane, by saying they will, youre moulding what it means to be a circle
@@Axenvyyhe's talking about adding another postulate. With this new postulate in place, the rationals-only system wouldn't satisfy the axioms and it wouldn't even be meaningful to talk about this system in that context.
If I recall correctly, that was in fact Hilbert's solution to the shortcoming. Indeed, I think it's kind of unfair to call this a failure on Euclid's part, as it's clear that he did in fact assume an axiom of continuity, but it merely didn't occur to him that the semantics of his statements needed to clarify what it means to "draw" a line or circle. It was a minor mistake, one that required inventing intentional, pathological counterexamples that violate the understood definitions of the terms to expose. I would be far more interested in identifying proofs that fail even if you hold the axiom of continuity.
Great video! Just wanted to say that I noticed these mistakes too. Similar mistake(I think. It's been long since I did elements) also occurs in his proof that two points in a circle when joined lies in the circle(basically proving convexity) and also I think the proof that two circles intersect at either 0, 1 or 2 points.
I would emphasize that even the modern axiom sets are not 'totally gapless' in the sense that there might still be geometric results which cannot be proven or disproven by them. It is merely the case that these systems can (we believe) justify the totality of the conclusions which are presented in a single specific, arbitrary book written by one human. That list of conclusions is not the universe of all possible geometric conclusions.
As far as I know, Hilbert's axioms were proven to be complete, meaning that any statement within that system can be proven or disproven. I don't remember where I heard it, so it could be wrong.
@@nivpearlman6514 You might be thinking of another concept that is also named after Hilbert: "Hilbert space is a complete inner product space, which means that every convergent sequence of vectors in the space converges to an element within the space."
7:18 but you switched from an analytic function to a non analytic discrete function. You also switched infinities to one strictly smaller. The points you say aren’t part of the discrete circles are part of the analytic continuation of the discrete function.
You don't seem to know what the terms "analytic", "discrete" or "analytic continuation" mean by this comment. (Just as an example, the rationals are not discrete in their standard topology, since there exist subsets of Q that are not open, take for example the set of rationals in [0,1].) Regardless, where in Euclid's postulates is asserted that circles are "analytic"? And where is it asserted that they have the cardinality of the continuum? Euclid would not even have heard of Cantor's ideas so there is no reason why he would have addressed this.
@@lock_ray I’m not a mathematician by training, so it’s highly likely that my mental model of the terms mentioned is not complete. What I’m referring to is the fact that the formula for a circle using the reals (uncountable) as parameters, is an analytic function. I conjure that the same formula restricted to the rationals (countable) is not. While such a formula may be continuous in the rationals, its cover over the reals is discontinuous. The uncountable infinity of the reals is strictly larger than the countable infinity of the rationals. The rationals are discrete in the reals. It is my understanding that an analytic continuation of the rational formula of a circle, will reveal the irrationals between the rationals and be equivalent to the formula given real parameters. Maybe analytic continuation is not the right term for extending a function beyond its domain?
@@StarLeagueus You can't fix the problems in Euclid by creating an even greater problem, the blatantly wrong claim that a Zeno machine - as computing calls the "real line" point reductionism can do any arithmetics and thus form a field. You are free to give a proof by demonstration that a randomly picked real number can function as an input to an arithmetic operation. If you can't give a constructive demonstration, why should anyone accept an arbitrary axiom as proof of anything? I could declare axiomatically, that every proposition I make is an absolute mathematical truth. Would you buy that? If not, why not? Every arbitrary "proof" by mere declarative axiom has the same truth value as my axiom.
@1:51: “each geometry system has its own parallel axiom” is false. Some systems do not have a parallel axiom at all. The preeminent example is absolute geometry: en.m.wikipedia.org/wiki/Absolute_geometry
Wait, so this plane cannot rotate a square root of 2 length? Wouldn't the contruction of its measure imply the plane must include at least most algebraic numbers instead of only rationals
No, Euclid's definitions of point and line have nothing to do with "undefined primitive notions". Euclid's definitions are very crisp, and the tersity of for didactic purposes, as the main purpose is to open the intuition of a student, and overexplaining doesn't work well in that respect. The main hint for getting the meaning is the common notion 5: "The whole is greater than the part". I grant that much confusion arises from the fact that Eulcid doesn't make a clear enough distinction between point (end of a ray or line segment) and node/vertex, a meet of two lines. Greek pure geometry does not do coordinate system neusis, because pure geometry rejects the neusis method as mere applied math. So criticism of Greek pure geometry by assuming neusis is not valid. Neusis is rejected because of Zeno's absurdity proofs showing that neusis leads to infinine regress, which if considered ontological would negate the possibility of movement and thus the constructive method of Greek pure geometry. Hilbert didn't improve on Euclid, he made a stinking pile of mess.
The bent definition of the plane to have only rational coordinates seems so bogus, such a plane cant even contain an equilateral triangles in the first place, I don't even feel Euclid made a mistake as to say, atleast not with his reasoning, here were just getting into technicalities which weren't even considered for millenias and just came about recently, cant really see this as a lapse at Euclid's end
Points (0,0), (1,0), (1,1) is an equilateral triangle. Just most rotations of it are non nonsensical in the trig we use. There is another version of Trig (I think called rational trig) that can rotate it within the rational domain. That system will also allow you to build triangles within modulo a prime in 2 dimensions (ie a 7 x 7 grid, integers mod 7 on it). Interesting but I really hate how the author of the book I have tries to say how regular trig is not really 'good' and he does not stop....
"cool little video combining a few disconnected ideas in the youtube education space." the goofy rabbit midway through:
Rabbit jumpscare
Cool video. I didnt know euclid made mistakes too. Makes me feel less insecure.
Would the circle problem be resolved if there were a postulate assuming lines and circles were continuous and contained all real numbers between their points?
Yes. In that case, they would intersect.
I think a better resolution is the algebraic one. Basically, viewing lines and circles as just equations, an intersection becomes a solution to when you set these equations equal to each other. Then you can define the _constructable_ plane as the set of solutions to these equations. Solutions exist because of field theory.
This leaves open the possibility for non-constructable points, such as pi or the cuberoot of 2.
I think there are a lot of ambiguous things that you're saying, I don't think the problem of circles not intersection in our modded "Rational coordinates only" plane simply by saying circles are continuous, in that plane they can't be, that is if you define circle to be the locus of points equidistant from a point, circles just won't be continuous curves in such a plane, by saying they will, youre moulding what it means to be a circle
@@Axenvyyhe's talking about adding another postulate. With this new postulate in place, the rationals-only system wouldn't satisfy the axioms and it wouldn't even be meaningful to talk about this system in that context.
If I recall correctly, that was in fact Hilbert's solution to the shortcoming.
Indeed, I think it's kind of unfair to call this a failure on Euclid's part, as it's clear that he did in fact assume an axiom of continuity, but it merely didn't occur to him that the semantics of his statements needed to clarify what it means to "draw" a line or circle. It was a minor mistake, one that required inventing intentional, pathological counterexamples that violate the understood definitions of the terms to expose.
I would be far more interested in identifying proofs that fail even if you hold the axiom of continuity.
Great video! Just wanted to say that I noticed these mistakes too. Similar mistake(I think. It's been long since I did elements) also occurs in his proof that two points in a circle when joined lies in the circle(basically proving convexity) and also I think the proof that two circles intersect at either 0, 1 or 2 points.
In taxicab geometry, the circles can intersect over an entire continuüm, making identification of point C really difficult.
I would emphasize that even the modern axiom sets are not 'totally gapless' in the sense that there might still be geometric results which cannot be proven or disproven by them. It is merely the case that these systems can (we believe) justify the totality of the conclusions which are presented in a single specific, arbitrary book written by one human. That list of conclusions is not the universe of all possible geometric conclusions.
As far as I know, Hilbert's axioms were proven to be complete, meaning that any statement within that system can be proven or disproven.
I don't remember where I heard it, so it could be wrong.
@@nivpearlman6514 You might be thinking of another concept that is also named after Hilbert: "Hilbert space is a complete inner product space, which means that every convergent sequence of vectors in the space converges to an element within the space."
Every good mathematician has made mistakes.
7:18 but you switched from an analytic function to a non analytic discrete function. You also switched infinities to one strictly smaller. The points you say aren’t part of the discrete circles are part of the analytic continuation of the discrete function.
You are absolutely correct. We mention this in our blog: starleague.us/blog/euclid-made-mistakes-too
You don't seem to know what the terms "analytic", "discrete" or "analytic continuation" mean by this comment. (Just as an example, the rationals are not discrete in their standard topology, since there exist subsets of Q that are not open, take for example the set of rationals in [0,1].)
Regardless, where in Euclid's postulates is asserted that circles are "analytic"? And where is it asserted that they have the cardinality of the continuum? Euclid would not even have heard of Cantor's ideas so there is no reason why he would have addressed this.
@@lock_ray I’m not a mathematician by training, so it’s highly likely that my mental model of the terms mentioned is not complete. What I’m referring to is the fact that the formula for a circle using the reals (uncountable) as parameters, is an analytic function. I conjure that the same formula restricted to the rationals (countable) is not. While such a formula may be continuous in the rationals, its cover over the reals is discontinuous. The uncountable infinity of the reals is strictly larger than the countable infinity of the rationals. The rationals are discrete in the reals. It is my understanding that an analytic continuation of the rational formula of a circle, will reveal the irrationals between the rationals and be equivalent to the formula given real parameters. Maybe analytic continuation is not the right term for extending a function beyond its domain?
@@StarLeagueus You can't fix the problems in Euclid by creating an even greater problem, the blatantly wrong claim that a Zeno machine - as computing calls the "real line" point reductionism can do any arithmetics and thus form a field.
You are free to give a proof by demonstration that a randomly picked real number can function as an input to an arithmetic operation. If you can't give a constructive demonstration, why should anyone accept an arbitrary axiom as proof of anything?
I could declare axiomatically, that every proposition I make is an absolute mathematical truth. Would you buy that?
If not, why not? Every arbitrary "proof" by mere declarative axiom has the same truth value as my axiom.
Great presentation!
@1:51: “each geometry system has its own parallel axiom” is false.
Some systems do not have a parallel axiom at all. The preeminent example is absolute geometry: en.m.wikipedia.org/wiki/Absolute_geometry
Your animations remind me of Morphocular's.
This looks like Manim, an animation program created by 3Blue1Brown fyi
Im more interested in the figures who came up with the original concepts that euclid used in his books
Wait, so this plane cannot rotate a square root of 2 length? Wouldn't the contruction of its measure imply the plane must include at least most algebraic numbers instead of only rationals
That would be the "Constructable Plane".
@@StarLeagueus Right, but isn't the Elements based on this? Because with just straight edge and compass there is inevitably irrational measures
No. "Rational Plane" is not the constructable plane. Consider the definitions of shapes as "Sets".
@@StarLeagueuswhat plane is he referring to?
No, Euclid's definitions of point and line have nothing to do with "undefined primitive notions". Euclid's definitions are very crisp, and the tersity of for didactic purposes, as the main purpose is to open the intuition of a student, and overexplaining doesn't work well in that respect.
The main hint for getting the meaning is the common notion 5: "The whole is greater than the part".
I grant that much confusion arises from the fact that Eulcid doesn't make a clear enough distinction between point (end of a ray or line segment) and node/vertex, a meet of two lines.
Greek pure geometry does not do coordinate system neusis, because pure geometry rejects the neusis method as mere applied math. So criticism of Greek pure geometry by assuming neusis is not valid. Neusis is rejected because of Zeno's absurdity proofs showing that neusis leads to infinine regress, which if considered ontological would negate the possibility of movement and thus the constructive method of Greek pure geometry.
Hilbert didn't improve on Euclid, he made a stinking pile of mess.
I like the use of Oliver Byrne edition of Euclid.
The bent definition of the plane to have only rational coordinates seems so bogus, such a plane cant even contain an equilateral triangles in the first place, I don't even feel Euclid made a mistake as to say, atleast not with his reasoning, here were just getting into technicalities which weren't even considered for millenias and just came about recently, cant really see this as a lapse at Euclid's end
Points (0,0), (1,0), (1,1) is an equilateral triangle. Just most rotations of it are non nonsensical in the trig we use. There is another version of Trig (I think called rational trig) that can rotate it within the rational domain. That system will also allow you to build triangles within modulo a prime in 2 dimensions (ie a 7 x 7 grid, integers mod 7 on it). Interesting but I really hate how the author of the book I have tries to say how regular trig is not really 'good' and he does not stop....
Yup. Greeks rejected neusis from pure geometry for very good reasons. The main reason being Zeno's Reductio ad absurdum proofs.
True. I have the same feeling as you. I dont think Euclid proof is an error.
Regarding the congruent triangle ABC, does the 4 points AICB look like a triangle to you?🤣
What time is this in reference to? Thanks.