A few thoughts on the comments below: First of all, props to @samueldeandrade8535 for his passion and desire to dig deeper. Second, one of the comments said that I'm proponent of the coordinate-free approach. That's true, but only insofar as I'm a proponent of all approaches - the more the better, and the more different the better - the goal being to recognize the relative strengths and weaknesses of each approach. Case in point, I'm a huge proponent of the coordinate approach. In fact, in this course, my emphasis on the coordinate-free approach is mean to help illustrate the need for coordinates and to be able to introduce them properly.
Third, some commenters are making the very mistake I mentioned at the beginning of the video - using different definitions from mine. I'm not using an abstract algebraic definition of length and then claim the classical definition (tape measure) as a special case. I'm only using the classical as-Euclid-would-have-understood-it definition of length and build on top of that. If you would like to clarify all of this for yourself, watch the first 5 or 6 videos of this following playlist: czcams.com/video/Ww_aQqWZhz8/video.html
Unrelated to the topic of this video, you need to turn off autofocus on your camera. Lock the focal distance to either the board or just in front of the board when you start recording. Narrow the aperture (set the f-stop to a higher number like f/4 or f/8, etc.) to increase the depth of field, if you have enough light.
I haven’t seen your other video, so you might have mentioned this. But this proves the law of cosines immediately, also it generalized easily to 3d involving the diagonal of a tetrahedron, and appears to give a 3d version of law of cosines
You people have no idea about what you are talking about, do you? For you people is news that we can DEFINE cosine of an angle by the Law of Cosines, just like we can define it using Euler's formula, just like we can defined it using its differential equation property ... Man ...
Your definition of the dot product does require the notion of a metric on your vector space. That metric in this case is pythagoras, however i believe in different vector spaces (like the space of cts functions f:[0,1]->R] with f•g=int(fg) ) we can derive the same pythagorean property. So i guess the point is that being in a hilbert space gives you pyhtagoras, but then euclidean pythagoras relies on the notion of the length of a vector (which you require for your dot product) and in this case youre measuring your vector with pythagoras. So in summary, unless im overlooking something, you have proved a very awesome property in a hilbert space, but to relate the norms to lengths in your euclidean vector space you need to define the lengths which are defined by pythagoras.
Good point, but I need to clarify (or rather reiterate) my present approach. Length is not defined by Pythagoras. Length is defined by picking up a tape measure and holding it to the segment. That's why I envoke an imaginary conversation with Euclid who lived at least a few decades before Hilbert.
First, thank you for help me understand math. In the future, if you can use rational trigonometry instead the standard trigonometry, I would appreciate.
Beautiful. It seems to me that much interest has been given to proving Pythagoras Theorem via trigonometry lately, as a result of the work of Johnson-Jackson challenging the claim of Elisha Loomis. I have made videos on this and on Jason Zimba's proof. We want to avoid circular reasoning.
The proof isn't circular, it's just that vectors and dot products are more advanced tools that pack a lot of geometric information into a small expression. For example, if you want to add two vectors u and v geometrically, you have to construct a parallelogram, which is a lot more "physical work" than just writing u+v. Each line of this proof has an equivalent geometric interpretation, it's just that it takes more work to draw it: we can interpret the dot product of vector u with itself (u*u) as the area of the square with its side length. If v is orthogonal to u, u*(u-v) can be interpreted as an area-preserving shear transformation. when you do c*c = (a-b)*(a-b) = a*(a-b) - b*(a-b) (distributing just once), you are effectively shearing the smaller squares into parallelograms and squishing them together, like the animation here: en.wikipedia.org/wiki/Pythagorean_theorem#Proof_by_area-preserving_shearing
I don't think so. This is not really the problem. The problem is that this guy thinks he proved classical Pythagoras Theorem with abstract notions of length and angle, without showing they correspond to usual classic length and angle. It is very easy to check this is the problem. Just change the dot product. The plane R² has an infinite number of possible dot products. Choose one different from the usual one.
@@samueldeandrade8535 But that can be proven pretty easily for this choice of dot product I think: for any vector v, we have defined the dot product of a vector with itself as (lenv)^2, which works for all v because the angle of a vector with itself is always 0. Then, you are correct that dr Grinfeld proved the "dot product equality" a*a+b*b=c*c, but these can be substituted with (lena)^2, (lenb)^2, (lenc)^2 for any choice of a,b, as pointed out above. QED
@samueldeandrade8535 But there are some problems with your analysis. 1. For all the abstract inner products that can be defined on R2, all of them satisfy the Pythagorean property. So all that is required is to prove that classical dot product is an example of the abstract dot product, which is what he does in this video. 2. He defines classical dot product in this video, NOT the general abstract one. Because he DEFINES it as length_a * length_b * angle_a_b.
@samueldeandrade8535 Proofs don't require motivation to be correct. I agree that the greeks would have probably called you insane, if you defined this dot product as length_a * length_b * angle_a_b. But just because a proof seems to come out of nowhere DOES NOT make it circular.
This is a great proof. I see no flaws. Was expecting to see a sneaky equivocation between the component formula for the dot product and the cos theta formula for the dot product, but since it does not have this I believe it is valid
What did you think it was proved? Certainly NOT the classical Pythagoras Theorem. You can't proof classical Pythagoras Theorem with some abstract notions of length and angle (or cosine of angle)! Man, do you people really study Math?
Man, someone has to be ... in a very special way to insist on such idea. Let's understand: given a field K, a vector space V and a symmetric bilinear form • : V×V → K we can consider the quadratic form associated with •, Q: V → K, Q(u) := u•u And for all that we can easily demonstrate, there is NOTHING new about this, the Pythagoras Theorem for • : "If u,v in V are such that u•v=0, then Q(u-v) = Q(u)+Q(v)" IF K and • HAVE MORE PROPERTIES, not going to tell all details here, you can also define length of a vector and angle between two vectors by |u| = √Q(u) and Angle(u,v) = (u•v)/(|u|*|v|) This is all fine. Now ... the problem with this guy's idea is that it seems he thinks he is demonstrating the classical Pythagoras Theorem, instead of demonstrating it for the abstract dot product. HE IS NOT. To do that he would have to establish a "correspondence" between triangles and pairs of linear independent vectors, in a way that the usual length from classic Geometry corresponds to the abstract length AND the usual angle correspond to Arccos(Angle). THIS is what makes the proof INVALID. If you want to call it circular, you can, because to show such correspondence you will need Pythagoras Theorem. So, for the Math of Euler, let's stop this insanity. It is extremely disappointing to see this whole situation. The guy thinks that using abstract functions he can prove THE theorem that is the reason for the definition of said abstract functions.
@@samueldeandrade8535 But he did choose a bilinear form. What he proved is that the space of "geometric vectors" (that is equivalence classes of line segments with direction) forms a vector space, in that vector space you have a bilinear function, the geometric dot product (using geometric length and angle, not some abstract length and angle), and it so happens that with this bilinear function as the inner product, the abstract notion of vector length coincides with the concrete, geometric vector length. So assuming that the proof of inner product space axioms doesn't require the Pythagorean theorem, you can certainly use the abstract Pythagorean theorem to prove the classical one.
@@gregc517 congrats for replying like a normal person, despite my attempts to provoke reactions. I will answer you in a normal way. He defined the dot product in this video, in the original "proof" he didn't. The main issue with this "proof" is: why define := u*v*cos(u,v)??? What motivates this definition? Without geometry, without the history of Mathematics, nothing motivates this "chosen" dot product. The definition of the usual dot product in the plane = xz+yw came from the problem to find the projection of a vector over another vector, which is equivalent to finding the cosine of the angle between two vectors. So, he can only invoke the dot product he did in this video BECAUSE mathematicians in the past invented Analytic Geometry and defined a thing called dot product, , satisfying = x²+y² = |(x,y)|² this last equation we know because of Pythagoras, and cos(u,v) = /(|u|*|v|) The funny thing is that in the first video, he shows a classical proof of Pythagoras, the one having the right triangle repeated 4 times inside a square with sides a+b, and says "I could never figure this out, because it requires a lot of ingenuity", then he presents this "proof", invoking a dot product that if didn't come from the historical, Pythagoras motivated, definition, then requires 1000x times more ingenuity. The feeling about this "proof" is like proving something by induction that doesn't need induction to be proved. For example, find an expression for the sum S(n) = 1+2+...+n You can solve this problem using induction to prove S(n) = n(n+1)/2? Of course you can. But people that never saw the expression n(n+1)/2 will think you magically came up with it. When we all know comes from S(n) = 1+2+...+(n-1)+n S(n) = n+(n-1)+...+2+1 2S(n) = (n+1)+(n+1)+...+(n+1)+(n+1) 2S(n) = n(n+1) S(n) = n(n+1)/2 Do you get it? His dot product only exists because of the Pythagoras Theorem. Because in the past Mathematicians considered (x,y) and (z,w) vectors, there is a multiple of (z,w), t(z,w), such that the points (0,0), (x,y) and t(z,w) form a right triangle. If that was the case, by the Pythagoras Theorem, we need to have |(x,y)|² = |t(z,w)|² + |(x,y)-t(z,w)|² This implies x²+y² = t²z²+t²w²+(x-tz)²+(y-tw)² = t²z²+t²w²+x²-2txz+t²z²+y²-2tyw+t²w² = x²+y²+2t(tz²+tw²-xz-yw) We obtained x²+y² = x²+y²+2t(tz²+tw²-xz-yw) which implies 2t(tz²+tw²-xz-yw) = 0 So, t=0, which is not the scalar we are looking for, or tz²+tw²-xz-yw = 0 or t = (xz+yw)/(z²+w²) Then mathematicians realised the expression in the numerator is obtained adding the products of the respective coordinates of the vectors (x,y) and (z,w). They admired it, saw it was good, gave it a notation, , and a name: DOT PRODUCT. And even before that they already realised it is related to length because |(z,w)|² = z²+w² = Ufa. So do you get it? The definition and study of dot products only exist because of Classical Geometry, in particular because of the Pythagoras Theorem. I recently watched a teacher saying the definition of cosines using the right triangle is worse than the one using the unit circunference, in which cosine of an angle is defined as the x coordinate of the angle in radians ... She says this definition is better because causes less confusion, because if kids learn cosine is the ratio between two lengths, they get confused when they learn cosine is negative for obtuse angles. So she prefers to ignore the historical development of the concept. It is the same here, ignoring the history of the concept. This is ... at least questionable. His "surprise" saying "I did doubt it because it is so simple" is as valid as the surprise people get with Cramer's rule: the variable x_i of the solution (if it exists and it is unique) of the system Ax=b is given by x_i = det(A_i)/det(A) where A_i is the matrix A but with b in the place of column i. Why this shouldn't be a surprise? Because the definition of determinant CAME FROM THIS. The expression that appears when we try to solve a general linear system is what motivated the notion of determinant!!! Anyways, if people want to treat Mathematics as some "game with rules given by the gods", forgetting its historical motivations, there is nothing I can do.
@@samueldeandrade8535 Completely agree that noone could've came up with this proof without already having the Pythagorean theorem. Still, there is a distinction between "making sense historically" and being correct logically. You could say that this proof is "circular in its ideas" in some sense, and you'd probabily be right, still, that is not what is usually meant by proofs being circular. Also sometimes there might be value in using tools that came later to simplify proofs of old theorems. This case might not be such a good example of that, since the classical proofs are simple enough, but take the example using group theory to prove Fermat's little theorem. Of course it wasn't proved with group theory originally, but group theory doesn't depend on it logically, and proving the result in "one line" (after having proven a number of theorems/lemmas) provides a nice example of the power of group theory, and as long as you have internalized how groups work, it is much simpler than more elementary proofs (or at least the ones I know of). And here, this proof has the one thing going for it, that if you're already familiar with the dot product, it simplifies the proof (at least somewhat).
@@gregc517 I like you. You are very careful with your ideas and words. I still won't accept this "proof". It is too artificial. And I don't think we should go against Math's history more than we already did. About the distinction between "making sense historically" and being correct logically, yeah, for sure. And the historical aspect IS more relevant than the "correct logically" aspect, because any axiom is proof of itself, isn't it? So, p |- p is logically correct proof if I include p as an axiom of my theory. It feels like that's what is happening in his "proof". "You could say that this proof is circular in its ideas ... that is not what is usually meant by proofs being circular" Yep, I know. And this makes me want to extend the notion of circular proofs. I just search for this and saw something I don't think we said here, that "circular proofs are logically valid". Haha. "Also sometimes there might be value in using tools that came later ..." Yeah. But this is not exactly the case. I mean, if the tool that came later is independent of the old theorem, ok. But, in this case, the tool didn't simply came later, it came later and it came because of the old theorem. It is the snake biting its tail. Motivation behing Math is something that we really shouldn't lose. And probably lack of motivation is the reason why so many people don't like Math. Most concepts are taught in an empty way. "And here, this proof has the one thing going for it, that if you're familiar with the dot product, it simplifies the proof" I disagree. I think that for a person to be familiar with the notion of dot product, they need to know its motivation. Even if you adjust the word, instead of "familiar" you use "to know", "to be familiar with the rules", etc, I still think the statement will be a mistake. Because the rules, any rules, by themselves are pretty much irrelevant in my opinion. An unjustified rule shouldn't be a rule. Rules are nothing, reasons are everything. Anyway, I don't even know what I am thinking anymore. I have to sleep. Have a nice day.
I will make a last, serious, argument against this: accepting this "proof" is basically the same as accepting to prove lim_{x→0} sin(x)/x = 1 using L'Hôpital's Rule. That's it.
A few thoughts on the comments below:
First of all, props to @samueldeandrade8535 for his passion and desire to dig deeper.
Second, one of the comments said that I'm proponent of the coordinate-free approach. That's true, but only insofar as I'm a proponent of all approaches - the more the better, and the more different the better - the goal being to recognize the relative strengths and weaknesses of each approach. Case in point, I'm a huge proponent of the coordinate approach. In fact, in this course, my emphasis on the coordinate-free approach is mean to help illustrate the need for coordinates and to be able to introduce them properly.
Third, some commenters are making the very mistake I mentioned at the beginning of the video - using different definitions from mine. I'm not using an abstract algebraic definition of length and then claim the classical definition (tape measure) as a special case. I'm only using the classical as-Euclid-would-have-understood-it definition of length and build on top of that. If you would like to clarify all of this for yourself, watch the first 5 or 6 videos of this following playlist: czcams.com/video/Ww_aQqWZhz8/video.html
Well no, it's triangular.
Unrelated to the topic of this video, you need to turn off autofocus on your camera. Lock the focal distance to either the board or just in front of the board when you start recording. Narrow the aperture (set the f-stop to a higher number like f/4 or f/8, etc.) to increase the depth of field, if you have enough light.
I haven’t seen your other video, so you might have mentioned this. But this proves the law of cosines immediately, also it generalized easily to 3d involving the diagonal of a tetrahedron, and appears to give a 3d version of law of cosines
Yes, that's a good observation!
You people have no idea about what you are talking about, do you?
For you people is news that we can DEFINE cosine of an angle by the Law of Cosines, just like we can define it using Euler's formula, just like we can defined it using its differential equation property ... Man ...
Your definition of the dot product does require the notion of a metric on your vector space. That metric in this case is pythagoras, however i believe in different vector spaces (like the space of cts functions f:[0,1]->R] with f•g=int(fg) ) we can derive the same pythagorean property. So i guess the point is that being in a hilbert space gives you pyhtagoras, but then euclidean pythagoras relies on the notion of the length of a vector (which you require for your dot product) and in this case youre measuring your vector with pythagoras.
So in summary, unless im overlooking something, you have proved a very awesome property in a hilbert space, but to relate the norms to lengths in your euclidean vector space you need to define the lengths which are defined by pythagoras.
Good point, but I need to clarify (or rather reiterate) my present approach. Length is not defined by Pythagoras. Length is defined by picking up a tape measure and holding it to the segment. That's why I envoke an imaginary conversation with Euclid who lived at least a few decades before Hilbert.
First, thank you for help me understand math. In the future, if you can use rational trigonometry instead the standard trigonometry, I would appreciate.
Beautiful. It seems to me that much interest has been given to proving Pythagoras Theorem via trigonometry lately, as a result of the work of Johnson-Jackson challenging the claim of Elisha Loomis. I have made videos on this and on Jason Zimba's proof. We want to avoid circular reasoning.
Very nice! Thank you.
The proof isn't circular, it's just that vectors and dot products are more advanced tools that pack a lot of geometric information into a small expression. For example, if you want to add two vectors u and v geometrically, you have to construct a parallelogram, which is a lot more "physical work" than just writing u+v. Each line of this proof has an equivalent geometric interpretation, it's just that it takes more work to draw it:
we can interpret the dot product of vector u with itself (u*u) as the area of the square with its side length. If v is orthogonal to u, u*(u-v) can be interpreted as an area-preserving shear transformation.
when you do c*c = (a-b)*(a-b) = a*(a-b) - b*(a-b) (distributing just once), you are effectively shearing the smaller squares into parallelograms and squishing them together, like the animation here:
en.wikipedia.org/wiki/Pythagorean_theorem#Proof_by_area-preserving_shearing
One thing I would add is that the elegance of this proof speaks to the effectiveness of vector algebra.
Your definition of dot product requires that the "length of the projection" being unique. That's the case only if your vector space is Euclidean.
I don't think so. This is not really the problem. The problem is that this guy thinks he proved classical Pythagoras Theorem with abstract notions of length and angle, without showing they correspond to usual classic length and angle.
It is very easy to check this is the problem. Just change the dot product. The plane R² has an infinite number of possible dot products. Choose one different from the usual one.
@@samueldeandrade8535 But that can be proven pretty easily for this choice of dot product I think: for any vector v, we have defined the dot product of a vector with itself as (lenv)^2, which works for all v because the angle of a vector with itself is always 0. Then, you are correct that dr Grinfeld proved the "dot product equality" a*a+b*b=c*c, but these can be substituted with (lena)^2, (lenb)^2, (lenc)^2 for any choice of a,b, as pointed out above. QED
@@alegian7934 tell me: what is the motivation behind his dot product?
@samueldeandrade8535 But there are some problems with your analysis.
1. For all the abstract inner products that can be defined on R2, all of them satisfy the Pythagorean property. So all that is required is to prove that classical dot product is an example of the abstract dot product, which is what he does in this video.
2. He defines classical dot product in this video, NOT the general abstract one. Because he DEFINES it as length_a * length_b * angle_a_b.
@samueldeandrade8535 Proofs don't require motivation to be correct. I agree that the greeks would have probably called you insane, if you defined this dot product as length_a * length_b * angle_a_b. But just because a proof seems to come out of nowhere DOES NOT make it circular.
This is a great proof. I see no flaws. Was expecting to see a sneaky equivocation between the component formula for the dot product and the cos theta formula for the dot product, but since it does not have this I believe it is valid
That's right!
What did you think it was proved? Certainly NOT the classical Pythagoras Theorem. You can't proof classical Pythagoras Theorem with some abstract notions of length and angle (or cosine of angle)! Man, do you people really study Math?
The only thing that's circular is the perfect shape of these videos! ❤🎉😊
Thank you!
@@MathTheBeautiful And mazel tov! Your work on all these videos is a true mitzvah. 😇
Wrong. If he thinks he proved classical Pythagoras Theorem, it is indeed circular reasoning.
Nicely done. Kudos for taking the time. I don't think I would have paid attention to the peanut gallery :)
That's my audience though!
What a chad
👍😁
It is circular !
Man, someone has to be ... in a very special way to insist on such idea. Let's understand:
given a field K, a vector space V and a symmetric bilinear form
• : V×V → K
we can consider the quadratic form associated with •,
Q: V → K,
Q(u) := u•u
And for all that we can easily demonstrate, there is NOTHING new about this, the Pythagoras Theorem for • :
"If u,v in V are such that u•v=0, then
Q(u-v) = Q(u)+Q(v)"
IF K and • HAVE MORE PROPERTIES, not going to tell all details here, you can also define length of a vector and angle between two vectors by
|u| = √Q(u)
and
Angle(u,v) = (u•v)/(|u|*|v|)
This is all fine. Now ... the problem with this guy's idea is that it seems he thinks he is demonstrating the classical Pythagoras Theorem, instead of demonstrating it for the abstract dot product. HE IS NOT. To do that he would have to establish a "correspondence" between triangles and pairs of linear independent vectors, in a way that the usual length from classic Geometry corresponds to the abstract length AND the usual angle correspond to Arccos(Angle). THIS is what makes the proof INVALID. If you want to call it circular, you can, because to show such correspondence you will need Pythagoras Theorem.
So, for the Math of Euler, let's stop this insanity. It is extremely disappointing to see this whole situation. The guy thinks that using abstract functions he can prove THE theorem that is the reason for the definition of said abstract functions.
And, sorry, he has to prove some correspondence, CHOOSING a bilinear form •, because it will be true ONLY for one of those!
@@samueldeandrade8535 But he did choose a bilinear form. What he proved is that the space of "geometric vectors" (that is equivalence classes of line segments with direction) forms a vector space, in that vector space you have a bilinear function, the geometric dot product (using geometric length and angle, not some abstract length and angle), and it so happens that with this bilinear function as the inner product, the abstract notion of vector length coincides with the concrete, geometric vector length. So assuming that the proof of inner product space axioms doesn't require the Pythagorean theorem, you can certainly use the abstract Pythagorean theorem to prove the classical one.
@@gregc517 congrats for replying like a normal person, despite my attempts to provoke reactions. I will answer you in a normal way.
He defined the dot product in this video, in the original "proof" he didn't.
The main issue with this "proof" is: why define
:= u*v*cos(u,v)???
What motivates this definition? Without geometry, without the history of Mathematics, nothing motivates this "chosen" dot product. The definition of the usual dot product in the plane
= xz+yw
came from the problem to find the projection of a vector over another vector, which is equivalent to finding the cosine of the angle between two vectors.
So, he can only invoke the dot product he did in this video BECAUSE mathematicians in the past invented Analytic Geometry and defined a thing called dot product, , satisfying
= x²+y² = |(x,y)|²
this last equation we know because of Pythagoras, and
cos(u,v) = /(|u|*|v|)
The funny thing is that in the first video, he shows a classical proof of Pythagoras, the one having the right triangle repeated 4 times inside a square with sides a+b, and says "I could never figure this out, because it requires a lot of ingenuity", then he presents this "proof", invoking a dot product that if didn't come from the historical, Pythagoras motivated, definition, then requires 1000x times more ingenuity.
The feeling about this "proof" is like proving something by induction that doesn't need induction to be proved. For example, find an expression for the sum
S(n) = 1+2+...+n
You can solve this problem using induction to prove
S(n) = n(n+1)/2?
Of course you can. But people that never saw the expression n(n+1)/2 will think you magically came up with it. When we all know comes from
S(n) = 1+2+...+(n-1)+n
S(n) = n+(n-1)+...+2+1
2S(n) = (n+1)+(n+1)+...+(n+1)+(n+1)
2S(n) = n(n+1)
S(n) = n(n+1)/2
Do you get it? His dot product only exists because of the Pythagoras Theorem. Because in the past Mathematicians considered
(x,y) and (z,w) vectors, there is a multiple of (z,w), t(z,w), such that the points
(0,0), (x,y) and t(z,w)
form a right triangle. If that was the case, by the Pythagoras Theorem, we need to have
|(x,y)|² = |t(z,w)|² + |(x,y)-t(z,w)|²
This implies
x²+y²
= t²z²+t²w²+(x-tz)²+(y-tw)²
= t²z²+t²w²+x²-2txz+t²z²+y²-2tyw+t²w²
= x²+y²+2t(tz²+tw²-xz-yw)
We obtained
x²+y² = x²+y²+2t(tz²+tw²-xz-yw)
which implies
2t(tz²+tw²-xz-yw) = 0
So, t=0, which is not the scalar we are looking for, or
tz²+tw²-xz-yw = 0
or
t = (xz+yw)/(z²+w²)
Then mathematicians realised the expression in the numerator is obtained adding the products of the respective coordinates of the vectors (x,y) and (z,w). They admired it, saw it was good, gave it a notation, , and a name: DOT PRODUCT. And even before that they already realised it is related to length because
|(z,w)|² = z²+w² =
Ufa. So do you get it? The definition and study of dot products only exist because of Classical Geometry, in particular because of the Pythagoras Theorem.
I recently watched a teacher saying the definition of cosines using the right triangle is worse than the one using the unit circunference, in which cosine of an angle is defined as the x coordinate of the angle in radians ... She says this definition is better because causes less confusion, because if kids learn cosine is the ratio between two lengths, they get confused when they learn cosine is negative for obtuse angles. So she prefers to ignore the historical development of the concept. It is the same here, ignoring the history of the concept. This is ... at least questionable.
His "surprise" saying "I did doubt it because it is so simple" is as valid as the surprise people get with Cramer's rule:
the variable x_i of the solution (if it exists and it is unique) of the system Ax=b is given by
x_i = det(A_i)/det(A)
where A_i is the matrix A but with b in the place of column i.
Why this shouldn't be a surprise? Because the definition of determinant CAME FROM THIS. The expression that appears when we try to solve a general linear system is what motivated the notion of determinant!!!
Anyways, if people want to treat Mathematics as some "game with rules given by the gods", forgetting its historical motivations, there is nothing I can do.
@@samueldeandrade8535 Completely agree that noone could've came up with this proof without already having the Pythagorean theorem. Still, there is a distinction between "making sense historically" and being correct logically. You could say that this proof is "circular in its ideas" in some sense, and you'd probabily be right, still, that is not what is usually meant by proofs being circular. Also sometimes there might be value in using tools that came later to simplify proofs of old theorems. This case might not be such a good example of that, since the classical proofs are simple enough, but take the example using group theory to prove Fermat's little theorem. Of course it wasn't proved with group theory originally, but group theory doesn't depend on it logically, and proving the result in "one line" (after having proven a number of theorems/lemmas) provides a nice example of the power of group theory, and as long as you have internalized how groups work, it is much simpler than more elementary proofs (or at least the ones I know of). And here, this proof has the one thing going for it, that if you're already familiar with the dot product, it simplifies the proof (at least somewhat).
@@gregc517 I like you. You are very careful with your ideas and words. I still won't accept this "proof". It is too artificial. And I don't think we should go against Math's history more than we already did. About the distinction between "making sense historically" and being correct logically, yeah, for sure. And the historical aspect IS more relevant than the "correct logically" aspect, because any axiom is proof of itself, isn't it? So, p |- p is logically correct proof if I include p as an axiom of my theory. It feels like that's what is happening in his "proof".
"You could say that this proof is circular in its ideas ... that is not what is usually meant by proofs being circular"
Yep, I know. And this makes me want to extend the notion of circular proofs. I just search for this and saw something I don't think we said here, that "circular proofs are logically valid". Haha.
"Also sometimes there might be value in using tools that came later ..."
Yeah. But this is not exactly the case. I mean, if the tool that came later is independent of the old theorem, ok. But, in this case, the tool didn't simply came later, it came later and it came because of the old theorem. It is the snake biting its tail. Motivation behing Math is something that we really shouldn't lose. And probably lack of motivation is the reason why so many people don't like Math. Most concepts are taught in an empty way.
"And here, this proof has the one thing going for it, that if you're familiar with the dot product, it simplifies the proof"
I disagree. I think that for a person to be familiar with the notion of dot product, they need to know its motivation. Even if you adjust the word, instead of "familiar" you use "to know", "to be familiar with the rules", etc, I still think the statement will be a mistake. Because the rules, any rules, by themselves are pretty much irrelevant in my opinion. An unjustified rule shouldn't be a rule. Rules are nothing, reasons are everything. Anyway, I don't even know what I am thinking anymore. I have to sleep.
Have a nice day.
I will make a last, serious, argument against this:
accepting this "proof" is basically the same as accepting to prove
lim_{x→0} sin(x)/x = 1
using L'Hôpital's Rule. That's it.
Oh my Euler ... how tiresome. Let's watch this ... questionable lecture.