Math is a tool of the mind to describe our world, imaginary numbers is only a extension of that tool to allow us to go beyond what mathematical logic prevents us to do, while still getting in the end a real number. Math, despite being powerful, is a flawed tool, so getting around its flaws by creating things like imaginary numbers isn’t absurd and doesn’t make the result any less real at the end.
On the other hand, I don’t think calling everything we don’t understand “magic” or the new trending words “supernatural” and “a miracle” and give god or anything else (like karma) credit for it would be more clever.
Back then, we didn’t understood the concept of thunder and interpreted it as god’s wrath. Now, we understand it’s a transmission of electricity from the negatively charged clouds to the neutral ground through ionized particles in the air. I don’t think that scientists now, despite referring to the same phenomena, are talking about the same thing as we did a long time ago.
So no, no scientist will discover the afterlife “but we’ll just call them “Post-Human Conciousness Wells” or something, and insist it totally isn’t the same thing as that ancient superstition.” as it won’t be.
Incorrect. You can construct an isomorphism between the even subalgebra of the 2D geometric algebra Cl(2) and the complex numbers that maps 1 to the unit scalar and i to the pseudoscalar: https://link.springer.com/article/10.1007/BF01883676
Are you interested in proving me wrong, or figuring out the right answer? If you actually read that article instead of just the title, you would have noticed at the end it says
This leads us to say a few words about the widely held opinion that, because complex numbers are fundamental to quantum mechanics, it is
desirable to “complexify” every bit of physics, including spacetime itself. It will be apparent that we disagree with this view, and hope earnestly that it is quite wrong, and that complex numbers (as mystical uninterpreted scalars) will prove to be unnecessary even in quantum mechanics
They literally say that “complex numbers are fundamental to quantum mechanics”. In other fields of physics complex numbers are just a convenient tool, but in quantum mechanics they are(as far as we know) fundamental, even if the author hopes that to be proved wrong at some point.
You seem like you know a bit about alternatives to complex numbers in other areas of physics, so it would be interesting to have a further conversation, as long as you stop being so defensive.
Complex numbers seem to be used either as 2d vectors or as representation of waves/circles in exponentials, is there an alternative that combines both of those uses?
Yep, there’s an alternative, i.e. equivalent mathematical formulation that does everything complex numbers do: the geometric algebra introduced in the article I posted earlier.
The fundamental object of GA is the “multivector”, which is essentially a sum of scalars, vectors, bivectors and higher grade elements. For instance, you could take the unit x-vector and add it onto some number, say 2, to get the multivector M = 2 + e_x. (To be precise, the space of multivectors is the direct sum over the n-th wedge of the base vector space, n = 0 to dim V).
Another important concept is k-vectors, which are essentially k-dimensional volume elements. For instance, a bivector is an area with a direction, and a trivector is a volume with a direction (in 3D there is only one possible “direction” for the volume, but in 4D spacetime volumes itself can be oriented like surfaces can be in 3D).
Then, you introduce the “geometric product” for two vectors a and b:
ab = a·b + a ∧ b
where a · b is the normal scalar product between the two vectors, and a ∧ b is the wedge product between them. The wedge product essentially is the plane spanned by the two vectors, and is antisymmetric (a ∧ b = - b ∧ a, because the orientation of the plane is reversed when exchanging the vector). For instance, the unit bivector in the x-y plane is given by
B_xy = e_x e_y = e_x ∧ e_y
Notice how the scalar product part of the geometric product is zero, and only the wedge (i.e. bivector part) remains
In 3D, there are four types (“grades”) of objects: scalars, vectors, bivectors (also known as 3D pseudovectors) and trivectors (or also known as 3D pseudoscalars). It’s already a very rich subject and has many advantages over classical vector calculus, but for replacing complex numbers, we’re mainly concerned with the 2D case.
In the 2D case, there are three types of objects: Scalars, 2D vectors, and bivectors/2D pseudoscalars. There is only one possible orientation for a 2D plane in 2D, so we just denote a bivector with area A as B = A I, where I = e_x ∧ e_y is the only unit bivector/2D pseudoscalar.
A nice thing we notice about the I is that it squares to -1 with the geometric product:
I^2 = (e_x ∧ e_y)^2 = (e_x e_y)^2 = e_x e_y e_x e_y = - e_x e_y e_y e_x = -e_x e_x = -1
The first step works because the scalar product part between e_x and e_y is zero. The second step is just writing out the square. The third step is e_y e_x = e_y ∧ e_x = - e_x ∧ e_y = -e_x e_y, which again works because e_x · e_y = 0. We see that the 2d pseudoscalar I behaves just like the “classic” imaginary unit i.
Because the geometric product is associative, and commutative if only scalars and bivectors are involved, the geometric notion of scalars and 2D pseudoscalars can fully replace the notion of complex numbers by making the substitution a + bi -> a + bI.
If you want to learn more about GA, I can recommend Doran, Lasenby: Geometric Algebra for Physicsists :)
Stating mathematics is a tool doesn’t answer if mathematics are real or not.
But I would say, from my humble experience, that mathematics is both unreal and perfectly tangible.
Mathematics is totally a real thing as it obeys strict rule in logic that are true in our real world, axioms, on which everything else is based so that it can’t be used to state things as being true out of the blue, without any justification before using those axioms, which you can translate into our real world.
But math also has its limits and has been used to demonstrate that it itself is incomplete, undecidable and inconsistant (mathematically, of course, it’s not our common definition here). Meaning, as mathematics are imperfect, it can’t describe our world perfectly and therefore isn’t real.
On the other hand, I don’t think calling everything we don’t understand “magic” … and give god or anything else (like karma) credit for it would be more clever.
i think quite a few theologians would agree with that point
And it’s great. Though, as a religious person, thinking this way is no more than shooting yourself in the foot, which is quite sad because religion has only two choice: either cultivating the ignorance but going against science, which is wrong, or cultivating knowledge but overtime, disappearing as a religion. Either way, nowadays it’s doomed.
i don’t see how it’s shooting yourself in the foot. one of the ideas behind the argument i linked to is that pitting god against science isn’t good theology. science will offer more compelling explanations for material phenomena, but that doesn’t necessarily exclude the existence of a god. the idea is to see god as more of an architect: something that made a world that has all these wonderful scientific rules and complex systems that we can discover.
i should mention that i’m not a religious person but i do think it’s an interesting thing to think about.
Religion and science are orthogonal. Science seeks to answer the question of “how?”, while religion seeks to answer “why?”
Understanding the Big Bang all the way through evolution doesn’t give an indication as to why all of this happened. Why are we here? What is our purpose? Science doesn’t have an answer for these questions because these questions are orthogonal to science. Any kind of answer to this kind of question would constitute a religion.
It’s really atheism (at least in the present iteration) that’s doomed to failure. It’s dependent on ignorance of basic philosophy, and attempts to derive any kind of morality based solely on science results in things like eugenics and an “the ends justify the means” kind of mentality. Atheist ethics have resulted in more deaths than all other religions combined. And yes, atheism is a religion, but the ignorance of atheists has resulted in them believing it isn’t a religion even when it exhibits all the properties of a religion. It’s just a shit religion, which is why it’s doomed to fail.
Religion and science are orthogonal. Science seeks to answer the question of “how?”, while religion seeks to answer “why?”
Religion doesn’t try to answer anything: it’s just blind faith. You’re not gonna try to tell me religious people are “looking for” anything. The definition of religion is “belief in a deity”. It doesn’t try to explain or find out anything.
It’s dependent on ignorance of basic philosophy, and attempts to derive any kind of morality based solely on science results
Since when atheism prevents philosophy? Haven’t you heard of atheist philosophers? They exist, they’re not fairies, you know.
About morality, it’s still a subject and a lot of philosophers have different opinion, with the subjective or objective moral, relativistic moral, etc…
And whatever you mean by “derive any kind of morality based solely on science results”, it’s still better than arbitrarily define a moral based on a book written by some people a long time ago to then enforce it for centuries, with violence if needed, and then when the bad atheists come to clean all the mess by making moral laws to have everybody end up agreeing on after few decades, claim it was just a misinterpretation of the texts or whatever, which is the dumbest excuse I’ve ever heard of.
Atheism isn’t a religion either: atheism is a lack of belief in the existence of an unproven (and certainly unprovable) entity.
So a lack of belief certainly didn’t kill anybody.
And atheism was never the reason or the foundation of the sentence “the end justify the means”, it existed long before atheism was even a concept.
i completely agree with the first two paragraphs, but i don’t quite understand what you mean in the third paragraph. could you elaborate on what you mean by the present iteration of atheism, how its like a religion, and why you think it’s doomed to fail? it sounds interesting and i haven’t heard much about it.
Math is a tool of the mind to describe our world, imaginary numbers is only a extension of that tool to allow us to go beyond what mathematical logic prevents us to do, while still getting in the end a real number. Math, despite being powerful, is a flawed tool, so getting around its flaws by creating things like imaginary numbers isn’t absurd and doesn’t make the result any less real at the end.
On the other hand, I don’t think calling everything we don’t understand “magic” or the new trending words “supernatural” and “a miracle” and give god or anything else (like karma) credit for it would be more clever. Back then, we didn’t understood the concept of thunder and interpreted it as god’s wrath. Now, we understand it’s a transmission of electricity from the negatively charged clouds to the neutral ground through ionized particles in the air. I don’t think that scientists now, despite referring to the same phenomena, are talking about the same thing as we did a long time ago.
So no, no scientist will discover the afterlife “but we’ll just call them “Post-Human Conciousness Wells” or something, and insist it totally isn’t the same thing as that ancient superstition.” as it won’t be.
Quantum mechanics, as far as we know, requires imaginary numbers.
nope, they’re just one mathematical construct out of many (e.g. 2D vector calculus or geometric algebra), and they just happened to stick
Nope, you’re just wrong. Quantum mechanics without complex numbers(real quantum theory) is less predictive than complex(regular) quantum theory. https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/
Incorrect. You can construct an isomorphism between the even subalgebra of the 2D geometric algebra Cl(2) and the complex numbers that maps 1 to the unit scalar and i to the pseudoscalar: https://link.springer.com/article/10.1007/BF01883676
Are you interested in proving me wrong, or figuring out the right answer? If you actually read that article instead of just the title, you would have noticed at the end it says
They literally say that “complex numbers are fundamental to quantum mechanics”. In other fields of physics complex numbers are just a convenient tool, but in quantum mechanics they are(as far as we know) fundamental, even if the author hopes that to be proved wrong at some point.
You seem like you know a bit about alternatives to complex numbers in other areas of physics, so it would be interesting to have a further conversation, as long as you stop being so defensive.
Complex numbers seem to be used either as 2d vectors or as representation of waves/circles in exponentials, is there an alternative that combines both of those uses?
Yep, there’s an alternative, i.e. equivalent mathematical formulation that does everything complex numbers do: the geometric algebra introduced in the article I posted earlier.
The fundamental object of GA is the “multivector”, which is essentially a sum of scalars, vectors, bivectors and higher grade elements. For instance, you could take the unit x-vector and add it onto some number, say 2, to get the multivector M = 2 + e_x. (To be precise, the space of multivectors is the direct sum over the n-th wedge of the base vector space, n = 0 to dim V).
Another important concept is k-vectors, which are essentially k-dimensional volume elements. For instance, a bivector is an area with a direction, and a trivector is a volume with a direction (in 3D there is only one possible “direction” for the volume, but in 4D spacetime volumes itself can be oriented like surfaces can be in 3D).
Then, you introduce the “geometric product” for two vectors a and b:
ab = a·b + a ∧ b
where a · b is the normal scalar product between the two vectors, and a ∧ b is the wedge product between them. The wedge product essentially is the plane spanned by the two vectors, and is antisymmetric (a ∧ b = - b ∧ a, because the orientation of the plane is reversed when exchanging the vector). For instance, the unit bivector in the x-y plane is given by
B_xy = e_x e_y = e_x ∧ e_y
Notice how the scalar product part of the geometric product is zero, and only the wedge (i.e. bivector part) remains
In 3D, there are four types (“grades”) of objects: scalars, vectors, bivectors (also known as 3D pseudovectors) and trivectors (or also known as 3D pseudoscalars). It’s already a very rich subject and has many advantages over classical vector calculus, but for replacing complex numbers, we’re mainly concerned with the 2D case.
In the 2D case, there are three types of objects: Scalars, 2D vectors, and bivectors/2D pseudoscalars. There is only one possible orientation for a 2D plane in 2D, so we just denote a bivector with area A as B = A I, where I = e_x ∧ e_y is the only unit bivector/2D pseudoscalar.
A nice thing we notice about the I is that it squares to -1 with the geometric product:
I^2 = (e_x ∧ e_y)^2 = (e_x e_y)^2 = e_x e_y e_x e_y = - e_x e_y e_y e_x = -e_x e_x = -1
The first step works because the scalar product part between e_x and e_y is zero. The second step is just writing out the square. The third step is e_y e_x = e_y ∧ e_x = - e_x ∧ e_y = -e_x e_y, which again works because e_x · e_y = 0. We see that the 2d pseudoscalar I behaves just like the “classic” imaginary unit i.
Because the geometric product is associative, and commutative if only scalars and bivectors are involved, the geometric notion of scalars and 2D pseudoscalars can fully replace the notion of complex numbers by making the substitution a + bi -> a + bI.
If you want to learn more about GA, I can recommend Doran, Lasenby: Geometric Algebra for Physicsists :)
There’s actually some controversy on that one:
https://www.scientificamerican.com/article/is-the-mathematical-world-real/
Stating mathematics is a tool doesn’t answer if mathematics are real or not. But I would say, from my humble experience, that mathematics is both unreal and perfectly tangible. Mathematics is totally a real thing as it obeys strict rule in logic that are true in our real world, axioms, on which everything else is based so that it can’t be used to state things as being true out of the blue, without any justification before using those axioms, which you can translate into our real world. But math also has its limits and has been used to demonstrate that it itself is incomplete, undecidable and inconsistant (mathematically, of course, it’s not our common definition here). Meaning, as mathematics are imperfect, it can’t describe our world perfectly and therefore isn’t real.
There is an excellent video from Veritasium on the subject of the limits of math: https://piped.video/HeQX2HjkcNo
Here is an alternative Piped link(s): https://piped.video/HeQX2HjkcNo
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I’m open-source, check me out at GitHub.
Man I really love Lemmy for that kind of shit. I’ll change the link right away
i think quite a few theologians would agree with that point
And it’s great. Though, as a religious person, thinking this way is no more than shooting yourself in the foot, which is quite sad because religion has only two choice: either cultivating the ignorance but going against science, which is wrong, or cultivating knowledge but overtime, disappearing as a religion. Either way, nowadays it’s doomed.
i don’t see how it’s shooting yourself in the foot. one of the ideas behind the argument i linked to is that pitting god against science isn’t good theology. science will offer more compelling explanations for material phenomena, but that doesn’t necessarily exclude the existence of a god. the idea is to see god as more of an architect: something that made a world that has all these wonderful scientific rules and complex systems that we can discover.
i should mention that i’m not a religious person but i do think it’s an interesting thing to think about.
Religion and science are orthogonal. Science seeks to answer the question of “how?”, while religion seeks to answer “why?”
Understanding the Big Bang all the way through evolution doesn’t give an indication as to why all of this happened. Why are we here? What is our purpose? Science doesn’t have an answer for these questions because these questions are orthogonal to science. Any kind of answer to this kind of question would constitute a religion.
It’s really atheism (at least in the present iteration) that’s doomed to failure. It’s dependent on ignorance of basic philosophy, and attempts to derive any kind of morality based solely on science results in things like eugenics and an “the ends justify the means” kind of mentality. Atheist ethics have resulted in more deaths than all other religions combined. And yes, atheism is a religion, but the ignorance of atheists has resulted in them believing it isn’t a religion even when it exhibits all the properties of a religion. It’s just a shit religion, which is why it’s doomed to fail.
Religion doesn’t try to answer anything: it’s just blind faith. You’re not gonna try to tell me religious people are “looking for” anything. The definition of religion is “belief in a deity”. It doesn’t try to explain or find out anything.
Since when atheism prevents philosophy? Haven’t you heard of atheist philosophers? They exist, they’re not fairies, you know. About morality, it’s still a subject and a lot of philosophers have different opinion, with the subjective or objective moral, relativistic moral, etc… And whatever you mean by “derive any kind of morality based solely on science results”, it’s still better than arbitrarily define a moral based on a book written by some people a long time ago to then enforce it for centuries, with violence if needed, and then when the bad atheists come to clean all the mess by making moral laws to have everybody end up agreeing on after few decades, claim it was just a misinterpretation of the texts or whatever, which is the dumbest excuse I’ve ever heard of.
Atheism isn’t a religion either: atheism is a lack of belief in the existence of an unproven (and certainly unprovable) entity. So a lack of belief certainly didn’t kill anybody.
And atheism was never the reason or the foundation of the sentence “the end justify the means”, it existed long before atheism was even a concept.
i completely agree with the first two paragraphs, but i don’t quite understand what you mean in the third paragraph. could you elaborate on what you mean by the present iteration of atheism, how its like a religion, and why you think it’s doomed to fail? it sounds interesting and i haven’t heard much about it.