Was it just mathematicians trying to find some patterns then we found applications in which they are useful?

Beware Cat Shirt $21.68 |

Beware Cat Shirt $21.68 |

Skip to content
# What is the logical essence behind cross and dot product?

Was it just mathematicians trying to find some patterns then we found applications in which they are useful?

Beware Cat Shirt $21.68 |

Beware Cat Shirt $21.68 |

Comments are closed.

Pure mathematics doesn't have any bearing on practicality or instantiation in reality. Hell half the mathematicians who developed the math in e.g. quantum electrodynamics would be rolling in their graves to learn that what was essentially joke math is being used in physics.

The fact that mathematics is astonishingly applicable to the physical world is a major topic in philosophy, and has been since at least Plato.

So what fundamentally distinguishes pure mathematics from pseudomathematics? also I'm interested to know what is that topic in philosophy, as I have some tendency toward philosophy; I'm sure that it complements science.

not the guy you talked to, but for the love of you not becoming a moron, steer clear of hegel, fricker was a wacko

>pseudomathematics

don't bother, you are talking with a finitist, they are like the special needs child of philosophy and mathematics

You both believe pseudomathematics exists and don't know it's distinction from pure mathematics?

this is why self stufy guide IQfycels will never amount to anything.

where do you think calculus came from? physics.

where do you think cross and dot products came from? physics.

they are useful tools in the real world. if work, torque and other physical shit didn't work like that they would be just another operation between vectors

>Just post a restatement of the problem

Your post doesn't amount to anything. This is a discussion of *how any why* it is the case that pure mathematics is both logically and historically prior to applications in physics.

>physicists find pattern

>mathematicians formalize it

>WOOOOW HOW WERE THOSE EPIC MAETHEMATICIANS ABLE TO INVENT SHIT FROM THIN AIR THAT MAGICALLY.APPLIED TO THE REAL WORLD???

grow up

>where do you think calculus came from? physics.

>where do you think cross and dot products came from? physics.

Math is primary. Pure platonic abstractions of reality create reality in the first place.

This has been a relatively mainstream view for a long time but you need to understand that something like this obliterates a lot of philosophical commitments.

>This has been a relatively mainstream view for a long time but you need to understand that something like this obliterates a lot of philosophical commitments.

Pic related. If your philosophy does not adhere to reality, it's not useful. A similar thing could be said to all of the theoretical particle models where it's just physicists creating all these elaborate theories like 11 dimensions, charm quarks, and whatever other drivel that can't be used as proper prediction models.

What are you on about? Quantities, geometries, mathematical abstractions exist without people needing to exist. Math is discovered and is already in reality, otherwise there would be no such thing as math.

>Pure platonic abstractions of reality create reality in the first place.

If a tree falls in and no one is around to hear it, did it make a sound? Plato's theory of forms is completely anthropocentric. Subjective perspective has no bearing on objective reality.

steer clear of heidegger too

guy was a cuck

no, really

his wife got pregnant by another man and he was totally ok with it

cuckpilled and based but his philosophy is too cucky for my taste

what the frick does self study have to do with not understanding vector spaces?

He meant that college studies would give a more in-depth experience, but he didn't formulate his opinion in a good manner.

To give actual insight into their origin as opposed to whatever is going on right now: Once it was discovered that complex numbers allow you to represent rotation in two dimensions, attempts were made to find a 3D equivalent.

These failed until Hamilton engaged in a bit of vandalism on a bridge with the discovery of quaternions, which are the 4D equivalent to complex numbers, defined by [math]i^2=j^2=k^2=ijk=-1[/math].

You can represent a vector in 3D as a quaternion with no real part (and each axis corresponding to a different quaternionic component), and if you multiply two such quaternions together...

[math](ai+bj+ck)(di+fj+gk) = adi^2 + afij + agik + bdji + bfj^2 + bgjk + cdki + cfkj + cgk^2[/math]

[math]= -ad + afk - agj - bdk - bf + bgi + cdj - cfi - cg[/math]

[math]= -(ad+bf+cg) + (bg - cf)i + (-ag + cd)j + (af - bd)k[/math]

Observe that the dot product of our vectors is the same as the real part of this answer, just negative, and that the vector we get as a cross product corresponds to the nonreal part of the answer.

>cross product

Definetely!

Whole magnetism shit is confused cause they exacly did that. Magnetism is just a relativistic side product of coulomb law (Grant&Phillips Electromagnetism). But they needed to have use for the novel operator. Another is torque.

But! Wu experiment. Cobalt 60 wants to shoot decay product (electron) to certain (magnetic field) direction. This means the things in nucleus rotate certain way.

Anon, I will answer your question very simply.

The cross product and the dot product are two different ways of solving the problem of multiplying two objects (vectors) that aren't just numbers.

We say that a vector is a quantity that has a magnitude and a direction. We know how to multiply magnitudes (normal multiplication) but we don't know what it means to multiply a direction. To answer that problem, mathematicians found two solutions :

Solution 1 (the dot product) : The idea behind the dot product is to project the second vector onto the first vector. Once its done, you basically come back to a situation in 1 dimension, which corresponds to regular multiplication of two numbers.

Solution 2 (the cross product) : In that case, you want to conserve the fact that there are more than 1 dimension, so you want to drag the second vector across the length of the first vector, which will give you the shape of a parallelogram. A parallelogram is a generalization of a rectangle. This is important to keep in mind because when you multiply two regular numbers geometrically, you drag the width of a rectangle across the length of the rectangle which will give you the area of that rectangle.

When I multiply 3 x 2 = 6, 6 is a length but it is also the area of the rectangle that has a length of 3 and a width of 2. The cross product is similar : the length of the cross product is also the area of the parallelogram created by the two non perpendicular vectors connected at their origin.

No need for fricking moronic complex numbers or geometric algebra to understand that

this guy is right

dot product arises naturally if you are doing trigonometry/geometry

cross product is actually kind of weird and i think it first was derived when Hamilton was working out quaternions

As I said in my post, cross product is just generalizing multiplication to two line segments that are not perpendicular like we have for scalar multiplication. It has to factor in a non 90 degree angle and orientation. There's nothing mystical about it.

No to your statement OP, dot and cross product are geometric properties that were later formalized into mathematics. The idea that mathematics is created as a "game" and then an application is "found" is largely a myth that's propagated by people who failed out of their PHD program. Topology and Riemannian manifolds (work of Poincare, Riemann, Maxwell et al) were all branches of physics that had some mathematical application later.

Even fields like axiomatic set theory and the Hilbert program are very closely linked to developments in Analysis, which began as a branch of physics. (see this Rudin lecture)

The most popular mathematical systems (rational, real, complex numbers, linear algebra, etc...) are popular and useful through centuries of trial and error. There have been plenty of mathematical offshoots that have fallen out of fashion. The branches of math we use the most tend to be battle-tested that way.

>a myth that's propagated by people who failed out of their PHD program

>spacing

people get PHDs in formal math to play games now?

Linear thinking.

Because sphere tensors are hard to think about and do shit about.

Shit's hard, and e = ln(e)

completely moronic

Thanks all, for this fruitful discussion.

hey, it certainly beats the 8735737the iteration of the "imaginaries not real", "me no like sets" or "0.9... is not 1" bait threads, this one was a bit of fresh air which some actual decent information about the topic at hand

I agree, as questions with sound intentions necessarily lead to fruitful discussions and fulfilling answers, manifesting that every participant has to fill his knowledge gaps.