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?

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  1. 1 month ago
    Anonymous

    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.

    • 1 month ago
      Anonymous

      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.

      • 1 month ago
        Anonymous

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

      • 1 month ago
        Anonymous

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

      • 1 month ago
        Anonymous

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

    • 1 month ago
      Anonymous

      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

      • 1 month ago
        Anonymous

        >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.

        • 1 month ago
          Anonymous

          >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

      • 1 month ago
        Anonymous

        >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.

        • 1 month ago
          Anonymous

          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.

          • 1 month ago
            Anonymous

            >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.

            >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.

            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.

        • 1 month ago
          Anonymous

          >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.

      • 1 month ago
        Anonymous

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

        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

      • 1 month ago
        Anonymous

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

        • 1 month ago
          Anonymous

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

  2. 1 month ago
    Anonymous

    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.

  3. 1 month ago
    Anonymous

    >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.

  4. 1 month ago
    Anonymous

    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

    • 1 month ago
      Anonymous

      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

      • 1 month ago
        Anonymous

        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.

  5. 1 month ago
    Anonymous

    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.

    • 1 month ago
      Anonymous

      >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?

  6. 1 month ago
    Anonymous

    Linear thinking.
    Because sphere tensors are hard to think about and do shit about.
    Shit's hard, and e = ln(e)

  7. 1 month ago
    Anonymous

    completely moronic

  8. 1 month ago
    Anonymous

    Thanks all, for this fruitful discussion.

    • 1 month ago
      Anonymous

      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

      • 1 month ago
        Anonymous

        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.

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