I ask this in relation to ” /(x/y) ” = ” *(y/x) ”
My mathematical ignorance does not allow me to perceive exactly what it is that confuses me about these manoeuvres and so perhaps my question is vague.
I have no difficulty with it as a technique; as something through which I can put an expression, and out at the other end the right result will appear. What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.
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**EDIT:**
It was very rewarding for me to read all of your comments. Thank you most kindly for enlightening me.
For those interested in the cause of my previous confusion:
The gaps in my understanding of going from y*x=z to y=z/x were definitions of the equal sign and division.
I can see now that I previously considered the = sign to mean «result» or «answer» in some sort of final sense, like a conclusion; I now see that it only states that this is equal to that.
Following this fundamental piece of knowledge, I can belatedly understand what an equation is. From there, via the definition of division as the opposite of multiplication, I can see that if I divide something while also multiplying it with the same number, these actions cancel each other out.
And so the magical spell between y*x=z and y=z/x is the logic above expressed mathematically as x/(y*x)=z/x.
In: 25
The comment herein is a different approach, trying to respond to the reason why the question was asked rather than merely justify or explain the procedures involved. If you only care about math, please skip it.
>What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.
It works as mathematical (logical) procedure because it is a kind of “magic spell”, and it is a kind of magic spell because it always works even if, and even though, we don’t know why it always works.
Consider the following equivalency:
Meta = above and beyond
Super = above and beyond
Physics = the universe
Nature = the universe
Therefore metaphysical isn’t logically different from supernatural.
Why mathematics works is a deeply philosophical and, in truth, unanswerable question. Descartes reasoned that it could only be answered by presuming that a benevolent God (a supernatural agency) was kind enough to provide a rational (logically consistent) universe for us to exist in.(This relies on an “inverse teleology” of God’s intention.) More contemporary mathematicians generally (but not exclusively) tend to believe that a better answer is the *anthropic principle*, a “reverse teleology” of natural selection: if we didn’t exist in a rational universe (one where math works) then we wouldn’t exist at all to begin with and wouldn’t be here to wonder why.
Most mathematicians (which might include Decartes, I think, if he had lived in a post-Darwin world) prefer the reverse teleology because it doesn’t require a supernatural agency, but it isn’t *logically* different from the inverse teleology, it simply assumes the rationality of metaphysics (that math always works, even when it isn’t human calculation but executed by automated computers) without a supporting supernatural explanation of *why* metaphysics is rational. This explains why many hyper-rationalists ascribe to a “brain in a jar” worldview of the universe as a simulation. This is supposedly a coherent and “logical” idea, but isn’t really. The “simulation theory” requires that our universe is ostensibly in a computational system *intentionally designed by an extra-universal [metaphysical/supernatural] agency*, so in reality it takes the weakest part of Descartes inverse teleology and combines it with the weakest part of the anthropic reverse teleology and still doesn’t address the original question of *why* mathematics is metaphysical in nature, or why metaphysics is mathematical in nature. It simply accepts the assumption as correct because it is “logic!” and leaves it at that, using circular logic to deny the fact it is circular logic. It works because everything in the objective universe (outside of humanity and our own thoughts, perhaps) behaves logically, according to mathematical laws of physics, which cannot be broken despite the absence of any enforcement mechanism or agency.
Ultimately, “why it works” has only one real answer, described as *the ineffability of being*. The meaning and purpose of it working is unknowable (the meaning is epistemically uncertain and the purpose is metaphysically uncertain,) all we can know is *that it does work*, every time, without fail, almost as if it is magic.
Thanks for your time. Hope it helps.
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