/sci/ - Science & Math

you're a faggot

>>8456537
If infinity isn't a number, then how can I say how many guys have banged your mom?

>>8457971

>>8456537
>tfw ejaculate breaks the light barrier

>>8456537
Infinitesimals are real, but are not "real" in the sense of being real numbers.

For instance, in the ring [math]\mathbb{R}\left[ \varepsilon \right]/\left\langle {{\varepsilon ^2}} \right\rangle [/math], epsilon is an infinitesimal.

>>8458547

>>8458573

In non standard analysis, infinitesimals are rigourous.

>>8458573
>dual numbers
>infinitesimals

yea sure

>>8460524
It isn't obvious how it is connected to like calculus unless you look at the geometry.

For instance, consider a closed immersion of schemes [math]X \to Y[/math] defined by an ideal [math]\mathcal{J}[/math].

An infinitesimal thickening is a factorization [math]X \to Z \to Y[/math] where Z is a scheme with the same underlying top. space as X but defined by the ideal [math]\mathcal{J}/{\mathcal{J}^2}[/math].


So let [math]X = \operatorname{Spec} k\left[ x \right][/math] and [math]Y = \operatorname{Spec} k\left[ {x,y} \right][/math]. Denote [math]f:X \to Y[/math] for the closed immersion.

[math]Z = \operatorname{Spec} \frac{{k\left[ {x,y} \right]}}{{\left\langle {{y^2}} \right\rangle }}[/math] defines a thickening [math]\operatorname{Spec} k\left[ x \right] \to \operatorname{Spec} \frac{{k\left[ {x,y} \right]}}{{\left\langle {{y^2}} \right\rangle }} \to \operatorname{Spec} k\left[ {x,y} \right][/math].

Under [math]{\mathcal{O}_Y} \to {f_*}{\mathcal{O}_X}[/math] we recover the polynomials [math] p\left( {x,0} \right)[/math], however under [math]{\mathcal{O}_Y} \to {f_*}{\mathcal{O}_Z}[/math] we also recover [math]\left( {\partial p/\partial y} \right)\left( {x,0} \right)[/math].


So this switch to looking at dual numbers is infact giving us infinitesimal information.

>>8456537
>pic
kek, I made that. Glad to see it got saved

>>8460624
i don't understand it

>>8460625
You weren't there maaan. The original image was just a waterfall of shit and the thread was about how you get to the girl shit free or parodies of such. I turned it into a Trolley problem.

>>8460622
>algebraic geometry actually makes sense to some people
you are my hero

>>8461079
>you are my hero
its ez

>>8456537
clitoris!=penis

>>8461863
i wish
i know all the definitions and things, and i've worked with commutative algebra some, but i just don't have the intuition to be able to manipulate things like sheaves or specs efficiently

i keep reading through hartshorne hoping that something will eventually click, but it's been almost a year now and i still suck at it

>>8462229
>but i just don't have the intuition
>i keep reading through hartshorne

There is your problem. Hartshorne is a great book, but horrible for intuition. Try Eisenbud and Harris.

>>8456537
>infinitesimals aren't real
they are mental constructs, Leibniz

>>8463089
prove it faggot