Abstract Algebra is Impossible Without These 8 Things

Important note: for the Descartes rule of signs, there are actually 3, not 2, sign changes. But in the summary document below the error is fixed.

PDF Summary

This video was made with the help of this Abstract Algebra book

--
Al-Khwarizmi Video (it's from awhile ago but it explains that concept well!)

Galois Theory Video
---

😎 Become a member to have exclusive access:

🔔 Subscribe:

🥹 Consider supporting us on Patreon:

[jimcoughlin]

I think your example of Descartes' rule of signs is incorrect. Your text says there are 2 changes of sign in the given polynomial, but there are 3. Indeed if you graph the polynomial there are 3 positive roots. Please correct this so as to not confuse students.

[Khashayarissi-obyj]

With luck and more power to you.
hoping for more videos.

[Prof_Michael]

Keep it moving
I like your series cos I’m a mathematician as well

[Khashayarissi-obyj]

With luck and more power to you.
hoping for more videos.

[ravikantpatil]

Very good content as well as presented ❤

[steaminglobster]

As EE engineer, I like the way you guys explaining math. Should keeping this system level way, top to bottom approach in the future! I also think any recommendations for great books on math will be helpful too.

[MrBeen]

Yes, I want to know more about Abstract Algebra from the ground up

[KeithKessler]

Would love some recommendations for texts for self-study to learn the basics of abstract algebra, especially some that are free or in the public domain.

[ygfddgghhbvdx]

Can you do video on how greek and roman mathematician did math with roman numaral, such as euclids division algorithm.

[ramaronin]

Finalmente um canal que apresenta álgebra abstrata de forma didática. e ainda por uma dupla de br!

[sumdumbmick]

you open by claiming that the Naturals have been used since the beginning of mathematics, but this is a lie. the Naturals were invented in the late 19th century as a part of Logicism.

this is significant, because the definition of the Naturals is self-contradictory, and this self-contradiction is the source of Godel Incompleteness. the simple fact that Greek Geometry is not subject to Godel Incompleteness should be sufficient to convince you that what I'm saying is basically correct, but since I know that you're unconvinced, consider what an absolute value is.

an absolute value, or a norm, of a vector is supposed to be the magnitude of that vector. being a magnitude it shouldn't have a unit, since a combination of a unit and a magnitude is a vector, not a magnitude. the problem comes here, because an absolute value always has a unit, since it is positive. a positive value is a vector, not a magnitude, because its direction is opposite that of a negative value. this means that we're obfuscating magnitudes and 1-dimensional positive vectors. and we can show that this is specifically the self-contradiction at the heart of the paradox of the Naturals which begets Godel Incompleteness in any mathematical system which admits the Naturals.

so now let's consider where the Naturals come from within Logicism. here we get hand-wavy arguments about the Successor function mapping to the Naturals, and with this we apply the Axiom of Extensionality (if using ZF(C)) to conclude that the Successor function generates the Naturals. however, this necessarily means that the Naturals must be ordered, since the Successor function generates an ordered set by way of its construction. yet, the Naturals are supposed to correspond to bare magnitudes, like the 7 that appears in 7 elephants, 7 bottles of water, and 7 days in a week. when asking what's the number part of 7 elephants it must be 7, and only 7, yet when asking what's the number part of -7 we always say that -7 is a number, and similarly we say that 7 is +7. this allows us to claim that the Naturals are the positive Integers, while also claiming that the Naturals are fundamentally unsigned.

and this obfuscation of what's happening, of sometimes leaving the unit off of what we call numbers, and sometimes permitting it to be an integral part of what we call numbers, is at the heart of the Godel Incompleteness theorems, since Godel computes Godel numbers as signed, ordered, ordinals, but then via the same implicit violation of rigor suddenly claims that they're unsigned, unordered, nominals. and this is so true that the second Incompleteness theorem specifically demonstrates that this approach is nonsense. yet, somehow, absolutely nobody's noticed what's actually happening, and instead simply carried forward with the Logicist assumption that the Naturals are not a completely nonsensical, self-contradictory notion that breaks everything they're incorporated into.

so you're opening with a claim that you should be ashamed to have ever believed in, not proudly broadcasting your unwaiveringly mindless faith in.