/sci/ - Science & Math

https://youtu.be/R1awrN9NOEY?t=5m
Can anyone explain what kind of fungus or bacteria grew at the start of minute 5? Like, that shit is not normal. That is not how a cake grows. It really looks like a fungus or some shit like that. I refuse to believe that is normal.

Recently I've been reading about the life of Paul Erdos and how he would abuse coffee and amphetamines and considered those two things fundamental to being able to produce new theorems.

I am right now training to be a mathematician and I really don't want to miss out on what soon may be my best years. Right now I only drink one cup a day in the mornings to wake up as university starts pretty early but I wouldn't have a problem with increasing this dose. I don't really care that much about my health, I just want to be a great mathematician.

How much coffee should I be drinking daily?

I just finished reading Hardy's A Mathematician's Apology and I am a bit perplexed.

I want to discuss the title of that book. Why did he call it an apology? From the title it feels like he's going to say:

>Okay, I know that I did nothing good in my life for other people and instead decided to focus on finding knowledge only I and a few others care about for my own personal glory. I am sorry for that, I should have used my genius to be a doctor instead. Sorry.

But then in the book he basically justifies the attitude's of a mathematician. First he begins by debunking the notion of "using your genius for something else" by saying that men are BORN good in at most one thing and that if you do not quickly train that one and only one skill you have you will forever be some mediocre faggot working at Mc Donald's.

And I don't mention "men" for no reason. Throughout the book he says "men" in contexts where it is way more appropiate to say "person".

For example:
"Ambition has been the driving force behind nearly all the best work of the world. In particular, practically all substantial contributions to human happiness have been made by ambitions MEN."

Going back to my original point, he also justifies his own self-indulgence by saying how by studying mathematics he met the best men in the world (talking about Ramanujan and Littlewood).

He then says, right at the end, that mathematics is good because when you get to the level of abstraction he was on, math is so useless that it can't even be used for war.

So what was the point of this book? I am a student of mathematics from that perspective that book is really good as an insight into the mind of a past genius. He talks about his motivations and achievements in a relatable manner. But then why not call it "A Mathematician's Motivation"?

Other than that he also talks of the beauty of mathematics, so why not call it "A Mathematician's circlejerk"?

How is this an apology?

I was thinking about an interesting question:
If two functions f and g are equal at a point x, does that mean their derivatives are equal?

At first I thought yes, this is intuitive, but then I realized that if the two functions are only equal at x, then this doesn't hold. Then I thought, what if they are equal inside of an interval. Say [a.b]. Will it hold. Well, I haven't done analysis but I have taken calculus so I wanted someone to check my proof.

Theorem: Let f and g be two functions that are at least equal in the interval [a,b] and differentiable in the interval (a,b). Their derivatives will also be equal at least in every point inside (a,b).

Proof: First, lets pick an arbitrary c inside of (a,b). We know that the following limit must exist:
[math] \lim_{h\to 0}\frac{f(c + h) - f(c)}{h} = k[/math]

As f(c) = g(c) I will immediately replace it to get:
[math] \lim_{h\to 0}\frac{f(c + h) - g(c)}{h} = k[/math]

Now this, by definition, means:
[math] \forall \epsilon > 0 \exists \delta > 0[/math] such that [math] 0 < |h| < \delta \implies | \frac{f(c + h) - g(c)}{h} - k | < \epsilon [/math]

So this means that there exists some mapping from epsilon to deltas. Lets suppose
[math] \delta = w( \epsilon ) [/math]
Now consider the set:
[math]E = \{ \epsilon [/math] such that [math] w( \epsilon) < min(|-c+b|,|-c+a|) \} [/math] where min takes the minimum of the two numbers.

And define the new mapping
[math] w_2 ( \epsilon ) =
\left\{
\begin{array}{ll}
w( \epsilon ) & \mbox{if } \epsilon \in E \\
min(|-c+b|,|-c+a|) & \mbox{if } \epsilon \notin E
\end{array}
\right. [/math]

From this definition we get that [math] w ( \epsilon ) \geq w_2 ( \epsilon ) [/math] for all epsilons.

Which means that we can freely replace the new deltas found by this mapping and the limit will still hold. So:

[math] \forall \epsilon > 0 \exists \delta = w_2 ( \epsilon ) > 0[/math] such that [math] 0 < |h| < \delta \implies | \frac{f(c + h) - g(c)}{h} - k | < \epsilon [/math]

Cont.

How can proof by contradiction be a valid method? In systems like ZFC that are not known to be consistent then how do we know that if a sentence S is true then its negation Z must be false? I mean, the definition of consistency is that the theory does not contain contradictions. So if we do not know that ZFC is consistent (unprovable) then how can we assume that it can't contain contradictions? For all we know, ZFC could contain contradictions. Many.

Just because Z is false, that does not mean S is true. If this was true for all Z then ZFC would be consistent, which is an unprovable statement inside ZFC. And if ZFC is the foundation for all of mathematics does this not mean that the method of proof by contradiction is invalid for all areas of mathematics, unless reformulated in smaller, self contained theories that can be proven to be consistent?

Do you think it is morally fine to discriminate based on how good at person is at mathematics? Just to be clear, I don't mean like bash someone for not knowing topology as maybe they haven't reached that level. I mean more in the sense of people who understand new concepts quickly vs people who take fucking ages to understand something. Is it okay to discriminate based on that?

I ask because the deeper I go within mathematics the more I see other people as disgusting dogs. Lesser beings that are clearly below me and I feel like maybe I should act on those thoughts.

Imagine a society like this: We teach kids arithmetic and some algebra. We give them a good idea of what numbers are and how operations work. Then at the earliest age possible we give them a test:

We define a field (in the algebraic sense) and then ask them to solve equations in that field. Maybe ask them to prove certain theorems about that field. Then we measure their pure mathematical skill with their results. This would be good because no one would have previous knowledge about the field, so it would be all about how quickly they can understand the objects and operations of the field so that they can start solving problems.

Then we give the top 10% of students and their family free healthcare, free housing, free food, special privileges when it comes to choosing their university and other things. All funded from the new taxes we would implement for the lower 90% of kids and their family. They would be basically second class citizens.

What do you think of a society like this? I think this would definitely save society. Smart kids born in poor families would immediately be upgraded to high status wealthy citizens who would stand on top the filthy dogs below them.

Do you have alternative plans? Do you think this would be immoral (explain why)?

Time for a /sci/ story, bois.

>Be me
>Finish my freshman year in mathematics, great grades
>Decide I am going to prove the goldbach conjecture in my summer time
>Find a new way of approaching the problem
>Start working
>Get distracted
>Start fantasizing about going on national television to talk about my proof and becoming famous, etc.
>Go back to work
>All the "manual labor" is done, time to get some theorems
>Google "theorems about prime numbers"
>A whole wikipedia list of theorems
>Read all of them
>None of them strong enough to help me
>Actually, all those theorems are really fucking weak. They barely say anything.

Why are number theorists such a bunch of light weights? Holy fuck. Those theorems are SO fucking weak.

>Decide number theorists are gay
>Forget their theorems
>Going to prove this through sheer force of will, using nothing but my own work
>Keep fantasizing about getting invited to a top uni to finish my degree and then get 50 PhDs
>Keep working
>Realize Goldbach conjecture is too hard to prove by sheer force of will. Decide to make a weaker conjecture related to my method
>Try to prove it
>My weaker conjecture is too strong, get an EVEN weaker conjecture and try to prove that
>Nothing
>Finally realize my method is potentially flawed and I need to look for another direction
>Go back to wikipedia page about prime numbers, give it another look
>NOTHING
>Gay weak ass theorems
>Decide I still need to prove it from scratch, just need another direction

FUUUUUUUUUUUUUUUUUUUUUCK.

3000 years of number theory for this shit.

>Uh, uhhh there is at least 1 prime number between n and 2n

NO FUCKING SHIT MOTHERFUCKKKEERRRRRRR. YOU CAN'T PROVE ANYTHING WITH THAT.

Seriously number theorists, what a fucking gay field. Once I prove the goldbach conjecture I'm moving to algebra because I don't want to be associated with faggots.

If the Goldbach conjecture were ever solved, what practical application would it bring?

For those who don't know what it is, the Goldbach conjecture states every even integer greater than 2 can be expressed as the sum of two primes. It has never been proven.