Topic: Mathematics isn't unbelievable.

Mathematics is not intuitive for the very reason mathematics works so well: Abstraction. Many believe we evolved or were "given" an innate ability to understand mathematics. Some argue that mathematics is unreasonably effective because we can understand it. I disagree. It may be unreasonably effective, but I reject the notion that it is unreasonably effective because it was "made for us" or that it's intuitive. Here's a very simple example:


\(
{ x^2 } = 4 \\
=> { \sqrt{x^2} } = { \pm \sqrt{4} } \\
=> x = \pm { \sqrt{4} } \\
x = 2 \\
or \\
x = -2
\)


But, remember that:


\(
\sqrt{4} = 2 \\
and \\
- \sqrt{4} = -2
\)


There's no plus or minus, positive is implied outside of the context of quadratics, except when we explicitly add a negative in front. Try it on your calculator, if you like. There is no intuitive way to know that quadratics have two solutions. You have to know that ahead of time. That is, at minimum you have to know that this is the case:


\(
\sqrt{ x^2 } = | x |
\)


Which means we don't know weather x is going to be positive or negative, because it will always present as positive in this context. How can that be intuited? How can one derive that? The only reason quadratics work at all is because when "we" developed math, we tweaked it so it works with the rest of mathematics. In this case, if we failed to interpret x as an absolute value, then it would break the definition of a function. We would get -x and x, two outputs given the same input. There is no proof for the outcome of an absolute value, it is simply defined in mathematics as so. [1]


There are plenty little rules in mathematics similar to this, in that one can't possibly "just know" without either being taught them, or going through a long series of mental gymnastics to make an educated guess as to what the standard definition is. To my knowledge, the difference between principle square root and square root wasn't mathematically derived, so much as defined so that the rest of mathematics could work. Allow me to explain.


\(
(-3)^2 = 9 \\
but \\
-3^2 = -9
\)


Why? Because:


\(
(-3)^2 = 9 \\
=> -3 * -3 = 9 \\
and \\
-3^2 = -9 \\
=> -3 * 3 = -9
\)


So the parenthesis make all the difference here. The square only applies to 3 in the first example, and it treats the negative as a separate term, whereas the square applies to both the negative and the 3 using the parenthesis, because -3 is basically -1* 3. Note the difficulty and mental gymnastics one would have to jump through to simply intuit all of this. One cannot know intuitively how to interpret parenthesis, they are simply mathematically defined. In order to pull something like that off, you would need to conceptually re-invent mathematics in a sense. We skipped over something, however. In the above example, how do we even know -3 * -3 is equal to positive 9? Again, it's simply a definition, and that definition wasn't the product of a proof, but instead the product of necessity:



So, in my opinion, mathematics isn't unbelievable. It isn't an intuitive part of human nature. It is the product of thousands of years of analytic, grueling, step by step thinking. Mathematics was and is built on previous steps, and when mathematics was originally being constructed, sometimes we simply had to define rules to construct a consistent language. These days, new rules are rarely needed, as the consistent framework of mathematics has already been built. The lower levels of logic and definitions allow the higher levels to flow and create accurate, consistent, logical and provable mathematics. To those who are adept at math, to experts, things may seem to work too well. Everything fits together perfectly, it all makes sense, and it produces useful results. Consider, however, how irrational it all seems to someone who doesn't know the underlying rules or the motivation behind these rules. It seems totally arbitrary, and, in a sense, that argument could be made. The motivations for underlying definitions being necessity for consistency.


[1] http://math.stackexchange.com/a/258958

Re: Mathematics isn't unbelievable.

I still suck balls at it though...

"Remember, misery is comfortable. It's why so many people prefer it. Happiness takes effort."

Re: Mathematics isn't unbelievable.

Stop using weather instead of whether.

It's been long enough.

Also, I think of quadratics as a parabola, and see the solutions as it intersecting the x-axis, and think of the square root of a parabola as a graphical transformation on the parabola. It seems to make intuitive sense to me, unless you just don't see the graph or something and think of it like a linear equation, then it doesn't make sense.

However, not being able to understand everything in math with intuition all the time is a given. There have been physicists wrt quantum physics or some shit who have said so: lol just know the rules, it's too hard for it to intuitively "make sense" in your head! There are more examples, I think especially in linear algebra. But perhaps being able to understand math with intuition is good because it gives a deeper understanding and makes it harder to forget?

Often shit makes sense as long as you just accept complicated axioms that have been built upon for a long time and are too annoying to remember and just build from there. Like the Pythagorean theorem. Guess it is pretty common in trigonometry... other stuff in trig might make sense if you accept that the Pythagorean theorem works, but guess what? I don't fucking understand the Pythagorean Theorem. I don't know why it works. There are proofs that I've tried to look at but they were annoying and shit.

I also forgot how to derive the quadratic formula, yet I passed Multivariable Calculus and Linear Algebra (big ass matrix eigenvector eigenshits) both with a B somehow.
I could probably derive the quadratic formula with enough fucking around...

Then there's the Monty Hall door problem, which highly looked upon mathematicians have been wrong about: https://en.wikipedia.org/wiki/Monty_Hall_problem It is a VERY unintuitive problem which doesn't make sense unless you apply a bunch of annoying stats/game theory/discrete math? rules. Definitely a good way to argue that math is not easy, but being able to do stuff with math or just to "learn" math that has already been discovered (which does take some effort and can be rewarding, even if you're not rediscovering everything?) is not THAT hard if u just accept da formula into ur anus : ^(

Also, imaginary numbers in electrical engineering cancel out or something. And then they have useful properties... somehow? I have no idea.

muh einstein fucking imagination iz moar importent than knowledoeg

fug

Guess one should ask famous mathematicians who are still alive, how they feel about math... or some shit...

sloth wrote:

Comfy does not provide challenge, challenge provides success, success provides happiness. Our world is not comfy, although we tried to make it so. Slaves of our own inventions, yada, yada. Not only on a technological level, also on a social and political level. Nothing more but apes. Apes with psychosomatic disorders.

Re: Mathematics isn't unbelievable.

Idk, I read a thing on quora by some graduate math guy with published research who said after you get to a certain point, trying to visualize math is a huge waste of time. See: tesseracts on up in higher dimensional euclidean math. I don't think just visualizing it or intuition is always the best idea in terms of doing or knowing math, I'm just making the argument that math, in general, is not inherently intuitive. A lot of people argue that it has like, transcendental origins, but it seems to me (someone bad at math) that the rules we have derived are not things we derived through some special knowledge humans are born with, we came up with it by looking at nature, trying to model it, trying to build rules, and eventually making enough rules and tweaking those rules so they all fit together and form a language. It doesn't seem to me like it came easily or naturally. Based on a lot of arguments I've read, a lot of people seem to be under the impression that it did come easily.


Imaginary numbers is a great example for this.

\(
\sqrt{ -1 } = i
\)


Obviously breaks the "rules". What I'm basically proposing is that math isn't inherently something we as "humang beinz" can understand because of properties of the universe and our minds. It's something we can understand because we spent a lot of time building it. I'll have to go and find the documentary / arguments where people were making these claims.

Re: Mathematics isn't unbelievable.

AFAIK math is just the scientific extension of human logic. Then you have the philosophic/theoretic and the combination of both that is conceived as practical logic outcomes. Math defines, Theory explains, Practical applies.

Always quantifying and qualifying stuff before putting them into practice.

I guess the idea that math is natural is from the simple fact that it is a universal "language" among humans as it relies on our perception and experiences. One item with another same item will always be seen as 1+1 or perhaps 1 batch of 2(and the similar reverse applications via subtraction and division).

So I guess that would mean human logic functions on the basic premise of addition and multiplication and their opposites. IMO the more complex mathematical concepts are simply quantifications of existing qualified logic. Defining using a more complex system to make it universally recorded and understandable.

"Creepy crazy fucking idiot Nr. 873894532"-aCol

Wes wrote:


^^ funny
this guy
the most well written and verbose shitposter on the internet

Re: Mathematics isn't unbelievable.

loon_attic wrote:

Often shit makes sense as long as you just accept complicated axioms that have been built upon for a long time and are too annoying to remember and just build from there.


I really cannot bring myself to "just accept" axioms. I don't know what it is about me. I can't get myself to move on from a problem until I feel like I *really* understand it. It's not that I have like, a drive to understand it, it's that I literally won't remember how to perform operations or have any applicable ability to do the math without mostly understanding all of the components. It just doesn't fit together in my head and I can't "get" it until I know how it's all working together. I really hate the idea of just accepting things like that, too. I envy people who can just sort of memorize all the steps without sort of recursively thinking down through the whole array of components. I don't get how that works in your head though. How do you ever remember how to do it? When you run into weird problems, how do you know what steps to take to solve it without the underlying theory being firmly understood? Just memorizing *more* steps? That seems at least as hard as getting a good feeling for everything involved. It's like, even if I can solve the problems, if I don't get why a certain thing works, I can't really move on or I will forget it within a day.

Re: Mathematics isn't unbelievable.

Literally a lot of calculus exists just because algebra breaks when you divide by zero. The concept of a limit is a workaround. Idk how people think this shit is magic when it's patched together like this via trial and error and discovery.

Re: Mathematics isn't unbelievable.

because it's complicated and looks arcane with all those symbols and shit that are also complicated

sloth wrote:

Comfy does not provide challenge, challenge provides success, success provides happiness. Our world is not comfy, although we tried to make it so. Slaves of our own inventions, yada, yada. Not only on a technological level, also on a social and political level. Nothing more but apes. Apes with psychosomatic disorders.

Re: Mathematics isn't unbelievable.

loon_attic wrote:

because it's complicated and looks arcane with all those symbols and shit that are also complicated

I'd probably advance in math if there was a class that just taught me what the fuck all those symbols meant. Seriously, that's the spot I'm stuck at. They'll show examples of its use, proof of how it works, but I don't need that. I need, in a simplest possible way, an explanation of wtf the thing I'm looking at is intended to represent.

THEN I'll be interested in how it works or where to use it. The last class of math I took in college, before dropping it, the prof was going over the lessons and introducing new symbols and shit and I was just like "where the hell did that come in from?!" because somehow I was supposed to know.

Eh...I don't really need math anymore. I'll work on it one day, when I'm rich enough not to bother working daily.

"Creepy crazy fucking idiot Nr. 873894532"-aCol

Wes wrote:


^^ funny
this guy
the most well written and verbose shitposter on the internet