Seemingly crazy scientific ideas

[Surlethe]

Also, while I probably have grossly misunderstood what it actually means, doesn't Godel's Incompleteness Theorem completely deep-six the notion of the universe being as Neo saw it?

Why would it? Godel's incompleteness theorem simply says that, given any axiomatic system complicated enough to encode the natural numbers, there are propositions which cannot be proven without adding additional axioms. So when you add the additional axioms, you can now prove the proposition, but now there are more propositions whose truth you can't settle.

As an example, the ZF axioms of set theory straightforwardly lead to the natural numbers. Chief among the propositions that fall 'outside the cloth' is the Axiom of Choice, which is equivalent to such results as the Tychonoff Theorem(1), Zorn's Lemma(2), the Well-Ordering Theorem(3), and the theorem which states that if between two sets there exists a surjection, there is an injection the other direction. Formulated, the Axiom of Choice seems innocent: Given any collection C of nonempty sets, there exists a set A such that A is composed of one element from each element of C. So, if you have a bunch of piles, you can pick one thing from each pile. But the things you can prove with it include the Banach-Tarski Paradox (look it up).

1. Any product of compact spaces is compact
2. Any set with a partial order in which every totally-ordered subset is bounded above has a maximal element
3. Given any set, there exists an ordering such that under that order every subset has a least element

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[brianeyci]

Awesome! I was hoping that might be the case. I have such trouble wrapping my head around a few physical concepts, relativity chief among them. I decided to just stop trying for the time being, and try again once I have a better grasp of calculus.

Honestly you are not alone. I thought I was so great, moving into university with my pittance of understanding that a good Grade Eight student could defeat. Then I took a few math classes and declared myself awesome, just because I had a smallish basic understanding of how to be a number monkey. Only in the past year or so have I truly understood how ignorant I am. I bragged about being a math major when the best math I knew was first-year Calculus -- it is embarassing thinking back. I am taking a year or two break from completing a degree, after several embarassing setbacks. I have only one or two semesters of math classes left but I've hit a roadblock I can't defeat... for now.

It didn't help that I switched midstream from an English major to a mathematics major. The scary part is some people never, ever come to this revelation that Mike talks about, how to reduce the physical world into numbers. It takes an immense amount of education and failures just for someone to realize how ignorant he is.

If you want my suggestion, I suggest not giving up entirely but keeping in contact with some math majors in college. The kind who can't stop talking about math.

[Nephtys]

To be fair, to understand how you can model more or less everything, you need to know the relations between electrical, thermal, fluid, translational and rotational systems, as well as how to couple and model them.

When I learned state-spaces, that just blew my mind pretty crazy that you could model pretty complex-seeming systems. Most people who don't go down a science/engineering route of education are boggled enough at newtonian physics.

[Spin Echo]

Probably the most mind-blowing revelation in the development of any young aspiring scientist or engineer's mind is the revelation that the world really can be turned into a bunch of numbers and equations in your mind, and that this actually works better than the subjective comprehension you've used all your life up to that point.

Awesome! I was hoping that might be the case. I have such trouble wrapping my head around a few physical concepts, relativity chief among them. I decided to just stop trying for the time being, and try again once I have a better grasp of calculus.

The problem is, you often come out the other side and realise how many seemingly simple systems we can't describe with equations and model. Yay for complex materials!

[Kuroneko]

Okay, you've completely lost me. Please explain what you mean by the universe being "pointless" in the mathematical sense?

See my first post on the second page, and PM me if you still want more detail.

[Adrian Laguna]

The problem is, you often come out the other side and realise how many seemingly simple systems we can't describe with equations and model. Yay for complex materials!

I've heard about how trying to model things like water flowing down a slide can give engineers aneurysms.

[Paolo]

The problem is, you often come out the other side and realise how many seemingly simple systems we can't describe with equations and model. Yay for complex materials!

I've heard about how trying to model things like water flowing down a slide can give engineers aneurysms.

Well, there are plenty of systems (if in fact not almost all of them) people can't model exactly, but most can be reduced to linear approximations of a sort. Water flowing down a slide, for example, would start with an exact solution of an exactly solvable term describing a zero viscosity bulk flow and then approximate the nonlinear contributions with an infinite series of infinitely increasing order derivatives. These series will generally converge, although there's a way to handle divergent ones I've never bothered to look up. This is called perturbation; basically, the behavior of a system is described in terms of a simpler one summed with some deviation computed from its derivatives.

[Darth Wong]

In general, engineers have no problem modeling small but predictable complex systems numerically, using experimentally derived correlations. That sort of thing is more likely to cause problems for scientists and anyone who insists upon first-principles derivation.