Ummm...what was that Jim?? Hmmmmmmmm?Originally Posted by [b
Ummm...what was that Jim?? Hmmmmmmmm?Originally Posted by [b
"Fox terriers are born with about four times as much original sin in them as other dogs." - Jerome K. Jerome
Now did I mention anything specifically? You can misinterpret that in any way you would like! It could be the weather.... yeah... that's right... the weather....Originally Posted by [b
That's because you're probably being made to do useless proofs. :) Showing that lines intersect at right angles isn't very fun, but there are a number of very powerful proofs that are much more interesting. Such as the proof that pi is an irrational number (IE there is no way to write pi as a quotient of two whole numbers,) or that you can't divide an angle into three perfectly equal parts. There are all sorts of crazy things that we only know about through proofs, such as a three-dimensional shape with infinite surface area and finite volume. There's even a proof that shows that once you can represent proofs within the system you're doing proofs on, the whole thing breaks down and either leaves unanswerable truths or contradictions among your theorems. That's why I think it's unfortunate that proofs recieve so little treatment in the early stages of education; if you hadn't had your formative years of math bogged down entirely in computations, you'd probably be ready to tackle the interesting and worthwhile stuff by now. But first you have to work on the trivial stuff to get a handle on the practice. Your proofs probably seem like more work than they're worth because there's a serious emphasis in contemporary American education on reciting what you're told rather than making deductions from evidence. Proofs are just solid, painfully detailed arguments of why something can or cannot be true, but when you're trained to think without justification, justifying your reasoning to others becomes very difficult. Most people don't make it much past there and then write off math as useless and dull... It's a conspiracy, I tell you!Originally Posted by [b
~Joe
o//~ Livin' like a bug ain't easy / My old clothes don't seem to fit me /
I got little tiny bug feet / I don't really know what bugs eat /
Don't want no one steppin' on me / Now I'm sympathizin' with fleas /
Livin' like a bug ain't easy / Livin' like a bug ain't easy... o//~
I wouldn't mind hearing some of these book recommendations as well.
Give me an example of what you're trying to prove. I'm not sure where your misunderstanding is.Originally Posted by [b
Generally speaking, proofs work by changing your premises (the axioms, or rules of your system, and your assumptions, or the ideas that you're exploring in the system) into theorems (conclusions about your system) through an operation called direct consequence. The direct consequences of a theorem are the other theorems that are a single proof-step away.
So, say that I know that if theorem P is true, theorem Q is also true. This might be written symbolically as P -> Q, or P implies Q. If I then assume that P is true (written P; P is false would be something like -P or not P,) then a direct consequence of P and P -> Q is Q, because if P were true and Q were not, then it would not be true that P implies Q. This direct consequence follows from the formal definition of implication in symbolic logic. Symbolic logic, AKA mechanical logic, has strict, unwavering definitions of truth and falsehood for all the basic logical operations; And, Or, Implies, Not, For All, and others. You can find these definitions online. The nice thing about symbolic logic is that there's no guesswork to it; it's called mechanical logic because a machine could do it, exploring all possible theorems by manipulating one string of logical symbols into another according to a table of rules. These manipulations are called Rules of Inferrence.
Logic isn't all operations, though. Like the Ps and Qs we need variable values to represent the notion of truth and falsehood. There are propositions (which is what P and Q are in the example above) which hold the value of true or false and often stand in place for ideas about the properties of something. For example, P could stand for "Joe grows carnivorous plants" and Q for "Joe grows plants." Then P -> Q could be interpreted as "If Joe grows carnivorous plants, Joe grows plants." You're probably also using predicates, which are like propositions that are more generic. Predicates take abstract objects, referred to as variables, as input and return truth or falsehood. For example, equality is a binary predicate, meaning that it takes two variables and decides on truth or falsehood depending on the values given. In the case of equality, truth is returned only when the given objects are identical. We could have a 'grows' predicate G, and if I were object j and carnivorous plants were object c, then we could write G( j, c ) or Gjc for "Joe grows carnivorous plants." There are also predicates that have one, three, four, or even undefined amounts of variables. In any case, prepositions and predicates by themselves are referred to as atoms; when logical operators are used to join them, the resulting theorems are called molecules. Written out entirely in operators of formal logic along with predicates and propositions, theorems are known as formulas, in the same way that y = x + 1 is a formula of algebra.
So a proof proceeds by starting with an assumption and axioms, all in written in terms of logical formulas, and then, by using rules of inferrence, you create a chain of direct consequences that lead to your conclusion. Oftentimes, however, you start with information that isn't in the form of formal logic. From here, you have two options. You can translate your axioms, assumptions and wanted conclusions into symbolic logic by formally defining all of the predicates and theorems involved. However, this is often an impractical way to proceed; you must rigidly define the truth values for every predicate, formalize all the relevant axioms, and after all that work the mechanical deduction may be very complicated and/or drawn out.
The other option is to conciously practice the virtue which the method of symbolic logic gives us for free, which is to leave no possibility unexplored or vaugely argued. You don't use logical operators in terms of the symbols and rules for manipulating them, but it helps to model your arguments in the same way, because the rules of inferrence all come from readily evident logical truths. If your proof is about some object, you must make sure to deal with every possible type of object it could be. So, if I'm trying to prove some property of square numbers, I must make sure that my deductions are valid for even squares as well as odd squares. This basically amounts to not making unnecessary assumptions. Never be satisfied with, "It can't be this way," or, "It must be that way." Every argument, every property or circumstance you discuss, must have a valid and obvious justification. It may seem stupid to only work with ideas that are obvious consequences of one another, but after three or four steps, you can get to some pretty un-obvious conclusions. At first, doing proofs in this manner is very difficult, but if you can practice going between formal proofs (with symbols and the rules of inferrence) and 'real' proofs (the informal, longhand proofs that most math uses) then the most complicated part of proofs becomes writing them out in English in such a way that people won't misinterpret your grammar.
Hope that helps some.... I know it's a lot to take in all at once.
Best luck,
~Joe
o//~ Livin' like a bug ain't easy / My old clothes don't seem to fit me /
I got little tiny bug feet / I don't really know what bugs eat /
Don't want no one steppin' on me / Now I'm sympathizin' with fleas /
Livin' like a bug ain't easy / Livin' like a bug ain't easy... o//~
It's a bit off-topic, but I have to asnwer:
Well, the idea is that the cold was a minor second problem, while the big issue was dust choking out the light and impeding photosynthesis. That would mean less plants, which would mean less herbivores, which means less carnivores, with the big things and the warm-blooded things (since both large size and high metabolism increase food requirements) being the first to starve.Originally Posted by [b
Also, it's technically inaccurate to say that ectotherms survived unscatched; many species of crocodylians and other reptiles perished, including the pterosaurs and marine reptiles. They just didn't get screwed nearly as badly as the dinosaurs did.
There is, of course, still debate. Other alternatives include Bakker's theory, volcanic eruptions, or disease. It's complciated by the fact we've never directly witnessed a large asteroid strike on a rocky planet, thus must rely on historical evidence to infer the effects. Hopefully if that situation changes, it'll involve the moon or some rocky planet other than Earth.
In part, yes. Big animals have some advantages (efficent metabolism and walking, protection from small predators, mass-based inertial effects for temperature, salinity, etc,), but suffer the dual problems of long generation time and small numbers. In 100 years, there are only about 8 or so generations of elephants, while there are 100 or more of mice, meaning that if natural selection is working on both, the mice will respond sooner (this is why bacertia and viri are so problematic; they evovle on a scale of weeks, while we take centuries to evolve resistance to them). Also, if a disaster kills 50% of the population, there are still several thousand mice, who can breed and repopulate, and the chances of valuable alleles being permanently lost because the animal carrying them died are minimal, as arre inbreeding effects. In contrast, there's probably only a handful of elephants left after the 50% loss, meaning that rare alleles were likely lost (reducing genetic diversity) and that inbreeding will become a factor in the future (which is very, very bad for a species survival).Originally Posted by [b
Mokele
\"With malleus aforethought, mammals got an earful of their ancestor's jaw.\"
--J. Burns, on the evolution of auditory ossicles.
I've read that cockroaches will outlive us all. I'd be willing to bet that bacteria and viruses will outlive the cockroaches.
Here's wonderfuly poor logic: Cats have green eyes. My wife has green eyes. Therefore, my wife is a cat. Or... does that mean that I am married to a bazillion green-eyed cats? * *Yeah, *P->Q, does not mean that Q->P!
I love this thread!
Droserae will inherit the earth.