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Ok you Mathamaticians!

Ok, you guys Please explain the theory behind Indirect proofs!

you try to asume somethings wrong to prove that youre assumtions write about it being wrong and you prove that youre assumtion was wrong?

Thats like saying that the world is square and trying to prove it!!
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JIM!!!
 
Indirect proof is basically testing the logical consequence of something, rather than the fact itself, possibly because the fact is inaccessible.

An excellent example is whether dinosaurs were warm-blooded or not. Metabolism itself does not fossilize, so we cannot examine this directly. Instead, we say "If dinosaurs were cold-blooded, they could not have inhabited polar regions. We have found dinosaur fossils in polar regions. Therefore they were not cold blooded."

It should not be confused with affirming the consequent, which is a logical fallacy. That would be "If dinosaurs were cold-blooded, they could not have inhabited polar regions. We have not found dinosaur fossils in polar regions. Therefore they were cold blooded." The first example is true because it would be impossible to exist in polar climates without warm blood, while this example is flawed because there are many other possible reasons for the absence from polar regions (historical constraints, inability to populate the area due to lack of migration routes, poor fossilization conditions, etc.).

Another fallacy is denying the antecedent, which would take the form "If dinosaurs were cold-blooded, they could not have inhabited polar regions. Dinosaurs were warm blooded, therefore they inhabited polor regions." Ignoring that we have no way of establishing dinosaur metabolism, even if we did know they were warm blooded, there are numerous reasons they might *not* inhabit polar regions.

Hopefully this helped.

Mokele
 
Formal logic to the rescue! Let's positive thinking! In mathematical logic, indirect proof refers to the taking the negation of your hypothesis (by assuming the theorem you're testing is false) then working towards a contradiction from the direct consequences of your axioms and assumption. This method is commonly known as the Latin reductio ad absurdum or "reduce to nonsense," because any contradiction (showing that some theorem is both provably true and false within your system of inferrence) implies an inherent contradiction in the assumptions at work. So long as you're working with logically sound axioms and only taking one assumption at a time, you know that your hypothesis is true if taking its negation leads to contradiction, because it is the only theorem which could be at odds with the axioms.
The idea that if a statement is not false, it must be true is known as the law of the excluded middle, meaning that there is no third 'middle' value between truth and falsehood. The intuitionist philosophy denies the excluded middle on the grounds that there often is no way of constructing the objects described by theorems derived by indirect proof, and thus we cannot with certainty say what such objects are, even if our proofs assert that they exist.
Wikipedia on reductio ad absurdum.
Here's a sketch of the method:
1) I want to prove that it's true that some undefined force hereto be referred to as gravity causes some objects to fall.
2) The negation of an assertion that some objects have a certain property is that all such objects do not have that property, so assume that gravity causes no objects to fall.
3) Accepting that my senses are all accurate and I'm not somehow decieved into making an incorrect observation, and assuming the above is true, I hold a pen in the air and then drop it. I observe the pen to fall to the ground.
4) If the assumption in #2 is correct and all things do not fall, then pens, being things, also do not fall.
5) I have proven that pens fall, and also concluded through consequences of my assumption that pens do not fall.
6) I have a theorem (pens fall) which is true, while it's negation is also true. But by the definition of negation, the negation of a true statement is false, and thus in this logical system, true statements are equal to false statements. Reductio ad absurdum.
If you have any further questions, please post them or PM me; I spent a year of studies slaving away at formal and mechanical logic and I'm always glad to make use of it.
Best luck,
~Joe
 
And I thought my head hurt b4 hand!

Collage professers shouldnt be teaching online summer school classes for High Schoolers!!!!

Thanks, you both cleared it up a little bit!
 
Hehe, I'm happy to be of assistance. After doing logic and lots of other proof-based math, I've come to believe that logic and proofs are vastly underemphasized in modern education. Formal logic is very simple once you learn to read the lingo; I'm fairly sure that children could begin working with logic in early grade school. Basic logic is far easier than memorizing multiplication tables.
I guess what I'm trying to say is that you're fortunate to have the chance to ask these questions now; at my high school I had to take precalculus and calculus as independant courses with not-so-attentive professors 'teaching' me (IE telling me to read out of a textbook) in spare minutes between their regular classes. I never really got to work with proofs in school before college, beyond some very weak work in analytic geometry. My textbooks mentioned proofs, but lessons were all about crunching numbers, which really isn't what math is about, as it turns out. In any case, I certainly never had proofs properly explained to me. Even in my freshman year of college, taking an advanced proof-based math course, my understanding of how to go about writing a fully justified proof was weak, to say the least. It wasn't until I took a course on logic and computability in my sophomore year that I was given a chance to examine and practice proofwriting. However, it has helped me in all aspects of my education; logic is a classic component of philosophy, applicable to language, science and art alike. If you can stomach it, I would recommend reading up a bit on formal logic and proof. I wish I had a book I could point you to, but my logic professor wrote all of our textbooks for that course and they aren't in wide print. They're too dense, anyways. You can probably find some good resources on that Wikipedia page, or from the excellent reference Mathworld.
~Joe
 
[b said:
Quote[/b] (nepenthes_ak @ July 20 2006,12:09)]And I thought my head hurt b4 hand!
A little late on the scene and both Joe & Henry answered the question. Trust me, though, math logic will give you the same right answer every time, at least at this level. Hey, it's a lot easier to figure out than other things we know and love - and I don't mean the plants!
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[b said:
Quote[/b] (Mokele @ July 19 2006,10:52)]An excellent example is whether dinosaurs were warm-blooded or not.  Metabolism itself does not fossilize, so we cannot examine this directly.  Instead, we say "If dinosaurs were cold-blooded, they could not have inhabited polar regions.  We have found dinosaur fossils in polar regions.  Therefore they were not cold blooded."
And yet scientists insist that it was a meteor impact that caused massive global cooling which killed them off yet left crocodiles, turtles, snakes, lizards, frogs, toads and salamanders (i.e. all the cold-blooded creatures) more or less untouched. Now explain that one to me?!?

As far as I am concerned Bakker had the right idea. The meteor was not the reason, the dinos were already on the way out.
 
[b said:
Quote[/b] (Pyro @ July 20 2006,9:37)]
[b said:
Quote[/b] (Mokele @ July 19 2006,10:52)]An excellent example is whether dinosaurs were warm-blooded or not.  Metabolism itself does not fossilize, so we cannot examine this directly.  Instead, we say "If dinosaurs were cold-blooded, they could not have inhabited polar regions.  We have found dinosaur fossils in polar regions.  Therefore they were not cold blooded."
And yet scientists insist that it was a meteor impact that caused massive global cooling which killed them off yet left crocodiles, turtles, snakes, lizards, frogs, toads and salamanders (i.e. all the cold-blooded creatures) more or less untouched. Now explain that one to me?!?

As far as I am concerned Bakker had the right idea. The meteor was not the reason, the dinos were already on the way out.
Would that have something to do with smaller critters having more time to adapt / evolve than the larger ones?
 
[b said:
Quote[/b] (jimscott @ July 20 2006,9:53)]Would that have something to do with smaller critters having more time to adapt / evolve than the larger ones?
Which part? The cold-bloods surviving in place of the dinos or Bakker's theory?
 
  • #10
I just think the proof's are more work than what they are usefull for, Thats probably cause I still dont quite understand them but im gradualy getting them better. I just hope I dont have to use them as much in my math class this comming year, I would more than likely fail again.
 
  • #11
[b said:
Quote[/b] ]Hey, it's a lot easier to figure out than other things we know and love - and I don't mean the plants!

Ummm...what was that Jim?? Hmmmmmmmm?
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  • #12
[b said:
Quote[/b] (PlantAKiss @ July 20 2006,11:23)]
[b said:
Quote[/b] ]Hey, it's a lot easier to figure out than other things we know and love - and I don't mean the plants!

Ummm...what was that Jim??  Hmmmmmmmm?  
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Now did I mention anything specifically?
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You can misinterpret that in any way you would like!
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It could be the weather.... yeah... that's right... the weather....
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  • #13
uhmmm, how do I use these in writing a proof though, I thoght I under stood but I guess I just don under stand how IM suposed to write a proof with it I guess.

I guess I though understanding the theory would help me write the proofs, but... yea.
 
  • #14
[b said:
Quote[/b] (nepenthes_ak @ July 20 2006,7:42)]I just think the proof's are more work than what they are usefull for, Thats probably cause I still dont quite understand them but im gradualy getting them better. I just hope I dont have to use them as much in my math class this comming year, I would more than likely fail again.
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!
~Joe
 
  • #15
I wouldn't mind hearing some of these book recommendations as well.
 
  • #16
[b said:
Quote[/b] (nepenthes_ak @ July 20 2006,11:06)]uhmmm, how do I use these in writing a proof though, I thoght I under stood but I guess I just don under stand how IM suposed to write a proof with it I guess.
Give me an example of what you're trying to prove. I'm not sure where your misunderstanding is.
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
 
  • #17
It's a bit off-topic, but I have to asnwer:

[b said:
Quote[/b] ]And yet scientists insist that it was a meteor impact that caused massive global cooling which killed them off yet left crocodiles, turtles, snakes, lizards, frogs, toads and salamanders (i.e. all the cold-blooded creatures) more or less untouched. Now explain that one to me?!?

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.

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.

[b said:
Quote[/b] ]Would that have something to do with smaller critters having more time to adapt / evolve than the larger ones?

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).

Mokele
 
  • #18
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?  
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 Yeah,  P->Q, does not mean that Q->P!
 
  • #19
jim stay out of those interspecies relation ships!!
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Thanks, I understand it better now, thanks, Im just waiting for it to be graded to see how I did... illl let you all know
 
  • #20
I love this thread!
 
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