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Thread: Ok you Mathamaticians!

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    nepenthes_ak's Avatar
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    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!!

    JIM!!!

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    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
    \"With malleus aforethought, mammals got an earful of their ancestor's jaw.\"
    --J. Burns, on the evolution of auditory ossicles.

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    Let's positive thinking! seedjar's Avatar
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    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
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    nepenthes_ak's Avatar
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    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!

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    Let's positive thinking! seedjar's Avatar
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    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
    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//~

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    Tropical Fish Enthusiast jimscott's Avatar
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    Quote Originally Posted by [b
    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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    N=R* fs fp ne fl fi fc L Pyro's Avatar
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    Quote Originally Posted by [b
    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.
    'My love was science- specifically biology and, more specifically, when placed in a common jar, which of two organisms would devour the other.'

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    Tropical Fish Enthusiast jimscott's Avatar
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    Quote Originally Posted by [b
    Quote[/b] (Pyro @ July 20 2006,9:37)]
    Quote Originally Posted by [b
    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?

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