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After the statement is proved, can it be refuted?

  1. Jun 10, 2010 #1
    Economist milton freidman: We can never prove sth; we can only confirm its existence.
    But in mathematics, it seems that after the statement is proved, it is 100% true.

    Statement1: If i wake up at 10a.m, i will get an A in that day's exam.
    Is this statement true? I dunno. Can i prove this? This statement will be refuted on the day i get up at 10a.m but get a B in that day's exam.

    Statement2: If n is an even integer, 5n is also an even integer. Sure we can prove this. Is it 100% true? It seems yes as it is derived from the axiom of real no. system. Will someone refute this statement by finding a counterexample?

    Last edited: Jun 10, 2010
  2. jcsd
  3. Jun 11, 2010 #2


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    Of course, any "proof" is only valid under certain assumptions.
    Usually, in mathematics, these assumptions are implicit (for example, we are working in a system of integers which satisfy Peano's axioms).

    In more "practical" areas of science, including physics, the assumptions are usually of the type "suppose that <something> works <in this particular way>", and then we can prove that <something else> works in <some other particular way>.

    The main difference, IMO, is that in mathematics the assumptions (axioms) are not tested against reality, so anything we prove under their validity is completely and undeniably true (if the proof is correct, you cannot find a counter-example). You might be able to prove a different result under different axioms, which is equally valid because I cannot argue that your axioms are "wrong" and mine are "right".

    However, the "axioms" of physics or everyday life actually have some real-life meaning. We can show that if energy is conserved a ball will move this way, or if everything is pre-destined then I will get an A whenever I get up at 10 am. But the moment we find that the ball is not moving like predicted, or you get up at 10 am and get a B on the test, clearly your model of reality should be adjusted. Provided that the proof you gave is (logically) valid, that means that apparently our assumptions were incorrect.
  4. Jun 11, 2010 #3
    Yup, maths is about learning what you can conclude given a certain set of assumptions. Most sciences is about finding new conclusions and correcting erroneous assumptions.
  5. Jun 13, 2010 #4
    If you do refute a statement after it has been proven, every mathematician's reality will come breaking down.
  6. Jun 13, 2010 #5


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    At least that of the mathematician who has proven it.
    Because it would mean that his proof was incorrect.
  7. Jun 13, 2010 #6
    No it would mean there's a contradiction exists in mathematics and the whole of it has to be revised.
  8. Jun 14, 2010 #7
    An economist? There's your problem....

    In formal logic, any statement can be be true or false. If you can come up with some proof of the statement, then the statement is true. If you can come up with some proof of the statement's negation, then the statement is false.

    Early work in logic posed the question whether or not logic was consistent. That means that no statement is both true AND false. Depending on what assumptions you make, your logic may or may not be consistent.

    The problem with your first example is that it's not true on its own. It might be true in some narrow context (for example, the teacher promised you an A if you woke up at 10am, and he always keeps his word). But otherwise, the premise does not entail the conclusion.

    In the second example, it does. (Of course, assuming the usual context... that integers are even when they are divisible by two and that you DO IN FACT mean that 5n means 5 times n and not some other weird operation).

    Once a statement is formally proved, it is guaranteed to be true -- you have a proof for it. The only problem you will run into is that there might ALSO be a proof that the opposite of the statement is also true. In other words, you may be working with an inconsistent logic.

    In the case that your logic IS inconsistent, weird crap happens. Every statement becomes both true and false (because anything follows from an absurdity). In other words, your counter examples are integers that are both even *and* odd. Because of this craziness, we do our best to make sure our axioms are all consistent. (Which ironically can't be proven, a la Godel).
  9. Jun 15, 2010 #8


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    You could liken it to a game of chess or a football match:

    If a particular situation is declared to be legally possible, can it then again be declared in violation of the rules?

    The only ways this could happen is:

    a) The basic rules are inconsistent, allowing an identical situation to be both legal and illegal. (both contestants right in their proof)

    b) The one declaring something to be possible in the first way made a logical error somewhere

    c) The one declaring the same thing to be illegal in the second instance made a logical error somewhere

    d) The case cannot be determined to be either legal or illegal (both contestants wrong in their proof)

    d) is perhaps the strangest case of all, yet Gödel proved that statements could be made within the framework of standard mathematics that could be neither proven or disproven within the system. This is called the feature of "incompleteness"
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