What is Mathematical proof: Definition and 61 Discussions

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

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    Mathematica There is no twin paradox - mathematical proof

    For the twin paradox to be considered a true paradox the framing of the scenario must be stringent, that is to say we cannot permit assumptions to be ignored. Therefore I must start with a short description of the twin paradox followed by identification of the inherent assumptions. From...
  2. A

    Integral as Antiderivative: A Mathematical Proof

    Is there a mathematical proof that can prove that the integral is the antiderivative?
  3. D

    Mathematica Mathematical Proof For Lorenz Coordinate Transformation

    Hi, I have a pretty in depth understanding of special relativity. Recently I have been searching for mathematical proof of the Lorenz transformation. I found some information about it, but to tell the truth I didn't understand much of it. Maybe one of you guys can shed some light on the proof...
  4. B

    Mathematica How do you write a mathematical proof?

    I've figured out a faster way of doing simple mixture problems; how do I mathematically prove it consistantly works? Edit: Is my question badly worded somehow? 30 people have looked at this; 0 have replied. Kind of awkward, but if you don't know what I'm trying to ask, or I need to give...
  5. D

    Mathematica A question concerning density, mathematical proof.

    Current science describes a solid, liquid, and gas, as having different densities. Solids have the highest density, then comes liquids, and finally gases. They say in between the particles lies "empty space." Can someone please describe to me the properties of "empty space" because as fas as I...
  6. D

    Mathematica A question concerning density, mathematical proof.

    Current science describes a solid, liquid, and gas, as having different densities. Solids have the highest density, then comes liquids, and finally gases. They say in between the particles lies "empty space." Can someone please describe to me the properties of "empty space" because as fas as I...
  7. C

    Mathematica Proving Statement: If m^2 is 4k+3, Then m is 4k+3

    I'm given the statement: if m^2 is of the form 4k+3, then m is of the form 4k+3. I don't even know how to begin proving this. I'm guessing by contraposition.
  8. C

    Mathematica Prove If m Not Form 4k+3, m^2 Is Not Form 4k+3

    I'm given the statement: if m^2 is of the form 4k+3, then m is of the form 4k+3. I don't even know how to begin proving this. I'm guessing by contraposition.
  9. C

    Mathematica Mastering Mathematical Proofs: Solving the Universe's Integers with a-3b and a+b

    We are given the following statement: The universe is all integers. If a-3b is even, then a+b is even. I began off with saying that 3 cases exist: a is odd and b is even, a is even and b is odd, both a and b are odd, and both a and b are even. After this point I got really confused and lost...
  10. B

    Proving $\Psi$ and $\Psi'$ Normalizable: A Mathematical Proof

    I am trying to prove that \frac{d\langle p \rangle}{dt} = \langle -\frac{\partial V}{\partial x} \rangle I am done if I can just prove that \left[ \Psi^*\frac{\partial^2 \Psi}{\partial x^2} \right]_{-\infty}^{\infty} = 0 \left[ \frac{\partial \Psi}{\partial x} \frac{\partial...
  11. BasketDaN

    Mathematica Mathematical Proof That a 45 Degree Angle Launch is Best for Displacement

    Does anybody know how to mathematically prove that an object launched at a 45 degree angle will have the largest possible displacement?
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