Proof is a general term for providing evidence or demonstrating the truth of a claim, while mathematical proof specifically refers to the rigorous logical reasoning and analysis used to establish the truth of mathematical statements.
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Proof and mathematical proof are both methods of demonstrating the truth of a claim, but there are distinct differences between the two. A proof is a general term that can refer to any type of evidence or argument used to establish the truth of a statement. For example, a lawyer may present a proof in court to support their client’s case, or a scientist may present proof of a new discovery based on experimental data.
On the other hand, mathematical proof specifically refers to the rigorous logical reasoning and analysis used to establish the truth of mathematical statements. A mathematical proof consists of a series of logical steps that must follow from the axioms and previous theorems. If there are any errors in the reasoning, the proof is considered invalid.
As famous mathematician Paul Erdős said, “A mathematician is a device for turning coffee into theorems.” Mathematical proof involves a great deal of creativity, critical thinking, and perseverance. Even the most brilliant mathematicians may spend years working on a single proof.
Here are some interesting facts about mathematical proof:
- Some famous open problems in mathematics, such as the Riemann Hypothesis and the P vs. NP problem, have yet to be solved despite decades of effort from mathematicians all over the world.
- The concept of proof has been around for thousands of years. Ancient Greek mathematicians such as Euclid and Pythagoras developed the foundations of modern proof techniques.
- The four-color theorem, which states that any map can be colored using only four colors so that no two adjacent regions have the same color, was first conjectured in the 1850s. It took over a century for mathematicians to finally come up with a proof that was widely accepted.
- In 2013, mathematician Yitang Zhang made a breakthrough in the study of prime numbers by proving that there are infinitely many pairs of primes that are less than 70 million units apart. His proof was the first major result on the subject in years.
- There are many different types of mathematical proofs, including direct proofs, indirect proofs, proofs by contradiction, and proofs by induction. Each type of proof is best suited to different types of problems.
Table: Comparison of Proof and Mathematical Proof
|Can refer to any type of evidence or argument||Specifically refers to the rigorous logical reasoning and analysis used to prove mathematical statements|
|May not always follow the same logical process||Consists of a series of logical steps that must follow from the axioms and previous theorems|
|Can be used in a variety of fields||Used exclusively in mathematics|
|May be less rigorous||Requires absolute precision and logical accuracy|
|Can be subjective at times||Is objective and universally agreed upon if valid|
Watch related video
In this video, four basic proof techniques used in mathematics are introduced: direct proof, proof by contradiction, proof by induction, and proof by contrapositive. The speaker emphasizes the importance of understanding definitions in proofs and provides a step-by-step explanation of how to use each proof technique. The example used to demonstrate these techniques is proving the statement that the sum of any two consecutive numbers is odd. Each technique is explained using this example, and their various strengths and weaknesses are discussed. While these techniques are fundamental, there is still some subtle reasoning involved in each problem.
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Proof – Higher A mathematical proof is a sequence of statements that follow on logically from each other that shows that something is always true. Using letters to stand for numbers means that we can make statements about all numbers in general, rather than specific numbers in particular.
Typically in physics when someone “derives an equation” I’ve found this to mean they took other physics equations that they think approximate reality to some extent and then mathematically manipulated them in some way to come up with the equation they “derived”. Where by a physics equation I mean a mathematical relationship between physical quantities that scientists have defined (e.g. distance or mass) which can be measured by multiples of units (e.g. meters or kilograms)
Now in so far as you accept those original equations as axioms, embedded in whatever formal system allowed them to make those mathematical manipulations or inference rules, then yes its a proof in that system. But this proof says nothing about whether or not what you derived accurately represents reality which I assume was the goal – as that would depend on the accuracy of the original equations.
For example, let x be your mass, let y be the mass of earth, and let z be the mass of the sun. Now with the (clearly ina…
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