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True, False, and the Mathematician's Logic

Mathematical Truth

What is truth and how can we find it? All by itself that is a question for philosophy.

In mathematics we are concerned with mathematical truth (and falsity) based on proof that use the rules of logic. (See also True, False, and the Mathematician's Logic.)

No statement or assertion can be taken as true unless it has been rigorously demonstrated using logic. Even the most plausible, believable, and well-accepted notions remain conjectures until they are proved. A statement can seem true, and for many years people try and cannot find and reason for it to be false. Such a statement does not become a theorem in mathematics until it has been proved.

Sometimes this results in a complex relationship between mathematical knowledge and knowledge gained from intuition, guessing, and experience in the world. For instance, pursuing a notion with mathematical rigor can lead to contradictions or paradox.

True, False, and the Mathematician's Logic

It is not unusual to make a mathematical statement which, after numerous trials or examples, seems entirely not only plausible, but impossible to contradict. Even so, these statements often cannot be considered true from a mathematical standpoint because they have not been rigorously proved. A rigorous proof does not have to be a series of inscrutable statements filled with arcane terminology and unfamiliar symbols. It need only be a clear description of why something has to be true. A proof must take into account all possibilities.

The experience of thinking that something must be true, but not being able to actually prove it "beyond the shadow of a doubt", may lead students to believe that mathematical thinking places unnecessary restrictions on drawing conclusions about the world. The contrary is actually true.

Because of such stringent criteria for establishing and accepting something as true, there is no danger that a revolutionary new idea will appear that shakes the foundations of mathematics, causing mathematicians to have to reject things that they once though were true and start over. New discoveries in mathematics--new truths--are added to what is already known. New ideas may open up whole new fields of inquiry, but they will never cause anything to be removed from the body of knowledge that mathematicians recognize as true.

The play in this section illustrates another aspect of the power of mathematical rigor. We fill our everyday conversation with plenty of information about what we are describing so that our listeners can understand it the way we do. But in mathematics, when we begin asking questions about a collection of objects that we have invented, there are no rules of ettiquite that say these objects should be polite and generous about giving information about themselves. The situation is more like the one at Unusual School.

Mathematicians have to make the most of every little piece of information that they have when they are trying to draw conclusions from what they already know. A common strategy is to try to find out exactly what the possibilities for solutions and conclusions are, and to try to eliminate the ones that are impossible. Sometimes the possibilities are infinite, or not all of them can be eliminated, and mathematicians must use other strategies. But the strategy of listing possibilities and eliminating them one by one will work well for figuring out who's who at Unusual School.

Date: 2015-12-17; view: 522

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