Math is often referred to as “the science of rigorous proof,” which means that in order to find out if something is certain, you have to check for any problems that can occur in proving a theory. However, proofs alone are not enough to make sure that a concept is true. In order to consider if a mathematical statement is true or not, we can use the formal system, developed by Euclid. This model of reasoning includes three key elements: axioms, deductive reasoning, and theorems. To reason formally, you must accomplish these steps in this order.
The simplest form of proof is using the axiom system, which is a basic assumption. This is a form that is a firm foundation for knowledge in the mathematical sense in which statements are assumed to be true. Also, with this way of knowing you would get caught in an “infinite regress” if you tried to prove everything. This means proving A in terms of B, so on and so on. Therefore, there is no such thing as absolute proof. However, there can be absolute proof if you assume an axiom is true, which is in fact not absolute.
The next step in the formal system is deductive reasoning, which is a form of reasoning that moves from the general statement to the particular statement, known as a syllogism. A syllogism consists of two premises, a conclusion, three different terms that all occur twice, and words that tell us the quantity, known as quantifiers. An example of a syllogism is: “All dogs are mammals. Fido is a dog. Therefore Fido is a mammal.” This form of reasoning can be useful and lead us to a reasonably certain statement, but on the other hand they can go down the wrong road because if either one of the premises are false, the conclusion will be false as well. Another way it can go wrong is that one of the two premises could be wrong and get a valid conclusion. Another element called “belief bias” comes into play and will mess up the validity of a statement. This term means that we believe in the validity of a statement...
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