Intuition plays an important role in all the areas of knowledge. It provides the foundation on which our understanding of each area of knowledge is built. These core intuitions are the fundamental basis for everything we know. Both reason and perception are dependent on intuition. Because many of the areas of knowledge rely on these two ways of knowing, it can be said that they also rely on intuition. Three of the areas that rely on intuition are mathematics, natural science, and ethics. In mathematics intuition is the basis of our theories. Intuition plays the same role in the natural sciences, such as physics. Our ethics are directly formed from our intuitions about what we observe in society.
Mathematics is an area of knowledge that relies mainly on reason to show that things are true. This in turn means that mathematics must be based on intuition. There are several models of reasoning on which we base our mathematical knowledge. One of these models was developed by Euclid and is known as the formal system. His system has three key elements. These elements are axioms, deductive reasoning, and theorems. The axioms are the systems “starting points or basic assumptions” (Lagemaat). These axioms are considered to be the self-evident truths that provide the foundations for mathematical knowledge, and show that it is based on intuition. The second element of the formal system is deductive reasoning. Deductive reasoning is an important part of this system. It is built from two or more premises that lead us to a conclusion. It is another fundamental law of reason that can only be justified with intuitive knowledge. This leads to the third element, which are the theorems. These simple theorems are derived from the other two elements. If we know our theorems are true then we must also know that our axioms and deductive reasoning are correct. The first two elements can only be consider true knowledge if we say that our intuitions are correct. If we tried to prove them...
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