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1.1.7.1 Á PrioriVersion 1.3 March 2013 (Previous Version) We know some things are true á priori – before we have to look at the world itself. · For instance, a triangle has three sides. We know that because that’s what ‘triangle’ means. · Two plus two equals four. 2 + 2 = 4. That comes out of the laws of arithmetic. Á priori knowledge arises out of definitions and how we use language. Á priori propositions can't be proved right or wrong by looking at the world. We call this being NOT contingent, or we say they are necessary. We don't know á priori that the sky is normally blue (except when it is night, cloudy, sandy or smoky). We discover the sky’s colour by looking at it. If the world had been different the sky might have been green. It is not necessary that the sky be blue. The sky’s colour is contingent on the world and the natural laws of our universe. A tautology is always true. For example: it is raining and it is raining. A contradiction is always false: it is raining and it is NOT raining (at the same place and time). These are simple examples, but these rules can become useful in more complex situations. more Mathematics is true á priori. It involves symbols and manipulation of symbols that are unrelated to the real world. It is amazing that we can use mathematics to describe the world, as we do in physics, chemistry, engineering, biology and medicine. In a way the link between mathematics and reality is a happy coincidence that we had no right to expect, but fortunately it works. Not all mathematics has immediate applicability to the real world. The German mathematician George Riemann wondered what would happen if the parallel line axiom in Euclidean geometry wasn’t true – if lines that are as parallel as we can make them eventually meet. This is what happens when we try to draw straight lines on a ball, like the lines of longitude drawn on a globe of the world: they are parallel at the equator, but meet at the poles! This idea didn’t have much practical use until Einstein realized that we can explain the behaviour of light and time, mass and energy, using Riemann’s mathematics, which is why in the theory of relativity we say space-time is curved (but it's more like a saddle than a ball). Some people have hoped that one day we would find a way for all knowledge to be like á priori knowledge, or somehow derived from it, so that we would know for certain what is always true. For instance, the philosopher Kant claimed his categorical imperative (act as if the maxim of your actions is universal) was compelling, and derived from pure reason. But this quest has been in vain: unfortunately life is too complex. The things that we know á priori are relatively trivial and don't directly help with life’s major questions or to find meaning or purpose. Although mathematics and logic are essential tools, they are only useful when we link them to the real world, practical problems, such as how much to pay for goods, how to best educate our children, whether to invest in the stock market, or how to synchronise signals from orbiting satellites. We phrase our current conclusion on this issue as follows: 1.1.7.1 What we know for certain – a priori, analytic, necessary, tautological or contradictory propositions – are very useful but tell us nothing about the world itself, only about how definitions of ideas link to each other. more
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