I haven't read "Gödel, Escher, Bach" but based on its description it sounds like it beat me to this idea. That said I'm going to explain this in a single post (much shorter than a book) and making it as easy to understand as I can. That said it's probably still going to be pretty thick.
I'll start by paraphrasing Gödel's first incompleteness theorem: "All systems must have at least 1 axiom." An axiom is something that is accepted to be true without being proven. Axioms are core basic beliefs that everyone has many of such as "I exist," "other people are not my imagination and they exist in the same way I do," and "I have senses which let me measure the world I live in." Axioms can't be proven since you have to use something as a basis for your argument. Using an axiom to prove itself is circular reasoning. Although it is possible for an axiom to agree or disagree with another axiom there must be at least 1 that is assumed to be true on which all other claims are based.
"How do I know what axioms to trust? You can't prove to me that I exist!" The fact that you can read this article shows that you already have chosen some axioms. While there are some basic axioms that everyone needs to survive how are the rest chosen? "I'm going to choose axioms that meet this criteria." That would make that criteria the axiom on which the others are based. Since axioms are so fundamental there's no getting around the fact that they are indeed "accepted without being proven" or even recommended. This is to say that all axioms are chosen arbitrarily, at random, and without reason. "I like axiom A more than axiom B" that's an example of being chosen arbitrarily. "Axiom A is consistent with my other axioms" then those other axioms and the desire for consistency are you base ones. It's fine to change axioms or to have an axiom demoted to being based on others.
Have I lost you yet? I'll explain it from a different angle. Let's suppose you have some bit of knowledge that isn't an axiom such as "you need to stir the noodles so they don't stick to the pan." Consider what that knowledge is based on. And what is that based on. And what is that based on etc. Eventually you can follow it back to at least 1 bit of knowledge which is not based on any other knowledge. That most fundamental knowledge is an axiom. This is the "first mover" argument except with knowledge it is impossible to claim that your knowledge is an infinite series backwards. Unless you believe that you have infinite knowledge (in which case why are you reading this?) then you must believe that going backwards will eventually reach an end (an axiom or a group of them).
"Of course I trust that my senses can detect the world. There's no other way to live." But how much do you trust them on a scale of 0 to 100%? You can't trust them 0% because if you don't trust pain at all then there's no reason for you to eat or breath which leads to a quick death. However you also shouldn't trust them 100% of the time because senses can be tricked by slight of hand or hallucinogens. 100% trust is much less dangerous than 0% but it can still lead to trouble (people tricking you or a drug addiction). So you need to trust your senses only sometimes. How do you know when to trust them? It's based on something else you trust. "Of course a person can't just disappear, it's some kind of stage magic" is based on axioms. But obviously you can't use your senses to decide which knowledge should be chosen as a basis for when to trust your senses.
Consider also how much you trust other people. It can't be 0% if you plan to live around people and interact: "if I work for you how do I know that you'll pay me?", "if I pay you how do I know you'll let me walk out with these groceries?", and "if I get near you how do I know you won't try to kill me?". But it shouldn't be 100% either (evil people exist). "Who you should trust" is a more interesting question than "when to trust." Growing up most people trust their parents by default who in turn instill many ideas. Later in life those ideas can be questioned but the ideas we grow up with are likely foundational. Therefore what we grow up with (whether it matches our parents or not) may stick our entire life since they were the original axioms (assuming they survived teenager years).
"I could try to find axioms that don't contradict." First of all your desire for no contradiction and the criteria you use to judge that are based on axioms. Secondly there are plenty of axioms that don't contradict so you'll need to chose from among them arbitrarily. Thirdly in the case of contradiction you'll need to decide which one to keep arbitrarily.
Have I driven home the point that all knowledge is arbitrarily based yet? This argument is a logical consequence of Gödel's first incompleteness theorem. Of course this article depends upon axioms so you may disagree given different axioms (or even the same ones). But since you must have at least 1 axiom the point remains: that axiom was chosen without reason.
I'd like to end by giving this article a practical application (as left field as it may be). Given that all knowledge is arbitrarily based there's room to be compassionate for even the craziest or most offensive ideas/opinions. If you hear someone who is for or against some hotly debated topic (eg abortion, homosexuality) know that they just have very different axioms than you and the way they chose those axioms was just as valid as how you chose your's. Therefore even in strong disagreement you can have a calm loving conversation. If someone think aliens, witchcraft, and ghosts are a good explanation/reason don't call them crazy or criticize because from their point of view you seem just as crazy.
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