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Just a note from your friendly philosophy degree holder: Axioms are arbitrary. Use the axioms that are the most useful.
by scoofy 2y ago
Just a note from your friendly philosophy degree holder:
Axioms are arbitrary. Use the axioms that are the most useful.
- Spivak 2y agoDefinitions are neither true nor false. They're either useful or not useful. The question of whether or not the integer 1 is a prime doesn't make sense. The question is is it useful to define it as such and the answer is a resounding no.
- gblargg 2y agoAgreed. Definitions are made to differentiate things in a way useful for some goal. The question "Is X an M?" without a context or goal basically picks up whatever vague goals or purposes a person has lingering below the surface of consciousness, differing from what other participants have below theirs, leading to different answers, with no way to select the best one. In the case of what is considered prime, it's a matter of what definition simplifies the things that use it. It could be that two concepts are better, one including 1 and the other not including it. Since it's just a language shorthand, it makes no fundamental difference other than efficiency and clarity in communication about math.
- NoahZuniga 2y agoWhile axioms are in some sense arbitrary, it is helpful if they are consistent (informally: you can't prove something that "is false"; formally: you can't prove p and not p). Also other people like it if your axioms feel obvious.
- scoofy 2y agoMy point is that axioms "feeling obvious" is exactly a signal that they will be useful. The point of deductive reasoning based on axioms is that it is a shortcut to fill in problems of induction, which is what happens when we use pure empiricism. If you really want to go down the road of solipsism, read Karl Popper.
- LudwigNagasena 2y agoYou can’t axiom your way out of 1 apple and 1 apple being 2 apples together. So axioms are not really that arbitrary.
- scoofy 2y agoThey are, by definition. The reason why we choose them is exactly to map a deductive framework onto an inductive reality.
- LudwigNagasena 2y agoThat doesn’t seem to match the definition of “arbitrary”.
- scoofy 2y agoWe can choose whichever axioms we want. There is still arguments over the axiom of choice, but nobody cares because it’s entirely helpful.
- amavect 2y agoYou implicitly used an axiom to ignore the differences between the apples. Someone else could use different axioms to talk about the sizes of the apples (1 large + 1 small = ?), or the color of the apples (1 red + 1 green = ?), or the taste of the apples (1 sweet + 1 sour = ?). People "axiom" their way out of 1+1=2 in this way: by changing the axioms, they change the topic, so they change the conclusion. I observe this pattern in disagreements very often.
- LudwigNagasena 2y agoI have used appropriate axioms, not arbitrary axioms. If you want to talk about size or color or taste, you would use “axioms” appropriate for you case.
- edanm 2y agoAnd as is demonstrated by this article, arguing about axioms is a very useful way of doing math exposition :)