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From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic an
by pitaj 10y ago
From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic and math can lead us to useful conclusions.
- FrostyElsa 10y agoDidn't it take some guy like 200 pages to prove logically that 2+2=4?
- redial 10y agoBut isn't reason is the result of logic? as in something is reasonable only when it is logically sound?
- pitaj 10y agoA line of thought can only be reasonable if it is logically sound. Reason is built on top of logic. All of this IIRC.
- lgas 10y agoWhere can I find this standard set of axioms?
- cmrx64 10y agoOne widely-used system is Higher Order Logic (with various extensions sometimes), one core of which you can find here: http://sourceforge.net/projects/hol/files/hol/kananaskis-10/kananaskis-10-logic.pdf/download http://sourceforge.net/projects/hol/files/hol/kananaskis-10/... This is based on type theory (and has a model in set theory), but isn't as powerful as the logics used in Coq and similar theorem provers (the Calculus of (Inductive) Constructions). You might be interested in metamath: http://us.metamath.org/ http://us.metamath.org/, which lists the axioms used in its largest body of work: http://us.metamath.org/mpegif/mmset.html#axioms http://us.metamath.org/mpegif/mmset.html#axioms
- baddox 10y agoI'm not sure what the parent commenter means by "reason and science" giving us axioms, but for mathematics the most common axiomatic system is ZFC set theory. Some of the axioms are easy to understand even with a mathematics background: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
- lgas 10y agoI would consider these to be mathematical axioms. I was looking for what pitaj was describing as coming from outside of math. Maybe I'm just misunderstanding what pitaj was saying.
- pitaj 10y agoBy axioms, maybe I used the wrong word, but what I was referring to is things like the concept of "burden of proof" in reason.
- 613style 10y agoI don't think I would agree with the general statement that science gives us axioms. In normal usage, "axioms" are not the same as scientific "givens" or "data" even though it makes sense to interpret the word that way. In math and logic, the axioms we use are typically very foundational. We might say, "the empty set exists," or "for all x, it holds that x = x." There's no need to invoke science for these sorts of axioms and we can make many useful conclusions without it.
- marcosdumay 10y agoYou can create any set of axioms you want, but unless they carry a reference to something that happen "outside of Math", at the "Real World", they are only useful for nerd-sniping.
- Koshkin 10y agoThing is, there is no universally accepted "standard set of axioms" in mathematics (let alone in all of "science"). Not since around the early 1900s anyway. Even assumptions that may seem obvious to most people, such as the famous Axiom of Choice, are only accepted provisionally in modern mathematical texts. In the extreme case of what is known as the "intuitionistic logic", even some of the most widely accepted laws of logical reasoning are rejected. If you add the fact (Goedel's Incompleteness Theorem) that given enough axioms there always be statements that are impossible to prove or disprove, the whole situation with logic will become, well, complicated.
- cscurmudgeon 10y agoYou can't do science without logic. They are all highly intertwined.