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If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered The magic part for me is that some axioms have been cho
by danck 6y ago
If you only look at mathematics I think it's simply:
- Axioms are invented
- Conclusions are discovered
The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.
- vanderZwan 6y agoIsn't that simply because axioms that don't lead to consistent conclusions are rejected?
- danck 6y agoYou can invent and pick axioms in many ways that (probably) won't lead to inconsistencies. But they won't all be powerful enough or relevant in the real world.
- vanderZwan 6y agoWell, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored. Or in other words: the constraints on maths are imposed from outside of maths.
- gibspaulding 6y agoDoesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.
- red75prime 6y agoArithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.
- kevin_thibedeau 6y agoIt is a system for modeling concepts invented by man. Everything that falls out of such a system is a product of the invention. Numbers don't inherently exist. Everything derived from that concept can't be a "discovery".
- red75prime 6y ago> Numbers don't inherently exist How do we know that it is true?
- kevin_thibedeau 6y agoThey are symbols that we assign arbitrary meaning to. They are useful because of the axiomatic framework constructed to support them.
- mmmBacon 6y agoThis is an excellent point. When I took algebra as an undergraduate I was blown away by the fact that you can choose any axioms and then derive an algebra based on those axioms. I was blown away because prior to that course I just assumed that our “standard” axioms were immutable.
- Koshkin 6y ago> choose any axioms and then derive Sound almost like "jump off the roof and see what happens."
- deleted 6y ago[deleted]
- symplee 6y agoIsn't it the other way around? Axioms are chosen because there are no observable counter examples in the real world.
- karmakaze 6y agoNot at all, pure math is in part about exploring axiomatic systems that may or may not have a physical counterpart. The latter is immaterial.
- symplee 6y agoCan you give some examples of axioms in pure math that run completely counter to our physical world? For example: It is NOT possible to draw a straight line from any point to any other point. It is NOT possible to extend a line segment continuously in both directions. etc... or Things which are equal to the same thing are NOT equal to one another. If equals are added to equals, the wholes are NOT equal. The whole is LESS than the part. Note that the original forms of the above axioms "make sense" to us because everything in our physical experience agrees with them. So when you said that the "physical counterpart ... is immaterial", I was curious to see an example of a "physically impossible" axiom.
- red75prime 6y agoMost of large cardinal axioms.
- naasking 6y ago> If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered How would you revise this statement if we lived in a "Mathematical Universe", like Max Tegmark's hypothesis. > The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world. It's actually hard to avoid Turing completeness, and once you have that, any recursively enumerable function is calculable. All you need is addition and multiplication on numbers.
- pfortuny 6y agoOh no: the axioms come much much later. The order is exactly the reverse one.
- danck 6y agoWell, I have to agree. From a practical perspective.