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There is a precise, consistent definition of what it means for a set to be infinite. How is this a bad abstraction more than other mathematical definition?
by syops 5y ago
There is a precise, consistent definition of what it means for a set to be infinite. How is this a bad abstraction more than other mathematical definition?
- amelius 5y agoPrecision doesn't matter. Software abstractions, once coded, are precise. The problem is with human thinking. If the mind can't grasp at all times all the relevant technicalities that come with an abstraction, then the abstraction is bad.
- syops 5y agoWhat technicalities with the definition of an infinite set can't be grasped by the mind? The definition is quite simple and easy to understand (for a mathematician).
- CrazyStat 5y agoThis is a very anthropocentric view of mathematics. Why should we expect mathematics to be amenable to a puny human mind grasping all relevant technicalities of a concept?
- amelius 5y agoMost of mathematics is abstractions invented by humans to make math more comprehensible by humans. E.g. you can do differential equations without complex numbers, but they make life so much easier. Infinity is a mathematical concept invented by humans to make thinking about large numbers easier in some cases, but it's deceptive in other cases.
- simonh 5y agoWhich cases? Is there mathematics of infinity that you're privy to that are unknown to mathematicians? Also I disagree that thinking about infinity is useful when thinking about large numbers. That seems to me to be a fallacy. Any finite large number is inconsequential, and might as well be a 'small' number for any practical purposes, compared to any infinity.
- syops 5y agoThe set theoretic definition of infinity did not come about in order to make thinking about large numbers easier. You are wrong in your perception on this.
- deleted 5y ago[deleted]
- morelisp 5y agoThe set-theoretic definition of infinity came about to make thinking about small numbers easier. As trivial examples, if we reject this definition of cardinality, we can no longer speak of the natural numbers as a set (nor therefore even finite subsets of them), we would lose key set-theoretic definitions of both natural and real numbers, we could not talk set-theoretically about the domain/range of most interesting functions, etc. Honestly, that would be a much more confusing mathematical world than one with Hilbert's hotel or Banach-Tarski spheres.