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> While saying "almost all" real numbers are irrational... is saying that real values are all irrational, except for... the entire infinitely large set of ratio

> While saying "almost all" real numbers are irrational... is saying that real values are all irrational, except for... the entire infinitely large set of rational numbers. You're claiming that the precise mathematical definitions of these terms are ridiculous, and then just repeating those mathematical definitions of the terms to support your claim. > That's my whole point -- that meanings are being mixed up, that this way of classifying infinite sets shouldn't be conceptualized as "size" at all because the analogies break down, well, instantly. The whole point is that when you're comparing infinite sets, common sense terms like "size," "almost all," etc. don't make any sense at all, so mathematicians create precise definitions for the terms that are then used among all mathematicians and do make sense. These terms are common among mathematicians and they do make sense. The fact that these definitions don't precisely match the definitions used by non-mathematicians is precisely the goal, because those definitions make no sense whatsoever when referring to infinite sets. It's okay to complain about using jargon in discussions not related to mathematics, although that's not what is happening here (and it's difficult to imagine a conversation about infinite sets that is not also about mathematics). This complaint is comparable to complaining that the definitions of "sharp" and "flat" in music theory don't at all match the definitions of "sharp" and "flat" used in conversations not about music. I suppose another way to approach this is to ask what you would propose the term "almost all" to mean when referring to an infinite set? For that matter, what do you propose that the term "almost all" means to a layperson referring to small finite sets? What percentage do you need in order to use "almost all"?

robbedpeter 5y ago
Mathematics is a language unto itself, and some concepts are more difficult to map to natural language than others. Even if you had a much broader English vocabulary, with unique words assigned to every currently known mathematical abstraction, the intuitive sense in which we speak and read and understand will clash with the behavior and features of mathematical symbols. Natural language maps onto an embodied experience designed by evolution. Math transcends what our brains are built to expect and predict. Infinities, exponentials, quantum entanglement, and an infinite array of other very real things simply exist outside what we are capable of understanding intuitively, without some special conjunction of neural wiring, training, or happenstance. The nature of infinities might seem like navel gazing, but it helps us understand the features of different types of numbers, and those are the building blocks of proofs. Knowing to which infinity class a number belongs might help inform the optimization of real world engineering of chips or data storage, or drug design algorithms, and so on. It could eventually be part of figuring out practical quantum computers, solving p vs np, or maybe just a piece of a better factorization algorithm that improves cryptography.

robbedpeter 5y ago

Mathematics is a language unto itself, and some concepts are more difficult to map to natural language than others. Even if you had a much broader English vocabulary, with unique words assigned to every currently known mathematical abstraction, the intuitive sense in which we speak and read and understand will clash with the behavior and features of mathematical symbols. Natural language maps onto an embodied experience designed by evolution. Math transcends what our brains are built to expect and predict. Infinities, exponentials, quantum entanglement, and an infinite array of other very real things simply exist outside what we are capable of understanding intuitively, without some special conjunction of neural wiring, training, or happenstance. The nature of infinities might seem like navel gazing, but it helps us understand the features of different types of numbers, and those are the building blocks of proofs. Knowing to which infinity class a number belongs might help inform the optimization of real world engineering of chips or data storage, or drug design algorithms, and so on. It could eventually be part of figuring out practical quantum computers, solving p vs np, or maybe just a piece of a better factorization algorithm that improves cryptography.