Math's Arbitrary Non-Necessary Necessarily-Disconnected Conventional Link to Reality

That's not really an issue. Most mathematicians will tell you that mathematics is not about the real world.You seem to be confused by the fact that mathematicians sometimes say that their mathematics is real. However, as Rrhain tried to point out, that's more of a philosophical issue.The major philosophies of mathematics are:

• Platonism: mathematical objects exist in a world of ideal platonic forms (sometimes loosely described as "Plato's heaven".
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• Realism (or mathematical realism): this is just another name for platonism.
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• Fictionalism: mathematical objects (such as numbers) are useful fictions.
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• Intuitionism: mathematical objects exist only in the intuitions of people (particular, of mathematicians).
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• Constructivism: mathematics is a science of what we can construct mathematically.
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• Formalism: mathematicians make meaningless formal marks, and play rule-based games with these marks.

In practice, most mathematicians are platonists. A few are fictionalists, but you would probably have difficulty distinguishing between platonists and fictionalists. constructivists and intuitionists have much in common. In particular, they tend to be skeptical of much that platonists say about infinite sets. I am not convinced that there are any actual formalists, so formalism is mostly a fall back position used to explain to skeptics (such as you) what it is that mathematicians do.I consider myself a fictionalist.I have never asked an intuitionist about whether 0.9999... is equal to 1. That's partly because I don't actually know any intuitionists in real life, though I know of some. My best guess is that intuitionists and constructivists would agree that 0.9999... is equal to 1. That they are equal is a matter of well accepted convention.

Interesting. Thanks.ABE: No big deal, and not rude at all. (I asked for reinstatement so I could comment in this thread).Edited by nwr, : add comment on edited change in the post to which this was responding.

You fit more as a constructivist or intuitionist. The intuitionists take the view that there are no infinite decimal expansions, only finite ones. Fictionalists don't have a problem dealing with fictions, so most would not have a problem with infinite decimal expansions.In the particular case of 0.9999..., what we have is a recurring decimal expansion. And that's different from an infinite decimal expansion. We can actually have a notation for recurring decimals that only requires writing down finitely many symbols. While a platonist would take that notation as a shorthand for an infinite decimal expansion, a constructivist or intuitionist might take it as a separate notation in its own right which does not require anything infinite. I think that's what cavediver's constructivist friend was implying (see Message 8).

I'll note that you replied to me, but quoted text from cavediver. As far as I know, constructivists don't have a principled objection to calculus as calculus. But they do object to some of the thing that are done in calculus. They would object to examples of a nowhere continuous function or a nowhere differentiable function, because those examples are non-constructive.Perhaps cavediver can further comment, since he has a constructivist friend. The closest I came was attending a seminar by Errett Bishop, and that was many years ago.

You have just bumped into a variation on the sorites paradox. It isn't a problem with the mathematics. It's an example of why mathematicians tell you that mathematics is not about reality.