"*Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist
Menschenwerk*"
("God made the integers, all else is the work of man"). –Leopold Kronecker.... of course, Kronecker was wrong, and Cantor was right. But the quote is an excellent dis. :-) On Sun, Dec 29, 2019 at 9:41 PM Andrew Barnert via Python-ideas < python-ideas@python.org> wrote: > On Dec 29, 2019, at 18:20, Chris Angelico <ros...@gmail.com> wrote: > > > > On Mon, Dec 30, 2019 at 11:47 AM Steven D'Aprano <st...@pearwood.info> > wrote: > >> > >> On Mon, Dec 30, 2019 at 08:30:41AM +1100, Chris Angelico wrote: > >> > >>>> Especially since it fails quite a few commonsense tests for whether or > >>>> not something is a number: > >> [...] > >>>> The answer in all four cases is No. If something doesn't quack like a > >>>> duck, doesn't swim like a duck, and doesn't walk like a duck, and is > >>>> explicitly called Not A Duck, would we insist that it's actually a > duck? > >>> > >>> Be careful: This kind of logic and intuition doesn't always hold true > >>> even for things that we actually DO call numbers. The counting numbers > >>> follow logical intuition, but you can't count the number of spoons on > >>> a table and get a result of "negative five" or "the square root of > >>> two" or "3 + 2i". > >> > >> That's because none of those examples are counting numbers :-) > >> > >> My set of "commonsense tests" weren't intended to be an exhaustive or > >> bulletproof set of tests for numberness. They were intended to be > >> simple, obvious and useful tests: if it quacks, swims and walks like a > >> duck, it's probably a duck. The silent Legless Burrowing Duck being a > >> rare exception.[1] > > > > Exactly my point! Counting numbers follow logical intuition; but you > > attested that you could use logical intuition to figure out if > > something is a "number". Not a "counting number". Logical intuition > > does NOT explain all the behaviours of non-counting numbers, and you > > can't say "oh this is illogical ergo it's not a number". The logic of > > logical intuition is illogical. :) > > Counting numbers are intuitively numbers. So are measures. And yet, > they’re different. Which one is the “one true numbers”? Who cares? Medieval > mathematicians did spend thousands of pages trying to resolve that > question, but it’s a lot more productive to just accept that the intuitive > notion of “number” is vague and instead come up with systematic ways to > define and compare and contrast and relate different algebras (not just > those two). > > Are complex numbers numbers? Sure, if you want. Or no, if you prefer. But > they’re still not real numbers, much less natural numbers. That’s obvious, > and nearly useless. What you really want to know is which properties of the > reals also hold of the complex numbers, and that’s a lot less obvious and a > lot more useful. > > And the same is true for IEEE binary64. You can say they’re not numbers, > or that they are, or that some of them are and some of them aren’t, but > they’re not the rationals (or the reals or the affinely extended reals or a > subalgebra of any of the above); what you really want to know is which > properties of the rationals hold under what approximation regime for the > IEEE binary64s. > _______________________________________________ > Python-ideas mailing list -- python-ideas@python.org > To unsubscribe send an email to python-ideas-le...@python.org > https://mail.python.org/mailman3/lists/python-ideas.python.org/ > Message archived at > https://mail.python.org/archives/list/python-ideas@python.org/message/6OMWP4HWHGH6ES2VD4HIWS4Y6KDFA3SA/ > Code of Conduct: http://python.org/psf/codeofconduct/ > -- Keeping medicines from the bloodstreams of the sick; food from the bellies of the hungry; books from the hands of the uneducated; technology from the underdeveloped; and putting advocates of freedom in prisons. Intellectual property is to the 21st century what the slave trade was to the 16th.
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