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> in English whole numbers mean 0, 1, 2, ... According to Wikipedia [1], the term is ambiguous. Its talk page [2] has plenty discussion about it though. [1] https://en.wikipedia.org/wiki/Whole_number [2] https://en.wikipedia.org/wiki/Talk:Whole_number
Huh, I never knew there was an ambiguity. To me it makes total sense to include negative integers. "Whole" means "without a fractional part".
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I can't seem to find a reference for it now, but a math professor once told us that some early cultures didn't even have a proper word/symbol for the number "one", and the act of counting didn't kick in until there were two items.
Greek geometers (e.g. in Euclid's Elements, Book 7) define a "unit" to be 1 and "numbers" (arithmoi) to start from 2. https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/book... But in practice there are a bunch of propositions/proofs where 1 is treated as a number just like any other.
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Meh, well different strokes for different folks I guess. I got into higher math while reading Ian Stewart's book in grade school but I guess some people are going to want to go the standard way. My problem with the book we are discussing is that it seems rather prosaic -- it doesn't really give a sense of the true reason to practice math: the asking of interesting questions and creating new universes. It's just the s…
The language comes first then the applications. There’s a reason the order of topics evolved the way it has.
The language is a baby compared to the applications.
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In German, it is not and never was ambiguous, so it is not a mistranslation. What the meaning of "Whole number" in English is, is not really relevant, except for explaining the mistake in the book.
> In German, it is not and never was ambiguous I’m not sure about that ”and never was”. Mathematicians used to have fairly loose definitions for all kinds of things and different fields sometimes have incompatible definitions for a term, so I think it’s not impossible “ganzzahlig” was used ambiguously for ℤ or ℕ (in- or excluding zero) for a while in some corners. When was the phrase “natural number” even invented? T…
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Well, I'm just going by empirical evidence: what has worked for me and what has worked for many of my fellow colleauges that have done actual research in mathematics.
Before one can do advanced mathematics (I have too have done research in math) one needs to learn the basics. Reading Smullyan did not in any way help you learn advanced mathematics. It may have helped you to get motivated to learn math and want to learn advanced math but it didn’t help you accomplish this. There’s a reason just about every mathematics department teaches classes on how to do proofs and on basic set t…
As a person interested in self-learning Mathematics, i have read and amassed a lot of "popular mathematics" books by authors like Ian Stewart, W.W.Sawyer, E.T.Bell, George Gamow etc. all of which were great motivators but none of which taught me the basics of "Modern Mathematics" which i could only get from Textbooks. The quality of Textbooks are of course all over the map and so i am always on the lookout for the simplest, clearest and yet rigourous explanations available. The book under discussion seems to check all such boxes for a beginning student.
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Huh, I never knew there was an ambiguity. To me it makes total sense to include negative integers. "Whole" means "without a fractional part".
Is i a whole number?
No, it is not even a real number, but every whole number surely is a real number.
Yes, it is a "Gaußsche ganze Zahl".
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The language comes first then the applications. There’s a reason the order of topics evolved the way it has.
It's weird that everyone from Euclid to Gauss disagrees with you. The language is a baby compared to the applications.
How do you propose one do applications of point set topology without knowing about sets and mappings? Before the applications one must know the language. We don’t teach the quadratic formula and solving simple velocity problems before teaching students how do the basics of manipulating algebraic expressions.
One must first be a baby before being an adult.
I was prepared to hate this book as much as I've hated any other "introduction" to math, since I've found that mathematicians generally suck at introducing or explaining things. But the first little section that breaks down set notation into plain english was fantastic. That's exactly the kind of thing I need personally, as notation is extremely off-putting and confusing for me.
and then there's Grant Sanderson