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All of the ways is quite the ask. There are 2 main main approaches: explicit constructions and axiomatic constructions. Dedikend cuts and Von Neuman numbers a
by gizmo686 2y ago
All of the ways is quite the ask.
There are 2 main main approaches: explicit constructions and axiomatic constructions.
Dedikend cuts and Von Neuman numbers are both examples of explicit constructions. You can point to exactly what a given number "is", and then prove properties about you construction.
The other approach is the axiomatic construction, where you define what properties you want your numbers to have. You then (hopefully) prove that such a system exists and is, in some sense unique.
Taking the integers, the most common explicit construction is the set-theoretic Von Neumon construction, where 0={}, 1={0}, 2={0,1}, 3={0,1,2}...
You can also imagine defing 0={}, 1={0}, 2={1}, 3={2}
Note that while we say these are equivalent definitions, they produce different results in the sense that, for example, 0∈3 in the former, but not in the latter. [0].
You can also define natural numbers as a subset of the cardinals. In this construction, 3 represents the collection of all sets that can be put in a 1-to-1 correspondance to the set { {}, {{}}, {{{}}} }. Here, the exlusion of various infinities is a rather arbitrary part of the definition.
You can also define integers as a subset of combinatorical games, given by 0={|}, 1={0|}, 2={1|}. This naturally leads to negative numbers, -1={|0}, and a zoo of "numbers" not seen outside of combinatorical game theory, such as ↑ = {0|{0|0}}. Still, if you restrict yourself to the integer looking numbers and follow the game theory rules of addition, you get something equivalent to the natural numbers.
You can also define them as strings constructed from the alphabet of a single character, so 0="", 1="a", 2="aa". (This is the free monoid over a single set)
Lambda Calculus gives us the church encoding, where:
* 0 = λf.λx.x
* 1 = λf.λx.f x
* 2 = λf.λx.f (f x)
Under the axiomatic definition of numbers, the classical definition is given by the 9 Peano axioms, which amount to assuming the existence of 0 and a "succesor" function S(n) informally corresponding to S(n) = n+1.
All of the above examples are things I have actually seen and that arise naturally in their respective branch of mathametics.
[0] This contradiction is not actually an issue, because no one ever actually says 0∈3, even though it is technically true in some constructions.
- anon291 2y agoThese are all constructions though. They don't say what the natural numbers are. The natural numbers is a set with a starting element and that has, for each element, a thing immediately following. Things like 0 in 3 make no sense since 0 refers to the first thing in whatever representation you have, not a particular one. I love formal theorem proving and provers, but this shouldnt lead us to believe that the natural numbers are fundamentally any particular representation N is defined by its properties not its representation.
- lmm 2y ago> These are all constructions though. They don't say what the natural numbers are. A construction is exactly what would tell you what they are. > The natural numbers is a set with a starting element and that has, for each element, a thing immediately following. OK, but, like, which one? There are lots of sets that this describes. > Things like 0 in 3 make no sense since 0 refers to the first thing in whatever representation you have, not a particular one. If 0 and 3 are sets then we can legitimately ask whether 0 is a member of 3 or not. If 0 and 3 aren't sets then what are they? If the natural numbers are a set then 0 and 3 should also be sets, since set theory only talks about sets. > N is defined by its properties not its representation. Then what is it? If you say it's a set with those properties, then it should be a set and we can ask set questions about it. If you say it's somehow all of the sets with those properties, or some other concept that somehow encompasses them, then you still haven't explained what it actually is.
- BalinKing 2y agoNot a direct reply to your question, but it seems to me that GP is describing a "structuralist" view (cf. https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_m..., on which I'd guess much has been written over the years (both for and against).
- anon291 2y agoI think so yes. The article here espouses a constructivist view of mathematics While I like that view personally, I think it's wrong to act as if it's the only one like this article does. Many people have different views.
- lupire 2y agoIf I choose to wear a fedora, is a fedora an essential part of who I am? Or an arbitrary choice that be changed? This question goes back to Plato. You use Von Neuman naturals. I use Peano naturals. Which are the real naturals? Can they both be the naturals, even though they are clearly different? The natural numbers are not any one construction. They are the abstract idea of whose properties are what all the constructiona have in common. Math is like God. We use feeble human language to approximately talk about something that is beyond our ability.
- simpaticoder 2y agoWe must remember that these constructions retcon the actual human experience with numbers, which is firmly rooted in counting and measurement, that always come after this experiential basis. Formal definitions appear driven by an aesthetic consensus, mostly about minimalism and axiom flavor. Individual human experience with numbers reflects human history and at both scales 'counting and measurement' is more fundamental in the sense that it always comes first, and therefore all constructions derive from it, must maintain consistency with it, and conflicts must be resolved within it, and it is safe to ignore inconsistencies that can't be expressed within it.
- staunton 2y agoIn those terms, the foundations (of any topic or field) are never fundamental. I think people in this thread are using a different meaning of "fundamental".
- lupire 2y agoWithout formal constructions, humans have a nasty habit of proving things that wrong or not even wrong. Our ancestors dealt with this problem in the 19th Century by creating tools to help distinguish truth from balderdash.
- simpaticoder 2y agoYour statement is consistent with my point which I mean to point out the psycho-social origin of such constructions and therefore the limits of their utility. I do not say they are not useful at all.