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Numbers Are Leaves

christo.sh

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Re: Numbers Are Leaves

#21

Post by TZubiri »

Etheryte wrote: post #19
TZubiri wrote: post #9

> I would like to understand why numbers looks like leaves 1st of all they don't. The graph doesn't look like pinnatids or palmatids. There is some resemblance of an alternating disposition of leaves, but that's not the shape of the leaf itself but the distribution of them, and it's a stretch. Secondly, I'll take the generous interpretation of the question which is, why the graph looks mathematically like leaves, and…

I think the last point doesn't really hold its own in any way. Many discoveries throughout history have started from someone just playing around with an idea, toying with it at first, but eventually becoming obsessed. The thing is, there's no way to tell beforehand. It might be a toy with no ultimate use or meaning, or it might lead to something entirely novel somewhere down the line. That's why play, in a very broad…

If 1 out of 10 chmessicians discover something useful, the discoverer had good taste and deserves credit.

There is no insurance redistributing credit amongst all of the pointless searches.

The guys who invented imaginary numbers or eigenvectors weren't just throwing darts at a board and got "lucky".

Re: Numbers Are Leaves

#22

Post by epgui »

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well.

> "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements."

Why? What's the motivation here?

It seems to me like `next(x) = {x}` is simpler than `next(x) = x ∪ {x}`, and I'm not totally clear on what the extra complexity buys us.

I am of course familiar with the structure `next(x) = x ∪ {x}`, having seen it in textbooks and in a set theory / mathematical foundations class, but I feel like I've never really understood what insight this structure captured. It seems like it's always presented matter-of-factly.

Anyone?

Re: Numbers Are Leaves

#23

Post by tromp »

epgui wrote: post #22

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well. > "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements." Why?…

Representing the natural number n by a set of size n turns out to be rather useful. And the obvious choice for a set of n elements is the representations of the n natural numbers 0..n-1.

Re: Numbers Are Leaves

#24

Post by epgui »

tromp wrote: post #23
epgui wrote: post #22

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well. > "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements." Why?…

Representing the natural number n by a set of size n turns out to be rather useful. And the obvious choice for a set of n elements is the representations of the n natural numbers 0..n-1.

I think I can see why... But the obvious question to me becomes:

- What do the operations of addition and multiplication look like with this structure? Doesn't this seem a bit complex?

Re: Numbers Are Leaves

#25

Post by xanderlewis »

epgui wrote: post #22

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well. > "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements." Why?…

Look up ‘transitive sets’ if you haven’t heard of them already.

The primary answer to your question is that in the von Neumann definition the ordinals are transitive and well ordered by the epsilon (membership) relation, which is a pair of stipulations that can then be used as a definition of the ordinals if you like. This in turn is nice for many other reasons!

Also, you can make convenient definitions like defining the supremum of a set of ordinals to be the union of that set. The union of two of your ordinals usually won’t be an ordinal.

In general, it’s nice to have an ordinal simply be the set of its predecessors, which is something that this definition implies.

Re: Numbers Are Leaves

#26

Post by noam_k »

epgui wrote: post #22

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well. > "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements." Why?…

Not a professional mathematician, but you have the benefit of set operations mapping to functions you're familiar with.

For example, set union becomes the max function.

Re: Numbers Are Leaves

#27

Post by epgui »

noam_k wrote: post #26
epgui wrote: post #22

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well. > "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements." Why?…

Not a professional mathematician, but you have the benefit of set operations mapping to functions you're familiar with. For example, set union becomes the max function.

That one is neat, but how do you define addition and multiplication on these structures?

Re: Numbers Are Leaves

#28

Post by epgui »

xanderlewis wrote: post #25
epgui wrote: post #22

Setting aside the rest of the article, there is one thing I've never really understood the motivation for, and I think this article really highlights it well. > "Well congratulations this works! We can represent numbers using singleton sets (sets with one element). However, it would be nice if our sets had some more structure. Specifically we would like the set corresponding to the number n to have n elements." Why?…

Look up ‘transitive sets’ if you haven’t heard of them already. The primary answer to your question is that in the von Neumann definition the ordinals are transitive and well ordered by the epsilon (membership) relation, which is a pair of stipulations that can then be used as a definition of the ordinals if you like. This in turn is nice for many other reasons! Also, you can make convenient definitions like defining…

Thanks for the pointers!

Re: Numbers Are Leaves

#29

Post by bpoudev »

epgui wrote: post #24
tromp wrote: post #23

Earlier quoted context omitted.

Representing the natural number n by a set of size n turns out to be rather useful. And the obvious choice for a set of n elements is the representations of the n natural numbers 0..n-1.

I think I can see why... But the obvious question to me becomes: - What do the operations of addition and multiplication look like with this structure? Doesn't this seem a bit complex?

I think it's (like so many things) a question of tradeoffs. Programmers often think of complexity (and hence performance) of operations, but that is not important to a mathematician.

The fundamental operation, the successor function, does not look much different, S(n) = n ∪ {n} vs S(n) = {n}. Mathematics usually defines addition in terms of this function, so that n + m = 1 + (n-1) + m and 0 + m = m. This can be done via induction and works equally well, regardless of which "implementation" we choose. Similarly, multiplication is repeated addition. Seen in this way, both "implementations" of natural numbers leads to horribly inefficient, but ultimately very similar, addition and multiplication operations.

However, the representation S(n) = n ∪ {n} leads to a very simple definition of "a finite set of size n". It is simply a set, which has a bijection between it and n. This, in turn, leads to a much easier arithmetic. Instead of manipulating a specific set representing a given number, we can say that any set of size n can represent the number n. Then addition simply becomes disjoint union, and multiplication becomes Cartesian product, from which things like associativity and commutativity can be proven much easier than in the inductive definition.

Re: Numbers Are Leaves

#30

Post by xanderlewis »

epgui wrote: post #27
noam_k wrote: post #26

Earlier quoted context omitted.

Not a professional mathematician, but you have the benefit of set operations mapping to functions you're familiar with. For example, set union becomes the max function.

That one is neat, but how do you define addition and multiplication on these structures?

You use (transfinite, if your ordinals are large enough) recursion! Just define a + 1 to be the successor of a — succ(a) — and then, assuming we’ve defined a + b, define

a + (b + 1) := (a + b) + 1 = succ(a + b)

(it’s only slightly more complicated for infinite ordinals)

You can do a similar thing for multiplication, and exponents, and so on.

Technically, you have to use induction to prove that this definition indeed works to define the operations for all ordinals.

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