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What Even Is a Number?

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Re: What Even Is a Number?

#11

Post by cobbzilla »

msla wrote: post #4
dandare wrote: post #3

> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…

> was there ever an attempt to do math with something like y=1/0 or another illegal construct? Wheels are a type of algebra where division by 0 is defined. https://en.wikipedia.org/wiki/Wheel_theory https://math.stackexchange.com/questions/994508/wheel-theory... To call wheels "obscure" is to make them out to be better-known than they are, however.

I had not heard of Wheel theory, quite fascinating.

But eek! From the wikipedia article:

    0x ≠ 0 in the general case
    x − x ≠ 0 in the general case
Like, you get the ability to divide by zero, but at what cost? Multiplying by zero is also not zero! Seems bonkers.

Are there any practical uses for this kind of algebra?

Re: What Even Is a Number?

#12

Post by 12elephant »

xamuel wrote: post #8

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…

Your castle is built on sand. What, then, is {}?

Re: What Even Is a Number?

#14

Post by [deleted] »

cobbzilla wrote: post #10

I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea. You cannot show just "one" unattached to anything else. The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of…

[deleted]

Re: What Even Is a Number?

#15

Post by mjburgess »

cobbzilla wrote: post #10

I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea. You cannot show just "one" unattached to anything else. The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of…

That's just "Exist as physical objects exist"

Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties.

Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!".

"Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does not imply any particular properties.

"To Exit" in general is "to be found in some domain"

2 can be found in the domain of numbers

Re: What Even Is a Number?

#16

Post by fredliu »

_bxg1 wrote: post #7

It's extremely obvious that the author is a computer scientist

Exactly! I didn't know the author, but couldn't help but check on his background after reading the blogpost.

It's almost a dead giveaway for excerpts like 'The notion of a counter is not that of a single concrete class of object, it is an abstract description of the behaviour of a system providing certain operations satisfying certain rules.' , and 'A number is not any one thing, it is any one of any number of things that implement some operations (and there are different types of number depending on what operations you want to implement).'

Re: What Even Is a Number?

#18

Post by xamuel »

12elephant wrote: post #12
xamuel wrote: post #8

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…

Your castle is built on sand. What, then, is {}?

One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset.

Other axioms of set theory are used to formalize other steps in my post. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... For example, the Axiom of Infinity is used to formalize the handwavy part where I said "In general, the natural n+1 is {0,...,n}".

Re: What Even Is a Number?

#19

Post by dahfizz »

12elephant wrote: post #12
xamuel wrote: post #8

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…

Your castle is built on sand. What, then, is {}?

> the natural 0 is the empty set.

Re: What Even Is a Number?

#20

Post by _bxg1 »

fredliu wrote: post #16
_bxg1 wrote: post #7

It's extremely obvious that the author is a computer scientist

Exactly! I didn't know the author, but couldn't help but check on his background after reading the blogpost. It's almost a dead giveaway for excerpts like 'The notion of a counter is not that of a single concrete class of object, it is an abstract description of the behaviour of a system providing certain operations satisfying certain rules.' , and 'A number is not any one thing, it is any one of any number of things…

And the very first thing he did was define natural numbers as a grammar of operations with rules for interpretation
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