How inevitable is the concept of numbers? • Hacker News Board

Re: How inevitable is the concept of numbers?

#21

Post by ezconnect » Tue, May 25, 2021, 7:27 PM UTC

paganel wrote: post #10

ddlutz wrote: post #3

Earlier quoted context omitted.

Isn't multiplication by 0 natural? If i'm at a party and everybody wants 2 beers, then if there is 1 person I need 2 beers (1 * 2), 2 people need 4 beers (2 * 2), but 0 people need 0 beers (0 * 2).

“Zero people” and “needing” is an oxymoron, nothing (zero) can’t be associated to any natural thing (like “needing”), that’s what I was trying to say in my not so clear comment above. Writing this down I realized we’re still trying to write in fancier words what the pre-Socratics had a clear understanding of 2500+ years ago, and speaking of the Greeks is too bad that Wolfram didn’t mention Plato by name in the first…

When you combine understanding language and numbers you will have a hard time.

Re: How inevitable is the concept of numbers?

#22

Post by agumonkey » Tue, May 25, 2021, 7:27 PM UTC

ezconnect wrote: post #6

When my two year old can't count the items in front of him he just says it's "many".

One two infinity

Re: How inevitable is the concept of numbers?

#23

Post by dogecoinbase » Tue, May 25, 2021, 7:38 PM UTC

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

I would highly recommend Paul Benacerraf's article "What Numbers Could Not Be" for a philosophical if somewhat technical take on the subject (PDF link): http://michaeljohnsonphilosophy.com/wp-content/uploads/2015/...

The thesis, broadly, revolves around contemplation about whether an number is an object, but the larger question is whether it's possible to have any small part of mathematics, even a single integer, which doesn't imply the whole (or at least, a significant structure).

Re: How inevitable is the concept of numbers?

#24

Post by pharrington » Tue, May 25, 2021, 7:40 PM UTC

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

How much do you add to five to get two?

edit: another intuition is that ordering is a property of numbers. Positive numbers are ordered in one direction. Can numbers be ordered in other directions?

Re: How inevitable is the concept of numbers?

#25

Post by aaroninsf » Tue, May 25, 2021, 7:40 PM UTC

"The brain does much more than just recollect; it inter-compares, it synthesizes, it analyzes. It generates abstractions.

The simplest thought like the concept of the number one has an elaborate logical underpinning; the brain has its own language for testing the structure and consistency of the world."

Thank you, Carl. Sit down Stephen.

Re: How inevitable is the concept of numbers?

#26

Post by gorgoiler » Tue, May 25, 2021, 7:45 PM UTC

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

It’s fun to posit these ideas by going in the intuitive direction, then going in the reverse direction and seeing what happens.

I recently found this to be helpful when investigation fixed and floating point numbers. If ”1011” means 2^3 + 2^1 + 2^0, which is 11 the “1011.101” means the same thing but with an additional 2^-1 + 2^-3 aka 5/8. Negative powers seem weird to begin with but it kind of just is there to just discover, due to the symmetry.

This kind of arithmetic is far less fundamental than what you and the article are talking about — it is just representation really, in computers — but I think it is a good example of how if you can tread a path in one direction then turning around and coming back to where you started then carrying on in the other direction is a useful tool for teaching and learning.

Re: How inevitable is the concept of numbers?

#27

Post by hcarvalhoalves » Tue, May 25, 2021, 7:47 PM UTC

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

I believe the concept of negative number doesn't arise naturally from counting, you need equations – even though we teach it all at once to kids when introducing the number line.

Re: How inevitable is the concept of numbers?

#28

Post by contravariant » Tue, May 25, 2021, 7:49 PM UTC

SquibblesRedux wrote: post #4

While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism. Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas. I would rather see de…

Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.

Re: How inevitable is the concept of numbers?

#29

Post by reggieband » Tue, May 25, 2021, 7:50 PM UTC

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

I'm not sure if it is what you want, but Intuitionism [1] is one area that challenges modern fashions in Mathematical thinking. It suffered greatly under the formalist approach lead by David Hilbert and still has little main-stream support despite Gödel and his incompleteness proofs. Veritasium's latest video on Gödel's incompleteness [2] gives a pretty fair account of how we settled on the current fashionable foundations of Math (including nods to Cantor and Hilbert). For a more formal history there is a book "The Philosophy of Set Theory" [3] that sketches out how we got to where we are.

I have always been unsatisfied with the current foundations of math and their obtuse basis in Set Theory. Although I must admit, Category Theory has alleviated that quite a lot, especially with the relaxing of equivalence compared to equality.

1. https://en.wikipedia.org/wiki/Intuitionism

2. https://www.youtube.com/watch?v=HeQX2HjkcNo

3. https://www.maa.org/press/maa-reviews/the-philosophy-of-set-...

Re: How inevitable is the concept of numbers?

#30

Post by bluenose69 » Tue, May 25, 2021, 7:50 PM UTC

My guess is that the aliens of which Wolfram speaks would know about numbers. After all, animals do (see e.g. https://www.bbc.com/future/article/20121128-animals-that-can...).