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How inevitable is the concept of numbers?

writings.stephenwolfram.com

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Re: How inevitable is the concept of numbers?

#61

Post by feanaro »

hcarvalhoalves wrote: post #27
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.

You don't really need equations, you just need missing elements, or debt. Do I have extra bricks, just enough (0 extra) of them or am I missing some? The case of missing bricks is naturally modelled with negative numbers.

Re: How inevitable is the concept of numbers?

#62

Post by eigenket »

breck wrote: post #57

All structures in our universe are trees^. Numbers are a metalanguage for describing those trees. It is possible that there are other independent universes that we can't perceive, but I'd expect any aliens that we eventually interact with to be operating only in the treeverse, and so also be fluent in a language like our numbers. ^ Perhaps there are other independent universes out there that we don't perceive, but hu…

I genuinely don't understand what this means. How a beach ball a tree? What about the symmetry group of the beach ball?

Re: How inevitable is the concept of numbers?

#63

Post by geijoenr »

"We take in some visual scene. But when we describe it in human language we’re always in effect coming up with a symbolic description of the scene."

Human mind works in a qualitative manner, we need symbols to translate qualitative perception into a quantitative abstraction. This started as an economic and social organization need (geometry), but later evolved into Mathematics and got to overcome the shortcomings of natural language to describe reality (philosophy became science).

Numbers are just symbols that map human perception to a reality that is inherently quantitative.

I fail to grasp what Mr. Wolfram tries to explain here, but it looks to me as if he is regressing into philosophy.

Re: How inevitable is the concept of numbers?

#64

Post by coder-3 »

slver wrote: post #16
ManBlanket wrote: post #7

‘If a lion could speak, we could not understand him’. - Wittgenstein So even things we understand as fundamental truths of the universe are so deeply rooted in human experience that it's possible another living entity could be so different than ourselves that the capacity to sympathize with one another about certain things may simply not exist. Would a lion understand a corsage? Would an alien know what 12 x 12 means…

When we train neural networks, we don't tell them what to think and how to think about it. We mostly focus on overall structure, layers and we want useful outcomes. And yet we see lots of patterns that repeat ways in which we think. I think if a lion could speak, we'd understand him, because we're much closer to "a speaking lion" ourselves than we realize. Our intelligence didn't just happen at the last mile between…

The way I see the universe is that all this complexity is just the interplay of a finite few axioms (the fundamental laws of physics). Given enough time and scale, the seemingly unique complexity converges back into a few (relatively speaking) patterns. If that wasn't the case the universe would be pure chaos.

Re: How inevitable is the concept of numbers?

#66

Post by eigenket »

joe_the_user wrote: post #59
Aardwolf wrote: post #47

Earlier quoted context omitted.

Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs

It's not. It's a single proof about a set, a set that's assumed to be uncountable in standard ZF set theory. The "axiom system" that (supposedly) contain a countable number of axioms. But these too are constructs of set theory. We still create proofs one by one of theories about axiom systems with infinite axiom - so we have a countable/enumerable set of such theories. The proof systems to we can see or touch involve…

I think you have a slightly stricter definition of "a proof" than me. I would consider a proof that all the numbers in (0,1) are positive to also be a proof that the number 0.5 is positive, as well as the number 1/e, and Champernowne's constant.

Since the original question was about uncountably many mathematical truths I would say we have one proof that proves uncountably many mathematical truths.

Re: How inevitable is the concept of numbers?

#68

Post by paganel »

dogecoinbase wrote: post #23
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, whi…

Thanks a lot for the link/reference, that's the sort of stuff I was looking for.

Re: How inevitable is the concept of numbers?

#69

Post by breck »

eigenket wrote: post #62
breck wrote: post #57

All structures in our universe are trees^. Numbers are a metalanguage for describing those trees. It is possible that there are other independent universes that we can't perceive, but I'd expect any aliens that we eventually interact with to be operating only in the treeverse, and so also be fluent in a language like our numbers. ^ Perhaps there are other independent universes out there that we don't perceive, but hu…

I genuinely don't understand what this means. How a beach ball a tree? What about the symmetry group of the beach ball?

> How a beach ball a tree?

      _ 
    /   \
    |   |
    \ _ /

Re: How inevitable is the concept of numbers?

#70

Post by eigenket »

breck wrote: post #69
eigenket wrote: post #62

Earlier quoted context omitted.

I genuinely don't understand what this means. How a beach ball a tree? What about the symmetry group of the beach ball?

> How a beach ball a tree? _ / \ | | \ _ /

That is neither a beach ball, nor a tree.
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