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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?

#81

Post by reggieband »

eigenket wrote: post #60
reggieband wrote: post #29

Earlier quoted context omitted.

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 founda…

I am somewhat familiar with the standard foundation of maths in set theory, and slightly less familiar with the program to ground maths in Category theory (although I do know a fair bit of category theory). Maybe I am biased but I do not find the category theory foundations any less obtuse than the standard formulations. Like its pretty cool that you can do this stuff with category theory, but I think there is a reas…

I didn't meant to imply that Category Theory addressed the obtuseness of Set Theory. Rather, I was alluding to work in Category Theory that helps to redefine our ideas of equality and equivalence. A discussion of that is available in Quanta Magazine [1].

1. https://www.quantamagazine.org/with-category-theory-mathemat...

Re: How inevitable is the concept of numbers?

#82

Post by marcodiego »

There is a brazilian tribe (piranhã) that knows no concept of quantity besides one, two an d many. Also, in their language, verbs are not flexed related to time. This is probably du e to their lifestyle that needs no long planning, discussions about the past or managing m ultiple instances of the same resources.

Re: How inevitable is the concept of numbers?

#83

Post by jcranmer »

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…

I think I understand what you're trying to say. Let me try to motivate algebra in less explicitly algebraic terms for you:

Zero is an algebraic concept of nothing. While it refers to no physical thing, its existence in algebra is necessary to describe several mathematical laws, and several properties it has in algebra inherently derives from its algebraic concept of nothing.

Let's define both addition and multiplication [1]. Addition is an abstract representation of combination: you combine a pile of 2 things and a pile of 3 things to get a pile 5 things. Now zero is the model for what you can combine with anything else that doesn't do anything: combining a pile of 0 things (that is, nothing) with a pile of 3 things leaves you a pile of 3 things. Negative numbers represent undoing a combination: combine a pile of 5 things with a pile of -3 things (i.e., "take 3 things from the pile") leaves with a pile of 2 things. Negative numbers and zero numbers may not necessarily have a direct physical analogue, but by introducing them for algebraic purposes, things actually become simpler: you use the same terminology and logic to deal with both adding and removing things, or perhaps coming to the conclusion that in the end there's no net effect.

Now, multiplication is scaling. A half a pile of 2 things is 1 thing. Scaling and combining interact with each other, too. Taking half of a pile of 3 things and half of a pile of 5 things is the same taking half of a pile of 8 things. Or I can say that doubling a pile and then adding another of the original is the same as tripling a pile (i.e., 2x + x = 3x).

This is where things get interesting. We can do nothing by adding a pile of something and immediately taking it away, leaving us with what we started (i.e., 0 = a·x - a·x). From above, we can also see that that is the same as adding a pile whose scale is 0 (i.e., a·x - a·x = (a - a)·x). Simplifying the equation a bit, we end up with 0 = 0·x: multiplying by 0 must yield 0 to make both addition and multiplication make sense. So the concept of nothing times anything yielding nothing isn't a requirement of nothing itself, but it's a requirement of how addition and multiplication works, and how nothing itself interacts with those operations.

Incidentally, the deeper you dive into mathematics, the more important you realize the concept of 0--of nothing--actually is. The most powerful ways to describe operations are based on how they arrive at doing nothing in interestingly nontrivial ways. And things that don't have ways to do nothing tend not to be very interesting structures to look at.

[1] I'm alluding to a vector spaces here, although by glossing over the difference between scalars and vectors, it could also be viewed as rings instead.

Re: How inevitable is the concept of numbers?

#84

Post by Tainnor »

marcodiego wrote: post #82

There is a brazilian tribe (piranhã) that knows no concept of quantity besides one, two an d many. Also, in their language, verbs are not flexed related to time. This is probably du e to their lifestyle that needs no long planning, discussions about the past or managing m ultiple instances of the same resources.

Basically every single claim about Pirahã (not Piranha) needs to be taken with a huge grain of salt. There's just not enough people who have studied the language and the claims are so strong and unparalleled that we really ought to have more evidence between making any conclusive statements.

Re: How inevitable is the concept of numbers?

#85

Post by diegoperini »

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…

At t0, you say zero.

At t1, you say one.

...

At t7, you say seven.

Positive seven is what you say while your clock says t7. Negative seven is what you plan and "will" say at t7 while your clock still says t0.

Re: How inevitable is the concept of numbers?

#86

Post by Someone »

ddlutz wrote: post #3
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…

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).

Given how long it took to figure out that zero exists I don’t think it is natural, or at least a lot less natural than 1, 2, etc.

https://en.wikipedia.org/wiki/History_of_ancient_numeral_sys...: “Abstract numerals, dissociated from the thing being counted, were invented about 3100 BC”

https://en.wikipedia.org/wiki/0#History: “By 1770 BC, the Egyptians had a symbol for zero in accounting texts”

That’s over a thousand years, and that’s only the use of zeroes as a placeholder inside numbers. The use of a lone ‘zero’ symbol for the number zero seems to have taken over 2,000 more years (https://en.wikipedia.org/wiki/Brahmagupta#Zero)

Re: How inevitable is the concept of numbers?

#87

Post by osrec »

marcodiego wrote: post #82

There is a brazilian tribe (piranhã) that knows no concept of quantity besides one, two an d many. Also, in their language, verbs are not flexed related to time. This is probably du e to their lifestyle that needs no long planning, discussions about the past or managing m ultiple instances of the same resources.

Interesting. I would have thought that just by looking at their fingers and toes they'd arrive at a set of 1 to 10 at least...

Re: How inevitable is the concept of numbers?

#88

Post by Aardwolf »

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…

Do you mean the proof that 0.25 >= 0 and the proof that 1/e >= 0 count as the same one, because there's a more general proof that a set of values including those is >= 0? But then where do you draw the line? When can you consider 2 proofs different enough to count as different ones?

Re: How inevitable is the concept of numbers?

#90

Post by swagasaurus-rex »

mrfox321 wrote: post #73
slver wrote: post #8

I like that he keeps it open-minded, but I've thought a lot about this and to me the case for numbers being inevitable is decisively yes. Every adaptable system evolves to adapt to a changing environment by detecting "modes" (categories) and adapting to each mode. Then we start noticing categories come in instances. There's a tree, there are more trees. So now it's useful to count them... Then it's useful to have fra…

Your argument is tautological. Do we really know if adaptation is responding to discrete classifications of environmental pressure? There could be other ways to describe how systems adapt. You are ascribing a lot to how you think it emerged. I think your opinions are great, but I just have a hard time having beliefs about things that are as fundamental as numbers.

Another perspective could be that we are creatures emergent out of the physical and chemical processes of the universe.

There is some similarity or equivalence to some of these processes with mathematics, which allows us to model a subset of physics/chemistry with mathematics.

Because physics/chemistry can create us using rules that follow math (or even some undiscovered rules), this same physics/chemistry gives us the faculties of reason that lets us think about and apply math in some universal way

The physical, chemical, and mathematical substrate to which we are born allow us to even discover that the rules of math can be applied to some subset of the laws (or chaos?) of the universe.

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