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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?

#191

Post by Koshkin »

esarbe wrote: post #173
Koshkin wrote: post #130

Earlier quoted context omitted.

> Did we really know what “5” was then? What is five-ness really? Ah, but we do, and it’s very simple: it is what five cows an five fingers have in common in the most obvious sense. There’s no mystery there. Or, stated in another way, “fiveness is something you’ll be very sad not to see when you look at your hand.”

This presumes the existence of a category like 'cow' or 'finger'. I'm pretty sure there's nothing intrinsic about 'cows' and 'fingers'; these are artifacts of human cognition. I base reality, there are no such things as 'cows' and 'fingers'. Just systems with similar properties. And if you have to be very pedantic, there are not even discrete systems, since everything is linked by information. You cannot separate a r…

I don't know about that. Reality is strongly suggestive. (Of course you can argue that "reality" also is a figment of our imagination or, for example, it's not a thing in the first place, but I personally wouldn't go that far.)

Re: How inevitable is the concept of numbers?

#192

Post by Koshkin »

Radim wrote: post #176
esarbe wrote: post #173

Earlier quoted context omitted.

This presumes the existence of a category like 'cow' or 'finger'. I'm pretty sure there's nothing intrinsic about 'cows' and 'fingers'; these are artifacts of human cognition. I base reality, there are no such things as 'cows' and 'fingers'. Just systems with similar properties. And if you have to be very pedantic, there are not even discrete systems, since everything is linked by information. You cannot separate a r…

> You cannot separate a river from the ocean - it's part of the same system. > discrete things - these also just exist in our imagination What leads you to separate "imagination" from "exists"? Isn't imagination also "part of the same system"? Imagination definitely has the capacity to affect the world, so it's not obvious in what way it's "not real". Why do you draw a hard line there?

More to the point, the idea that "you cannot separate a river from the ocean" seems to be ostensibly wrong - otherwise we wouldn't have the (very useful in practice) notions of 'river' and 'ocean' in the first place. If you look at a map, the differences stand out pretty clearly - e.g., a river has a starkly different topology from that of the ocean. So, no, these differences are not just figments of our imagination, as, in particular, everyday practice shows.

In general, the failure to perceive emergent phenomena as something different from the particular substrate, and consider it separately - for instance, the failure to see how nature is not just a bunch of atoms moving around, has a name - reductionism. It is a form of intellectual blindness (not to be confused with the ability to think abstractly).

Re: How inevitable is the concept of numbers?

#193

Post by mejutoco »

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…

From what I read, negative numbers took a while to be accepted for the same reason, so you are not alone ;)

IMO negative numbers, complex numbers for circles, etc can be thought of as practical overloads. If we agree that the symbol minus means a debt, we can use it like that. We could agree, on *5 as meaning something, and if enough find it useful in 200 years we will teach it in school as obvious.

I think the negative number is obviously useful (you can add up positive and negative numbers and get a balance, it has an easy way to be plotted, etc)

P.S. another example is the Orbifold notation, from Conway. https://en.wikipedia.org/wiki/Orbifold_notation

Re: How inevitable is the concept of numbers?

#194

Post by dalmo3 »

esarbe wrote: post #189
Radim wrote: post #186

Earlier quoted context omitted.

But "world", "separate" and "parts" are all language concepts too. I feel the inconsistency in your (circular) argument is not getting through. That divisions are arbitrary and "exist just in our imagination" doesn't line up with your admission that imagination is real, and using words to describe it. That line between "real" vs "arbitrary / imaginary" is not as clear cut as you (unconsciously, apparently) draw it.

> I feel the inconsistency in your (circular) argument is not getting through. I guess it's not, since I'm not convinced that mine is a circular argument. My argument - to put is simply - is that we are all part of the same system and that there are no divisions. Without divisions, no numbers. I don't need any distinction between 'real' and 'arbitrary' for this to hold, that's a dichotomy you assume on your part.

In your world without numbers, do letters exist? Does the letter "n" exist? Is there any relation between "n" and "nn" and "nnn"? If so, how would you describe that relation?

Re: How inevitable is the concept of numbers?

#195

Post by tlb »

hypertele-Xii wrote: post #100

Saying that the universe is irreducible with pockets of reducibility is a total oxymoron. It's not "irreducible" if parts of it are reducible. "This wall is impenetrable, except for the holes there. Don't mind them I'm making a point here!"

He's using a technical definition of irreducibility [https://en.wikipedia.org/wiki/Irreducibility] for which the claim is true. It doesn't mean "cannot be expressed more concisely at all".

Re: How inevitable is the concept of numbers?

#198

Post by Radim »

esarbe wrote: post #189
Radim wrote: post #186

Earlier quoted context omitted.

But "world", "separate" and "parts" are all language concepts too. I feel the inconsistency in your (circular) argument is not getting through. That divisions are arbitrary and "exist just in our imagination" doesn't line up with your admission that imagination is real, and using words to describe it. That line between "real" vs "arbitrary / imaginary" is not as clear cut as you (unconsciously, apparently) draw it.

> I feel the inconsistency in your (circular) argument is not getting through. I guess it's not, since I'm not convinced that mine is a circular argument. My argument - to put is simply - is that we are all part of the same system and that there are no divisions. Without divisions, no numbers. I don't need any distinction between 'real' and 'arbitrary' for this to hold, that's a dichotomy you assume on your part.

Yes, we may be all part of the same whole. But there definitely are divisions – you are manifesting them with your own words. QED.

Let me try differently. Your premise seems to be that "base reality" (your words) is a single teeming interconnectedness, indivisibly unique, from which it follows "there are no discrete systems", no categories, from which it follows that counting discrete things is an "artifact of human cognition".

Correct? Did I get your position right?

I simply pointed out that human cognition / imagination, including language and categories and logic and numbers, is as much a part of the base reality (that same teeming interconnectedness) as anything else. You manifest your words = they capture a pattern, patently recognizable from other patterns, transmittable (e.g. to me), with a potential to affect me and others and our future.

The world being interconnected doesn't mean it's undifferentiated. All work is still ahead of you in showing that categories and words are somehow "not intrinsic" (again, your words). Yes they may be a teeming that refers to other teeming, but you haven't shown that's anything special, worthy of singling out as extrinsic.

Re: How inevitable is the concept of numbers?

#199

Post by dhosek »

andai wrote: post #140
dhosek wrote: post #135

When I used to teach a class called “Mathematics for Liberal Arts Majors,” the first class began with having students “draw six.¹” I refused to give more explicit instructions. It was fascinating to see the results that came out of this, along with the ideas about numbers that it sparked. 1. I have to confess that the choice of six was not arbitrary.

Why six?

Mathematically, 6 is the most interesting small number. For example, one might represent it as

  XXXXXX

  XXX
  XXX

  XX
  XX
  XX

  X
  XX
  XXX
or the sides of a cube, hexagons or six-spoked wheels. Occasionally someone would draw a rectangular prism with sides of length 1, 2 and 3. I often had students who might also draw 六 or 陸 or ٦ or ৬ (I had a diverse student body).

Re: How inevitable is the concept of numbers?

#200

Post by vageli »

esarbe wrote: post #173
Koshkin wrote: post #130

Earlier quoted context omitted.

> Did we really know what “5” was then? What is five-ness really? Ah, but we do, and it’s very simple: it is what five cows an five fingers have in common in the most obvious sense. There’s no mystery there. Or, stated in another way, “fiveness is something you’ll be very sad not to see when you look at your hand.”

This presumes the existence of a category like 'cow' or 'finger'. I'm pretty sure there's nothing intrinsic about 'cows' and 'fingers'; these are artifacts of human cognition. I base reality, there are no such things as 'cows' and 'fingers'. Just systems with similar properties. And if you have to be very pedantic, there are not even discrete systems, since everything is linked by information. You cannot separate a r…

> This presumes the existence of a category like 'cow' or 'finger'. I'm pretty sure there's nothing intrinsic about 'cows' and 'fingers'; these are artifacts of human cognition. I base reality, there are no such things as 'cows' and 'fingers'. Just systems with similar properties. And if you have to be very pedantic, there are not even discrete systems, since everything is linked by information. You cannot separate a river from the ocean - it's part of the same system.

When I cut my finger the cow does not share in my pain. When we kill the cow for its meat, we do not share in its pain. That the cow becomes part of us through its consumption does not seem to invalidate this point—discrete systems do exist in our experience.

Of course, if you zoom out far enough you might refer to the sum of those discrete parts as some singular, complex system, but it seems the human experience is fairly limited in exposing this subtlety (not to mention that it is often useful to discuss the parts themselves without considering their relationship to the entire universe).

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