Earlier quoted context omitted.
But we can make new words.
Only an (unbounded) finite number of them.
how so?
How inevitable is the concept of numbers?
171–180 of 211 posts
Re: How inevitable is the concept of numbers?
#172Earlier quoted context omitted.
Almost all of modern mathematics is not about numbers, although a lot of the objects studied can eventually be related to numbers in some way.
It's not about numbers, but it's built on concepts which wouldn't exist without numbers as a foundational abstraction. There may be entire categories of representational concepts which we abstract poorly, or don't abstract at all, or possibly don't perceive at all - because from our point of view the relationships are so complex and remote they're effectively invisible.
I've been thinking about this since I read about the Pirahã people. The concept of number enables engineering and business (accounting) (and perhaps timekeeping). Someone unfamiliar with numbers would find it very difficult to function in modern society and a group of such people wouldn't build modern civilization (unless they discovered numbers, which someone did at some point, of course). Now, here's an example of my crazy dream: imagine the possibility of currently inaccessible ways of perceiving or thinking, that, if they were to become available, would enable a four year old child to gain an understanding of, for example, elementary particle physics from zero to phd level in mere minutes. Of course, that way of thinking might be so different that said understanding might make no use of (our current) mathematics or the concept of elementary particle at all.
Re: How inevitable is the concept of numbers?
#173I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
> 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 river from the ocean - it's part of the same system.
So, given that there are no categories to put discrete things into - they just exist in our imagination - and there are probably not even discrete things - these also just exist in our imagination - the question of five-ness is not really answered.
Re: How inevitable is the concept of numbers?
#174Earlier quoted context omitted.
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.
It is an abstract proof of a generator for concrete proofs of specific assignments to variables. The potential is uncountable, but only a countable subset will ever be invoked.
I don't know what it really means for a proof to be invoked, and I also don't really like the idea of separating proofs into concrete and abstract proofs. Either it proves something or it doesn't.
Re: How inevitable is the concept of numbers?
#175I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
Yes; the difference between token and denoted concept. They are not the same thing (as René Magritte probably wanted to say with his "This is not a pipe" painting). In computer contexts: each representation of 5, in whatever base or language would point to the same element in a lookup-table.
But for us humans, the tokens and concepts blend to some degree: maybe not so much with regards to numerals (although there seem to be exceptions ...). So even if 'Amour', 'Love' and 'Liebe' essentially point to the same concept, we might conceive of them differently, ever so slightly, depending on the language we chose to use.
Which gives an inkling of the vast difference between computers and humans, at the current stage.
Re: How inevitable is the concept of numbers?
#176Earlier 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…
> 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?
Re: How inevitable is the concept of numbers?
#177Earlier 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…
Also, the concepts "artifacts", "human", "human cognition" and "artifacts of human cognition" are all example artifacts of human cognition.
As is "circular", "reasoning" and "circular reasoning", as well as "irony".
Re: How inevitable is the concept of numbers?
#178I guess one of the most fundamental difference between Wolfram’s model for fundamental physics and traditional physics is that Wolfram’s doesn’t have the concept of measure or of continuum at the fundamental level. Space and time, according to Wolfram’s model of the universe, are emerging properties of ‘the network’. Without such things as space and measures, there is no numbers in the fundamental “equations” that dr…
It's so sad to see that he is taking an interesting question and turns it into a sales pitch of his computational universe theory. It's as if Stephen Wolframs mental horizon starts getting more and more restricted over the years since he tries to frame everything he sees in terms of his physical theory.
Or maybe he has an interesting theory he understands deeply, and think its valuable to show how this theory can give insight to the problem in question. Why be so cynical?
Re: How inevitable is the concept of numbers?
#179Earlier 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?
I'm not drawing a hard line at all; yes, imagination is part of the same system. And of course it has the capacity to affect the world.
I'm just pointing that without arbitrarily dividing the world into separate parts, numbers don't arise.
Re: How inevitable is the concept of numbers?
#180Earlier quoted context omitted.
> 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?
I'm not drawing a hard line at all; yes, imagination is part of the same system. And of course it has the capacity to affect the world. I'm just pointing that without arbitrarily dividing the world into separate parts, numbers don't arise.
> I'm just pointing that without arbitrarily dividing the world into separate parts, numbers don't arise.
I agree. Though I think a stronger statement is also true: Without "arbitrarily" dividing the world into separate parts, non-trivial thought is not possible.
Assuming by "parts", you mean "categories", I think my statement is still true. And, once we have categories, we can use numbers to compress information, which in terms allow us to perform more complex computation/though with our limited computational power.
As to how arbitrary our categories are, one could argue that some are hardwired to our DNA, as claimed by Chomsky(1).