The Natural Base?

[BT Industries]

I've been thinking about this for a while and my question is:

What is your opinion on there being a natural base or counting system?

Taking a Platonic view of the universe, is there any 'system' to the universe regarding numbers, not just 'there are x apples'?

I mean, we might just be measuring the universe wrong.

Most people would probably default to either base one or binary (base two), but why? Is there any reason that the numbers one and two are special? This brings us to my second questions:

Integers. Why are they considered different and get precedence over other numbers? What is even the definition of an integer, I mean even though the number 5 doesn't have any decimals in base ten, if written in base Pi, it's irrational. It might also be that x whole apples are necessarily an integer number of apples, but the operating word is whole and it might just be us that are centred around whole, full and integer, even though they might not be different  or special in any way.

-T

 

(P.S. Maths counts as science in this forum right?)

[Bill Phil]

What's more important is the idea of numbers and counting with them. The bases and systems used are arbitrary, so long as they can be successfully used.

Since logarithm base e is called the natural logarithm, then the natural base would be e... But that's a base in the context of exponents and logarithms.

[LN400]

1 hour ago, BT Industries said:

I've been thinking about this for a while and my question is:

What is your opinion on there being a natural base or counting system? [Q1]

Taking a Platonic view of the universe, is there any 'system' to the universe regarding numbers, not just 'there are x apples'? [Q2]

I mean, we might just be measuring the universe wrong.

Most people would probably default to either base one or binary (base two), but why? Is there any reason that the numbers one and two are special? [Q3] This brings us to my second questions:

Integers. Why are they considered different and get precedence over other numbers? [Q4] What is even the definition of an integer, I mean even though the number 5 doesn't have any decimals in base ten, if written in base Pi, it's irrational. It might also be that x whole apples are necessarily an integer number of apples, but the operating word is whole and it might just be us that are centred around whole, full and integer, even though they might not be different  or special in any way.

-T

 

(P.S. Maths counts as science in this forum right?)

My 2.14632... cents:

A1: I don't think any number is more natural than others, What it comes down to is notation and only that. The value of a number is independent from whichever symbols we choose. A base number is there for no other reason than to allow a consistent system of notation.

A2: If the physicists are correct then there is a system. By that I mean there are laws of nature that are the same wherever you go. That's system right there. The maths in science is not about describing some kind of absolute reality but to allow us to predict future events when the initial conditions are known. They create models of the real world, models with a limited field of applications. So far, physics have shown some staggering promise and many mathematicians and physicists are themselves amazed at how well maths can be used to describe and predict. This speaks for a systematic universe.

 A3: 1 and 2 are not special in base 2. 0 and 1 are. There are 2 symbols for a very good reason. If you think of them not as numbers but 2 different  logical statements, then you have here the minimum number of logical statements that still allows conveying of information. Everything can be expressed using 2 and only 2 symbols, the complete work of Shakespeare can be written in 0s and 1s. Less than 2 statementss and it is impossible to convey information. 2 is therefor the lowest number of digits one can construct a consistent number system from. It doesn't mean it's more natural or easier to use though,

A4: To look at integers, one need to look at 2 other values first: The Unit and no units. An integer is nothing more than a value constructed entirely of units. If pi is your base, then pi/pi becomes the unit and since that is constructed of units, then it is an integer and 5 isn't. Pi/pi can then be written using the symbol 1 but 5 would be a symbol for an irrational number just like pi is in base 10. Integers aren't more special than that but once you have a set of integers, then they will share some properties not shared by any onther, non-integer number.

Another thing is, we have created a universe of mathematics based on how we percieve the world. We see 1 apple, 34 sheep. We count units because that's how our brains and sensory organs are wired. It allows us to function in a society. We understand that 1 sheep is not 2 sheep, we understand that if we have some sheep and lose a few then we will have fewer sheep than before. We count units and we created systems that allow us to note down on clay or paper how much of this and that we have. One such system is the base 10 system we use, with symbols slowly getting acceptance before being agreed on they work for our purposes.

 

Anyway, just my take on it.

[kerbiloid]

Because apples have clear borders. When there are hundreds of apples, we count them by weight and not necessary in integer numbers.
Integer numbers mark clicks when you cross a border.

Edited September 15, 2016 by kerbiloid

[wumpus]

Since base 2 can't be reduced, it would presumably count.  But that is hardly the only way to represent numbers.

Consider the Godel base system: describe a number by multiplying prime numbers together (I think his system used only single primes and thus was sparse, perhaps some sort of inductive means to describe the numbers of each prime...).  Also using only ones and zeros hardly forces you to use binary.  Binary is simply the most efficient means to code numbers with ones and zeros if you want to describe integers equally.  But try looking up arithmetic coding (typically used in data compression, especially compared to Huffman compression): you can encode different bases (not just powers of 2) efficiently in binary (of course there will be leftovers for finite numbers when not using powers of 2).

I really don't think this can be answered (unless you really pull off a base-e system.  And then only if you manage to pull it off without "cheating" by using arithmetic compression).  But I see no reason to believe that adding various powers of a single number times various coefficients is the only 'natural' number system.

[cubinator]

4 is the base for life on Earth, so that might be something.

[magnemoe]

43 minutes ago, wumpus said:

Since base 2 can't be reduced, it would presumably count.  But that is hardly the only way to represent numbers.

Consider the Godel base system: describe a number by multiplying prime numbers together (I think his system used only single primes and thus was sparse, perhaps some sort of inductive means to describe the numbers of each prime...).  Also using only ones and zeros hardly forces you to use binary.  Binary is simply the most efficient means to code numbers with ones and zeros if you want to describe integers equally.  But try looking up arithmetic coding (typically used in data compression, especially compared to Huffman compression): you can encode different bases (not just powers of 2) efficiently in binary (of course there will be leftovers for finite numbers when not using powers of 2).

I really don't think this can be answered (unless you really pull off a base-e system.  And then only if you manage to pull it off without "cheating" by using arithmetic compression).  But I see no reason to believe that adding various powers of a single number times various coefficients is the only 'natural' number system.

Base 2 is not very practical for brains like ours, its perfect for digital stuff who is on or off. Note that flash memory tend have four values in an cell making it base 4. 
For us its base 10 as we started with counting on fingers, most cultures tend to use base 5, know from roman numbers but lots of civilizations had the same base 5 system. 
Base 10 is better as it gives shorter numbers while its easy to remember the numbering. Roman numbers and the other systems are also hopeless for doing math. 
Was the Arab numbers developed by traders and accountants in India? makes some sense. 

An alien is likely to follow this, number of fingers and you want an reasonable high number, you could get away with three fingers if flexible like three thumbs however base 3 would be low so you go for 6 or perhaps 12. 
OT / OT Fermi tread crossover/  could an bad numbering system hurt you say you discover printing and push for literacy while using something as impractical for calculation as roman numbers. 
Or would people working with numbers develop something more like the arab numerals base 8-20 is irrelevant for their own use, probably convert before doing public presentation like US scientists use the metric system outside of public stuff. 

[LN400]

8 hours ago, magnemoe said:

Base 10 is better as it gives shorter numbers while its easy to remember the numbering. Roman numbers and the other systems are also hopeless for doing math. 

The huge difference between base anything systems as we know them, and Roman numerals is we have a positional system and the Romans, at best, had a pseudo-positional system. That's the difference that makes it possible for us to do calculus. Any Roman trying to invent calculus would have be droven mad.

Another thing is, there are societies today where the people traditionally had no concept of numbers greater than say 3. To them, the entire number universe consisted of these numbers: One, two and three. Any number greater than 3 was not considered a number but was to them "many" a bit like we use infinity. Negative numbers, don't be ridiculous. Fractions, forget it, has nothing to do with numbers, they would think. They are still modern humans but their culture took some turns that eventually led to this.

EDIT: Perhaps another interesting question would be: Are positional systems the way to go, or could there be other systems that inherently includes the properties of a positional system while also possessing other properties that could be extremely useful in describing the world around us?

Edited September 16, 2016 by LN400