Would numbers exist if no one had ever invented them?

[polymath]

I know most people don't like math, but this thread is about the philosophy of mathematics.

Math can be approached from two different philosophical viewpoints:

1. Mathematics is a man-made logical construction, and numbers exist only as far as someone thinks about them.

2. Numbers are natural objects with independent existence, and the laws of addition, multiplication etc. governing the behavior of numbers are of similar status as the laws of physics governing the behavior of matter and energy.

One can also think of math in a utilitarian way, i.e. it's a useful tool that can be used to describe natural phenomena.

Both viewpoints have their proponents... I personally would like to think that numbers are natural and would exist even without anyone thinking about them. The reason for this is that even though numbers are supposedly man-made, we often have no a priori knowledge of how they behave in a particular situation.

Let me demonstrate this with a relatively familiar example: Let us choose some complex number c and then construct a sequence of numbers z₀, z₁, z₂, z₃, ... such that z₀=0 and z_n=(z_n-1)²+c . Depending on the choice of number c, the sequence can either grow without bound, with numbers z_n getting larger and larger with increasing n, or they can stay within some bounds.

Already in the 19th century, it was known that it's very hard to characterize the set of numbers c for which this sequence doesn't grow without bound. They couldn't draw a picture of the set in the complex plane, because they didn't have computers. Only in the mid-20th century it became possible to draw a picture:

Everyone has seen this picture... It's the familiar fractal Mandelbrot set. Probably everyone who has done some programming has once written a code that draws a picture of MS.

Before Benoit Mandelbrot got the the idea to draw this with a computer, people had no idea how complex the structure of MS would be. If one zooms to the border of the set, one finds ever finer details at all scales.

Finding out that numbers can exhibit this kind of unexpected and complex behavior feels a lot more like studying a phenomenon of nature than playing with a construction you invented yourself. (BTW in nature there are lots of chaotic phenomena that are related to fractals)

Of course this is not a proof that numbers are natural, it's just an attempt to make it seem plausible.

What do you think, are numbers artificial or natural?

 

[I NUK3D U]

Looking to music and further into audible frequencies will tell you that numbers and integer mathematics exist in nature without human interpretation.

 

[Psyduck]

I think only natural numbers are "real." The rationals can be constructed from the naturals, Q = NxN/~, and the reals can be constructed as limits of cauchy sequences of rationals. And the complex numbers can be constructed as the algebraic closure of the reals.

“God made natural numbers, all the rest are made by Man.” Leopold Kroneker (1823 - 1891)

 

[polymath]

“God made natural numbers, all the rest are made by Man.” Leopold Kroneker (1823 - 1891)

I remember some famous mathematician paraphrased that as "God created power series, all the rest is work of Man". Can't remember right now who it was...

 

[jamaica0535]

In an exploration of the idea's in mandala and the act of creating them, you start to realise that these forms follow certain ratios and proportions and that geometry can be seriously trippy shit. Great form of meditation though, an artistic outlet for the practice. Try to let go, stop trying so hard to make something specific, every great painting started with a line or brushstroke. Basically you pay attention to detail, symmetry, and go one piece at a time to allow it to become whatever it becomes. The result is a sort of over elaborate Rorschach, instead of you interpreting random patterns into things you have something that came from your hand.

 

[qwe]

The reason for this is that even though numbers are supposedly man-made, we often have no a priori knowledge of how they behave in a particular situation.

we explore our language and semantics just like we explore our mathematics and numbers. we have no a priori knowledge of what sort of picture our words will paint in a particular syntax structure, and we often go into, say, a debate, not knowing where the words will lead us.

i would say that mathematics is essentially a language, and a language is essentially a computational tool for our interaction with reality. without humans or other computational structures, numbers don't exist in and of themselves.

 

[Quantum Perception]

I think their artificial. They come from the way our minds work.
What is a ONE? How can things be individuated if all wholes are really pieces all the way through?
The way i see it, numbers are arbitrary markers for a computational mind. They cant exist on their own because their nature is to be observed and used.

 

[Quantum Perception]

we explore our language and semantics just like we explore our mathematics and numbers. we have no a priori knowledge of what sort of picture our words will paint in a particular syntax structure, and we often go into, say, a debate, not knowing where the words will lead us.

i would say that mathematics is essentially a language, and a language is essentially a computational tool for our interaction with reality. without humans or other computational structures, numbers don't exist in and of themselves.

I agree. Some say we dont need a posteri knowledge of numbers so that makes them a priori. This would mean the days of the week would also fall into this category because they do not exist out there, and we come to understand them in the mind.

 

[polymath]

we explore our language and semantics just like we explore our mathematics and numbers. we have no a priori knowledge of what sort of picture our words will paint in a particular syntax structure, and we often go into, say, a debate, not knowing where the words will lead us.

i would say that mathematics is essentially a language, and a language is essentially a computational tool for our interaction with reality. without humans or other computational structures, numbers don't exist in and of themselves.

I'd argue that mathematics differs from ordinary language... The need for language was already there before it was developed. On the other hand, many seeminly abstract, artificial and 'useless' inventions have many times been made in the field of pure mathematics, only to find practical use in physics decades or centuries later. An example of this is noncommutative algebra, which was developed in the 18th century IIRC. Much later, in early 20th century Werner Heisenberg found out that the basic postulates of quantum mechanics have to be represented in terms of noncommutative quantities.

 

[alasdairm]

What is a ONE? How can things be individuated if all wholes are really pieces all the way through?

i don't think it's that hard to understand the handy concept of referring to the collection of atoms in a discrete apple as "one apple". it's easily understood and it's useful.

if we were not here, this many apples:

plus this many apples

makes this many apples

.

and this many

plus this many

makes this many

the fact is that we just happen to have a label for this many

which is called "two".

alasdair

 

[Quantum Perception]

i don't think it's that hard to understand the handy concept of referring to the collection of atoms in a discrete apple as "one apple". it's easily understood and it's useful.

if we were not here, this many apples:

plus this many apples

makes this many apples

.

and this many

plus this many

makes this many

the fact is that we just happen to have a label for this many

which is called "two".

alasdair

But these ones and twos are arbitrary, they don't exist with out an observer contemplating them. What makes the apple a whole and not a piece of the tree? An observer not anything in-of-itself.

 

[L2R]

natural order =/= "numbers"

 

[ebola?]

Being as such never exists as we know it in the absence of our intervention within it.

ebola

 

[echo off]

like the cat in the box
they would both exist and not exist
but we can't say for sure which

 

[alasdairm]

But these ones and twos are arbitrary, they don't exist with out an observer contemplating them.

that's basically the question of objective reality (or otherwise). i think that's a different discussion.

What makes the apple a whole and not a piece of the tree?

well, it's both. as i said, i don't think it's that hard to understand the handy concept of referring to the collection of atoms in a discrete apple as "one apple". it's easily understood and it's useful and it's a lot shorter than having to discuss the entire nature of being every time one wanted to buy/eat/etc. "an apple".

alasdair

 

[Quantum Perception]

^Well the question from the op is related to objective reality. And sure its easy for us to see things the way they are to us, but when the question is asked about how things are without us, the topic will get deep.

 

[modern buddha]

If it hadn't been existed to this point, it would be thought of now. Something so natural will enter a language sometime.

 

[Pindar]

Well there are two theories on the theory on number.

The one given by Bertrand Russell:

"We may now go on to define numbers in general as any one of the bundles into which similarity collects classes. A number will be a set of classes such as that any two are similar to each other, and none outside the set are similar to any inside the set. In other words, a number (in general) is any collection which is the number of one of its members; or, more simply still:

A number is anything which is the number of some class. [Collections of twos, threes, and fours]

Such a definition has a verbal appearance of being circular, but in fact it is not. We define “the number of a given class” without using the notion of number in general; therefore we may define number in general in terms of “the number of a given class” without committing any logical error.

Definitions of this sort are in fact very common. The class of fathers, for example, would have to be defined by first defining what it is to be the father of somebody; then the class of fathers will be all those who are somebody’s father. Similarly if we want to define square numbers (say), we must first define what we mean by saying that one number is the square of another, and then define square numbers as those that are the squares of other numbers. This kind of procedure is very common, and it is important to realise that it is legitimate and even often necessary."

He asserts that number has no being outside of it being but a mere mathematical convenience, or as he called it "a logical fiction", since Being itself is under erasure ("The first thing is to realise why classes cannot be regarded as part of the ultimate furniture of the world. It is difficult to explain precisely what one means by this statement...If we had a complete symbolic language, with a definition for everything definable, and an undefined symbol for everything indefinable, the undefined symbols in this language would represent symbolically what I mean by “the ultimate furniture of the world.")

The second theory is by Alan Badiou:

"The analysis of numbers, then is not merely a matter of formal representation, it is a matter of realities. A number is neither part of a concept, nor an operational fiction, nor an empirical given, nor a constituent or transcendental category, nor a syntax, nor a language game nor even an abstraction from our idea of order. Number is a form of being. From this it follows that we are either presented by a one or a multiple, zero being itself void and since being must necessarily be reduced to one from the multiple it cannot be that this is the point of genesis."

I'd like to think that this all goes back to the concept of whether there exists an objective reality outside of our minds, since number is defined the way some may say language is defined, as a series of assemblages, the zero existing by way of logical contradiction or existing solely because the number system has no way of portraying said contradiction except by way of the number zero.

 

[echo off]

Pindar, you should probably provide the source for what seems to surely be a cut and paste job.

and then it'd be nice to see at least some summary... i like the second theory by alan badiou more. i mean, not like more... i believe... the first theory is a bit of mumbo jumbo to me (too convoluted to read).

as i see it, numbers existed since the beginning of everything. and now that i've had to read a convoluted reply to your question i find myself questioning your question... and it almost seems less aimed at 'would numbers exist if no one had ever invented them?' and more aimed at 'how do umm i exist?" there was nothing, and then there was something, which could be counted as 1 x "everything" or sub-divided into several kinds of categories of data such as 1/2 of "everything" or "how many words i said used to figure out 1/2 of everything"... and whether humans exist to put labels on this isn't really important... unless you are asking a different question, as I suggested it feels like you are.

now that i've confused people XD if something created everything, then perhaps numbers were invented, at which point you probably shouldn't trust them as much as you feel you can, unless you have faith that whatever created numbers wants to favor you, at which point go for broke, but i think you're a bit egotistical, and will partake no further in this conversation... with myself... good day, self.

 

[Pindar]

I apologize, just wanted to portray the philosophical basis for each viewpoint. The first is the basically that numbers are just collections that have no basis in objective reality, like 5 books and 5 pencils are similar to each other but not similar to 6 books or 4 pencils; they are just used to explain objects more better. The second, which reflects a whole different vein, from analytic philosophy of Russell to the structuralist/post-structuralist of Badiou, states that all structure emulates Being, therefore number as a structure should be treated in the same manner.