What is a number?

[Toadenalin]

I made a comment on this thread to try and establish that some mathematical proofs were not beyond question. I made a bit of a throwaway comment to the effect of "No mathematician knows what a number is". I was, quite rightly, called on this by RubberTrucky, and I thought it might be interesting to discuss because;
- It has nothing to do with God (for a change)
- It probably won't degenerate into people flinging the word 'logical fallacy' around without really understanding what they're doing
- I don't really understand it fully, so there's every chance someone will be able to actually help me reach an opinion.

The Sorites Paradox
I'm going to claim that number systems are inherently subject to what is called the 'Sorites Paradox', and as such we can never say "This concept is a number, and that concept is not a number". The best we might be able to do is something like "I believe this concept is closer to being a number than not a number, but you have every right to disagree"

"The Sorites Paradox arises out of vague predicates" - if I say to you, "Here is a heap of sand", you could legitimately say to me "I don't think that is a heap, I think that is a pile (or 'a few grains' or 'a mountain')". To put it another way, if I kept adding grains of sand to a table, when would you tell me I had gone from a collection of grains of sand on a table to a 'heap' of sand. Two grains? Ten grains? One hundred grains? Clearly, this is not a question with a rational answer - if you say "Forty" and I say "Fourty-one" there is no external judge we can go to. From this, I conclude that (in the sense I want to use it in this thread) we do not know what a 'heap' actually is. Sure, we can describe it in terms of synonyms - "It is like an arrangement, but more haphazard" or "Similar to a collection, but less intentional", or we can use it in speech, but we have no, specifically, logically designating way of distinguishing a 'heap' from a 'non-heap'.

The Sorites Paradox applied to numbers

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.
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Natural numbers, integers, rational numbers (skip this if you understand the definitions)
So if you agree with the above paragraph (and I should probably point out here, there's no particular reason why you have to, it just seems pretty rational to me), you might already see where this is going in terms of numbers.

Sure, we can all quite easily concieve of the 'natural numbers' as being numbers (1, 2, 3 .... etc). These are quite nice because they correspond to things we already can see and touch in the real world
- How many eyes do you have (2)
- How many friends do you have (maybe 10 or 15 close ones)
- How many atoms are there in the universe (I don't know, but I'm damn sure it will be a big positive number)

We can probably make sense of 'integers' too (the 'natural numbers' and all the negative numbers). These are less like something that corresponds to the real world, but we can still make sense of being "£1000 in debt", so it shouldn't bother us unduly. 'Rational numbers' are one number divided by another (or 'fractions') and present a similar kind of problem - we don't use them every day, but they correspond to more or less familiar things (Betting on the toss of a coin is an application of 'rational numbers')

.Irrational numbers, complex numbers (This is where the mindfuckery begins)
Irrational numbers are where things get brutal. These are any number on the 'number line', and include 'pi', 'the square root of -2' and so on. Human cannot correctly concieve of these numbers, since they go on forever without repitition. The best we can say is something like "Pi is a bit smaller than 3.15, but bigger than 3.14" and repeat this for different levels of abstraction.

This means that when we talk about the number 0.999... (recurring, meaning the 9's continue forever), we are actually expressing the same concept as the number '1'*. Does this mean we have two different numbers, '1' and '0.9999...' expressing the same concept? Does it mean we have two identical numbers written differently? If they are identical, why do we concieve of them differently?

*Proofs

Complex numbers are any number which involves an 'imaginary' component. I hate them, and they are the bane of my existence doing A-level maths. They are the outcome if you square a negative number, and as far as I can see do not really correspond in any meaningful way to concepts we would use when we are talking about numbers.

Really un-numberish concepts

If 'The biggest number I can write in ten seconds' is a number (which I think it must be, based on what I've described above), then since I can write "The biggest number I can write in ten seconds, plus one" in less than ten seconds, is that a number? It would seem to be infinitely big, since if the biggest number I can write in ten seconds is (say) 10, then the biggest number I can write in ten seconds, plus one is 11. But since 11>10, 10 is not the biggest number I can write in ten seconds, 11 is, and so on ad infinitum.

If 'the number of people on the earth at the moment' is a number, then 'the square root of the number of people on the earth' should also be a number. Though the actual number of people on the earth keeps changing, 'the square root of the number of people on the earth' will always refer to something real and tangible. In fact, you could write this as F(x), where F is a function denoting 'the square root of x' and 'x' is 'the number of people on the earth. But if we agree that x can take any value whatsoever (and why not?) then it looks very much like we are admitting a function over a domain as being a number.

Finally, wikipedia has some more types of numbers that I don't even begin to understand at the bottom of its page on a number. While I wouldn't recommend it as a real source, it might serve as a stepping stone if any of you would like to look at this in more detail

Tl;dr: Start reading from 'Really un-numberish concepts'. If even that is too long for you, go somewhere else.

[RubberTrucky]

I will have to leave now, so I'll make it quick and I will repeat probably what is posted here.

Natural numbers= everything you can count, like the number of apples in a basket, or the amount of people in a city. They continue one by one. So if you have a natural number N, you can always have 1 more of it and the result will be N+1 (the successor)

Integers: now take the numbers and you can sum them. This goes like 5 people and another group of 6 people make a total of eleven people. One might ask if I have a group of 5 people, how many people do I need to make a group of 11 people? Of course, these are 6 people. So we can count the absence of people and call it -6. Substraction will then result in counting absences and basically every natural number gets a converse partner, which cancels them out. N gets -N so that N+(-N)=0.
(note, zero, oddly enough is defined years after people were counting)

Rationals: Now we're talking fractions. A part of a given object, a half, or maybe a third or 1/3669. We can easily imagine them. This gives rise to the rational numbers (together with their 'absence' counterpart. Special is that if you combine a lot of them, you will always find an integer.

Real numbers:
Originated from The Pythagorean theorem. They found that for numbers a and b as sides of a triangle a number c would make the third side such that a^2+b^2=c^2. Then they noticed that if a=b=1 a number should exist such that it's square becomes 2. However you can prove no rational can ever exist to have this property. So there were 'gaps'. You could always estimate the root of 2 by a decimal in which you cut of the tail. So you can just fill in ever cipher in the tail and get something closer. In the limit you get the root of 2.

So a construct for the real numbers is taking a sequance of rationals such that the points lie closer to eachother as you reach further in the sequence (Cauchy sequences) such a sequence can be represented as a real number. In that way you fill in all the gaps in the rationals and get something called metrical complete, which we call the real numbers.

As for ciomplex numbers, you invent a number i such that i^2=-1. then you write every complex number as a+ib where a and b are real numbers. This results in the set of complex numbers.

But I should be going now, I'll certainly will enter again later.

[Blue-Bullet-Bill]

Ow.....
I hope you guys can pay for the clean-up of exploded brain fragment surounding my computer.

[RubberTrucky]

At 6/17/09 12:40 PM, Toadenalin wrote:
This means that when we talk about the number 0.999... (recurring, meaning the 9's continue forever), we are actually expressing the same concept as the number '1'*. Does this mean we have two different numbers, '1' and '0.9999...' expressing the same concept? Does it mean we have two identical numbers written differently? If they are identical, why do we concieve of them differently?

Here is something interesting happening (it's partially in a proof on wikipedia). Read my post earlier. remember. Read what I say about the construction of the real numbers as Cauchy sequences. This is what happens. When you write 0.999999... you're implying that 1 is the limit of the sequence
0.9
0.99
0.999
0.9999
0.999...
So you're just representing 1 as this Cauchy sequence. Naturally an easy way to make a rational Cauchy sequence for a rational number q would be to take the sequence (q,q,q,q,q,...) this constant sequence is clearly Cauchy and it 'converges' to 1. Two sequences x_n, y_n are taken to be the same as the sequence (x_n-y_n) converges to zero. In case of the constant sequence of 1 and the sequence 0.999999... it is clear that it does so. So mathematically the equality is senseful.

Complex numbers are any number which involves an 'imaginary' component. I hate them, and they are the bane of my existence doing A-level maths. They are the outcome if you square a negative number, and as far as I can see do not really correspond in any meaningful way to concepts we would use when we are talking about numbers.

Complex numbers are very interesting. they are constructed such that EVERY polynomial has a complex root. So you can write every polynomial of degree n as
(z-z_1)(z-z_2)(z-z_3)...(z-z_n).
And this is a key feature in mathematics (I mean this is REALLY big!!!)

Besides this, the notation exp(ix) is also very important in solving oscilatory problems and differential equations (like for instance currents on an AC net in electricity)
Exponentials work better then goniometric functions.

[Toadenalin]

At 6/17/09 05:03 PM, RubberTrucky wrote:

At 6/17/09 12:40 PM, Toadenalin wrote:

So mathematically the equality is senseful.

That's actually really interesting - I'd reccomend people read your first post first though, it explains rationals better than I do.

It seems like defining numbers in terms of a Cauchy sequence (and therefore limits) might run into problems though. Although it works great for finding the limit of a function, I can think a case where limits might not be helpful for defining a 'number'.

The limit of the sequence (n_2) = 1/(n_1) + 1 is 1. No problem there - as n becomes larger and larger, 1/n limits towards 0, so the sequence converges on 0+1, which is one. But imagine an even more simple sequence (n_2) = (n_1) + 1. This limits to infinity, which is clearly not a number in the same way '1' is. For starters, no matter what value of (n_1) you use, you will always be infinitely far away from your limit. Does that mean that (n_2) = (n_1) + 1 isn't coherant? Equally, a function like sin(x) will not converge at all.

This seems to me like the concept which is manipulated in a sin-type function cannot be a number in the way you describe it, because both functions resist Cauchy analysis.

Just to muddy things further, does the phrase "7 during the daytime and 8 during the nightime" express a number?

I'm not a mathematician, so please forgive any glaring errors!

[RubberTrucky]

At 6/17/09 06:47 PM, Toadenalin wrote: Although it works great for finding the limit of a function, I can think a case where limits might not be helpful for defining a 'number'.

No it does work for defining a number. it all comes down to approximating it, describing which numbers are really close to it, eventually zooming in on it totally.

The limit of the sequence (n_2) = 1/(n_1) + 1 is 1. No problem there - as n becomes larger and larger, 1/n limits towards 0, so the sequence converges on 0+1, which is one.

Hmm, this is recursion, it defines a sequence, but it's not really a necessary way to construct a sequence. Mind that. (so if I took random numbers, i could also get a sequence. But yeah, it can be recursive.
Note however that your convergence as a recursion doesn't approach 1. it should go to the golden section.

But imagine an even more simple sequence (n_2) = (n_1) + 1. This limits to infinity, which is clearly not a number in the same way '1' is.

And here, I bring in the definition of a Cauchy sequence: the distance between points in the 'tail' of the sequence need to become arbitrarily small. The given sequence does not have this property, since the distance between two points is always at least 1.
Formaly, a sequence x_n is called a Cauchy sequence, if for every strict positive number e, you can find n_0 such that x_n-x_m in absolute value is at most e for every choice of n and m bigger then n_0.
So if you fix an arbitrarily small positive number, after some point the distance between two arbitrary point in that tail becomes less than e.

This guarantees 'convergence'. The thing however is that rationals often miss those limits and so aren't 'complete'. that's why the real numbers are so special. they fill those gaps.

Equally, a function like sin(x) will not converge at all.

This seems to me like the concept which is manipulated in a sin-type function cannot be a number in the way you describe it, because both functions resist Cauchy analysis.

Just to muddy things further, does the phrase "7 during the daytime and 8 during the nightime" express a number?

I'm not a mathematician, so please forgive any glaring errors!

[zendahl]

while i can openly say that anyone who understands half of this is way better at math than me, I have to ask how this aplies to politics?

[RubberTrucky]

At 6/18/09 09:00 AM, zendahl wrote: while i can openly say that anyone who understands half of this is way better at math than me, I have to ask how this aplies to politics?

It could make for a debate, if more people are willing to join. Toadenalin proposed some problems with mathematics as a not so perfect theory. I asked what those were and he argues this over here.

The thing is, however I'm afraid, that although this has the making of a quite interesting discussion on math and the statement, that maybe not a lot of people on NG gives a rat's ass about mathematics. But then again, it's not because a person doesn't like math, that a discussion about it anywhere is stupid. If we're taking that course, we might as well close down all of politics, since there is bound to be a majority who doesn't care for any of the threads in politics.

[Ericho]

Numbers or at least numerical value exist because they are used to apply to real-life things and are considered "real" because they serve a purpose, just like religion. Hee hee.

[Toadenalin]

At 6/18/09 12:41 PM, Ericho wrote: Numbers or at least numerical value exist because they are used to apply to real-life things and are considered "real" because they serve a purpose, just like religion. Hee hee.

Sure, this is what I'm trying to get at. I have a different conception of a 'number' to RubberTrucky - RubberTrucky is a much better mathmo than me, and can talk about a 'number' being the limit of a sequence. I can't deal with that level of abstraction, so I think of a 'number' as being something I could, in principal, hold in my hand. As a logician, I have to think of 'numbers' in terms of set theory, that NEVER apply to things in the real world - I get in masive trouble with my tutor if I don't write '3' as 'sss0' in my proofs. A really first class mathematician might be able to concieve of imaginary numbers as easily as RubberTrucky can think of limits of sequences or I can think of tangible objects. Which one of us (if any) are right? Is there a right answer? Perhaps most interestingly, does it actually matter?

[RubberTrucky]

Up to the real numbers, everything is imaginable in the real world.

Pythagoras's theorem says so.

Like I said, take an orthogonal triangle with sides which do not form a natural kwadrature (like 3,4->5) and you will have a real number.

Circles will produce pi and any kind of exponential phenomenon will relate to the natural logarithm e.

Hence the term 'real' numbers.

Complex numbers are less prevalent. This is because they are actually just a trick to solve equations.
A big example of this is quantum theory. Quantum theory relies on complex number (for instance to apply the Spectral theorem on its operators)
Basically, states are presented as wave functions, with an amplitude. These are contained in complex numbers with a radius and a phase: exp^(x+iy)=exp^x (cos y+ i sin y).
Because the space of wave functions can be seen as a vector space over elementary wave functions, observables are seen as linear operators on this space.
But what do we measure? These are its eigenvalues. For an operator A, an eigenvalue a is a number, such that for an element v=/= 0 it occurs that Av=av, simply said.
To get that all these eigenvalues are real numbers, the operator needs to be Hermitian and then the spectral theorem works to receive all eigenvalues.

[Ericho]

It's been said imaginary numbers are "real" in that they serve a purpose in mathematics. Why can't you divide by zero and make a number for that if you can take the square root of a negative number and make up a number for that?

[RubberTrucky]

At 6/19/09 11:43 AM, Ericho wrote: It's been said imaginary numbers are "real" in that they serve a purpose in mathematics. Why can't you divide by zero and make a number for that if you can take the square root of a negative number and make up a number for that?

Oh, that's not so hard I think.

the reason why we can't take the root of a negative number is because each real number squared is positive by definition of the square. (-1)(-1)=1
So we just need a new number such that i*i=-1 and we're ready to go.

As to division by zero, this is a lot deeper. 0 is defined as an element such that a*0=0 for every number a. So righting someting as a/0, would be the number you have to multiply with 0 such that it becomes a.
Dang it, I don't have time to reason further. I will have to leave you at that and come back.

Excersise: Suppose F is a number such that F*0=1. Find out how this behaves towards other numbers (sum and product)

It is partially done by taking F= 'infinity'. (not so much to be taken litterally)
But you can see by the exersise that this entity isn't compatible with the standard notions of product and sum onto the set of numbers (the complex numbers however, are)

[lapis]

At 6/17/09 06:47 PM, Toadenalin wrote: It seems like defining numbers in terms of a Cauchy sequence (and therefore limits) might run into problems though.

If you want to get technical, I don't think 0.999... really qualifies as being a "number", let alone the "definition" of 1. It's just a mathematical expression, a function with a fixed independent variable in some sense, that can be evaluated and which happens to yield 1 as the answer. Namely, lim_{x->infinity} sum_{1}^{x} 9*10^{-i}, which can be evaluated and which yields (and is therefore equal to) 1. Like how 1+1 equals 2. And 2+1=3.

I'm trying to figure out what you're getting at in this thread but here goes: these collections of pixels that we call numbers exist solely for the sake of ease (or even possibility) of notation. sin(3) is so rarely stumbled upon that it would be redundant to come up with a fancy symbol for it. You could, though, and introduce ten symbols, AA BB CC DD EE FF GG HH II and JJ, just to denote sin(i), i=0...9 (with sin(0) equal to our 0). Then you could write the number 1 as arcsin(AA). Like in: that guy has arcsin(AA) child. It's just not practical.

I think all you have to do is to ontologically accept that there is a difference between existence and non-existence, and denote those states with 1 and 0. From there, you can express the ontological concept of plurality (in binary notation), although I think all civilisations throughout history sooner had a need to express plurality than to express non-existence. Anything else that you want to use to express (in a a sense numerical) concepts that you experience and want to put on paper you can express using words or notation for the sake of brevity. Then again, the words "heap" and "bunch" simply serve to express the subjective notion of "thinking that some quantity is large". They need no precise definition. Where the border between "large" and "not remarkable" or "small" lies? Don't know.

We can accept that we (assuming you're not a millionaire) would buy a copy of GTA IV, if we were looking for one in a city with several game shops, for 1 eurocent and not for EU10,000. So where's the edge? For what price x would we buy the game and for which price x + 1 eurocent would we decide not to? Would you buy the game if it was priced EU37.99 but not if it was priced EU40.00 (assuming that a psychological barrier would naturally lie near a value that could cause a few numbers on the lefthand side of the number to shift)?

Or something. I'm not sure if I'm getting across what I want to say.

[satanbrain]

he proved that in the wrong way. infinity is a process, not a number.

[Ericho]

At 6/19/09 12:49 PM, RubberTrucky wrote: As to division by zero, this is a lot deeper. 0 is defined as an element such that a*0=0 for every number a. So righting someting as a/0, would be the number you have to multiply with 0 such that it becomes a.

Wow, I didn't know mathematics had such philosophy behind it.

[RubberTrucky]

At 6/19/09 12:49 PM, RubberTrucky wrote:
Excersise: Suppose F is a number such that F*0=1. Find out how this behaves towards other numbers (sum and product)

Along the way, I found the solution.

It's the 'magic math' paradox, we all know from the youtube vids. Suppose F*0=1. Then we have for each real number a that
F*(0*a)=f*0=1=(F*0)*a=a.

this is a contradiction. If we were to introduce F, it does not behave well with the standard product * or sum +.
For infinity, we get inf*a=inf for every a real, non-zero.
Also inf*a=inf
and inf*0 is undefined.

[sparrow]

I believe my maths teacher once showed me an equation which as 1 = 2 but I might be imagining things. I'm sure he proved it somehow. Does anyone know what i'm on about?

[killxp]

At 6/21/09 05:39 PM, zrick wrote: I believe my maths teacher once showed me an equation which as 1 = 2 but I might be imagining things. I'm sure he proved it somehow. Does anyone know what i'm on about?

Yeah, the fallacy in the "proof" is in the step where you divide by a '0' Division on both sides of the equation, as long as its the same thing, is fine, as long as its not zero. You can pretty much say that any number equals any other number if you have divided by 0 somewhere. The tricky part is that the '0' is disguised as some expression (a-b, where you previously stated that a=1=b). You can just search for classic math proof fallacy and you should get something. Have fun. Late.

[RubberTrucky]

At 6/21/09 05:39 PM, zrick wrote: I believe my maths teacher once showed me an equation which as 1 = 2 but I might be imagining things. I'm sure he proved it somehow. Does anyone know what i'm on about?

Of course, division by 0 is 1 profound way to trick people into believing equalities, there are far more interesting, not so overused cases of such fallacies.

[ThoArtNoobish]

Number:the property possessed by a sum or total or indefinite quantity of units or individuals; "he had a number of chores to do"; "the number of ...
a concept of quantity involving zero and units; "every number has a unique position in the sequence"
act: a short theatrical performance that is part of a longer program; "he did his act three times every evening"; "she had a catchy little routine"; "it was one of the best numbers he ever did"
phone number: the number is used in calling a particular telephone; "he has an unlisted number"
numeral: a symbol used to represent a number; "he learned to write the numerals before he went to school"
total: add up in number or quantity; "The bills amounted to $2,000"; "The bill came to $2,000"
issue: one of a series published periodically; "she found an old issue of the magazine in her dentist's waiting room"
give numbers to; "You should number the pages of the thesis"
a select company of people; "I hope to become one of their number before I die"
enumerate; "We must number the names of the great mathematicians"
a numeral or string of numerals that is used for identification; "she refused to give them her Social Security number"
count: put into a group; "The academy counts several Nobel Prize winners among its members"
a clothing measurement; "a number 13 shoe"
count: determine the number or amount of; "Can you count the books on your shelf?"; "Count your change"
the grammatical category for the forms of nouns and pronouns and verbs that are used depending on the number of entities involved (singular or dual or plural); "in English the subject and the verb must agree in number"
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Where Did Numbers Originate?

Thousands of years ago there were no numbers to represent two or three. Instead fingers, rocks, sticks or eyes were used to represent numbers. There were neither clocks nor calendars to help keep track of time. The sun and moon were used to distinguish between 1 PM and 4 PM. Most civilizations did not have words for numbers larger than two so they had to use terminology familiar to them such as flocks of sheep, heaps of grain, or lots of people. There was little need for a numeric system until groups of people formed clans, villages and settlements and began a system of bartering and trade that in turn created a demand for currency. How would you distinguish between five and fifty if you could only use the above terminology?

Paper and pencils were not available to transcribe numbers. Other methods were invented for means of communication and teaching of numerical systems. Babylonians stamped numbers in clay by using a stick and depressing it into the clay at different angles or pressures and the Egyptians painted on pottery and cut numbers into stone.

Numerical systems devised of symbols were used instead of numbers. For example, the Egyptians used the following numerical symbols:

The Chinese had one of the oldest systems of numerals that were based on sticks laid on tables to represent calculations. It is as follows:

From about 450 BC the Greeks had several ways to write their numbers, the most common way was to use the first ten letters in their alphabet to represent the first ten numbers. To distinguish between numbers and letters they often placed a mark (/ or %uFFFD) by each letter:

The Roman numerical system is still used today although the symbols have changed from time to time. The Romans often wrote four as IIII instead of IV, I from V. Today the Roman numerals are used to represent numerical chapters of books or for the main divisions of outlines. The earliest forms of Roman numeral values are:

Finger numerals were used by the ancient Greeks, Romans, Europeans of the Middle Ages, and later the Asiatics. Still today you can see children learning to count on our own finger numerical system. The old system is as follows:

From counting by means of %uFFFDflocks%uFFFD to finger symbols our current numerical system has evolved from the Hindu numerals to present day numbers. The journey has taken us from 2400 BC to present day and we still use some of the old numerical systems and symbols. Our system of numerics is ever changing and who knows what it will look like in 2140 AD. Will we still count using our fingers or will mankind invent a new numerical tool? Sanscrit letters of the 11. Century A.D. Apices of Boethius and of the Middle Ages Gubar-numerals of the West Arabs Numerals of the East Arabs Numerals of Maximus Planudes. Devangari-numerals. From the Mirror of the World, printed by Caxton, 1480 From the Bamberg Arithmetic by Wagner, 1488. From De Arts Supp- urtandi by Tonstall, 1522

This chart shows the change of numbers from their ancient to their present-day forms

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Number examples 1,2,3,4,5,6,7,8,9,10 basic
Number: a never ending, on going count Forever!!!

Got that Fool damn (=

[killxp]

At 6/23/09 10:47 PM, ThoArtNoobish wrote: Got that Fool damn (=

So you basically just copied text and pasted it into here huh? Or did you actually read the previous posts? Doesn't look like it because you aren't referring to what they have mentioned. Thanks for you input anyways. Late.