The Nature and Behaviour of Infinity
Feb. 5th, 2011 03:57 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
purplecatAs you do, I remarked in passing to my in-laws that infinity plus one was not the same as one plus infinity. This gem of wisdom was duly repeated by my niece and nephew to their mathematics teacher who retorted that I was wrong and, moreover, there was no such thing as infinity. I was therefore requested, in turn, to provide a one page explanation* which could be shown to said mathematics teacher.
Now, it must be said, I don't like to undermine the fine teachers of mathematics who, I suspect, have a hard enough job as it is performing their task without random aunts interfering. On the other hand, a challenge has been laid down.
I am an inveterate hoarder so even though I do not own any of Wittgenstein's works** I do nevertheless still have my handwritten undergraduate essay `Inifinity is not a huge number' (WITTGENSTEIN). Is this a blunder or an important insight. I will spare you the essay-writing prose of my younger self but to summarise in brief. Quantum theory (among other things) leads us to believe that we live in a finite universe. Things do not go on getting smaller and smaller ad infinitum but stop at a defined smallest amount of stuff. Therefore, since this universe has finite space and things can only get so small, there is no such thing as infinity. Wittgenstein maintained that we were simply misinterpreting the phrase `and so on' as in "You count up one, two, three, four, five and so on" to imply that there was something that actually existed at the end of the and so on.
So, physically speaking, the mathematics teacher is correct. There is no such thing as infinity. However I bet he's going to teach his class about imaginary numbers at some point and they don't exist either.
At the same time (well previously) mathematicians were interested in defining numbers in a fully formalised way. The standard construction currently runs thus (NB. for this to seem like a remotely good idea you have to assume that a set is a simpler concept than a number, in this mathematicians may diverge from the general population - please bear with them):
Numbers defined in this way are referred to as ordinal numbers.
The mathematical fraternity arguably then got a bit carried away and discovered/invented*** the inductive set. An inductive set is one which contains zero and if some number, n is in the inductive set then so is n + 1 so basically it contains all the numbers (and possibly some other stuff as well). Then, bless them, the worthies of mathematics announced that they assumed an inductive set existed - this is called the axiom of infinity. Having decided that these things existed (much to the later irritation of Wittgenstein) they decided to call the smallest such set omega. Omega is where you end up (assuming you end up anywhere which Wittgenstein says you can't) when you count for ever. So when I blithely started making pronouncements about infinity to my relatives it was omega I was talking about.
To go back to our notation using sets we can write omega as {0, 1, 2, ...} where ... means "and so on". There is a less hand wavy definition but I'm assuming that if I start going on about the limit of a function as it tends to infinity most of my readers will bail.
Since mathematicians are inveterate categorisers they divided ordinal numbers up into three groups: zero, successor ordinals (numbers like 1, 2 and 3 which are sets containing lower numbers - or, if you like, numbers that are equal to some other number plus one) and limit ordinals which are all the others (like omega).
Assuming you are a mathematician and not an ordinary sensible person, the moment you have discovered/invented numbers and infinity you want to discover/invent a whole load of other useful things. Addition springs effortlessly to mind here.
This is where things, if they weren't already technical enough, get even more technical so I'm going to try hand waving even more wildly. I'm trying to give a general idea here and the technical detail that makes it work is omitted. There are various ways to define addition this but let's say you add two numbers in this weird set notation by "gluing" them together end to end and then "relabelling" the second number as appropriate (which basically means starting from the end of the first number plus one (see below)). So:
1 + 2 =
{0} glued to {0, 1} =
{0, 0, 1} =
{0, 1, 2} =
3
(Can you see I relabelled the second number by starting from where the first number ended, so 0 became 1 and 1 became 2).
Leaving aside the apparent eccentricity of this approach to something perfectly normal and every day like addition, this process works exactly as you would expect for zero and successor ordinals, the kind of numbers you meet in the normal run of things.
What happens if we do it with a limit ordinal?
1 + omega =
{0} glued to {0, 1, 2, 3, ....} =
{0, 0, 1, 2, 3, ...} =
{0, 1, 2, 3, 4, ...} =
omega
So 1 + omega is equal to omega which is kind of freaky but, if you think about it, makes a sort of sense. If you have an infinite number of things and you get another one then you still have an infinite number of things.
On the other hand
omega + 1 =
{0, 1, 2, 3, ...} + {0} =
{0, 1, 2, 3, ..., 0} =
omega with {0} after the dots.
We can't relabel the 0 at the end because we don't know what the last number in omega is, in fact there is no last number in omega. omega plus 1 is just omega plus one - weird, counterintuitive in lots of ways, but true if you happen to be using this particular set up to define numbers and infinity. Lots of mathematicians find this kind of thing cute and exciting****.
You may think that all this just means that gluing-then-relabelling is a silly way to define addition. All I can say is that it has it's uses and no other way actually behaves any better.
At this point I suspect many of you will have some sympathy with Wittgenstein's assertion that this is all demonstrable nonsense and mathematicians have let themselves get entirely too carried away with all the "and so on" and ... business into thinking something exists which they can then do more maths with. Just because it's cute and exciting, doesn't make it true. This is a valid point and remember that the mathematicians can only do this if they assume the existence of an inductive set, they've not been able to prove that one exists. However, transfinite mathematics of this kind continues to be studied, used and developed and so, philosophical qualms aside, I feel entirely justified in asserting that infinity plus one is not the same as one plus infinity.
* I'm being lazy and assuming that Facebook will remorselessly suck this explanation into itself and then my relatives can print it off and present it to their maths teacher.
** except I discover, a twenty-year-old photocopy of pages 2-17 of The Blue Book.
*** depending upon your preferred philosophical standpoint on the nature of mathematics.
**** when I say `lots of mathematicians' here, obviously I mean me. I'm just assuming that a lot of other people got into maths for the same sorts of reasons I did.
Now, it must be said, I don't like to undermine the fine teachers of mathematics who, I suspect, have a hard enough job as it is performing their task without random aunts interfering. On the other hand, a challenge has been laid down.
I am an inveterate hoarder so even though I do not own any of Wittgenstein's works** I do nevertheless still have my handwritten undergraduate essay `Inifinity is not a huge number' (WITTGENSTEIN). Is this a blunder or an important insight. I will spare you the essay-writing prose of my younger self but to summarise in brief. Quantum theory (among other things) leads us to believe that we live in a finite universe. Things do not go on getting smaller and smaller ad infinitum but stop at a defined smallest amount of stuff. Therefore, since this universe has finite space and things can only get so small, there is no such thing as infinity. Wittgenstein maintained that we were simply misinterpreting the phrase `and so on' as in "You count up one, two, three, four, five and so on" to imply that there was something that actually existed at the end of the and so on.
So, physically speaking, the mathematics teacher is correct. There is no such thing as infinity. However I bet he's going to teach his class about imaginary numbers at some point and they don't exist either.
At the same time (well previously) mathematicians were interested in defining numbers in a fully formalised way. The standard construction currently runs thus (NB. for this to seem like a remotely good idea you have to assume that a set is a simpler concept than a number, in this mathematicians may diverge from the general population - please bear with them):
- We call the empty set `zero': 0 = {}
- We call the set containing zero (the empty set) 'one': 1 = {0} = {{}}
- We call the set containing zero and one `two': 2 = {0, 1} = {{}, {{}}}
- And so on... (did you see what I did there?)
Numbers defined in this way are referred to as ordinal numbers.
The mathematical fraternity arguably then got a bit carried away and discovered/invented*** the inductive set. An inductive set is one which contains zero and if some number, n is in the inductive set then so is n + 1 so basically it contains all the numbers (and possibly some other stuff as well). Then, bless them, the worthies of mathematics announced that they assumed an inductive set existed - this is called the axiom of infinity. Having decided that these things existed (much to the later irritation of Wittgenstein) they decided to call the smallest such set omega. Omega is where you end up (assuming you end up anywhere which Wittgenstein says you can't) when you count for ever. So when I blithely started making pronouncements about infinity to my relatives it was omega I was talking about.
To go back to our notation using sets we can write omega as {0, 1, 2, ...} where ... means "and so on". There is a less hand wavy definition but I'm assuming that if I start going on about the limit of a function as it tends to infinity most of my readers will bail.
Since mathematicians are inveterate categorisers they divided ordinal numbers up into three groups: zero, successor ordinals (numbers like 1, 2 and 3 which are sets containing lower numbers - or, if you like, numbers that are equal to some other number plus one) and limit ordinals which are all the others (like omega).
Assuming you are a mathematician and not an ordinary sensible person, the moment you have discovered/invented numbers and infinity you want to discover/invent a whole load of other useful things. Addition springs effortlessly to mind here.
This is where things, if they weren't already technical enough, get even more technical so I'm going to try hand waving even more wildly. I'm trying to give a general idea here and the technical detail that makes it work is omitted. There are various ways to define addition this but let's say you add two numbers in this weird set notation by "gluing" them together end to end and then "relabelling" the second number as appropriate (which basically means starting from the end of the first number plus one (see below)). So:
1 + 2 =
{0} glued to {0, 1} =
{0, 0, 1} =
{0, 1, 2} =
3
(Can you see I relabelled the second number by starting from where the first number ended, so 0 became 1 and 1 became 2).
Leaving aside the apparent eccentricity of this approach to something perfectly normal and every day like addition, this process works exactly as you would expect for zero and successor ordinals, the kind of numbers you meet in the normal run of things.
What happens if we do it with a limit ordinal?
1 + omega =
{0} glued to {0, 1, 2, 3, ....} =
{0, 0, 1, 2, 3, ...} =
{0, 1, 2, 3, 4, ...} =
omega
So 1 + omega is equal to omega which is kind of freaky but, if you think about it, makes a sort of sense. If you have an infinite number of things and you get another one then you still have an infinite number of things.
On the other hand
omega + 1 =
{0, 1, 2, 3, ...} + {0} =
{0, 1, 2, 3, ..., 0} =
omega with {0} after the dots.
We can't relabel the 0 at the end because we don't know what the last number in omega is, in fact there is no last number in omega. omega plus 1 is just omega plus one - weird, counterintuitive in lots of ways, but true if you happen to be using this particular set up to define numbers and infinity. Lots of mathematicians find this kind of thing cute and exciting****.
You may think that all this just means that gluing-then-relabelling is a silly way to define addition. All I can say is that it has it's uses and no other way actually behaves any better.
At this point I suspect many of you will have some sympathy with Wittgenstein's assertion that this is all demonstrable nonsense and mathematicians have let themselves get entirely too carried away with all the "and so on" and ... business into thinking something exists which they can then do more maths with. Just because it's cute and exciting, doesn't make it true. This is a valid point and remember that the mathematicians can only do this if they assume the existence of an inductive set, they've not been able to prove that one exists. However, transfinite mathematics of this kind continues to be studied, used and developed and so, philosophical qualms aside, I feel entirely justified in asserting that infinity plus one is not the same as one plus infinity.
* I'm being lazy and assuming that Facebook will remorselessly suck this explanation into itself and then my relatives can print it off and present it to their maths teacher.
** except I discover, a twenty-year-old photocopy of pages 2-17 of The Blue Book.
*** depending upon your preferred philosophical standpoint on the nature of mathematics.
**** when I say `lots of mathematicians' here, obviously I mean me. I'm just assuming that a lot of other people got into maths for the same sorts of reasons I did.