The Nature and Behaviour of Infinity

[purplecat]

As 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):

• 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.


This entry was originally posted at http://purplecat.dreamwidth.org/34109.html.

Comments

[lukadreaming.livejournal.com]

I have a copy of the Very Short History of Mathematics and am wondering how much of it I will understand. An O Level grade C from the dark ages is not a useful starting point. I am doing OK with Alex's Adventures in Numberland. So far *g*.

I did manage to explain to someone the relevance of the hat. More or less *g*.

[reggietate.livejournal.com]

I just about scraped up a passing grade in CSE maths. To this day I've no idea how I did it :-)

No doubt this is why I can handwave so blithely when it comes to things like Cutter's matrix.

[purplecat]

It must be said that I usually find I have to explain things several times to different people before I hit on a way of explaining it that most people will follow. It's quite frustrating....

[lukadreaming.livejournal.com]

I understood some of what you were saying above, but you're on a loser with me from the start. I remember numbers, but my brain doesn't seem to assimilate mathematical concepts. I know part of it was because the teaching at the school I went to was lamentable. Hence why I am taking to reading these books to see if I can remedy the gaps.

[reggietate.livejournal.com]

*attempts to understand*.... *brain asplodes*

Maths is so not my subject! :-D

[purplecat]

Oh dear! I'm sorry!

[inamac.livejournal.com]

I failed maths O Level (twice), but if only it had been presented as philosophy/logic I might have had a prayer of understanding it.

This is all much more interesting than arithmetic (if of less practical use when paying for the shopping). Thank you.

(Icon for teacher, not you!)

[purplecat]

I was a little surprised at the teacher's response. Wittgenstein's argument is fairly firmly in Philosophy of Mathematics and not something taught in most undergraduate maths degrees, whereas the various constructions of infinity (of which this is just one) are pretty standard stuff. In fact, if anything, I'd expect a school maths teacher to assume that the existence of infinity was something proved by Cantor's diagonal argument (*ahem* maybe another time!) rather than proclaiming point-blank that there was no such thing.

[sheldon stevens (from livejournal.com)]

Speaking as one of those rare biologists who explored maths all the way up to A-level (though admittedly scraping a bare pass), and having read New Scientist's recent article on infinity (taking about 20 minutes - I had to read several bits twice and I confess my lips moved) I can grasp that there are infinities of infinities, but I can't help sympathising with Wittgenstein and Gwendolyn's teacher as it doesn't seem exactly in touch with reality.

[purplecat]

The problem is, if you are interested in the formalisation of mathematics, that there is an awful lot of stuff that patently works and is incredibly basic and useful that we simply can't prove without assuming the existence of an inductive set.

That said, without inductive sets, mathematics is far, far better behaved (and complete, which is nice). You just can't do much with it.

I didn't really get into the many infinities here. That's something more aligned to a different discussion which centers around a separate construction of numbers, called cardinal numbers in which, for the purpose of simplicity, obviously, omega is referred to as aleph zero.

[munchkinofdoom.livejournal.com]

I can safely say I didn't underatand any of that. *nods*

I dropped maths after year 10, after being the bane of my Advanced Maths teacher's life. He refused to be the only teacher who failed me for anything. Music didn't count - we all failed. *veg*

[purplecat]

Oh dear! I am sorry. As I said to Luka above, I do find I need to explain things several times to different people before I really get the hang of the best way to do it.

[purplecat]

I think it is, but I've always been attracted to the abstract.

[lil-shepherd.livejournal.com]

Oh, I understand that mathematicians say this, and what their reasoning is (and sat through a Horizon on the subject that had me yelling at the TV) but I still think that 'infinity' is simply a useful concept in mathematics - rather like Shrodinger's Cat is a useful concept in quantum mechanics - but does not actually exist. Just, actually, as any number does not exist - there has to be one of a thing. It is an abstract concept. You cannot have one, or an infinity. You must have one atom or an infinite number of atoms. The first is not abstract - the second, on the other hand, is.

[purplecat]

Well we get into the complicated question of what mathematics is about which is very difficult. If it is about some abstract realm of abstract objects - some version of Plato's idealised concepts - then we need a good explanation of how we actually know or learn anything about this realm. On the other hand, if it is just, some kind of notational convenience, symbols pushed around a piece of paper according to certain rules, then we need to explain why it has such predictive power and in particular why concepts like imaginary number and infinity which, in this interpretation, really are just squiggles on paper, are so damn useful.

However, I very much doubt that the teacher wanted to open a discussion of the nature of mathematical reality when he asserted that I was wrong. In some ways I was more irritated that, having successfully got the children excited and interested about some (quite advanced and abstract) feature of mathematics the teacher's reaction was obviously to make them feel ignorant and stupid. Moreover, as a teacher of mathematics, he really shouldn't be asserting that such a cornerstone of large areas of mathematics doesn't exist without explaining the philosophical standpoint behind it. I mean, I've lost touch with the A level syllabus, but I assume they still teach integration and differentiation which is difficult without coming to terms with some concept of infinity.

[lil-shepherd.livejournal.com]

I don't see how its predictive power is affected by whether or not it is a useful abstract. Again, quantum is very similar. You can use quantum mechanics and mathematics to predict with great accuracy, but they are tools that work in particular circumstances.

[lsellersfic.livejournal.com]

If it's just a human invented squiggle of some sort and doesn't relate to anything real then it's power to predict reality ought to be random at best. If, on the other hand, it does relate to something real, say some abstract principle then abstract principles must be "real" in some sense and the difficulty comes in defining in what sense they are real and how they interact with the concrete. It's an entirely open question, as far as I'm aware, with massive holes on both sides of the argument. The "it's a human invention" people currently have the upper hand in the philosophical debate but they definitely struggle to explain how something we just made up, which doesn't relate to anything that can be described as real, turns out to be useful.

[lil-shepherd.livejournal.com]

I don't understand this argument. There are a good many abstracts - for instance the way our brains process our senses - that are not 'real' in a meaningful sense. Colour is simply a mental reaction of certain wavelengths of light - it is an abstract, and none of us can be sure we mean the same thing by, say, 'red', but this does not matter. Being able to differentiate between certain wavelength of the energy spectrum is survival positive. That is is not in any way 'real' is irrelevant.

It is useful. It works.

It may be that mathematics will be found to be only a partial descriptive and predictive method, rather like Newtonian physics, and quantum, I suspect, will almost certainly be only a partial explanation, but all these things work. I don't see that whether number is 'real' in any sense apart from the mental affects this. Abstracts are useful.

[lsellersfic.livejournal.com]

Well of the nature of mental concepts is a philosophical minefield as is the question of to what, exactly, the words we use for them refer.

In the case of mathematics the general intuitionist view is that the rules of mathematics are a human invention - Wittgenstein (or possibly Dummett) described them as a `psychological quirk' and then rather unconvincingly (in my opinion) posited a race of aliens who considered the value of an object to be it's area in contact with the ground and who saw nothing odd in an object changing it's value if you put it on its side but that is by the by.

Leaving aside the awkward question of what mental states are and how they relate to the reality beyond our senses, we can say that `red' is a fuzzy concept, a categorisation of a continuum that has no actual discrete segmentation, but which it is convenient for people to use in discourse and has. Infinity and imaginary numbers are worse than this, insofar as they refer to anything, they refer to something that definitely doesn't exist. The rules of mathematical reasoning are even harder to justify since it's difficult to even semi-convince yourself they might refer to some abstract object.

Moreover, unlike red, and its fuzziness of reference around the edges there is relatively little dispute, in practical situations, over the meaning of the terms infinite, imaginary number, rule of deduction. We rely on these terms to describe, predict and guarantee things about our environment which seems a lot of expect of a psychological quirk or a discursive convenience unless they capture some truth about the way physical reality works (much as red, admitedly imperfectly, captures some range of the light spectrum) but since these concepts have no physical reality we are forced to start talking about an abstract reality (in a way that red doesn't force us to do) and most current philosophers find that very objectionable.

[lil-shepherd.livejournal.com]

But they don't have to be 'real' to describe a 'truth'. Really, they don't.

[lsellersfic.livejournal.com]

Abstracts should only be useful if they capture something, even imperfectly, about reality.

Since we know there is no such thing as infinity or imaginary numbers (or even something they approximate to or imperfectly capture) then we are left with an awkward question of what it is they really do refer to.

Of course we may, in the future, come up with mathematical theories that do not require us to use these concepts in order to predict reality (though I find that very unlikely, given things like Godel's incompleteness theorem) but even so, we are left with trying to explain what Modus Ponens (if A implies B and A is true then B is true) refers to and without that we can't really reason at all.

[lil-shepherd.livejournal.com]

Logic and language, like mathematics, are human tools, and closely related, as far as I can see.

And I happen to think philosophy is pretty much a useless subject, but there you go.

[foradan.livejournal.com]

You say that imaginary numbers aren't real. Obviously they are not in the set R, but I assume that you mean a less formal definition of 'real'. So my question for you is: are negative numbers real? And if so why do they have more validity than imaginary ones?

[purplecat]

Well, to be honest, I consider all kinds of numbers equally abstract (with a large "don't know" hanging over what I actually think abstract concepts are), but in terms of something that we know doesn't physically exist imaginary numbers are an easy and obvious choice while you can always handwave about debt and such like for negative numbers. I was tempted, given the context, to reference real numbers with infinite expansions since, in a finite universe, they no more exist that infinity does but I thought that was a rather complex point to make. I just wanted something easy to point at to say "look standard school mathematics! if you want to say infinity doesn't exist, then you have to say this thing doesn't exist either!".

[foradan.livejournal.com]

I was surprised to learn that both negative and imaginary numbers were accepted as being as valid as positive numbers by the general mathematical community around the same time (18th century). In school maths, imaginary numbers are taught much later than negative numbers.

your LJ posts

[jhgowen.livejournal.com]

Louise, I always enjoy reading your intellectual LJ posts, even if sometimes I don't understand them all. Please keep them up. Perhaps I should try and do the same with my research area, but the details of silicon crystal growth may not be of such broad interest.

Re: your LJ posts

[purplecat]

To be honest, I've been quite surprised how interesting people have found stuff I consider fairly obscure. A lot of it, to be honest, harks back to my undergraduate material or stuff I've read randomly since without it being centrally about my research (though I suppose I do also post about accepted papers - but usually in a lot less detail). Nearly all the recent stuff has arisen out of offline conversations of one sort or another and then sort of built a momentum of its own.

Basically, I'd say give it a go, but maybe pick things you've found interest people in general conversation.

infinity and beyond

[jhgowen.livejournal.com]

I think it is entirely the wrong, lazy, kind of maths teacher who cannot cope with a pupil asking about infinity. My mother would have made a valid attempt to introduce a concept of infinity to her class, but then she was of the old school more concerned with educating pupils than ticking boxes. I still remember meeting "e^i.pi = -1" for the first time, and thinking how beautiful it was. My mother always wanted me to do maths at university.
I have never heard of omega. What about aleph-1?

P.S. Why a _photocopy_ of the Blue Book? Had you left your original copy somewhere?

Re: infinity and beyond

[purplecat]

Well to be fair to the maths teacher I don't know exactly what happened. But the children obviously felt dismissed and wanted something to back up their argument. Omega, broadly speaking, is equivalent to aleph₀ and it depends on how you are defining numbers. Cardinal numbers define numbers in terms of the size of set (unlike defining them in terms of actual sets like the ordinal numbers do). Aleph₀ is the size of the set of natural numbers (and rational numbers), 2^aleph₀ is the size of the set of real numbers. Cantor showed that every cardinal number has a "next one up" so aleph₁ is the next cardinal number after aleph₀. It's an open question whether aleph₁ is the same thing as 2^aleph₀ - i.e. whether the real numbers form the next largest set after the natural numbers. This is called the continuum hypothesis. Last time I looked (which was about twenty years ago) it had been shown that most of mathematics was consistent whether the continuum hypothesis was true or whether it was false which suggests it may be undecidable in some sense.

Re: infinity and beyond

[purplecat]

As to why a photocopy, I've no idea. I assume that either Somerville library didn't have a copy and either my tutor provided one or I photocopied the one in the Philosophy sub-fac library (which IIRC wasn't open at weekends so I may have been in some kind of Friday essay panic) or possibly the Somerville copy got recalled and I hadn't finished with it.

[king-pellinor.livejournal.com]

I like your complicated posts too :-)

I have lots of questions that I'm going to ignore, ont he grounds that you've already answered them with waving hands :-)

But the bit about not being able to relabel the 0 in Omega+1 as you don't know what the last bit of Omega is... doesn't the same argument apply to all the dots? You're basically saying that with each dot up to and including the end of Mega, that you're going to shift them one place over and give them the value of the number to their right. Couldn't you do that with the +1?

What if you say OK, we don't know what the last number in Omega is, but whatever it is we'll shift it (and all preceding numbers, calling the 0 at the start 0 again for the sake of argument) over to the left to make room for the +1's 0, and then renumber from the left like we did with 1+Omega?

You're going to wave your hands at me, aren't you? :-D

[purplecat]

I am going to have to wave my hands I think yes. Broadly its all in the definition of "relabel" but if you shift all the numbers in omega one to the left there is still no "end point" because there is an infinite number of them.

The problem basically arises because there's a sort of anti-symmetry in the notation {0, 1, 2, 3, ....} we know where it starts on the left but it doesn't actually stop anywhere on the right and this means it behaves differently depending whether you are trying to do things to it on the right or the left (I omitted from the explanation that actually these sets are ordered from smallest to largest). The way ... is captured formally preserves that anti-symmetry and so attempts to define operations like addition which rely on doing things at one end or the other of the number start doing strange things. To be honest, I don't know if any serious attempts have been made to find a set-theoretic construction of numbers which have infinites behaving more intuitively, my guess would be that there are other consequences which are undesirable but I don't know off hand.