highest

1.1.2

The Highest Number

" In Eternal Ignorance there is an infinite capacity for knowledge "

-- Sbiis Saibian

"We don't mind where the question lead us. We know nowhere is better than everywhere" -- Tim Exile

No answer will ever be satisfying, because people always want more.

What is the highest Number? This seemingly innocuous question has been asked numerous times across the web and around the world in a variety of forms (ie. what's the biggest/largest/greatest/last number?). One get's the sense that the questioner urgently wants the matter settled once and for all. Just what IS the highest number?! There must be one after all right? Once children get a good handle of the concept of number they often ask this question too. It's a common question asked on mathematics forums, and yahoo answers. It's also a question that seems to garner glib or pat responses: sometimes disinterested, sometimes annoyed, and sometimes patient but patronizing. Apparently people have got this one figured out, so there's really no need to ask it ... one more time. Yet no matter how many times and ways the question is answered, still it returns: What is the highest Number?

The question arises in a fairly natural manner once the mind has a sufficient grasp of Number. Even without having names, or symbols for the numbers it is still possible to formulate the question. One can imagine a small multitude, then imagine a larger multitude,then a much larger one, then a much much larger one still and then ... the mind quickly reels and grasps for a question to make sense of this: Where does this process end? At the highest number presumably. At least that's what the question itself suggests. It can clearly be seen to contain the presupposition that there is a largest number, and if only we could find it, we could bring an end to the madness; and so the questioner, unable to come to the answer himself (or unwilling to accept the one they suspect but dread) seeks to find the answer by consulting an expert (usually a mathematician). As fate would have it for our intrepid questioner, the experts are ready to weigh in on the question with scripted answers on hand. It would seem our questioner has nothing to fear, because there IS an answer to the question.

So our questioner pops the question at his local mathematics forum, and the mathematician is all to happy to answer this one: "I'm glad you asked. The answer is: There is no highest Number, for if there was I could just add one to it, and that would be higher still! ". For some this answer comes as a relief. They had already expected this but were unsure whether anyone else had come to the same conclusion. Perhaps their intuition had told them there is no highest Number, but they couldn't think of a convincing proof of this. For those of this mindset the mathematicians response has all the finality of a judges gavel. Ah, now the mysterious has been reduced to the obvious. With that their minds are released from the search: after all why seek out something that doesn't exist.

But for others this answer is only a source of growing anxiety. Something terrible is coming, something incomprehensible and monstrous, something our intrepid questioner is afraid to let into his mind. Here is where the discussion get's very confusing indeed.

The questioner objects "well if you add one then that would become the largest number".

"Nope" responses the mathematician without even a pause "that would not be the largest number either because you could also add one to that".

The questioner not to be put off speculates "well, then maybe the highest number is constantly changing".

At this point the mathematician may pause for a moment to comprehend the gulf in thinking, and realises that the questioner has completely missed the point. "No the highest number can't change. Take a set of any numbers. The highest number of any such set is one which is greater than every other member of the set. For example take the set of one, two, and three. Three is the highest member of this set simply because it is greater than one or two, the only other members of the set. This can not change-"

but before our mathematician can swing the conversation back to his main point the questioner thinks he's got the mathematician now: "Ah, but see every set would have a hightest member then. Simply look for the member with the property of being the greatest".

The mathematician beginning to grow impatient counters "No, that is a false assumption. Not every set must have a highest member. The set of Counting Numbers is an example of such a set, because every member has a member in the set that is greater than it",

"except the highest member",

"there is no highest member",

"but that makes no sense! You'd run out of numbers",

"No you wouldn't",

"Yes you would. How could you keep coming up with names and symbols for them. Eventually would exhaust all the possibilities".

"Ah", now the mathematician thinks he's figured out the difficultly: "When I say numbers I don't mean the forms that we use to express them, I mean the numbers themselves. It's true that we might exhaust the forms: for example there is only a limited number of decimal expansions containing ten or less digits. But even if we can't express it, the number still remains. When I say there is no highest number, I mean to say that there is no end to the numbers themselves, not that there is no end to the forms we have". This the mathematician confidently feels, should settle the matter.

"But how can the numbers have no end?",

"What do you mean how can they have no end! I've just explained there is a difference between 'numbers' and the words and symbols used to express them. I will even concede that there is only a limited number of numbers that we can express. I merely contend that the numbers themselves continue forever".

"But nothing continues forever, everything must end at some point if your willing to wait long enough".

The mathematician may feel that he now knows the root of the problem, and will attempt to topple this assertion. "What makes you claim that nothing continues forever. How do you know this?",

"Because no one has ever discovered an actual infinity. Everything we know of has an end".

"So you gather from experience that things tend to have an end. People die, buildings crumble, stars explode, etc. However, just because we have never observed an actual infinity, and in fact we could never "observe" such a totality even if it existed, we can not automatically conclude that it doesn't exist. Absense of evidence is not evidence of absense"

"But infinity doesn't make any sense. How can something not have an end. If you wait long enough something has to give, even if we're just talking about numbers".

This should strike the mathematical ear as incomprehensible, but for argument's sake let's say our mathematician has enough mental flexability to see across the gap in thinking and understand what the questioner means by this cryptic statement: "When you say something has to give, and this is how we know infinity is not possible, your basing this of physical examples. Physical objects may decay and come to an end, but numbers are ideas. Ideas aren't subject to decay"

"Sure they are. Ideas from long ago are forgotten or change over time. Even ideas don't stay the same"

If the mathematician isn't outright frustrated at this point, he either has incredible patience or he's never had to put this much effort into the question and is perhaps partly amused. "Perhaps idea was a poor choice of words. If by idea you mean, something inside someones head, then yes, ideas can be forgotten and 'die'-"

"Exactly" says the questioner, again thinking the mathematician has only proven his point. "There has to be a highest number because there are only so many thoughts the human mind can have. Even if you can't name or write the number down, you can still have an idea about it. There must be a highest number someone could imagine"

"maybe ... but let me clarify-"

"Now you said imagine the highest number and then add one to it. But you can't! Because you haven't imagined it!"

"What do you mean you haven't imagined it?! That's the point, you can't. If you think you have the highest number, you can always make a larger one"

"But you can't make it larger if you don't have it in the first place. What is this hypothetical largest number that you are claiming can't be the largest..."

"It's irrelevant. It could be anything. If it would make the argument easier to understand you could use an example number as the largest: say a hundred. Then I could add one to it and get a larger number"

"All that proves is that hundred isn't the largest number. So you would have to look for a larger one"

"But no matter how large the number you start out with you can always add one"

"But we don't know that. We only tried it for a hundred. Maybe there's a number so big you can't add one to it"

Perhaps at this point the mathematician could be forgiven for thinking that the questioner can't be entirely serious. But the questioner is no troll. He means it when he says there maybe a number you can't add one to.

"But that's not how numbers work, every number has a successor, a number greater than itself"

"You said earlier that absense of evidence is not evidence of absense. Just because every number we know has a number greater than it, doesn't mean it's true for all numbers. Therefore your argument is simply based on the assumption that the hypothetical highest number has the same properties of all the numbers we are familiar with"

At this point it may seem the mathematician is beginning to lose ground in this debate. This is sure to seem apparent to the questioner at least, who feels they are making a valid point and have gotten to the heart of the matter. However what we are actually approaching is a kind of impasse. The reason the mathematician and our questioner are failing to reach an agreement is because they are beginning from a different set of assumptions. These assumptions are mutually exclusive. The mathematician concludes there is no highest number because 1. The set of all Counting Numbers share certain universal properties, and 2. Because there is no end to the Counting Numbers. The Questioner concludes there must be a highest number on philosophical grounds, despite his inability to demonstrate it, because 1. We can't know all numbers have the same properties and 2. Because everything must come to an end.

So the mathematician, with his basic premises now being questioned is forced to now go on the offensive:

"Alright then. So what is the highest number?"

"I don't know. In fact, no one can know. Because if someone thought of it, they could add one to it, and it wouldn't be the highest number"

The great irony here is that the questioner has now turned the mathematicians technique against him. So the number claimed to be the highest in the original argument, simply isn't, making it invalid. At this point the mathematician's patience has surely run out and he has already decided that the questioner is either 'stupid' a 'crank' or both.

"So you mean to tell me you think numbers end somewhere. Wouldn't that be arbitrary. Do you suppose there is a limit to the number of things that could exist. Maybe there exists an infinite number of universes. That would be impossible if there wasn't a 'number' for each and every one of them!"

"An infinite number of universes couldn't exist"

As a last ditch attempt to convince the questioner that there is no last number, the mathematician is forced to get ultra platonic:

"Alright, fine. There can't be an infinite number of universes, or time, or whatever. So what? Numbers don't need to be physically represented to have meaning. There may only be 10⁸⁰ particles in the observable universe, but 10¹⁰⁰ still has meaning-"

"because you can write it down"

"Yes but there would even be numbers we could not write down, yet we could still imagine them. Just imagine a number so big that there is no way for us to express it. There we've imagined something without any way of expressing it"

"But what number have you imagined. That's like imagining things going on forever. How much is that? It makes no sense, it's not specific, it's just the idea of adding more and more"

"It can be made specific. For example say the smallest number larger than any number expressible with a hundred decimal digits."

"That's not a good example, we already showed you could express that easily"

"It was just an example. We would also say there is a smallest number larger than any expressible using 10¹⁰⁰ standard mathematical symbols"

"But your still expressing the number somehow. Even if you can't express it using less than 10¹⁰⁰ your still expressing it in some manner. How can you claim that there exists something you can't express. That's contradictory"

So with about every bridge burned the mathematician has no other choice than to simply assert:

"Numbers are abstractions! They don't exist in the same way that you or I do, or even the same way symbols on paper or ideas in my head do. They are simply truths that exist outside of time and space. They can be forgotten, but they can always be rediscovered again. If there is any life out in the universe they too would come to the same mathematical conclusions that we have. Now this is incontestable: If we accept that one is a natural number, and the successor of every natural number is a natural number, then we must conclude there is no greatest number. If you want to start from some other assumptions and arrive at another conclusion, fine. But it doesn't change the consequences of following these premises to their natural conclusion"

"How can you say it's incontestable when you haven't even bothered to check. I agree that one is a natural number, and that adding one to a natural number gives you another natural number. Okay so I guess two is natural, and so is three, and four, and so on. But no matter how far you've gone, assuming you haven't already reached the highest number, you can't know whether or not the process has an end or not. As you yourself said, we can't observe an actual infinity, so then there is no way to directly prove that it exists"

"We don't need to prove it exists, it's axiomatic."

"That's just claiming it's obvious. I disagree"

"Ad infinidum it would seem" jeers the mathematician ironically.

And with that our battle of wits has reached an end, or rather an impasse. The thinking of the questioner is just too different for any agreement to be reached. The mathematicians frame of reference is just so different than his arguments fall on deaf ears.

You may now be thinking that my intrepid questioner is just a silly rhetical device. Believe it or not, there seem to be people who seem positively allergic to infinity, which is that which never ends, ... ever. I must admit a certain discomfort with it myself, though it does not extend to outright disbelief. Although I may be accused of painting a rather sympathic view of the questioner, it must not be concluded that my ideas are synomous with his. In many respects I'm more on the side of the mathematician. There is no reason to reject the infinite outright. All arguments against it seem to rest in the presuppostion that all things must come to an end. Why?

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There are a few common responses to this question, and while each is a logically self-consistent response, none of them really satisfies the impulse that caused the question to be asked in the first place, ... at least not for some. I'll explain what I mean by that sardonic comment in a bit, but first let's examine the common responses to this question, and deconstruct them while we're at it.