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Department of Anomalous Mathematics Weekly Colloquium - Lecture 1
Dr. Conway, head of the Department of Anomalous Mathematics, approaches the podium
Ahem. Okay everyone! If everyone could please be seated. Thank you.
Rustling of chairs as audience takes their seats
Thank you everyone for joining us today. The formation of the Foundation's specially dedicated Department of Anomalous Mathematics has long been overdue and I'm glad to see that we're all getting to know each other a little better. Let me be the first to welcome everyone. As you're all aware, over the next few weeks, some of our senior mathematicians are going to share what they're working on. Hopefully this will foster collaboration between researchers working on similar problems and give some of our junior researchers an opportunity to see what progress has been achieved in containing and studying anomalous mathematics. I'm sure we'll all learn a lot from these weekly sessions.
Now, without further delay, let me now introduce our first speaker.
Looks down at paper
Professor Hutchinson is one of our most esteemed researchers. Before being scooped up by the Foundation, he produced a number of remarkable insights, particularly in number theory and, less conventionally, analytic philosophy, for which he won a number of awards. Since he's been with us, he's led the group studying SCP-033, one of our earliest examples of anomalous mathematics. I think it's fair to say that he's been a key figure in developing and studying anomalous mathematics. If you'll please, Professor Hutchinson everyone.
Gestures towards an elderly man sitting at the front who stands up and takes the podium. Polite clapping as Dr. Conway takes a seat.
Thank you John for the kind words. Let me first say that it's a great honour to be the first speaker of our new department's weekly sessions. I look forward to getting to know you all a little better.
As John said, my group has primarily focused on studying and containing SCP-033. I realise that most of you have probably already read the unclassified documentation but to be honest, that article could really do with an update so let me briefly go over what SCP-033 is. I'm sure we'll get round to updating the article any day now.
Knowing grins from many members of the audience
SCP-033 is the missing integer. That's right, there's an integer which is missing. No it's not some obscenely large monstrosity. It's a small positive integer less than 10. And no, before you ask, I can't be more specific than that. It's exact size is classified for reasons which will become apparent. Now some of you are asking yourself 'how that's possible? Surely the Peano axioms can be used to define all the positive integers sequentially? How could we skip one?' Well it turns out that things are a little more subtle than that. If we're going to be rigorous, the Peano axioms define the 'natural numbers'
Draws a large $\mathbb{N}$ on the blackboard behind him
Now for the purposes of this discussion, when I say 'integer', I mean a number whose fractional part equals 0. More precisely, a number is an integer if it is equal to its own integer part. We can of course use the natural numbers to define a particular closed group of integers,
Draws a large $\mathbb{Z}$ on the blackboard beside $\mathbb{N}$
but that doesn't mean there aren't any other integers. Let's call $\mathbb{Z}$ the set of classic integers. It turns out that it's surprisingly easy to show that in the real numbers
Draws a large $\mathbb{R}$ on the blackboard below $\mathbb{Z}$
there exists another integer, theta prime, which isn't a member of $\mathbb{Z}$.
Adds $\theta'\in\mathbb{R}$ and $\theta'\notin\mathbb{Z}$ to the board
To be clear, the number doesn't belong to $\mathbb{Z}$ but is still an integer because it's fractional part equals 0. I wish I had a pure existence proof to share but it turns out that all the proofs are necessarily constructive in nature so showing them would be a breach of classified information. What I can share with you is that the axiom of completeness is necessary to generate a set of numbers containing theta prime. Fields with cardinality less that $\mathbb{R}$ won't contain it.
Okay. So far so good. The sharper members of the audience however have probably started wondering to themselves what's so anomalous about this number? I mean, it's weird for there to be a small integer which everyone missed but is it really any weirder than say the Banach-Tarski theorem? Every mathematician worth his salt knows that strange things happen in the continuum.
Well there are two ways to answer this question. The superficial answer is the empirical observation that physical substrates which have this number 'derived' on them tend to disintegrate. The more 'uniform' the medium, the faster the disintegration happens. This is true whether the derivation is hand written or stored electronically. Of course this has made storing derivations of theta prime a pain but to be honest, it's far from the most difficult anomaly the Foundation has contained. This is also why details of the exact value are classified. We don't want a 'universal disintegrator' loose on the web.
A more sophisticated answer requires delving into the relationship between theta prime and nature. Aristotle claimed that nature abhors a vacuum. Well as far as we can tell, nature really abhors theta prime. We have never ever observed a physical system which was characterised by theta prime. Let me clarify, every mathematician knows in their heart that numbers really exist in some abstract sense. At the same time, we can't deny that they are also often useful for describing physical phenomena. The natural numbers are useful for describing sets of physical objects. A trivial use of the negative numbers is to describe debts. Even the complex numbers are needed to describe the quantum wave function. Well theta prime describes nothing. No set ever contains theta prime objects, no force ever applies theta prime Newton's, no pair of objects are ever theta prime meters apart. And believe me, we've tried really hard to design systems which would necessitate theta prime. No doubt this is one of the reasons it took so long to discover.
By way of analogy, it's closely related to the large cardinal $\aleph_{1}$ and the continuum hypothesis. We all know that the cardinality of $\mathbb{R}$ is strictly greater than that of $\mathbb{N}$ but nobody knows whether it's cardinality is $\aleph_{1}$ or some other large cardinal. Going a step further, it's been proven that the answer is undecidable in ZFC so it can't be proven either way using the ZFC axioms. But this is surprising. I mean, all we need to do to show that it's cardinality is greater than $\aleph_{1}$ is construct an uncountable subset of $\mathbb{R}$ with cardinality less than $\mathbb{R}$. So what we've actually proven is that it's impossible to construct such a subset using ZFC but that such subsets might exist anyway. Now it seems clear to me that if such subsets were needed to describe a physical phenomenon, then even if we couldn't construct them, that would resolve the continuum hypothesis. But no such phenomenon has ever been discovered. If the continuum hypothesis happens to be false, we might conclude that 'nature abhors $\aleph_{1}$' but of course, it's unconstructable character guarantees that we will never observe it's abhorrence. Theta prime, in contrast, is readily constructable which is why we have no difficulty observing it's disintegration property so easily.
I see I've begun losing many of you. Let me add one final point before we move onto questions. If the above reasoning is sound, then we reach an interesting philosophical conundrum, namely is theta prime absent from nature because theta prime 'disintegrates' all systems which it characterises or is theta prime absent because nature prevents it from manifesting? That is, is it theta prime which is anomalous or the universe? Our original research took for granted the former but over the last few years, I've begun taking the latter possibility more seriously. I mean, from a purely mathematical point of view, theta prime is barely stranger than pi. All the standard operations are well defined for it and in fact a number of very beautiful identities can be written down using it. On the other hand, as everyone who works for the Foundation knows, the universe has a harsh and capricious character. I wouldn't put it outside the realm of possibility that nature itself has it out for theta prime. It would hardly be the worst thing 'the universe' has done.
Anyway, enough from me, I think now would be a good time for questions.
Several audience members raise hands
If you don't mind stating your name and area of research before asking, that might speed up getting to know each other. Yes, the gentleman in the three piece suit.
Hands are lowered while an over dressed man awkwardly looks around before speaking
Um, oh, thank you for the very interesting talk. I'm Dr. Alvin and work on analytic numerology. My work occasionally brings me into contact with semioticians, you know, those guys who study symbols and things. Anyway, I once had a rather lively discussion with one of them in which she insisted that the idea of a missing integer was incoherent and that SCP-033 is nothing more than a rune which disintegrates matter while projecting a memetic belief on observers that the symbols convey a number. Um, what do you think of such a theory?
Ah yes. The good old 'rune theory'. No offence to semioticians, I mean some of my best friends study semiotics, but your typical semiotician couldn't even list Euclid's axioms of geometry let alone anything as sophisticated as the Peano axioms or ZFC. I mean, if you can't even define the classical integers, why on Earth do you think your qualified to have an opinion on whether anomalous integers exist? Basically, as far as I'm concerned, the math is needed to derive theta prime is no different to that for any other real number so if you want to insist that theta prime is actually a rune as opposed to a number, it seems to me that you'd have to reclassify every number as nothing more than a rune. I know of some rather extreme semioticians who in fact argue for that but to be frank, I've never really understood why that isn't just inventing new words for the same thing. As long as you don't tell me how I can and can't do math, I don't really care what you call it.
Next question.
Hands are raised again.
Yes, the young woman sitting on the aisle.
Lowering of hands.
Hi, junior researcher Natali. Just started working on Algebraic Zoology. I was wondering if you could perhaps give some properties of theta prime. You know, is it odd or even, prime or composite, etc?
Okay. Yes, I see why you might have that confusion. Let me again emphasise that theta prime is a member of $\mathbb{R}$ and not $\mathbb{Z}$. Just think of it as a typical member of $\mathbb{R}$. Is the square root of 2 even or odd? Is pi prime or a composite? These concepts are defined for the classic integers. We don't have even or odd real numbers. So to answer your question, theta prime is neither even nor odd, it is neither prime nor composite.
Tall man in a lab coat asks out loud
Is it irrational?
Proffesor Hutchinson looks around for the source of the voice, eventually settling on the interruptor
You didn't say your name or area of research. In any event, like I said earlier, $\mathbb{R}$ is the smallest group which contains theta prime so yes, theta prime is not a rational number. Maybe I should spell it out, theta prime is also not an algebraic number or even a computable number. Of course this is true for most reals so it's hardly all that surprising when you think about it for a moment.
Okay. I think we have time for one more question.
A few hands are raised
Yes, you in the rather amusing T-shirt.
Hands are lowered while the man in the T-shirt looks rather smug
Duncan Kemp from Anomalous Probability and Statistics. Is theta prime the only anomalous integer? If not, do we know how many there are or anything about their distribution amongst the reals?
Ah. Great question. The answer is we just don't know. Theta prime is the only anomalous prime we know of but existence proofs have evaded us making it difficult to even conjecture on whether other anomalous integers exist.
Tall man in the lab coat loudly asks
Can't you just add 1 to theta prime to get another anomalous integer?
Professor Hutchinson rolls his eyes
Really? You don't think any of us studying theta prime thought of that?
Professor Hutchinson sighs audibly
You're still thinking of theta prime as a member of $\mathbb{Z}$. The characteristic feature of the classical integers is that they're closed under addition but again, theta prime isn't in $\mathbb{Z}$. So no, adding 1 to theta prime does not give you an integer in the same way that adding 1 to almost all real numbers doesn't give you an integer. You think that's weird? Well that's why it's an SCP, okay?
Department head, Dr. Conway, approaches the podium and takes over the mike
It looks like that's all we have time for. Thank you again Professor Hutchinson for a very interesting talk. I'm sure he'll be happy to answer any other questions afterwards. And thank you all for coming. I look forward to seeing everyone again next week when Dr. Buckhanon will tell us about his research into SCP-$\mathbb{Y}$.
