Tangibility of the transcendental 2 - updated.

I'm re-posting this blog entry with a crucial correction to the λ function, which previously wasn't properly defined. 


"I was doing a bit of reading about transcendental numbers today, and had an idea of constructing my own well defined transcendental. I was inspired by the Liouville constant. Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:

m_p =  0.2357911131719232931...
or the natural numbers:
m_N = 0.12345678910111213...
The construction proceeded as follows. First we define the function λ:→. Intuituively λ(n) gives the number of digits in n, i.e. the "length" of n in base 10.

Now we can express the numbers as infinite sums:

Where p_n is the n'th prime.

Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.

To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My m_N is in fact a constant previously formulated by D. G. Champernowne. And it turns out that m_p is already known as the Copeland-Erdos constant:

Notice the striking similarity in the formulation - I have formulated m_p completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my λ function acts a lot like the floor of base 10 logarithm of y. In fact the relation is the following:

Which makes the sums in the exponents of the definitions of m_p and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, so we need to add k back to the sum. And consequently,

Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both m_N and m_p are irrational, but now I also know that m_N is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which I suspect would be a herculean task."

A simple idea.

Physicists disagree about what exists, but they tend to agree that nothing is just the lack of that which exists. Note that this is physical nothing, or physical impossibility. Nothing "contains", as it were, all the physically impossible objects, but there are none, so nothing is "empty", as it were. There is one more interesting point to add, which will play a crucial role later on - sure physicists will agree that nothing is the lack of any physical thing, but since what counts as a physically possible object needn't be congruent across physical theories (it's not), physicists will disagree about the list of stuff that nothing is meant to denote the lack of.

Is it really that difficult to extend this idea to logical impossibility? I think not. Analogously, we extend the idea of physical nothing to logical nothing as the lack of what can exist. Here, nothing "contains", as it were, all the logically impossible objects, but there can't be such things, so nothing is "empty", as it were. Furthermore logicians will tend to agree that nothing possible is the lack of any possible thing, but since what counts as a logically possible object needn't be congruent across logical theories (it isn't), logicians will disagree about the list of stuff that nothing possible is meant to denote the lack of.

Clearly, in some set theoretic ontological model nothing possible can be identified with the empty set. That is we think of the empty set Ø_K(L) as the object associated with all the impossible objects in the L theory, K. We can thus speak of the order of the empty set, here called the rarity, defined as the cardinality of all the K(L) impossible objects.

Illustrating some of Ludwig Wittgenstein's ideas from the Tractatus.

It Goes Without Saying.

After a decade of a comfortable tenure as reader in baroque logics, Lev felt the onset of an uninspiring impasse coming on. He could sense the cold, judgemental gaze of his best work in the field of non-normal world conjuration, staring at him from the old, dust covered journal volumes, stacked neatly on the shelf near his desk.

Some say that it must have been the mixture of mundane routine and the onset of his flourishing that led him to leave the academy. Before his seven year disappearance many of his close associates reported that Lev’s long cherished, yet latent interest in eastern philosophy – Zen Buddhism in particular – took the form of an obsession. He would constantly talk of the higher jhanas, emptiness and other eastern concepts. Subsequently his interest in quietist philosophy grew. It is believed that he wrote this haiku just weeks before his now legendary departure.

a field of wheat

waving in the breeze

whispers

It is commonly agreed that he spent most of the time in India, Tibet, China and Japan. But it was the series of events which unraveled over the next seven years following his return, that made Lev one of the pivotal figures of 21^st century philosophy – in particular the establishment of the field that has become to be known as radical quietism. Although a term which Lev never himself used, the foundation of radical quietism has been unanimously attributed to him by philosophy historians.

His seminal publication in the Hush! Quarterly, is nowadays considered as the turning point in Lev’s philosophical career – this 37 blank page tour de force established him once and for all as the founding father of this new approach, or as some say – style, to philosophical inquiry. His associates and peers agreed that this indeed was the most that he didn’t say in decades. This revolutionary publication, or radical quietism manifesto as it is often referred to, received an immediate non-reply of awe and the highest acclaim from the Hush! Quarterly editors and the quietist community at large.

More papers followed. In the subsequent seven highly productive years, Lev published over thirty papers, each no less brilliant than the preceding ones; each with an equal clarity to the initial gem; each beaming with equal passion and fervor of Lev's intrepid genius. Also given that each subsequent article referenced precisely the previous ones, including page numbers, a complete body of work emerged over that decade which rightfully so is unanimously considered as the foundation and the purest source of not only radical quietism, but quietism in general.

The Poet, the Chemist, and the Physicist.

The trio met one day over some wine and discussed matters of rhyme, thyme and time. Soon the conversation turned to the notions of nothingness and emptiness - what is empty? The poet finished his glass, and pointed to it --- as far as I am concerned this glass is empty, as it is devoid of wine - the sparkling grape, the drink of gods. Hold on a minute! --- exclaimed the chemist --- surely it's not empty as it contains air - we'd have to pump all the air out of the glass, creating a vacuum in it, and then and only then would it be empty. It wouldn't make any difference to me --- replied the poet, shrugging his shoulders. Please let me interject at this juncture dear fellows --- interjected the physicist --- and let me put an end to your obvious confusion. Vacuum, as you describe it doesn't cut it at all, since we know that even empty space is a breeding ground for virtual particles whose immediate annihilation results in what we observe and call vacuum energy. The poet looked at the physicist with a frown of suspicion --- I know nothing when I see it good man, and I won't let anyone tell me otherwise - let the bartender settle this matter - haloo, good fellow! Another round please!

Zeno's Apology

We know that the ancient Greeks loved their wine. So must have Zeno, the Greek philosopher who among other things is known for devising numerous paradoxes intended to endorse the views of Parmenides.

One of those paradoxes, called "The Dichotomy" claimed that you could never traverse a finite stretch of space, for in order to get to the destination one would have to first get to the half-way point, but before that one would have to reach the quarter-way point, and so on. So since traversing a finite stretch of space entailed performing an infinite number of tasks, reaching the destination, Zeno argued was impossible. 

We could rephrase this paradox in a way to claim that finishing a goblet of wine is impossible. For before one drinks the whole lot, one need's to drink half first etc. 

Now, either Zeno was also a very unusual Greek insofar as he abstained from wine - which is very doubtful, and would surely be insulting to even assume - or when it came to wine drinking he could perform the impossible! 

I'd take that as a valid defense, if Zeno chose to use it.

Much Ado About Nothing

AN EXERCISE IN INTUITION
Suppose we have two physical vessels of varying size, U and u, and suppose further that they're both empty (in the usual sense - we've run out of wine). Should we consider the degree of their emptiness as the same (i.e. even talk of such degrees is nonsense!), or can we say in a meaningful way that their degrees of emptiness differ?

AN ARGUMENT
Consider two urns in a marble shop. The shopkeeper labelled them in the following way - urn 1 holds only (and can only hold) white marbles, whereas urn 2 can hold both white and black marbles.

Now consider a new assistant that has just been offered an apprenticeship in the shop of his dreams (he loves playing marbles), and is not as yet familiar with the arcana of the labeling system, i.e. he hasn't as yet been told by the shopkeeper what the restrictions on the contents of the urns are.

Now consider the statements 'urn 1 contains no white marbles' and 'urn 2 contains no white marbles' - they entail different things, depending who is exposed to that information - the shopkeeper or the assistant. In particular, to the shopkeeper it means that 'urn 1 is empty'.

On a more technical side, we can substitute the terms 'has no white marbles' and 'is empty' interchangeably salva veritate in the context of urn 1 (charity permitted, for naturally one could generate oddly sounding sentences). This cannot however be done in the context of urn 2.

The above argument, and exercise in intuition do not exactly express the same ideas, but overlap on what is essentially the general idea.