What are dreams, mum?

I was a very inquisitive child, or at least so I've been told, and the beginning of my ability to speak was marked with an onset of a daily torrent of interrogations, which I subjected my mother to. I used to ask my mum -- the all knowing oracle -- about basically anything and everything.

"What are dreams, mum?" -- I inquired one day. This question, as she later told me, had stumped her at the time -- she didn't expect it coming from a six year old, any more than she knew how to explain it to one. 

"Well" -- stalled the oracle a little as she gathered her thoughts -- "they're a kind of recapitulation, night-time thoughts if you will, a replay of the experiences you had during the day -- that includes the people you may have encountered during that day, such as your friends and your teachers". 

"So if I were to encounter a monster" -- I pried (I was obsessed with dinosaurs and monsters) -- "I may dream about one the following night?"

"Yes, although as you know there really are no monsters" -- my mum took the opportunity to reassure me.

"So if dreams are night thoughts about the things I encounter during the day" -- I persisted -- "why do I sometimes dream about monsters?"

Here, my mum was a little stumped, and didn't know what else to say -- after all she didn't have a degree in psychology. After a little while she refined her answer -- she always came through, being the oracle -- "You may also dream about any thoughts that you may have during the day -- why, you may even dream about your auntie who lives in America!"

"What if I think hard enough about something I like" -- the merciless torrent continued -- "I may dream about it too?" -- I asked, brimming with excitement about having devised an idea for an experiment to be conducted over the coming days and nights.

Now, having looked back, I realize that I've never actually stopped asking such questions.

Friendship

One of the most beautiful qualities of true friendship is to understand and to be understood.
Lucius Annaeus Seneca

It's difficult to overstate the importance of this.

Is necessary absence the same as impossible presence?

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Ruminations on Everything and Nothing

Common relevance - situational assumptions.

The common-use ontological notions of nothing/emptiness are context dependent, with enthymematic information filling out the gaps concerning what kind of absence is under consideration, and as such relevant absence. This isn't very different from notions of universal nature, such as all/everything, where what counts as anything at all, is conditioned on the relevant context. I will argue that the empty class is a relative notion, and is conditioned by what is considered as possible/impossible in any given context, whether it be an informal one (situation) or formal (formal theory).

For example "she ate everything" (in most everyday cases) certainly doesn't refer to some paradoxical Leviathan who has devoured the entire world and would have to eat itself also, if the statement was to be considered as is, without qualification. Rather, almost always, the context dictates what the scope of quantification is restricted to - and in most cases the type of food stuff contents of, say a plate, or fridge etc.

Similarly in the cases of common expressions such as "the house is empty", or even ones like "the house is absolutely empty", or "there's nothing left" the scope of quantification is restricted to the things that count as ontologically relevant e.g. people, furniture, etc.

Relevant scope of quantification can be thought of as relevant possibility (context conditioned/indexed possibility). When I say "Put down your pencils everyone!", in a room full of students somewhere in one of the University of Queensland St. Lucia campus lecture rooms, at the end of a multivariate calculus quiz, this address does not include the security guard checking id's in the Biological Sciences Library, nor does it include my friend who at that very moment is sailing the Mediterranean Sea - they're outside the scope of relevant quantification, since they're not even enrolled in MATH 1052 (arguably, a non-controversial minimal requirement). This would still hold if by some strange circumstances my sailor friend somehow acquired a copy of the quiz and was solving the problems simultaneously with my students. Not being a relevant someone in that context, she is a relevantly impossible someone. 

Of course there is a world where she arranges with the school to be allowed to take the test remotely, and as such participate in the quiz via some appropriate video-phone technology. However in most cases of common language use, such peculiar circumstances do not tend to be part of enthymematic content (common situations don't imply uncommon information about them, i.e. our reasoning in most cases is governed by a ceteris paribus set of assumptions), but rather carefully explicated instead.

Furthermore, if after 15 minutes or so, having marked "all" (the ones handed out in the classroom) the quizzes, I said "No one got full marks.", and if it turned out that my sailor friend in fact got all the answers right, I would still be telling the truth, for my friend fails to be in the scope of the relevant quantification (indexed by that informal context). 

Formally, we can think of my friend as being an element of the empty set (here indexed by the above quiz scenario), as far as "someones" are concerned. But since empty sets have no elements, instead we think of the sailor friend being in some sense associated with the empty set indexed by the quiz scenario. Since such relevantly impossible objects can be thought of as a collection, we can also talk of the cardinality of that collection. But since we're interested in a particular relation such a collection has to the empty set indexed by the quiz context, rather than using the term cardinality, we use the term rarity, which captures the intuitive meaning of the cardinality in question.

Meta relevance - logical assumptions.

There seems to be no reason why the above arguments, and their form, cannot be generalized, and as such extended to formal languages, logical systems and their theories. That is, what counts now as relevantly possible in some theory (a theory being the situation analogue of the above quiz scenario), is just what is possible in the logic on which the theory is based.

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.