"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Wednesday, December 18, 2013

How empty is { }? Different orders of the Empty Set. Coursera Forum discussion.

Here is a raw cut-and-paste of the exchange on this topic from a Coursera forum.

I'd like to thank those who provided constructive criticism, or just picked at the idea with an unbiased razor of skepticism. Apparently most of the interlocutors were inspired, and some have even admitted to having been led to think in a new way --- to a philosopher, there's no greater reward :)

It's this sort of exchange that allows the idea to mature, and hopefully catch on :)
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Mariusz Popieluch· 2 months ago
Let me introduce the discussion with a little humor.

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!

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The above humorous scenario pertains to natural language and some theories of varying degree of conceptual refinement, but the idea extends nicely (I believe) to formal languages, theories and logics too, thus generalizing the idea of emptiness. In particular it intends to do to the notion of emptiness and it's various formalizations such as the set theoretic ∅, what Cantor has done with the notion of infinity - show that there is structure within it, and varying orders of it.
I know this is a radical idea, but rest assured that the idea of varying orders of infinity wasn't initially taken lightly either - even today many still think it's preposterous. But interestingly enough, there seems to be a demand for such a hierarchy - hinted at in the literature - but I'm not aware of a formal theory of various orders of the empty set being developed.
The general claim is that what counts as empty is dependent on the strength of a logic underlying the theory in which it is defined and the expressibility (richness) of the language of the theory. That is, in some theories there are more objects that fail to be members of the empty set (e.g. to the physicist there is more to the contents of the glass than wine and air), thus rendering the empty set somehow emptier (to put it simply).
Or to put it another way - we could just define the order of the empty set by the cardinality of all the impossible objects in a given language/logic/theory. Those impossible objects are not actually members of the empty set, but rather are assigned to it in some sense - by saying that some object (this includes linguistic objects) is an element of the empty set (in a given theory), is just another way of saying that this object is impossible (in that theory).
Example.
Consider classical logic CL and some paraconsistent logic, say LP. In LP contradictions are not considered as impossible. Now define a property G: of being green all over, and not being green all over. Now, in CL any object with that property is considered as an impossible object, take the object a, so in CL we obviously have Ga ∈∅, but in LP we have Ga ∉∅. Hence the suggestion to index ∅ by logics - this may become crucial when talking of various theories in some metalanguage/meta theory.

Philosophically speaking "{ }" is an object with a fixed extension across theories (it has no elements), yet varying intension. What the above aims to show is that the varying intension can be accompanied with a varying quasi-extension, i.e. rarity as opposed to cardinality.

Mariusz Popieluch· 2 months ago

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.

David M. Kaziska· 2 months ago
In your first post the set, {wine in the glass}, is empty after the wine is consumed (neglecting any residual wine).  For the poet's own purposes it is empty but we may his or her statement that it is empty as colloquial in light of the chemist's and physicist's later statements.  I'm not seeing mathematical consequences which would require redefining the notion empty sets.  {wine in the glass} may be an empty set but {air in the glass} may be nonempty.  

In your later post, the shopkeeper reasons as follows.  (P1)  All marbles in Urn 1 are white, (P2)  There are no white marbles in Urn 1, therefore (C)  Urn 1 is empty.  The assistant has (P2) but not (P1) and cannot draw the conclusion.  Urn 1 is still empty, though, it's not a matter which is relative to the observer.

Mariusz Popieluch· 2 months ago
Thank you for your reply David. I'll do my best to address your observations, and further clarify the idea. :)

The later post intends to illustrate the idea that in some theory T based on some logic L1 some formula F may express an impossibility (which we can express as that formula being "an element" of, or assigned to, the empty set), whereas the exactly same formula in an analogue to T based on a logic L2 which is stronger than L1 may not come out as impossible, and as such cannot be expressed as "belonging"/being assigned to the empty set.

Note: "element of" and "belonging" are terms I tentatively use to denote some kind of assignment of F to the empty set (or empty sets, since I'm arguing that they're needn't be unique across analogous theories based on distinct logics). Note also that for each single theory the empty set still remains a unique object/element, as theories are usually based on a single logic.

The epistemic states of the protagonists of the marble shop illustration (informal context) correspond to the assumptions/axioms/conditions/semantics of theories based on distinct logics in the formal context.

Some logics distinguish semantically the propositions "p & ~p" and "p & q". In particular they consider the first proposition to take a fixed value for all valuations (contradictions are always false, and only false in classical logic). Some other logics do not make that distinction.

Likewise the shopkeeper distinguishes the two propositions concerning the urn contents, whereas the assistant doesn't. Try to think of the urn labelling system and the protagonists' distinct epistemic states as an analogy for the distinct meta-linguistic propoerties of analogous theories based on distinct logics, and the propositions concerning the urn content as object language entities, i.e. formulae.

But if the informal analogies muddle, rather than clarify the idea, I'll be happy to talk more formally about it. :)


Louise Craven· 2 months ago
Every set is a subset of the relevant domain of discourse. Thus if we are only talking of natural numbers, the empty set is the set containing no natural numbers. The poet was thinking of drinks when he said his glass was empty, so it truly was. The chemist and physicist tried to confuse him by exploiting the fact that he had not explicitly stated the domain of discourse, and they were also right about the nature of the empty set within their individual domains of discourse.  In the marble example, the difference between the shopkeeper and the assistant is not in the 'size' of the empty set, but in the inferences they can draw from the fact of emptiness, which depends on other information.

I am, however, intrigued by your suggestion and it may be that in some applications of mathematics to real-world problems, the size of the empty set in the model may need to vary according to the significance of 'nothing' in the area modelled. For instance, complete extinction of a species is of much greater significance than the remaining existence of a number of breeding pairs much smaller than the population at a previous time.

I shall go now and give it more thought....

Mariusz Popieluch· 2 months ago
Hi Louise - thank you for taking the time to read the OP.

The informal anecdotes are mere illustrations of the general idea.

In the first scenario I intended to show, via the relevant domains of discourse illustration,that given the same language (in this case natural language), but distinct theories, identical propositions (formulae, i.e. linguistic objects) entail different things - in particular, where in one theory "no wine in glass" does entail "glass is empty", in other theories it doesn't. I stress again, that this is merely an illustration of meta theoretic properties, rather than a discussion concerning restricted quantification to relevant domains of discourse per se.

Think of the Poet, the Chemist and the Physicist as analogues of theories of "The stuff on the table, at the local restaurant.", albeit based on dictinct logics. In the poet's logic "there's no wine in the glass" can be associated with the empty set - this however isn't true in the chemist's or physicist's logics. Likewise in the chemist's logic "there is no liquid and no gas in the glass" can be associated with the empty set - but again, this isn't true on the physicist's logic.

As for the second scenario, please see my above reply to David. And please do not shy away from addressing the formal content of the OP.

Andrew Kelley· 2 months ago
First let me say that I enjoyed reading this post. It made me think in a way I hadn't thought before.

I think you may be on to something with recognizing different orders of the empty set. However I am as of yet unable to think of any examples where the knowledge of such different orders is relevant to solving a problem. Are you?

Mariusz Popieluch· 2 months ago
It may serve as the ontological foundation for a theory of comparative impossibility.

Francisco Vasconcelos· 2 months ago
I think that the analogy between empty set and infinite set is not the most correct one. The evident oppositions would be between empty and complete set, and infinite and infinitesimal amounts.

So on the empty/complete side, I'm not aware that there are different orders of completeness. Both these terms are usually used as binary categories applied to other sets, both finite and infinite. Yes, their meaning varies with language context, but so do everything else, such as the number "2", the operator "+", or the word "number".

On the other hand, I think it's easier to make the claim that there are different orders of infinitesimal quantities. For example, think about the probability of picking the number "5" at random from the set of all natural numbers. Since the set is infinite, the probability is 0. However, it seems that this "0" is still bigger than the "0" probability of random sampling the number "5" from the set of all real numbers, since the pooling set is infinite to a higher degree than the previous one.

Andrew Kelley· 2 months ago
I'm not buying that you can say that one 0 is bigger than another 0. Zero is zero. Can you prove it?

Mariusz Popieluch· 2 months ago
Francisco, thank you for your reply -- you're right that it's not the most correct analogy, as it's not intended to be a directanalogy, i.e. a dual of sorts. It's a weak kind of analogy, in the sense that in both cases of infinity and nothingness/emptiness we're dealing with concept which apparently don't admit to degrees. Whereas Cantor developed the idea of bijection as the criterion for equinumerosity, I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅L.

As I said before, in the above response to Louise, the contexts of relevant discourse analogy is again merely an illustrationof the more precise idea. See the above reply.

The observation you expressed in your last paragraph puts you in good company. In the context of probability theory the idea of different orders of “zero” has been hinted at by Andrey Kolmogorov and Bruno de Finetti as a possible candidate to solving some probability theory paradoxes – “Like Kolmogorov, de Finetti is occupied mostly with probabilities defined directly on arbitrary uncountable sets; but he views additivity differently, and is led to such anomalies as an unlimited sequence of layers, like an onion, or different orders of zero probabilities that add up to one, etc. ” (E.T. James: Probability Theory, the Logic of Science, 2011, p.656).

Francisco Vasconcelos· 2 months ago
Andrew, try not to read my "0" as the integer number 0, but as an infinitesimal quantity that is infinitely close to zero, something like the result of limx→∞1x. As infinity can have different sizes, the result of this limit quotient should be able to have different sizes too.

Mariusz, thanks for the info

" different orders of zero probabilities that add up to one"

this is definitely interesting and worth checking out.

Regarding your emptiness orders:

"I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅L."

I get it now. So I guess you're trying to say that in different theories the empty set can be connected to a set of propositions that can have a different finite or infinite size.
Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?

Andrew Kelley· 2 months ago
We can mathematically prove that infinity can have different sizes by using bijections. But we cannot mathematically prove that limit of 1/x as x approaches infinity is not equal to the integer zero. In fact, we *can* prove that, for example, 0.999999 repeating is exactly equal to 1. Not a number which has an infinitesimally small difference than 1. 1.

Francisco Vasconcelos· 2 months ago
Andrew I'm still trying to figure out what different kind of "0" can mean, it seems that this concept raises some issues in probability theory, but lets forget that for a moment and define the following:

For any functions f(x) and g(y), assume that
limx→∞f(x)=0
limy→∞g(y)=0

Now suppose the following hypothesis:
For any arbitrary ϵ, there is NOT a bijection between all possible values of g(y) and f(x), for x,y≥ϵ

If this proposition is true, then the limit of g(y) and f(x) being "0" might have different meanings, since we know that for any given ϵ, one of the sets will always be larger than the other.


Mariusz Popieluch· 2 months ago
@Francisco: "Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?"

Well, personally I'm not entirely on-board with that idea, for the reasons I stated in my reply to David above: "Note also that for each single theory, the empty set still remains a unique object/element, as theories are usually based on a single logic."

My position is such, due to what I mean by "different orders of emptiness". But this is not to say that I'm not open to the variation of this idea, which you and the company of famous thinkers suggest. :)

Andrew Kelley· 2 months ago
Francisco, thank you for this example. I think you may be right but I am finding it extremely hard to wrap my brain around it.

Hayden VanIderstine· 2 months ago
Francisco, in order for the hypothesis that there exists no bijection between all possible values of g(y) and f(x) when x,y>ϵ, then it is necessary that the cardinality of the set of all possible g(y) with y>ϵ be different from the set of all possible f(x) with x>ϵ. For this to be the case, then both sets cannot both be of any of the following cardinalities:

Finite
Countably infinite (set of naturals)
Uncountably infinite 1 (set of reals)
Uncountable infinite 2 (set of all functions defined from the set of reals to the set of reals)

What I am wondering, is what is the domain and range of f and g, because if they both have a range being a subset of the real numbers (or the entire set of real numbers), then there necessarily exists a bijection between them.

Francisco Vasconcelos· 2 months ago
Hayden,

I'm not sure I follow you. The way I see it, for there to be no bijection it is only required that the domain of f(x),x>ϵ has a different cardinality of g(y),y>ϵ. So, for example if the domain of f(x) is a subset of the rational numbers, and the domain of g(y) is an interval of the real numbers, then there is no bijection.


Hayden VanIderstine· 2 months ago
In my opinion, excellent post Mariusz. I am thinking in a new way now thanks to you.

What we consider as empty is dependent on what we acknowledge exists.

{n∈N | 1 < n < 2} =  ∅
Exactly  |N| elements fail to be in the set {n∈N | 1 < n < 2}

But exactly |R| elements fail to be in the set {x∈R | 1 < x < 2}.

According to your definition of the rarity of some empty set, would the rarity of the empty set in a logic which isn't strong enough to construct the reals, but is strong enough to construct the naturals, be |N|, because there are only a countably infinite number of sentences like 1 < n < 2 which could be made which are impossible?



(btw, I'm not well versed in formal logic, nor am I sure that there exists a logic which is strong enough for the naturals, but not enough for the reals).

Mariusz Popieluch· 2 months ago
Hayden - thank you for your insightful reply and interpretation of the ideas in this thread. Your observations, and question also made me think more carefully about what I'm proposing.

The natural numbers, or Peano arithmetic, is a first order theory, whereas the it is not possible to characterize the reals with first-order logic alone since the supremum axiom of the reals quantifies over subsets of the naturals, and is therefore a second-order logical statement. See the list of axioms here: https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach

The stronger logic in which the axioms of R are expressed is second order logic, which is stronger than first order. So to answer your question directly - first order logic is the logic "which is strong enough for the naturals, but not enough for the reals", for the reasons given above.

(Note on the terminology: Logic B is stronger than logic A iff all theorems of A are B therorems, and there exists some B theorem that's not an A theorem. Or equivalently B is a stronger logic than A iff the set of A theorems is a proper subset of B theorems.)

Mariusz Popieluch· 2 months ago
Also, to clarify - "What we consider as empty is dependent on what we acknowledge exists." is not entirely correct.

Rather "What we consider as empty is dependent on what we acknowledge can exist.".

Pedro Forquesato· 2 months ago
Hello Mariusz,

Nice post! It brought a good discussion, and that is the purpose of this forum. I think your stories are good examples of the importance of clear and rigorous (for example mathematical) sentences in philosophy (and other studies), and the danger of "spoke communication".

Let's try to solve it by translating to "mathematical language" what they are saying. In the case of the urns, for example, when the apprentice says "the urn is empty", he means (lets define U as urn and W as the set of white marbles):
U = { }
While when the master says "the urn is empty", rigorously he means:
U intersection W = { }

So while in English what they say is the same, actually they are saying different logical propositions, and thus it is not paradoxical that the second might be true while the first is not. It is not the empty sets that are different, it is the translation from English to logic that differs. (Naturally the same argument is valid for the poet's glass).

Mariusz Popieluch· 2 months ago
Thank you for your reply Pedro - if you're interested in a mathematical treatment of the idea, don't look at the intuitive illustrations of the marble shop, and the trio drinking wine. Instead focus on the content of the OP that follows the phrase "The general claim is...". :)

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To be continued...

Sunday, December 8, 2013

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.


Friday, November 22, 2013

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.

Thursday, September 26, 2013

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.



Saturday, August 24, 2013

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:
mp =  0.2357911131719232931...
or the natural numbers:
mN = 0.12345678910111213...
The construction proceeded as follows. First we define the function λ:\mathbb{N}\mathbb{N}. 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 pn 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 mN is in fact a constant previously formulated by D. G. Champernowne. And it turns out that mp is already known as the Copeland-Erdos constant:



Notice the striking similarity in the formulation - I have formulated mp 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 mp 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 mN and mp are irrational, but now I also know that mN is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which I suspect would be a herculean task."

Friday, August 16, 2013

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 ØK(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.

Monday, May 27, 2013

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 after his return, that made Lev one of the pivotal figures of 21st 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!

Sunday, April 21, 2013

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.

Saturday, April 20, 2013

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.

Saturday, April 6, 2013

A Sneeze


"It might be an interesting literary exercise to try to describe a sneeze. And all the feelings that precede it." - Oliver Sacks.
Challenge accepted. 
With a latent and barely perceptible tickle of the sinus, comes the announcement of this explosive fanfare of the face. As this initially innocent phantom tingling spreads and overwhelms us with its intensity, pushing tears aside in its tyrannical claim to our nasal cavity, an inner revolt brews.  This somatic protest elevates rapidly, as we tense up and inhale spasmodically liters of air, only to expel them all again, extraditing the unbearable intruder in a convulsive wave of euphoria.

Friday, April 5, 2013

What is your calling?

We make shit up. That's all we ever do, at best. At worst we just repeat the shit made up by others. We need the shit others made up, mind you, in order to make up our own. This is not to be confused with stirring shit up - that's an altogether distinct, and somewhat conditional activity - for in order to do any shit stirring, there already needs to be some shit around - shit that has already been made up, naturally. There are ways out of ever having to do anything with making shit up, or shit stirring. Those who have adopted the two most common evasive strategies, either don't do shit, or don't give a shit.

Breeding Unicorns

It is illegal to breed unicorns - illogical even, or ill founded according to some. It's safe to conclude that breeding unicorns is an ill activity.

Saturday, March 30, 2013

Epidiosyncragyrology

Epidiosyncragyrology - the science dealing with stirring patterns resulting from human activity intended to dissolve sugar in tea or coffee, by stirring with a teaspoon. Most epidiosyncragyrologists consider the recent emergence of this interdisciplinary field as nothing short of a revolution in the study of the human mind.



Wednesday, March 13, 2013

Gecko commotion!

Two gecko's are either fighting, playing or mating outside my window - hmm, can't tell. There's plenty of commotion and a cacophony of gecko-squeaks. Oh wait! - I can see one has got hold of the other's tail with it's jaws. Whoa! - the gecko binary system has just swung haphazardly, pivoted on the hind legs of the biter, coming close to dropping off the window altogether. Nope - I still don't know what to make of it.

Friends

My flatmate's dog Russell, came to greet me last night when I got back from a tutorial. After trotting giddily downstairs he paused politely at the threshold, looking up at me, until I said - hello there, come in! After I turned the TV on and sat down, he sat next to me. A minute or two later, while we were sitting there, watching the tube, he glanced over, and looked into my eyes - it was then, that I had a thought which put that moment in perspective - "so, my friend - here we are, two living beings sharing a moment in eternity."

Saturday, February 16, 2013

A fragment of an essay on wonder.


With our fangs of taxonomy muzzled with wonder, as the alluring fragrance of mystery provides no substance for their bite to seize, we are drawn toward its promising source. During that earliest stage of our encounter, we dare to look closer, and quietly step forward, slowly becoming aware of a latent form emerging out of the formless, secret garden, as a cloud would gather out of a homogeneous haze, slowly transforming into a barely perceptible,  faintly budding, chimerical sprout – as we step closer, the quivering penumbra of this autogenetic presence remains unstartled, as the zephyr of wonder conceals our scent. Every such step is what we call metaphysics.

Captivated by this animated apparition, as it blossoms into a seductively incandescent form, we pause mid stride, reach out, and grasp it. As the phantasmagorical fog of indeterminacy recedes, we remain transfixed by the now tangible, floral epiphenomenon, quivering weakly on our palm, its vivid fragrance stirring a familiar, sweet melody – of the lullaby we loved so much as children. As the abandoned blossom quietly withers before our eyes, its poignant odor reverberates into the somber motif of a Requiem for metaphysics.

We’ve clumsily drawn on the eternal sands of mystery’s boundless shore the laughable content, and modus operandi of our mind – the sad part lies in the inevitability of that outcome – we carry the stigma of our finitude with us all the time, and before long, its viral character infects any realm of being, totally.

Saturday, January 5, 2013

The link between Heisenberg's Uncertainity Principle and the 2-nd Law of Thernodynamics.

I met the New Scientist article which I read this morning with mixed feelings. This is the bitter-sweet taste of being right. I guess this is what intellectualists generally feel when they eventually see their (good) ideas finally vindicated, yet initially thought of by the community of experts (and purported, self-proclaimed experts) to be at best proposterous if not nonsensical.

Below is my idea, with justifications of a physics freshman (at the time I had first year thermodynamics and quantum physics under my belt), and the audacity of a philosopher, which I posted years ago in the Philosophy Forums. Needless to say, as it is clear, my idea was met with scorn to say the least.

My Philosophy Forums entry, from 4th March 2010 (emphasis added):

* * *

Subject: Uncertainity and Entropy 04/03/10

This rumination does not purport to be scientific so I guess it may be free of the objection of pseudoscience. This disclaimer I think is useful in order to appreciate the creative aspect/potential of the idea behind the post and will hopefully make it more digestable to all the physycists cringing at the eclectic application of physical concepts in what follows.

The main idea has the following conceptual chain: uncertainity*knowledge*information*entropy. Hence a connection between Heisenberg's principle (quantum mechanics) and the second law of thermodynamics (classical mechanics).

Some useful quotes from Knight: as for photons - "The fact that waves are spread out makes it meaningless to specify an exact frequency and an exact arrival time simultaneously. This is an inherent feature of weaviness that applies to all waves", for matter particles - "Our knowledge about a particle is inherently uncertain. Why? Because of the wave-like nature of matter. The "particle" is spread out in space, so there simply is not a precise value of its position x. Similarly, the de Broglie relationship between momentum and wavelength implies that we cannot know the momentum of a wave packet any more exactly than we can know its wavelength or frequency".

Now, insofar as knowlwdge is knowledge of something, i.e. information (see Shannon), hence there is, I suggest, a positive correlation between knowledge and entropy.

I conclude that the uncertainity principle is (also) a classical statement about the entropy of a system in which the agent's knowledge (neural firing pattern(?)) is taken to be part of the experimental system. The random nature (conforming to some wavefunction, where position and momentum are random variables) of the "potential observable" within the broader system "potential observable + brain of scientist" would proceed to a lower entropy state if the experimenter yielded more precise information then the uncertainity principle allowes, and thereby violate the second law of thermodynamics.
* * * 

And here is the link to the patronising, stubborn and contemptous replies I received from the pseudo-experts on the forum (I think one of them is even a moderator in the matters of physics).


And below is the New Scientist article which both did and didn't make my day this morning ;)

From the New Scientist 23rd June 2012 (emphasis added):
Stephanie Wehner and Esther Hänggi at the National University of Singapore's Centre for Quantum Technology have taken a new tack, recasting the uncertainty principle in the language of information theory.

First, they suggest that the two properties of a single object that cannot be known simultaneously can be thought of as two streams of information encoded in the same particle. In the same way that you can't know a particle's momentum and location to an arbitrarily high level of accuracy, you also can't completely decode both of these messages. If you figure out how to read message 1 more accurately, then your ability to decrypt message 2 becomes more limited.

Next the pair calculate what happens if they loosen the limits of the uncertainty principle in this scenario, allowing the messages to be better decoded and letting you access information that you wouldn't have had when the uncertainty principle was in force.

Wehner and Hänggi conclude that this is the same as getting more useful energy, or work, out of a system than is put in, which is forbidden by the second law of thermodynamics. That is because both energy and information are needed to extract work from a system.

To understand why, imagine trying to drive a piston using a container full of heated gas. If you don't know in which direction the gas particles are moving, you may angle the piston wrongly and get no useful work out of the system. But if you do know which way they are moving, you will be able to angle the piston so that the moving particles drive it. You will have converted the heat into useful work in the second scenario, even though the same amount of energy is available as in the first scenario.

Being able to decode both of the messages in Wehner and Hänggi's imaginary particle suddenly gives you more information. As demonstrated by the piston, this means you have the potential to do more work. But this extra work comes for free so is the same as creating a perpetual motion machine, which is forbidden by thermodynamics (arxiv.org/abs/1205.6894v1).

"The second law of thermodynamics is something which we see everywhere and basically no one is questioning," says Mario Berta, a theoretical physicist from the Swiss Federal Institute of Technology in Zurich, who was not involved in the work. "Now we know that without an uncertainty principle we could break the second law."

Jessica Giggs "To be quantum is to be uncertain", New Scientist v214 n2870 (online here)

Ironically, being a philosophy major, I identify more with being part of a community which includes peolple like Stephanie and Esther, rather than one that gives authoritarian roles (as moderators on Philosophy Forums) to such pseudo-intellectuals as the ones that have been shown (above) to be resistant to new ideas.