"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

Saturday, October 25, 2014

There's no "hard way" of solving a problem --- an anecdote about John von Neumann as told by Eugine Wigner.

The following problem can be solved either the easy way or the hard way.

Two cyclists 40 miles apart are riding toward each other on a straight track; each one is going at a speed of 20 miles per hour. A swallow starting above one of one of them flies back and forth between them at a rate of 50 miles per hour. It does this until the cyclists meet. What is the total distance the swallow has flown?

The swallow actually flies back and forth an infinite number of times before the cyclists meet, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the cyclists are 40 miles apart and each cyclist is going 20 miles an hour, it takes one hour for the cyclists to meet. Therefore the swallow was flying for one hour. Flying at a rate of 50 miles per hour, it must have flown 50 miles. That's all there is to it.

When this problem was posed to John von Neumann by Max Born, Neumann immediately replied, "50 miles!"
"It is very strange," said Born, "but nearly everyone tries to sum the infinite series."
"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

Source: John von Neumann Documentary starting at approximately 18 minutes into the film.
Based on the version of the anecdote from "Math Jokes".

Wednesday, October 8, 2014

Best of imperfect worlds.

Lev somewhat dissatisfied with his previous cosmic project, decided to embark on a more careful enterprise of world creation, and turn down the perfection parameter from maximum. This time he decided to play with the parameters of fundamental values, and observe how they'd influence the hedonistic dynamics. The idea was to run a simulation where hugs are set as having principal value, and consequently become the sought after currency -- in other words, hugs in that world were to be the sole wealth determinant. But in what sense 'hugs'? -- one may ask. Receiving them of course! And naturally they'd have to be of genuine sincerity; neither bought nor forced in any way. That is, hugs have value only if they're sincere and welcomed. But then how does one accumulate such wealth, given that hugs are such ephemeral phenomena? Surely, one can't be in possession of a great number of hugs. The only way one can accumulate wealth of this kind, that is, become a prosperous hugee, is to guarantee and maintain the existence of those willing to give those hugs, i.e. huggers (aka ready-to-hug beings). In other words, one can attain a high flux of hugs, and wealth would be interpreted as maintaining a high hug-flux. This can be done in many ways of course, and Lev calibrated the simulation with no limit on the degrees of freedom concerning valid hug-acquisition. Naturally those who attain the ability to reach and maintain a high hug-flux steady state, aka hugagogues  are sought after as raw models for guidance, whose wisdom would guarantee the attainment of wealth in that world. Lev has hypothesized that such a world would be among those that are the closest to being perfect without giving rise to any absurd consequences, which inevitably accompany perfection.

Wednesday, August 27, 2014

Explicit solution (formula) to the "truth table" recurrence relation.

If anyone has done any introductory logic, or has been introduced to representing sets in terms of their characteristic function, then the following binary matrix will look familiar. Each row is a distinct combination of the elements of some set. This is the matrix for a three element set. Call it matrix A.

1 1 1 ...
0 1 1
1 0 1
0 0 1
1 1 0
0 1 0
1 0 0
0 0 0
:
:

Some of the most common applications/interpretations of this matrix are:

i) For any natural number N (of columns), the rows of the matrix (of which there is 2N) represent the set of all functions N→{0,1}., were N is a finite subset of the Natural numbers.

ii) The rows of a table thus generated, exhaust all truth value assignments to a propositional variable appearing in a formula of Propositional Calculus.

iii) Equivalently each row can be interpreted as the image of the characteristic function of a power set of some set S (we don't need to assume the axiom of countable choice since here we're dealing with finite sets). In other words each row of the matrix corresponds to a distinct combination of the elements of S. In fact the map : (S)→ ROWS (of the matrix) is a bijection: f(X)=r iff r(i)=1 iff i∈X, where we identify the matrix column indecies with the wellordering of the elements of S.

So clearly this matrix is a big deal, and subsequently the result here is an important one, since it basically compresses the entire matrix to a simple function of its rows and columns.

The way to generate the matrix in a way to ensure that all combinations are exhausted, is to follow the obvious pattern for column generation --- single iteration, double iteration, quadruple iteration, etc. In general, each column i has a 2i-1  iteration of 1's and 0's (starting with 1's in each column).

Below is the proof for the explicit formula of such matrices recurrence relations, i.e. as a function of the matrix'  row and column number. That is, given only the row and column numbers, the formula gives the value that appears in the matrix on those 'coordinates'. That is, the formula takes the following functional form:

                                                                       θ : 2n×→ {0,1}

EXAMPLE: θ(k,i), where the row number is = 7 and column number is= 2, i.e. θ(7,2) = 0.
Now for the proof, which I have only given a sketch of in a previous post. Subsequently I lost the proof, and only had the formula, which bothered me, so I re-proved it last week. So here we go. Most of the pattern recognition which underlies the solution is obviously to be found in the matrix itself, and the diagram below intends to capture and make salient those patterns which may not be so obvious at first glance. The integer numbered columns, i.e. columns 1, 2, 3, 4,..., are the actual columns of the matrix A. 
To the right of each actual matrix column i, I list k(mod 2i), which is the first pattern that ought to be observed. Can you see how a single cycle of k(mod 2i) matches the length of the pair of iterated 1's and 0's, in column i? The next key observation, and perhaps the crucial one, is that the pattern of out actual matrix columns is basically a function of k(mod 2i). More precisely, note how each i'th actual column has the same pattern as k(mod 2i)k(mod 2i-1), with the exception of being slightly 'misaligned' and having some product of 2 in the place there 1's ought to be. I have indicated the k(mod 2i)k(mod 2i-1) rows with a yellow heading. Have a look for a while, and the pattern* should 'jump out'. Once you're convinced that it is so, all we need to do is (i) to rectify the 'misalignment', and (ii) the multiples of 2 integers instead of 1's. By noting that the 'misalignment' is also a function of column number, we see that adding 2i-11 to k, does the trick. Finally divide all values in each i'th column by 2i-1 and we're basically done. That is, our formula can be expressed as the function:
This formula is good enough and does the trick, but using the identity below, relating the mod and floor functions, it can make it more concise.
Giving the formula its final form. I guess it could be rendered even more concise and elegant with further algebraic fiddling, but I shall leave it there.

*Admittedly, a more rigorous proof is required than merely conjecturing that this pattern ought to hold in general. I may get around to it soon. But it does seem obvious that the proof is correct, since no surprises will arise in the relationship of the values of k(mod 2i) and k(mod 2i-1as both k and i increase.

If we switch the order of 1's and 0's in the table, i.e. if we let 0's precede 1's, like so (call this matrix B):

000...
100
010
110
001
101
011
111
:
:
Then the same reasoning yields a more elegant formula:






Tuesday, April 22, 2014

On the (nontrivial) non-uniqueness of the empty collection.

Consider a property : "is green all over, and is not green all over".

Alice and Bob are friends. Bob reasons about the world, and accordingly conditions his understanding of what is or isn't logically possible to classical logic. In particular, not only there are no objects with property P, but there cannot be any such objects since P is a logically impossible property. Alice on the other hand, having travelled far and wide, and having seen curiosities such that no classically (as per classical logic) minded philosopher has even dreamed of, accepts paraconsistent logic as the correct way to reason about the world and its many wonders.


Alice and Bob agree that 'empty' (or 'the empty collection', denoted (e)) is an absence of any-thing, i.e. it is an absence of any object (or simply 'a set with no elements'). They also adopt a covention whereby saying that 'some-thing is contained by the empty collection' means the same as saying that 'that some-thing does not exist', e.g. 'four sided triangles are contained by the empty collection' is just another way of saying that 'four sided triangles don't exist'. This is the convention both Alice and Bob adopt, and agree on. Nota bene, impossibilia (impossible objects) are also contained by the empty collection, for since they can't exist, in particular they don't actually exist.


The duo however disagree about what counts as an object. Whereas Bob considers any object with property P as impossible, i.e. an object that cannot possibly exist, Alice doesn't. As a consequence, within their discussion about the world and its many wonders, the term 'empty collection' despite being correctly understood by both friends as having the same extension, is incorrectly assumed to have the same intension. The extension is the same since it is an absence of any object, but its intension is distinct, due to our protagonists' differences in reasoning about the world, which in turn bear on what counts as possible and what doesn't.


In particular, for Bob the empty collection has the property of containing all "objects" with property P, since that amounts to saying that such objects don't exist (since they can't exist!). Alice would disagree with Bob with regards to such a property --- she would negate it having such a property outright, by saying that it's not the case that the empty collection contains all objects with property P, since (after all) the existence of such objects is consistent with her view of the world. So if she were to accept that the empty collection had such a property, it would amount to saying that the empty collection is not empty after all. Instead, Alice asserts the negation, i.e. that there are some objects with property P that are not contained by the empty collection.


What does this mean? That Alice and Bob aren't talking about a unique empty collection, but rather two distinct ones. To eliminate confusion they decide to introduce yet another convention whereby they index those distinct empty collections by what makes them distinct --- in this case the distinction is conditioned on who is considering the empty collection, i.e. (e)-A, and (e)-B. But since the differences in the intension of those two terms are in virtue of the reasoning system (logic) adopted by either friend, we can conclude that the intension of "empty collection" is logic relative.


Unlike the act of conditioning the meaning of 'the empty collection' to contexts which deliver contingent distinctions, e.g. what a caveman, a poet, a chemist or a physicist would consider as 'empty', the difference in the meaning of "the empty collection" in the case of Alice and Bob arises out of how such concepts can be reasoned about in principle. As such it is a distinction in the meaning of 'the empty collection' in principle, i.e. in principle that concept has no unique meaning.


Also this is a nontrivial claim, since it's possible (conceivable) that Alice and Bob never met, and so never discovered the discrepancy in the meaning of 'the empty collection'. In other words, it's a nontrivial result since it's conceivable that there could be rational agents confined to a single reasoning system only, thereby not being capable (lacking the necessary epistemic condition, which is the act of abstracting away from the preferred reasoning system) of seeing the fundamental non-uniqueness of the notion of 'the empty collection'.


To sum up: Alice's notion of 'the empty collection' is not the same as Bob's notion of it is. Concisely speaking (e)-A is not identical to (e)-B, which yields the truth of the claim '(e)-LP is not identical to (e)-classical'.


But let's assume per impossible that (e)-LP is identical to (e)-classical; hence x is in (e)-LP iff x is in (e)-classical. Also pick the object a (such that Pa) as an LP-possible object (a is an LP possibilia). This is a fair assumption since not all contradictory properties need be impossible in LP, thus allowing objects with such properties to be legitimate possibilia. Hence a is not in (e)-LP, according to the convention adopted by Alice and Bob. But according to the same convention, all x such that Px are elements of (e)-classical, i.e. all x such that Px are classically impossible (are classical impossibilia). In particular a is in (e)-classical, but that means that a is in (e)-LP given the hypothesis (identity assumption), but we assumed that a is not in (e)-LP, which yields a contradiction.


Therefore, adopting classical logic as the one governing this proof (the metalogic here), and reductio ad absurdum as a valid proof method, it follows that (e)-LP is not identical to (e)-classical, as required.


The key idea of the above discussion can be compressed to saying that although the extension of classically impossible objects can be expressed as the extension of the set (CI) of objects that satisfy some inconsistent property, since those two are necessarily coextensive in classical logic, but the extension of paraconsistently impossible objects cannot be expressed as the extension of CI. Hence, the intension of 'no possible objects' is logic relative (obviously?).

Note: 'element of' and 'belongs to' and 'is in' are terms expressing binary relations that I tentatively use to denote some kind of association of an impossibilia with the empty collection, or the empty set (or empty sets, since I'm arguing that they're needn't be unique across distinct theories). 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. (This assumpton needn't be correct.)


One may rightfully object, and some have, that there's a fundamental problem with asserting that anything is an element of the empty collection. But suppose I claimed that the objects it 'contains' are non-existent, but by doing so were I to wish to retain the meaningfulness of such 'objects' I would be committing to some kind of non-eistic ontology, which posits non-existent objects. But those non-existent objects are objects of some kind after all, and thus cannot be said to be elements of the empty collection, which contains no objects whatsoever. So suppose I reject non-eism, and say that when I say 'a is an element of the empty collection', in no way do I wish to commit to a's existence --- not even as a non-existent object --- i.e. there is no a. But then what is it that I am talking about when I refer to a? I'm not refering to anything apparently, by definition.


I choose the second route, i.e. the one that doesn't commit to an non-eistic ontology. 


(To be continued.)