My very first post

The emphasis of this blog is meant to be on the "Beyond" arising from maths. Being a professional mathematician, as I started studying data science about a year ago, I was surprised to discover connections with maths that I had not anticipated. Surely, I knew that data science was in a way all about maths. I've heard people say that it was just a fancy word (or two) for applied statistics. What surprised me was thus not that maths was involved, but rather that I've encountered areas of maths which had nothing to do with statistics and thus I didn't expect to run into them in connection to data science. Like noncommutative lattices, for example.

But more about this in the posts to follow.

Logical puzzles

In this very first post I would like to share another "beyond", also connected to noncommutative lattices, an application of maths that's obsessed me for quite some time. Let's call them 'noncommutative logical puzzles', if you want. Hold on, let's go slowly here. What do we mean by 'noncommutative logical puzzles'? Let's decide on what we mean by 'logical puzzles' first.

There is one name that comes to mind when talking about logical Puzzles: Raymond Smullyan. Remember his inspiring books like Alice in the Puzzle-Land or What Is the Name of this Book - the title of which is itself a fine Smullyan puzzle. Many of his famous puzzles feature knights (who always tell the truth) and knaves (who never do). A typical riddle would give you a couple of statements uttered by the natives on the Island of knights and knaves, and you would then have to tell the type of the person(s) speaking. Or, more interestingly, determine whether there was gold on the island. 

So, what would a noncommutative logical puzzle be? Or rather, what's so 'commutative' about the usual Smullyan-type puzzles? Well, a short answer is that they are based on classical logic, which has many properties, one of them (or two, rather) being commutativity. We won't go deep, and please feel free to skip the following section, but let's just recall the very basics of classical logic.

Classical logic is commutative

In everyday life as well as in classical mathematics we are used to reason in terms of classical logic. Every sentence is either true or false, and we can use certain connectives to connect simple sentences into more complicated ones. For instance, the negation ¬p (read: 'not p') of a sentence p is true if and only if p is false, the conjunction p∧q (read: 'p and q') is true only in the case that both p and q are true, while their disjunction p∨q (read: 'p or q') is true whenever at least one of p, q is true. Moreover, by a convention the implication p→q (read: 'p implies q', or 'if p then q') is said to be true in all cases except when p is true and q is false. Denoting the value true by 1 and the value false by 0, we can see ¬, ∧, ∨ and → as operations on the set {0, 1}. The operations are determined by the tables below:

p	¬p
0	1
1	0

p	q	p∧q
0	0	0
0	1	0
1	0	0
1	1	1

p	q	p∨q
0	0	0
0	1	1
1	0	1
1	1	1

and

p	q	p→q
0	0	1
0	1	1
1	0	0
1	1	1

So what's commutative about that, you might ask. Oh, that's very easy. Simply the fact that

p and q = q and p,

and likewise,

p or q = q or p.

But that's obvious, you might argue, how could it be any other way? Well... that's were noncommutative logic and noncommutative structures come into play.

Is natural language always commutative?

Although commutativity might seem obvious and natural, there are ways to observe the noncommutative nature of the connectives 'and', 'or' in natural language. Consider the following examples:

Sentence 1: Alice found gold and ran away.

Sentence 2: Alice ran away and found gold.

Our perception about the two sentences is not the same. While Sentence 1 probably refers to a situation when Alice ran away with gold (maybe stealing it or running away from an intruder), the Sentence 2 refers to a situation when Alice first ran away from something or someone, and then accidentally found gold.

Another example would be the following pair of sentences:

Sentence 1: I drank the wine and filled the glass.

Sentence 2: I filled the glass and drank the wine.

In the situation of the first sentence I probably drank more than one glass of wine, while in the second one we only know of one glass of wine. Moreover, at the end of the action described by each sentence, the glass is full after Sentence 1 and empty after Sentence 2. The two sentences are thus certainly not equivalent.

Both above examples have been discussed in the paper below, where we studied algebraic properties of such constructions:

KCV, M. Sadrzadeh, D. Kartsaklis and B. Blundell, Non-commutative logic for com-positional distributional semantics, in: J. Kennedy and R. J. G. B. de Queiroz (eds.), Logic, Language, Information, and Computation, Springer, Berlin, volume 10388 of Lecture Notes in Computer Science, 2017 pp. 110–124, doi:10.1007/978-3-662-55386-28, proceedings of the 24th International Workshop (WoLLIC 2017) held at the University College London, London, July 18 – 21, 2017.

If you want yet another example, how about the title of this blog:

Maths & Beyond

Consider an alternative title:

Beyond & Maths

Would that have the same meaning? It is not just that the second version is somewhat bizarre, it also has a different meaning (if it is not simply meaningless). It seems to refer to some abstract "Beyond" (of what?) and then also some maths, while the original title refers to maths and beyond (of math).

So, how about noncommutative logic puzzles? Let's start with them in the next post.