# Asia Day 0

On kind of a whim, I decided to go to a bunch of places in Asia. I never did go on a cool trip after I finished my undergrad. I kind of figured if I were going to go check out Asia, it’d have to be now, in what’s likely my last significant chunk of time I have where I can do what I want without feeling bad.

So I decided to go to a bunch of places. How did I decide? Well, I definitely want to go to Japan, so I decided to be in Tokyo during Comiket because I am a gigantic nerd. And if I was going to Japan, I should probably go to Hong Kong because I likely wouldn’t go there otherwise and family and all. And I decided to go to Singapore because just going to HK and Tokyo seemed kind of lame for a trip and Singapore seems like a cool place that I could survive in with English and poor Cantonese skills. And I decided to go to Kuala Lumpur because it’s close enough to Singapore and also seems like a cool city. I like cities. So the plan is Singapore, Kuala Lumpur, Hong Kong, and Tokyo.

I got up bright and early to get to Pearson because US security takes hours or something. It turns out that didn’t really help except maybe to beat the horrible 401 rush hour traffic. They do this thing now where they queue you in customs by departure time, so I basically spent two hours between check-in and customs. Apparently this is because of US budget cuts or something. THANKS OBAMA.

The first flight was a tiny one to Chicago and was fairly pleasant. Wifi was allegedly offered on the flight but I never got it to work. Oh well, it was short enough that it didn’t really matter and I got some chores done in Animal Crossing. O’Hare was okay I guess except that they don’t offer free wifi. THANKS OBAMA.

I don’t fly much, so this is my first significant experience on a long flight. It sucks. Being on an airplane is neat. Being on an airplane for fifteen hours without Internet is not so great. Especially when assigned next to a baby. I dabbled around on my 3DS and iPad for a bit before realising I could check out the on board entertainment. I ended up watching the first half an hour of House of Cards and Wreck-it Ralph before settling on two episodes of Top Gear and Utada Hikaru’s second singles collection. Deliverance eventually arrived and I got to explore HKG.

First, I’d like to mention that the HK security people don’t seem intimidating at all. They all seemed like laid back Cantonese sitcom people. After the quick checkpoint, I looked for food and decided to get congee because the airline food was kinda gross. One very nice thing about HKG is that they have these wonderful charging stations that include USB outlets. Sadly, there only seemed to be one for an entire stretch of gates.

The flight from HK to Singapore was a lot nicer since the actual flight was much shorter and I was seated beside grown men instead of screaming and pooping children. And that brings us to Singapore.

# Challenges in Combinatorics on Words (Day 5)

My knees were killing me, but that wouldn’t stop me from going to a talk that relates algebra and automata theory!

• Kiran Kedlaya gave two talks on Christol’s theorem. Christol’s theorem says that a formal Laurent series is algebraic over the field $\mathbb F_q(t)$, where $q$ is a prime power, if and only if it is automatic. The second of his two talks was about a theorem of his which generalized Christol’s theorem to apply for general power series.
• Eric Rowland gave a neat talk on characterizing $p$-automatic sequences using 1-dimensional cellular automata. There’s actually a lot more algebra to cellular automata than I would’ve expected was possible (and even some connections to Kedlaya’s talk on Christol’s theorem). Then again, I don’t really know much about cellular automata other than Conway’s Game of Life.

So even though I didn’t really contribute at the workshop and I was kind of wandering around as a lone graduate student, it was a really interesting experience. At the very least, I got to meet some interesting people working on interesting things and I have a pile of interesting things to look up over the summer before I head to Kingston and ramp up into hardcore math research mode.

And now, some miscellany.

• Lunch notes: tried Mother’s Dumplings again and opted for a double order of dumplings this time around. I went with boiled again, because I couldn’t justify paying a bit more to get a bit less, even if the steamed dumplings were supposed to be amazing. Maybe that’ll happen if I’m there with other people in the future.
• Commuting notes: I got a ride to and from Fields, since it probably wouldn’t be a good idea to be on the TTC for long periods of time with my knees in their current state. The morning trip involved going across Eglinton, which is an absurdly wide road and I don’t really know why people are mad about LRTs on that road when it’d essentially be replacing a lightly used HOV lane (or maybe even not). Once we hit the DVP, traffic got heavy and Bloor was pretty bad. We went down Sherbourne, where I saw the Minnan-Wong bike lanes and we continued on Carlton and College.
• The evening commute was also interesting. My dad usually takes Lake Shore through to The Beaches and up Kingston, but apparently, there’s some construction going on there, so we went along Gerrard through the east end of old Toronto instead. We ended up cutting across to Southwest Scarborough on Danforth and back up to STC to pick up a new phone. This was a much better route than the one proposed by my dad, who wanted to go up to the DVP.
• Replacement phone notes: Got a new phone and restoring service was pretty easy. The tough part was restoring from iCloud backups since, Apple, in their sometimes questionable wisdom, decided that you could only restore iCloud backups when you first set up your phone, which the Koodo lady zipped past while we were at the booth. So I had to reset the iPhone, which was baffling, since it required downloading the latest iOS update, which I’m pretty sure I’d already downloaded. But after all of that, my phone was in pretty much the same state as I had it in yesterday.

# Challenges in Combinatorics on Words (Day 4): Bus theft edition

So today was an adventure for reasons unrelated to exciting developments in combinatorics on words.

• More talks and pretty proof heavy, which I thought I’d enjoy, but for someone who’s not in the field, it was kind of tedious. It was interesting to see that conjectures do get proven, I guess.
• Theoreticians in CS love complexity measures, so we had two today! Antonio Restivo defined a complexity measure based on periodicity and Jörg Endrullis talked about comparing two different infinite words by using transducers. The transducer thing was pretty interesting because it’s more automata stuff and because there are so many natural questions that arise that haven’t been worked on very much yet.
• Also, problems were getting solved during the workshop. Steffen Kopecki mentioned that him and others had solved some cases of Thomas Stoll’s problem, which asked if there are infinitely many odd $n$ such that $s_2(n^2) = s_2(n) = k$, where $s_2(n)$ is the sum of binary digits of $n$.
• I finally got an experience of stereotypical Malvern life, in which my phone got stolen on the bus right as the hooligans were leaving the bus. I chased them down and I guess I was faster than I looked because they looked back and went “oh shit” and one of them decided they needed to stop me so they pushed me.
• I chased them a bit longer but stopped because I was feeling tired and I realized my knee was actually bleeding really badly, which one guy who was walking home pointed out. That guy was good people and let me into his home to treat my wounds, provided wifi to see if I could track my phone down, and a phone for contacting people.
• My dad picked me up a bit later and we decided the cut on the knee was pretty nasty so we went over to the hospital, which is my first experience with the Canadian healthcare system after being politically aware. Since my injuries weren’t that bad, I started keeping track of the dreaded wait times. It took about an hour before the doctor saw me and half an hour to treat and get stitches and maybe another half an hour for followup with cleaning and stuff, so it took almost two hours on the dot. That seemed reasonable but maybe I’ve been socially engineered by the communism. Also, didn’t pay anything.

# Challenges in Combinatorics on Words (Day 3)

A short day today, with interested persons off to a visit at the ROM.

• Now that open problem presentations are over, it’s mostly just talks and problem solving time. I don’t know if it was intentional, but today’s talks (other than the plenary talk) dealt mostly with algorithmic aspects of strings, which aren’t really my thing.
• There was one talk which was particularly interesting, which was Florin Manea’s talk on finding hidden repetitions, which introduced the idea and motivation behind “hidden” repetitions. We want to check for repetitions of a particular factor $x$ or $f(x)$, where $f$ is an involutive (anti-)morphism. This problem actually comes out of the DNA setting, where words are taken over $\{A,C,G,T\}$ and taking the Watson-Crick complement of a word is the involutive antimorphism we’re interested in.
• Jason Bell gave two plenary talks, one yesterday and one today, on algebraic aspects of $k$-automatic sequences. I’d read about $k$-regular sequences before out of interest but didn’t really retain much of it and I’m glad that I got a chance to have someone actually explain how to derive them from the idea of $k$-automatic sequences and also what the $k$-kernel is.
• For lunch, it was raining and I didn’t feel like walking all the way down Spadina to King’s Noodle so I decided to try Kom Jug Yuen. It was more expensive and not as great as I was expecting. I’ll just walk to King’s or Gold Stone again next time.
• The Fields Institute keeps all of its mathematicians and visiting mathematicians very well caffeinated and fed throughout the day. I think they have a scheduled coffee break at 3pm-ish because a bunch of people that I didn’t recognize were always around the coffees and foods and talking about math that I didn’t recognize. For coffee, they’ve usually got some combination of Starbucks and Timothy’s. For food, they have a wide selection of fruits and cakes. For breakfast, they have a platter of baked goods. I have also seen a platter of pita wedges and some kind of nice bread with various delicious dips.

# Challenges in Combinatorics on Words (Day 2)

More open problems and talks!

• There were two talks and a bunch of open problems by Aleksi Saarela and Juhani Karhumäki on $k$-abelian equivalence. So you have your alphabet $\Sigma = \{a_1,…,a_m\}$ and two words $u,v \in \Sigma^*$ and $u = v$ if they’re the same, which is obvious. We have this notion of abelian equivalence, where we have $u \equiv v$ if $|u|_a = |v|_a$ for every $a \in \Sigma$. That is, $u$ and $v$ have the same number of each letter ($aaabba$ and $ababaa$ are abelian equivalent since they both have 4 $a$s and 2 $b$s). We generalize abelian equivalence to $k$-abelian equivalence by saying that $u \equiv_k v$ if $|u|_x = |v|_x$ for every factor $x$ of length up to $k$. A lot of the problems that were posed were questions of how to extend properties that we know for the normal case and the abelian case to this new $k$-abelian setting.
• The room that we’re in at the Fields Institute has a neat projector setup, where it uses two screens. The right screen displays whatever is currently displayed on the computer, while the left screen displays what was previously the current screen. This is really useful, because speakers tend to go through their slide decks a lot faster than I can write and often refer to definitions and theorems stated on the previous slide. However, the system has an interesting quirk: it somehow detects when the screen changes, which works for most presentations, since they’re static slides, but there was one presentation where the speaker had a slide with an animated gif or something on it and the left screen kept updating.
• There was a problem from Štepán Starosta that dealt with extending things we know about palindromes to generalizations of palindromes. So instead of considering the mirror or reverse operation, what you’d do is consider a finite group of involutive morphisms and antimorphisms (an involutive function $\Theta$ has the property that $\Theta^2$ is the identity function).
• I met a prof who happened to do his undergrad at Waterloo and is currently at a university overseas. We had a nice chat about various flavours of CS double majors and students chasing trends to make monies (as it turns out, CS/C&O wasn’t always popular).
• I think I can articulate now why I feel useless in the problem solving sessions. While I know a bunch of results and definitions, combinatorics on words really is a pretty different field from automata theory or formal languages. So, since the basic “language” of the two fields is the same, if someone walked me through a proof or something, I’d be able to follow it. But when it comes to working on problems, even though the two are related, there’s an intuition to these kinds of problems that I haven’t developed yet nor do I really have a feel for how results are connected or what certain properties might imply.

# Challenges in Combinatorics on Words (Day 1)

Workshop blogging! Very exciting, since I’ll try to reconstruct some highlights from my awful note-taking for each day.

• One of the neat things about this is that unlike a “real” conference, there aren’t many results presented. Instead, the focus is on open problems and working on those problems. Essentially, what you have is very smart people coming up and talking about a problem that they had trouble solving and all of the things they tried before they got frustrated and gave up or something.
• Luca Zamboni presented an open problem involving factors of an aperiodic infinite word and palindromes. An interesting tidbit was when he described how to use palindromes to describe a word. Basically, you say that certain positions of a word are palindromes and you try and fill in those spots with letters. So the question that comes up is trying to figure out what is the fewest number of palindromes that’s necessary to describe a particular word.
• Eric Rowland talked about $p$-automatic sequences and deriving an automaton from some polynomial which describes a $p$-automatic sequence. We know how to get the polynomial from an automaton, but coming up with an efficient algorithm for the other direction is tricky.
• Neil Sloane gave a neat talk about curling numbers. If we have a word $S = XYYYY \cdots Y = XY^k$ with factors $X$ and $Y$ for some finite $k$, then the curling number of $S$ is $k$. We can define a sequence where the $n$th term of the sequence is the curling number of the previous $n-1$ terms. There are a bunch of interesting conjectures that arise from this sort of thing.
• Also, he really likes sequences, which I guess is what you’d expect from the guy who started the Online Encyclopedia of Integer Sequences. I was in his group for dinner and he spent much of the time introducing various sequences to us and we spent a lot of it playing around with them.
• I was pretty much useless during open problem solving sessions because of a number of reasons (tiredness, lack of experience in the field and problems, overheating from sitting in the sun). But, I did watch a group work on a problem having to do with unbordered words and factors and it was nice knowing that a bunch of international experts worked on problems in much of the same way I do: by writing stuff on the board and staring at it for a while.

# How Toronto’s War on the Church™ ended up going

I should start by noting that I’m by no means an expert on planning or urbanism at all and I’ve only been following this issue whenever it’s popped up. I’ve tried to go through whatever I can of the respective documents and files, but there’s only so much I can understand as an amateur. Basically, the best I can do is read and try to follow what happens at council so that’s what I’ll be focusing on.

So a few months ago, I noticed this thing scary graphic being passed along on Facebook about how Toronto was declaring war on the church or some such thing. Basically the war was conducted on two fronts:

1. The harmonized zoning bylaw
2. The Toronto District School Board raising rents on religious groups by some inordinate amount

I’ll be focusing on the harmonized zoning bylaw. First, a bit of history is necessary to put everything in context.

As you may or may not remember, what we now know as the City of Toronto used to be a two-tier municipality consisting of six different cities (Scarborough, North York, East York, York, Etobicoke, and Toronto) and a regional layer of government on top called Metropolitan Toronto. Everyone was happy with this arrangement, so obviously, the provincial government needed to wreck it.

Sometime during the first term of Mike Harris’ Common Sense Revolution™ Progressive Conservative provincial government, it was thought that Toronto (and a few other cities around the province) could run more efficiently as one giant city instead of a bunch of different cities. And so, in 1998, the Government of Ontario decided to smush all of these cities together even though no one wanted it to happen and here we are today, with one giant City of Toronto. Obviously, having to merge six different governments together into one giant government is not a trivial task and even now, 15 years later, we’re still trying to work out the kinks. One of those kinks is planning and zoning regulations.

You see, because Toronto used to be six autonomous cities, this means that all of those cities each had their own sets of planning regulations. This is obviously not ideal. So in order to try and simplify things (or at least make them less unwieldy), the city tried to work on harmonizing the bylaw across the city. This has been a work in progress for many, many years and almost happened in 2010 but it kind of blew up and everyone went back to the drawing board and here we are again.

Now a few months ago, someone found out that the newest version of the proposed draft zoning bylaw turned out to severely restrict the zones where places of worship could be established. Even though the immediate reaction was over the top spiritual war rhetoric, the concerns weren’t unjustified. Essentially, places of worship were limited to select commercial and institutional zones and that was all, no residential or industrial zones. Was it intentional? Was it a mistake? Who knows? But it’s helpful to remember that planning staff was trying to go through an incredibly complex set of regulations and trying to make everything somehow work together.

For those of you who aren’t familiar with how City Council works, essentially, stuff starts out in the various committees before heading out to the council floor. And so, religious leaders and groups went and got in touch with their councillors with their concerns and went and deputed at Planning and Growth Committee meetings and the bylaw was revised into something much more reasonable before being shipped off to council. In other words, there was something that was overlooked by someone, interested and concerned parties gave input and worked with their representatives, changes were made, and civic governance worked as intended.

With that done, we fast forward to the April 3 meeting of Toronto City Council. At this particular council meeting, the mayor made the zoning bylaw and Hero Burger at Nathan Phillips Square his two key items. Most people will remember the second one because councillors basically argued about whether to put a Hero Burger in Nathan Phillips Square for two or three hours and it featured the deputy mayor reading a selection of items from the menu of Hero Burger among other things. But before the Hero Burger shenanigans started, a pretty healthy chunk of time was spent dealing with the zoning bylaw and within that debate, there was a substantial amount of stuff dealing with places of worship.

As was mentioned earlier, initially, places of worship really were significantly impacted negatively by the proposed changes. However, all of that got significantly reworked and when it hit the council floor, it was something much more reasonable. The proposed bylaw now allowed for places of worship in all residential and commercial zones, various institutional zones, and industrial office zones. This was mostly fine, except for some fighting over whether to allow places of worship in light industrial zones.

The context behind this particular scrap is that a common thing for smaller churches to do is to rent or establish their church in areas which are zoned for industrial use because the cost of doing so is a lot cheaper. However, this puts them at odds with the city’s attempts at trying to preserve its industrial lands. What tends to happen is because the land value in these zones tends to be cheaper, condo developers often buy up these lands and try to get zoning changes on the lands. The result is that there are fewer and fewer places in the city where industry can set up operations.

Of course, churches don’t tend to be huge developers or speculators buying up cheap land to flip over to developers, so what’s the problem with letting them on industrial lands? The problem is that places of worship still negatively impact the use of industrial lands for industry. The reason for this is because a place of worship is classified as a sensitive use under provincial regulations and so the surrounding industry has to restrict their industrial activities, even though they’re in an industrial zone.

The particular motion (Motion 3, Part 3) to allow places of worship on light industrial zones was eventually lost and the initial recommendations from Planning and Growth Management were basically passed. Of course, this doesn’t meant that churches are suddenly getting booted from industrial lands. Most churches that are already there legally based on whatever bylaw was in effect before gets grandfathered and gets to stay there. It’s just new churches won’t be able to move into industrial zones.

So tl;dr, everything worked out in the end and it’s going back to staff for one more go-over before being finalized for reals.

# Twitter bots, twitter bots, twitter bots, folks

After the incredible success of @SomeHonMembers, I decided to create @HonSpeakerBot, which was not nearly as popular, but whatever.

The lack of any transcripts or any data made a bot for #TOpoli difficult. But now, there are two. You may have seen them around as I was testing them.

## Mayor Robot Ford hates streetcars

This idea is based on a popular Japanese twitter bot by the name of @akari_daisuki. What Akari does is takes a random Wikipedia article title or something from her timeline. Let’s call this thing $x$. Every fifteen minutes, she tweets the Japanese equivalent of “Yay $x$, Akari loves $x$”.

The idea behind @MayorRobotFord is similar. He takes a random Wikipedia article title or something from the #TOpoli stream and tweets, in his familiar way, “$x$, $x$, $x$, folks. We’re getting $x$.” or “People want $x$, folks. $x$, $x$.”, depending on how long $x$ is.

How does it work? Well, the Wikipedia portion is easy enough. We just get access to the API to grab a list of ten random pages. The more complicated part is pulling stuff off of #TOpoli.

Since this is a twitter app, it’s not too hard to get a bunch of tweets off of #TOpoli. We just use the API and we’ve got the fifteen latest #TOpoli tweets ready to be used. The difficult part is extracting noun phrases, or NPs, which is where that graduate class on computational linguistics comes in handy.

So how do we do this? Well, first of all we have to identify the parts of speech in a given tweet. So we tokenize it first and split it up into words and punctuation. Then, we use a part-of-speech tagger to go through and figure out what each little piece is. The POS tagger that I used was the default one that came with the Natural Language Toolkit. Normally, you’d need to train a tagger on a corpus. This default one was trained on the Brown corpus, which is a body of text which was hand tagged for training purposes.

So now our tweets are all tagged and we assume that they’re properly tagged. There’s obviously going to be some slight errors here and there, but whatever, we want to make a twitter bot, so it’s not that important. But we only have our parts of speech. We want to be able to relate the different parts of speech into phrases. So we need some kind of parsing or chunking to put these pieces together into larger phrases that make sense.

For this, I used a bigram chunker trained on the conll2000 corpus. Like the Brown corpus for tagging, the conll2000 corpus is manually parsed and chunked for training purposes. What a bigram chunker does is it analyses every consecutive pair of words in a sentence to come up with a statistical model. It uses this to come up with the most likely NPs to arise from the sentence. We can then just pluck out all of the NPs the chunker identifies.

Once we have all of our NPs, we stick them in a list with our Wikipedia titles and randomly select one to use in our tweet. The Wikipedia API has a limit of 10 titles per call and the twitter API grabs 15 tweets per call. Thus, the chance of getting a Wikipedia title is at best somewhere around 2/5 of the time. However, that’s not taking into account removing entries that are too large. That quick calculation also assumes that there’s only one NP per tweet when there could be many, so in reality, the chance of grabbing something from #TOpoli is much more likely, which might be for the best if you want weird accidental metacommentary.

## The Core Service Review

One day, I decided to look through the City of Toronto’s open data catalogue and happened upon an interesting entry called the Core Service Review Qualitative data.

Lo and behold, it was exactly that.

After some fiddling around with an Excel module for Python and figuring out how to split tweets that are larger than 140 characters, I let it go.

@TOCoreService will tweet one entry, randomly chosen, from the 12000 submissions, or close to 58000 answers. These range from short answers like “transit” or “taxes” to fairly lengthy responses.

So what’s the point of this bot? Well, the data is up there for anyone to read, which is nice for transparency and engagement. Of course, whether anyone who’s not city staff would want to read 13000 responses is another matter. But here, we pretty decent collection of opinions on what our priorities should be from real citizens. It’d be a shame if the only people who read them were city staff.

# Toronto City Council, 2012

2012 has been a hell of a year, especially if you’re into the city council scene in Toronto. Basically, the year in Toronto politics can be summed up by the following neat graph.

Yikes.

What you’re looking at is a similarity graph of recorded votes in city council, from October 2011 to October 2012. An edge is drawn between two councillors if they voted the same way 90% of the time. The edge is coloured blue if they voted together 92.5% of the time and it’s green if they voted together 95% of the time. Remember, the last time we did this, the graph looked kind of like this:

Let’s refresh our memory of the first year of the Ford council. Most councillors were willing to work with Ford in the face of his relative popularity at the time. Right-wing and centrist councillors tried to position themselves to gain the mayor’s favour and the mayor had a pretty easy time getting his agenda through. With little effort, he was able to repeal the vehicle registration tax and put an end to Miller’s Transit City plan. It seemed like we were in for a long four years.

But then, something happened over the summer. In his quest for efficiencies, the mayor had actually dove into the realm of cutting services. People didn’t like that. After all, the mayor had promised he could reduce spending without cutting services. And it’s here where the mayor and the citizens diverged, on where the line between finding efficiencies and cutting services was drawn. And so, the mayor’s popularity dropped.

And then there was the hilarious Port Lands thing, but whatever.

Anyway, fast forward to January 2012, when the vote on the budget is taking place. Via some fascinating political manoeuvring, a majority of councillors were able to reverse some of the mayor’s planned cuts. In February, a majority of councillors, again, reverse the mayor’s plans and performed some necromancy on the Transit City LRT lines. Council had realized sometime in the preceding months that the mayor was no longer the threat that he was at the beginning of the term and his refusal to compromise on some very reasonable points made him look worse.

And so, 2012 has played out, with Council taking the task of governance into its own hands, without the guidance of the mayor.

So how different does the dynamics of council look after 2012? Here’s a graph that takes all of the data from the beginning of Ford’s term up until the last council meeting on October 4, 2012.

What will this graph look like by the end of the term, in 2014? It’s hard to say. Remember, we were all expecting 2011 to be the new normal, until 2012 hit. Who knows what could happen in another year. The alliances at council are always shifting and there’s always the temptation for some councillors to go out on a limb and inadvertently blow something up in the process.

Well, there is one thing, which is that the data could look significantly different because of a structural change made at the last council meeting. In Ford’s council, the mayor insisted that every single vote be a recorded vote, in an effort to improve accountability and transparency. Of course, what this means is a blowup in the number of recorded votes for things like speaking extensions. During the last council meeting, it was decided that speaking extensions would be done away with. This will likely affect the data because most councillors usually just vote yes. Well, except for the one councillor who always votes against speaking extensions: the Midnight Mayor, Mark Grimes.

## Bonus: Miller, 2009-10

Toronto’s council voting records go all the way back to the beginning of 2009. Since I was already clicking endlessly to download the voting records for the year, I thought it’d be neat to see how different council was back in the final days of the Miller council. The time period represented here is from January 2009 to the final council meeting in August 2010.

The most striking thing is, of course, where the former councillor for Ward 2 can be found.

It was kind of tough to figure out a good threshold for this dataset because the differences in voting were much, much greater, almost certainly caused by Ford’s insistence on recorded votes for speaking extensions. Here, an edge is drawn between two councillors if they voted together at least 75% of the time. If they voted together more than 85% of the time, then the edge is orange, and the edge is red if they voted together more than 90% of the time. Of course, the colour scheme was chosen to reflect the evil New Democratic Communists running council at the time. Since I didn’t really pay attention to council during those years, I don’t have much to say, but I’m sure that if you did, you’ll find some interesting quirks.

# God in $n$-space

So here’s a question about the nature of God that’s probably atypical. But I should probably preface this by saying that this is purely an academic exercise and thought experiment and that I’m not really looking to establish any deep theological truths. It’s entirely possible that I’m horribly wrong.

One of the things Christians do when describing God’s eternal nature is to say that because he has no beginning and no end, he exists outside of time.

I’ve never really understood what this meant.

The rationale for this kind of explanation is that our finite minds can’t comprehend infinity. As a mathematician, that notion seems kind of silly. Here, I’ll give an example of something we’re all familiar with that doesn’t have a beginning or end: $\mathbb Z$, the set of integers. We’re even able to distinguish between different cardinalities of infinity and have developed useful number systems in which we can, yes, divide by zero and get infinity as a legitimate result. So what’s the problem?

To me, the notion that God exists outside of time is like saying God exists outside of space. No one seems to have a problem with the second one, after all, omnipresence is one of God’s attributes. This is an idea we can use.

I’m sure we’re all familiar with the concept of 3-space, or $\mathbb R^3$, which is how we describe the three physical dimensions of space and all. So God’s omnipresence in 3-space would just mean that he’s present in every point in $\mathbb R^3$.

But mathematicians aren’t satisfied with stopping at $\mathbb R^3$. We like to generalize, which is where we get into things like $\mathbb R^n$ for some integer $n$. Or how about even $\mathbb C^n$? So now we’ve got $n$-dimensional space to deal with. That’s hard to wrap your head around if you try to think of it in analogous physical terms (because there aren’t any). Anyhow, we don’t even have to stop at finite-dimensional spaces, we can extend things to infinite-dimensional spaces.

Whether or not these things actually physically exist isn’t that important. We’re just concerned with this: how does God’s omnipresence translate when we extend space to however many dimensions? It’s simple, he’s still present at every point in space.

So what if we take one of those dimensions to be time? I mean, a lot of people often like to think of time as the fourth dimension.

Then God is present at every point in time as well. For me, thinking about it this way actually answers a question I did have for a while: what is meant by God’s unchanging nature? This is one of those questions that the outside of time thing was meant to “answer” but it doesn’t actually answer anything, since it really just handwaves it away. But with the dimensionality angle, we can say that God is the same entity at every point in time.

I’m sure there are plenty of other questions that arise from thinking about it like this, but, at least for me, the advantage in this approach is that it’s analogous to ideas we’re already comfortable with, namely God’s omnipresence in $\mathbb R^3$. It explains why God can change his mind and direct things at multiple points in time if he wanted to.

So this thought experiment led an interesting question on prayer. One component of prayer is that Christians often petition God to act in some way, in the present or future. But if God is all-powerful and ever-present, does it make sense to pray for things that occurred in the past?