Tuesday, January 27, 2009


A friend of mine sent me this blog post on why google web drive won't kill windows or anything else. To be honest, I'm surprised by the author's short-sightedness.

First, Scott mentions DropBox as a pre-existing replacement for GDrive. He then points out that Google plans to tie the GDrive in to Google Docs and that DropBox doesn't currently have that functionality. However, he doesn't see that as a game changer. What he doesn't comprehend is that Google has more than just Google Docs. Google has Gmail, Calendar, YouTube, Blogger, and an ever growing number of other sites. They also have an operating system (Android). So, you'll be able to turn on your netbook and have it sync your email, documents, favorite shows and blogs, etc... immediately from the cloud. Don't think that will happen? GMail is already offering an offline syncing mechanism through Gears through GMail Labs How much longer before they expand the syncing mechansim to work with other things like Google Docs, YouTube, etc... Google has consistently been able to deliver on big ideas and this one is one of the biggest.

Scott also mentions the trust issue. Who wants Google to store their most personal documents? I think this will become less of an issue over time. Already people are using services such as carbonite to back thier computer up online. How much different is it to trust an encrypted Google cloud? I think this issue will stay a hot topic for a few years, but in 4 or 5 years when everyone is using the cloud more and more it will become a non-issue except for the most sensitive documents. Google is already heavily advertising its security features.

Another issue is downtime. What happens when the cloud goes offline? Once again, I think this will be less of an issue over time as the cloud becomes more stable. Even now, how often does GMail go down? I think my internet provider goes down more often than GMail does. Moreover, I can't get much done without an internet connection anyway, so offline availability doesn't really help me out much. I think the more Google convinces people that the cloud technology is stable the more they will flock to it and use it. After all there are many benefits to cloud technology such as redundency, multi-computer availability, etc...

Scott is right in saying that there are currently alternatives in DropBox and Windows Live Sync, but only Windows Live Sync has the capability of rivaling GDrive. Microsoft has services in its hotmail mail service and its online office suite. They will have to continue to integrate those into the cloud to keep up with Google. Not only that, but if they could integrate their next XBox platform into the cloud so that you could store your games on the cloud (or just download them directly from the cloud) then that would be a big plus and something that Google can't currently rival. Having a home media system synced to Microsoft's cloud could promote using the Microsoft cloud for other things such as mail and documents.

I certainly believe in Microsoft's ability to beat back the Google threat, but I'm not narrow minded enough to think that GDrive is not a threat. It is the backbone of the internet operating system that Google is building to take on Microsoft.

Monday, January 26, 2009

The Google Threat

Disclaimer: I work for Microsoft on their Live Search product. In no way is what I'm blogging about representative of the actual views of Microsoft. I'm much too low level to have any input or insight into Microsoft's thought process.

I see a lot of people on various message boards comment on how Microsoft should stop funding Live Search and Online Services. Many people say that Microsoft should just focus on what they do best (developer tools, operating systems, and office) and ignore this whole "Internet Thing (TM)". Microsoft loses millions of dollars a year competing with the Google Behemouth with very little to show for it in terms of market share. Why not just cut your losses, spin off the division, and let it die a horrible death?

I'll tell you why. Google is Microsoft's biggest threat. Google threatens Microsoft's operating system dominance, their developer tools dominance, and their office dominance. If this were the Civil War, Google would be the North. They are the ones competing on Microsoft's home turf. They are trying to burn Atlanta, and some would say they are succeeding. As a Microsoft employee, I am not at all scared of Apple. They will continue to tinker with their iPhone and Macs and a few hardcore Apple fans will always be there to keep them going. Don't get me wrong, I think the iPhone is a huge improvement. I own one. I like it (but don't love it). However, Apple isn't making inroads into Microsoft's core. Objective C is not the Visual Studio killer and OS X won't kill Windows. Apple wants to produce the "perfect system". And that's fine. They'll end up creating something very beautiful that will be mimiced by others, but it will take them years to do so and they'll do so single mindedly. They won't be diverse enough to kill the Microsoft behemouth. Google, on the other hand, is thinking big. They have a plan that goes after all of Microsoft's core competencies and they are taking the fight to MS.

Let's look at a few examples:

1. Office vs Google Apps for Business - Take a look at a number of business that are evaluating or have switched to Google Apps for Business. Now a few things have to be said. Many of these companies are using Google Aps in addition to MS Office. They are using Office for thier internal communications and confidential emails and Google Aps for less sensitive material. For now, that is the best they can do. In addition, many of the companies are switching from other systems such as sendmail, so that's not a direct gain against MS. Nevertheless, how long is it before Google provides encrypted email and a guarantee of privacy? How long before they win a big MS client and other companies start looking at the cheaper Gmail system with reduced administrative costs? This is an obvious attack and one that is gaining momentum.

2. Visual Studio vs Google Web Toolkit - This is a bit of a misclassification. Really, Google Web Toolkit is more of an attack on Microsoft's Azure Platform than on Visual Studio itself. However, Visual Studio is included in the assault. With Eclipse, Google Web Toolkit, and many libraries such as jUnit, Guice, etc... Google has teamed up with the Open Source community to take on MS. Google wants developers to develop on its "platform" just like Microsfot wants developers to develop on its "platform". That is why Microsoft developed Silverlight. Silverlight allows developers to take their .NET familiarity and transfer it to the browser. This keeps people in the Microsoft environment. Google takes the same approach. They want people to use its services, so they provide the Google Web Toolkit to keep people in their environment. It's also a good testing ground for larger conquests, providing computing power and frameworks for the enterprise. Once developers become comfortable with the Google environment, it will be an easy transition to open up their cloud to businesses and allow them to develop and deploy on it.

3. Windows vs Android - Shouldn't this be Windows Mobile vs Android? No, definitley not. Android is a direct attack on Windows and the desktop. Already many people are speculating that Android will be released on netbooks soon.

See Android notebook coming early next year?, Android netbooks? Wouldn't it be lovely, and Android netbook is a possibility. Google will use their operating system to keep your information and applications in the cloud and you will be able to access them from any computer, especially your Android netbook. You'll log in and immediately see your desktop with your applications that are stored on the cloud. When you click on an application, it will download and begin running immediately (though many applications will still work through the web, like GMail, blogger, etc...). Netbooks are already taking a chunk out of Microsoft's sales and having Google's name on it will only increase sales. Also see this article on how Netbooks sales are killing Microsoft.

I haven't tried to paint bleak picture on purpose. Instead, I was trying to show that Google is taking the battle to Microsoft and Microsoft must respond. Search, in particular Live Search is a key component to that. But, it is not the only component. Windows 7 and Azure are other key components. In the future, people's computers will live on the cloud. Search will go beyond finding web pages. Instead, you'll perform searches for applications that will fit with your current settings. You might even rebuild your OS components in the cloud specifically to fit your library versions. You'll search for a song and not only get an mp3, but also a list of movies that you own that have that song in them. You might, if your search options are set correctly, even get a list of similar songs, or songs you played before or after that song. Search is the backbone of the next generation computer and the next generation HCI. Google, I believe, backed into this and is now expanding it to its inevitable conclusion. Microsoft realized it after the fact and is rushing to catch up. Regardless, we need competition and I believe that Microsoft has the resources and the dedication to provide that competition.

Needless to say, the things I outlined above are not going to happen overnight, but Google is taking a long term view. So is Microsoft. Google is using their advertising revenue to subsidize their desktop pursuits. Microsoft is using their desktop revenue to subsidize thier advertising pursuits. Both are in it for the long haul. It will take years for Google to implement some of the things I discussed above. I expect 8-10 years will pass before all of our data and programs live in the cloud. I expect another 10 years will pass before most businesses data and programs live in the cloud. Both companies will be here after that time has passed, which company will be the dominant one? I have no idea, but it will be exciting to find out!

Saturday, January 10, 2009

Browser Toolbars

Recently, Microsoft announced a deal with Dell to distribute the MSN toolbar with new computer purchases. This comes after previous announcements with Sun and Lenovo. While I am excited about the traffic this will bring to Live Search, I have to say that I hate browser toolbars. Seriously, what good are they? Most browsers already have the functinality that the toolbar provides: search box, term highlighting, etc... The MSN Toolbar does provide a few interesting twists like automatically launching Live Messenger, but on the whole, who needs them? They just use up space and memory. My wife's computer has both the Yahoo! toolbar and the Google toolbar on it. I have to go take them both off now...

You know, what I'd really like is a browser that prohibits toolbars, that I could use. And while I'm on that topic, over Christmas I installed extra RAM in my father-in-laws computer because it was going slow (it only had 256 MB of RAM). I also noticed that he hadn't upgraded from IE 6 and had some Yahoo! toolbar that provided virtual tabs of some sort. Why hadn't the browser updated itself to IE7? Seriously, why doesn't our software just fix itself? Why don't my drivers automatically update? ARRRGGG!!!!

Ok, I feel better.

Friday, January 09, 2009

New Layout

As you can tell, I picked a new layout from blogger. Let me know what you think.

Thursday, January 08, 2009

Rules for running an IT organization

The last post was how to run a research organization. In this post, I'm going to set my sights even higher and tell how to run an IT organization. These rules have been picked through my studies of various companies.

1. Hire the best people - I've harped on this before. So have others and others and still others. If you don't know by now that there is at least a 10x difference between the top programmers and everyone else then you haven't been paying attention. I can't say much on this topic that hasn't already been written, but I will point out that you need to be proactive about hiring the right people. It's not just the free food. It's about providing employees with incentives to do great work. If you just need a warm body, then provide him with a steady paycheck. But if you work in IT, you don't want warm bodies, they are not in the 10x group. You want the best, so you have to make it worth it. You have to make them feel important and wanted. You have to put effort into finding them at the top colleges, recruting them, and retaining them. If you are a small company, you can provide stock options. A large company? You can provide free food :-) Either way, you have to provide challenging problems, a sense of ownership, and other fantastic people to work with. Remember, if you have ten of the 10x group that is equivalent to one-hundred of everyone else. If you don't think that is true because of parallelism, etc... you are wrong. Parallelism doesn't apply because the communication pathways increase leading to lower overall productivity. Just like you don't get a linear increase with most parallel algorithms, the same applies for programmers. The communication overhead gets in the way. Therefore, with ten 10x programmers you can be more productive than with one-hundred other programmers. And you don't even have to pay them 10x as much, though you will have to pay some extra. With the savings, you can provide more stock options or free food :-)

2. Make the source available to all - Everyone in the company should have access to all the source code. At Google, the designers of the library can easily update it, because they can open everyone else's code up in eclipse, choose refactor, and then save and recommit it. They are not blocked. In addition, an open code base implies group accountability. Too often, programmers get too attached to their work. They don't see the flaws that are inevitable in their programs. An open code base allows everyone to view and comment on everyone else's work. It forces programmers to realize that they are mortal and make mistakes. Eventually, they will come to know that the group is better than the individual and their work improves due to peer review. Forcing an open code base allows this to happen sooner and with more acceptance.

3. Dedication to infrastructure - Who is responsible for the build system? Who writes the core infrastructure components? Who sets up the distributed computing system? Who determines which dependency injection framework should be used? All of these are core infrastructure decisions and they all matter. As we'll see in a future bullet point, standardizing the infrastructure is vital. You want your application programmers to focus on their individual application. If every application programmer has to focus on their infrastructure, then you are losing 3 to 6 months out of every project. If it takes a week to set up an automated build system and a week to attach the test harness and another week to set up the distributed key/value store, then you have lost 3 weeks of your project to things that have to occur for nearly every project! Instead, you want application programmers to think about their application. You want them to worry about delighting their customers. They need an automated build system, but they shouldn't have to think about which one to use, how to set it up, or what happens when it isn't working. That is infrastructure. In the past, there were extra teams responsible for hardware, networking, and OS setup. These were not the responsibility of programmers. Now, we're adding additional layers. The build system, test framework, and distributed computing platform are additional infrastructure components that must be standardized and managed elsewhere. There could be additional infrastructure components for your organization as well. For instance, Google uses dependency injection so often that they wrote their own framework for it. Notice that I didn't say every team wrote their own framework. Instead, one team wrote it and maintains it and everyone else uses it. This consistency and willingness to use other people's work makes for a successful company. As another example, Google has a Java collections library that the company uses. Every team can take advantage of this without having to rewrite it. In other words, the goal of the infrastructure group is to find and eliminate duplication throughout the company. This could be with hardware, applications, or libraries. Regardless, duplication is the enemy of the IT organization and it must be eliminated!

4. Repeatability - Everything that is put into production must be a repeatable process. Not only that, but it must be repeatable by someone else! This means both applications and documentation must be written and available that show how to repeat it. Once again, we're trying to seek out and eliminate duplication. Programmers in the future shouldn't have to reverse engineer or re-create your application. If it is worth putting into production, then it is worth documenting and ensuring repeatability.

5. Enforcing standards - You want programmers to feel empowered, but you also want productivity. You need standards to ensure the latter, but you need only the right standards to ensure the former. Standardizing on one build system (perhaps per language) ensures that everyone can access and build everyone else's code. Standardizing on a common base class ensures pandamonium. You want to standardize tools, not techniques. Not only does this allow programmers to quickly move from one project to the next, but it also provides continual feedback and improvement on your internal tools. When a programmer says, "Tool X doesn't provide capability Y so I'll write my own" you are in for disaster. Now, every project begins with a 3 week tool writing cycle. Instead, if the programmer would just fix Tool X then everyone else can take advantage of capability Y. Libraries work the same way. Pick a dependency injection library and use it across the organization. It doesn't matter if one team happens to like Spring over Guice. They are both open source and you can alter it to suit your needs. Just pick one and be consistent. Get everyone moving in the same direction. That way, when they increase others' velocity they move in that direction even faster.

6. Don't start from scratch - If nothing exists, then you have to start from scratch. However, if you have a working product, then don't start rewriting it from scratch. The only thing you do is create new bugs that you don't know about instead of fixing the old ones you already knew about. Now, that doesn't mean that you shouldn't take a troublesome component and rewrite it. That doesn't even mean that over the course of a year you don't eventually rewrite 90% of the application. It does mean that you work piecemeal, with legacy code. You create tests, if you don't have them. You write documents, if you don't have them. You continually improve the product you have. Then, when you are ready, you begin refactoring. You improve its structure a little bit at a time. Just enough to add your new functionality with new tests. Then a little more and a little more. Eventually, your crufty working system blossoms into a beautiful piece of artwork. Well, you'll think so anyway, but the next developer won't understand why you did X, Y, and Z and he'll want to rewrite it from scratch. That's the problem and the point. When we don't understand how something works, we want to rewrite it so that we do. However, the next maintainner doesn't understand it either, so the rewrite circle continues and is vicious. If something works, use it, clean it, modify it. Don't rewrite it.

7. Be dedicated to testing - The best organizations have dedicated testers. This is not a coincidence. Recognizing quality as a core attribute of a product is vital to having a quality product. Manufacturing companies have known this forever. Why is it that IT shops think they can ignore quality and still have a quality product? Google's Chrome has unit tests for highly ranked web pages, automated UI testing, and random input testing. Not only that, but they also ran other test suites against it. For instance, it passes 99% of webkit's layout tests and passes all but 2 of jQuery's unit tests. Quality is a core attribute. At Microsoft, there is a Software Development in Test job role. Each team is assigned one or more of these resources to ensure they deliver a quality product. These people are top notch programmers that love to break things. They are not random guys off the street. They can code just as well as the SDE's and it is vital that they can do so. Testing today is about coding. It is about double entry bookkeeping. It is about automation and repeatability. People expect thier products to work, out of the box. They expect future releases to be backwards compatible. They expect a quality product. To delivery that, quality must be a feature. The organization must be dedicated to quality - having a special testing division is one way to commit to that level of dedication.

8. Metrics - "In God we trust, all others bring data." Metrics are at the core of process improvement. How can you possibly know if you are improving if you don't measure that improvement? Would you be satisfied with your water treatment plant if it told you the water quality was improving because it looked a bit clearer? No. You'd want to know the Ph levels, the amount of sediment in the water, etc... You expect a quality product to have metrics that back up its quality. If speed is a feature, then you'd expect metrics around how fast the product will go in certain conditions. If accuracy is a feature, then you'd expect benchmarks showing the accuracy against other products or human judges. In all cases, metrics are vital to a product in order to show improvement and achievement of goals. The one caveat is to be sure you are measuring the right thing. It is inevitable that programmers will find a way to improve the metrics. If the metrics are measuring the right thing, then that is great. If the metrics are measuring the wrong thing, say lines of code, then you have a recipe for disaster. Make sure that your metrics are measuring externally visible things, not internal ones. You want improvements to your metrics to affect the customer, not your programmer's salary. Everyone cheats the system, you want those cheats to have a positive impact on your final product.

Well, there you have it. 8 rules for running an IT organization. Hopefully they will make your company the next IT powerhouse!

Tuesday, January 06, 2009

Applied Research

At one point in my career, I had the experience of being on an applied research team. In fact, I was one of its founding members. At the time, I wasn't sure what applied research was; even now, I can't say for sure I know.

For our team, applied research was new product development. Honestly, I think that had disastrous effects. We were in limbo between delivering a new product and maintaining our "research" mantra. In the end, we wanted product teams to adopt and maintain our product. This, too, led to tension. Product teams want to feel comfortable with what they will eventually own. They want to use and support things they have used and supported in the past. Research products are wild-cards and not to be used unless nothing else like it exists. Even then, most product teams will choose to rewrite it. Take, for instance, Pig. Pig is a dataflow language created by Yahoo! Research to run on Hadoop. It was created to prove a point, but it was also created to be used. However, once it transfered to a product team in Yahoo!, it was scheduled for rewrite. I'm not sure the cause. It could be Not Invented Here syndrome. It could be that research teams are not focused on the long term, so they deliver a short term product. It could be that research teams don't know how to code or that product teams and research teams speak a different language, so they can't see eye to eye. Regardless, having a research team deliver to a product team is doomed to failure, if you count failure as an eventual rewrite of the product or technology.

To help guide my thoughts on this matter, I had the good fortune to speak to Hector Garcia-Molina of Stanford University. If you don't know who this guy is, you are missing out. Stop and go read the wikipedia entry linked to above. He recounted to me his description of what a research organization should do. I'll recount here and embelish a little.

First, a research organization should publish. The benefit of publishing is that it brings notariety/publicity to your organization. The company's sales force can go to battle with slides that reference conference proceedings. Of course, no sales person would read the proceedings, but it does make their presentation look legit. In addition, presenting at conferences allows the researches to make the aquantence of people like Dr. Garcia-Molina and bring new insight and innovations back. Finally, publishing creates an attractive employment environment. New graduates from top schools want to go to a place that has a publishing history so that they can continue their research. To get a graduate of MIT or Stanford, you need a publication record. I'll add one more to his list. I think a research department that publishes shows a commitment to innovation by the company; a commitment that roots itself in the culture and makes the company a hotbed of innovation.

Second, a research organization can be used as a SWAT force. Tackling a hard problem or subset of the problem. I think this can be the area where the most "good" can be done internally. Instead of creating a product, create an extension to an existing product. There are always "next version" features that never get created. These features either provide too little value or are too difficult. It is that latter segment where the research department can really shine. Since they are not constrained by time to market, etc..., the research department can think outside the box and take the time to create an "academic" solution. For example, PageRank was an academic solution to the problem of which web sites were more popular. It is a beautiful recursive algorithm that just happens to produce great results. Is it a perfect algorithm? No, of course not. Was it better than the "engineering" algorithms of the time? You betcha! It was what happened when two academics got together and had the time to think about the problem. They realized the probelm was similar to that of research paper citations, so they devised an algorithm that treated the problem thusly. Had they had to meet an arbitrary deadline so that their employer could make the trade show deadline, they would not have come up with PageRank. That is the benefit of a research team. Not to create some fancy product for the trade show, but to take the existing product the next step. To finish out the "next version" features and do so elegantly.

The final task for a research organization is to benefit the company at large. There are many ways to do this. One is to create best practices and spread them out to the other development teams. Another is to investigate various technologies and report on how they could/should be used throughout the company. An additional way could be to create the infrastructure to ensure developer productivity if no other team is responsible for it. Do developers have access to distributed key/value stores? If not, install MemcacheDB or CouchDB, and help developers to connect to it by creating modules in various languages. Do developers understand and have access to technologies like Hadoop and Pig? If not, create demos and give a roadshow. All of these things lay in the "best practices" umbrella and often organizations don't have the teams in place to create and distribute the "know-how" required to use them.

In all three cases, the applied research team stayed away from the product team's core product. Instead, the research team focuses on small, manageable pieces that fit nicely into the strategy already outlined by the product team. This creates trust and will foster future collaborations between the product and research teams.

Good luck and let me know of your experiences dealing with research teams!

Thursday, January 01, 2009

Facebook lament

I have too many facebook friends. I lament it. I only have about 5 that I actually want to monitor and communicate with; however, when people request friendship I don't want to say 'no, I care nothing for you.' I really want a "preferred facebook" view where I can just track and interact with those I care to. One would think that it is not that big of a deal, but it really is. For instance, I just went to superpoke a friend of mine and it took far too long because I had to find that person in my list of friends (and I don't have *that* many). It actually reduces the amount of time I spend on facebook.

Less is more.

Infinities and Series

Those of you who read this blog (all 1 of you!) know that the number of integers is infinite and the number of irrational numbers is infinite, but the latter infinity is greater than the former infinity. I tend to think of this as mathematical shenanigans. The idea that one infinity is larger than another just doesn't meet the "beauty" test that accompanies mathematics. In the past, I've argued that infinities don't exist, so we can make contradicting remarks about them to our hearts content. It's similar to asking the question "Can God make a rock so big even he can't lift it?" The words make sense and are in the right order, but the semantics of the sentence are off. They produce a barber paradox that belongs to the realm of meaninglessness. They show that Goedel is alive and well and relevant for today's meta-mathematical problems, if only we'd heed his words. Ok, back to infinities.

First, can we create a more intuitive reason for why one infinity might be greater than the other, a reason that doesn't fall back on establishing a one-to-one correspondence with the integers?

First, let's establish that the integers exist in one-dimensional space. In fact, they exist on a number line which is the definition of one-dimensional space :-) But, let's consider a slightly different definition of dimensionality. In this definition, we'll look at the number of infinite dimensions. In the case of integers, the length of the integer is finite. No matter how big the integer becomes, there is always a finite number of digits. Even if the integer explodes to a googolplex digits, there is still a finite number of them. Therefore, the only infinity is in how many integers there are, the size of the actual integer is finite. So, there is only one dimension of infinity.

Let's now look at rational numbers. In the case of rational numbers you might say that there are two dimensions of infinities. The first dimension is the number of rational numbers. There are an infinite number of rational numbers. The second dimension is that some rational numbers have an infinite decimal expansion. For example, 1/3 has an infinite expansion of 0.333333333... Therefore, rational numbers have two dimensions of infinity and should be larger than integral numbers, right? Not so fast. That's just their decimal expansion. If you keep them in their functional form, then we get a different story. All rational numbers can be expressed as the division of two integers. Moreover, we know that both integers have a finite number of digits. Finally, we know that adding two integers with a finite number of digits will produce a third integer with a finite number of digits. Therefore, there is a representation that is finite in length for every rational number. That leads to the logical conclusion that all rational numbers have one dimension of infinity.

Next up, irrational numbers. Irrational numbers extend to positive and negative infinity, giving one dimension of infinity. In addition, they have an infinite expansion in every base, which gives them a second dimension of infinity. So, we can easily see why there are more irrational numbers than rational numbers, because we allow irrational numbers to have an infinite expansion!

So, if we can't represent irrational numbers with a fixed number of rational numbers, what can we represent them with? Why, an infinite number of rational numbers, of course! For example, PI can be represented by the following series (one of many): PI = 4 * (SUM[k=0 to inf] (-1^k)/(2k+1))

So, in essence, we have an infinite set of numbers each composed of an infinite set of numbers. Two dimensions of infinity!

But wait! If we can create a series to represent each irrational number, does that mean that there exists a representation that is not infinite and therefore we can count them, similar to the rational numbers above? No. With rational numbers there was a finite number of integers that created the rational number. With irrational numbers there is an infinite number of rational numbers.

So, it is true that the number of irrational numbers exceeds the number of rational numbers, right? Well, maybe. I think it is fair to say that an infinitely expanded irrational number doesn't exist, so we're back to the land of Goedel. It is worth thinking of an irrational number as a function of an infinite number of rational numbers much the same as a rational number is a function (division) of two integers. Functions are often useful in mathematical manipulations, but that doesn't make them "real". The number 3 is a physical, real number. You can count out 3 things. The number PI is not a physical, real number. We can only approximate it. In fact, complex analysis is based around the number i, which is a function (sqrt) applied to -1. As long as we don't expand the function, we can do mathematics with it, but if we ever need to expand it our equations blow up. The same thing is true with irrational numbers, they are mathematical niceties. Abstractions that we can manipulate as long as we don't look too closely. Similarly, questions about abstractions such as if one abstraction is infinitely larger than another abstraction requires you to look to closely, so you get crazy results, just like if you had really taken the sqrt of -1 or divided by infinity (another function).

So, the next time you see PI, take it for what it is, a function that can be evaluated to the necessary precision. A mathematical abstraction that can be admired from afar. Just don't get too close.