Do you believe in absolute efficiency?

Weekend read. Hence I post this a bit later on Friday.
Lately, I've been fascinated by infinity. And in infinity, some weird algebra pops up. Yet that weirdness is very much akin to what our business stakeholders want, driven by what our clients demand, and hence our KPIs drive us. Do more with less. And that is what absolute efficiency means.

I told you, this is weekend reading, so don't run away just yet. Save this post and read it when you have time.

In an area of mathematics called Category Theory, it is important to pay attention to the axioms of something called "set theory." And one of the axioms in "set theory" is the "axiom of choice."

Before we go further, what does this have to do with business requirements and workloads? Well, like the axiom of choice leads to some strange conclusions, so does cherry-picking from LEAN, DevOps, AI, and even traditional organizational models. Especially when you base it all on pure mathematical and linear extrapolations. But the main purpose of this article is to have a bit of mind-fun on the weekend. 

Let me first explain what an "axiom" is. An axiom is a basic assumption or a self-evident statement that serves as a foundation for a system of knowledge, such as mathematics or physics. Axioms are used to establish the basic principles of a theory and to deduce logical consequences from these principles. In mathematics, axioms serve as the starting point from which theorems can be deduced using logical reasoning.

Axioms are usually chosen for their simplicity and intuitive appeal, as well as for their ability to generate a rich and consistent body of knowledge. In mathematics, axioms are typically expressed in symbolic language, making it possible to reason rigorously about mathematical objects and structures.

It's important to note that axioms are not derived from experience or observation; rather, they are taken as given and serve as the basis for logical deduction. The validity of an axiomatic system depends on the consistency of the axioms and the rules of inference used to deduce theorems.

Stay here; there is a business point at the end.

In summary, axioms are fundamental statements that serve as the starting point for logical reasoning and the development of a consistent body of knowledge in a particular field of study.

The axiom of choice works with sets that are not empty. E.g.; sets of work and sets of priorities. The axiom of choice states that if you have a number of sets (work, priorities, resources, etc) and none of these sets are empty, meaning you have minimally one piece of work, one priority, and one resource, then a choice function will allow you to select an element from each set at once.

Now, this axiom comes in several incarnations. There are, in fact, three well-known equivalent forms:

  • The well-ordered principle is the assumption that any set can be ordered, yet there are many mathematical sets that do not have a minimal element (in math, we're talking about subsets of real numbers) e.g.; a set or priorities all equal to "top."
  • Zorn's Lemma: Given a partially ordered set in which every chain has an upper bound, there exists at least one maximal element.
  • The Hahn-Banach Theorem: Given a vector space over the real numbers and a linear function defined on a proper subspace, it is possible to extend the function to the entire space while preserving its norm.

Now the real question is, do you believe the axiom of choice? Given the definition of what an axiom is, you probably do (and I have to add here: in your own reality, based on research done by many analysts like Charlie Veich, but equally Broockman and Kalla, Wolfe and Williams, and even published cartoons as the duckrabbit and the Rubin vase.) But that belief has consequences.

If you believe this axiom, then that leads to some very interesting conclusions. One is the conclusion that trans-finite induction works over real numbers. Without going deeper into this, I probably already lost the majority of my readers; it would prove, very counter-intuitively, that you can use non-intersecting (disjoint) circles to cover every portion of 3-dimensional Euclidean space (that is, the space we live in (or I should say perceive to live in.) Just try to visualize this, and you'll see the issues.

It becomes even crazier. Another consequence of the Axiom of Choice is the Banach-Tarski paradox. This paradox states that you can take ONE sold ball, with a finite volume and using rigid motions (things we can do, like cut a ball with a knife), to re-arrange the pieces so that you get TWO balls that are the exact same size and the exact same volume. In other words, do twice as much with the same resource by doing some smart splicing.

Do you still believe this mathematical axiom to be true?

Regardless of your answer, brains much bigger than mine decided that they needed to add axioms to Set Theory to keep things together. Alexander Grothendieck introduced the idea of a number so big that it cannot be accessed by any theories within Set Theory. That number, however, allows us to contemplate a category set of all sets that is bounced by that number, but we'll never reach that number. So we may as well make our category smaller and call it quits.

This leads back to my high school experience. I had many run-ins with my religion teacher and my math teacher where I refused to accept that "something" knows everything and that infinity-infinity=infinity. I'm born in the late '60s, hence high school in the early '80s. Today we would probably say that that is unknowable (at that level of education.)

Today we'd say that different infinities exist based on answers from different people (see personal reality comment earlier). Infinity is a construct, like poetry. Students revolt when some professors ask them to imagine doing something an infinite number of times. The answer from some students is simply "No." Like students rejecting "Hilbert's infinite hotel." It is, however, an interesting mathematical construct. But as these students say, do not confuse constructs with what can be achieved.

And yet, the example of an ape who has a typewriter that will eventually get to a piece like Hamlet, written by Shakespeare, cannot be proven wrong. We can say that the chance of this happening would require a period longer than the age of the universe. That would probably mean that it is impossible, given that the same ape would not live long enough to conceive of sequences that would lead to Hamlet. And the next ape would not continue on from the previous one for obvious reasons (and neither would a human.)

Now I do grant that at this point, we get into the philosophy of math. The point here is that if you give something infinite time, then some random event will happen.

We could continue this train of thought into the continuum hypothesis, but honestly, that is so far beyond my understanding that no more text is needed here. And beyond that, math experts can't even prove if that hypothesis is true or not, writing in 2023. Hence we're entering the realm of meaning and thus the foundations of later math work.

But that is so far beyond what we have to live with today.

So where does this bring us in relation to IT Operations?

Let me bring you back to the Axiom of Choice and the consequence of the Banach-Tarski paradox.

Given finite sets of

  1. business requirements
  2. resources
  3. day-to-day operations (including incidents)
  4. new service/client releases

There does exist a selection that you can choose that maximizes the use of resources that satisfies the above criteria. That is mathematically proven.
But there does not exist a solution that satisfies all criteria 100% without happening into the paradox. That has some very real business repercussions.

Equally, many efficiency programs purport to provide certain savings based e.g.;  on small chunks of time shaved off in a process chain. Each small saving is then added up, and you end up with a large number, a bit like the cut-up ball in the paradox. We have seen over the past decades that this is not always valid thinking; certain process chains do benefit, but many don't unless you add more elements to the mix. 

Agile has attempted to address the resource and workload issue by adding in metrics such as the velocity of a team. The velocity indicates the rate of progress in software development from one sprint to the next. Used properly, this may lead to an increase in output by smartly slicing up the work items. This leads to a shorter time to market. In other words, do more with the same resources.  Hah! The paradox is no more! Not quite. There are limits to how much can be sliced and diced, and as more releases happen, the risk of incidents increases. So, like the math wizards before, we need to add more mitigating items,  

Give that some thought. It is the reason why we have our IT and business prioritisations processes. Do them well, and savings can be made, just not all the way to infinity.

If you want to discuss more on how this applies to your company, contact me via, and let's have a conversation.


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