Here’s a really neat overview of mathematics. Enjoy.

# Category Mathematics

Mathematics in any form

# Is the universe infinite?

I have been returning to this question lately – and I see three possible answers:

- The universe is finite
- The universe is infinite
- The universe is “infinitely finite”

Option 1 introduces an “edge problem” where the particles at the end of the universe will have interacting forces on only one side. If this option is true, the universe started out as point-like Big Bang, satisfying the requirements for a Black Hole.

If Option 2 is true, the universe has always been infinite since nothing can go from finite to infinite (or vice versa). It started out as infinitely large and very dense at the Big Bang, satisfying the requirements for a Black Hole at all areas of space.

Option 3 would be similar to moving on Earth’s surface – if you move straight in one direction, you eventually circle the Earth and end up where you started. The universe could be a 3 dimensional space residing in a higher dimensional space – if you travel in one direction, you would never reach an edge. Instead you can end up back where you started (given that the higher dimensional space is a uniform “sphere”). The universe could have started out as a small 4D+ space.

I can’t for the moment see other options. Please pitch in with your own views.

One question that often pop up with an infinite universe is this: “If the universe is infinite, would everything that can happen be bound to happen – and an infinitely amount of times?”. The usual answer when you Google this is “Yes.” The answer is the same for “If you throw a dice an infinite number of times, must you eventually roll a six? Must you in fact roll an infinite number of sixes?”

While it may be intuitively correct to answer “yes” to these questions, the answer is in fact wrong. Here’s why:

*Consider the natural numbers 1, 2, 3, …*

There are infinitely many of them … so 2 must show up more than once, right? Manifestly wrong.

But say we are talking about states of matter in a finite region. This would be modeled by using finitely many numbers, 1, 2, 3, say, and making an infinite list.

1, 2, 3, 1, 3, 1, 3, 1, 3, 1, 3, …

You say 2 must appear again … but it doesn’t. If you have finitely many states and infinitely many trials, all you can say for sure is that at least one state must reappear infinitely many times. But any particular state, such as the state that defines “you” or a pink elephant or a galaxy; might appear zero, one, 47, or infinitely many times.

*It’s amazing how many otherwise smart people are fooled into thinking that “in an infinite universe, everything must happen.” This is manifestly false.*

So even in an infinite universe, a chance of something specific happening is undecided. This is related to the equation

which is mathematically undecided.

The question of whether the universe is finite, infinite or something else poses some interesting questions. And perhaps some interesting answers may arise.

# Hidden risk of outsourcing IT

There is a potentially undersold risk as a company considers outsourcing IT development or operations: The loss of internal productivity to mentor outsourcing consultants are often difficult to recuperate.

When we learn complex systems, our competence usually follows a Sigmoid curve (also known as “S-curve” or “Logistic curve”).

“Many natural processes, such as those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time.” (1)

“In this case the improvement of proficiency starts slowly, then increases rapidly, and finally levels off.” (2)

When a company is looking to outsource the development or operations of proprietary complex IT systems, the consultants will usually follow this learning curve. But in order to eventually become productive, the consultant will rely on mentors to learn the ropes.

An internal competent developer or system administrator is assigned as a mentor to the external consultant. The mentor will experience a drop in productivity, and regain the productivity concurrent with the consultant.

Spreading the burden among several mentors may make the productivity loss less visible, but the combined loss of the mentors may even be greater.

According to a recent survey I did, the productivity loss of the mentor was at least 50%. The time needed from scratch to a fully productive developer was 24 months. The question is “How long would it take for the productivity of the consulatant to make up for the productivity loss of the mentor?”. Or in other words “How long until this scenario goes break-even?”.

To calculate this, we turn to the Sigmoid function:

Productivity of the consultant (p) is the Sigmoid function over time (t). We adopt the function to go from 0 to the time needed to become fully productive (T).

Then we adopt the function for the mentor’s productivity (P) starting from his dropped productivity and back to full productivity after T time. The drop (D) is the fraction of his full productivity (1).

The reason for the slight difference in the equations (the factors “7” and “8”) represents the fact that even after the time “T”, the consultant would on average still be a notch lower in productivity than the mentor.

The two curves combined with “T = 24” and “D = 0.5”:

The accumulated productivity of the consultant over time is the area under the blue curve, i.e. the integral of p(t).

To get the accumulated loss of productivity of the mentor over time, P(t), we first invert the mentor’s productivity to get his productivity loss, Q(t).

And the integral of Q(t).

The big question is “At what time (t) does the consultant’s productivity make up for the mentor’s lost productivity?”

With:

… we get:

…which reduces to:

Graphically represented:

The question is so big that WolframAlpha cannot display the numeric result within its standard computational time. But with the help of my trusted old HP-41 calculator, the answer was achieved: It takes 19 months of mentoring for the outsourcing project to break even.

The hidden risk is that ** if the consultant quits before that time, the outsourcing is a losing proposition**.

So, if a company considers outsourcing IT to let’s say a Baltic company, one must be very certain that the turn-over of their consultants is above this break-even by a good margin.

The risk management: First figure out how long it usually takes a new employee in the company to get up to full production speed. Add some time if the consultant speaks a different language, is of a different culture and especially if the mentoring is done from a distance. This will be your “T” time.

Then, by a few short pilots, figure out the mentor’s productivity loss. This will be your drop “D”. Along with WolframAlpha and an HP-41, this is all you need to calculate the break-even for the outsourcing project. With the use of some employment statistics from the outsourcing company or the IT industry of that country, you will have a pretty clear picture of the risk involved.

One can, to some degree, mitigate this risk through effective Knowledge Management. A competent Knowledge Manager with an excellent company wiki solution and efficient training setups could shorten the time to break-even by perhaps 20%. Nevertheless, it’s a serious risk to consider – especially since tacit knowledge from years of experience in the company is hard to transfer. Add to this the risk of the mentor quitting or is put on other tasks. Thus the consultant’s stay should exceed “T” with a good margin.

**Update (2015-01-18)**

An approximation formula will suffice for quick gain/loss calculations. This formula gives the net gain (if positive) or loss (if negative) for any given time (“t”):

# Infinity…

Some quick questions up for discussion:

- Is infinity pluss one the same as infinity?
- Is infinity times two the same as infinity?
- What is infinity minus infinity?
- What is infinity divided by infinity?
- Is the reciprocal of infinity equal to zero?
- Can something infinite have a beginning?

# A remarkable book

I place on the top book shelf, albeit a notch below “Jonathan Livingston Seagull“, among books that has had deep impact on my views – the remarkable “Logicomix“.

An epic search for truth, through the eyes of Bertrand Russel rendered in an epic form. The blending of the foundational quest in mathematics and the aesthetics of great comic artwork. It is easy to understand why this book has been given awards across the boards – it presents deep concepts in an ingenious and simple way.

We are taken through the basic concepts of foundational logic – from Aristotle to the modern masters such as Bertrand Russel, Alfred North Whitehead, Ludwig Wittgenstein and Kurt Gödel.

The genius of Wittgenstein was news to me. He represents some fascinating insight into the foundations of so called reality and its limitations.

… because:

And the quest culminates with the profound realization og Gödel in his incompleteness theorems. I have covered this before, but a clearer summary of what I consider to be Man’s greatest intellectual achievement to date would be along these lines:

- If the system is consistent, it cannot be complete.
- The consistency of the axioms cannot be proven within the system.

Gödel’s first incompleteness theorem showed that a system of logic could not be both consistent and complete. According to the theorem, within every sufficiently powerful logical system, there exists a statement G that essentially reads, “The statement G cannot be proved.” Such a statement is a sort of Catch-22: if G is provable, then it is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.

Gödel’s second incompleteness theorem shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement “there are no contradictions in the system H” cannot be proven in system H unless there are contradictions in the system (in which case it can be proven both true and false).

This is precisely why the “*the meaning of the world does not reside in the world*“. Which in essence gives a foundation for free will.

# Skills and arrogance

“*Could you explain what ‘Knot Theories in N-dimensional space’ is?*“, I asked while we walked down the stairs to the ground floor and down the long corridor to the soda vending machine. The Chemistry Department at Oslo University was the venue for the weekly meeting in the role-playing association. It wasn’t much fun to have John Rognes as one of the players in my role-playing world. He was far above anyone I’ve known when it came to problem solving and getting the player characters out of a tight spot. It seemed to passify the other players. But that night I at least got to pick his brains about the passion that brought him mathematical fame. At age 18, he had won prizes in several European countries for his theories that only a handful of people would understand. He was a mathematical genius at the age of three and excelled in math and natural sciences since.

“*Sure*“, he said, “*It’s easy*“. He then went on to explain his theories in less than 10 minutes with a simplicity that even my grandmother could follow. I was stunned. I still am. And on top of his obvious genius, he was a fun and social guy. And bereft of arrogance.

I sometimes wonder why Brendan doesn’t display any arrogance. He has a remarkable background with amazing stories from Northern Ireland, plays golf like a pro, can easily make a living as a street entertainer with juggling and magic, competed in the World Championships in Foosball, beats the crap out of me at the pool or snooker table, is the most excellent instructor I’ve met, runs half marathons… etc. Everything the guy touches becomes a product. And he is a social and fun guy to be around.

Maybe the lack of arrogance is because Brendan doesn’t need to prove anything. Just like John. And so many other guys with great skills who are just confident at what they do. Not looking confident and having to prove it, but just *being* confident.

# An early inspiration for my interest in mathematics

Tonight I was showing Anette a wide variety of music that has inspired me. From Clannad, Isao Tomita, Vangelis, Klaus Nomi and Andreas Wollenweider. I stumbled across a work by one of Norway’s truly great composer, the late Arne Nordheim. I remember fondly how I discovered Nordheim at the age of 13. I used to sit at the main library in downtown Oslo working my way through number theory and calculus while I was listening to this: