One of the affable characteristics of mathematics I enjoy is that it forces you to think about the scope of a problem, variables, and anomalous states or contradictions that may occur. Math helps me think clearly and leads me to consider scenarios that would not be clear under the guise of non-mathematical thinking.

Long term versus short term rewards/utility create very interesting problems for intelligent life. If we use a greedy heuristic to guide us through the decision space — we choose the most preferable action at every decision step — we will, for many problems, arrive at overall sub-optimal solutions. Imagine if at every meal you ate candy and fatty foods because you like, for example, hot dogs over broccoli; you wouldn’t end up being a very healthy person over time, which if you value your health, is a sub optimal overall solution to your life. Much of wisdom is about navigating such decision spaces by avoiding local optima pitfalls.

This problem of relating local decisions/structure to global decisions/structure is actually quite universal across the mathematics I’m interested in — which falls under the umbrella of Artificial Intelligence. Convolutional neural networks are fascinating mathematical objects that capture the dynamics between global and local relationships by using successive applications of convolutional kernels in a feed-forward manner such that they build up successive representations of progressively macro structural relationships as the depth of the network increases. Many state of the art artificial intelligence applications use convolutional neural networks due to this power of expressing the relationships between local and global decisions/structures. In fact the ancient game of GO was functionally solved by using these networks.

How do we deal with this situation in our everyday lives? We use the notion of progress by figuring out our goals — overall utility AKA objective function — and what steps are needed to get there. Progress connects the notions of global and local utility. In simple problems, like going outside and driving a car to the supermarket, we know precisely what steps are needed to achieve our goals. Progress in achieving the objective in these trivial scenarios is simply a proportion of done steps / total steps; however, even in such a simple scenario, if we use time as a measure for progress instead of the proportion mentioned, we cannot arrive at precise estimates in a dynamical world. If we say our shopping trip is going to take 10 minutes, and five minutes into our shopping we proclaim we are half way done, we may feel silly when we get stuck in traffic that extends our trip time. In short, when we measure progress by time and we make statements about completion percentage or time left till completion, we are always assuming we know how long steps in the future will take. Computer Scientists have known this for a long time and have invented a notation for measuring the complexity/performance of algorithms in respect to varying magnitudes of parameters/inputs called Big O Notation.

What about the case of problems/goals that are seemingly well defined objectively, but in terms of procedures to achieve them are ambiguous? We have all gotten lost in this scenario at some point and act or feel like we are at sea without a compass. Examples would be “become a musician,” “play rock music,” “make my wife happy.” There are naturally multiple ways of combining variables to achieve the same outcomes, but if you do not combine them with a particular motif or consistency heuristic, you will create a drawn a quartered solution that tries to go in too many solution directions and explodes. “Breakfast in bed,” “I want to play rock music similar to AC/DC,” “I want to be the first flutist of the New York Philharmonic,” are all examples of particular motifs that help us produce cohesive solutions… and it should be clear that in complex situations we will have hierarchies of related motifs. Without historical precedents or role models, measuring progress of such fuzzy objectives is as impossible as knowing how far you need to go before you reach land when you are lost at sea; however, consistency can act as a compass that prevents you from toiling in a loop ad infinitum.

I’d also like to say one last thing about our perception of progress in others. People often use time/effort as a metric; yet telling me how long you’ve spent doing something, or how much hard work you’ve put into it, gives me no good indication of progress — as discussed above. It doesn’t tell me what steps you have taken and where you are in your journey. Many people make this critical mistake and become content with their supposed progress by virtue of equating time and effort expended to true progress. Rational people evaluate progress in respect to milestones or results that indicate the “step” one is on and in the case of fuzzy objectives at least approach things systematically. We tend to assume if someone spends a long time doing something they are probably good at it because we assume they have been measuring their progress properly and assume that someone will only spend a long time doing something if they are making progress, but as we all know from experience there are countless people that fall into common pitfalls of measuring progress and get stuck going nowhere.

At some point I will write about how language is a huge contributor to tricking ourselves. There are many cases where the same thing is simply said over and over again with different words and flavor sufficiently to be a novel solution/idea impostor, and we often consider this wasteful cycle progress.

So how do we avoid vain progress? Avoid using time as a metric, use historical milestones, and act with consistency in the face of the unknown.