Tomorrow’s Mathematician

The Dramatic Shift in Objective that Occurs Between Basic and Higher Mathematics

Once one becomes sufficiently advanced at mathematics they stop calculating. They stop shuffling symbols around to manipulate an expression. They stop doing word problems. They stop using math to express a scenario’s mechanism.

They stop doing everything you were taught mathematics is.

This has nothing to do with mathematical proficiency. It has to do with the dramatic shift in objective that occurs between basic and higher mathematics.

Higher mathematics is *entirely* about a specific approach to abstraction. It is about *choosing* the most analogically sound exemplar of the system of interest.

It is akin to choosing the retriever as the best example of a friendly family dog, or the pizza as the best example of Italian-style cuisine.

The “dog” and “pizza” are mathematical objects. They have, as all such objects in math do, a precisely defined set of properties.

Higher mathematics is a game of connecting the lines, to see how well one can draw the connections (purely analogical) between the system of interest and the chosen mathematical object.

Once the connection has been made, all that is left to do is make a logical argument to defend the choice of object as the best exemplar. Here, instead of premises being worded statements they are mathematical expressions (more objects).

This is why published papers never have calculations. Not because they were done in the background, but because they weren’t done at all.

All the calculating, symbol shuffling, word problems, and mechanistic chain-of-events you were taught in school are not carried over into research, meaning the application of math to the real world is nothing like what most people think it is. Not even close.

There are still engineering disciplines that run the same old calculations to confirm a trajectory, calculate the load-bearing capacity of a steel beam, or some other quality control. But these are fully established textbook calculations. They haven’t changed in decades, and they represent an increasingly dated version of engineering; fully deterministic physical things whose sizes and speeds have been all but maxed out.

Engineering is now headed towards synthesis. A blending of countless factors that lead to nonlinear, nondeterministic, and almost entirely informational properties.

What is needed now is not explicit calculations or symbol shuffling, which would prove useless regardless, but logical arguments made using an advanced understanding of mathematical objects and their properties. This is how we spearhead progress, both theoretical and applied, as our complex creations take over the internal guts of machinery, and express *only* surface-level properties we can reason about.

So, what is the skill we should be looking for when it comes to the advanced, and only, version of mathematics now relevant? One thing is for sure; it has absolutely nothing to do with speed of calculation or chain-of-event reasoning. Nothing to do with expression manipulation or aptitude for word problems. Nothing to do with how mathematics is currently taught.

Pattern recognition. Analogizing. Deep intuition. That’s the game now.

The kind of people who can make a rapid and creative connection between entities. Those who perceive, and feel, the essence of a thing, and spin their intuitions into logical narratives.

Those who can make a precise argument for the best family dog or Italian cuisine.

That is tomorrow’s mathematician.