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Puzzles: Inside the Mind of a Mathematician, Part III

There is little else that reveals a true mathematician in a recreational context quite like a puzzle. Whether it’s a brain teaser, a card game, an object freer, or a crossword, all good puzzles are an extension of the “if-this-then-that” argument. They require patience and deduction, and mental trial and error, and are all designed to test the strength of a person’s logic.

Logic is a profound concern that we interact with every day. Our ability to engage in logical reasoning is, to our knowledge, unprecedented. And for the mathematicians among us, and others who engage with high-level mathematics on a daily basis, logic is not only critical and productive, but it is a stunningly convenient reality underpinning much of our natural world.

To: Little System, From: Big System

As creatures who create systems as tools for governing our world, the fact that much of the natural world runs on logical systems itself is a great gift to us. Perhaps this is no accident. Perhaps humans love logical systems because we’ve arisen from and adapted to a world that runs on one, too. Regardless of the which-came-first dilemma, the world runs on logic, and we make great use of this.

The basic idea of logic is that two opposing truths cannot be true at the same time. A palm sized object cannot be both in my left hand and across the room simultaneously. This fundamental axiom gives rise to other deductible arguments that allow us to form more complex conclusions: for one, about how a situation is behaving, and for two, about how a situation might behave if given certain inputs.

From the first discoveries of physics, we have made use of its core, dependable truths in building our civilizations. It’s not random whether a bridge can hold the weight of the traffic cycling across it; we simply need to find the parameters that determine the physical limits so that we can operate within them. Or, to take another example, light does not appear in a random fashion; it always originates from a source. This allows us to investigate, direct, and create sources of light so as best to serve our purposes.

If physics is logic-made-muscle, which we wield to build our world and keep it in motion, then mathematics is the logical nervous system behind it all. Just like the human nervous system is a network of highly similar neurons dictating instructions based on a predetermined set of rules, in mathematics, there are no inconsistencies. The fact that we can engage in gamified puzzles and apply “if-this-then-that” logic to a situation is only possible because mathematics underpin our world.

As alluded to earlier, this is remarkably helpful, as it means that there is an element of predictability to our world. We are more like free will agents operating on a predictable canvas. And we are not the only species that makes use of the world’s logical structure.

Crow Statistics

More than one animal species has been credited with the ability to use logic, and recent research has brought crows into that category1.

A group of crows was given a statistical puzzle. When presented with several pictures, choosing certain images over others would result in being given a treat. There was a small trick: some pictures came with a higher probability of receiving a reward. As such, it was up to the crows to figure out which combinations of pictures they should choose in order to maximize their likelihood of reward.

With a little bit of practice, the crows nailed this challenge, along with several variations (though not every variation proved successful). Aside from the degree of arguable self-awareness they demonstrated (the ability to understand that their actions could influence their environment and result in a more favourable outcome to themselves), the crows showed that they could engage in statistically-influenced decision-making in a non-random manner, even when positive results were not guaranteed. This is exactly the kind of behaviour we use constantly, and one that’s crucial to the mathematical mind.

Of course, despite their dark feathers, crows pale in comparison to human children in logic tests, and new research has shown logical reasoning emerging at an even earlier age2. When presented with a logical problem, toddlers as young as 19 months were able to demonstrate deductive reasoning. They were presented with a familiar object and an unfamiliar object, along with hearing the name of each object. In recognizing that the familiar object paired with the familiar word, they were able to deduce that the unfamiliar word must be associated with the unfamiliar object. This gives an incredible level of insight into how humans learn, especially in our earliest stages of development when absolutely everything is new. Our minds start out as puzzle boards with missing pieces and we are constantly evaluating new inputs that come in to see if they fit some of the unknown spaces and contribute to a more complete picture of the world.

A Mathematician’s Wheelhouse

With all of the power that the most basic logic and deduction can afford us, it’s no surprise that gifted mathematical minds leverage logical reasoning for next-level accomplishments. Logic and mathematics are inseparable, and mathematicians are constantly putting their weight on fundamental axioms to jump to rational conclusions.

Is logic in your wheelhouse, too? Do you love a good puzzle? Maybe you’re a bit of a mathematician, too. Let us know in the comments.

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