You Can’t Answer this [Math] Question

I’m sorry for the pithy title, but hey, that’s what people on Medium do, right? I swear I’m not a bozo trying to grab your attention to make a few bucks off of you and I really have no interest in you reading this other than to share some ideas I think are worth sharing. Let’s get right to it.

What’s the answer to this question

How many of the options below are correct answers to this question?

a. 75%
b. 25%
c. 25%
d. 50%

Well? What do you think? If you tell me it’s “a” then I would say 3 / 4 answers are correct, but you told me the answer is “a” so there must only be one answer. If you tell me the answer is “d” I would say 2 / 4 answers are correct, but you told me the answer is “d” so there must only be one answer. See where I’m going with this?

You can’t answer the question. But who cares? Well I played a bit of a trick on you and I only did so because I want to share something with you — the “cool” side of math. That might sound stupid, but bear with me. Math is cool, and for a reason you might not yet have imagined.

You see, math is one of those things that’s misunderstood and often taught in a really bad way in the modern era. If you’ve slogged your way through a 21st century grade school mathematics curriculum you’ve probably been inundated with questions asking you to apply “rules” and push around symbols to pop out a “right” answer, leaving you with the idea math is a cut and dry subject with questions that have binary, true or false, numerical answers.

But that isn’t all there is to mathematics. Not even close. In fact memorizing “rules” and applying them to scenarios with completely defined parameters is actually a very small part of the study and application of mathematics. Mathematical ideas are just as open to interpretation and critical analysis as ideas in the social sciences or what we typically consider “soft domains”. The challenge and amount of mental effort to make a mathematical statement meaningful is the same as it is with normal language and doing good work in other domains is just as challenging as it is in mathematics.

What’s more, one of the really interesting things about mathematics is that we don’t really know what it is or how or why it exists. Are numbers a “thing”? Are we inventing or discovering them? What do they mean? How do we claim to know so much about something and yet we have no real grasp of its incarnation?

Those are deep questions that don’t have straight forward answers and they touch on things that have real implications. The beauty of math is in the fascinating and exciting journey of figuring out “what” it is and what’s possible with it. And the only way we’re going to do that is by trying out new ways of doing math and seeing what’s possible with them. And trying new ways of doing math means thinking critically about what mathematics is and finding completely new, never before imagined ways of doing it.

If you can learn to question math deep enough you’ll learn to see it’s not arcane wizardry that only a few people in an ivory tower can make sense of. In fact it’s a lot more accessible than most people think.

Math is a rich and universally accessible set of mental tools that is rife with opportunities to apply the same type of critical thinking that you can in all other parts of life. The beauty is that there are no major barriers to doing math. You don’t need a particle collider or a sample of living tissue — just pick up a book, fire up your imagination and drill into the problem. There are no dearth of situations in the world around you which might benefit from the application of mathematical principles — especially in today’s era where you have the rich ecosystem of personal computers to support you!

There are enough problems to capture your imagination for more than your lifetime (at least for me).

Critical > Procedural

So why is the question I asked above interesting? Well first of all it’s an interesting problem because it exposes a fundamental truth — math, like other parts of life, requires critical thinking and intuition. It’s perfectly possible to say something in math that means nothing at all or that is totally inconsistent and strange. There is simply no master set of magical, fancy symbols that you can put together that solve all problems and being able to regurgitate the most confusing and difficult to interpret statement is not what makes you “good” at math.

What you have probably found in life is that a lot of times some people only care for seeming smart. And when you only care for the appearance of being smart you usually don’t sign on to doing the hard mental work to figure things out but nonetheless try to present the appearance of understanding things by… well, usually doing what you would think (if you hadn’t done much thinking) is impressive which is a ton of fancy and lengthy equations with many twists and turns and “gotchas”.

If you really want to understand life you will come to see complexity is the enemy and the length of an equation or amount of greek symbols in a mathematical statement is not a sign of it’s value. It is a misunderstood correlation. It’s not hard to spot when you actually understand the principles behind doing good work in mathematics and often times people who think “fancy” or “arcane” = impressive end up doing lot’s of complicated stuff that accomplishes nothing at all, or worse, simply creates confusion.

But I digress.

What do I mean about the question I asked above? Well, let’s go back to it

How many of the options below are correct answers to this question?

a. 75%
b. 25%
c. 25%
d. 50%

What’s the answer? Well if you were taught mathematics is a dry subject with no room for critical thinking or intuition then you probably tried to solve the problem like any other boring old math problem and found yourself caught in a paradox, or something of the sort. None of those answers could possibly be right because for the answer to be right a condition that makes the answer wrong would have to be true.

But I didn’t ask the question to analyze it, I highlighted the question as an example of a case where you have to think critically and question the framing of the question to make sense of what’s asked of you. When trying to answer that question you might have found yourself confused and ultimately led to the conclusion that you just couldn’t figure out the answer. But what if you took a different angle and stepped back to observe the phrasing of the question itself?

I mean, let’s ignore the numbers, what is the question trying to get at? Did you assume it was a reasonable thing to ask in the first place? If so, why?

See, the promise of mathematics is not in the largest, most numerous or most complex and difficult to figure out series of symbols that seem consistent, it’s in the relation of mathematics with our experience of life. The true beauty of math is in its ability to relate simply and elegantly to our real, lived experience.

The more time you spend trying to fit math into reality in a simple, direct and accessible way the more you realize that doing math well really does require nothing other than the simple, natural aspects of our personality everyone posses — curiosity, common sense and intuition. That’s part of the reason I think nurturing the positive parts of our personalities is so important. It’s not just pie in the sky fluff to say the way you think and feel matters — it really does. Treating yourself well is a exponential win. You should do it.

See, the question I asked above has a simpler, more direct answer that doesn’t involve any fancy mathematical knowledge at all. Instead, you could just use critical thinking and intuition to assess the situation and then, once you’ve properly contextualized what’s actually being asked, progress to analyze it mathematically or say you should not mathematically analyze it all because it makes no sense.

If you think about it, it’d kind of silly to ask a question that asks what the answer to the question that the question is asking is, isn’t it? That is effectively equivalent to me asking you “What question am I asking you?” and claiming that I just asked you a question. You can’t answer a question that didn’t actually ask a question. And if the question was what the answer to the question was the question got ahead of itself and asked for an answer before there was even a question!

Did I confuse you? Well, if you’re curious, think about it a bit. It’s kind of weird, right? A little bit funny, admittedly, but there’s something very true there. I didn’t do any fancy symbol pushing but I did point out something insightful about the question I asked. But I only did so when critically analyzing the framing of the question, not diving head first into trying to answer an unanswerable question with fancy ideas.

Do you see what I did there? I used critical thinking and intuition to analyze a mathematical statement and come out with an insight that shaped the way I thought about the situation. That is how you can unlock the promise of mathematics, or any other domain. That style of thinking (critical thinking and the construction of unique and original ideas) is what reveals the promise of knowledge.

Really knowing is about having clarity, not a volume of facts. It’s about honing your ability to assess an idea or situation and tease out the ideas that get you closest to the truth. We all have that ability and we all have the ability to practice and enhance that ability. There is no superpower that some people possess that makes them categorically smarter than anyone else. There is no standardized test that can tell you what’s possible when you apply yourself and there is no limit to how many new ideas we can come up with that can make life brighter for all of us.

How do I know that? Well, I just have carefully and deeply considered my experience of life and landed on the idea that what we think of as intelligence is universally accessible, capable of creating win-wins and possible to apply in almost all scenarios. I can see the hints of that brightness in everyone I talk to, even when they don’t seem to see it in themselves, and that’s one of the reasons I respect and appreciate people.

Concluding Thoughts

Anyways, if you want another cool mind bending problem that you can do with nothing more than two quarters check out https://www.scientificamerican.com/article/the-sat-problem-that-everybody-got-wrong/ and http://www.donaldsauter.com/rolling-circles.htm. It’s a fun and puzzling one I assure you.

If you want to hear someone I know little about but who has given lectures expressing similar ideas and who I understand is quite well regarded for their intellectual interests check out this lecture from Richard Feynman — https://www.youtube.com/watch?v=B-eh2SD54fM (I don’t think this is copyrighted but please let me know if it is I’ll take it down).

Anyways, I hope you enjoyed this, and I apologize for goading you into clicking on this. Congrats if you made it this far into my ramblings. I don’t expect many will (;