Would you answer truthfully if you were asked in a survey whether you consume cannabis? Maybe you do like to wind down with an edible after a long day. But the legal contradictions between federal and state laws make you wary. Besides, what will your kids' friends' parents think? The glare of judgment is blinding. Or, what if someone asked you whether you voted for Donald Trump in 2016? What about in 2020? The systematic polling errors in each of those US presidential elections live in infamy. …

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The 8-cell (Photo: Wikimedia Commons)

How many spatial dimensions do we live in? How many can we directly perceive? Are the two answers the same? If they are different, what effect, if any, do any higher dimensions have on us?

What Is a Dimension?

We exist in physical space. In this space, every point or location can be labeled by a set of coordinates in a coordinate system. There can be more than one coordinate system for the same space. But once we choose a system, two points have the same coordinates if and only if they are the same point in space. The number of independent coordinates needed to label a point is the number of spatial dimensions. This dimensionality is a geometric property of space and is the same in every coordinate system for that space. …

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Photo by Dylan Nolte on Unsplash

Keep drawing integers from a random number generator (or a customized lottery machine) until you get a number that is smaller than the previous pick. If your run ends after three or fewer draws (including the last picked smaller number) then you pay $10. Otherwise, you receive $20. Will you accept this bet? Think about it.

Your instincts may tell you it’s a good bet. You need to pick at least two numbers before your run could possibly end. So it seems plausible that the average length of such draws would be larger than three. …

But Their Preponderance Is Not

A utopia, in which everyone is a cooperating member of society, cannot exist (at least not for long). Why? Because it is unstable. If everyone plays by the rules, all it takes is one individual to break the rules, gain an advantage over others, and attract more people to break the rules.

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Short of utopia, is there something else we can aspire to as a society? Indeed. We can dissuade antisocial behavior. Behaviors can be shaped by incentives. The specific incentives in a society lead to a particular balance of behaviors. That particular mix is not what is optimal for everyone. But it is a stable equilibrium. Equilibrium means that if society is in that state, it will remain in that state unless something changes. Stable means that if society’s composition deviates from the equilibrium, it’s in the interest of some people to change their behaviors (for better or for worse), gain personal advantage and drive society back towards that stable equilibrium. …

Let’s dispel the myth that a single vote does not mathematically make a difference in an election.

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Photo by Arnaud Jaegers on Unsplash

If a million ballots are cast in an evenly poised election between two candidates, what are the odds that your single vote could decide the election? Almost 0? 1 in 1,000,000? Well, actually it’s close to 1 in 1,000! Admittedly it’s still a small likelihood, but not as small as you thought, right? If someone gave me a 1 in 1,000 shot at directly choosing an elected official, and all I had to do was to cast a vote, I would take that in a heartbeat.

Have you thought much about the speed of light and its implications? If not, you really should. Therein lie marvels.

  1. We see into the past
  2. We all agree on the speed of light
  3. Simultaneity is in the eye of the beholder
  4. Time slows down
  5. Lengths shorten
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Photo by Casey Horner on Unsplash

We See Into The Past

Get a flashlight, stand one foot away from a wall, direct the flashlight at the wall and turn it on. The light beam will show up on the wall, seemingly instantly. Actually, it wasn’t instant. It took approximately a nanosecond. A nanosecond is 10⁻⁹ seconds i.e. a billionth of a second. …

What is a real number, really? What kinds of numbers make them up? What do they represent?

Natural Numbers: Counting

Starting at the most concrete level, a number is a count of how many objects there are. There are six bananas in my fruit basket. I have two children. There are eleven players in a cricket team. And so on. This notion of a number as a count is, for most of us, our first encounter with numbers. Such numbers are vividly reflected in the physical world. They are apparent even to some animals (NYT article).

Such a number that is a count of things is known as a natural number. It can be represented quite literally with tally marks equal to the count. So three tally marks represent a count of three objects. …

How many numbers of each type are there? Can two infinities be equal? Can one infinity be larger than another infinity?

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Real Numbers Line

Why Are You Reading This Right Now?

Of all the things you could have chosen to do at this moment, you chose to read this. Why? And after reading this, you'll choose another activity. Perhaps walk your dog or cook dinner. With every such choice, you traverse in time one branch in a tree of forking possibilities. How do you decide which branch to take?

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You’re reading this right now presumably because you think you’ll get some personal value from it. You can quantify that value, in principle, by assigning a real number called utility (from Economics) to it. The higher the utility, the more value you get. But while you're reading this you're also depriving yourself of the utility of doing one of the many alternate activities available to you. That is the opportunity cost of reading this now. In theory, if you had perfect knowledge of how much value you get from each of the infinite ways in which you can spend every moment of your time, you could define a personal utility function of time for each possibility. Then, the answer to the question "what should you do now" simply becomes do that which increases the utility to you the most. In other words, choose the path of steepest ascent in the space of utility functions. Actually things are more complicated. …


Vishesh Khemani, Ph.D.

Mindful Thinker | Software Engineer (Google, Amazon) | Theoretical Physicist (MIT) | Husband, Dad, Dog Dad

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