Archive for February, 2018

Making everyday conversation less fuzzy

February 18, 2018

In everyday conversation we use words and phrases that are, at best, ambiguous and vague. Yet, despite the fuzziness we somehow manage to communicate with each other and get things done.

A cool thing about mathematicians is that they work really hard to give words and phrases precise meaning. Today I learned how they give precision to the phrase infinite set. I think it’s mind-blowing.

Consider the set of positive integers, 1, 2, 3, 4, …

Clearly it is an infinite set. But wait! What does “infinite set” mean? That’s one of those ambiguous, vague phrases. So, how to define it? Let’s approach the problem by identifying a characteristic of infinite sets.

Notice that each positive integer can be mapped to an even integer by multiplying the positive integer by 2:

For every positive integer there is a corresponding even integer. There is a 1-for-1 correspondence between the two sets.

Therefore the two sets have the same number of members!

This result is surprising in view of the fact that the set of even integers is a proper subset of the set of positive integers (3, for example, is in the positive set but not the even set). We are accustomed to thinking of a set as being “larger” than any of its proper subsets, but here we are inescapably led to conclude that sometimes a set and a proper subset of that set may have the same number of members.

That is really unusual behavior!

When we examine the sets that exhibit this unusual behavior, we find that they are just the ones that we would intuitively call infinite.

DEFINITION: A set is infinite if and only if it is equivalent to a proper subset of itself.

Wow!

A New Goal: Aim to Be Less Wrong

February 16, 2018

At a conference last week, I received an interesting piece of advice:

“Assume you are wrong.”

The advice came from Brian Nosek, a fellow psychology professor and the executive director of the Center for Open Science. Nosek wasn’t objecting to any particular claim I’d made — he was offering a strategy for pursuing better science, and for encouraging others to do the same.

When Nosek recommended that I and other scientists assume that we are wrong, he was sharing a strategy that he’s employed in his own lab — a strategy for changing the way we offer and respond to critique.

Assuming you are right might be a motivating force, sustaining the enormous effort that conducting scientific work requires. But it also makes it easy to construe criticisms as personal attacks, and for scientific arguments to devolve into personal battles. Beginning, instead, from the assumption you are wrong, a criticism is easier to construe as a helpful pointer, a constructive suggestion for how to be less wrong — a goal that your critic presumably shares.

https://www.npr.org/sections/13.7/2018/02/12/585057058/a-new-goal-aim-to-be-less-wrong

With my mind I shall heal my body

February 6, 2018

When I was growing up my parents didn’t have money. When one of us kids got injured there wasn’t the option of going to a physical therapist or a massage therapist or a chiropractor or an acupuncturist. I read a lot of books on the power of the mind, so when I got injured I used my mind to heal myself. And it worked. I successfully healed myself of a heart murmur. When I was young, my mind was powerful and strong.

Now that I am an adult, I can afford to hire physical therapists and massage therapists and chiropractors and acupuncturists. 3 months ago, I injured my shoulder. Since then I’ve undergone physical therapy, massage therapy, chiropractic, and acupuncture. I’ve given them a lot of my money and time. The pain in my shoulder remains. Recently I remembered how, as a child, I used my mind to heal myself. Sadly, I’ve come to realize that my mind has become weak in terms of its power to heal myself. But, I am determined, my mind will be powerful again! Deep inside, I know how to heal my shoulder – do the deep relaxation that I used to do as a child. I have started doing this and my shoulder is already improving.

Hello darkness my old friend …

February 4, 2018

Hello darkness my old friend
I’ve come to talk with you again

That song was written by Art Garfunkel in honor of his friend Sanford Greenberg, who he befriended at Columbia University in the 1950s. The two made a pact that they would always help each other in times of need. During the school year Greenberg contracted an eye disease and went blind. Garfunkel helped him through that period, reading his school assignments, encouraging him. Greenberg graduated and moved to London. One day he got a call from Garfunkel who told him that he had been writing music with a fellow by the name of Paul Simon and needed \$400 to record their first album. Greenberg had \$404 in his bank account, wrote out a check for \$400 and sent it to Garfunkel.

It is a wonderful story, well worth listening to:

At 15 minutes, 56 seconds: http://www.bbc.co.uk/programmes/p05wxfkh

What is the meaning of life?

February 2, 2018

What is the meaning of life? I think people ask that question out of the assumption that meaning is something you can look for, “Oh I found it. Here’s the meaning. Here’s what it is. Here’s what I’ve been looking for.” But that doesn’t consider the possibility that, maybe, meaning of life is something that you create, you manufacture for yourself and for others. When I think of meaning in life, I ask, “Have I learned something today that I didn’t know yesterday?” Bringing me a little closer to knowing all that can be known in the Universe. Just a little closer. However far away all the knowledge sits, I am a little closer. If I live a day and I don’t know a little more that day than the day before, then I think I wasted that day.

— Neil deGrasse Tyson (6:53) https://www.youtube.com/watch?v=ltJbyS_DWIo