You want proof?

An image is worth a thousand words. And an equation is worth a thousand images. Yet, the value of pictorial proofs is astounding. I was reminded of this fact upon reading a delightful monograph by a brilliant mind, Sanjoy Mahajan, “Street-Fighting Mathematics”.

Let us examine one example (this comes from one of the exercises in Sanjoy’s book).

Consider the following sum:

You probably know that the general formula is: n(n+1)/2

But do you know why? Try it with a couple of examples to make sure that it works:

1+2+3=3x4/2=6

1+2+3+4+5=5x6/2=15

Proof 1: Group the first and last term, the 2^nd term and the one before last, etc.

q.e.d.

 (This is assuming that n is even. If n is odd, the last term in the second line is (n+1)/2+(n+1)/2).

Proof 2: By induction.

It is trivial to verify that S1=1.

Assume that 

is correct.

Then

q.e.d.

You probably have seen Proof 1 and/or Proof 2 at school at some point. And you may have forgotten about them. Here comes a neat graphical proof. Try to forget this one!

Proof 3: Pictorial

Here is another example.

You probably know Pythagoras theorem for right triangles:  where a,b are the sides and c is the hypotenuse. But do you remember the proof? Here is a pictorial proof. No words.

References

Sanjoy Mahajan. Street-Fighting Mathematics. The Art of Educated Guessing and Opportunistic Problem Solving. MIT Press. Cambridge, MA

Individuals with highly superior autobiographical memory have false memories too

The ability to remember events from the past constitutes one of the quintessential components of who we are. Take a moment to consider you and your life without those revered moments. Forming memories is likely to have played a central role during evolution by conferring species the power to learn from mistakes and to recall useful information such as directions to food sources.

Individuals differ significantly in their abilities to recall information. In particular, recent studies have identified elite players in the particular domain of autobiographical memories. How superior are these creatures in remembering their past? The results are quite impressive. For example, an investigator asked about the events that happened on October 19, 1987. If you are like me, I would not be able to report anything at all. After scrutiny, I would be able to report events that happened that year, and perhaps even pinpoint some of those events to specific months by combining memory and reconstructive logic. One of these individuals with superior autobiographical memory responded: “It was a Monday. That was the day of the big stock market crash and the cellist Jacqueline du Pre died that day.” Wow.

Intriguingly, these superior autobiographers are not better than you and me on normal non-autobiographical laboratory tests. For example, if you give them a long list of words and you test them an hour or a day later, their recall rate would be similar to that of age and gender-matched controls. Perhaps superior autobiographers keep amazingly detailed diaries, which they scrutinize over and over on a routine basis but this does not seem to be the case for all of them.

An elegant recent study examined individuals with superior autobiographical memories in tasks that involve the formation of false memories. Several studies have documented that memories are malleable and can be distorted. In one of the multiple paradigms developed to study false memories, subjects are presented with a series of slides that describe a story. After a delay of approximately one hour, subjects are presented with a narrative of the same story that introduces misinformation. Subsequently subjects are queried in terms of the story and they often report “having seen” events that were never shown during the slides but were falsely reconstructed from the narrative. It happens to the best of us. The formation of false memories has been a serious issue in court, where witnesses may faithfully believe that they remember events that never happened.

Back to the superior autobiographers, it stands to reason that if somebody can remember that the 19^th of October of 1987 was a Monday, that person will not be easily fooled into buying misinformation. Well, it turns out that this logical hypothesis is quite wrong. Individuals with superior autobiographical memories are as prone to form false memories (in laboratory tests of non-autobiographical information) as controls. They report remembering words that were never shown, they report seeing events that never happened and they may even tell you that they have seen footage of news events that does not exist.

Our memories are reconstructions, with a significant component of reality, and a few sprinkles of added fantasy, logical deduction, embellishment, emotional distortions and other tricks. Investigators can manipulate the formation of false memories by astutely implanting a few seeds of misinformation here and there. And to this date, nobody has found individuals who are immune to such memory distortions. Even amazing people who have extraordinary autobiographical recollections can make mistakes. Even if you can remember that the violinist Josephine du Pre died on Monday, October 19^th, 1987, you can be prone to forming false memories.

Reference:

False memories in highly superior autobiographical memory individuals.

Patihis L, Frenda SJ, Leport AK, Petersen N, Nichols RM, Stark CE, McGaugh JL, Loftus EF. Proc Natl Acad Sci U S A. 2013 Dec 24;110(52):20947-52. doi: 10.1073/pnas.1314373110. Epub 2013 Nov 18.