What are scientists researching about?

Plant research in space (NASA)

In previous posts I have mentioned some dangers faced by the future of science in the face of the growing obstacles suffered by basic research and the tendency of politicians to prioritize practical applications. The arguments I offered in those posts were qualitative. It is possible, however, to obtain real quantitative data, extracted from the journal Science News, which publishes a fairly complete summary of the results of research in all fields of science and technology. By analyzing 23,946 articles published during 30 years, I have obtained the following table, which shows the number of scientific news related to fourteen of the most researched topics in the fields of biology and medicine during that period of time. (More than half the research results belong to those two fields).

Daring to say “I don’t know”

I don’t know. It seems quite simple. Why so few people dare to say it?

Several years ago, when it became fashionable in popular newspapers to publish mini-surveys, answered by four or five people, about a current issue, I wondered at seeing that, whatever the question, not one of them ever answered I don’t know. Everyone was perfectly clear about what they should answer in every case.

Some of the questions had substance:

• How would you end the civil war in Yugoslavia?
• How would you solve the unemployment problem?
• How would you stop terrorism?

Five years in PopulScience

Albert Einstein

This week we celebrate a small anniversary: five years since this blog was created. In this time, 245 posts have been published. The Spanish version of the blog is a little older: it was created 30 weeks before, in January 2014, and has published 257 posts.

To mark the date by some kind of celebration, I have decided to compute the list of people most mentioned in the blog in these five years. The following table shows the names of the ten people most quoted and the number of times their name has been quoted:

Name	Times quoted in PopulScience
Albert Einstein	42
Isaac Newton	33
Stephen Hawking	20
C.S. Lewis	20
Aristotle	17
Charles Darwin	14
Isaac Asimov	14
Richard Dawkins	12
Plato	10
Ptolemy	9

Zero probability

In a previous post I mentioned that an event can happen once or several times, although the probability of its happening is zero. The probability of an event is defined as the ratio of the number of favorable cases to that of possible cases. Therefore, if the number of possible cases is infinite, while that of favorable cases is finite, the probability turns out to be zero.

At first glance it seems incredible that an event with zero probability can actually happen. I think the matter will be clearer with a simple example. Two friends, A and B, are talking, and what they say is this:

A: If I ask you to choose a number between 1 and 100, what is the probability that you choose a specific number, such as 25?

B: 1/100, obviously.

A: If I ask you to choose a number between 1 and 1000, what is the probability that you choose 25?

B: 1/1000.

A: If I ask you to choose a number between 1 and 10,000, what is the probability that you choose 25?

B: 1/10,000.

A: If I ask you to choose a positive integer number, what is the probability that you choose 25?

B: Zero, for the set of integers has infinite elements, and one divided by infinity is equal to zero.

A: Choose any number among all the positive integers and tell me which number you have chosen.

B: I choose 2²⁵⁰⁰-1.

A: You have just made an event with zero  probability.

Thinking a little you’ll see that the probability of choosing, among all the integers, any finite set, however large, is also zero. For instance:

A: If I ask you to choose ten different numbers between one and one hundred, what is the probability that you choose precisely the numbers between 11 and 20? (their order does not matter)

B: 1 / 17,310,309,456,440

A: And if I ask you to choose ten different numbers among all the positive integers, what is the probability that you choose precisely the numbers between 11 and 20?

B: Zero.

I leave to the curious reader to compute why the probability of choosing numbers 11 to 20 among those from one to one hundred is precisely what B has stated.

To finish this post, I’ll propose a few more exercises for the reader. Whoever solves them has the opportunity to write a comment explaining how they arrived to the solution.

1.      What is the last digit of 6²⁵⁰⁰?

2.      What is the penultimate digit of 6²⁵⁰⁰?

3.      What is the penultimate digit of 6^1,000,000?

4.      What is the probability that the last digit of 6ⁿ is odd?

5.      What is the probability that the penultimate digit of 6ⁿ is odd?

By Vincent Pantaloni, CC BY-SA 4.0, Wikimedia Commons

The same post in Spanish
Thematic Thread on Statistics: Previous Next

Manuel Alfonseca

Happy summer holidays. See you by mid-August

Mathematical theology

Ernst Zermelo

Ernst Zermelo (1871-1953) was a famous mathematician of the early twentieth century. Among his achievements, the following can be mentioned:

• In 1899 he discovered Russell’s paradox, two years before Russell. Although he did not publish it, he did comment it with his colleagues at the University of Göttingen, such as David Hilbert. Russell’s paradox proved that Cantor’s set theory is inconsistent, since it makes it possible to build the set of all sets that don’t belong to themselves. There are sets that don’t belong to themselves, such as the set of even numbers, which is not an even number. Others do belong to themselves, such as the set of infinite sets, which is an infinite set. Now we can ask ourselves: Does the set of all the sets that don’t belong to themselves belong to itself? This question leads us to a paradox: if it does belong, it must belong; and if it doesn’t belong, it mustn’t belong.
• In 1904 he proved the well-ordering theorem as the first step towards proving the continuum hypothesis, the first of Hilbert’s 23 unsolved problems. The well-ordering theorem states that every set can be well-ordered, which means that any non-empty ordered subset must have a minimum element. To prove it, he proposed the axiom of choice, which we will discuss later.
• In 1905 he began to work on an axiomatic set theory. His system, improved in 1922 by Adolf Fraenkel, is a set of 8 axioms, which today is called the Zermelo-Fraenkel (ZF) system. Adding the axiom of choice to this system, we obtain the ZFC system, which is most used today in set theory.

Travelling to the past?

S.Augustin, by Louis Comfort Tiffany
Lightner Museum

In his Confessions (Book XI, chapter 14), St. Augustine wrote these words, still valid today:

What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.

In the current situation of our scientific and philosophical knowledge, we still don’t know what time is.

·         For classical philosophy and Newton’s science, time is a property of the universe. Therefore, time would be absolute.

·         For Kant, time is an a priori form of human sensibility (i.e. a kind of mental container to which our sensory experiences adapt).

·         For Einstein, time is relative to the state of repose or movement of each physical object. There is, therefore, no absolute time.

·         For the standard cosmological theory, there is the possibility to define an absolute cosmic time for every physical object, measuring the time distance since the Big Bang to the present.

·         For the A theory of time (using J. McTaggart’s terminology) the flow of time is part of reality. The past no longer exists. The future does not yet exist. There is only the present. If the A theory is correct, travel to the past is impossible, because you cannot travel to what does not exist.

·         For the B theory of time, the flow of time is an illusion. Past, present and future exist simultaneously, but for each of us the past is no longer directly accessible, and the future is not yet accessible. Einstein adopted the B philosophy of time. In a condolence letter written to someone who had lost a beloved person, he wrote the following:

The distinction between past, present and future is only a stubbornly persistent illusion.

The symbol of death

Azrael, the angel of death
Evelyn De Morgan (1855-1919)

For an educated classical Greek, the number 8 represented death. Why? Let’s see what this funeral assignment was based on.

1. Multiply by 8 the first 8 natural numbers.
2. Add the digits for each result.
3. If the total obtained has more than one digit, we add those digits again.

Multiply	Add digits	2nd addition
1×8=8	8	8
2×8=16	1+6=7	7
3×8=24	2+4=6	6
4×8=32	3+2=5	5
5×8=40	4+0=4	4
6×8=48	4+8=12	1+2=3
7×8=56	5+6=11	1+1=2
8×8=64	6+4=10	1+0=1

Observe that we obtain the sequence 8,7,6,5,4,3,2,1. For the Greeks, this succession starts at 8 and descends to die at 1. That is why number 8 represented death.