Paul Davies, popularizer of science

Paul Davies

Paul Davies came to the fore among scientists who devote time to popular science with his 1992 book The Mind of God, written in response to Stephen Hawking’s final words in his popular best-seller A Brief History of Time. In another post I talked about another of his popular books, The Eerie Silence. Here I am going to discuss two other books he has written.

The Last Three Minutes (1994): This book on popular science is a little behind the times, as it predates the standard cosmological model, but explains well the state of cosmology when the book was published, and many of the things it says are still valid. It says something very interesting: that the Big Bang theory by Lemaître (whom Davies does not name) should have been accepted long before its two surprisingly accurate predictions gave it a boost in the sixties, because there is another argument supporting it, that scientists of the 19th century should have noticed, but didn’t: If the universe were infinitely old, it would have died by now. It is evident that something that moves to a stop at a finite rate cannot have existed from all eternity. By the way, there is an error in this paragraph: Davies ignores the difference between what is eternal and everlasting, which was solved fifteen centuries ago by Boethius. And there is a major flaw when he says that the radius of the visible universe is 15 billion light-years, because he does not take into account the expansion of the universe. The correct radius is about 43 billion light-years.

My 10 Favorite Scientific Discoveries of the 20th Century

In a post published two weeks ago, I commented on an article in Science News that tried to answer this question: which were the ten most important scientific discoveries of the last century? Some of my readers asked what is my personal opinion. This is my answer.

To begin with, I will point out that scientific research can advance in four different ways:

1. Theoretical science, which tries to discover fundamental laws in the universe.
2. Experimental science, which confirms or falsifies theories by carrying out experiments.
3. Observational science, which instead of experimenting, observes. Astronomy, for instance, uses these methods, as experimentation is almost never possible.
4. Technology, the practical application of science, whose goal is to build devices that work.

The limits of mathematics

Kurt Gödel

In the last decades of the nineteenth century, Friedrich Ludwig Gottlob Frege, a professor in the university of Vienna, undertook an ambitious goal: formalizing the arithmetic in a set of axioms and deduction rules, in such a way that every true theorem would be deductible from the axioms by a finite number of applications of the deduction rules. The result was a monumental book, Grundgesetze der Arithmetike (1893-1903), which introduced, among other things, a basic formalization of set theory and a cumbersome notation, quickly replaced by Peano’s, which we are using now.

Unfortunately for Frege, when the second volume of his book was about to be published, he received a letter from Bertrand Russell, proving that his formulation of set theory entails an inconsistency. In Frege’s set theory, some sets are not member of themselves (as the set of all integers, which is not an integer), while other sets are members of themselves (as the set of all infinite sets, which is an infinite set). Russell then defined this set: the set of all sets that are not members of themselves. It is easy to see that this set leads to a paradox: if it is a member of itself, it cannot be a member of itself, and vice versa. Russell’s paradox wreaked havoc with Frege’s work, who had to add a hasty appendix to his book and then abandoned his research on the fundamentals of mathematics.

The three laws of Robotics

Isaac Asimov

Isaac Asimov was a prolific science fiction and popular science writer who published in the 40s a series of stories about robots, later compiled in the I, Robot collection. In these stories he invented a word that has become a part of the technological vocabulary, as the name of a discipline: Robotics. He also formulated the three famous laws of Robotics, which in his opinion should be implemented in every robot to make secure our interactions with these machines that, when Asimov formulated the laws, were simple future forecasts.

The three laws of Robotics are the following:

First Law: A robot may not harm a human being, or through inaction allow a human being to come to harm.

Second Law: A robot must obey any order given by a human being, except those that conflict with the first law.

Third Law: A robot must protect its own existence as long as such protection does not conflict with the first two laws.

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.

Gödel and realism

Kurt Gödel

Kurt Gödel (1906-1978) was one of the most important mathematicians of the 20th century. In 1931, when he was 25, he rose to fame with his mathematical proof that the attempt to build a complete axiomatic system, from which one can deduce all the arithmetic of natural numbers or any equivalent system, is doomed to failure.

His first incompleteness theorem says the following:

Every consistent formal system as powerful as elementary arithmetic is not complete (it contains true undecidable propositions).

Let us look at a simplified informal demonstration:

Let theorem G say the following: This theorem G cannot be proved from the axioms and rules of system S.

• If we assume that Theorem G is false, system S is inconsistent, since a false theorem can be proved from the axioms and rules of S.
• Then if S is consistent, G must be true, and therefore cannot be proved from the axioms of S.
  

Gödel’s theorem shows that every axiomatic formalization of arithmetic is either inconsistent (it allows false theorems to be proved), or incomplete (it contains true theorems that cannot be proved).

What does physics tell us about time travel?

In the previous article we considered a few paradoxes that could bring us to doubt the possibility of time travel. But what does physics say about this? Is there any theory that would make time travel possible? Is it true, as some say, that Einstein’s special theory of relativity implies that it will be possible to travel in time?

First of all, we must refute a fairly widespread misconception. We often hear people saying something like this:

If it were possible to travel at speeds greater than the speed of light, we would travel backwards in time, because the passage of time would become negative.

Is this true? Consider the equation that defines the relationship between proper time and external time for a body moving with a uniform rectilinear speed, according to the special theory of relativity:

Where t is the time experienced by travelers who move at speed v; t₀ is the equivalent external time (the time experienced by an object at rest); and c is the speed of light. 

We can see that, for v < c, the term inside the square root is positive and less than 1, its root would also be less than 1, and therefore t < t₀ (the time experienced by the travelers is shortened).