As the title suggests, the main theme of the book represents the existence of various factors that describe how we should approach mathematics. The conclusion drawn has two things: formalism (claiming that Maths is invented by the human mind) and Platonism (regarding mathematics as an a priori universal language whose truths are merely discovered).
Being a senior astrophysicist at the Hubble Space Telescope Science Institute and author of a few other math books aimed at the general public, Mario Livio has written a short, accessible, and in many ways profound exploration of the nature of mathematics. He centers his book around two questions:
1. "Is mathematics ultimately invented or discovered?" and
2. "Why is mathematics so effective and productive in explaining the world around us that it even yields new knowledge?"
2. "Why is mathematics so effective and productive in explaining the world around us that it even yields new knowledge?"
He frames his inquiry with what physicist Roger Penrose describes as the triple mystery. The idea is that there are three worlds that people experience: the world of physical reality, the world of our minds, and the abstract world of mathematics. Then the mysteries are as follows:
1. why would world of physical reality give rise to our minds that perceive the reality?
2. why would our minds give rise to abstract mathematics? and
3. why does mathematics so effectively describe the physical reality in which we exist?
1. why would world of physical reality give rise to our minds that perceive the reality?
2. why would our minds give rise to abstract mathematics? and
3. why does mathematics so effectively describe the physical reality in which we exist?
There are several occasions whereby the author proves the existence of maths that has been 'discovered' - just like Shakespeare did not 'invent' Hamlet. There is stress on the statistical part of the Mathematics and syllogism at the beginning. Aristotle, Archimedis, Dante, Descartes, Galileo, Copernicus, Kepler, Tesla, Bernoulli's, Mendle, Gauss,....everyone's contribution has been dealt with suitably in a lucid manner.
Each chapter discusses important topics like geometry, logic, topology, statistics and probability theory, as well as major breakthroughs in adjacent fields – such as physics or astronomy. The narrations are easy to follow and the overall tone is objective. Unlike many popular science books that tend to get tedious or uninteresting after the first few chapters, it has a good structure and can keep the reader engaged.
Talking about the Natural Selection (of Darwin fame), the author argues that if there was no society and everyone was mere re-producer (of the next generation), then there would be no selection at all!
That would mean that a marriage in a society allows the genes to 'sense' the order and promote genes that is not defective for the next generation.
The next beautiful thing I liked about was the 'golden ratio' concept. There are many examples, including the human hand and the face where one part is longer than the other (in a single entity) and a mere mathematical division gives the value of 1.6xx which is abundant in nature (just like the pi).
Discussing about the relationship between Mathematics and Logic, there is good mention about the "barber paradox" by Bertrand Russel. The most excellent example is one around De Morgan, a Mathematician. When asked about his age, his reply was "I was x years old in the year x^2". It turns out that his age was 43 and 43x43 gave 1849. 1849 - 43 thus gave 1806, the year in which he was born!
George Boole, on whose adjectives we have based our Computer Science Syllabus gets his due with a brief mention about his logic using mathematics. The famous 'Gordian Knot' is used to describe the logic and perturbation of various knots. I had to explore a new science behind the knots. (Haven't we come across various knots undone by magicians? But then we never thought there is Mathematics behind it).
A mathematical model of the atom that turned out to be wrong turned into the pure mathematics of knot theory, which then yielded the key to understanding the structure of DNA. As Livio surveys the field, he exhibits a charming sense of surprise at each unexpected turn in the problems he’s describing.
Slowly, the impact of language is brought in to suggest how language is useful in making one understand the logic behind every action. "You cannot repair a Hoover Dam, using a chewing gum" - a blistering example of how easy it is for us to understand the sentence and the 'logic' behind it. Einstein thus insisted that mathematics is a creation of the human mind, abstracting from messy reality to invent a language whose implications can be unspooled in an imaginary realm of perfection.
He ends with the following quote from Bertrand Russell's "The Problems of Philosophy":
"Thus to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy
contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good."
contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good."
