Plato was right

Today's guest post is by Marek Wolf

When in 1997 I visited Isaac Newton Institute in Cambridge I bought a book "Men of Mathematics" written by E.T. Bell, edition from 1937 [1]. In the five story bookshop there were a few shelves with classics: "Mathematical Principles of Natural Philosophy" by Isaac Newton, "On the Origin of Species" by Charles Darwin, "Principles of Quantum Mechanics" by Paul A.M. Dirac from 1958 and the above-mentioned book by E.T. Bell.

Chapter nine of Bell's book  is devoted to Leonhard Euler (1707--1783). On page 147, citing de Morgan, Bell tells an anecdote, about Euler who was a deeply religious Calvinist. At the same time there lived Denis Diderot, who, as a progressive philosopher,  was atheist. The two men  met one day and Diderot asked Euler if he has a proof that God exists. Euler wrote some formula and said: "Hence God exists. Sir, please reply". Diderot embarrassed quickly walked away.

We do not know what the formula was, but here, I will present two formulas in the same spirit, which I often expose  students to during the first calculus lecture. The first is following: 

 Add two first odd numbers: 1+3 = 4, we added two numbers and the result is 4=22. Next we add five: 1+3+5=9, the summations of three first odd numbers gives 9=32. Next we add 7 and obtain 16=42, again the square of a number of added terms. It continues to be true up to to infinity. In formula we can write it as

The simplest proof is by induction.

The second example is following:

 We add cubes of consecutive natural numbers. Thus we have

1 + 23 = 1 + 8 = 9= 32 = (1+2)2 ,

1 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2 ,

1 + 23 + 33 +  43 = 100 = 102 =  (1+2+3+4)2 : the sum of cubes of the consecutive integer numbers is a square of the sum of these numbers.

And again so on to infinity.

Isn't it miraculous and mysterious? It seems to hide some deep secret. As an equation we write

This identity is sometimes called Nicomachus’s theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). About four centuries ago, Johannes Faulhaber (1580--1635) developed formulas for the power sums. He published many papers on this topic, initially he found particular sums up to exponent 7 in 1614. Subsequently Faulhaber gave a formula expressing the sum of the k-th powers of the first n positive integers. In no other case is a sum of powers of consecutive natural numbers equal to square of another sum. In [2] Sheila M. Edmonds proved that no higher power of n sums would be expressed as the square of the sum of powers of 1, 2, ...,, n. More precisely she proved that if

then p=3 and q=1.

These formulas (1) and (2) are the absolute and eternal truth. It means that they were true even before mankind appeared on the Earth. They were true also before the Big Bang (if you believe in it). It means that there is a world containing such mathematical truths independent of our real Universe. This idea was first put forward by the Greek philosopher Plato (428/427 or 424/423 – 348/347 BC). This system of beliefs in philosophy is called "Platonism". 

 What is the reality of mathematical theorems? Has the theorem existed anywhere until the moment it is formulated by a mathematician, or is it only the appearance of it in the head of a human that creates the reality it concerns?

Many mathematicians consider themselves Platonists.

Today Euler could argue using the axiom of choice. It is a kind of mathematical joke: choice axiom is equivalent to the existence of God. The choice axiom is a statement from set theory and was formulated in 1904 by Ernst Zermelo. The axiom of choice says that given any collection of non--empty sets (even if the collection is uncountable), it is possible to take one element from each set and construct a new set from these elements. Assuming the axiom of choice,   Zermelo proved that each set can be well-ordered. It means that in each set there exists the first element with respect to some ordering relation. In particular a set of all causes can be well ordered.  Hence there exists the first cause. 

 According to Saint Thomas Aquinas the first cause of everything was God. In the opposite direction it is simpler: God is an omnipotent being and thus can choose from each set by one element and the axiom of choice follows. In 1963 Paul Cohen (1934 -- 2007) proved that the axiom of choice is independent of other axioms of the set theory. It means it is possible to have consistent mathematics with or without the choice axiom. Therefore it "means" it is possible to have a world with God and a world without God.

German mathematician Don Zagier in his habilitation lecture in 1977 said "... upon looking at prime numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation." [3]


[1] E.T. Bell, "Men of Mathematics. The lives and achievements of great mathematician from Zeno to Poincare ", A touchstone book, 1937, New York, London, Sydney

[2] Sheila M. Edmonds, Sums of Powers of the Natural Numbers, Mathematical Gazette, Vol. 41, No. 337 (Oct., 1957), pp. 187-188

[3] D.Zagier, The first 50 million prime number,  Math. Intellig. no. 0 (1977), p.8, left column

Marek Wolf