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:
The second example is following:
We add cubes of consecutive natural numbers. Thus we have1 + 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.
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
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
00:16:12"... Penrose draws inspiration from the concept of platonic mathematical entities suggesting that mathematical truths exist independently of human minds and are somehow woven into the fabric of the universe. ..."
Roger Penrose: "Quantum Mechanics Wrong And Consciousness Isn't computation"
P.S.2. (A.J) Concerning Plato himself see The Cosmic Context of Greek Philosophy: Part Eleven.
P.S.3. (A.J) And another somewhat "lighter" quote:
... “By the end of the eleventh century,” Edmond said, “the greatest intellectual exploration and discovery on earth was taking place in and around Baghdad. Then, almost overnight, that changed. A brilliant scholar named Hamid al-Ghazali—now considered one of the most influential Muslims in history—wrote a series of persuasive texts questioning the logic of Plato and Aristotle and declaring mathematics to be ‘the philosophy of the devil.’ This began a confluence of events that undermined scientific thinking. The study of theology was made compulsory, and eventually the entire Islamic scientific movement collapsed....”Dan Brown, "Origin", Transworld, 2017
P.S.4 (A.J) See also Ivars Peterson, "Deriving the Structure of Numbers", Science News 2004
“... I love working on difficult math,” Linda Westrick says. “For hard problems, I make lists and draw pictures and look for patterns that I believe are there, and I find the work worthwhile even when I don’t make tangible progress.”
“Whether I’m proving theorems, designing robots, or inventing new math, my goal is the same: to create something beautiful,” she adds. “I hope to add to the beauty and simplicity of existing theories and create beautiful math of my own. I hope to inspire others by what I create.”
Yes if wasn't for number patterns I would not have become so interested in physics since the actual overall math (algebra, etc.) is beyond my abilities. One I recently noticed is that the self dual middle grade of a Clifford algebra Cl(n) (for an even n), is the sum of squares of for the Pascal triangle row for half the n. Thus the middle 4 grade for Cl(8) has 1^2+4^2+6^2+4^2+1^2=70 dimensions. I noticed it because via Tony Smith, I use a cellular automata rules table as a simplified Hodge Star map and the self dual 4-grade showed up as a bunch of square blocks and it just occurred to me recently that they would be a sum of squares.
ReplyDeleteConcerning the "eternal mathematical truths": I think there are "interesting truths" and "non-interesting truths". Which ones of the uncountable infinity of "truths" are "interesting"? It is the human mind that decides it. And the human mind is just a part of a much wider concept, that of a "cosmic consciousness".
ReplyDeleteWe, human beings, have the ability to "focus" our minds and decide "this is so beautiful". "But is it really "true"? In which sense? And what is "truth" anyway?"
It may be that the very concept of "sets", "formal systems", "axioms" is an invention of human mind. Other intelligence may well operate on a different bases. They may have no concept of "sets consisting of elements". There reasoning may well be all "fluid" and "wavy". They know nothing about rigid axiomatic systems. So what we call "mathematics" may be completely alien for them.
ReplyDeleteSince our human mathematics are built on a foundation, perhaps there is a universal realization/foundation/truth required for any intelligence to make progress, i.e. a starting block. One such statement I can think of is the strangely recursive "truth exists," where the act of giving a meaning to truth is, in itself, also a truth. What can we deduce from it? Well, there is a "timeless" aspect to it, challenging our preconceived notion of time. Maybe there is a structure which is always present, existing beyond our little lives which are commonly thought to have a definite start and end.
DeleteNow, is there a fully objective path which can be followed to understand reality without making any subjective assumption? Or is subjectivity a necessary condition–even if only temporary–for progress?
The conflict is as old as time: between materialism and spirit: “The mind of the spirit vs. the mind of the flesh.” Indeed, it seems that "truths" exist independently of human minds. And if that is so, the so do ideas which are composed of information. And if information/ideas are at the bottom or beginning of it all, then we may think that the division or opposition exists even there.
ReplyDeleteHere is a nifty article about the metaphysical status of evil.
https://ponerology.substack.com/p/supernatural-evil-and-ponerology
In physics there are a limited number of spacetime dimensions, particles and forces so you can think that theoretically fundamental information bits might handle this. For complex conscious things like materialism vs. spirit or good vs evil., it is hard to imagine fundamental bits handling this in the same way as physics though maybe it makes sense for bits to associate with certain complex consciousness patterns like with personality model factors.
ReplyDeleteContinuing my previous comments about math, truth, and alien intelligences: here is a relevant quote from "A World Without Time" by Palle Yourgrau, Basic Books 2005:
ReplyDelete" The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick
to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers. (Enthusiasts of artificial intelligence were not amused.)"
Yes, formalized systems and machines do help, but to know the the Truth it is necessary to go beyond these limits and learn how to read the non-digital information waves.