Photo by NASA on Unsplash

Adventure

​With his two Incompleteness Theorems, the late mathematician Kurt Gödel "showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved." So Carl Boyer explains in his History of Mathematics.

Even as non-mathematicians may not understand Godel's logical formalisms, we may well sense that the theorems function poetically as well as mathematically. They are an invitation for all of us to accept the incompleteness of life: the fact that, no matter how certain we may feel about our convictions, there's always something more, something novel, something incomplete. This is not bad news; it is good news. It means that life, like pure mathematics, is an adventure.

- Jay McDaniel, 11/20/2

Beyond Deduction
​

"Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions … It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced."

Carl B. Boyer, History of Mathematics

​

A Scholarly Conversation
​ on Godel's Theorems

BBC's In Our Time
​

"Melvyn Bragg and guests discuss an iconic piece of 20th century maths - Gödel’s Incompleteness Theorems. In 1900, in Paris, the International Congress of Mathematicians gathered in a mood of hope and fear. The edifice of maths was grand and ornate but its foundations, called axioms, had been shaken. They were deemed to be inconsistent and possibly paradoxical. At the conference, a young man called David Hilbert set out a plan to rebuild the foundations of maths – to make them consistent, all encompassing and without any hint of a paradox. Hilbert was one of the greatest mathematicians that ever lived, but his plan failed spectacularly because of Kurt Gödel.

"Gödel proved that there were some problems in maths that were impossible to solve, that the bright clear plain of mathematics was in fact a labyrinth filled with potential paradox. In doing so Gödel changed the way we understand what mathematics is and the implications of his work in physics and philosophy take us to the very edge of what we can know.

​With:
​

• Marcus du Sautoy, Professor of Mathematics at Wadham College, University of Oxford;
• John Barrow, Professor of Mathematical Sciences at the University of Cambridge and Gresham Professor of Geometry
• Philip Welch, Professor of Mathematical Logic at the University of Bristol.

​
Incompleteness Theorems 0:36David Hilbert 5:04
The Axioms of Euclid's Geometry 8:44
Set Theory 9:36
The Barber Paradox 12:51
Is There an Infinite Set between the Set of Counting Numbers and the Set of Infinite Decimal Numbers 29:41
Debate about the Difference between Proof and Truth in Mathematics 35:05

Neither Proved nor Disproved

'Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system FF within which a certain amount of arithmetic can be carried out, there are statements of the language of FF which can neither be proved nor disproved in FF. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).' 

- Stanford Encyclopedia of Philosophy​

Incompleteness
​
Notes on Whitehead and Kurt Gödel

"There remains the final reflection, how shallow, puny, and imperfect are efforts to sound the depths in the nature of things. In philosophical discussion the merest hint of dogmatic certainty as to finality of statement is an exhibition of folly." (Alfred North Whitehead)
​
Kurt Gödel is one of the most important mathematicians of the twentieth century, known especially for his two Incompleteness Theorems. These theorems concern the limits of provability in formal axiomatic theories.  In what follows I offer some incomplete notes on Whitehead and Gödel, hoping my reflections might be springboards for further thought on the part of non-mathematicians and, for that matter, non-philosophers. I write for people who are curious, pure and simple.

A word of warning: many mathematicians insist that Gödel's theory does not apply to topics outside mathematics and that, even within mathematics it applies only to axioms in number theory. They would disapprove of my notes on Whitehead and Kurt Godel. I apologize in advance for being presumptuous. 

​Whitehead himself was a mathematician, but much less influential than Gödel in the world of mathematics. However, Whitehead's outlook on life complements Gödel's overall philosophy in at least three ways. 

First, Whitehead supports the idea that life itself is "incomplete" at any given moment, because novelty is part of the very makeup of the universe. We humans and the rest of creation are never finished: individually, socially, intellectually, or spiritually. This does not mean that we are always progressing or that things are improving. But it does mean that they are changing. The universe itself is, as Whitehead puts it, a "creative advance into novelty." 

Second, Whitehead's outlook includes the idea, which Gödel embraced, that mathematical entities are not reducible to human constructs. They are truly part of the makeup of the universe, and they existed as potentialities long before human beings evolved and imagined them. Gödel was a Platonist, and in his way Whitehead was, too. Whitehead did not see eternal objects as more real than the actual world, but he did see them as real apart from human construction. We discover them, we do not create them.

Third, Whitehead's philosophy resonates with the religious side of Gödel's life. Here is the account in Wikipedia:

"Gödel believed that God was personal, and called his philosophy "rationalistic, idealistic, optimistic, and theological". Gödel believed in an afterlife, saying, "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of.... But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."...In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza." Of religion(s) in general, he said: "Religions are, for the most part, bad—but religion is not". According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning", while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."

In Process and Reality, Whitehead affirms a personal God who loves the world with "a tender care that nothing be lost" and who is a "fellow sufferer who understands." And several Whiteheadians, David Griffin in particular, use Whitehead's thought to make sense of life-after-death.

On, then, to the mathematics. My primary source for understanding Gödel's theorems, or trying to understand them, is an essay by Solomon Feferman, of the Institute for Advanced Study at Princeton, written for Gödel Centenary Program in 2006, called "The nature and significance of Gödel’s incompleteness theorems." Click here. Feferman studied and met Gödel.

"I was a member of the Institute in 1959-60, a couple of years after receiving my PhD at the University of California in Berkeley, where I had worked with Alfred Tarski, another great logician. The subject of my dissertation was directly concerned with the method of arithmetization that Gödel had used to prove his theorems, and my main concern after that was to study systematic ways of overcoming incompleteness. Mathematical logic was going through a period of prodigious development in the 1950s and 1960s, and Berkeley and Princeton were two meccas for researchers in that field. For me, the prospect of meeting with Gödel and drawing on him for guidance and inspiration was particularly exciting. I didn’t know at the time what it took to get invited. Hassler Whitney commented for an obituary notice in 1978 that “it was hard to appoint a new member in logic at the Institute because Gödel could not prove to himself that a number of candidates shouldn’t be members, with the evidence at hand.” That makes it sound like the problem for Gödel was deciding who not to invite. Anyhow, I ended up being one of the lucky few." [2]

So what do the two theorems say? The general idea is that, within a given mathematical branch, there will always be some propositions that can't be proven either true or false using the rules and axioms of the branch at issue. In this sense the branch will be incomplete as will be our thinking within the branch. We must create new axioms, thinking outside the branch, to move forward, and these new axioms, within the new branch, will also be incomplete. There is no finality.

A recognition of incompleteness did not lead Gödel to distrust mathematics. He was himself a mathematical Platonist, believing that mathematical entities have reality independent of the human mind.

"Gödel was a Platonist – an extreme conceptual realist: for him, the numbers of arithmetic and the systems of mathematics have a reality independent of anyone experiencing or thinking about them. Since to him mathematics was not a human construct, it was natural for him to regard proof as distinct from truth. As his Incompleteness Theorem itself seems to demonstrate, he felt that we could intuitively grasp truths we could not prove. Constructivists, who believe that the truths of mathematics are constructed through human reasoning and do not have a human-independent reality, lack such sublime certainty: for a mathematical constructivist everything must be proved if it is to be fully grasped." [2]

Gödel believed we could intuit mathematical relationships we cannot prove. In Whitehead's language, we can "conceptually prehend" truths we cannot prove. Truth is more than proof.

What does it come to? When it comes to mathematics, for Godel and for Whitehead, we must be both humble and creative. Humble, in that mathematics are incomplete, and creative in that, if we want to prove things, they must create new axioms that lie outside the branch at issue. 

I say axioms reluctantly. Whitehead, himself a mathematician, did not think that philosophy should proceed from, or be deduced from, formal axioms. We cannot deduce the nature of the world and ourselves within it by relying on axioms in our minds. We must turn to experience itself. Whitehead writes that the sole justification of any thought, even mathematical thought, is the illumination of immediate experience. Whitehead puts it this way: "The elucidation of immediate experience is the sole justification for any thought, and the starting-point for thought is the analytic observation of components of this experience."

*

Many scholars, especially mathematicians, believe that Gödel's theorems are best left to mathematics, and that discussions of their implications for other areas of life are vague and often misguided. See the essay by Rebecca Goldstein (click here): 

"Like Heisenberg’s uncertainty principle, Gödel’s incompleteness theorem has captured the public imagination, supposedly demonstrating that there are absolute limits to what can be known. More specifically, it is thought to tell us that there are mathematical truths which can never be proved. These are among the many misconceptions and misuses of Gödel’s theorem and its consequences. Incompleteness has been held to show, for example, that there cannot be a Theory of Everything, the so-called holy grail of modern physics. Some philosophers and mathematicians say it proves that minds can’t be modelled by machines, while others argue that they can be modelled but that Gödel’s theorem shows we can’t know it. Postmodernists have claimed to find support in it for the view that objective truth is chimerical. And in the Bibliography of Christianity and Mathematics (yes, there is such a publication) it is asserted that ‘theologians can be comforted in their failure to systematize revealed truth because mathematicians cannot grasp all mathematical truths in their systems either.’ Not only that, the incompleteness theorem is held to imply the existence of God, since only He can decide all truths."

There is no need here add to the confusion. But it is worth noting that, for process theologians, the idea that our thinking is limited and that we can discover new ideas rings true, not only mathematically but existentially. And for us, the universe is itself a non-mechanistic process: a creative advance into novelty. Not only in math but in life, we best be humble and creative,

And let it be noted that, for process theologians, God does not decide all truths. The truths of the universe, what happens at all levels, are decided by entities in the universe. not by God. Whitehead believes that decision, the cutting off of potentialities and the actualizing of others, is the essence of actuality: be it microscopic, macroscopic, terrestrial or galactic, human or divine. And yet the decisions of the world, including human decisions, are embodiments of timeless potentials which mathematicians explore. Our understandings of these timeless potentials, these eternal objects, are never complete, But always, in pure mathematics, they are interesting and beautiful. There is no need to grasp all the truths. It is enough to look for them and enjoy small tastes, spiritually and mathematically, as did Kurt Gödel.

​- Jay McDaniel, 11/10/22

[1] Tucker McElroy (2005). A to Z of Mathematicians. Infobase Publishing. p. 118. ISBN 978-0-8160-5338-4.
[2] http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf