FOS COMMISSION: JOHN TAYLOR 9

One of the greatest, and perhaps most baffling, results from the history of modern mathematics are the two incompleteness theorems of Kurt Goedel. 

Published in 1931, they exposed what has been described as a “hole in the foundations”1 of mathematics, setting off an avalanche of philosophical speculations in turn.2 

Reflecting on Goedel’s theorems, many metaphysicians, including Wolfgang Smith, have argued that they illuminate the human mind’s capacity to comprehend mathematical truths without brute force reasoning. A most striking vindication of what scholastics know as the “intellect”—if there ever was one! 

However, as will become apparent, Kurt Gödel’s incompleteness theorems concern far more than the status of human cognition, as they carry enormous implications for the ontology of mathematics itself. 

To explore these implications, this essay will unravel in three parts. First, part I will trace the major milestones in mathematics that preceded Gödel’s theorems, before turning to their eventual development and formulation. Next, part II will argue that Gödel’s theorems have two major implications for an ontology of mathematics. First, that they concord with and uphold the two principles of Smithian science. Second, that they provide a springboard for interpreting mathematics by way of irreducible wholeness. Finally, part III will show how Gödel’s incompleteness theorems relate to Cardinal Newman’s Illative Sense and how they open our hearts and minds to God. 

Part 1: What are Goedel’s Incompleteness Theorems?  

It goes without saying that Kurt Gödel was a genius of unprecedented proportions, whose contributions stretched into fields as far and wide as set theory, the foundations of mathematics, philosophy, physics and even legal theory. 

However, by far the most consequential of Goedel’s intellectual contributions are his two incompleteness theorems. In what follows, I will now explain the context that preceded these theorems (part 1.1), their formulation (part 1.2), and their implications for consciousness (part 1.3). 

Part 1.1 Some Context 

In the early 1920s mathematics’ top man was David Hilbert, famed for his enormous contributions to invariant theory and the foundations of geometry.3 However, much like its counterpart physics, mathematics was also beginning to “lose its grip on reality”.4 Since, it was starting to incorporate increasingly more abstract and infinite objects into its theoretical machinery.5 

Broadly speaking mathematics can be divided into “finitary” and “infinitary” mathematics. (It was “infinitary mathematics” that was getting out of control by the 1920s).6 

In short finitary mathematics refers to mathematics that is built using finite constructions.7 To this end, it assumes no “actual infinities”.8 For example, the set of all natural numbers is not assumed to be an object, in finitary mathematics. But instead, is treated as an indefinitely extendable process: for any given natural number, one can construct the next, without requiring the existence of a completed infinite totality of all natural numbers. In Aristotelian terms infinity is treated as a potentiality, not as an actuality, in finitary mathematics.9 Assuming this conception of the infinite, finitary mathematics can prove theorems using inference rules that operate only on finite mathematical objects and finite strings of symbols. Importantly, such proofs do not rely on completed infinite sets, or entities, but instead proceed through a finite sequence of logically valid steps that can, in principle, be checked or verified by a finite procedure.10 Some examples of finitary mathematics include: elementary number theory, combinatorics, and finite number theory. 

By contrast, infinitary mathematics refers to mathematics based on actual infinities. From the outset, it treats infinite collections as complete, well-defined objects.11 For example, the set of all natural numbers, the set of all points on a geometric line, and the set of all real numbers can each be regarded as actual infinities. Like finitary mathematics, infinitary mathematics also involves proving theorems through the use of axioms and rules of inference. However, the axioms and inference rules employed in infinitary mathematics are often regarded as more philosophically or mathematically exotic because they explicitly invoke infinite objects.12 For instance, the axiom of infinity affirms the existence of an infinite set (the set of natural numbers),13 while transfinite induction elongates the principle of mathematical induction to infinite ordinal numbers.14 Some examples of infinitary mathematics include set theory and real analysis. 

Ignoring the finer details surrounding finitary and infinitary mathematics, Hilbert and his contemporaries were concerned that infinitary mathematics was too far removed from the concrete. As Hilbert correctly observed, “the infinite is nowhere to be found in reality”.15 

For this reason, Hilbert only regarded finitary mathematics as “real mathematics”, whose objects are rooted in an abstract Platonism,16 while regarding infinitary mathematics as “ideal mathematics,” whose objects are only useful fictions.17 However, despite this, Hilbert did not reject infinitary mathematics outright. On the contrary, he considered infinitary mathematics to be incredibly useful, proclaiming famously of Cantor’s set theory that “no one shall drive us out of the paradise that Cantor created for us”.18 

At the same time, however, Hilbert also wanted to find a way to prevent mathematics from shooting off to infinity and beyond. For all its mathematical utility, the concept of the actual infinite had generated a plethora of paradoxes that demanded containment, ranging from: Cantor’s paradox to Hilbert’s Infinite Hotel. In response, Hilbert proposed a program whose central aim was, in essence, to reduce infinitary mathematics to finitary mathematics: to demonstrate that the former could ultimately be justified on the secure foundations of the latter.19  

Achieving this auspicious goal, however, first required establishing that finitary mathematics itself constituted a secure foundation to which the infinitary could legitimately be reduced. According to Hilbert, this required demonstrating two points. First, that every expressible statement within finitary mathematics could, in principle, be proved using finitary methods.20 Second, that finitary mathematics was capable of proving its own consistency.21 The reason for the second was that if finitary mathematics cannot prove its own consistency and theorems, then it certainly cannot prove all infinitary theorems and the consistency of infinitary mathematics.22 In that case infinitary mathematics could not be reduced to finitary mathematics, thereby, undermining the entire telos of Hilbert’s Program. 

To this end, the precise mechanism which Hilbert believed infinitary mathematics could be reduced are finite mathematical machines known as “formal systems”; specifically formal systems of arithmetic. We will now cover the nature of these formal systems, how these systems relate to Goedel’s two Incompleteness Theorems, and in turn their connection to consciousness.  

Part 1.2. The Incompleteness Theorems and their Implications for Consciousness 

Having presented his program in the early 1920s, optimism was rife that Hilbert’s dream would eventually be realized. However, this promise was sorely tested with the publication of Gödel’s two incompleteness theorems in 1931.  

I will now explain these two theorems, along with their proofs, before considering their implications for consciousness. We will proceed in three stages: first I will explain what a formal system is, second, I will state and explain the two incompleteness theorems, third I will break down the meaning of the two theorems for consciousness.  

Stage 1: What are formal systems? 

In short, formal systems are systems of reasoning made up of the following components: 

“Axioms. These are basic rules or statements that we accept as true without needing to prove them. For example, that “0 is a number” or that “no natural number comes before zero”. 

Symbols. A notation to write down statements. For example: 0 for zero, +, ×, = for addition, multiplication, and equality, ∃ for “there exists”, ∀ for “for all”, → for “implies”, and ¬ for “not”. 

Theorems & rules of inference - statements & rules for how to make new true statements from old ones. These are things like: If P and P → Q both are true, then Q is true. Induction: If a statement is true for 0, and if the statement being true for X implies it’s true for X + 1. Then it’s true for all natural numbers.”23 

Now, from a logical perspective there are various properties which we would like formal systems to have and are relevant to Goedel’s incompleteness theorems. 

These properties include:  

Soundness: Every theorem that can be proved is true within a formal system.24  

Completeness: Every statement that is true can be proved within a formal system.25 

Consistency: A statement and its negation cannot be proved within a formal system.26 

When a “formal system can carry out a certain amount elementary arithmetic”27, this means it can “express the basic facts of arithmetic, such as multiplication and addition.”28 It is precisely these formal systems which David Hilbert wanted to reduce mathematics to.   

An example of such a formal system is Peano Arithmetic (PA). 

PA’s axioms are: 

“ 

1. Zero is a natural number. (Some formulations use 1 instead of 0 as the "first" natural number) 
2. Every natural number x has a successor s(x) in the natural numbers. 
3. Zero is not the successor of any natural number. 
4. If the successor of two natural numbers is the same, then the two original numbers are the same. 
5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.  ”29 

Its symbols include: “0, 1, 2, 3, …, the +, -, and other signs, and some formulations, for example, denote the successor of a number x by S(x)”.30  

Its inference rules include: “modus ponens, universal generalization, and mathematical induction.”31 

With PA, one can prove theorems such as “for all natural numbers X, X + 0 = 0.”32 

Stage 2: What are Goedel’s incompleteness theorems? 

In 1931 Goedel proposed the following two theorems about incompleteness:  

The First Incompleteness Theorem states: No sound formal system that can carry out a certain amount of elementary arithmetic is complete.33 

The Second Incompleteness Theorem states: No formal system as described in the first can prove its own consistency.34 

Let us now consider the meaning of these two theorems and on what terms they can be established. 

For Goedel’s first incompleteness theorem: “for any sound formal system Z, capable of carrying out a certain amount of elementary arithmetic, for example PA, we always find a sentence that is true yet not provable. This sentence is aptly called the Goedel sentence (Gs). In short Gs, via a special encoding known as Goedel numbering, roughly says ‘Gs is not provable in Z’.”35 This is a blatant catch 22! For if you could prove Gs, in Z, Gs would ipso facto be false. Since the conditions for G’s truth are dependent on its unprovability (within Z).36  

Now because we know Z is “sound, and also that Gs is false, this means Gs cannot be proved using the machinery of Z. Since to do so would be to prove something that is false, which Z’s soundness does not permit. However, by elimination this implies that Gs is true.”37 So, voila we have something that is true, yet unprovable, which also makes Z incomplete. 

Now for his second incompleteness theorem Goedel was able to establish that if you could prove the consistency of a formal system such as Z you could also prove Z’s Gs.38 However, we know that Gs cannot be proved. So, it follows that no formal system, as described, can prove its own consistency. 

To sum things up Goedel showed that certain formal systems will always contain an unprovable sentence Gs and, because of this their consistency also cannot be proved. 

Stage 3: What do the incompleteness theorems mean for consciousness? 

What separates Goedel’s incompleteness theorems from more ordinary mathematical theorems is the vertiginous ensemble of metaphysical repercussions attributed to them.  

The most famous of these repercussions is that the theorems prove consciousness is non-computational. In other words, that consciousness does not reduce to step-by-step, or algorithmic, reasoning. This conclusion was primarily reached by Roger Penrose and John Randolph Lucas,39 and was later formalized in what became known as the Penrose-Lucas argument:  

Premise 1: Via Goedel’s incompleteness theorems human mathematicians can recognize truths that cannot be proved within a formal system. 

Premise 2: Formal systems cannot recognize truths without proofs. 

Conclusion: The human mind is more than a formal system.40 

From a Christian standpoint the Penrose-Lucas argument supports non-reductive modes of knowing. For example, the: intellect, the nous, and as we shall see Cardinal Newman’s very important concept of the Illative Sense. 

Part 2: Goedel’s Theorems, Mathematics and the Principles of Smithian Science  

Having presented Goedel’s two incompleteness theorems, and seen their implications for consciousness, we are now in a position to investigate how they might comport with the two Principles of Smithian science (part 2.1) and with a general metaphysics of mathematics (part 2.2). 

As Essay 3 reveals the two principles of Smithian science are: 

Principle 1: Irreducible Wholeness is the essential element that enables any science to work. 

Principle 2: There exists a grounding-to-emergence relationship between Irreducible Wholeness and the structural form of all scientific theories. 

Part 2.1. 

Regarding these two principles the first observation I’d like to make is that because of its self-referentiality the Goedel sentence is clearly an IW. By effectively saying, “I cannot be proved,” its information content is not contained in the sum of its individual components. But, instead, “emerges” from a reflexive self-negation that transcends the sentence’s constituent parts and operates at the level of the sentence as an irreducible whole. 

From this, we can draw two conclusions. 

First, that irreducible wholeness is demonstrably present in formal systems of arithmetic and thereby mathematics at large. Second, that the Goedel sentence is an example of a grounding to emergence in Mathematics. For, it is an example of irreducible wholeness appearing at the top firmament of a scientific theory—albeit in a formal-mathematical one. Remember that the Goedel sentence refers to itself and so in a sense is the “highest statement” that a formal system can have. 

Likewise for consistency this property is also an IW. Since, by also being unprovable this “macro property” of formal systems transcends a finite series of steps and so is an IW by not being reducible to a proof. 

Once again, consistency matches with the first principle of Smithian science. Since, the property of consistency is also a macro or higher-level property, not reducible to a proof, and thus comports with Principle 2 of Smithian science. 

Similarly, for Principle 1, the idea that “irreducible wholeness is the essential element that enables” any science to function means that, in formal systems, IW is what undergirds their stability as wholes—even when the statements they produce appear reducible to mere “sums of parts.” 

To grasp what this means, consider how we recognise properties such as consistency in the absence of a formal proof. We recognise them through exemplification: by encountering the system as a whole in its concrete instances and seeing how its properties are manifested through the coherent operation of its parts. 

In other words, we can tell that a formal system is consistent just by using it! Evidently, such a transfer of knowledge could only be possible if a formal system were first couched in irreducible wholeness. Since, without IW underlying a formal system we could not intuit from one particular instance of consistency that consistency would transfer to all of other instances. Thus, Goedel’s incompleteness theorems lend support to Principle 1: that IW is the essential element which enables any science, such as formal systems, to work. 

Part 2.2 

Having shown how Gödel’s two incompleteness theorems connect with the two principles of Smithian science, I would like to make a few pedestrian observations on the relationship between IW and mathematics at large. 

In perfecting his ontology of physics, Wolfgang Smith became aware, that this ontology might also be applicable to the field of mathematics.41 Indeed, I distinctly remember him saying as much to a group of us during a Zoom call! 

As we have seen, Prof. Smith’s ontology of physics hinges upon the breakdown of irreducible wholeness (IW) and the movement between the different levels of IW. For example, the semi-creation of SX depends upon the destruction of X, which is itself an IW, into a quantitative object.42 Likewise, in quantum theory, there is a transition between the IW of the SX domain and the hidden IW of the transcorporeal domain.43 

Might there, then, be a similar ontology of mathematics? And, if so, what might Gödel’s theorems have to do with it? 

As we have seen, the mathematical analogue of the dividing line between the classical and quantum domains is the distinction between finitary and infinitary mathematics. Once again, what arguably separates these two domains is irreducible wholeness. We have good reason to think this is the case because Gödel’s theorem points to an irreducibility of infinitary mathematics to the finitary. It is therefore not unreasonable to conjecture that IW grounds mathematics as a whole, enabling the mathematician to glide between the finite and the infinite. Indeed, when we are able to apprehend and manipulate entire infinite sets as wholes, how could this be understood as anything other than an instance of irreducible wholeness? 

So, perhaps the solid foundation that Hilbert was looking for to ground mathematics, at large, is not to be found in any formal theory but instead is found in irreducible wholeness itself! What is needed next, then, is to extend these preliminary conjectures about IW into more advanced areas of mathematics and to examine whether the same structure of irreducible wholeness can be discerned there too. 

Part 3: Cardinal Newman’s Illative Sense and Goedel’s Theorems  

Cardinal John Henry Newman (1801–1890) was a renowned theologian, writer, and priest. He was a leading figure in the Oxford Movement before converting from Anglicanism to Catholicism in 1845. In 1879, Pope Leo XIII made him a Cardinal. 

One of Newman’s lesser-known contributions is his epistemological concept of the Illative Sense. In simple terms, the illative sense refers to the mind’s ability to arrive at truth without relying solely on formal deductive or inductive reasoning.44 Instead, the mind reaches a conclusion through the subtle and often unconscious assimilation of numerous facts, experiences, and pieces of evidence that, when taken together, point in a particular direction.45 

For example, a person might come to believe in God not because of a single apologetic argument, but through the convergence of many different experiences. They might encounter a profound act of mercy, become aware of the wretchedness of sin, and experience the beauty and order of nature. Each of these experiences is distinct and may not, by itself, constitute a conclusive argument for God’s existence. However, when considered together, the mind can assimilate them into a broader whole and arrive at the conclusion that God exists. For Newman, this kind of reasoning helps explain how human beings can arrive at genuine knowledge through the cumulative force of experience and evidence, even when that knowledge cannot be reduced to a formal syllogism. 

What is particularly interesting about Goedel’s theorem is that the kind of knowledge involved in discerning the consistency of a formal system bears a striking resemblance to the kind of knowledge Newman describes through his concept of the Illative Sense. Recall that the consistency of such systems is discerned through engagement with the system itself and the application of its rules. Could this not be understood as an example of Newman’s Illative Sense at work within mathematics? If so, Goedel’s theorems would demonstrate more than the non-mechanical character of the human mind. They would also suggest that the mind operates according to a mode of knowing that bears a striking resemblance to the kind of reasoning Newman regarded as indispensable to religious knowledge.46 

Conclusion: 

In closing, we have explored the monumental role that Goedel's theorems have played in mathematics, as well as their broader ontological implications for the discipline. We have also considered their intriguing connection to Cardinal Newman’s concept of the Illative Sense. But might Newman’s concept carry further significance in relation to these ideas? That is the question we shall explore in the next essay, which will examine the connection between Smith’s work and Cardinal Levada’s New Apologetic. 

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