Godel and Wittgenstein; same goal, different approach

 The current issue of Philosophy Now (Issue 169, Aug/Sep 2025) has as its theme, The Sources of Knowledge Issue, with a clever graphic on the cover depicting bottles of ‘sauces’ of 4 famous philosophers in this area: Thomas Kuhn, Karl Popper, Kurt Godel and Edmund Gettier. The last one is possibly not as famous as the other 3, and I’m surprised they didn’t include Ludwig Wittgenstein, though there is at least one article featuring him inside.
 
I’ve already written a letter to the Editor over one article Challenging the Objectivity of Science by Sina Mirzaye Shirkoohi, who is a ‘PhD Candidate at the Faculty of Administrative Sciences of the University Laval in Quebec City’; and which I may feature in a future post if it gets published.
 
But this post is based on an article titled Godel, Wittgenstein & the Limits of Knowledge by Michael D McGranahan, who has a ‘BS in Geology from San Diego State and an MS in Geophysics from Stanford, with 10 years [experience] in oil and gas exploration before making a career change’, without specifying what that career change is. ‘He is a lifelong student of science, philosophy and history.’ So, on the face of it, we may have a bit in common, because I’ve also worked in oil and gas, though in a non-technical role and I have no qualifications in anything. I’ve also had a lifelong interest in science and more recently, philosophy, but I’m unsure I would call myself a student, except of the autodidactic kind, and certainly not of history. I’m probably best described as a dilettante.
 
That’s a long runup, but I like to give people their due credentials, especially when I have them at hand. McGranahan, in his own words, ‘wants to explore the convergence of Godel and Wittgenstein on the limits of knowledge’, whereas I prefer to point out the distinctions. I should say up front that I’m hardly a scholar on Wittgenstein, though I feel I’m familiar enough with his seminal ideas regarding the role of language in epistemology. It should also be pointed out that Wittgenstein was one of the most influential philosophers of the 20th Century, especially in academia.
 
I will start with a quote cited by McGranahan: “The limits of my language mean the limits of my world.”
 
I once wrote a rather pretentious list titled, My philosophy in 24 dot points, where I paraphrase Wittgenstein: We think and conceptualise in a language. Axiomatically, this limits what we can conceive and think about. This is not exactly the same as the quote given above, and it has a subtly different emphasis. In effect, I think Wittgenstein has it back-to-front, based solely on his statement, obviously out-of-context, so I might be misrepresenting him, but I think it’s the limits of our knowledge of the world, that determines the limits of our language, rather than the other way round.
 
As I pointed out in my last post, we are continually creating new language to assimilate new knowledge. So, when I say, ‘this limits what we can conceive and think about’, it’s obvious that different cultures living in different environments will develop concepts that aren’t necessarily compatible with each other and this will be reflected in their respective languages. It’s one of the reasons all languages adopt new words from other languages when people from different cultures interact.
 
Humans are unique in that we think in a language. In fact, it’s not too much of a stretch to analogise it with software, remembering software is a concept that didn’t come into common parlance until after Wittgenstein died in 1951 (though Turing died in 1954).
 
To extend that metaphor, language becomes our ‘operating language’ for ‘thinking’, and note that it happens early in one’s childhood, well before we develop an ability to comprehend complex and abstract concepts. Just on that, arguably our exposure to stories is our first encounter with abstract concepts, if by abstract we mean entities that only exist in one’s mind.
 
I have a particular view, that as far as I know, is not shared with anyone else, which is that we have a unique ability to nest concepts within concepts ad infinitum, which allows us to create mental ‘black boxes’ in our thinking. To give an example, all the sentences I’m currently writing are made of distinct words, yet each sentence has a meaning that transcends the meaning of the individual words. Then, of course, the accumulation of sentences hopefully provides a cogent argument that you can follow. The same happens in a story which is arguably even more amazing, given a novel (like Elvene) contains close to 100k words, and will take up 8hrs of your life, but probably over 2 or 3 days. So we maintain mental continuity despite breaks and interruptions.
 
Wittgenstein once made the same point (regarding words and sentences), so that specific example is not original. Where my view differs is that I contend it also reflects our understanding of the physical world, which comprises of entities within entities that have different physical representations at different levels. The example I like to give is a human body made up of individual cells, which themselves contain strands of DNA that provide the code for the specific individual’s construction and functioning. From memory, Douglas Hoffstadter made a similar point in Godel Escher Bach, so maybe not an original idea after all.
 
Time to talk about Godel. I’m not a logician, but I don’t believe you need to be to appreciate the far-reaching consequences of his groundbreaking theorem. In fact, as McGranahan points out, there are 2 theorems: Godel’s First Incompleteness Theorem and his Second Incompleteness Theorem. And it’s best to quote McGranahan directly:
 
Godel’s First Incompleteness Theorem proves mathematically that any consistent formal mathematical system within which a certain amount of elementary arithmetic can be carried out, is incomplete – meaning, there are one or more true statements that can be made in the language of the system which can neither be proved nor disproved in the system.
 
He then states the logical conclusion of this proof:
 
This finding leads to two alternatives: Alternative #1: If a set of axioms is consistent, then it is incomplete. Alternative #2: In a consistent system, not every statement can be proved in the language of that system.
 
Godel’s Second Incompleteness Theorem is simply this: No set of axioms can prove its own consistency.
 
It’s Alternative #2 that goes to the nub of the theorem: there are and always will be mathematical ‘truths’ that can’t be proved ‘true’ using the axioms of that system. Godel said himself that such truths (true statements) might be proved by expanding the system with new axioms. In other words, you may need to discover new mathematics to uncover new proofs, and this is what we’ve found in practice, and why some conjectures take so long to prove – like hundreds of years. The implication behind this is that our search for mathematical truths is neverending, meaning that mathematics is a neverending endeavour.
 
As McGranahan succinctly puts it: So knowing something is true, and proving it, are two different things.
 
This has led Roger Penrose to argue that Godel’s Theorems demonstrate the distinction between the human mind and a computer. Because a human mind can intuit a ‘truth’ that a computer can’t prove with logic. In a sense, he’s right, which is why we have conjectures like the ones I mentioned in my last post relating to prime numbers – the twin prime conjecture, the Goldbach conjecture and Riemann’s famous hypothesis. However, they also demonstrate the relationship between Godel’s Theorem and Turing’s famous Halting Problem, which Gregory Chaitin argues are really 2 manifestations of the same problem.
 
With each of those conjectures, you can create an algorithm to find all the solutions on a computer, but you can’t run the computer to infinity, so unless it ‘stops’, you don’t know if they’re true or not. The irony is that (for each conjecture): if it stops, it’s false and if it’s true, it never stops so it’s unknown. I covered this in another post where I argued that there is a relationship between infinity and the unknowable. The obvious connection here, that no one remarks on, is that Godel’s theorems only work because mathematics is infinite. If it was finite, it would be 'complete'. I came to an understanding of Godel’s Theorem through Turing’s Halting Problem, because it was easier to understand. A machine is unable to determine if a mathematical ‘truth’ is true or not through logic alone.
 
According to McGranahan, Wittgenstein said that “Tautology and contradiction are without sense.” He then said, “Tautology and contradiction are, however, nonsensical.” This implies that ‘without sense’ and ‘nonsensical’ have different meanings, “which illustrates the very language problem of which we speak” (McGranahan using Wittgenstein’s own language style to make his point). According to McGranahan, Wittgenstein then concluded: “that mathematics (if tautology and contradiction will be allowed to stand for mathematics), is nonsense.” (Parentheses in the original)
 
According to McGranahan, “…because in his logic, mathematical formulae are not bipolar (true or false) and hence cannot form pictures and elements and objects [which is how Wittgenstein defines language], and thus cannot describe actual states of affairs, and therefore, cannot describe the world.”
 
I feel that McGranahan doesn’t really resolve this, except to say: “There would seem to be a conflict… Who is right?” I actually think that if anyone is wrong, it’s Wittgenstein, though I admit a personal prejudice, in as much as I don’t think language defines the world.
 
On the other hand, everything we’ve learned about the world since the scientific revolution has come to us through mathematics, not language, and that was just as true in Wittgenstein’s time as it is now; after all, he lived through the 2 great scientific revolutions of quantum mechanics and relativity theory, both dependent on mathematics only discovered after Newton’s revolution.
 
The limits of our knowledge of the physical world are determined by the limits of our knowledge of mathematics (known as physics). And our language, while it ‘axiomatically limits what we can conceive and think about’, can also be (and continually is) expanded to adopt new concepts.