Can Consciousness Be Calculated on a Napkin? (Short version)

A Friendly Exploration of Big Ideas

Imagine you’re sitting in a diner. You scribble “2 + 2 = 4” on a napkin, and your friend smiles, “That’s right!” Now, imagine you write down a complicated equation or even a design for a new machine. If you follow a clear set of instructions, you’ll eventually reach an answer. This is what we call computable math-a series of clear, step-by-step instructions that any computer (or a very patient person) can follow.

But here’s a curious thought: If math on a napkin can be broken down into clear steps, could our own minds-our consciousness-be nothing more than a set of instructions? Or is there something extra, something that can’t be fully captured by a computer program?

Computable vs. Non-Computable: What’s the Difference?

Computable Things:

  • These are problems or tasks that can be solved by following a clear, finite set of steps (an algorithm).
  • Examples include basic arithmetic (like our “2 + 2 = 4”), solving equations, and even many complex calculations.

Non-Computable Things:

  • Some problems have no complete set of rules that will always lead to an answer.
  • A famous example is Turing’s Halting Problem, which shows that there’s no single method that can decide, for every possible computer program, whether it will eventually stop or run forever.

Almost everything we do with computers falls into the computable category. But a few puzzles in mathematics show us that not everything can be broken down into simple steps.

What Does This Have to Do With Consciousness?

Physicist and mathematician Roger Penrose suggests that our conscious minds might not work like a computer. His idea is inspired by discoveries in mathematics that show limits in formal systems (the set of rules and steps used in computers). Here are a couple of key points:

Gödel’s Incompleteness Theorem

In the 1930s, Kurt Gödel showed that any system that is complex enough (like the math behind computers) has some true statements that it just can’t prove using its own rules.

  • Why it matters:
    • When mathematicians see these “unprovable” truths, it seems like they are stepping outside the rigid system of rules.
    • If our minds can see truths that a strict algorithm cannot, maybe our thinking isn’t just following a set of instructions.

The Chinese Room Thought Experiment

Imagine a person in a room with a rulebook for responding to messages in Chinese. Although they can produce correct answers by following the book, they don’t truly understand Chinese.

  • The idea:
    • A computer might be very good at following rules and producing correct outputs, but does it really “understand” what it’s doing?
    • Many argue that true understanding or consciousness involves more than just manipulating symbols.

AI and Consciousness: Are They the Same?

Today’s computers and AI can perform incredible tasks-solving complex problems, translating languages, and even playing chess at superhuman levels. However, many experts believe:

  • AI is powerful but not conscious.
    • It can process data and follow rules, but it doesn’t experience a “Eureka!” moment or have a sense of self.
  • Human consciousness might involve something beyond just computation.
    • It could be that our brains use some yet-to-be-understood process-possibly even involving quantum mechanics-to achieve what we call awareness.

Critiques and Counterpoints

Not everyone agrees that our minds are beyond algorithms. Some argue:

  • Our “intuition” might simply be very advanced computation.
    • What feels like a leap of insight might just be our brains processing vast amounts of information quickly.
  • We might be switching to a higher-level set of rules.
    • When we “step outside” a problem, we might just be applying a more complex algorithm rather than using a mysterious non-computable process.

In other words, while the idea that consciousness can’t be fully computed is intriguing, it’s not a settled issue. The debate continues, with brilliant minds on both sides.

In Conclusion: The Ongoing Mystery

The journey to understand consciousness is like following a trail with many interesting stops:

  • Math and computers show us that some problems can be solved step by step, while others resist such simple treatment.
  • Our minds might be more than just a set of instructions, giving rise to genuine understanding and self-awareness.
  • AI, for all its power, might always lack the true “spark” of consciousness.

So, can we write an algorithm on a napkin that fully captures consciousness? The current answer is: probably not. But that very mystery is what makes the exploration so exciting. As we learn more about both the brain and the limits of computation, we may yet uncover new principles that transform our understanding of ourselves.

Pull out a napkin, grab a pen, and keep exploring-there’s a vast, fascinating world waiting to be discovered!

Longer Version

Imagine you’re sitting in a diner. You scribble a simple equation on a napkin - “2 + 2 = 4” - and hand it to your friend. She nods, “Yes, that’s correct.” Now you write down a complicated integral, or maybe the design for a new system. If you followed the standard recipe of mathematical or computational steps, you’d eventually arrive at a result. That’s computable math: it’s what machines (or sufficiently patient humans) can do, step by logical step, guaranteed to finish in finite time.

So you might ask: if we can do so many brilliant calculations on a napkin, or on a supercomputer, what about our consciousness? Could it be just another algorithmic recipe waiting to be written down?

Recently, I’ve been diving into the various interviews of one of my favorite mathematicians and physicists, Roger Penrose - great theoretical physicist and one of the minds behind the Penrose–Hawking singularity theorems-says “Nope, not so fast.” He repeats in many interviews that our awareness and understanding can’t be fully captured by any algorithm. And if we can’t pen down an algorithm for consciousness, we sure can’t code it into a computer-no matter how powerful or super-duper advanced. That’s the big claim: AI might become a monstrous chess champion, but it still won’t genuinely understand chess.

Let’s briefly view some ideas - and try to explain why it matters.

Computable vs. Non-Computable: The Two Realms of Math

Computable: If you can write a finite set of instructions - an algorithm - so that a machine (or a relentless human) can carry it out and eventually finish, you’re in the realm of computable math.

Example: Arithmetic, standard algebra, solving certain classes of equations (linear, polynomial), enumerating prime numbers, etc.

Non-Computable: There’s no such universal, always-terminating recipe. Present the same problem to the world’s mightiest supercomputer, and it might crunch away till the end of the cosmos, never definitively knowing if or when it should stop.

Example: Turing’s famous Halting Problem states that no single algorithm can decide if any arbitrary program will run forever or eventually halt. This is the all-time champion example of non-computability. In practice, 99.9% of what you do with computers (from streaming cat videos to deciphering gene patterns) is computable-thank goodness! But Penrose reminds us there’s a deep corner of mathematics that is provably beyond any mechanical, step-by-step procedure.

Here are a few more examples of such problems, but feel free to skip them

(A) Non-Computable Math

A1. Non-Computable Numbers

Non-computable real numbers are those that cannot be approximated with arbitrary precision by any finite algorithm. While they exist mathematically, their exact values cannot be fully calculated. Examples include:

  • Chaitin’s Constant ($\Omega$): A number representing the probability that a randomly generated program will halt. It is definable but non-computable because calculating its digits would require solving the halting problem for infinitely many programs.
  • Busy Beaver Sum: Defined as $ \sum_{i=1}^\infty 2^{-\Sigma(i)} $, where $ \Sigma(n) $ is the Busy Beaver function. Since $ \Sigma(n) $ grows faster than any computable function, this sum converges but cannot be computed beyond a few known digits.
  • Truth-Teller Sequences: Numbers constructed by encoding the truth values of arithmetic statements (e.g., “the $n$-th Turing machine halts”) into binary expansions. These are non-computable due to their reliance on undecidable properties.
A2. Non-Computable Functions

These functions cannot be implemented by any algorithm:

  • Halting Function: Determines whether a Turing machine halts on a given input. Proved non-computable by Alan Turing due to self-referential paradoxes.
  • Busy Beaver Function ($\Sigma(n)$): Outputs the maximum number of steps a Turing machine with $n$ states can take before halting. Its growth rate outpaces all computable functions.
  • Hilbert’s 10th Problem: A function that determines whether a Diophantine equation has integer solutions. Matiyasevich’s theorem proved its non-computability.
A3. Non-Computable Problems in Geometry and Logic

Certain problems inherently resist algorithmic solutions:

  • Tiling Problems: Determining whether a set of polygonal tiles can cover an infinite plane. Specific instances are linked to non-computable real numbers.
  • Truth in Arithmetic: The function that labels arithmetic statements as true or false is non-computable (Tarski’s undefinability theorem).
  • Rice’s Theorem: States that all non-trivial semantic properties of programs (e.g., “does this program compute a prime?”) are undecidable.
A4. Philosophical and Practical Implications
  • Limits of Mathematics: Non-computable entities highlight boundaries in formal systems, as shown by Gödel’s incompleteness theorems and the halting problem.
  • Approximation vs. Exactness: Non-computable numbers like $\Omega$ can be approximated, but their exact values remain unknowable. This contrasts with computable numbers like $\pi$, which can be calculated to arbitrary precision.

  • Generative Definitions: Some non-computable functions (e.g., forcing in set theory) rely on non-constructive methods involving infinite processes.

  • Non-computable math reveals the interplay between definability and computability.
  • Logical paradoxes (e.g., self-reference) and undecidable properties underpin many non-computable results.
  • These concepts challenge the intuition that “definable” implies “computable,” exposing fundamental constraints in mathematics and computation.

Gödel’s Great Escape: Stepping Outside the System

Here’s the gem that set Penrose down this path-Kurt Gödel’s Incompleteness Theorem. In the 1930s, Gödel showed that any rich-enough logical system contains statements that the system can’t prove true or false; yet from the outside, a mathematician might see the statement is obviously true. That means the system’s internal rules are too cramped to capture all the truths about numbers.

Penrose zeroes in on what that means about us, the mathematicians or thinkers. We can step outside the system and say, “Ha! We see that statement is true,” even though the formal system, bound by mechanical rules, can’t prove it. This “outside-the-system” vantage is akin to a consciousness that does more than just churn through an algorithm. If our minds were purely algorithmic, We could say, “we’d be stuck in the same box as that formal system.” But we’re not. So either:

Our consciousness is not algorithmic, or we’re just fooling ourselves about that outside glimpse of truth.

You might be thinking: “Okay, so Gödel proved formal systems have blind spots. But how does that automatically mean the mind is outside this formal machinery?” Great question! The argument says if a purely algorithmic system (like a computer) can’t prove certain truths but a human can see them, that suggests our cognitive process isn’t just step-by-step computation.

But to make that a statement about consciousness, we need to show that awareness-the sense of “Aha, that’s true!”-requires some non-algorithmic leap. Critics might say we’re just layering more algorithms on top or jumping to a higher-level formal system. So the core question becomes: Does this ‘outside-the-system’ vantage reflect genuine self-awareness (meta-cognition), or is it simply a larger, more complex algorithm playing out?

If it’s the former, it implies consciousness has access to something beyond mechanical rules. If it’s the latter, well, then we’re still inside the machine. A deeper dive here would explore how that meta-cognitive “I see I’m using rules” step might, or might not, go beyond computability.

Vs. The AI Optimists

No matter how many GPUs you harness or how sophisticated your neural networks become, you won’t get a truly conscious system. Why? Because at the end of the day, it’s still following algorithmic rules-glorified symbol-shuffling, if you will. Sure, you might see mind-blowing feats of mimicry-like a language model that can write poetry or a future digital brain that aces every test-but understanding isn’t the same as rolling through a set of programmed steps.

Imagine an AI as the most brilliant “math student” in existence: it can solve impossible integrals at lightning speed, regurgitate formulas, and even produce surprisingly clever proofs. Yet it doesn’t “see” the conceptual landscape the way a deep-thinking mathematician does. It’s deftly manipulating symbols, but there’s no “aha!” spark of self-awareness or the capacity to jump outside its own rulebook and say, “Oh, I see why this must be true.”

The Chinese Room Thought Experiment

A classic illustration of this gap is John Searle’s Chinese Room scenario. Suppose you’re in a sealed room with a rulebook for Chinese. People outside slide Chinese questions under the door. You use the rulebook to craft perfectly valid Chinese answers, then pass them back. To an outside observer, you “speak Chinese.” But in truth, you don’t understand a word of it; you’re just pushing symbols around according to set instructions. That’s the worry with AI: it can appear to “know” but might just be blindly following orders.

Danger Without Understanding

Does this mean AI is harmless? Definitely not. A highly advanced system could wreak havoc-controlling power grids, manipulating financial markets, or designing cutting-edge weapons. In that sense, it’s as “dangerous” as an automated nuclear reactor gone rogue: huge power, zero comprehension or moral insight. That’s why people fret over letting black-box algorithms loose in critical roles.

Still, “dangerous” and “conscious” aren’t the same. A super-powerful system can do colossal damage without any awareness of what it’s doing. It’s the distinction between raw computational muscle and genuine understanding-and that gap might remain firmly in place, no matter how fast our machines get.

Deep Dive: The Quantum Hunch

Some researchers take things a step further by suggesting quantum mechanics (or possibly some not-yet-understood physical principle) might be key to explaining how the brain accomplishes non-computational thinking. One well-known proposal, developed with Stuart Hameroff and called the “Orch-OR” theory, posits that quantum coherence in microtubules inside neurons could play a crucial role. It’s undeniably speculative-skeptics point out that no one has clearly demonstrated quantum effects of this kind persisting in the warm, noisy environment of the brain.

Yet, the broader point these theorists emphasize stands apart from the quantum details: they argue that purely classical, rule-following computation simply can’t replicate what conscious minds do. If we ever manage to pinpoint the precise workings of the brain, they expect to find a hint (or more) of non-computable physics lurking inside.

A Quick Run Through the Arguments

Gödel: Any formal system has unprovable truths, yet we (from outside) see they’re true.

Turing: Computation = formal system manipulations, so it inherits these limitations.

Human Minds: We routinely step beyond those limitations when we do mathematics-thus, the suggestion that we’re not purely algorithmic. Consciousness: Tied up in that ability to “step outside,” to understand rather than follow rules blindly.

AI: Even the best AI you can dream up-still a Turing machine (or a network built from computational elements). So if Penrose is right, it’s missing the “understanding” dimension.

“Can Consciousness Be Calculated on a Napkin?”

Try writing a fully universal “consciousness algorithm” on your diner napkin-We could say you can’t, because such an algorithm would eventually slam into those Gödelian walls. If you think you have the final list of steps, the final code, the final procedure that claims “I do it all,” there’s always some statement that your code can’t confirm but which a mindful mathematician can see is true.

In short: No-consciousness can’t be reduced to a neat set of instructions, not on a napkin, not in a supercomputer. It’s something bigger, or at least different, than mechanical symbol-shuffling.

Maybe It’s Not “AI” at All?

Look, there’s a bit of a mix-up in how we throw around the term “AI.” We say Artificial Intelligence, and right away, folks imagine a thinking machine-some digital brain that’s aware of what it’s doing. But that’s not really what we’ve built, at least not so far. What we have are fancy pattern-finders, supercharged calculators, or data-crunchers that can do tasks so quickly and seamlessly it looks magical.

But consciousness? That’s a whole different story. Consciousness isn’t just about juggling billions of data points or beating us at chess. It’s about knowing that you exist, understanding what a chessboard is, and intuitively grasping why you’re moving that rook. That’s not the problem these systems are solving. And it might not even be the right question for them.

If we keep chasing the idea of “machine consciousness,” we’re barking up the wrong tree. Instead, let’s embrace the brute-force brilliance these systems already have-and leave the mysteries of conscious experience for deeper exploration in physics, neuroscience, and maybe someday, some new field we haven’t even dreamed up yet.

So, Where Does That Leave Us?

Consciousness is a slippery fish: every time we try to tie it neatly to a set of rules, it wriggles away. We can build awesome machines-computers that can whip through millions of data points, outplay us at chess, or even detect diseases better than a top specialist. Yet many folks argue that the feeling of truly understanding something remains out of reach for a purely step-by-step system. A laptop might checkmate grandmasters, but it never has that electrifying “Eureka!” moment-the flash of insight we humans experience when we see a solution.

You might say, “So what if it’s just fancy symbol-pushing? Results are results, right?” And for many practical matters, that’s correct. But some contend that true comprehension-knowing that you know-isn’t captured by mechanical shuffling of symbols. This difference could be huge if our goal is to understand us-why we’re aware, why we have that subjective sense of experience when we, say, solve a math problem or watch a sunset.

A Quick Tour of Consciousness Theories

Scientists and philosophers bring a grab-bag of ideas to the table. Here’s a snapshot, plus a sense of whether they lean toward a “computable” viewpoint or hint at something beyond algorithms:

Global Workspace Theory (GWT)

Key Idea: Consciousness arises when information is broadcast to a “global workspace” accessible to various brain modules.

Computability: Generally computable-like a central blackboard system. Straightforward to model with algorithms.

Integrated Information Theory (IIT)

Key Idea: Consciousness corresponds to how strongly integrated and differentiated a system’s information is. The magic number here is “phi.”

Computability: In principle it’s computable, but calculating phi for real systems gets complicated. Debates continue about whether high phi truly means subjective experience.

Higher-Order Thought (HOT) Theories

Key Idea: A mental state is conscious when there’s a higher-order thought about that state-basically, “I’m aware I’m seeing red.”

Computability: Quite computable in theory-an AI could be programmed with meta-representations of its processes. But does that guarantee genuine first-person experience? Unclear.

Predictive Processing

Key Idea: The brain is a prediction machine, continuously guessing what’s “out there” and updating based on errors. Consciousness emerges from the interplay of these predictions with reality.

Computability: Heavily relies on Bayesian models and neural networks-very algorithm-friendly. Whether it captures the feeling of being aware is still up for debate.

Quantum Mind Theories

Key Idea: Some argue quantum coherence in neuronal structures might produce non-computable processes tied to consciousness.

Computability: These views explicitly suggest there’s more than classical algorithms at work. Many scientists remain skeptical, but it’s a bold claim that definitely pushes beyond ordinary computation.

Key Idea: Consciousness might be a basic property of the universe, not something that emerges from just the right arrangement of matter.

Computability: If consciousness is embedded in the fabric of reality itself, it’s not something you simply write code for-unless the entire universe is an algorithm (a wild notion on its own!).

The Frontier of Mystery

The main question is: Could there be some deep principle-maybe even new physics-that bridges the gap between mere symbol manipulation and the subjective glow of consciousness? Some say yes, some say no. For sure, it’s a wake-up call: maybe the mind isn’t just a glorified computer.

If we keep scratching at this puzzle, we might stumble onto new logic systems or an “X-factor” in the brain that sidesteps standard algorithms. Or we might find that real consciousness, by its nature, can’t fit into any neat mechanical box. That doesn’t mean we should quit investigating-on the contrary, it means there’s a genuine mystery here, one that could transform how we see ourselves and the universe.

Don’t Toss the Tech!

But wait-none of this is to say that artificial systems are useless. Far from it! They are and will revolutionizing healthcare, climate prediction, language translation, and plenty more. Advanced algorithms are a tremendous asset. However, being brilliant at pattern crunching or game-playing doesn’t equal being conscious. And that might be okay: we don’t need introspective search engines or existentially pensive thermostats to benefit society.

Still, a system can be non-conscious yet extraordinarily influential-sometimes in risky ways. Power without true understanding can be a volatile cocktail. So even if algorithms aren’t “aware,” they can shape the world around us in unforeseen and dangerous ways.

Why Humans Might Not Need to “Step Outside” Formal Systems: Critiques of the Gödelian Argument

Let’s face it: not everyone buys the idea that the human mind leapfrogs formal systems in some magical, non-algorithmic way. Here are some of the biggest counterarguments:

We Can’t Prove Our Own Consistency

Some arguments claim that humans spot the truth of Gödel sentences because we “see” the formal system is consistent. But here’s the snag:

  • Gödel’s Second Incompleteness Theorem states that no sufficiently strong formal system can prove its own consistency.
  • If our thinking is algorithmic, we’re stuck in the same logical boat-meaning we can’t absolutely prove our own consistency either.

Translation: If we can’t prove our consistency, how are we “magically” stepping outside to see Gödel truths? Maybe we’re not, and we’re just assuming it.

Intuition Might Be Algorithmic Too

A big chunk of the non-algorithmic claim leans on “human intuition.” But critics say:

  • Maybe that intuition is really heuristic or probabilistic reasoning, like a fancy guess-and-check method that could be coded in principle.
  • Humans do trial-and-error learning or reinforcement-like learning.
  • Complex “meta-cognitive” loops might simulate higher-level insight without anything supernatural happening.

Sure, we feel we’re doing something mysterious, but advanced algorithms can also look surprisingly “insightful” once they’re big and clever enough.

Human Reasoning Might Be Inconsistent

Gödel’s theorems require the system to be consistent. But what if humans aren’t exactly perfect logic machines?

  • We sometimes hold contradictory beliefs-then revise them when we catch ourselves.
  • There’s a branch of logic called paraconsistency that handles contradictions without everything exploding into nonsense.

If we’re inconsistent, we’re outside Gödel’s framework anyway-so no reason to wave the “incompleteness” flag as proof of super-human logic powers.

Humans Are Just Shifting to Meta-Systems

When a mathematician “sees” a Gödel statement is true, critics note we might just be jumping to a more powerful formal system-like going from system S to system S+:

  • Formal systems can prove Gödel’s results once they move up to a stronger framework.
  • Bouncing to a higher-level system is still an algorithmic move, just a more sophisticated one.

It’s not that we’re mystically outside all systems; we’re just switching from one formal apparatus to another, each with its own limits.

Practical vs. Theoretical Boundaries

Even if some exotic non-computable processes exist in principle, critics say:

  • Bounded Rationality: Real human brains are resource-limited; we approximate, skip steps, and otherwise behave like any advanced but finite machine.
  • Evolutionary Adaptation: Our cognitive tools evolved to handle survival tasks, not to unlock the ultimate cosmic proof. That’s consistent with algorithmic, not otherworldly, processes.

In plain English: If there is some super-duper non-computable magic, we sure don’t use it 24/7 to figure out grocery lists and traffic routes.

Key Objections to “We Transcend Computation”

Criticism Implication
We can’t prove our own consistency Undercuts the claim we can “just see” Gödel truths
Intuition may be algorithmic We don’t need magic to explain human insight
Human reasoning can be inconsistent Gödel’s theorems don’t apply if the system is inconsistent
We rely on meta-systems “Stepping outside” might just mean shifting to a more complex algorithm
We have practical, not infinite, power Real cognition likely fits within computable, bounded frameworks

When you look under the hood, there’s a lot of debate about whether we really escape the same constraints that bind formal systems. Maybe we’re just stacking or swapping algorithms-nothing truly non-computable in sight. Or maybe we’re logically messy enough (inconsistency, paraconsistency, etc.) that Gödel’s tidy framework doesn’t apply. Critics caution that while Gödel’s theorems are profound-exposing the limits of formal systems-they don’t necessarily crown the human mind as a special non-computational wizard.

So the question remains on the table: Are we powered by a unique spark that outruns any algorithm, or are we just very complicated-sometimes inconsistent-machines ourselves? Whichever side you land on, the conversation isn’t over yet, and that’s precisely what keeps things exciting.

Wrapping Up: The Adventure Continues

We’ve roamed across Turing’s Halting Problem, Gödel’s mind-boggling insights, quantum speculations, and every shade of AI optimism and skepticism. So is your consciousness just a big old flowchart? Or do you have a secret sauce no computer can replicate?

Truth is, nobody knows for sure. We do know that formal systems can’t prove every statement, and that some parts of math are fundamentally non-computable. But does that map perfectly onto the human mind? Maybe, maybe not. Meanwhile, AI becomes more powerful by the minute-stunning us with data-mining prowess but stopping short of awareness.

That’s precisely what keeps this topic so fascinating. We’re poking at the heart of what it means to be us-why we experience that gut-level sense of “I am.” Whether we turn out to be elaborate machines or something altogether more mystical, one thing’s certain: we’ve got a lot of exploring left to do. And that’s fantastic news for curious minds-there’s a vast frontier waiting, from advanced logic systems to quantum brain experiments, all searching for that final piece of the puzzle.

So pull out another napkin if you like-but don’t be surprised if consciousness won’t fit neatly on it. The quest goes on, and every new insight just deepens the mystery. Let’s keep that sense of wonder alive-and see where the next breakthrough leads us.