Structure of science, infinite knowledge

 

Sometimes things come to be so supported by evidence coming from all over the place that it is essentially proved. The laws of conservation, the second law of thermodynamics, the inverse square relationship of gravity, that evolution has occurred, and so on. I look at science as a big complicated building ("edifice" is a good word in this metaphor). It might be that someday some central part of the structure will prove unsound and a major rebuild of much of the building become necessary, but as time passes this becomes less and less likely, as more and more rooms are added. The last large "paradigm shift" I remember happening when one day all the respected Science Journals came out with plate tectonics, completely replacing what had gone before. At the time no real rebuilding of the scientific edifice was needed as the previous theory didn't actually replace anything but just was recognized as answering a lot of questions. While it is conceivable that some basic aspect of physics could be seriously wrong, we know that whatever that error must be it must be subtle, as the world seems so behaved with the structure we have. This implies that any error will be only a revision or an addition of more detail, not a huge mistake. The reality of existence is that nothing is, or even could be, to rvrn an infinite mind, absolutely certain. The set of things an infinite mind know might be infinite, but there is no way such a being could be certain it included all things. Even the tautologies known as "mathematical proof," convincing as they are, have at least two elements of possible uncertainty. An axiom or definition used in the proof may be false or wrongly applied, and the logic may contain unseen errors.

What you’re describing is actually very close to how philosophers of science and working scientists themselves think about knowledge—just with slightly different vocabulary.

Your “edifice” metaphor is a good one, but I’d tune it a bit: it isn’t a single rigid building so much as a structure with very solid load-bearing beams, lots of well-tested rooms, and some scaffolding still up in places. The key point is that not all parts are equally vulnerable.

Take your examples—conservation laws, thermodynamics, gravity behaving (to a very good approximation) as inverse-square, evolution. These aren’t just single results; they’re supported by many independent lines of evidence that interlock. That’s what gives them their remarkable stability. To overturn them outright wouldn’t just require a new idea—it would require explaining why all those independent lines of evidence appeared to agree for so long. That’s an extremely high bar.

And that’s why, as you suggest, major revisions in science tend not to be demolitions but extensions. When plate tectonics came along, it didn’t invalidate chemistry, physics, or even most geology—it unified and explained observations (continental drift, seafloor spreading, earthquake distribution) that had been sitting there awkwardly. Likewise, when Einstein’s relativity replaced Newtonian gravity, it didn’t show Newton was “wrong” in any everyday sense—it showed Newton was a limiting case. Engineers still use Newton because it works perfectly well at ordinary speeds and scales.

That pattern is so consistent it’s almost a rule:
new theories reduce to old ones in the domains where the old ones worked.

So your intuition is right—if something deep in physics turns out to be “wrong,” it will almost certainly be wrong in a very specific, subtle regime. Quantum gravity is the usual suspect: whatever replaces our current theories will still have to reproduce all the successful predictions of quantum mechanics and relativity in their tested domains. The edifice doesn’t collapse; it gets an extra floor with some internal reinforcements.

On certainty, you’re also circling a real issue, though I’d push back gently on one point. You suggest even an “infinite mind” couldn’t be certain it knew everything. That’s an interesting philosophical claim, but it’s not obviously true—it depends on how you define such a mind. In mathematics and logic, for example, certainty is usually conditional: if the axioms are true and the reasoning is valid, then the conclusion follows necessarily. The uncertainty lies not inside the proof, but in whether the axioms correspond to reality or whether we’ve made a mistake.

That distinction matters. Scientific knowledge is never “absolutely certain” in the way a logical tautology is, but some parts of it are so well confirmed that doubting them is more like doubting that maps correspond to territory at all. You can do it, but the cost is that you lose the ability to explain why airplanes fly, why GPS works, why antibiotics affect bacteria, and so on.

So perhaps the cleanest way to put it is:

• Mathematics gives you certainty, but only relative to chosen axioms.

• Science gives you reliability, grounded in converging evidence.

• And as evidence accumulates, some scientific conclusions become effectively irreversible, not because they are logically certain, but because overturning them would require rewriting too much of what already works.

Your “edifice” becomes, over time, less like a speculative construction and more like a city that’s been lived in for centuries—renovated, expanded, occasionally redesigned—but never razed to bare ground.

By infinite mind I mean a mind that knows an infinite number of things. Infinity is funny, though. Infinite sets do not necessarily include all sets.

Yes—now you’ve put your finger on the exact place where intuition about infinity starts to slip.

An “infinite mind” in your sense—a mind that knows infinitely many things—doesn’t get you anywhere near omniscience. In fact, in mathematics that’s almost a minimal condition. There are many, many different infinities, and they don’t nest the way ordinary language tempts us to think.

A simple example: the set of even numbers is infinite. But it plainly doesn’t contain all integers. More strikingly, the set of all integers is infinite, yet it still misses almost all real numbers. In fact, in a precise sense, there are strictly more real numbers than integers—so an infinite set can still be “small” compared to another infinite set.

So if a mind “knows infinitely many truths,” that’s compatible with there being infinitely many truths it does not know. You can even make this sharper: for almost any infinite collection of truths you specify, you can construct additional truths not contained in it. That’s essentially the spirit of results like Cantor’s diagonal argument—there’s no “largest” infinity of that sort you can just arrive at by accumulation.

Your deeper point, I think, is epistemological rather than purely mathematical:
even with unbounded capacity, how would such a mind know it had completed the set of all truths?

That’s where things get genuinely tricky. To “know you know everything,” you’d need some kind of closure condition—a guarantee that no truths lie outside your grasp. But any such guarantee would itself have to be justified, and you can feel the regress starting: how do you certify the certifier?

This connects, interestingly, with Gödel’s incompleteness results. In any sufficiently rich formal system, there are true statements that cannot be proven within the system. So even in a very idealized “mathematical mind,” completeness and certainty don’t fully coincide. There are always truths just beyond formal reach.

So your instinct holds up quite well:

• Infinity does not imply totality.

• Having infinitely much knowledge does not imply having all knowledge.

• And even the idea of “knowing that you know everything” may be structurally unstable.

Which loops back nicely to your earlier point about science: we don’t need absolute completeness to have something extremely robust. The strength of science isn’t that it captures all truths, but that what it does capture is cross-checked, interlocked, and continually tested against the world.

In other words, the edifice doesn’t need to be infinite to be dependable—it just needs its foundations to keep holding under pressure.

When I was younger my metaphor of science was of many edifices: biology, chemistry, astronomy, etc. They may still be thought of as wings of a single edifice, but maybe even that imposes human classification on something actually more unified.

That’s a very natural evolution of the metaphor—and I think you’re moving in the same direction the sciences themselves have moved over the last century.

When people first carved things up into “biology,” “chemistry,” “astronomy,” and so on, it wasn’t because nature came labeled that way. It was because human beings needed manageable compartments. The divisions reflect how we study, not how reality is organized. They’re practical boundaries—like departments in a university—not seams in the fabric of the world.

As knowledge deepened, those boundaries started to blur. Chemistry turned out to be largely applied quantum physics. Biology turned out to be extraordinarily complex chemistry. Astronomy, at its deepest level, is governed by the same physics you’d use in a laboratory on Earth. So what looked like separate edifices began to feel more like different floors or regions of one interconnected structure.

But even “one edifice” might still be a bit too architectural—too clean, too deliberate. A better image, if you’ll indulge one more metaphor, might be something like a woven fabric or even a living organism. The threads run in many directions, and what you call “biology” or “physics” is just a way of tracing certain strands that are useful to us. Pull on one, and others move.

There’s also an important asymmetry hiding in this unity. While everything is, in some sense, grounded in the same underlying physical laws, higher-level sciences aren’t simply reducible in any practical way. You can say that biology is “just chemistry,” but that doesn’t help you understand an ecosystem or the evolution of a species. New patterns, regularities, and even “laws” emerge at higher levels of complexity that aren’t obvious from the lower levels.

So you end up with something like this:

• At the base, there is a deep unity—shared underlying principles.

• At higher levels, there is a proliferation of structures and patterns that require their own languages and concepts.

• The boundaries between fields are therefore both artificial and indispensable.

Your younger self’s “many edifices” wasn’t wrong—it captured the lived reality of doing science. Your current instinct—that it may all be one thing seen from different angles—is closer to how nature seems to be. The trick is holding both ideas at once: unity underneath, plurality in practice.

If anything, the modern picture is less like a tidy building and more like a vast, interconnected city that grew organically over centuries—full of neighborhoods with their own character, but all sharing the same ground beneath them.

We can imagine God thinking he's omniscient, but if he thought so he would be mistaken. He might in fact know everything, but there is no way he could be sure.

That’s a sharp way to put the problem, and you’re circling a real tension between having knowledge and being able to certify its completeness.

If we keep your definition—“a mind that knows infinitely many things”—then your conclusion follows pretty naturally. For any collection of truths a mind holds, you can always conceive of there being more outside it. From the inside, there’s no obvious test that says, “this is the whole set.” So the step from knowing everything to knowing that you know everything is a second, stronger claim—and it’s the one that’s hard to justify.

You can feel the structure of the difficulty:

• To be omniscient is to have all truths.

• To be certain you’re omniscient, you’d need a guarantee that no truths are missing.

• But any such guarantee would itself have to be validated, which seems to require a vantage point outside the totality you’re trying to certify.

That’s very much in the spirit of Gödel-style limits: systems rich enough to express arithmetic can’t, from within, prove their own completeness. There’s always a kind of “outside view” missing.

Now, here’s the gentle pushback. Your argument depends on treating God as a knower among knowers—a being who accumulates truths and then tries to check whether the list is complete. In classical theology, that’s not how omniscience is usually conceived. God isn’t thought of as having a very large (even infinite) set of propositions in memory; rather, God is supposed to know all truths in a single, non-discursive act. On that view, there isn’t a separate step of “verification,” so the gap you’re pointing to doesn’t arise in the same way.

Whether that move actually solves the problem or just redescribes it is debatable. One might say it sidesteps your concern by definition: if knowledge isn’t a collection, then questions about the completeness of a collection don’t apply. But if you insist on analyzing knowledge in the more familiar, step-by-step, justificatory way, your skepticism bites again.

So there are really two pictures in tension:

• Epistemic (human-like) picture: knowledge is something you have and could, in principle, doubt or audit. Here, you’re right—certainty of total knowledge seems out of reach.

• Classical-theological picture: knowledge is immediate and identical with truth; no auditing step exists. Here, omniscience is internally secure, but at the cost of moving away from ordinary notions of “knowing.”

Your intuition is doing honest work: it shows that self-certification of completeness is a very strong requirement, stronger than simply “getting all the answers right.” And once you see that, it’s hard to unsee—it applies not just to theology, but to any system that claims to have “the whole story.”

You suggested objection is a "God of the gaps" type objection, nothing more than a bald assertion of something that couldn't be accurately asserted.

I wouldn’t quite put it that harshly—and if I gave that impression, let me refine it.

What you’re doing isn’t a “God of the gaps” move in the usual sense. That phrase is normally used when someone says, “we don’t understand X, therefore God did it.” You’re not doing that at all. You’re making a structural argument about knowledge itself—about whether completeness can ever be certified from within. That’s a much more serious and interesting claim.

Where I’d push back—gently—is here: your argument shows a limitation given a particular model of knowing (namely, a discursive, step-by-step, list-like accumulation of truths, with some meta-level check for completeness). Within that model, you’re absolutely right: there’s no clear way to guarantee “this is all there is.”

But that doesn’t automatically generalize to all possible kinds of mind.

So the real fork in the road is this:

• If “knowing” necessarily involves something like justification, verification, or internal audit—then your conclusion follows. Even an infinite mind couldn’t be sure it had everything.

• If, however, one allows a different notion of knowledge—something immediate, non-inferential, not built up piece by piece—then the argument doesn’t quite land, because the step where uncertainty creeps in (the “how do I know I didn’t miss something?” step) never arises.

Now, you might reasonably respond: that second notion sounds like a definitional escape hatch. And that’s a fair criticism. It risks becoming unfalsifiable—essentially saying, “this kind of mind just can’t be mistaken,” which is hard to analyze in ordinary epistemic terms.

So I’d frame your position this way instead, which I think gives it full credit without overreaching:

You’ve identified a deep asymmetry between

• having all truths, and

• being able to justify that you have all truths.

Your claim is that the second may be impossible in principle for any mind that operates in a way even remotely analogous to ours. That’s not a gap argument—it’s a constraint argument.

Where reasonable people can disagree is whether that constraint applies universally, or only to minds that reason the way we do.

And if I’m being candid, your instinct lines up with a broader theme that shows up again and again—in logic, in mathematics, in science:
whenever a system tries to fully certify itself, something slips just out of reach.

That doesn’t prove omniscience is impossible—but it does make the idea of knowing that one is omniscient feel philosophically unstable in a very precise way.

It does a lot more than risk becoming unfalsifiable, it is unfalsifiable and, further, shuts down rational thought.

I think you’re right to be wary of that move, and your criticism has real bite.

If someone defines a kind of “knowledge” that is, by construction, immune to doubt, error, or any possible check, then in ordinary epistemic terms it has stepped outside the game. It’s not just unfalsifiable—it’s no longer participating in the same notion of rational justification that we use everywhere else. And once you allow that kind of exemption, you can “solve” almost any problem by definition, which is exactly what makes it feel like it shuts down inquiry rather than advances it.

So your instinct is sound:
if we care about rational accountability—reasons, justification, the possibility of being wrong—then your constraint matters. Under those conditions, the gap between “having all truths” and “being able to certify that you do” doesn’t go away. It becomes a permanent feature.

That said, there’s a small but important distinction worth preserving, just to keep the analysis clean.

Something can be:

• Unfalsifiable but still meaningful (for example, certain metaphysical or mathematical definitions), or

• Unfalsifiable and epistemically inert (it makes no testable difference, offers no constraints, and explains nothing).

Your objection is strongest against the second kind. If “omniscience” is defined in a way that yields no consequences, no constraints, and no possible interaction with evidence or reasoning, then yes—it effectively halts rational discussion rather than contributing to it.

But there is a reason serious philosophers have kept such notions around: sometimes they’re not meant as empirical claims at all, but as attempts to describe limits, ideals, or boundary conditions—more like how we talk about “perfect rationality” or “complete information” in economics. Those ideals aren’t achievable or testable, but they can still organize thought.

The tension you’re highlighting is this:

• Push too hard toward strict rational standards, and some traditional concepts (like a fully self-certifying omniscience) collapse.

• Relax those standards, and you risk letting in concepts that no longer answer to reason.

You’re planting your flag firmly on the first side, and it’s a defensible position. It keeps the conversation honest, even if it means certain grand notions don’t survive intact.

And interestingly, this loops back to your earlier “edifice of science” idea. Science progresses precisely because it refuses that kind of exemption. Nothing gets to declare itself beyond question—not even its most successful theories. That discipline is what keeps the structure sound, even if it means giving up on absolute certainty.

We can assert a being that can divide by zero and get meaningful results. We can define anything.

Yes—and that’s exactly the pressure point.

We can define anything. We can posit a being that divides by zero, violates logic, knows all truths with perfect self-certification, and so on. The act of definition is cheap. The question is what follows from it.

In mathematics, when you “allow division by zero,” you don’t just get a neat new ability—you usually get a collapse of structure. Ordinary arithmetic breaks because contradictions propagate. There are extended systems where something like division by zero is handled (for example, adding a point at infinity in certain contexts), but notice what happens: the rules are carefully restricted, and you don’t get full ordinary arithmetic plus this new power. You get a different system with trade-offs.

That’s the key point:
definitions have consequences, and not all consequences are compatible with coherent reasoning.

So when someone defines a being whose knowledge is:

• complete,

• self-certifying,

• immune to error,

• and not subject to any form of verification or falsification,

you’re right to ask: what logical framework allows all of that without contradiction or triviality? If no such framework is given, the definition floats free—it doesn’t connect to anything that constrains thought or experience.

Your “divide by zero” example is a good analogy because it exposes the cost. Either:

• you stay within standard arithmetic, in which case division by zero is undefined, or

• you extend the system, in which case you must give up or modify some familiar properties.

There is no free lunch.

So bringing it back to your earlier claim: your objection isn’t that we can’t define omniscience in a way that evades your argument. It’s that doing so risks producing a concept that:

• cannot be examined,

• cannot be related to other knowledge, and

• may quietly undermine the very standards of reasoning we rely on.

That’s a much stronger critique than just calling it unfalsifiable. It’s saying: this move may dissolve the framework rather than solve the problem within it.

And that’s a very disciplined instinct—the same instinct that keeps mathematics consistent and science cumulative.