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One way to see Asimov's "infinity of wrongness" is perhaps as a fractal. You could view the bulbs in the mandelbrot as being a kind of knowledge, and the "main bulb" occupying the majority of the area belonging to the set as the set of truths known about our universe. The mandelbrot set is infinite in complexity, however its area is finite and bounded!

Or as ironing out the wrinkles on a great big t-shirt, where each wrinkle is sub-wrinkled with smaller wrinkles and so on. We've "ironed out" the biggest wrinkles, there are infinitely more but they are much smaller. We're perhaps over half-way ironed, in a quantitative sense.

I disagree fundamentally. You may as well ask me to imagine Earth as a disc, with multiple rotating spotlights shining down on it and a giant ice wall around the edges. I understand what the image is trying to convey, I simply do not agree that this is the shape of the thing I experience.

I think you missed the entire point of the essay. You should actually read it.

Science is incremental, revolutions in science mostly just adjust the edges of our knowledge, at more and more extreme corner cases (extreme high energies, extreme high/low temperatures, etc). No, we absolutely don't know it all, but as always, new knowledge and theories will only affect those edges, and refine the predictions for the nth+1 decimal place.

By and by, the science that directly affects our daily lives has remained stable and most progress has been in the engineering to put all this knowledge to practical and efficient use.

I like how in English one can make it appear if one is skilfully and logically dismissing an argument without actually even trying to. I don't presume that this was your intent, but the phenomenon is quite interesting and I quite confidently believe dangerous (in that it contributes to some degree to inaccurate models in the minds of those who ingest such text, and those models are what drive action, much of which is harmful...which is easy to see in {choose your outgroup}, but far less easy to see in one's ingroup).

I think your use of infinity isn't particularly helpful here as it leads to the contradiction that knowing more doesn't lessen the knowledge gap, whereas it does appear to do so.

Maybe, a better interpretation would be that as we learn and understand more, we approach the limits of knowledge. Now it may take an infinite amount of knowledge to actually reach the limits of knowledge (c.f. an infinite series can approach a finite value, but takes forever to get there), but it can still be shown that we are getting nearer.

The other aspect is that as we understand more, we appreciate that there's even more to understand, but that can be thought of as our precision increasing and looking at the available knowledge in greater detail.

There is no contradiction because there is no limit of knowledge.

The limit of y = e^x is infinity. You can keep increasing x and y will increase exponentially. So you plot the function, let's say with the x axis going up to 10 and the y axis going up to e^10. The graph shows very clearly that, while there was progress before, we have even more progress now. Exponentially more, even! What came before is dwarfed by what we have now; if you look at the range of y for x values 9-10 you can see how little of a difference all those others values (1-8) had between one another, compared to the changes we have now. The rate of change is so high that we're basically in an era of semi-complete knowledge. We must be at some kind of inflection point, this is truly a unique era of understanding.

Then you repeat. Set the x axis to 100, and the y axis up to e^100. Oh wait, it's the same graph. That's because it's always the same graph. It's scale-invariant. The slope at every point is always whatever y is.

We're always at right at the limit of explaining the "essential nature" of the universe because the "essential nature" of the universe can only be (to us) what we can understand it to be. We chose e^10 as the limit of the y-axis in our first exercise because that's all the knowledge we knew about. We chose e^100 as the limit of the y-axis in our second exercise because that, too, was all the knowledge we knew about. Choosing these random values as the limits of our function (i.e. the limit of the "essential nature of the universe") leaks information into the visualization that will always paint a picture showing that we're at the most transformative time there ever was.

When we do it that way, we will always come to the same _wrong_ conclusion. We will always dwarf what came before and be dwarfed by what comes after. To think that we're actually living in an inflection point is hubris, it's wishful thinking, it's the sour grapes of mortality.

> There is no contradiction because there is no limit of knowledge.

I don't think we know enough to be able to state that definitively. It's feasible that the universe behaves mathematically (it seems to so far) and thus possible to gain a thorough understanding of the underlying principles, if not the specific facts (c.f. with understanding how to produce integers yet not "knowing" all the integers).

Even if the universe doesn't have underlying rules to be discovered, there's still a limit to number of configurations available to particles etc. within our visible universe. Although that number might appear to be infinite to us, it's actually drastically closer to zero than to infinity.

So, if there is indeed some finite limit, then using y = e^x would be the wrong function as that doesn't approach a finite value.

This leads to a more fundamental question: What is the universe?

Is the optimal move in an a given chess board considered knowledge? If so, can't we create entirely new sets of knowledge from the emergent properties of an arbitrary set of rules called a "game"? If we can create an infinite set of arbitrary combinations of rules and states (games), then knowledge should be infinite. Maybe not all knowledge is scientifically applicable, but we have learned a great deal about science and engineering from studying chess. In fact, we are starting to learn more about learning as a process and not as some magical thing that human beings can do, just from studying the best way to make decisions in this totally-contrived and scientifically-useless game.

Taking this a step further, let's look at the animal kingdom. If learning about the intricacies of the mating habits of birds can help an arbitrary bird increase its impact on the future gene pool, is that knowledge not worth something to the bird? To bird society? Are the things we learn about ourselves knowledge? They certainly have utility. Is there any limit to what we can learn about ourselves, about the stochastic process of life? Is life not part of the universe?

Is computer science even knowledge? It seems if we're more directly concerned with the physical nature of the universe, we ought not to care about what the system of a computer actually does; we only need to care about what it is, about its physical structure. Except, that's not actually how we pursue knowledge or science at all.

In my view, Asimov's sentiment can be reduced to a complete tautology: we're at the point where we know almost everything there is to know about the things we think we can know.

There aren't an infinite number of chess positions, moves or even games, so that's not a good example. It's possible to come up with a number game that could have infinite possibilities, but that doesn't mean that the universe could even contain some of the options within our visibility. Our current state of knowledge about the universe strongly suggests that there's a finite limit to the available knowledge (I.e. between the Planck scale and the visible horizon due to the speed of light).

A googolplex looks to be the first number we've found that is too big to be contained in our universe.

You're right--chess is a decidedly finite game. Even so, we have not "solved" this simple, finite game--not even close! If we're not close to solving such a trivial game, how can we be close to the limit of the knowledge of the universe?

A googolplex is "too big to be contained" in our universe yet here we are talking about it. We can perform operations on this number, compare it to other numbers, and even come up with mathematical proofs showing that it's too big to exist. There are an infinite amount of numbers larger than a googolplex and we could have an infinite amount of conversations about them. The material limit of the universe does not limit our ability to create information, to learn things.

There isn't enough space in the universe for an infinite series, either, yet we can (and do) still use them, we reason about them, we learn from them. We can even reduce some infinite series to a finite number. The material bounds of the universe are not a limit of knowledge.

I think you're mistaking the map for the territory. A googolplex is a representation of a number, but not the number itself, although it's simple enough that we can get away with using the representation as it's obvious what the form of the number would be. However a number such as tree(3) is unimaginably bigger, but more crucially, we don't know anything about the form of the number beyond its size and we can't sensibly use it in calculations.

Now both of those numbers are finite and we could try to figure out how many numbers we could "describe" such as tree(3), but that would be limited by the number of symbols (i.e numbers, operators, letters and words) that could be used (i.e we would have less than a googolplex different numbers that could be represented using maths, language and thought). That's still going to be a finite number.

If the Universe is the territory than Knowledge is the map. I'm not at all mistaking the map for the territory: I'm pointing out that the set of maps that can describe a given territory are virtually infinite. Asimov is saying the map is almost complete and I'm saying there are an infinite number of maps left to go.

Cartographers in the 18th centuries were "basically done" mapping out the Earth. In the 20th century we were able to use satellite imagery to get the "full picture". Does that mean we have perfect knowledge of the Earth? Absolutely not. There is never a final frontier of knowledge.

> The gap between what we know and what is true might have decreased immensely, but it is still infinite.

The other two commenters have taken different approaches to infinity, but it seems that your argument doesn't hold even for a plain-as-in-real-numbers infinity.

Being satisfied with finite knowledge gains, I have no hope to achieve 1% of infinity (or any other fraction of infinity).

The universe is infinite in size, another assumption. If I'd fly on vacation to Tenerife, a quantifiable shift of my position by mere thousands of miles would be "zero in relation to infinity". Yet, it's not unimportant for my rest. Talking about infinity doesn't automatically cancel all the finite measurements and bring them to zero.

One might plausibly assert that while the particulars of the universe may be infinite, the fundamental rules which govern the universe are finite and thus at least in principle entirely knowable. While I don't think the 20th century makes an air tight case for the latter, I think it isn't an unreasonable conclusion to draw from 20th Century Physics either.

It's surprising that we find ourselves in a universe which does appear to obey certain laws. There's a whole bunch of assumptions made to help us understand how things work and it turns out they're mostly correct. i.e. It's more astounding that we CAN understand the universe and how well maths can act as a model/language to understand it.

Not that surprising. A universe without predictable laws would be unlikely to host life. I mean physics really only knows how to do statics and oscillations and those things work because many systems are at equilibrium or just slightly perturbed from it, probably because of the mysterious low entropy condition which defines the past.

Well, we can start by asserting that we do not know everything. We can assert this from the contra positive: If we did know everything, then we would have an acceptable solution to all of our problems. Since we do not, it stands to reason that we don't know everything.

Once we accept that the assertion is valid, then it raises the likelihood that our understanding of the world is incomplete in some way. And furthermore is incomplete to different degrees along multiple dimensions of knowledge. Whether incomplete or wrong is a word choice, it doesn't change what's missing. So with each new discovery, our understanding improves, our wrongness decreases.

> If we did know everything, then we would have an acceptable solution to all of our problems. Since we do not, it stands to reason that we don't know everything.

This can be proven false by contradiction: it may be possible to _know_ that one of our problems has no solution.

> This can be proven false by contradiction: it may be possible to _know_ that one of our problems has no solution.

This claim requires a bit more explanation, not sure if possible. Can you elaborate, or use an example?